Answers upon several heads in philosophy first drawn up for the private satisfaction of some friends : now exposed to publick view and examination / by William Marshall, Dr. of physick of the colledge of physicians in London.
         Marshall, William, 17th cent.
      
       
         
           1670
        
      
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             Answers upon several heads in philosophy first drawn up for the private satisfaction of some friends : now exposed to publick view and examination / by William Marshall, Dr. of physick of the colledge of physicians in London.
             Marshall, William, 17th cent.
          
           [24], 237 p., 2 folded leaves of plates : ill.
           
             Printed by T.L. for Nathaniel Brooke ...,
             London :
             1670.
          
           
             Errata on preliminary p. [23]
             Pages 1-14 bound out of order.
             Reproduction of original in the University of Illinois (Urbana-Champaign Campus). Library.
          
        
      
    
     
       
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           Science -- Early works to 1800.
        
      
    
     
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           ANSWERS
           upon
           several
           heads
           in
           Philosophy
           ,
           FIRST
           Drawn
           up
           for
           the
           private
           satisfaction
           of
           some
           Friends
           ;
           now
           exposed
           to
           publick
           view
           and
           examination
           .
        
         
           By
           
             William
             Marshall
             ,
             Dr.
          
           of
           Physick
           of
           the
           Colledge
           of
           Physicians
           ,
           in
           London
           .
        
         
           
             Dij
             bene
             fecerunt
             in
             opis
             me
             ,
             quodque
             pusilli
             .
          
           
             Finxerunt
             animi
             ,
             raro
             &
             per
             pauca
             loquentis
             :
             .
          
           
             Hor.
             
          
        
         
           LONDON
           ,
           Printed
           by
           
             T.
             L.
          
           for
           
             Nathaniel
             Brooke
          
           ,
           at
           the
           Angel
           in
           Cornh●●
           near
           the
           Royal
           Exchange
           and
           at
           
             Gresham
             Colledge
          
           .
           1670.
           
        
      
       
         
         
         
           To
           the
           right
           Honorable
           William
           ,
           Lord
           Viscount
           Bronker
           President
           ,
           and
           the
           rest
           of
           the
           learned
           members
           ,
           of
           the
           Royal
           Society
           .
        
         
           To
           the
           Incomparably
           learned
           Dr.
           
             Francis
             Glisson
          
           President
           ,
           and
           the
           rest
           of
           the
           learned
           fellows
           of
           the
           ,
           Colledge
           of
           Physicians
           in
           London
           .
        
         
           To
           the
           worthily
           famous
           Professors
           of
           Gresham
           Colledge
           in
           London
           .
        
         
           The
           Author
           wisheth
           all
           happines
           ,
           submitting
           these
           his
           Philosophical
           answers
           to
           their
           grave
           censure
           .
        
         
           THe
           things
           here
           presented
           ,
           and
           which
           shall
           be
           hereafter
           presented
           of
           the
           like
           nature
           ,
           will
           vindicate
           mee
           from
           both
           suspition
           in
           you
           ,
           and
           expectation
           in
           the
           Reader
           ,
           of
           a
           dedicatory
           piece
           of
           flattery
           :
           I
           every
           where
           manifestly
           using
           a
           Philosophical
           freedome
           .
           
           〈…〉
           so
           ill
           a
           servant
           either
           to
           〈◊〉
           Honours
           ,
           or
           to
           noble
           truths
           ,
           which
           are
           the
           choice
           food
           of
           refined
           minds
           ,
           to
           make
           this
           address
           unto
           you
           for
           the
           protection
           of
           any
           thing
           which
           may
           deserve
           a
           black
           censure
           .
           Let
           every
           line
           bear
           its
           own
           fate
           in
           its
           own
           bosom
           .
           Only
           being
           well
           assured
           ,
           that
           these
           things
           in
           difference
           ,
           and
           thus
           discussed
           between
           me
           and
           some
           friends
           ;
           as
           they
           have
           been
           contrarily
           taught
           by
           persons
           worthily
           had
           in
           singular
           veneration
           ,
           both
           in
           the
           present
           and
           former
           ages
           ;
           so
           they
           concern
           some
           heads
           of
           Philosophy
           ,
           which
           are
           of
           no
           small
           consequence
           unto
           true
           learning
           ;
           and
           besides
           ,
           lying
           deeper
           in
           the
           pit
           of
           obscurity
           then
           the
           first
           fadom
           ,
           they
           ought
           to
           receive
           the
           ablest
           decision
           ;
           I
           knew
           not
           therefore
           how
           more
           candidly
           to
           strip
           my self
           of
           any
           concern
           pervicatiously
           to
           defend
           any
           thing
           in
           these
           papers
           ,
           then
           by
           laying
           them
           and
           their
           fate
           at
           the
           feet
           of
           so
           noble
           and
           able
           Judges
           .
           Possibly
           the
           thanks
           of
           private
           
           friends
           may
           have
           begot
           in
           me
           a
           〈◊〉
           opinion
           of
           some
           of
           these
           things
           the●
           they
           may
           deserve
           :
           your
           judgment
           may
           correct
           the
           errour
           of
           their
           indulgence
           .
           As
           all
           that
           desire
           to
           learn
           ,
           ought
           to
           wait
           upon
           you
           and
           your
           honourable
           societies
           ,
           as
           being
           together
           with
           the
           famous
           Universities
           and
           our
           Law
           Colledges
           ,
           I
           mean
           our
           Innes
           of
           Court
           ,
           as
           the
           soul
           and
           spirits
           of
           the
           Nation
           ;
           so
           these
           Papers
           wait
           upon
           you
           ,
           not
           presuming
           in
           the
           least
           to
           inform
           ,
           but
           submissely
           craving
           rather
           to
           learn
           of
           you
           what
           they
           ought
           to
           think
           of
           themselves
           :
           according
           to
           the
           noble
           method
           of
           Philosophy
           in
           some
           of
           your
           societies
           lately
           instituted
           and
           pursued
           ,
           not
           seeking
           for
           a
           precarious
           allowance
           ,
           as
           a
           Rogue
           and
           Felone
           may
           find
           refuge
           and
           safety
           at
           an
           holy
           sanctuary
           ;
           but
           (
           however
           without
           the
           overweening
           dotage
           of
           self
           opinion
           )
           either
           by
           their
           own
           fair
           truth
           and
           clearness
           to
           be
           justifyed
           ,
           or
           else
           willingly
           to
           embrace
           the
           flames
           ,
           which
           their
           errours
           
           have
           deserved
           .
           I
           know
           you
           will
           pardon
           the
           hasty
           rude
           dresse
           ,
           or
           rather
           nakednes
           ,
           (
           best
           suiting
           the
           pursuances
           of
           truth
           )
           in
           which
           these
           Answers
           to
           the
           questioning
           letters
           of
           some
           private
           friends
           ,
           were
           first
           of
           all
           conceived
           :
           which
           without
           falsiifyng
           could
           not
           afterwards
           be
           altered
           :
           and
           their
           highest
           ambition
           then
           was
           only
           private
           satisfaction
           ,
           though
           now
           ,
           except
           I
           will
           leave
           those
           ,
           for
           whose
           satisfaction
           they
           were
           first
           written
           ,
           unsatisfied
           ,
           they
           must
           proceed
           to
           put
           themselves
           upon
           the
           test
           for
           publick
           ,
           either
           justification
           ,
           or
           disallowance
           .
           The
           which
           I
           am
           the
           more
           easily
           perswaded
           to
           ,
           observing
           it
           a
           method
           allow'd
           and
           practiced
           both
           by
           antients
           and
           moderns
           to
           preserve
           and
           deliver
           scattered
           points
           of
           Philosophy
           by
           way
           of
           Dialogues
           and
           Epistles
           .
           And
           indeed
           this
           method
           hath
           this
           special
           advantage
           in
           it
           ;
           we
           may
           in
           a
           word
           present
           what
           we
           have
           to
           say
           that
           is
           new
           ,
           without
           loading
           the
           Reader
           with
           an
           unsavoury
           
           crambe
           ,
           which
           hath
           been
           a
           thousand
           times
           before
           in
           the
           press
           .
           As
           if
           these
           paters
           be
           able
           ,
           which
           I
           leave
           to
           your
           serious
           censure
           ,
           to
           justify
           themselves
           :
           it
           is
           scarce
           possible
           for
           them
           to
           fall
           into
           the
           hands
           of
           any
           Reader
           ,
           who
           shall
           not
           herein
           ,
           if
           he
           read
           them
           diligently
           and
           with
           understanding
           ,
           find
           ,
           not
           only
           somewhat
           which
           is
           new
           ,
           but
           as
           well
           that
           is
           memorable
           ,
           and
           beyond
           all
           doubting
           ,
           certain
           ,
           and
           perhaps
           neither
           by
           himself
           formerly
           thought
           of
           ,
           nor
           possibly
           every
           where
           in
           that
           vast
           Ocean
           of
           Books
           ,
           which
           are
           in
           the
           world
           ,
           to
           be
           met
           with
           .
           In
           these
           philosophical
           oppositions
           ,
           till
           I
           be
           called
           to
           account
           ,
           I
           have
           for
           the
           honour
           I
           bear
           them
           ,
           spared
           those
           great
           and
           venerable
           names
           both
           moderns
           and
           antients
           ,
           which
           I
           have
           herein
           opposed
           :
           it
           being
           my
           method
           in
           private
           ,
           to
           pursue
           what
           is
           publickly
           in
           you
           exemplary
           Viz.
           to
           weigh
           controversies
           and
           questions
           in
           philosophy
           ,
           not
           by
           parties
           and
           factions
           ,
           time
           and
           continuance
           ,
           
           or
           number
           and
           opinion
           ,
           but
           as
           near
           as
           possibly
           I
           can
           ,
           by
           observing
           how
           the
           scales
           bear
           between
           the
           opposite
           reasons
           :
           and
           I
           wish
           the
           method
           might
           be
           so
           happy
           ,
           as
           to
           become
           more
           common
           .
           For
           the
           same
           reasons
           I
           have
           suppressed
           the
           names
           of
           my
           ingenious
           friends
           ,
           in
           opposition
           to
           some
           of
           whose
           thoughts
           these
           things
           were
           first
           of
           all
           written
           .
           Every
           day
           make
           your
           names
           more
           illustrious
           for
           clearness
           of
           judgment
           in
           the
           mysteries
           of
           nature
           .
           Being
           the
           desire
           of
           all
           ,
           and
           doubtless
           expectation
           of
           most
           and
           particularly
           of
           your
           Orator
           and
           servant
        
         
           
             William
             Marshal
             .
          
        
      
       
         
         
           To
           his
           worthy
           and
           learned
           friends
           with
           whom
           these
           things
           were
           first
           debated
           ;
           the
           Authors
           apology
           for
           the
           publication
           of
           these
           his
           papers
           .
        
         
           TO
           some
           of
           you
           the
           name
           of
           an
           Apology
           might
           seem
           almost
           unintelligible
           ,
           or
           little
           less
           then
           mockery
           :
           your
           interest
           being
           no
           less
           in
           the
           birth
           ,
           then
           in
           the
           first
           conception
           of
           these
           philosophical
           discourses
           .
           Your
           quaerying
           curiosity
           at
           first
           impraegnated
           the
           mind
           ,
           with
           what
           your
           present
           importunity
           will
           not
           rest
           without
           committing
           to
           the
           Press
           .
           To
           the
           rest
           of
           you
           ,
           my
           wish
           is
           ,
           that
           with
           the
           favour
           of
           others
           ,
           I
           might
           not
           have
           stood
           in
           need
           of
           this
           Apology
           .
           However
           let
           not
           the
           publication
           be
           mis-construed
           ,
           as
           if
           I
           accounted
           all
           here
           debated
           to
           be
           peremptorily
           concluded
           ,
           or
           all
           that
           I
           have
           opposed
           ,
           conquered
           .
           The
           yet
           dissenting
           of
           several
           of
           you
           from
           several
           of
           the
           things
           here
           by
           me
           own'd
           ,
           might
           make
           a
           more
           hardy
           confidence
           then
           ,
           I
           hope
           ,
           I
           shall
           ever
           be
           guilty
           of
           ,
           ●o
           become
           modestly
           jealous
           
           of
           its
           own
           judgment
           .
           If
           the
           things
           here
           made
           publick
           should
           not
           be
           able
           to
           indure
           the
           piercing
           aire
           of
           this
           knowing
           age
           ,
           however
           our
           free
           and
           mutual
           intercourse
           upon
           such
           and
           the
           like
           philosophical
           doubts
           cannot
           want
           its
           justification
           .
           The
           publishing
           sometimes
           of
           a
           mistake
           is
           an
           occasion
           of
           leading
           others
           into
           the
           right
           way
           .
           And
           because
           every
           one
           cannot
           see
           cross
           the
           Mediterranean
           ,
           we
           are
           not
           therefore
           to
           walk
           either
           with
           hooded
           eyes
           ,
           or
           to
           dis-own
           the
           seeing
           of
           any
           thing
           at
           all
           ,
           or
           what
           appearances
           things
           make
           unto
           us
           .
           Though
           the
           mind
           ,
           which
           is
           the
           souls
           bed
           ,
           be
           neither
           in
           all
           of
           the
           samesize
           ,
           nor
           in
           all
           equally
           luminous
           ,
           or
           conceptive
           ,
           yet
           each
           is
           equally
           concerned
           diligently
           according
           ,
           to
           the
           proportion
           of
           his
           window
           ,
           to
           take
           a
           view
           of
           the
           goodly
           scenes
           of
           nature
           .
           In
           these
           things't
           is
           better
           to
           erre
           then
           to
           be
           idle
           :
           as
           I
           am
           sure
           you
           are
           equally
           perswaded
           with
        
         
           
             Your
             faithfull
             friend
             and
             servant
             WILLIAM
             MARSHAL
             .
          
        
      
       
         
         
           TO
           THE
           READER
           .
        
         
           THe
           publishing
           these
           papers
           hath
           now
           made
           thee
           my
           judge
           and
           censor
           :
           and
           they
           were
           no
           otherwise
           designed
           ,
           but
           for
           every
           one
           freely
           to
           exercise
           his
           judgment
           about
           them
           .
           Most
           of
           these
           Problemes
           are
           more
           frequently
           upon
           the
           Anvile
           then
           they
           are
           found
           easily
           malleable
           into
           fair
           ,
           clear
           ,
           doubtless
           conclusions
           .
           Therefore
           if
           we
           misse
           the
           goale
           ,
           it
           is
           but
           allowing
           one
           more
           in
           these
           sublime
           things
           to
           run
           a
           planetary
           course
           .
           I
           shall
           in
           a
           word
           tell
           the
           worst
           ,
           which
           can
           justly
           be
           laid
           to
           their
           charge
           ;
           some
           of
           the
           few
           things
           here
           discoursed
           ,
           have
           left
           the
           commonly
           troden
           path
           ;
           not
           as
           affecting
           novelty
           ,
           against
           which
           upon
           other
           oc●asions
           I
           appear
           for
           the
           antients
           ,
           
           but
           as
           chusing
           to
           follow
           reason
           rather
           then
           the
           name
           of
           any
           Authour
           .
           First
           therefore
           examine
           their
           reasons
           ,
           and
           then
           give
           unbiassed
           judgment
           for
           the
           truth
           ;
           which
           for
           the
           truths
           sake
           I
           shall
           be
           ready
           to
           accept
           ,
           though
           against
           me
           .
           For
           if
           those
           to
           whom
           and
           against
           whom
           ,
           though
           in
           a
           vayled
           way
           ,
           these
           papers
           were
           first
           written
           ,
           are
           supposed
           not
           to
           be
           exempted
           from
           errour
           ,
           there
           is
           no
           room
           left
           but
           only
           for
           weak
           minds
           in
           these
           secrets
           of
           nature
           to
           arrogate
           unto
           themselves
           an
           infallibility
           .
           If
           any
           complain
           of
           obscurity
           ,
           as
           both
           hast
           ,
           and
           brevity
           ,
           and
           the
           subject
           matter
           ,
           and
           the
           manner
           of
           writing
           ,
           and
           the
           quality
           of
           the
           persons
           for
           whom
           these
           things
           were
           first
           of
           all
           pen'd
           ,
           may
           make
           them
           thereunto
           lyable
           enough
           ,
           however
           I
           shall
           be
           ready
           to
           give
           both
           my
           account
           ,
           and
           if
           valuable
           ,
           my
           assistance
           to
           any
           who
           shall
           des●re
           it
           relating
           either
           to
           the
           obscurity
           ,
           or
           intricacy
           
           of
           any
           passage
           ;
           and
           to
           accept
           from
           them
           any
           reasons
           they
           shall
           be
           pleased
           to
           produce
           to
           the
           contrary
           ,
        
         
           
             Thine
             W.
             M.
             
          
           
             From
             my
             Study
             in
             Nags
             head
             Court
             in
             Gray's
             Church
             Street
             .
          
        
      
       
         
         
         
           A
           SYLLABE
           of
           the
           CONTENTS
           of
           these
           two
           SECTIONS
           .
        
         
           
             In
             the
             first
             Section
             .
          
           
             
               In
               the
               first
               Answer
               ;
               is
               treated
            
             
               Concerning
               the
               several
               Originals
               of
               Springs
               .
            
             
               Concerning
               the
               irregular
               Ebbings
               and
               Flowings
               of
               diverse
               particular
               Springs
               .
            
             
               Of
               Chymical
               multiplication
               ,
               or
               the
               encreasing
               of
               the
               quantity
               of
               liquours
               by
               destillation
               .
            
             
               Of
               the
               possibility
               of
               a
               persons
               learning
               ,
               in
               an
               hour
               or
               two
               ,
               to
               write
               his
               mind
               in
               a
               language
               he
               is
               ignorant
               of
               :
               so
               as
               ,
               what
               hee
               writeth
               in
               the
               unknown
               language
               ,
               shall
               be
               constantly
               without
               errour
               in
               the
               Grammatical
               concerns
               of
               it
               ,
               and
               the
               sense
               good
               and
               continued
               .
            
          
           
             
               In
               the
               second
               Answer
               ;
               is
               treated
            
             
               Concerning
               sympathyes
               between
               several
               parts
               of
               Animal
               bo●yes
               .
            
             
               
               Particularly
               ,
               of
               the
               sympathy
               which
               is
               between
               the
               breasts
               and
               womb
               :
               and
               whether
               it
               be
               founded
               upon
               inosculations
               of
               the
               Epigastrick
               and
               Mammary
               veins
               and
               arteryes
               .
            
             
               Whether
               from
               view
               of
               the
               Breasts
               ,
               judgment
               may
               be
               made
               concerning
               the
               sex
               of
               an
               unborn
               conception
               ?
            
             
               Of
               the
               Sympathetick
               relation
               of
               parts
               of
               the
               same
               side
               .
            
             
               Of
               the
               forming
               of
               the
               Phases
               and
               Shapes
               of
               the
               Moon
               from
               its
               mutuatitious
               light
               ,
               and
               the
               proportion
               in
               which
               it
               hath
               its
               situation
               to
               the
               eye
               :
               and
               why
               the
               Moon
               appeareth
               to
               the
               eye
               rather
               plane
               then
               Sphaeral
               .
            
          
           
             
               In
               the
               Third
               Answer
               .
            
             
               Fresh-water
               Springs
               are
               asserted
               at
               the
               bottom
               of
               the
               Sea.
               
            
             
               Whether
               by
               frequent
               transhaping
               ,
               and
               working
               upon
               Mercury
               ,
               somewhat
               be
               not
               lost
               irrecoverably
               ,
               beyond
               all
               possibility
               and
               hope
               of
               reduction
               ?
            
             
               That
               the
               doctrine
               of
               the
               four
               Elements
               as
               uncompounded
               bodyes
               ,
               cannot
               acquit
               it self
               ,
               against
               many
               rational
               doubtings
               and
               material
               controversyes
               ,
               to
               be
               a
               Doctrine
               unquestionable
               .
            
             
               That
               the
               four
               Elements
               cannot
               be
               the
               first
               Elements
               .
            
             
               Of
               the
               difficulty
               of
               bringing
               the
               firs●
               Elements
               to
               view
               .
            
             
               
               That
               ,
               in
               the
               Doctrine
               of
               the
               Ancients
               ,
               seemeth
               a
               greater
               number
               of
               first
               Elements
               asserted
               ,
               then
               four
               .
            
             
               What
               may
               have
               been
               the
               true
               rational
               intendment
               of
               the
               Ancients
               ,
               in
               their
               introducing
               the
               Doctrine
               of
               the
               four
               Elements
               .
            
          
           
             
               In
               the
               Fourth
               Answer
               ;
               is
               treated
            
             
               Of
               the
               Sutures
               in
               the
               skull
               :
               and
               their
               distinctness
               according
               to
               their
               several
               kinds
               ,
               number
               ,
               and
               order
               .
            
             
               How
               in
               the
               distinction
               of
               the
               futures
               ,
               both
               as
               to
               their
               number
               and
               order
               ,
               is
               a
               certain
               latitude
               ,
               admitting
               variety
               in
               the
               method
               of
               their
               account
               ,
               and
               the
               Anatomical
               Doctrine
               of
               them
               .
            
             
               The
               like
               latitude
               is
               observable
               in
               several
               parts
               both
               of
               Concrete
               ,
               and
               Abstract
               Mathematicks
               .
            
             
               Whether
               the
               praeeminence
               of
               right-side
               parts
               above
               the
               left
               ,
               be
               natural
               or
               arbitrary
               .
            
          
           
             
               In
               the
               Fifth
               answer
               ;
               is
               treated
            
             
               Of
               the
               consistency
               of
               immense
               volatility
               with
               immense
               ponderousnes
               .
            
             
               Of
               making
               tinctures
               by
               materation
               without
               addition
               .
            
             
               Whether
               predictions
               may
               be
               concerning
               Comets
               before
               their
               appearance
               ?
            
             
               What
               is
               her●in
               to
               be
               judged
               ,
               of
               Comets
               which
               are
               by
               new
               amassement
               ?
            
             
               What
               is
               her●in
               to
               be
               judged
               ,
               of
               Comets
               which
               are
               
               supposed
               to
               be
               permanent
               Bodyes
               ,
               and
               not
               to
               be
               new
               amassements
               .
            
          
           
             
               In
               the
               Sixth
               Answer
               ;
               is
               treated
            
             
               Of
               the
               motion
               of
               the
               Celestial
               Bodyes
               about
               their
               particular
               Axes
               .
            
             
               Whether
               such
               their
               Circumaxall
               motion
               afford
               a
               convictive
               demonstration
               and
               proof
               of
               the
               Copernican
               Systeme
               ?
            
             
               That
               the
               Earths
               motion
               is
               not
               sufficiently
               proved
               ,
               upon
               the
               Hypothesis
               granted
               ,
               That
               Comets
               are
               nearest
               the
               Earth
               ,
               when
               they
               are
               in
               opposition
               unto
               the
               Sun.
               
            
          
        
         
           
           
             In
             the
             Second
             SECTION
             .
          
           
             
               In
               the
               Seventh
               Answer
               ,
               is
               treated
            
             
               COncerning
               Angles
               of
               Contact
               .
            
             
               In
               order
               thereunto
               ,
               are
               given
               several
               definitions
               of
               Plane
               Angles
               ,
               and
               distinct
               names
               to
               them
               ,
               according
               to
               those
               diversityes
               of
               their
               kinds
               and
               constitutions
               ,
               which
               ,
               in
               this
               Question
               ,
               are
               more
               especially
               material
               and
               necessary
               to
               be
               distinguished
               .
            
             
               The
               state
               of
               the
               Question
               is
               set
               forth
               .
            
             
               It
               is
               shewn
               ,
               that
               Recto-convexe
               Angles
               of
               Contact
               are
               truly
               Angles
               according
               to
               the
               definition
               of
               Plane
               Angles
               .
            
             
               That
               Recto-Convexe
               Angles
               of
               Contact
               are
               neither
               destitute
               of
               Quantity
               ,
               nor
               their
               sides
               coincident
               .
            
             
               In
               what
               several
               wayes
               ,
               one
               Quantity
               may
               be
               greater
               then
               another
               .
            
             
               Of
               Homogeneity
               .
            
             
               What
               kind
               of
               Homogeneity
               it
               is
               ,
               that
               is
               requisite
               for
               proportionals
               ?
            
             
               That
               Angles
               have
               Figuration
               ,
               as
               well
               as
               Quantity
               .
            
             
               That
               in
               Angles
               is
               observable
               an
               Homogeneity
               or
               
               Heterogeneity
               which
               is
               Mathematical
               ,
               and
               also
               an
               Homogeneity
               or
               Heterogeneity
               which
               is
               not
               Mathematical
               ,
               nor
               any
               way
               concerning
               their
               Quantity
               ,
               but
               only
               their
               figuration
               ,
               &c.
               
            
             
               How
               three
               Angles
               being
               all
               Equal
               ;
               the
               Equality
               between
               the
               first
               and
               second
               ,
               may
               be
               more
               absolute
               then
               the
               Equality
               between
               the
               first
               and
               third
               being
               only
               respective
               .
            
             
               How
               some
               Angles
               are
               constituted
               by
               the
               Concrescency
               ,
               Composition
               ,
               and
               Annexion
               of
               several
               parts
               and
               Angles
               ,
               which
               are
               Heterogeneal
               each
               to
               other
               .
            
             
               VVhether
               like
               Angles
               ,
               in
               unequal
               Circles
               and
               in
               like
               segments
               of
               Circles
               be
               alwayes
               equal
               ;
            
             
               That
               all
               Equal
               Angles
               are
               not
               like
               .
            
             
               That
               all
               like
               Angles
               are
               not
               equal
               .
            
             
               VVhether
               this
               Controversy
               about
               the
               Recto-Convexe
               Angle
               of
               Contact
               ,
               belong
               to
               Concrete
               or
               Abstract
               Mathematicks
               ?
            
             
               VVhat
               was
               the
               first
               mistake
               ,
               that
               first
               ministred
               occasion
               for
               the
               starting
               of
               this
               controversy
               ,
               which
               hath
               been
               so
               long
               reteined
               ,
               in
               these
               Learnings
               ,
               in
               which
               nothing
               is
               more
               monstrous
               and
               unusual
               then
               pertinacious
               controversy
               ;
            
          
        
      
       
         
         
           These
           Escapes
           of
           the
           Press
           crave
           the
           help
           of
           the
           Readers
           Pen.
           viz
           ,
        
         
           PAg.
           10.
           line
           .
           11.
           read
           .
           aerial
           .
           p.
           17.
           l.
           2.
           r.
           heat
           .
           l.
           16.17
           .
           r.
           reason
           .
           p.
           18.
           l.
           10.
           r.
           easily
           .
           p.
           14.
           l.
           19.
           expunge
           the
           semicolon
           .
           p.
           21.
           l.
           28.
           r.
           Sympathy
           .
           p.
           22.
           l.
           15.16
           .
           r.
           tumultuous
           .
           l.
           24.
           r.
           those
           .
           p.
           23.
           l.
           29.
           r.
           concerning
           .
           p.
           24.
           l.
           18.
           r.
           falls
           .
           p.
           29.
           l.
           7.
           r.
           confessedly
           .
           p.
           33.
           l.
           penult
           .
           r.
           flushing
           .
           p.
           35.
           l.
           13.
           r.
           that
           that
           .
           l.
           28.
           r.
           fire
           .
           p.
           36.
           l.
           13.
           r.
           us
           .
           p.
           39.
           l.
           1.
           expunge
           the
           colon
           :
           l.
           16.
           r.
           foetid
           .
           l.
           26.
           r.
           their
           .
           l.
           27
           r.
           the.
           p.
           41.
           l.
           7.
           r.
           with
           the
           string
           .
           p.
           43.
           l.
           12.
           r.
           to
           the
           violating
           .
           p.
           44.
           l.
           2.
           and
           7.
           r.
           inanimates
           .
           p.
           49.
           l.
           3.
           r.
           seame
           p.
           56.
           l.
           9.
           r.
           certainly
           .
           p.
           57.
           l.
           15.
           for
           is
           regularly
           .
           r.
           irregugularly
           .
           p.
           61.
           l.
           7.
           for
           if
           seeming
           .
           r.
           it
           seeming
           .
           l.
           19.
           r.
           exact
           .
           p.
           69.
           l.
           5.
           r.
           limited
           .
           p.
           71.
           l.
           3.
           r.
           the
           Copernican
           .
           p.
           75.
           l.
           13.
           r.
           Jago
           .
           p.
           80.
           l.
           19.
           r.
           Lordships
           .
           p.
           82.
           l.
           17.
           r.
           like
           .
           p.
           86.
           l.
           5.
           r.
           BA
           .
           p.
           89.
           l.
           16.
           for
           mooving
           by
           .
           r.
           by
           mooving
           .
           p.
           90.
           l.
           2.
           r.
           an
           isoclitical
           .
           l.
           28.
           r.
           Be
           again
           .
           p.
           92.
           l.
           30.
           for
           AHH
           .
           r.
           AHF
           .
           p.
           100.
           l.
           6.
           r.
           even
           .
           l.
           19.
           r.
           say
           .
           p.
           101.
           l.
           1.
           r.
           one
           or
           each
           .
           p.
           124.
           l.
           29.
           r.
           recto-concave
           .
           p.
           155.
           l.
           26.
           and
           28.
           r.
           crooked-lined
           .
           p.
           160.
           l.
           18.
           r.
           not
           other
           .
           p.
           187.
           l.
           26.
           r.
           a
           mixt-lined
           secant
           angle
           or
           of
           p.
           200.
           l.
           26.
           r.
           crooked-lined
           .
           p.
           232.
           l.
           1.
           r.
           constant
           .
        
         
           The
           faults
           of
           the
           Orthography
           are
           referred
           to
           the
           Readers
           ingenuity
           .
        
      
       
         
         
           
        
         
         
           
        
      
    
     
       
         
           
             
             
             
               The
               First
               SECTION
               of
               PHILOSOPHICAL
               ANSWERS
               CONTEINIHG
               The
               first
               Six
               Answers
               upon
               several
               heads
               in
               Philosophy
               .
            
             
               By
               WILLIAM
               MARSHAL
            
             
               LONDON
               ,
               Printed
               by
               
                 T.
                 L.
              
               for
               
                 Nathaniel
                 Brooks
              
               ,
               at
               the
               Angel
               in
               Cornhil
               near
               the
               Royal
               Exchange
               .
               1670.
               
            
          
        
         
           
             
             
             
               ANSWERS
               Upon
               several
               heads
               in
               PHILOSOPHY
            
             
               
                 The
                 first
                 Answer
                 .
              
               
                 
                   Concerning
                   the
                   original
                   of
                   Springs
                   :
                   that
                   all
                   Springs
                   have
                   not
                   the
                   same
                   nor
                   the
                   like
                   original
                   :
                   that
                   all
                   Springs
                   are
                   not
                   from
                   the
                   Sea
                   :
                   that
                   several
                   things
                   may
                   by
                   the
                   way
                   contribute
                   as
                   originals
                   to
                   the
                   same
                   Spring
                   ,
                   whether
                   by
                   way
                   of
                   dissolution
                   ,
                   or
                   condensation
                   .
                   Also
                   of
                   the
                   ebbings
                   and
                   flowings
                   of
                   particular
                   Springs
                   :
                   in
                   some
                   more
                   especially
                   observed
                   to
                   be
                   in
                   an
                   unaccountable
                   irregularity
                   .
                   Also
                   of
                   Chymical
                   multiplication
                   ,
                   or
                   the
                   distilling
                   of
                   liquors
                   ,
                   so
                   as
                   by
                   distillation
                   to
                   encrease
                   their
                   quantity
                   to
                   any
                   given
                   proportion
                   .
                   That
                   Mercury
                   is
                   not
                   explicable
                   by
                   the
                   Doctrine
                   of
                   the
                   four
                   Elements
                   .
                   That
                   in
                   an
                   hour
                   or
                   two's
                   time
                   a
                   person
                   of
                   indifferent
                   parts
                   ,
                   able
                   to
                   read
                   and
                   write
                   his
                   native
                   language
                   only
                   ,
                   may
                   be
                   taught
                   to
                   write
                   his
                   mind
                   in
                   a
                   forein
                   language
                   ,
                   so
                   as
                   what
                   he
                   writeth
                   shall
                   be
                   true
                   in
                   the
                   forein
                   language
                   ,
                   and
                   the
                   sense
                   good
                   and
                   continued
                   .
                
              
               
                 AS
                 Nature
                 in
                 generation
                 worketh
                 the
                 first
                 beginnings
                 of
                 things
                 usually
                 in
                 a
                 very
                 dark
                 loom
                 ,
                 not
                 permitting
                 common
                 eyes
                 to
                 have
                 a
                 view
                 of
                 its
                 un●aught
                 art
                 in
                 ordering
                 and
                 warping
                 its
                 first
                 filaments
                 ,
                 covering
                 with
                 shells
                 and
                 membranes
                 and
                 divers
                 veils
                 the
                 mysterious
                 nature
                 
                 of
                 first
                 productions
                 :
                 The
                 same
                 judgement
                 may
                 be
                 of
                 Springs
                 ,
                 which
                 though
                 every
                 where
                 obvious
                 to
                 the
                 eye
                 ,
                 by
                 their
                 pleasant
                 murmurings
                 ,
                 crystalline
                 pureness
                 ,
                 and
                 perpetual
                 flux
                 ,
                 delighting
                 the
                 mind
                 ,
                 as
                 well
                 as
                 the
                 sense
                 ;
                 yet
                 whence
                 these
                 Rise
                 ,
                 and
                 how
                 they
                 are
                 fed
                 ,
                 and
                 wherewith
                 furnished
                 ,
                 is
                 more
                 obscure
                 and
                 aenigmatical
                 than
                 to
                 be
                 resolved
                 ,
                 as
                 many
                 easily
                 imagine
                 ,
                 in
                 a
                 word
                 .
                 And
                 yet
                 I
                 suppose
                 ,
                 an
                 erroneous
                 Hypothesis
                 may
                 be
                 the
                 chief
                 ground
                 of
                 difficulty
                 in
                 this
                 Question
                 ,
                 while
                 many
                 generally
                 expect
                 that
                 all
                 Springs
                 should
                 be
                 of
                 the
                 same
                 or
                 like
                 original
                 .
                 But
                 why
                 is
                 that
                 more
                 reasonable
                 than
                 to
                 expect
                 that
                 all
                 Springs
                 should
                 send
                 forth
                 waters
                 of
                 the
                 same
                 taste
                 ,
                 colour
                 ,
                 or
                 virtue
                 ?
                 Too
                 many
                 Instances
                 ,
                 Examples
                 and
                 Experiments
                 may
                 be
                 produced
                 of
                 Springs
                 owing
                 their
                 original
                 to
                 the
                 Sea
                 ,
                 percolated
                 thorow
                 the
                 Earth
                 ,
                 to
                 deny
                 ,
                 or
                 call
                 in
                 question
                 so
                 evident
                 a
                 Truth
                 ,
                 without
                 a
                 manifest
                 crazing
                 of
                 our
                 own
                 judgement
                 :
                 yet
                 that
                 the
                 Sea
                 by
                 such
                 percolations
                 is
                 not
                 the
                 original
                 of
                 all
                 Springs
                 ,
                 is
                 as
                 manifest
                 ,
                 as
                 true
                 ,
                 and
                 confirm'd
                 by
                 as
                 many
                 and
                 weighty
                 Experiences
                 ;
                 not
                 only
                 of
                 fresh
                 Springs
                 near
                 the
                 Sea
                 ,
                 and
                 in-land
                 salt-springs
                 ;
                 but
                 especially
                 if
                 we
                 consider
                 the
                 numerous
                 Springs
                 ,
                 which
                 are
                 in
                 the
                 tops
                 ,
                 or
                 sides
                 of
                 several
                 Mountains
                 ,
                 vastly
                 above
                 the
                 Seas
                 level
                 ;
                 and
                 therefore
                 not
                 possible
                 in
                 any
                 Channel
                 to
                 be
                 naturally
                 raised
                 so
                 high
                 ,
                 without
                 all
                 stay
                 ,
                 or
                 impediment
                 
                 of
                 percolation
                 .
                 Of
                 which
                 I
                 have
                 observed
                 a
                 special
                 instance
                 in
                 a
                 low
                 ,
                 somewhat
                 rocky
                 ,
                 wedge-like
                 Mountain
                 ,
                 situate
                 along
                 the
                 brim
                 of
                 the
                 full
                 Sea
                 ;
                 but
                 on
                 the
                 Land-side
                 ,
                 at
                 a
                 little
                 distance
                 ,
                 set
                 about
                 with
                 numerous
                 and
                 vast
                 Mountains
                 :
                 in
                 which
                 little
                 wedge-like
                 Mountain
                 ,
                 though
                 most
                 part
                 immediately
                 encompassed
                 with
                 a
                 Plain
                 ,
                 very
                 considerably
                 above
                 the
                 level
                 of
                 the
                 highest
                 ,
                 aequinoctial
                 ,
                 or
                 other
                 Tides
                 ;
                 was
                 a
                 perpetual
                 Spring
                 of
                 most
                 lympid
                 ,
                 clear
                 water
                 ,
                 not
                 unfamed
                 for
                 its
                 singular
                 medical
                 vertues
                 .
                 It
                 is
                 scarce
                 consentaneous
                 to
                 sense
                 ,
                 reason
                 and
                 experience
                 ,
                 to
                 derive
                 such
                 Fountains
                 out
                 of
                 the
                 Sea
                 only
                 by
                 way
                 of
                 percolation
                 ;
                 which
                 though
                 it
                 be
                 allowed
                 to
                 contribute
                 ,
                 in
                 some
                 places
                 ,
                 to
                 the
                 sweetning
                 of
                 the
                 fountainous
                 flux
                 ,
                 cannot
                 be
                 conceived
                 apt
                 to
                 raise
                 the
                 water
                 to
                 an
                 higher
                 level
                 than
                 it
                 had
                 before
                 .
                 Not
                 disallowing
                 the
                 former
                 therefore
                 in
                 its
                 place
                 ;
                 there
                 is
                 yet
                 besides
                 it
                 some
                 other
                 Original
                 of
                 Springs
                 to
                 be
                 inquired
                 after
                 .
                 And
                 how
                 perpetual
                 Springs
                 should
                 come
                 to
                 be
                 in
                 the
                 tops
                 of
                 the
                 highest
                 Mountains
                 of
                 all
                 ,
                 as
                 the
                 case
                 of
                 most
                 difficult
                 explication
                 ,
                 cannot
                 with
                 greater
                 reason
                 and
                 clearness
                 be
                 explain'd
                 ,
                 than
                 by
                 bringing
                 the
                 waters
                 thither
                 ,
                 not
                 in
                 a
                 watry
                 form
                 ,
                 but
                 as
                 vapours
                 and
                 exhalations
                 ,
                 at
                 leastwise
                 the
                 most
                 constantly
                 ,
                 and
                 in
                 the
                 greatest
                 proportion
                 ;
                 after
                 the
                 manner
                 of
                 a
                 Distillation
                 ,
                 ordered
                 and
                 managed
                 by
                 the
                 institutes
                 of
                 Nature
                 ;
                 
                 and
                 this
                 without
                 Espousing
                 the
                 Tenets
                 of
                 our
                 new
                 Platonists
                 ,
                 that
                 will
                 have
                 several
                 perpetuous
                 continued
                 Orbes
                 of
                 Fires
                 ,
                 diversly
                 graduated
                 ,
                 for
                 special
                 Theological
                 ends
                 ,
                 contained
                 still
                 between
                 two
                 and
                 two
                 perpetuous
                 and
                 continued
                 Crusts
                 ,
                 or
                 sphaeral
                 shels
                 of
                 Earth
                 ,
                 the
                 one
                 above
                 the
                 fire
                 ,
                 the
                 other
                 beneath
                 it
                 ,
                 in
                 respect
                 of
                 the
                 Earths
                 Center
                 .
                 And
                 those
                 that
                 know
                 how
                 under
                 the
                 surface
                 of
                 the
                 Earth
                 ,
                 whether
                 plain
                 ,
                 or
                 mountainous
                 ,
                 are
                 frequent
                 stores
                 ,
                 both
                 of
                 Materials
                 ,
                 and
                 naturally
                 formed
                 Cavities
                 ,
                 and
                 other
                 Instruments
                 ,
                 easily
                 accommodable
                 to
                 such
                 a
                 work
                 ,
                 will
                 not
                 conclude
                 that
                 to
                 be
                 a
                 conjecture
                 of
                 fancy
                 ,
                 which
                 not
                 only
                 for
                 its
                 possibility
                 ,
                 but
                 the
                 high
                 degree
                 of
                 its
                 probability
                 ,
                 may
                 justly
                 claim
                 to
                 be
                 entertain'd
                 as
                 a
                 most
                 real
                 and
                 undoubted
                 physical
                 Truth
                 .
                 For
                 what
                 doubt
                 can
                 be
                 made
                 of
                 subterraneous
                 heats
                 and
                 fires
                 ?
                 and
                 hot
                 Baths
                 and
                 Springs
                 ,
                 attest
                 the
                 ,
                 not
                 only
                 warmth
                 ,
                 which
                 is
                 sufficient
                 for
                 our
                 purpose
                 ,
                 but
                 even
                 the
                 actual
                 ,
                 and
                 frequently
                 intense
                 ,
                 heat
                 ,
                 and
                 ebullition
                 of
                 liquors
                 within
                 the
                 Caverns
                 of
                 the
                 Earth
                 ,
                 and
                 the
                 perpetuity
                 ,
                 or
                 constant
                 continuance
                 of
                 such
                 ebullitions
                 :
                 and
                 as
                 the
                 Earth
                 is
                 well
                 known
                 to
                 be
                 furnished
                 with
                 many
                 natural
                 Caverns
                 ,
                 so
                 in
                 those
                 Caverns
                 ,
                 for
                 shape
                 ,
                 proportion
                 ,
                 situation
                 ,
                 and
                 other
                 the
                 like
                 circumstances
                 ,
                 we
                 cannot
                 but
                 imagine
                 there
                 is
                 great
                 diversity
                 .
                 So
                 how
                 variously
                 and
                 plenteously
                 the
                 Earth
                 is
                 watered
                 
                 with
                 subterraneous
                 juyces
                 ,
                 and
                 moistures
                 ,
                 cannot
                 be
                 denied
                 by
                 the
                 observant
                 ,
                 who
                 almost
                 every
                 where
                 under
                 ground
                 find
                 lesser
                 veins
                 of
                 water
                 dispersed
                 up
                 and
                 down
                 the
                 Body
                 of
                 the
                 Earth
                 ,
                 and
                 in
                 many
                 places
                 large
                 concealed
                 Rivers
                 ,
                 elsewhere
                 unfathomable
                 Vaults
                 ,
                 and
                 Abysses
                 :
                 Sometime
                 seen
                 Rivers
                 in
                 their
                 proper
                 and
                 perpetual
                 Channels
                 ,
                 make
                 a
                 dark
                 course
                 and
                 discharge
                 of
                 their
                 waters
                 into
                 the
                 unseen
                 Bowels
                 of
                 the
                 Earth
                 .
                 Upon
                 the
                 whole
                 ,
                 in
                 the
                 method
                 of
                 this
                 explication
                 ,
                 waters
                 are
                 ,
                 in
                 the
                 same
                 manner
                 secretly
                 within
                 the
                 Earth
                 ,
                 raised
                 to
                 the
                 top
                 of
                 the
                 Mountains
                 ,
                 in
                 which
                 ,
                 to
                 our
                 view
                 ,
                 they
                 are
                 raised
                 in
                 the
                 open
                 Air
                 to
                 those
                 regions
                 ,
                 from
                 whence
                 they
                 fall
                 back
                 again
                 upon
                 us
                 in
                 the
                 form
                 of
                 rain
                 .
                 And
                 in
                 such
                 vapid
                 exhalations
                 ,
                 it
                 cannot
                 be
                 denied
                 but
                 in
                 some
                 places
                 ,
                 and
                 at
                 some
                 times
                 ,
                 the
                 Mountain
                 springs
                 ranging
                 in
                 a
                 much
                 higher
                 level
                 ,
                 may
                 by
                 possibility
                 be
                 furnished
                 from
                 the
                 percolated
                 sea-water
                 :
                 but
                 as
                 the
                 possibility
                 of
                 this
                 is
                 admitted
                 ,
                 so
                 the
                 universal
                 and
                 absolute
                 necessity
                 of
                 it
                 ,
                 may
                 not
                 be
                 averred
                 :
                 it
                 seeming
                 hard
                 to
                 say
                 ,
                 that
                 the
                 Mountain-springs
                 what
                 distance
                 soever
                 from
                 the
                 Sea
                 ,
                 have
                 no
                 other
                 way
                 of
                 being
                 furnished
                 ,
                 but
                 only
                 from
                 thence
                 .
                 And
                 ,
                 I
                 suppose
                 ,
                 it
                 will
                 as
                 uneasily
                 obtain
                 credit
                 ,
                 that
                 the
                 Sea
                 should
                 run
                 as
                 fast
                 every
                 way
                 ,
                 under
                 the
                 Earth
                 ,
                 to
                 the
                 Originals
                 of
                 Springs
                 ,
                 as
                 the
                 Springs
                 generally
                 do
                 in
                 seeking
                 out
                 their
                 way
                 unto
                 the
                 Sea.
                 I
                 doubt
                 not
                 ,
                 
                 but
                 besides
                 the
                 Seas
                 ,
                 there
                 are
                 under
                 the
                 earth
                 many
                 other
                 liquors
                 ,
                 contributing
                 to
                 the
                 origination
                 of
                 Springs
                 ,
                 some
                 having
                 their
                 first
                 rise
                 from
                 dissolution
                 ,
                 others
                 from
                 condensation
                 ,
                 whence
                 ,
                 and
                 from
                 the
                 passages
                 thorough
                 which
                 they
                 run
                 ,
                 and
                 are
                 percolated
                 ,
                 rise
                 very
                 often
                 those
                 special
                 vertues
                 and
                 dangers
                 of
                 some
                 peculiar
                 Springs
                 :
                 and
                 sometimes
                 in
                 a
                 seeming
                 praeposterous
                 way
                 ,
                 though
                 very
                 consentaneous
                 to
                 the
                 true
                 nature
                 of
                 things
                 ,
                 Springs
                 adjoyning
                 to
                 the
                 Sea
                 are
                 fresh
                 ,
                 and
                 at
                 a
                 vast
                 distance
                 from
                 the
                 Sea
                 ,
                 in
                 in-land
                 Countreys
                 sometimes
                 salt
                 .
                 But
                 besides
                 both
                 the
                 Seas
                 ,
                 &
                 those
                 other
                 waters
                 bred
                 under
                 the
                 earth
                 ;
                 it
                 s
                 more
                 than
                 probable
                 ▪
                 the
                 rains
                 falling
                 down
                 in
                 showers
                 from
                 Heaven
                 ,
                 adde
                 not
                 a
                 little
                 to
                 the
                 flushing
                 and
                 continuance
                 of
                 the
                 Springs
                 .
                 So
                 medical
                 Springs
                 after
                 rains
                 are
                 noted
                 for
                 a
                 while
                 to
                 be
                 of
                 less
                 virtue
                 ;
                 and
                 in
                 long
                 droughts
                 ,
                 't
                 is
                 usual
                 for
                 very
                 many
                 Springs
                 to
                 be
                 quite
                 dried
                 up
                 ,
                 till
                 rains
                 fall
                 again
                 :
                 and
                 where
                 ,
                 by
                 long
                 obscure
                 dens
                 ,
                 caverns
                 and
                 passages
                 under
                 ground
                 ,
                 access
                 may
                 be
                 had
                 to
                 subterraneous
                 Rivers
                 ;
                 such
                 Rivers
                 ,
                 though
                 lying
                 vastly
                 deep
                 under
                 the
                 Earths
                 surface
                 ,
                 they
                 are
                 sound
                 to
                 swell
                 at
                 after
                 rains
                 ,
                 and
                 by
                 their
                 accession
                 unto
                 the
                 wonted
                 stream
                 ,
                 frequently
                 do
                 make
                 a
                 most
                 hideous
                 and
                 horrid
                 noise
                 ,
                 full
                 of
                 terrour
                 to
                 those
                 in
                 the
                 Cave●
                 mouth
                 ,
                 and
                 at
                 other
                 times
                 unusual
                 .
                 From
                 all
                 which
                 ,
                 seriously
                 and
                 impartially
                 weighed
                 ,
                 is
                 made
                 undeniably
                 
                 apparent
                 ,
                 that
                 rains
                 ,
                 in
                 no
                 contemptible
                 proportion
                 furnish
                 forth
                 matter
                 to
                 be
                 ordinarily
                 by
                 channelly
                 veins
                 convey'd
                 ,
                 or
                 else
                 extraordinarily
                 by
                 a
                 natural
                 distillation
                 wrought
                 up
                 into
                 the
                 Springs
                 .
                 And
                 as
                 this
                 much
                 discussed
                 Question
                 is
                 resolved
                 in
                 these
                 easie
                 things
                 ,
                 every
                 where
                 offering
                 themselves
                 to
                 view
                 in
                 Nature
                 ;
                 so
                 I
                 doubt
                 not
                 ,
                 upon
                 the
                 same
                 principles
                 might
                 also
                 many
                 other
                 things
                 ,
                 seeming
                 at
                 first
                 sight
                 to
                 be
                 very
                 mysterious
                 ,
                 in
                 the
                 concern
                 of
                 Springs
                 ,
                 as
                 that
                 which
                 with
                 so
                 much
                 admiration
                 is
                 by
                 the
                 rude
                 ignorant
                 people
                 cryed
                 up
                 and
                 observed
                 in
                 some
                 ,
                 having
                 ,
                 as
                 they
                 term
                 it
                 ,
                 their
                 Ebbings
                 and
                 Flowings
                 in
                 such
                 irregular
                 frequency
                 as
                 by
                 no
                 art
                 can
                 possibly
                 be
                 reduced
                 to
                 any
                 certainty
                 of
                 account
                 ,
                 or
                 order
                 :
                 for
                 the
                 more
                 special
                 and
                 clear
                 observing
                 of
                 which
                 they
                 are
                 wont
                 to
                 receive
                 the
                 water
                 from
                 the
                 Spring
                 into
                 some
                 stone
                 ,
                 or
                 such
                 like
                 vessel
                 proportionately
                 bored
                 in
                 ,
                 or
                 near
                 the
                 bottom
                 :
                 all
                 which
                 in
                 truth
                 is
                 without
                 any
                 retrogradation
                 and
                 reciprocation
                 of
                 motion
                 in
                 its
                 channelly
                 veins
                 ;
                 being
                 nothing
                 else
                 but
                 an
                 inaequality
                 of
                 the
                 waters
                 issuing
                 from
                 the
                 Spring
                 in
                 equal
                 times
                 ,
                 either
                 by
                 reason
                 of
                 its
                 more
                 plenteous
                 filling
                 at
                 some
                 times
                 the
                 channelly
                 veins
                 of
                 the
                 Spring
                 as
                 it
                 passeth
                 ,
                 or
                 else
                 by
                 reason
                 it
                 passeth
                 with
                 a
                 greater
                 or
                 lesser
                 impetus
                 ;
                 or
                 possibly
                 in
                 some
                 cases
                 and
                 places
                 upon
                 both
                 grounds
                 :
                 which
                 what
                 it
                 hath
                 in
                 it
                 that
                 is
                 more
                 admirable
                 than
                 the
                 running
                 of
                 
                 any
                 ordinary
                 Brook
                 ,
                 sometimes
                 with
                 a
                 fuller
                 channel
                 ,
                 sometimes
                 with
                 a
                 stronger
                 stream
                 ,
                 I
                 do
                 not
                 yet
                 understand
                 :
                 and
                 how
                 easily
                 ,
                 upon
                 the
                 former
                 principles
                 ,
                 this
                 is
                 explicable
                 any
                 one
                 may
                 readily
                 perceive
                 :
                 that
                 we
                 may
                 more
                 justly
                 wonder
                 how
                 some
                 grave
                 Authors
                 came
                 to
                 be
                 so
                 transported
                 with
                 the
                 sight
                 or
                 fame
                 of
                 some
                 such
                 springs
                 ,
                 occurring
                 up
                 and
                 down
                 in
                 the
                 World
                 ,
                 as
                 in
                 their
                 reports
                 to
                 offer
                 them
                 to
                 the
                 thoughts
                 of
                 distant
                 Students
                 ,
                 and
                 succeeding
                 times
                 ,
                 as
                 containing
                 in
                 them
                 little
                 less
                 than
                 miracle
                 .
                 And
                 if
                 the
                 studious
                 would
                 observe
                 diligently
                 what
                 is
                 in
                 nature
                 ,
                 possibly
                 many
                 other
                 things
                 ,
                 seemingly
                 as
                 intricate
                 ,
                 might
                 be
                 capable
                 of
                 as
                 easie
                 explications
                 ;
                 especially
                 considering
                 the
                 vast
                 way
                 a
                 Spring
                 may
                 run
                 under
                 the
                 Earth
                 before
                 it
                 break
                 forth
                 ;
                 fairly
                 insinuated
                 to
                 us
                 by
                 the
                 pure
                 fresh
                 water
                 Springs
                 ,
                 bursting
                 up
                 a
                 great
                 way
                 within
                 the
                 floud
                 marks
                 of
                 the
                 Sea
                 ;
                 clearly
                 intimating
                 to
                 us
                 ,
                 how
                 two
                 Hills
                 at
                 a
                 considerable
                 distance
                 may
                 both
                 be
                 concerned
                 in
                 the
                 same
                 Spring
                 and
                 its
                 Course
                 .
                 And
                 whereas
                 we
                 number
                 and
                 place
                 Springs
                 ac-according
                 as
                 we
                 observe
                 them
                 to
                 break
                 forth
                 and
                 shew
                 themselves
                 unto
                 the
                 day
                 ;
                 in
                 the
                 true
                 aestimate
                 ,
                 and
                 upon
                 laborious
                 search
                 ,
                 it
                 hath
                 many
                 times
                 appeared
                 quite
                 otherwise
                 ;
                 that
                 the
                 original
                 of
                 the
                 Spring
                 ,
                 has
                 been
                 at
                 a
                 great
                 distance
                 from
                 the
                 eruption
                 ,
                 and
                 the
                 eruption
                 has
                 not
                 been
                 till
                 after
                 the
                 confluction
                 and
                 meeting
                 
                 of
                 ,
                 it
                 may
                 be
                 three
                 or
                 four
                 several
                 Springs
                 ,
                 every
                 one
                 carrying
                 in
                 it
                 sometimes
                 the
                 dissolution
                 ,
                 sometimes
                 the
                 spirit
                 ,
                 or
                 somewhat
                 of
                 the
                 first
                 rudiments
                 of
                 some
                 special
                 Mineral
                 ;
                 and
                 altogether
                 make
                 up
                 ,
                 not
                 seldome
                 ,
                 an
                 almost
                 inimitable
                 composition
                 .
                 Your
                 other
                 Quaery
                 concerning
                 Chymical
                 Multiplication
                 ,
                 or
                 the
                 distilling
                 of
                 Water
                 from
                 a
                 Pint
                 to
                 a
                 Quart
                 ,
                 if
                 my
                 Genius
                 fail
                 me
                 not
                 ,
                 when
                 rightly
                 proposed
                 and
                 understood
                 ,
                 cannot
                 want
                 some
                 affinity
                 with
                 the
                 matters
                 we
                 have
                 already
                 been
                 treating
                 of
                 .
                 For
                 if
                 any
                 would
                 impose
                 that
                 the
                 same
                 Water
                 as
                 by
                 way
                 of
                 expansion
                 ,
                 without
                 other
                 additional
                 water
                 or
                 matter
                 ,
                 might
                 by
                 distillation
                 be
                 brought
                 to
                 a
                 double
                 ,
                 treble
                 ,
                 and
                 so
                 a
                 thousand
                 times
                 as
                 large
                 dimensions
                 as
                 before
                 ,
                 as
                 Wines
                 ,
                 Vinegars
                 ,
                 and
                 other
                 Liquors
                 ,
                 have
                 in
                 them
                 ,
                 without
                 any
                 further
                 addition
                 ,
                 so
                 much
                 Phlegme
                 ,
                 and
                 so
                 much
                 Spirit
                 of
                 such
                 a
                 strength
                 ,
                 whether
                 fixed
                 ,
                 or
                 volatile
                 ;
                 it
                 would
                 require
                 a
                 very
                 credulous
                 judgement
                 to
                 allow
                 the
                 veracity
                 ,
                 or
                 possibility
                 of
                 such
                 a
                 Probleme
                 :
                 and
                 I
                 am
                 clearly
                 free
                 ,
                 to
                 disown
                 the
                 skill
                 or
                 power
                 of
                 any
                 such
                 Chymistry
                 ;
                 which
                 once
                 admitted
                 ,
                 were
                 it
                 not
                 for
                 the
                 shortness
                 of
                 humane
                 life
                 ,
                 and
                 the
                 hopelesness
                 of
                 obtaining
                 Vessels
                 ,
                 and
                 Instruments
                 large
                 enough
                 for
                 the
                 work
                 ,
                 and
                 conveniency
                 to
                 place
                 them
                 ,
                 the
                 world
                 might
                 be
                 in
                 danger
                 of
                 Drowning
                 from
                 the
                 lofty
                 Artist
                 ;
                 as
                 once
                 it
                 was
                 hypothetically
                 threatned
                 ,
                 with
                 
                 shaking
                 by
                 the
                 noble
                 and
                 ingenious
                 Mechanist
                 .
                 But
                 I
                 had
                 rather
                 understand
                 a
                 Probleme
                 favourably
                 ,
                 so
                 as
                 it
                 may
                 carry
                 truth
                 ,
                 possibility
                 ,
                 and
                 reason
                 along
                 with
                 it
                 :
                 and
                 so
                 methinks
                 may
                 this
                 be
                 interpretable
                 ,
                 to
                 wit
                 ,
                 subtilly
                 to
                 contrive
                 ,
                 find
                 out
                 ,
                 and
                 order
                 expedients
                 ,
                 ingenious
                 helps
                 ,
                 and
                 advantages
                 ,
                 to
                 condense
                 free
                 ,
                 and
                 open
                 exhalations
                 into
                 a
                 watery
                 form
                 with
                 dispatch
                 and
                 expeditely
                 ,
                 especially
                 at
                 any
                 season
                 ,
                 or
                 time
                 of
                 the
                 year
                 :
                 so
                 as
                 by
                 this
                 Art
                 out
                 of
                 the
                 arrial
                 vapours
                 within
                 a
                 limited
                 time
                 to
                 give
                 any
                 limited
                 measure
                 of
                 water
                 ,
                 which
                 reason
                 can
                 expect
                 from
                 Art
                 ,
                 or
                 without
                 abjuring
                 and
                 putting
                 off
                 its
                 own
                 nature
                 ,
                 handsomely
                 make
                 demand
                 of
                 .
                 And
                 that
                 this
                 is
                 the
                 Philosophy
                 of
                 Nature
                 appears
                 in
                 every
                 days
                 dewes
                 ,
                 the
                 trickling
                 moisture
                 hanging
                 on
                 the
                 chamber-side
                 of
                 the
                 Glass
                 ,
                 when
                 the
                 Air
                 is
                 very
                 sharp
                 ,
                 and
                 full
                 of
                 a
                 nitrous
                 coldness
                 without
                 :
                 and
                 we
                 cannot
                 rightly
                 conceive
                 the
                 Idea
                 of
                 that
                 famously
                 known
                 and
                 frequently
                 mentioned
                 and
                 in
                 hot
                 Countries
                 ,
                 though
                 rare
                 ,
                 yet
                 sometimes
                 by
                 sad
                 experience
                 attested
                 disease
                 ,
                 the
                 Diabetes
                 ,
                 without
                 admitting
                 both
                 the
                 possibility
                 of
                 such
                 a
                 Chymistry
                 ,
                 and
                 of
                 Expedients
                 for
                 the
                 more
                 ready
                 dispatch
                 of
                 it
                 :
                 for
                 without
                 an
                 aptness
                 to
                 imbibe
                 ,
                 attract
                 and
                 condense
                 the
                 vapid
                 exhalations
                 ,
                 which
                 successively
                 come
                 to
                 be
                 near
                 and
                 about
                 the
                 Patient
                 ,
                 and
                 that
                 with
                 very
                 singular
                 advantages
                 for
                 expedition
                 ,
                 how
                 possibly
                 
                 should
                 the
                 Patients
                 Urine
                 come
                 in
                 a
                 few
                 weeks
                 to
                 weigh
                 more
                 than
                 all
                 ,
                 both
                 the
                 drinks
                 and
                 meats
                 taken
                 by
                 the
                 Patient
                 all
                 along
                 that
                 time
                 ,
                 and
                 all
                 the
                 weight
                 of
                 the
                 Body
                 ,
                 whensoever
                 it
                 was
                 heaviest
                 during
                 that
                 time
                 besides
                 ?
                 To
                 contract
                 all
                 into
                 few
                 ,
                 the
                 possibility
                 of
                 the
                 Probleme
                 ,
                 according
                 to
                 this
                 explication
                 ,
                 needs
                 not
                 be
                 doubted
                 ,
                 being
                 an
                 every
                 where
                 obvious
                 work
                 of
                 Nature
                 ;
                 as
                 in
                 the
                 other
                 sense
                 it
                 seems
                 vain
                 and
                 frivolous
                 .
                 And
                 in
                 researches
                 about
                 expedients
                 for
                 dispatch
                 ;
                 it
                 is
                 possible
                 ,
                 here
                 ,
                 as
                 in
                 other
                 pieces
                 of
                 that
                 ingenious
                 Art
                 ,
                 several
                 Artists
                 may
                 have
                 several
                 methods
                 and
                 contrivances
                 ,
                 to
                 suck
                 in
                 still
                 more
                 vapours
                 in
                 a
                 continued
                 succession
                 ,
                 and
                 proportionably
                 to
                 condense
                 them
                 when
                 sucked
                 in
                 ;
                 also
                 to
                 dispose
                 the
                 vapours
                 ,
                 before
                 they
                 are
                 sucked
                 in
                 ,
                 for
                 more
                 expedite
                 and
                 easie
                 condensation
                 at
                 after
                 :
                 as
                 every
                 one
                 has
                 his
                 peculiar
                 Still
                 ,
                 his
                 peculiar
                 Solvent
                 ,
                 Bath
                 ,
                 Ferment
                 and
                 Furnace
                 .
                 And
                 that
                 this
                 point
                 of
                 Philosophy
                 were
                 a
                 little
                 more
                 cultivated
                 were
                 to
                 be
                 wished
                 ,
                 for
                 the
                 advantage
                 of
                 natural
                 Philosophy
                 ,
                 and
                 the
                 concernes
                 of
                 it
                 in
                 some
                 practical
                 Arts
                 :
                 for
                 so
                 hereby
                 the
                 difference
                 of
                 the
                 Airs
                 at
                 several
                 seasons
                 ,
                 or
                 at
                 the
                 same
                 season
                 in
                 several
                 places
                 ,
                 might
                 come
                 in
                 some
                 measure
                 to
                 be
                 known
                 ,
                 by
                 the
                 Taste
                 and
                 Eye
                 ,
                 as
                 well
                 as
                 by
                 the
                 Nostril
                 :
                 and
                 the
                 concerns
                 of
                 Rokes
                 ,
                 Fogs
                 ,
                 Mists
                 ,
                 and
                 Airs
                 any
                 wayes
                 altered
                 ,
                 or
                 infected
                 ,
                 may
                 be
                 hereby
                 
                 more
                 happily
                 discovered
                 in
                 relation
                 to
                 the
                 health
                 and
                 sickliness
                 of
                 Man
                 ,
                 and
                 so
                 of
                 other
                 Animals
                 :
                 so
                 what
                 difference
                 there
                 is
                 between
                 the
                 Breathings
                 of
                 several
                 Animals
                 of
                 the
                 same
                 ,
                 or
                 several
                 Sexes
                 and
                 Kinds
                 ,
                 or
                 of
                 the
                 same
                 Animal
                 at
                 several
                 times
                 of
                 its
                 age
                 ,
                 rest
                 ,
                 motion
                 ,
                 fear
                 ,
                 joy
                 ,
                 hope
                 ,
                 love
                 ,
                 anger
                 ,
                 health
                 ,
                 sickness
                 ,
                 peculiar
                 conditions
                 ,
                 or
                 evacuations
                 :
                 and
                 the
                 discriminations
                 between
                 simple
                 Airs
                 ,
                 and
                 the
                 Airs
                 altered
                 by
                 sprinklings
                 ,
                 strewings
                 ,
                 fumigations
                 ,
                 from
                 Minerals
                 ,
                 Vegetables
                 ,
                 Animals
                 ,
                 may
                 be
                 made
                 more
                 evident
                 .
                 Your
                 tying
                 up
                 Mercury
                 to
                 the
                 old
                 Elemental
                 Laws
                 ,
                 is
                 a
                 true
                 semblance
                 of
                 the
                 old
                 Mezentian
                 cruelty
                 ;
                 and
                 I
                 should
                 pity
                 the
                 poor
                 Fugitive
                 ,
                 were
                 I
                 not
                 ascertained
                 from
                 his
                 constant
                 course
                 ,
                 that
                 as
                 a
                 slippery
                 Hocas
                 ,
                 he
                 will
                 never
                 leave
                 transhaping
                 himself
                 till
                 ,
                 to
                 your
                 own
                 confession
                 ,
                 he
                 has
                 slipt
                 the
                 Collar
                 .
                 Four
                 qualities
                 will
                 assoon
                 solve
                 all
                 questions
                 ,
                 as
                 four
                 Elements
                 ;
                 which
                 some
                 of
                 the
                 learned
                 Antients
                 ,
                 vainly
                 hoped
                 ,
                 not
                 to
                 say
                 ,
                 foolishly
                 boasted
                 ,
                 to
                 do
                 .
                 As
                 Philosophers
                 distinguish
                 between
                 the
                 sights
                 of
                 Age
                 and
                 Time
                 ;
                 Age
                 first
                 sees
                 without
                 ,
                 and
                 after
                 not
                 without
                 Spectacles
                 :
                 Time
                 first
                 sees
                 with
                 them
                 ,
                 and
                 after
                 better
                 without
                 them
                 :
                 the
                 Elements
                 were
                 as
                 Spectacles
                 in
                 the
                 Beginnings
                 of
                 those
                 times
                 ,
                 when
                 men
                 first
                 set
                 themselves
                 about
                 the
                 studies
                 of
                 Arts
                 ,
                 that
                 helped
                 them
                 in
                 a
                 sort
                 to
                 conceive
                 somewhat
                 of
                 Bodies
                 ;
                 but
                 now
                 
                 the
                 Eye
                 of
                 time
                 grown
                 clearer
                 ,
                 seeth
                 much
                 more
                 happily
                 ,
                 as
                 is
                 by
                 many
                 supposed
                 ,
                 without
                 them
                 .
                 More
                 crass
                 and
                 thin
                 ,
                 solid
                 and
                 fluid
                 parts
                 are
                 not
                 denyed
                 ,
                 but
                 apparently
                 to
                 be
                 seen
                 ,
                 offering
                 themselves
                 to
                 view
                 in
                 several
                 plain
                 and
                 easie
                 methods
                 of
                 analysing
                 Bodies
                 ;
                 but
                 the
                 simplicity
                 of
                 those
                 parts
                 ,
                 obtained
                 in
                 such
                 Analysmes
                 ,
                 is
                 not
                 yet
                 demonstrated
                 .
                 The
                 strange
                 Monster
                 reported
                 with
                 so
                 much
                 averment
                 to
                 you
                 ,
                 which
                 seems
                 so
                 much
                 to
                 have
                 startled
                 your
                 patience
                 ,
                 to
                 wit
                 ,
                 that
                 a
                 person
                 of
                 parts
                 ,
                 educated
                 only
                 to
                 read
                 and
                 write
                 the
                 Language
                 of
                 the
                 Nation
                 wherein
                 he
                 was
                 born
                 ,
                 may
                 in
                 an
                 hours
                 time
                 be
                 taught
                 to
                 set
                 down
                 any
                 matter
                 ,
                 secret
                 ,
                 or
                 other
                 ,
                 in
                 another
                 language
                 ,
                 as
                 Latine
                 ,
                 Greek
                 ,
                 French
                 ,
                 &c.
                 and
                 the
                 Latine
                 ,
                 or
                 other
                 Language
                 to
                 be
                 true
                 ,
                 and
                 the
                 sense
                 good
                 and
                 continued
                 ;
                 believe
                 me
                 ,
                 though
                 it
                 be
                 true
                 ,
                 as
                 it
                 is
                 ,
                 yet
                 it
                 is
                 not
                 to
                 be
                 esteemed
                 worthy
                 your
                 least
                 discomposure
                 :
                 it
                 may
                 be
                 done
                 many
                 several
                 wayes
                 ,
                 and
                 yet
                 when
                 all
                 is
                 done
                 ,
                 it
                 is
                 but
                 a
                 slight
                 of
                 the
                 Brain
                 ;
                 like
                 many
                 of
                 the
                 mimical
                 Miracles
                 ,
                 performed
                 by
                 nimble
                 handed
                 Egyptians
                 .
                 That
                 there
                 may
                 be
                 no
                 room
                 for
                 distrust
                 of
                 the
                 truth
                 of
                 it
                 ,
                 if
                 affiance
                 in
                 me
                 may
                 so
                 far
                 prevail
                 ▪
                 you
                 may
                 be
                 assured
                 ,
                 I
                 have
                 seen
                 the
                 whole
                 Model
                 and
                 Systeme
                 of
                 it
                 ,
                 in
                 several
                 Languages
                 ,
                 besides
                 our
                 own
                 :
                 and
                 but
                 that
                 it
                 would
                 make
                 my
                 Letter
                 of
                 too
                 great
                 a
                 Bulk
                 ,
                 
                 should
                 herewith
                 have
                 sent
                 you
                 a
                 Transcript
                 of
                 one
                 of
                 them
                 .
                 In
                 this
                 ,
                 as
                 in
                 the
                 foregoing
                 Probleme
                 of
                 multiplying
                 Chymistry
                 ;
                 in
                 words
                 speaking
                 and
                 containing
                 truth
                 ,
                 something
                 seems
                 to
                 be
                 offered
                 to
                 the
                 understanding
                 of
                 the
                 hearer
                 ,
                 which
                 is
                 an
                 Herculian
                 shot
                 beyond
                 it
                 .
                 Pardon
                 the
                 length
                 in
                 which
                 I
                 have
                 laboured
                 discursively
                 to
                 send
                 back
                 all
                 the
                 Quaeries
                 and
                 Objections
                 of
                 your
                 last
                 fully
                 answered
                 .
              
            
             
               
               
                 The
                 Second
                 Answer
                 .
              
               
                 
                   Wherein
                   of
                   Sympathies
                   between
                   several
                   parts
                   of
                   the
                   same
                   Animal
                   Body
                   in
                   general
                   and
                   particularly
                   of
                   the
                   Sympathy
                   which
                   is
                   between
                   the
                   Breasts
                   and
                   the
                   Womb
                   :
                   and
                   that
                   the
                   ground
                   of
                   that
                   Sympathy
                   is
                   not
                   founded
                   upon
                   the
                   inosculations
                   of
                   the
                   Epigastrick
                   and
                   Mammary
                   Veins
                   and
                   Arteries
                   .
                   That
                   that
                   Sympathy
                   instructs
                   not
                   from
                   the
                   sight
                   of
                   the
                   Breast
                   concerning
                   the
                   Sex
                   of
                   the
                   unborn
                   Conception
                   .
                   Yet
                   parts
                   of
                   the
                   same
                   side
                   more
                   especially
                   Sympathize
                   one
                   with
                   another
                   ,
                   than
                   those
                   that
                   are
                   on
                   contrary
                   sides
                   .
                   That
                   to
                   found
                   the
                   Sympathy
                   between
                   the
                   Breasts
                   and
                   Womb
                   ;
                   is
                   no
                   necessity
                   of
                   vessels
                   passing
                   directly
                   between
                   the
                   one
                   part
                   and
                   the
                   other
                   :
                   but
                   the
                   grounds
                   of
                   their
                   Sympathy
                   may
                   be
                   several
                   other
                   wayes
                   clearly
                   explicable
                   ,
                   by
                   manifest
                   Arteries
                   ,
                   Sinews
                   ,
                   and
                   other
                   Vessels
                   .
                   That
                   the
                   several
                   shapes
                   of
                   the
                   Moon
                   have
                   their
                   original
                   ,
                   forming
                   ,
                   and
                   alteration
                   ,
                   from
                   the
                   parts
                   of
                   its
                   mutuatitious
                   light
                   ,
                   and
                   their
                   situation
                   in
                   respect
                   of
                   our
                   Eye
                   .
                   That
                   there
                   is
                   not
                   one
                   general
                   original
                   of
                   all
                   light
                   .
                   And
                   why
                   the
                   enlightned
                   part
                   of
                   the
                   Moon
                   is
                   still
                   toward
                   the
                   Sun.
                   And
                   why
                   the
                   Moon
                   appears
                   plane
                   and
                   not
                   sphaeral
                   .
                
              
               
                 I
                 should
                 wrong
                 the
                 truth
                 ,
                 if
                 I
                 should
                 not
                 acknowledge
                 ,
                 all
                 your
                 Arguments
                 and
                 Instances
                 prove
                 fairly
                 the
                 Sympathies
                 ,
                 which
                 nature
                 has
                 lodged
                 in
                 us
                 ,
                 between
                 some
                 special
                 parts
                 above
                 others
                 ;
                 to
                 which
                 may
                 be
                 added
                 by
                 way
                 of
                 Declaration
                 ;
                 that
                 though
                 no
                 part
                 in
                 the
                 whole
                 animal
                 Body
                 ,
                 but
                 is
                 of
                 affinity
                 and
                 
                 concern
                 unto
                 all
                 the
                 rest
                 ,
                 yet
                 some
                 ,
                 besides
                 their
                 general
                 relation
                 as
                 of
                 the
                 same
                 animal
                 Body
                 ,
                 have
                 sometimes
                 not
                 only
                 one
                 ,
                 but
                 several
                 other
                 more
                 special
                 grounds
                 of
                 Sympathy
                 ,
                 to
                 tye
                 &
                 link
                 them
                 one
                 to
                 another
                 ;
                 whether
                 from
                 homogeneity
                 of
                 substance
                 ,
                 or
                 commonness
                 of
                 original
                 ,
                 or
                 unity
                 and
                 concurrence
                 in
                 the
                 same
                 work
                 ,
                 or
                 concern
                 in
                 the
                 same
                 branchings
                 of
                 vessels
                 ,
                 whether
                 Arteries
                 ,
                 Sinews
                 ,
                 or
                 Veins
                 ,
                 of
                 what
                 kind
                 soever
                 ;
                 each
                 of
                 which
                 heightens
                 the
                 general
                 relation
                 which
                 is
                 between
                 the
                 parts
                 :
                 and
                 the
                 more
                 of
                 these
                 are
                 found
                 inter-curring
                 between
                 any
                 two
                 parts
                 ,
                 their
                 relation
                 is
                 so
                 much
                 the
                 more
                 fortifyed
                 .
                 But
                 of
                 all
                 other
                 ,
                 the
                 Sympathy
                 which
                 you
                 urge
                 ,
                 between
                 the
                 Breasts
                 and
                 Womb
                 ,
                 in
                 animals
                 is
                 too
                 universally
                 known
                 ,
                 and
                 variously
                 evident
                 to
                 be
                 denyed
                 ,
                 or
                 doubted
                 .
                 How
                 oft
                 ,
                 among
                 prying
                 and
                 suspitious
                 Women
                 ,
                 has
                 the
                 state
                 of
                 the
                 Womb
                 been
                 discovered
                 from
                 the
                 colour
                 of
                 the
                 Nipples
                 ,
                 and
                 condition
                 of
                 the
                 Breasts
                 ?
                 And
                 the
                 judgements
                 of
                 Physicians
                 justified
                 and
                 admired
                 ,
                 when
                 upon
                 the
                 manifest
                 intercourse
                 between
                 these
                 two
                 parts
                 ,
                 easily
                 transferring
                 humours
                 ,
                 whether
                 natural
                 ,
                 or
                 praeter-natural
                 ,
                 from
                 the
                 one
                 to
                 the
                 other
                 ,
                 they
                 have
                 sometimes
                 predicted
                 long
                 before
                 the
                 diseases
                 of
                 the
                 Breasts
                 ,
                 or
                 Womb
                 ?
                 So
                 upon
                 expectation
                 of
                 the
                 ones
                 evacuation
                 ,
                 the
                 cure
                 of
                 the
                 other
                 has
                 been
                 ordered
                 ,
                 and
                 the
                 event
                 and
                 time
                 of
                 recovery
                 foreseen
                 ?
                 and
                 observing
                 
                 how
                 they
                 grow
                 and
                 lessen
                 together
                 ,
                 heal
                 and
                 fade
                 together
                 ,
                 become
                 firm
                 &
                 flaccid
                 together
                 ,
                 replenish
                 and
                 empty
                 together
                 ,
                 except
                 when
                 the
                 one
                 drains
                 the
                 other
                 ,
                 the
                 intercourse
                 between
                 them
                 ,
                 and
                 their
                 common
                 concern
                 is
                 manifest
                 ;
                 however
                 the
                 grounds
                 and
                 passages
                 wherein
                 this
                 intercourse
                 is
                 founded
                 and
                 carried
                 on
                 ,
                 possibly
                 may
                 not
                 be
                 so
                 very
                 clear
                 ,
                 as
                 by
                 many
                 is
                 presumed
                 .
                 And
                 though
                 I
                 honour
                 you
                 for
                 honouring
                 the
                 Antients
                 ;
                 it
                 is
                 now
                 especially
                 less
                 clear
                 then
                 ever
                 ;
                 that
                 the
                 undeniable
                 consent
                 and
                 sympathy
                 ,
                 which
                 is
                 between
                 these
                 two
                 parts
                 ,
                 has
                 its
                 foundation
                 in
                 the
                 inosculations
                 of
                 the
                 epigastrick
                 veins
                 and
                 arteries
                 with
                 the
                 mammeryes
                 :
                 Except
                 we
                 will
                 so
                 farr
                 degrade
                 the
                 examinations
                 of
                 reasons
                 ,
                 and
                 trials
                 of
                 experience
                 ,
                 as
                 to
                 make
                 what
                 is
                 antient
                 ,
                 to
                 be
                 therefore
                 indisputably
                 authentick
                 ;
                 Concerning
                 the
                 sympathy
                 of
                 the
                 Breasts
                 and
                 womb
                 as
                 held
                 by
                 the
                 Antients
                 )
                 there
                 are
                 two
                 things
                 well
                 deserving
                 serious
                 &
                 further
                 consideration
                 :
                 For
                 hath
                 not
                 this
                 sympathy
                 been
                 rackt
                 beyond
                 its
                 nature
                 ,
                 and
                 tentered
                 beyond
                 the
                 truth
                 ,
                 of
                 what
                 it
                 is
                 able
                 to
                 bear
                 ,
                 or
                 discover
                 ;
                 when
                 ,
                 upon
                 the
                 view
                 of
                 the
                 Breasts
                 ,
                 declarations
                 and
                 predictions
                 are
                 made
                 ,
                 not
                 only
                 of
                 conceptions
                 ,
                 &
                 abortions
                 in
                 general
                 ,
                 with
                 a
                 probable
                 prognostick
                 of
                 the
                 time
                 and
                 event
                 of
                 the
                 feared
                 abortion
                 ;
                 but
                 also
                 from
                 the
                 difference
                 of
                 the
                 Breasts
                 on
                 each
                 side
                 ,
                 the
                 
                 sexe
                 of
                 the
                 conception
                 ,
                 or
                 abortion
                 is
                 peremptorily
                 limited
                 ;
                 and
                 when
                 twins
                 of
                 several
                 sexes
                 have
                 been
                 conceived
                 ,
                 the
                 twin
                 of
                 whether
                 sexe
                 shall
                 alone
                 be
                 aborted
                 is
                 foreshewn
                 ;
                 in
                 that
                 rare
                 &
                 unusuall
                 case
                 of
                 aborting
                 one
                 twin
                 ,
                 and
                 compleating
                 the
                 due
                 months
                 of
                 gestation
                 with
                 the
                 rest
                 ?
                 Not
                 to
                 alleadge
                 more
                 of
                 these
                 impertinent
                 medical
                 Enthusiasmes
                 ;
                 such
                 groundlesse
                 and
                 inconsequent
                 deductions
                 ,
                 as
                 they
                 are
                 casily
                 received
                 by
                 the
                 superstitious
                 ,
                 who
                 yawn
                 after
                 things
                 novel
                 and
                 strange
                 ,
                 and
                 blindly
                 swallow
                 what
                 is
                 offered
                 without
                 either
                 praeexamining
                 by
                 the
                 eye
                 of
                 reason
                 ,
                 or
                 proving
                 them
                 by
                 the
                 praemastications
                 of
                 experience
                 :
                 so
                 I
                 doubt
                 not
                 but
                 to
                 sober
                 minds
                 they
                 will
                 appear
                 an
                 apish
                 overhugging
                 the
                 truth
                 ,
                 to
                 the
                 ruine
                 and
                 destruction
                 of
                 it
                 :
                 an
                 overchurning
                 a
                 serious
                 ,
                 secret
                 ,
                 true
                 ,
                 observation
                 of
                 nature
                 into
                 ranckness
                 and
                 falsity
                 ;
                 and
                 a
                 praeposterous
                 grafting
                 a
                 wild
                 ,
                 wandering
                 fancy
                 upon
                 the
                 most
                 cultivated
                 stock
                 of
                 sober
                 ,
                 true
                 and
                 certain
                 observation
                 .
                 For
                 since
                 men
                 have
                 set
                 their
                 minds
                 to
                 learn
                 out
                 of
                 natures
                 unwritten
                 books
                 ,
                 as
                 well
                 as
                 out
                 of
                 the
                 writings
                 of
                 Authours
                 ;
                 what
                 hath
                 been
                 more
                 usual
                 than
                 to
                 find
                 male
                 and
                 female
                 conceptions
                 indifferently
                 on
                 the
                 one
                 or
                 other
                 side
                 ,
                 or
                 horn
                 of
                 the
                 womb
                 ?
                 And
                 though
                 against
                 experience
                 no
                 Argument
                 ought
                 to
                 avail
                 ;
                 we
                 shall
                 not
                 stiffe
                 what
                 they
                 seem
                 to
                 have
                 to
                 say
                 for
                 themselves
                 ;
                 
                 That
                 the
                 left
                 praeparing
                 vein
                 riseth
                 from
                 the
                 Emulgent
                 only
                 ,
                 which
                 may
                 be
                 suspected
                 ,
                 to
                 be
                 a
                 weakning
                 of
                 its
                 power
                 and
                 influxion
                 ;
                 whereas
                 the
                 right
                 praeparing
                 ,
                 or
                 spermatick
                 vein
                 riseth
                 immediately
                 from
                 the
                 great
                 discending
                 Trunk
                 ,
                 of
                 which
                 the
                 aforesaid
                 Emulgent
                 is
                 but
                 an
                 offset
                 .
                 All
                 this
                 is
                 generally
                 and
                 in
                 most
                 Bodies
                 true
                 :
                 and
                 in
                 former
                 Ages
                 ,
                 with
                 the
                 glossiness
                 of
                 its
                 probability
                 ,
                 carryed
                 away
                 the
                 minds
                 and
                 judgments
                 of
                 most
                 :
                 but
                 now
                 upon
                 more
                 curious
                 searches
                 made
                 into
                 the
                 Bowels
                 ,
                 and
                 heart
                 of
                 nature
                 ,
                 the
                 sophistry
                 ,
                 &
                 weaknes
                 of
                 the
                 former
                 Argument
                 manifestly
                 appears
                 :
                 the
                 spermatick
                 veins
                 ,
                 if
                 any
                 be
                 so
                 pleased
                 to
                 continue
                 that
                 name
                 ,
                 being
                 only
                 reducing
                 vessels
                 ,
                 and
                 carrying
                 forth
                 from
                 the
                 vital
                 fountain
                 nothing
                 at
                 all
                 to
                 the
                 Testicles
                 ,
                 womb
                 ,
                 Ovaries
                 ,
                 or
                 other
                 organes
                 of
                 generation
                 ;
                 as
                 may
                 appear
                 by
                 observing
                 ;
                 the
                 motion
                 of
                 the
                 juyces
                 contain'd
                 in
                 them
                 ,
                 notifyed
                 from
                 the
                 side
                 on
                 which
                 the
                 tumour
                 and
                 fulnes
                 suddenly
                 upon
                 an
                 intercepting
                 ligature
                 gathereth
                 .
                 And
                 the
                 different
                 rise
                 of
                 these
                 spermatick
                 veins
                 is
                 more
                 aptly
                 referred
                 to
                 the
                 indirect
                 situation
                 of
                 the
                 discending
                 Trunk
                 there
                 ;
                 which
                 usually
                 at
                 the
                 loins
                 inclines
                 a
                 little
                 rather
                 to
                 the
                 right
                 side
                 ,
                 making
                 the
                 left
                 Emulgent
                 accordingly
                 ,
                 to
                 be
                 commonly
                 a
                 little
                 longer
                 than
                 the
                 right
                 :
                 so
                 as
                 for
                 a
                 long
                 tender
                 venal
                 vessel
                 to
                 have
                 had
                 its
                 original
                 so
                 obliquely
                 remote
                 ,
                 
                 and
                 sloping
                 over
                 the
                 fore
                 bodies
                 of
                 the
                 Rack
                 bones
                 ,
                 might
                 not
                 seem
                 sufficiently
                 secure
                 ;
                 however
                 in
                 the
                 more
                 strongly
                 coated
                 arterial
                 vessels
                 ,
                 it
                 is
                 not
                 withstanding
                 otherwise
                 ;
                 the
                 right
                 and
                 left
                 spermatical
                 arteries
                 both
                 rising
                 constantly
                 from
                 the
                 descending
                 branch
                 of
                 the
                 Aorta
                 .
                 That
                 Argument
                 of
                 the
                 right
                 side
                 parts
                 being
                 ,
                 stronger
                 ,
                 hotter
                 ,
                 nimbler
                 ,
                 &c.
                 than
                 those
                 on
                 the
                 left
                 side
                 ;
                 annexeth
                 an
                 incohaerent
                 sequel
                 ,
                 to
                 a
                 Truth
                 which
                 is
                 meerly
                 accidental
                 :
                 the
                 special
                 strength
                 of
                 the
                 right
                 side
                 parts
                 not
                 being
                 any
                 natural
                 priviledge
                 ,
                 but
                 an
                 advantage
                 acquired
                 by
                 exercise
                 ,
                 use
                 and
                 education
                 ;
                 and
                 by
                 the
                 same
                 method
                 ,
                 as
                 easily
                 transferable
                 to
                 the
                 other
                 side
                 .
                 And
                 if
                 at
                 any
                 time
                 Twin
                 parts
                 be
                 not
                 equally
                 concern'd
                 ,
                 it
                 is
                 because
                 there
                 is
                 not
                 equall
                 relation
                 to
                 the
                 sides
                 :
                 so
                 the
                 Ovary
                 of
                 that
                 Side
                 to
                 which
                 the
                 Conception
                 is
                 more
                 principally
                 affixed
                 ,
                 in
                 many
                 females
                 is
                 of
                 a
                 fleshy
                 substance
                 ,
                 of
                 colour
                 like
                 Sanders
                 ,
                 or
                 somewhat
                 brighter
                 ,
                 the
                 other
                 all
                 the
                 while
                 remaining
                 without
                 any
                 alteration
                 as
                 a
                 cluster
                 of
                 watery
                 Blebs
                 &
                 vesicles
                 .
                 But
                 more
                 particularly
                 in
                 the
                 Copulations
                 of
                 mankind
                 ,
                 all
                 such
                 considerations
                 of
                 sides
                 must
                 of
                 necessity
                 be
                 obliterated
                 ;
                 the
                 right
                 side
                 of
                 the
                 one
                 Parent
                 being
                 applyed
                 to
                 the
                 left
                 of
                 the
                 other
                 ,
                 and
                 so
                 on
                 the
                 other
                 side
                 ,
                 the
                 left
                 to
                 the
                 right
                 .
                 In
                 all
                 this
                 we
                 deny
                 not
                 ,
                 but
                 right
                 side
                 parts
                 have
                 more
                 special
                 relation
                 to
                 right
                 side
                 parts
                 ,
                 where
                 
                 the
                 parts
                 twin
                 ,
                 and
                 are
                 double
                 :
                 but
                 to
                 parts
                 ,
                 which
                 are
                 single
                 ,
                 and
                 posited
                 in
                 the
                 middle
                 ,
                 Twin
                 parts
                 have
                 equall
                 relation
                 .
                 So
                 as
                 upon
                 the
                 whole
                 matter
                 ,
                 the
                 diseases
                 of
                 each
                 particular
                 Breast
                 are
                 more
                 prone
                 to
                 be
                 transferred
                 to
                 the
                 same
                 respective
                 side
                 of
                 the
                 womb
                 ,
                 and
                 those
                 of
                 either
                 side
                 of
                 the
                 womb
                 to
                 the
                 respective
                 breast
                 ;
                 of
                 which
                 are
                 most
                 apparent
                 footsteps
                 in
                 the
                 fabrick
                 of
                 nature
                 :
                 for
                 though
                 in
                 some
                 vessels
                 ,
                 as
                 Blood-veins
                 ,
                 water-veins
                 ,
                 chyle-veins
                 ,
                 arteries
                 ,
                 &c.
                 this
                 distinction
                 of
                 sides
                 be
                 again
                 lost
                 and
                 confounded
                 by
                 insertion
                 into
                 some
                 common
                 Trunk
                 ,
                 before
                 they
                 can
                 be
                 traced
                 from
                 the
                 one
                 part
                 ,
                 to
                 the
                 other
                 collateral
                 part
                 ;
                 yet
                 in
                 some
                 other
                 vessels
                 without
                 any
                 such
                 confusion
                 in
                 the
                 midway
                 ,
                 the
                 same
                 collateral
                 vessels
                 ramify
                 themselves
                 distinctly
                 unto
                 the
                 parts
                 all
                 along
                 on
                 the
                 same
                 side
                 :
                 as
                 is
                 most
                 manifest
                 in
                 the
                 visceral
                 sinew
                 ,
                 or
                 wandering
                 pair
                 :
                 but
                 yet
                 hereby
                 is
                 not
                 opened
                 any
                 window
                 to
                 see
                 into
                 the
                 sex
                 of
                 unborn
                 conceptions
                 .
                 And
                 ,
                 as
                 herein
                 I
                 desire
                 to
                 reverence
                 antiquity
                 ,
                 but
                 follow
                 reason
                 ;
                 so
                 upon
                 your
                 more
                 serious
                 thoughts
                 ,
                 I
                 doubt
                 not
                 but
                 you
                 will
                 judge
                 ,
                 with
                 me
                 ,
                 another
                 Point
                 upon
                 this
                 subject
                 very
                 well
                 worthy
                 calling
                 in
                 Question
                 :
                 and
                 that
                 is
                 the
                 much
                 noysed
                 ground
                 of
                 this
                 sympatly
                 between
                 the
                 Breasts
                 and
                 womb
                 ;
                 said
                 ,
                 and
                 formerly
                 thought
                 ,
                 to
                 be
                 the
                 inosculations
                 of
                 the
                 mammary
                 veins
                 and
                 Arteries
                 
                 with
                 the
                 Epigastriks
                 .
                 It
                 is
                 well
                 known
                 ,
                 how
                 many
                 lateral
                 heterogeneal
                 inosculations
                 ,
                 between
                 Arteries
                 and
                 Blood-veins
                 ,
                 have
                 been
                 introduced
                 by
                 closet
                 Anatomists
                 ;
                 while
                 ,
                 in
                 Bloudless
                 speculations
                 ,
                 they
                 formed
                 nature
                 according
                 to
                 their
                 reasonings
                 ,
                 to
                 solve
                 appearing
                 phaenomena's
                 ;
                 instead
                 of
                 forming
                 their
                 reasonings
                 according
                 to
                 what
                 they
                 ought
                 first
                 to
                 have
                 observed
                 in
                 nature
                 :
                 the
                 verity
                 and
                 authority
                 of
                 all
                 which
                 ,
                 among
                 Anatomists
                 at
                 sharp
                 ,
                 that
                 carry
                 their
                 Eyes
                 in
                 their
                 hands
                 ,
                 and
                 will
                 believe
                 no
                 more
                 than
                 they
                 see
                 ,
                 is
                 at
                 present
                 not
                 much
                 ;
                 however
                 formerly
                 solemnly
                 ,
                 and
                 for
                 singular
                 purposes
                 ,
                 placed
                 in
                 and
                 very
                 near
                 several
                 of
                 the
                 chief
                 Bowels
                 :
                 yet
                 the
                 fumultous
                 aestuation
                 which
                 this
                 Hypothesis
                 continually
                 placeth
                 nature
                 under
                 ,
                 like
                 the
                 meeting
                 of
                 two
                 contrary
                 seas
                 ,
                 may
                 be
                 sufficient
                 ,
                 in
                 reason
                 also
                 ,
                 to
                 decry
                 and
                 discard
                 it
                 :
                 for
                 which
                 reason
                 also
                 lateral
                 homogeneal
                 inosculations
                 of
                 bloud-veins
                 with
                 bloud-veins
                 ,
                 as
                 they
                 are
                 frequently
                 found
                 ,
                 so
                 they
                 may
                 as
                 easily
                 in
                 reason
                 be
                 admitted
                 .
                 But
                 the
                 inosculations
                 in
                 the
                 Question
                 being
                 final
                 ;
                 I
                 wave
                 these
                 that
                 are
                 lateral
                 .
                 And
                 final
                 inosculations
                 are
                 on
                 all
                 hands
                 ,
                 agreed
                 to
                 be
                 in
                 a
                 manner
                 impossible
                 to
                 be
                 shewn
                 ;
                 because
                 ,
                 when
                 granted
                 to
                 be
                 ,
                 yet
                 there
                 where
                 they
                 are
                 ,
                 the
                 vessels
                 are
                 so
                 minute
                 ,
                 and
                 slender
                 ,
                 that
                 it
                 neither
                 can
                 be
                 easy
                 for
                 the
                 quickest
                 sight
                 ,
                 nor
                 the
                 most
                 cautions
                 
                 hand
                 ,
                 either
                 ,
                 without
                 hurting
                 them
                 ,
                 to
                 come
                 at
                 the
                 vessels
                 ,
                 where
                 they
                 are
                 inosculated
                 ,
                 or
                 to
                 distinguish
                 the
                 vessels
                 when
                 heterogeneal
                 at
                 their
                 inosculation
                 ,
                 or
                 to
                 discern
                 a
                 final
                 inosculation
                 ,
                 whether
                 of
                 vessels
                 of
                 the
                 same
                 ,
                 or
                 of
                 different
                 kinds
                 .
                 So
                 that
                 this
                 whole
                 Question
                 about
                 final
                 inosculations
                 being
                 to
                 receive
                 its
                 decision
                 by
                 the
                 judgment
                 of
                 reason
                 ;
                 such
                 final
                 inosculations
                 as
                 are
                 homogeneal
                 ,
                 as
                 of
                 Arteries
                 with
                 Arteries
                 ,
                 and
                 Bloud-veins
                 with
                 Bloud-veins
                 of
                 the
                 same
                 Trunk
                 ,
                 such
                 as
                 in
                 this
                 Point
                 were
                 understood
                 and
                 intended
                 by
                 the
                 Antients
                 ,
                 are
                 no
                 way
                 capable
                 of
                 desence
                 ,
                 allowance
                 ,
                 or
                 justification
                 at
                 the
                 Barr
                 of
                 reason
                 :
                 for
                 it
                 experimentally
                 ,
                 and
                 to
                 the
                 eye
                 ,
                 appears
                 ,
                 that
                 if
                 any
                 such
                 inosculations
                 should
                 be
                 supposed
                 ,
                 the
                 same
                 Humours
                 at
                 the
                 same
                 time
                 ,
                 from
                 the
                 same
                 place
                 ,
                 should
                 move
                 two
                 contrary
                 ways
                 ,
                 to
                 wit
                 ,
                 both
                 to
                 ,
                 and
                 from
                 the
                 heart
                 ,
                 and
                 in
                 vessels
                 rising
                 from
                 ,
                 or
                 gathered
                 at
                 after
                 into
                 ,
                 the
                 same
                 trunk
                 :
                 in
                 which
                 account
                 ,
                 the
                 lung
                 vein
                 ,
                 the
                 hollow
                 vein
                 ,
                 and
                 the
                 port
                 vein
                 ,
                 though
                 all
                 of
                 them
                 bloud-veins
                 ,
                 yet
                 because
                 of
                 the
                 diversity
                 of
                 their
                 Trunks
                 ,
                 are
                 to
                 be
                 esteemed
                 as
                 heterogeneal
                 .
                 And
                 finall
                 homogeneal
                 inosculations
                 generally
                 ,
                 and
                 therefore
                 also
                 between
                 the
                 Mammaryes
                 and
                 Epigastricks
                 ,
                 being
                 thus
                 everted
                 ;
                 the
                 old
                 doctrine
                 oncerning
                 the
                 ground
                 of
                 the
                 sympathy
                 between
                 the
                 Breasts
                 and
                 womb
                 cannot
                 be
                 retained
                 .
                 
                 And
                 if
                 any
                 think
                 by
                 a
                 short
                 alteration
                 of
                 ,
                 and
                 putting
                 heterogeneal
                 inosculations
                 for
                 ,
                 the
                 rejected
                 homogeneal
                 ,
                 the
                 ground
                 of
                 this
                 sympathy
                 will
                 remain
                 firm
                 and
                 clear
                 as
                 of
                 old
                 ;
                 to
                 wit
                 by
                 making
                 humours
                 to
                 be
                 transferred
                 from
                 the
                 womb
                 by
                 the
                 Epigastrick
                 arteries
                 and
                 mammary
                 veins
                 to
                 the
                 Breasts
                 ,
                 and
                 by
                 the
                 mammary
                 arteries
                 ,
                 and
                 Epigastrick
                 veins
                 from
                 the
                 breasts
                 to
                 the
                 womb
                 ;
                 I
                 suppose
                 ,
                 that
                 though
                 such
                 inosculations
                 be
                 not
                 as
                 the
                 other
                 ,
                 impossible
                 ,
                 though
                 undemonstrable
                 to
                 sense
                 ,
                 yet
                 upon
                 mature
                 considerations
                 ,
                 these
                 will
                 not
                 be
                 judged
                 sufficient
                 to
                 found
                 this
                 sympathy
                 upon
                 .
                 If
                 we
                 lay
                 the
                 matter
                 seriously
                 in
                 the
                 ballance
                 ,
                 even
                 such
                 heterogeneal
                 and
                 final
                 inosculations
                 do
                 rather
                 hinder
                 the
                 transferring
                 of
                 humours
                 from
                 the
                 one
                 part
                 to
                 the
                 other
                 :
                 to
                 be
                 sure
                 ,
                 look
                 how
                 much
                 salls
                 ,
                 or
                 is
                 any
                 ways
                 drawn
                 into
                 their
                 capacities
                 and
                 cavities
                 ,
                 cannot
                 by
                 them
                 be
                 so
                 transferred
                 ,
                 but
                 is
                 otherways
                 disposed
                 of
                 :
                 for
                 the
                 mammarie
                 veins
                 carry
                 not
                 to
                 the
                 breasts
                 ,
                 but
                 to
                 the
                 subclavians
                 ,
                 and
                 so
                 to
                 the
                 heart
                 ,
                 and
                 the
                 mammary
                 arteries
                 carry
                 not
                 to
                 the
                 breasts
                 ,
                 but
                 to
                 the
                 Muscles
                 and
                 parts
                 on
                 the
                 forebelly
                 .
                 In
                 like
                 manner
                 the
                 womb
                 will
                 appear
                 equally
                 unconcerned
                 in
                 those
                 veins
                 and
                 arteries
                 :
                 for
                 the
                 Epigastrick
                 veins
                 carry
                 not
                 bloud
                 unto
                 the
                 womb
                 but
                 unto
                 their
                 collateral
                 Iliacks
                 ,
                 and
                 so
                 to
                 the
                 heart
                 :
                 and
                 the
                 Epigastrick
                 arteries
                 carry
                 not
                 
                 bloud
                 from
                 the
                 Iliacks
                 to
                 the
                 womb
                 ,
                 but
                 to
                 the
                 Muscles
                 lying
                 on
                 the
                 fore
                 part
                 of
                 the
                 belly
                 .
                 And
                 of
                 themselves
                 ,
                 neither
                 mammary
                 bloud-vessels
                 reach
                 the
                 womb
                 ,
                 nor
                 either
                 of
                 the
                 Epigastricks
                 the
                 breasts
                 :
                 so
                 as
                 ,
                 notwithstanding
                 any
                 thing
                 in
                 these
                 vessels
                 ,
                 we
                 yet
                 seem
                 to
                 be
                 in
                 the
                 dark
                 as
                 to
                 the
                 grounds
                 of
                 this
                 confessed
                 sympathy
                 .
                 Not
                 to
                 add
                 ,
                 that
                 in
                 many
                 Animals
                 some
                 ,
                 and
                 ,
                 in
                 some
                 all
                 ,
                 the
                 breasts
                 are
                 quite
                 out
                 of
                 the
                 way
                 of
                 the
                 Mammaryes
                 ,
                 on
                 the
                 same
                 side
                 of
                 the
                 Epigastricks
                 with
                 the
                 womb
                 ,
                 and
                 ,
                 on
                 that
                 side
                 ,
                 as
                 farr
                 removed
                 from
                 the
                 capillar
                 terminations
                 of
                 the
                 Epigastricks
                 and
                 mammaries
                 ,
                 as
                 the
                 womb
                 it self
                 .
                 And
                 professedly
                 I
                 understand
                 not
                 ,
                 upon
                 what
                 necessity
                 the
                 vessels
                 founding
                 ,
                 or
                 contributing
                 to
                 the
                 sympathy
                 of
                 these
                 two
                 parts
                 ,
                 must
                 needs
                 pass
                 directly
                 between
                 the
                 one
                 part
                 and
                 the
                 other
                 ;
                 and
                 may
                 not
                 as
                 well
                 here
                 ,
                 as
                 in
                 the
                 sympathy
                 of
                 other
                 parts
                 be
                 admitted
                 sufficient
                 for
                 this
                 purpose
                 ,
                 by
                 their
                 common
                 concern
                 and
                 relation
                 to
                 some
                 special
                 branch
                 ,
                 or
                 general
                 trunk
                 ,
                 And
                 vessells
                 thus
                 related
                 ,
                 touching
                 and
                 terminating
                 in
                 these
                 two
                 parts
                 ,
                 are
                 not
                 hard
                 to
                 be
                 found
                 in
                 several
                 kinds
                 ;
                 as
                 arteries
                 and
                 sinews
                 to
                 convey
                 like
                 matter
                 to
                 both
                 these
                 parts
                 :
                 veins
                 and
                 sinews
                 to
                 reduce
                 what
                 is
                 improper
                 ,
                 superfluous
                 ,
                 useless
                 ,
                 or
                 redundant
                 in
                 either
                 ,
                 or
                 meet
                 and
                 apt
                 to
                 be
                 transferred
                 from
                 the
                 one
                 to
                 the
                 other
                 :
                 in
                 which
                 both
                 cannot
                 but
                 be
                 assisted
                 by
                 
                 the
                 agreement
                 of
                 their
                 similar
                 attractions
                 ,
                 and
                 that
                 conformity
                 of
                 substance
                 ,
                 which
                 is
                 either
                 constant
                 ,
                 or
                 at
                 some
                 special
                 times
                 between
                 some
                 ,
                 or
                 all
                 of
                 each
                 of
                 their
                 parts
                 :
                 which
                 as
                 it
                 may
                 be
                 a
                 ground
                 of
                 the
                 maturation
                 &
                 exalting
                 of
                 these
                 parts
                 together
                 ,
                 so
                 thereby
                 the
                 Bodies
                 ,
                 pores
                 ,
                 and
                 passages
                 ,
                 of
                 both
                 ,
                 are
                 alike
                 open
                 ,
                 for
                 the
                 reception
                 of
                 like
                 humours
                 :
                 and
                 the
                 same
                 conform
                 substance
                 may
                 be
                 the
                 cause
                 why
                 those
                 vital
                 emanations
                 &
                 irradiations
                 which
                 the
                 generative
                 parts
                 receive
                 from
                 the
                 parts
                 which
                 are
                 principal
                 ;
                 when
                 ,
                 I
                 say
                 ,
                 those
                 vital
                 emanations
                 come
                 to
                 be
                 remitted
                 and
                 reflected
                 back
                 again
                 from
                 the
                 generative
                 parts
                 to
                 all
                 the
                 rest
                 of
                 the
                 body
                 ;
                 then
                 ,
                 above
                 all
                 others
                 ,
                 the
                 breasts
                 gather
                 into
                 themselves
                 powerfully
                 those
                 forces
                 of
                 nature
                 ,
                 which
                 are
                 so
                 reflected
                 back
                 again
                 from
                 the
                 generative
                 parts
                 universally
                 to
                 the
                 parts
                 of
                 the
                 whole
                 .
                 And
                 such
                 a
                 course
                 ,
                 as
                 I
                 have
                 hinted
                 ,
                 is
                 both
                 plain
                 by
                 view
                 ,
                 undeniably
                 certain
                 ,
                 and
                 familiar
                 in
                 the
                 usual
                 method
                 of
                 nature
                 ;
                 the
                 breasts
                 ,
                 above
                 other
                 parts
                 ,
                 being
                 exonerated
                 by
                 the
                 evacuations
                 of
                 the
                 womb
                 and
                 the
                 evacuations
                 of
                 the
                 womb
                 ,
                 more
                 especially
                 then
                 of
                 other
                 parts
                 ,
                 being
                 diverted
                 by
                 the
                 excretions
                 of
                 the
                 breasts
                 :
                 so
                 that
                 these
                 being
                 the
                 common
                 and
                 trodden
                 paths
                 of
                 nature
                 ,
                 there
                 cannot
                 but
                 remain
                 an
                 open
                 way
                 also
                 for
                 the
                 translation
                 of
                 other
                 matters
                 .
                 And
                 though
                 by
                 the
                 Mammaryes
                 and
                 Epigastricks
                 ,
                 
                 whether
                 inosculated
                 ,
                 or
                 not
                 inosculated
                 ,
                 I
                 have
                 above
                 shewn
                 ,
                 that
                 such
                 conveyances
                 cannot
                 be
                 made
                 ;
                 yet
                 nothing
                 hinders
                 ,
                 nay
                 't
                 is
                 very
                 apparent
                 ,
                 that
                 such
                 translations
                 may
                 be
                 aptly
                 enough
                 made
                 ,
                 nay
                 no
                 doubt
                 are
                 made
                 ,
                 by
                 the
                 thoracick
                 and
                 preparing
                 vessels
                 ;
                 to
                 wit
                 ,
                 by
                 the
                 Arteries
                 and
                 some
                 sinews
                 directly
                 ,
                 as
                 bringing
                 humours
                 directly
                 unto
                 the
                 parts
                 ,
                 and
                 by
                 other
                 sinews
                 and
                 veins
                 more
                 indirectly
                 ,
                 as
                 they
                 are
                 apt
                 to
                 carry
                 ,
                 or
                 not
                 carry
                 away
                 what
                 has
                 by
                 other
                 vessels
                 been
                 transferred
                 from
                 one
                 of
                 these
                 parts
                 unto
                 the
                 other
                 .
                 I
                 had
                 discoursed
                 these
                 things
                 shorter
                 ,
                 if
                 I
                 had
                 not
                 been
                 concerned
                 to
                 quit
                 my self
                 of
                 rashness
                 ,
                 with
                 which
                 you
                 had
                 attainted
                 me
                 for
                 seeming
                 herein
                 to
                 have
                 made
                 a
                 defection
                 from
                 the
                 Antients
                 .
                 I
                 should
                 assoon
                 defend
                 the
                 Moons
                 Eclipsses
                 to
                 be
                 a
                 celestiall
                 fainting
                 ,
                 and
                 planetary
                 sickness
                 ,
                 and
                 to
                 be
                 cured
                 by
                 Cymbals
                 ,
                 Drums
                 ,
                 Cornets
                 and
                 Trumpets
                 ;
                 because
                 the
                 Antients
                 thought
                 so
                 .
                 Yet
                 for
                 the
                 same
                 truths
                 sake
                 I
                 account
                 my self
                 bound
                 to
                 justify
                 the
                 old
                 Astronomers
                 in
                 the
                 causes
                 by
                 them
                 assigned
                 of
                 the
                 several
                 shapes
                 of
                 the
                 Moon
                 ,
                 as
                 depending
                 upon
                 its
                 mutuatious
                 light
                 ,
                 the
                 proportion
                 in
                 which
                 it
                 is
                 received
                 ,
                 and
                 the
                 positions
                 from
                 which
                 it
                 offers
                 its
                 self
                 to
                 our
                 view
                 ;
                 all
                 concerning
                 the
                 Earth
                 in
                 the
                 shapes
                 of
                 its
                 lucid
                 parts
                 ,
                 save
                 in
                 its
                 Eclipsses
                 sometimes
                 when
                 it
                 is
                 at
                 the
                 full
                 ,
                 being
                 the
                 product
                 merely
                 
                 of
                 ignorance
                 and
                 want
                 of
                 observation
                 .
                 I
                 am
                 not
                 of
                 opinion
                 with
                 many
                 ,
                 that
                 there
                 is
                 but
                 only
                 one
                 fountain
                 of
                 light
                 ,
                 which
                 is
                 the
                 sun
                 ;
                 to
                 which
                 by
                 imaginary
                 refractions
                 ,
                 and
                 reflections
                 ,
                 and
                 strange
                 labyrinthaean
                 windings
                 they
                 would
                 referre
                 all
                 other
                 lights
                 ;
                 sending
                 the
                 most
                 glorious
                 of
                 the
                 celestial
                 Bodies
                 a
                 borrowing
                 to
                 this
                 fountain
                 ;
                 while
                 evidently
                 in
                 these
                 inferiour
                 Regions
                 ,
                 in
                 the
                 mineral
                 ,
                 vegetable
                 and
                 animal
                 Kingdoms
                 ,
                 &c.
                 not
                 to
                 speak
                 of
                 the
                 caelestiall
                 flames
                 ,
                 several
                 special
                 fountains
                 of
                 light
                 have
                 been
                 ,
                 and
                 are
                 frequently
                 observed
                 ,
                 though
                 neither
                 so
                 full
                 ,
                 nor
                 durable
                 ,
                 and
                 perpetual
                 as
                 those
                 above
                 .
                 However
                 we
                 cannot
                 but
                 confesse
                 the
                 Sun
                 to
                 be
                 the
                 Fountain
                 of
                 the
                 principal
                 Lunar
                 light
                 ;
                 as
                 another
                 more
                 faint
                 light
                 discernable
                 in
                 all
                 parts
                 of
                 its
                 body
                 at
                 whatsoever
                 age
                 ,
                 so
                 it
                 be
                 out
                 of
                 the
                 Combustion
                 of
                 the
                 Beams
                 of
                 the
                 Sun
                 ,
                 which
                 lye
                 betwixt
                 us
                 and
                 the
                 Sun
                 ,
                 or
                 the
                 Beams
                 in
                 vicinity
                 to
                 them
                 ,
                 cannot
                 be
                 denyed
                 to
                 be
                 innate
                 unto
                 it
                 .
                 And
                 the
                 roundness
                 of
                 the
                 Moons
                 and
                 Suns
                 bodies
                 ,
                 being
                 evident
                 to
                 the
                 Eye
                 in
                 every
                 positure
                 and
                 revolution
                 ,
                 and
                 apparent
                 in
                 Solar
                 and
                 Lunar
                 Eclipses
                 ,
                 besides
                 several
                 other
                 demonstrations
                 of
                 their
                 rotundity
                 ;
                 it
                 follows
                 from
                 hence
                 by
                 necessary
                 consequence
                 that
                 the
                 Moons
                 body
                 cannot
                 be
                 guilded
                 all
                 over
                 ,
                 at
                 the
                 same
                 time
                 ,
                 with
                 such
                 luminous
                 beams
                 as
                 it
                 borrows
                 from
                 the
                 Sun
                 :
                 seeing
                 in
                 
                 any
                 two
                 sphaeres
                 ,
                 in
                 whatsoever
                 position
                 ,
                 the
                 one
                 without
                 the
                 other
                 ,
                 there
                 is
                 still
                 a
                 certain
                 part
                 of
                 the
                 surface
                 of
                 the
                 one
                 ,
                 unto
                 which
                 without
                 penetrating
                 its
                 body
                 from
                 no
                 point
                 of
                 the
                 other
                 can
                 be
                 drawn
                 a
                 straight
                 line
                 ,
                 as
                 is
                 easy
                 to
                 demonstrate
                 .
                 And
                 the
                 magnitude
                 of
                 the
                 bodies
                 of
                 these
                 two
                 Stars
                 ,
                 being
                 confusedly
                 unaequal
                 the
                 one
                 unto
                 the
                 other
                 ,
                 and
                 the
                 Suns
                 the
                 larger
                 ;
                 it
                 follows
                 in
                 like
                 manner
                 by
                 the
                 same
                 necessity
                 ,
                 that
                 though
                 all
                 the
                 Moon
                 cannot
                 be
                 at
                 once
                 enlightened
                 by
                 the
                 borrow'd
                 Solar
                 light
                 ,
                 yet
                 at
                 all
                 times
                 ,
                 more
                 than
                 half
                 of
                 the
                 Moons
                 sphaeral
                 surface
                 ,
                 is
                 so
                 enlightened
                 ;
                 except
                 in
                 those
                 cases
                 when
                 the
                 interposition
                 of
                 the
                 earth
                 ,
                 or
                 other
                 opake
                 bodies
                 ,
                 obstructs
                 the
                 solar
                 radiations
                 from
                 falling
                 upon
                 it
                 .
                 For
                 if
                 a
                 lesser
                 ,
                 and
                 a
                 greater
                 sphaere
                 be
                 put
                 and
                 adapted
                 into
                 an
                 Isoscelary
                 Cone
                 ,
                 whose
                 Axe
                 is
                 perpendicular
                 to
                 the
                 circular
                 Base
                 at
                 its
                 Center
                 ,
                 the
                 Lines
                 in
                 which
                 the
                 surface
                 of
                 the
                 Cone
                 toucheth
                 the
                 surfaces
                 of
                 the
                 sphaeres
                 ,
                 shall
                 be
                 two
                 Circles
                 less
                 than
                 the
                 respective
                 greatest
                 Circles
                 ,
                 each
                 of
                 its
                 own
                 Sphaers
                 as
                 is
                 not
                 hard
                 to
                 demonstrate
                 .
                 And
                 it
                 is
                 as
                 apparent
                 ,
                 that
                 the
                 Isorrhopical
                 center
                 of
                 that
                 part
                 of
                 the
                 Moons
                 surface
                 ,
                 which
                 is
                 so
                 solarly
                 enlightened
                 ,
                 is
                 still
                 the
                 point
                 directly
                 obverted
                 to
                 the
                 Sun
                 ;
                 thorow
                 which
                 point
                 a
                 straight
                 line
                 drawn
                 between
                 the
                 centers
                 of
                 these
                 two
                 Stars
                 passeth
                 .
                 So
                 as
                 the
                 enlightened
                 part
                 of
                 the
                 Moon
                 being
                 still
                 towards
                 
                 the
                 Sun
                 ,
                 no
                 wonder
                 ,
                 if
                 in
                 their
                 conjunction
                 the
                 Moon
                 cannot
                 be
                 seen
                 at
                 all
                 ;
                 as
                 chiefly
                 because
                 of
                 its
                 combustion
                 ,
                 so
                 also
                 because
                 the
                 enlightened
                 side
                 of
                 it
                 ,
                 is
                 turned
                 away
                 from
                 us
                 towards
                 the
                 Sun
                 :
                 that
                 though
                 ,
                 by
                 descending
                 into
                 deep
                 pits
                 ,
                 Stars
                 above
                 the
                 Horizon
                 may
                 be
                 seen
                 at
                 Noon
                 day
                 ,
                 yet
                 the
                 Moon
                 in
                 this
                 case
                 cannot
                 by
                 such
                 helps
                 be
                 in
                 like
                 manner
                 seen
                 ,
                 its
                 dark
                 side
                 being
                 towards
                 us
                 ,
                 and
                 the
                 most
                 that
                 our
                 Eye
                 can
                 at
                 any
                 time
                 at
                 one
                 view
                 see
                 of
                 the
                 Moons
                 surface
                 ,
                 being
                 ever
                 less
                 then
                 the
                 half
                 of
                 it
                 ,
                 as
                 is
                 demonstrable
                 .
                 Upon
                 the
                 same
                 grounds
                 ,
                 at
                 the
                 Opposition
                 ,
                 i.
                 e.
                 full
                 Moon
                 ,
                 the
                 enlightened
                 part
                 of
                 the
                 Moon
                 as
                 it
                 is
                 towards
                 the
                 Sun
                 ,
                 so
                 it
                 is
                 towards
                 us
                 also
                 ,
                 from
                 whence
                 follows
                 ,
                 that
                 all
                 ,
                 that
                 we
                 can
                 then
                 see
                 of
                 the
                 Moon
                 ,
                 is
                 enlightened
                 ,
                 and
                 that
                 much
                 more
                 of
                 it
                 is
                 enlightened
                 ,
                 than
                 it
                 can
                 be
                 possible
                 for
                 us
                 to
                 see
                 .
                 In
                 the
                 Quadratures
                 ,
                 whether
                 of
                 increase
                 ,
                 or
                 decrease
                 ,
                 the
                 state
                 and
                 positure
                 of
                 the
                 enlightened
                 ,
                 and
                 unenlightened
                 parts
                 of
                 the
                 Moon
                 ,
                 in
                 respect
                 of
                 the
                 Sun
                 and
                 us
                 ,
                 offer
                 themselves
                 in
                 a
                 less
                 entire
                 ,
                 divided
                 ,
                 middle
                 manner
                 :
                 to
                 wit
                 ,
                 the
                 parts
                 of
                 the
                 Moons
                 surface
                 obverted
                 unto
                 us
                 are
                 but
                 ,
                 as
                 it
                 were
                 ,
                 half
                 obverted
                 to
                 the
                 Sun
                 ,
                 and
                 the
                 parts
                 of
                 the
                 Moons
                 surface
                 obverted
                 to
                 the
                 Sun
                 are
                 but
                 ,
                 as
                 it
                 were
                 ,
                 half
                 obverted
                 to
                 us
                 :
                 so
                 as
                 what
                 is
                 obverted
                 to
                 us
                 in
                 the
                 Quadratures
                 is
                 ,
                 as
                 it
                 
                 were
                 ,
                 half
                 enlightened
                 ,
                 and
                 half
                 unenlightned
                 and
                 at
                 other
                 not
                 Cardinal
                 times
                 ,
                 the
                 nearer
                 that
                 the
                 Moon
                 is
                 to
                 the
                 full
                 ,
                 so
                 much
                 the
                 greater
                 part
                 of
                 the
                 Moons
                 surface
                 obverted
                 to
                 us
                 ,
                 is
                 by
                 the
                 Sun
                 enlightened
                 :
                 and
                 the
                 nearer
                 the
                 Moon
                 is
                 to
                 its
                 Change
                 ,
                 whether
                 past
                 ,
                 or
                 to
                 come
                 ,
                 so
                 much
                 the
                 lesser
                 part
                 doth
                 the
                 Sun
                 enlighten
                 of
                 the
                 Moons
                 surface
                 obverted
                 to
                 us
                 .
                 And
                 that
                 these
                 things
                 are
                 really
                 so
                 ,
                 may
                 be
                 from
                 hence
                 confirmed
                 :
                 at
                 whatsoever
                 .
                 Age
                 of
                 the
                 Moons
                 crescency
                 ,
                 or
                 decrescency
                 ,
                 if
                 you
                 observe
                 its
                 luminous
                 parts
                 seen
                 by
                 us
                 ,
                 however
                 horned
                 ,
                 or
                 bunched
                 ,
                 the
                 Sun
                 is
                 still
                 on
                 the
                 luminous
                 side
                 of
                 the
                 Moon
                 .
                 Now
                 the
                 cause
                 ,
                 why
                 the
                 Moons
                 luminous
                 parts
                 appear
                 plain
                 and
                 flat
                 ,
                 rather
                 than
                 bearing
                 out
                 ,
                 like
                 a
                 Sphaeral
                 surface
                 ,
                 or
                 a
                 portion
                 of
                 a
                 Sphaeral
                 surface
                 ,
                 is
                 because
                 the
                 inequality
                 of
                 the
                 visual
                 beams
                 ,
                 between
                 the
                 Eye
                 and
                 the
                 several
                 luminous
                 points
                 ,
                 is
                 not
                 so
                 proportionate
                 as
                 to
                 be
                 discernable
                 by
                 our
                 ordinary
                 sight
                 :
                 for
                 without
                 such
                 inequality
                 ,
                 to
                 be
                 discerned
                 in
                 the
                 visual
                 beams
                 ,
                 neither
                 sense
                 ,
                 nor
                 reason
                 can
                 conceive
                 true
                 Idea's
                 of
                 a
                 spheral
                 surface
                 .
                 And
                 upon
                 these
                 Hypotheses
                 and
                 principles
                 all
                 phaenomena's
                 concerning
                 the
                 Moons
                 shapes
                 ,
                 as
                 to
                 magnitude
                 ,
                 proportion
                 ,
                 continuance
                 ,
                 situation
                 ,
                 alteration
                 ,
                 alternation
                 ,
                 being
                 perpetually
                 and
                 accurately
                 solved
                 ;
                 the
                 Novelty
                 which
                 
                 would
                 dethrone
                 this
                 doctrine
                 so
                 apparent
                 ,
                 so
                 rational
                 and
                 unconvincible
                 of
                 the
                 Antients
                 ,
                 as
                 it
                 is
                 unworthy
                 the
                 acceptance
                 of
                 the
                 considerate
                 and
                 judicious
                 ,
                 so
                 it
                 cannot
                 but
                 be
                 unmeet
                 for
                 them
                 to
                 impose
                 and
                 obtrude
                 upon
                 others
                 ;
                 to
                 omit
                 the
                 grosse
                 inconsistency
                 involved
                 in
                 it
                 ,
                 which
                 I
                 shall
                 in
                 my
                 next
                 at
                 large
                 unfold
                 ,
                 if
                 these
                 things
                 fall
                 short
                 of
                 giving
                 Satisfaction
                 .
              
            
             
               
               
                 The
                 third
                 Answer
                 .
              
               
                 
                   That
                   there
                   are
                   fresh
                   watersprings
                   at
                   the
                   bottoms
                   of
                   the
                   most
                   antient
                   seas
                   .
                   Whether
                   Mercury
                   by
                   frequent
                   transhaping
                   it self
                   ,
                   and
                   often
                   reduction
                   loose
                   not
                   somwhat
                   of
                   its
                   powers
                   and
                   virtues
                   ?
                   That
                   the
                   Doctrine
                   and
                   being
                   of
                   the
                   four
                   Elements
                   ,
                   as
                   unmixed
                   bodies
                   by
                   their
                   mixture
                   making
                   up
                   all
                   other
                   bodies
                   ,
                   is
                   not
                   unquestionable
                   .
                   In
                   the
                   four
                   Elements
                   may
                   be
                   allowed
                   to
                   be
                   the
                   general
                   and
                   most
                   common
                   lodges
                   of
                   the
                   Worlds
                   first
                   Elements
                   ▪
                   but
                   themselves
                   at
                   the
                   most
                   can
                   but
                   be
                   allowed
                   to
                   be
                   only
                   secondary
                   Elements
                   .
                   How
                   nature
                   may
                   so
                   ballance
                   the
                   first
                   or
                   second
                   Elements
                   by
                   some
                   special
                   Symbolical
                   properties
                   among
                   them
                   ,
                   as
                   to
                   elude
                   all
                   the
                   endeavours
                   which
                   art
                   can
                   possibly
                   make
                   for
                   the
                   bringing
                   of
                   the
                   first
                   Elements
                   to
                   view
                   and
                   light
                   .
                   That
                   the
                   number
                   of
                   the
                   four
                   Elements
                   and
                   of
                   their
                   properties
                   or
                   qualities
                   singly
                   or
                   in
                   Conjugations
                   ascribed
                   to
                   them
                   seemeth
                   insufficient
                   to
                   vest
                   them
                   in
                   the
                   right
                   of
                   first
                   Elements
                   ;
                   upon
                   the
                   knowledge
                   of
                   whose
                   natures
                   ,
                   all
                   Physical
                   phaenomena
                   should
                   be
                   capable
                   of
                   explication
                   .
                   That
                   the
                   antients
                   seem
                   to
                   have
                   allowed
                   a
                   greater
                   number
                   of
                   first
                   Elements
                   .
                   A
                   conjecture
                   what
                   the
                   antients
                   might
                   rationally
                   at
                   first
                   design
                   at
                   their
                   first
                   introducing
                   the
                   Doctrine
                   of
                   the
                   four
                   Elements
                   .
                
              
               
                 
                   SIR
                   .
                
              
               
                 THat
                 fresh
                 water
                 springs
                 lye
                 at
                 the
                 bottom
                 of
                 the
                 seas
                 ,
                 both
                 frequent
                 ,
                 and
                 with
                 flashing
                 issues
                 ;
                 is
                 to
                 me
                 upon
                 several
                 observations
                 ,
                 not
                 undiligently
                 made
                 ,
                 as
                 absolutely
                 
                 certain
                 ,
                 as
                 to
                 you
                 it
                 seems
                 impossible
                 :
                 and
                 this
                 I
                 judge
                 not
                 only
                 in
                 the
                 seas
                 ,
                 which
                 have
                 made
                 inchroachments
                 by
                 inundation
                 upon
                 the
                 antient
                 bounders
                 of
                 lands
                 ,
                 which
                 before
                 were
                 plenteously
                 up
                 and
                 down
                 watered
                 with
                 springs
                 ;
                 but
                 as
                 well
                 ,
                 there
                 being
                 the
                 like
                 reason
                 ,
                 in
                 those
                 seas
                 ,
                 which
                 are
                 able
                 to
                 plead
                 the
                 highest
                 and
                 most
                 antient
                 praescription
                 ,
                 and
                 cannot
                 be
                 any
                 other
                 wayes
                 chargeable
                 ,
                 then
                 in
                 their
                 dayly
                 fluxes
                 and
                 refluxes
                 ,
                 with
                 the
                 least
                 new
                 invading
                 of
                 the
                 earths
                 bosome
                 ,
                 and
                 of
                 those
                 wells
                 of
                 coldboiling-natural
                 nectar
                 ,
                 with
                 which
                 it
                 is
                 usually
                 there
                 stored
                 .
                 And
                 why
                 should
                 natures
                 opening
                 a
                 vein
                 of
                 freshwater
                 into
                 the
                 sea
                 ,
                 seem
                 such
                 a
                 sea
                 monster
                 when
                 at
                 land
                 we
                 ordinarily
                 meet
                 with
                 divers
                 springs
                 ,
                 of
                 different
                 virtues
                 and
                 originals
                 ,
                 meeting
                 at
                 after
                 together
                 in
                 the
                 same
                 channel
                 ;
                 from
                 their
                 concourse
                 and
                 mixture
                 conceiving
                 secret
                 virtues
                 ,
                 manifest
                 alterations
                 ,
                 and
                 special
                 properties
                 ,
                 as
                 strange
                 to
                 the
                 illiterate
                 and
                 unexperienced
                 ,
                 and
                 generally
                 wondered
                 at
                 by
                 the
                 most
                 ,
                 as
                 the
                 boiling
                 heat
                 appearing
                 presently
                 in
                 the
                 suddenly
                 mixt
                 Oyles
                 of
                 Tartar
                 and
                 Vitriol
                 ,
                 though
                 cold
                 ,
                 when
                 poured
                 together
                 .
                 I
                 shall
                 not
                 urge
                 ,
                 that
                 some
                 ,
                 would
                 have
                 the
                 seas
                 ,
                 proportionately
                 to
                 their
                 depth
                 ,
                 fresh
                 at
                 the
                 bottom
                 .
                 I
                 only
                 move
                 ,
                 if
                 this
                 be
                 not
                 more
                 easy
                 ,
                 to
                 be
                 assented
                 to
                 ,
                 then
                 what
                 is
                 dayly
                 seen
                 ,
                 and
                 therefore
                 not
                 to
                 be
                 questioned
                 in
                 point
                 of
                 truth
                 ,
                 
                 that
                 the
                 sea
                 fishes
                 &c.
                 
                 Though
                 continually
                 living
                 in
                 that
                 briny
                 pickle
                 remain
                 still
                 however
                 fresh
                 :
                 many
                 times
                 the
                 sea-fowle
                 ,
                 that
                 most-what
                 fly
                 but
                 about
                 it
                 ,
                 sometimes
                 swim
                 in
                 the
                 surface
                 of
                 it
                 ,
                 senting
                 and
                 tasting
                 much
                 stronger
                 of
                 the
                 sea
                 ,
                 than
                 the
                 fish
                 that
                 live
                 deep
                 and
                 constant
                 in
                 it
                 .
                 But
                 Experience
                 being
                 the
                 grand
                 Umpiress
                 in
                 the
                 Question
                 ,
                 in
                 assurance
                 of
                 its
                 convictiveness
                 ,
                 I
                 forbear
                 at
                 present
                 .
                 For
                 your
                 reducing
                 of
                 Mercury
                 after
                 all
                 operations
                 of
                 fire
                 ,
                 whether
                 actual
                 ,
                 or
                 potential
                 upon
                 it
                 ,
                 I
                 have
                 no
                 reason
                 to
                 cherish
                 suspitions
                 of
                 the
                 truth
                 of
                 such
                 performances
                 :
                 but
                 that
                 reduction
                 shall
                 be
                 so
                 perfect
                 ,
                 as
                 to
                 give
                 back
                 the
                 Mercury
                 as
                 absolute
                 in
                 all
                 virtues
                 ,
                 as
                 it
                 was
                 ,
                 or
                 could
                 at
                 first
                 be
                 delivered
                 ,
                 will
                 not
                 be
                 easily
                 consented
                 to
                 ,
                 by
                 those
                 that
                 know
                 that
                 fire
                 burns
                 the
                 chief
                 wing
                 upon
                 which
                 in
                 Amalgames
                 Mercury
                 carryes
                 along
                 with
                 it
                 Gold
                 :
                 and
                 the
                 force
                 of
                 fire
                 upon
                 it
                 ,
                 is
                 in
                 the
                 nature
                 of
                 a
                 rape
                 robbing
                 it
                 of
                 that
                 virgin
                 treasure
                 ,
                 to
                 which
                 the
                 noble
                 mettal
                 is
                 so
                 sequacious
                 ;
                 which
                 once
                 lost
                 ,
                 can
                 never
                 be
                 restored
                 again
                 to
                 the
                 defloured
                 Mercury
                 :
                 as
                 in
                 all
                 volatiles
                 that
                 which
                 is
                 of
                 nimblest
                 wing
                 flyes
                 first
                 ,
                 and
                 the
                 highest
                 spirit
                 first
                 :
                 and
                 when
                 an
                 impregnation
                 is
                 to
                 be
                 made
                 by
                 fire
                 the
                 work
                 of
                 Philosophy
                 is
                 judged
                 chiefly
                 to
                 lye
                 in
                 the
                 governing
                 ,
                 of
                 the
                 firing
                 ,
                 that
                 ,
                 like
                 the
                 Sun
                 ,
                 it
                 may
                 give
                 enlivening
                 heat
                 ,
                 and
                 not
                 become
                 ,
                 as
                 a
                 destroying
                 Element
                 .
                 The
                 reason
                 ,
                 why
                 so
                 
                 few
                 are
                 acquainted
                 with
                 the
                 excellency
                 and
                 praerogative
                 of
                 this
                 pure
                 Mercury
                 ,
                 is
                 because
                 generally
                 it
                 is
                 a
                 fire-burnt
                 Mercury
                 ,
                 which
                 is
                 at
                 first
                 delivered
                 to
                 us
                 .
                 I
                 omit
                 that
                 in
                 a
                 thousand
                 instances
                 ,
                 after
                 Art
                 has
                 separated
                 the
                 natural
                 union
                 of
                 parts
                 ,
                 though
                 it
                 may
                 again
                 unite
                 them
                 ,
                 yet
                 that
                 re-union
                 will
                 in
                 many
                 points
                 fall
                 short
                 of
                 the
                 first
                 natural
                 union
                 .
                 But
                 why
                 does
                 it
                 seem
                 so
                 monstrous
                 to
                 you
                 ,
                 to
                 call
                 in
                 Question
                 the
                 existence
                 of
                 the
                 four
                 Elements
                 ,
                 it
                 being
                 a
                 doctrine
                 ,
                 which
                 in
                 all
                 Ages
                 has
                 been
                 attended
                 with
                 doubts
                 ?
                 if
                 you
                 think
                 you
                 can
                 shew
                 as
                 Air
                 ,
                 and
                 Earth
                 and
                 water
                 ,
                 yet
                 what
                 shew
                 can
                 you
                 make
                 of
                 Elemental
                 fire
                 ?
                 And
                 since
                 the
                 dissolution
                 of
                 the
                 Orbes
                 ,
                 there
                 being
                 no
                 Concave
                 of
                 the
                 Moon
                 ,
                 what
                 region
                 is
                 designed
                 and
                 intended
                 for
                 it
                 ?
                 I
                 presume
                 both
                 Hearths
                 ,
                 and
                 Altars
                 ,
                 are
                 too
                 low
                 ,
                 to
                 be
                 the
                 proper
                 Sphaere
                 of
                 this
                 high
                 Element
                 .
                 And
                 upon
                 the
                 whole
                 ,
                 as
                 little
                 can
                 be
                 said
                 for
                 any
                 of
                 the
                 other
                 three
                 as
                 for
                 this
                 :
                 the
                 State
                 of
                 the
                 Question
                 and
                 controversie
                 being
                 rightly
                 understood
                 .
                 For
                 it
                 is
                 not
                 fire
                 that
                 is
                 denied
                 ,
                 or
                 Earth
                 ,
                 or
                 Ayr
                 ,
                 or
                 water
                 ,
                 but
                 all
                 as
                 Elemental
                 .
                 How
                 great
                 a
                 share
                 and
                 proportion
                 these
                 have
                 in
                 making
                 up
                 this
                 part
                 of
                 the
                 worlds
                 systeme
                 ,
                 is
                 too
                 evident
                 to
                 sense
                 to
                 be
                 called
                 in
                 Question
                 :
                 many
                 leagues
                 of
                 Earth
                 ,
                 or
                 sea
                 ,
                 answering
                 to
                 each
                 single
                 degree
                 :
                 but
                 that
                 these
                 four
                 are
                 the
                 first
                 uncompounded
                 compounding
                 
                 bodies
                 ,
                 of
                 which
                 all
                 others
                 by
                 their
                 mixtion
                 consist
                 ,
                 and
                 are
                 made
                 ;
                 seems
                 ,
                 from
                 hence
                 ,
                 to
                 follow
                 by
                 a
                 very
                 sickly
                 consequence
                 .
                 Not
                 without
                 solid
                 consideration
                 ,
                 and
                 sufficient
                 cause
                 ,
                 did
                 the
                 Antients
                 honour
                 the
                 Earth
                 with
                 the
                 style
                 of
                 the
                 All-feeding-Earth
                 :
                 what
                 innumerable
                 vegetables
                 and
                 Animals
                 spring
                 ,
                 live
                 and
                 grow
                 in
                 its
                 bosome
                 ,
                 not
                 to
                 touch
                 at
                 the
                 inestimable
                 treasures
                 lodged
                 in
                 its
                 bowels
                 ?
                 Others
                 ,
                 and
                 of
                 the
                 worlds
                 sages
                 ,
                 with
                 honours
                 not
                 unequal
                 ,
                 assign
                 unto
                 the
                 water
                 not
                 only
                 Beauty's
                 birth
                 ,
                 but
                 the
                 Rise
                 and
                 original
                 of
                 all
                 things
                 :
                 and
                 every
                 moment
                 convinceth
                 the
                 use
                 and
                 necessity
                 of
                 Air
                 &c.
                 
                 These
                 are
                 Arguments
                 of
                 their
                 universal
                 use
                 ,
                 concern
                 and
                 excellency
                 ,
                 but
                 do
                 not
                 prove
                 their
                 Elementalness
                 ,
                 no
                 more
                 then
                 that
                 one
                 Tree
                 is
                 the
                 Element
                 of
                 another
                 ,
                 because
                 the
                 one
                 is
                 engrafted
                 into
                 the
                 other
                 ,
                 or
                 that
                 the
                 mother
                 is
                 the
                 Element
                 of
                 the
                 child
                 ,
                 while
                 it
                 lives
                 annexed
                 unto
                 her
                 in
                 the
                 womb
                 .
                 That
                 in
                 these
                 are
                 the
                 general
                 and
                 most
                 common
                 Lodges
                 of
                 the
                 worlds
                 Elements
                 ,
                 from
                 which
                 each
                 nature
                 may
                 furnish
                 it self
                 with
                 what
                 is
                 convenient
                 for
                 its
                 being
                 ,
                 nutrition
                 and
                 growth
                 ,
                 may
                 easily
                 be
                 consented
                 unto
                 :
                 but
                 that
                 these
                 are
                 the
                 very
                 Elements
                 unmixed
                 and
                 uncompounded
                 ,
                 will
                 be
                 an
                 Herculean
                 task
                 to
                 make
                 forth
                 by
                 any
                 Argument
                 ,
                 or
                 experience
                 ,
                 so
                 long
                 as
                 each
                 in
                 its
                 region
                 appears
                 replenished
                 with
                 all
                 variety
                 of
                 Beings
                 .
                 What
                 a
                 numerous
                 
                 diversity
                 of
                 Earths
                 are
                 to
                 be
                 found
                 in
                 the
                 Earth
                 ,
                 of
                 diverse
                 colours
                 ,
                 weights
                 ,
                 virtues
                 ;
                 some
                 healing
                 ,
                 some
                 scouring
                 ,
                 some
                 binding
                 ,
                 some
                 Alexipharmical
                 ,
                 besides
                 chalkes
                 ,
                 and
                 Marles
                 ,
                 and
                 several
                 clayes
                 ,
                 sands
                 and
                 gravels
                 ,
                 noble
                 quarries
                 ,
                 rich
                 mines
                 ,
                 coals
                 ,
                 bitumes
                 ,
                 marcasites
                 ,
                 salts
                 ,
                 minerals
                 and
                 metals
                 ?
                 so
                 in
                 water
                 ,
                 all
                 coagulable
                 vapours
                 and
                 exhalations
                 meet
                 condensed
                 with
                 the
                 alluvion
                 and
                 dissolution
                 of
                 various
                 salts
                 and
                 other
                 minerals
                 ,
                 together
                 with
                 manifold
                 subterraneous
                 Oyles
                 and
                 spirits
                 ;
                 whence
                 the
                 wonderful
                 difference
                 of
                 their
                 weights
                 ,
                 sents
                 ,
                 colours
                 ,
                 tasts
                 ,
                 consistence
                 and
                 operations
                 .
                 In
                 like
                 manner
                 the
                 Air
                 ,
                 what
                 is
                 it
                 but
                 an
                 uncertain
                 ,
                 unconstant
                 ,
                 randome
                 composition
                 of
                 all
                 sorts
                 of
                 fumes
                 and
                 vapours
                 ,
                 according
                 to
                 the
                 nature
                 and
                 position
                 of
                 the
                 Atomes
                 dispersed
                 in
                 it
                 sometimes
                 clear
                 ,
                 sometimes
                 cloudy
                 ,
                 sometimes
                 healthful
                 ,
                 sometimes
                 pestilent
                 ,
                 sometimes
                 delighting
                 ,
                 at
                 other
                 times
                 offending
                 the
                 senses
                 ?
                 And
                 as
                 difficult
                 will
                 it
                 be
                 to
                 find
                 and
                 shew
                 any
                 uncompounded
                 Fire
                 .
                 But
                 you
                 will
                 say
                 ,
                 that
                 though
                 these
                 be
                 not
                 Elemental
                 Earth
                 ,
                 Fire
                 ,
                 Aire
                 and
                 Water
                 ,
                 it
                 hinders
                 not
                 but
                 there
                 may
                 be
                 such
                 Beings
                 and
                 bodies
                 ;
                 I
                 answer
                 ,
                 I
                 have
                 no
                 abhorrency
                 against
                 the
                 opinion
                 ,
                 if
                 so
                 by
                 any
                 practice
                 ,
                 or
                 in
                 any
                 dissolution
                 these
                 Elements
                 could
                 but
                 be
                 shewn
                 ,
                 and
                 their
                 sufficiency
                 to
                 explain
                 the
                 phaenomena
                 of
                 nature
                 :
                 but
                 those
                 parts
                 ,
                 which
                 ,
                 generally
                 
                 in
                 dissolutions
                 :
                 are
                 offered
                 to
                 be
                 accepted
                 for
                 these
                 Elements
                 ,
                 appear
                 nothing
                 lesse
                 ,
                 and
                 ,
                 beyond
                 all
                 dispute
                 ,
                 very
                 distant
                 from
                 uncompounded
                 natures
                 ;
                 and
                 the
                 analogous
                 parts
                 in
                 several
                 dissolutions
                 ,
                 as
                 different
                 one
                 from
                 another
                 ,
                 as
                 one
                 of
                 the
                 Elements
                 can
                 be
                 from
                 another
                 Element
                 ;
                 which
                 is
                 repugnant
                 to
                 the
                 nature
                 of
                 first
                 Elements
                 ;
                 as
                 is
                 usually
                 and
                 truly
                 urged
                 against
                 salts
                 ,
                 sulphures
                 and
                 Mercuries
                 to
                 put
                 by
                 their
                 claim
                 in
                 first
                 Elementarinesse
                 ,
                 in
                 the
                 mixtion
                 of
                 bodies
                 :
                 Viz.
                 that
                 in
                 several
                 dissolutions
                 ,
                 the
                 analogous
                 parts
                 answer
                 not
                 in
                 the
                 least
                 one
                 to
                 another
                 ,
                 except
                 it
                 be
                 in
                 some
                 very
                 general
                 ,
                 and
                 external
                 conformity
                 ;
                 but
                 one
                 is
                 sowr
                 ,
                 another
                 sweet
                 ,
                 another
                 sharp
                 ,
                 another
                 faelid
                 ,
                 another
                 fragrant
                 ,
                 thus
                 expanding
                 into
                 all
                 manner
                 of
                 variety
                 ,
                 wherein
                 should
                 be
                 nothing
                 but
                 the
                 pure
                 simplicity
                 of
                 a
                 first
                 Element
                 .
                 In
                 a
                 good
                 sense
                 ,
                 and
                 with
                 fair
                 Explications
                 ,
                 I
                 have
                 been
                 ever
                 ready
                 to
                 acknowledge
                 their
                 interest
                 as
                 secondary
                 Elements
                 ,
                 as
                 a
                 doctrine
                 consentaneons
                 to
                 the
                 Sacred
                 Traditions
                 of
                 Theology
                 ;
                 or
                 as
                 before
                 ,
                 as
                 the
                 principal
                 Lodges
                 of
                 the
                 first
                 Elements
                 :
                 but
                 neither
                 that
                 which
                 fils
                 the
                 regions
                 ,
                 nor
                 that
                 which
                 remains
                 after
                 their
                 dissolution
                 of
                 mixed
                 bodies
                 ,
                 could
                 ever
                 yet
                 to
                 me
                 seem
                 capable
                 of
                 the
                 denomination
                 of
                 first
                 Elements
                 .
                 And
                 possibly
                 it
                 may
                 be
                 one
                 of
                 natures
                 mysteries
                 never
                 to
                 permit
                 
                 us
                 to
                 see
                 the
                 first
                 Elements
                 naked
                 so
                 curiously
                 ballancing
                 them
                 in
                 their
                 connexions
                 ,
                 that
                 all
                 attempts
                 of
                 Art
                 ,
                 for
                 their
                 discovery
                 ,
                 shall
                 be
                 eluded
                 :
                 so
                 any
                 Artifice
                 to
                 resolve
                 and
                 anatomize
                 a
                 natural
                 body
                 into
                 its
                 first
                 principles
                 ,
                 grounded
                 upon
                 the
                 fixedness
                 and
                 volatility
                 of
                 the
                 parts
                 ,
                 is
                 eluded
                 ,
                 if
                 first
                 ,
                 or
                 second
                 Elements
                 of
                 the
                 same
                 fixednes
                 ,
                 or
                 volatility
                 be
                 connexed
                 :
                 so
                 if
                 grounded
                 upon
                 dissolubleness
                 in
                 a
                 certain
                 Menstrue
                 ,
                 it
                 is
                 eluded
                 by
                 connexing
                 several
                 first
                 ,
                 or
                 second
                 Elements
                 aequally
                 dissoluble
                 in
                 that
                 ,
                 or
                 the
                 like
                 Menstrue
                 :
                 so
                 corrosious
                 by
                 separatories
                 ,
                 fumigating
                 ,
                 anointing
                 ,
                 cementing
                 are
                 all
                 eluded
                 as
                 to
                 the
                 bolting
                 forth
                 of
                 the
                 first
                 Elements
                 ,
                 if
                 they
                 meet
                 with
                 several
                 first
                 ,
                 or
                 other
                 Elements
                 aequally
                 passible
                 in
                 those
                 operations
                 .
                 And
                 the
                 very
                 number
                 of
                 the
                 Elements
                 has
                 not
                 seldom
                 rendered
                 them
                 unto
                 me
                 very
                 suspitious
                 ,
                 that
                 they
                 could
                 not
                 be
                 first
                 Elements
                 .
                 For
                 by
                 what
                 fair
                 Argument
                 shal
                 the
                 Quaternion
                 of
                 them
                 be
                 demonstrated
                 ?
                 if
                 ,
                 as
                 is
                 usual
                 ,
                 we
                 argue
                 from
                 qualities
                 ,
                 whether
                 singular
                 ,
                 or
                 in
                 symbolismes
                 ,
                 I
                 see
                 not
                 how
                 this
                 number
                 can
                 be
                 maintained
                 :
                 more
                 qualities
                 ,
                 and
                 more
                 conjugations
                 of
                 them
                 ,
                 ought
                 by
                 the
                 same
                 reason
                 to
                 prove
                 a
                 far
                 greater
                 number
                 of
                 Elements
                 And
                 alas
                 ,
                 how
                 vainly
                 light
                 do
                 they
                 render
                 themselves
                 ,
                 that
                 ,
                 by
                 four
                 conjugations
                 of
                 disputable
                 qualities
                 ,
                 seek
                 to
                 solve
                 all
                 the
                 phaenomena
                 
                 which
                 are
                 in
                 the
                 world
                 ;
                 of
                 the
                 Loadstones
                 pointing
                 northward
                 ,
                 its
                 drawing
                 untoucht
                 steel
                 ,
                 the
                 Load-touched
                 steels
                 drawing
                 of
                 other
                 steel
                 ,
                 the
                 bleeding
                 of
                 the
                 slain
                 at
                 the
                 presence
                 of
                 the
                 Murtherer
                 ,
                 the
                 moving
                 of
                 an
                 untouched
                 Lute-string
                 being
                 Unison
                 ,
                 or
                 in
                 a
                 strong
                 consonancy
                 string
                 that
                 is
                 moved
                 ,
                 the
                 sinking
                 of
                 persons
                 drowned
                 the
                 first
                 dayes
                 ,
                 and
                 then
                 floating
                 to
                 the
                 top
                 at
                 after
                 ,
                 they
                 have
                 layd
                 soaking
                 so
                 long
                 in
                 the
                 water
                 ,
                 the
                 Cramp
                 fishes
                 astonishing
                 the
                 Fishers
                 hand
                 at
                 the
                 long
                 distance
                 of
                 line
                 and
                 Angle
                 ,
                 the
                 strange
                 Ebbings
                 and
                 flowings
                 of
                 particular
                 seas
                 and
                 springs
                 ,
                 and
                 a
                 thousand
                 other
                 ?
                 Upon
                 the
                 old
                 principles
                 ,
                 how
                 lame
                 is
                 the
                 doctrine
                 of
                 solidness
                 and
                 fluidness
                 ,
                 of
                 opacity
                 and
                 transparency
                 ,
                 in
                 bodies
                 sometimes
                 of
                 the
                 same
                 solidnes
                 ,
                 sometimes
                 of
                 the
                 same
                 fluidnes
                 ,
                 besides
                 a
                 million
                 of
                 other
                 instances
                 ,
                 whose
                 resolution
                 mateth
                 the
                 doctrine
                 of
                 the
                 four
                 Elements
                 ?
                 However
                 mis-understand
                 me
                 not
                 ,
                 as
                 if
                 I
                 denied
                 Elements
                 ,
                 or
                 first
                 Elements
                 :
                 these
                 must
                 either
                 be
                 allowed
                 ,
                 or
                 no
                 mixtion
                 ;
                 of
                 which
                 we
                 have
                 lucide
                 examples
                 in
                 every
                 corner
                 of
                 the
                 world
                 :
                 for
                 it
                 was
                 a
                 Golden-rule
                 in
                 the
                 school
                 ,
                 which
                 now
                 I
                 am
                 opposing
                 ,
                 that
                 the
                 same
                 ,
                 or
                 like
                 ,
                 as
                 so
                 ,
                 working
                 upon
                 the
                 same
                 ,
                 or
                 like
                 ,
                 as
                 so
                 ,
                 cannot
                 make
                 any
                 alteration
                 :
                 that
                 I
                 suppose
                 I
                 have
                 good
                 reason
                 not
                 to
                 be
                 forward
                 to
                 embrace
                 the
                 new
                 received
                 
                 opinion
                 ,
                 so
                 much
                 cherished
                 by
                 some
                 persons
                 of
                 Eminency
                 and
                 parts
                 ,
                 that
                 there
                 is
                 only
                 one
                 Element
                 of
                 all
                 things
                 licked
                 into
                 several
                 external
                 shapes
                 and
                 forms
                 .
                 But
                 this
                 controversy
                 not
                 concerning
                 your
                 judgment
                 ,
                 it
                 were
                 trouble
                 now
                 to
                 pursue
                 an
                 impertinent
                 .
                 The
                 rest
                 I
                 suppose
                 ,
                 any
                 where
                 defensible
                 ,
                 save
                 at
                 Stagira
                 ;
                 where
                 Fire
                 must
                 burn
                 ,
                 Air
                 poyson
                 ,
                 Earth
                 bury
                 ,
                 and
                 water
                 drown
                 ,
                 whatsoever
                 shall
                 be
                 suspected
                 in
                 ,
                 or
                 alleadged
                 against
                 the
                 Traditions
                 of
                 the
                 imperial
                 Philosopher
                 .
                 However
                 my
                 thoughts
                 are
                 still
                 comported
                 to
                 yield
                 up
                 themselves
                 to
                 that
                 doctrine
                 upon
                 the
                 first
                 clear
                 and
                 experimental
                 demonstration
                 of
                 it
                 ;
                 if
                 any
                 such
                 might
                 be
                 hoped
                 .
                 In
                 the
                 interim
                 I
                 want
                 not
                 fair
                 footsteppings
                 in
                 the
                 Antients
                 to
                 induce
                 me
                 to
                 think
                 ,
                 that
                 when
                 they
                 make
                 Bodies
                 to
                 be
                 composed
                 of
                 that
                 which
                 is
                 hot
                 ,
                 and
                 that
                 which
                 is
                 cold
                 ,
                 of
                 that
                 which
                 is
                 dry
                 and
                 of
                 that
                 which
                 is
                 moyst
                 ;
                 they
                 mean
                 nothing
                 less
                 then
                 these
                 four
                 bodies
                 ,
                 or
                 rather
                 vast
                 Amassements
                 ,
                 vulgarly
                 now
                 known
                 by
                 the
                 name
                 of
                 the
                 four
                 Elements
                 :
                 but
                 they
                 rather
                 intend
                 ,
                 the
                 true
                 Elements
                 of
                 nature
                 ,
                 it
                 may
                 be
                 seldom
                 ,
                 or
                 never
                 seen
                 alone
                 ;
                 in
                 which
                 ,
                 such
                 qualities
                 ,
                 or
                 properties
                 are
                 specially
                 eminent
                 :
                 and
                 they
                 are
                 so
                 far
                 from
                 restraining
                 natural
                 compositions
                 to
                 the
                 at
                 present
                 received
                 Quaternary
                 of
                 Elements
                 ,
                 that
                 constantly
                 at
                 the
                 same
                 breath
                 they
                 hint
                 many
                 other
                 :
                 
                 adjoyning
                 in
                 their
                 recitals
                 of
                 natural
                 mixtions
                 ,
                 temperaments
                 ,
                 and
                 compositions
                 ,
                 to
                 that
                 which
                 it
                 hot
                 ,
                 cold
                 ,
                 moyst
                 ,
                 dry
                 ,
                 Elements
                 endowed
                 with
                 several
                 other
                 properties
                 ,
                 as
                 that
                 which
                 is
                 salt
                 ,
                 that
                 which
                 is
                 sowr
                 ,
                 that
                 which
                 is
                 bitter
                 ,
                 that
                 which
                 is
                 sharp
                 ,
                 that
                 which
                 is
                 sweet
                 &c.
                 and
                 the
                 extremities
                 and
                 distempers
                 intolerable
                 unto
                 nature
                 ,
                 whether
                 in
                 general
                 the
                 Animal
                 nature
                 ,
                 or
                 more
                 especially
                 the
                 Humane
                 nature
                 ,
                 are
                 made
                 to
                 be
                 ,
                 when
                 such
                 Elements
                 sever
                 from
                 their
                 mixture
                 ,
                 and
                 go
                 apart
                 by
                 themselves
                 to
                 violating
                 ,
                 and
                 breaking
                 up
                 that
                 sweet
                 proportionate
                 composition
                 and
                 harmony
                 ,
                 which
                 is
                 so
                 friendly
                 and
                 grateful
                 unto
                 nature
                 .
                 As
                 the
                 Pyrotechnists
                 analyse
                 bodies
                 ,
                 upon
                 the
                 consideration
                 of
                 several
                 differences
                 and
                 contrarieties
                 which
                 are
                 in
                 their
                 compositive
                 parts
                 ,
                 as
                 that
                 some
                 will
                 abide
                 the
                 fire
                 ,
                 others
                 not
                 ;
                 some
                 be
                 wrought
                 upon
                 by
                 such
                 dissolvents
                 ,
                 or
                 corroders
                 ,
                 others
                 not
                 :
                 so
                 the
                 Antients
                 to
                 reduce
                 all
                 inanimate
                 things
                 to
                 four
                 heads
                 ,
                 may
                 have
                 insisted
                 on
                 several
                 contrarieties
                 ,
                 paving
                 the
                 way
                 to
                 such
                 a
                 doctrine
                 :
                 Viz.
                 that
                 such
                 bodies
                 are
                 either
                 firmly
                 solid
                 ,
                 or
                 else
                 fluid
                 :
                 and
                 those
                 that
                 are
                 fluid
                 ,
                 are
                 such
                 ,
                 either
                 in
                 a
                 grosse
                 corpulent
                 matter
                 ,
                 or
                 in
                 fine
                 exhalations
                 :
                 and
                 those
                 exhalations
                 ,
                 either
                 apt
                 to
                 conceive
                 a
                 flame
                 ,
                 and
                 burn
                 ,
                 or
                 else
                 of
                 a
                 nature
                 ,
                 thereunto
                 indisposed
                 .
                 And
                 according
                 to
                 such
                 a
                 distribution
                 ,
                 many
                 various
                 bodies
                 
                 fall
                 aptly
                 enough
                 under
                 the
                 same
                 head
                 of
                 Amassment
                 :
                 and
                 all
                 inanimals
                 in
                 the
                 world
                 may
                 be
                 easily
                 reducible
                 to
                 some
                 one
                 of
                 these
                 four
                 heads
                 .
                 But
                 then
                 these
                 four
                 heads
                 are
                 not
                 particular
                 first
                 Elements
                 ,
                 uncompounded
                 ,
                 and
                 so
                 constitutive
                 of
                 things
                 in
                 mixtion
                 ;
                 but
                 general
                 notions
                 ,
                 unto
                 which
                 the
                 inanimals
                 of
                 this
                 lower
                 part
                 of
                 the
                 universe
                 ,
                 are
                 aptly
                 reducible
                 .
                 Favour
                 these
                 Conjectures
                 ,
                 which
                 obtrude
                 nothing
                 ,
                 only
                 labour
                 what
                 they
                 can
                 to
                 justify
                 the
                 methods
                 of
                 the
                 Antients
                 .
                 Send
                 back
                 my
                 instruments
                 and
                 the
                 last
                 transmitted
                 Problemes
                 with
                 your
                 first
                 convenience
                 .
              
            
             
               
               
                 The
                 Fourth
                 Answer
                 .
              
               
                 
                   Concerning
                   the
                   seams
                   of
                   the
                   Skull
                   ,
                   and
                   the
                   number
                   of
                   the
                   seams
                   by
                   which
                   the
                   Skullbones
                   are
                   joined
                   one
                   unto
                   another
                   ,
                   over
                   and
                   besides
                   the
                   three
                   principal
                   proper
                   and
                   true
                   seams
                   .
                   That
                   without
                   contradiction
                   ,
                   variety
                   may
                   be
                   admitted
                   both
                   in
                   the
                   number
                   and
                   order
                   of
                   such
                   seams
                   .
                   That
                   neither
                   abstruct
                   nor
                   concrete
                   Mathematicks
                   are
                   wholly
                   exempt
                   from
                   a
                   liablenes
                   to
                   such
                   varieties
                   .
                   That
                   the
                   pre-eminence
                   of
                   right-side
                   parts
                   is
                   no
                   way
                   natural
                   in
                   such
                   as
                   twin
                   but
                   only
                   at
                   pleasure
                   .
                
              
               
                 
                   SIR
                   .
                
              
               
                 THe
                 learned
                 men
                 were
                 riper
                 for
                 controversy
                 ,
                 than
                 became
                 the
                 gravity
                 of
                 their
                 profession
                 ,
                 or
                 else
                 upon
                 the
                 Point
                 ,
                 which
                 you
                 have
                 transmitted
                 to
                 me
                 ,
                 held
                 by
                 each
                 with
                 so
                 little
                 real
                 ,
                 though
                 a
                 manifest
                 nominal
                 difference
                 ,
                 it
                 had
                 not
                 been
                 easy
                 to
                 have
                 occasioned
                 the
                 hot
                 language
                 contained
                 in
                 your
                 report
                 .
                 Though
                 the
                 one
                 make
                 but
                 five
                 pairs
                 of
                 seams
                 ,
                 by
                 which
                 the
                 Temple
                 bones
                 are
                 joyned
                 to
                 the
                 wedge
                 bone
                 ,
                 and
                 also
                 to
                 the
                 two
                 Crown
                 bones
                 ,
                 and
                 those
                 (
                 besides
                 the
                 three
                 principal
                 seams
                 )
                 to
                 the
                 forehead
                 bone
                 ,
                 or
                 bones
                 ,
                 and
                 to
                 the
                 Nowl
                 bone
                 ;
                 and
                 the
                 other
                 ,
                 contrary
                 to
                 the
                 general
                 doctrine
                 of
                 the
                 Anatomick
                 School
                 ,
                 asserts
                 that
                 there
                 are
                 six
                 ;
                 and
                 both
                 make
                 their
                 appeal
                 to
                 Autopsy
                 and
                 the
                 Skeleton
                 ;
                 
                 I
                 see
                 not
                 wherein
                 either
                 is
                 disabled
                 to
                 maintain
                 his
                 Assertion
                 ,
                 as
                 both
                 agreable
                 to
                 truth
                 and
                 justifiable
                 upon
                 the
                 view
                 :
                 only
                 not
                 without
                 some
                 rational
                 ground
                 for
                 the
                 difference
                 ,
                 Viz.
                 either
                 hath
                 ,
                 as
                 I
                 suppose
                 ,
                 his
                 different
                 method
                 ,
                 upon
                 which
                 he
                 insists
                 in
                 making
                 up
                 the
                 account
                 of
                 the
                 seams
                 .
                 For
                 the
                 number
                 will
                 differ
                 according
                 to
                 the
                 ground
                 by
                 which
                 you
                 will
                 distinguish
                 and
                 make
                 out
                 the
                 unity
                 of
                 each
                 seam
                 :
                 as
                 also
                 according
                 as
                 the
                 seams
                 are
                 accounted
                 singly
                 ,
                 or
                 by
                 pairs
                 ;
                 from
                 which
                 last
                 manner
                 of
                 accounting
                 may
                 arise
                 also
                 a
                 third
                 different
                 number
                 of
                 them
                 ,
                 though
                 still
                 without
                 any
                 change
                 in
                 the
                 thing
                 and
                 object
                 .
                 Not
                 being
                 privy
                 to
                 the
                 terms
                 upon
                 which
                 this
                 controversy
                 was
                 between
                 them
                 managed
                 ;
                 this
                 Reconciliation
                 ,
                 though
                 unquestionably
                 true
                 ,
                 and
                 pertinent
                 to
                 the
                 Question
                 ,
                 as
                 by
                 you
                 in
                 general
                 proposed
                 ,
                 may
                 however
                 possibly
                 be
                 inpertinent
                 to
                 it
                 ,
                 as
                 by
                 them
                 stated
                 and
                 defended
                 :
                 though
                 I
                 understand
                 not
                 upon
                 what
                 other
                 terms
                 the
                 Assertions
                 of
                 both
                 can
                 be
                 capable
                 of
                 justification
                 :
                 and
                 I
                 am
                 not
                 willing
                 to
                 think
                 otherwise
                 ,
                 between
                 persons
                 ,
                 so
                 eminently
                 learned
                 .
                 It
                 cannot
                 be
                 denied
                 but
                 in
                 these
                 nicer
                 things
                 observed
                 in
                 the
                 head
                 ,
                 and
                 distinguished
                 usually
                 by
                 numbers
                 ,
                 as
                 the
                 seams
                 ,
                 and
                 holes
                 of
                 the
                 scull
                 ,
                 there
                 has
                 not
                 wanted
                 some
                 seeming
                 diversity
                 and
                 obscurity
                 :
                 while
                 it
                 has
                 been
                 thought
                 ,
                 that
                 merely
                 at
                 the
                 pleasure
                 of
                 Authors
                 
                 and
                 of
                 those
                 that
                 sit
                 in
                 the
                 Anatomical
                 chair
                 ,
                 sometimes
                 one
                 thing
                 ,
                 at
                 other
                 times
                 another
                 ,
                 and
                 not
                 constantly
                 the
                 same
                 thing
                 ,
                 is
                 made
                 to
                 give
                 the
                 order
                 ,
                 unity
                 ,
                 integrity
                 and
                 distinction
                 of
                 seams
                 .
                 So
                 the
                 Nowlebones
                 connexion
                 with
                 the
                 Crown
                 bone
                 ,
                 and
                 Temple
                 bone
                 of
                 the
                 same
                 side
                 ,
                 is
                 generally
                 construed
                 to
                 be
                 two
                 seams
                 ,
                 and
                 of
                 two
                 diverse
                 kinds
                 ;
                 the
                 one
                 true
                 ,
                 the
                 other
                 mendous
                 ,
                 or
                 defective
                 :
                 though
                 so
                 fair
                 be
                 the
                 continuance
                 of
                 the
                 one
                 seam
                 unto
                 the
                 other
                 ,
                 that
                 the
                 one
                 is
                 usually
                 known
                 by
                 the
                 name
                 of
                 an
                 Additament
                 unto
                 the
                 other
                 :
                 and
                 yet
                 in
                 other
                 seams
                 diversity
                 of
                 bones
                 doth
                 not
                 ,
                 with
                 all
                 ,
                 make
                 multiplicity
                 of
                 seams
                 ;
                 as
                 in
                 the
                 Lepidoidal
                 ,
                 or
                 scaleseam
                 where
                 the
                 Templebone
                 is
                 joyned
                 to
                 the
                 Crown-bone
                 and
                 to
                 the
                 wedge-bone
                 by
                 a
                 future
                 ,
                 generally
                 accounted
                 but
                 for
                 one
                 :
                 so
                 the
                 connexion
                 of
                 the
                 inner
                 process
                 of
                 the
                 forehead
                 bone
                 ,
                 to
                 four
                 several
                 bones
                 of
                 the
                 upper
                 chapp
                 ,
                 is
                 by
                 most
                 accounted
                 but
                 for
                 one
                 common
                 seam
                 .
                 The
                 like
                 ,
                 or
                 worse
                 perplexities
                 occurre
                 sometimes
                 in
                 the
                 account
                 of
                 the
                 holes
                 in
                 the
                 head
                 ;
                 while
                 some
                 of
                 the
                 holes
                 ,
                 which
                 are
                 common
                 to
                 several
                 bones
                 are
                 accounted
                 in
                 each
                 bone
                 :
                 others
                 ,
                 in
                 like
                 manner
                 common
                 only
                 in
                 one
                 of
                 the
                 bones
                 ,
                 whence
                 many
                 times
                 the
                 number
                 of
                 the
                 holes
                 pertinent
                 to
                 each
                 bone
                 is
                 left
                 confused
                 ,
                 and
                 imperfect
                 .
                 For
                 the
                 reverence
                 I
                 bear
                 unto
                 the
                 great
                 Artists
                 ,
                 in
                 whom
                 these
                 things
                 
                 may
                 be
                 ordinarily
                 observed
                 ;
                 I
                 am
                 not
                 willing
                 to
                 impute
                 them
                 any
                 otherwise
                 then
                 to
                 their
                 great
                 intentnesse
                 upon
                 matter
                 and
                 things
                 ;
                 by
                 which
                 they
                 seem
                 overborn
                 into
                 a
                 certain
                 negligence
                 of
                 names
                 and
                 appellations
                 .
                 And
                 those
                 whose
                 study
                 is
                 rather
                 to
                 be
                 Anatomists
                 ,
                 then
                 so
                 esteemed
                 ,
                 and
                 draw
                 their
                 Institutes
                 and
                 observations
                 from
                 the
                 Body
                 ,
                 more
                 then
                 out
                 of
                 Books
                 ,
                 will
                 easily
                 consent
                 with
                 me
                 that
                 in
                 this
                 whole
                 Doctrine
                 ,
                 of
                 the
                 seams
                 ,
                 and
                 holes
                 of
                 the
                 head
                 ,
                 names
                 and
                 numbers
                 are
                 of
                 less
                 moment
                 ,
                 so
                 there
                 be
                 a
                 clear
                 understanding
                 of
                 the
                 things
                 themselves
                 .
                 So
                 none
                 ,
                 who
                 has
                 truly
                 made
                 use
                 of
                 his
                 own
                 Eyes
                 can
                 deny
                 some
                 parts
                 ,
                 or
                 sides
                 of
                 the
                 true
                 seams
                 of
                 the
                 skull
                 to
                 be
                 mendous
                 ,
                 and
                 some
                 parts
                 ,
                 or
                 sides
                 of
                 some
                 of
                 the
                 mendous
                 to
                 be
                 in
                 a
                 sort
                 true
                 :
                 that
                 in
                 several
                 of
                 them
                 the
                 distinction
                 of
                 true
                 from
                 mendous
                 ,
                 and
                 sutures
                 from
                 harmonyes
                 ,
                 seems
                 to
                 be
                 oft
                 more
                 matter
                 of
                 name
                 imposed
                 at
                 pleasure
                 ,
                 without
                 clear
                 and
                 sufficient
                 ground
                 for
                 real
                 and
                 peremptory
                 distinction
                 .
                 Things
                 thus
                 standing
                 ,
                 in
                 the
                 contest
                 you
                 mention
                 ;
                 how
                 easy
                 is
                 it
                 to
                 make
                 ,
                 on
                 the
                 same
                 side
                 ,
                 the
                 seams
                 ,
                 by
                 which
                 the
                 Temple
                 bone
                 and
                 wedge
                 bone
                 are
                 joyned
                 one
                 to
                 another
                 ,
                 and
                 (
                 as
                 before
                 )
                 to
                 the
                 Nowl
                 bone
                 ,
                 Crown
                 bone
                 and
                 Forehead
                 bone
                 ,
                 to
                 be
                 at
                 pleasure
                 ,
                 either
                 five
                 in
                 number
                 ,
                 or
                 if
                 you
                 please
                 six
                 ;
                 For
                 the
                 one
                 ,
                 or
                 the
                 other
                 number
                 follows
                 ,
                 according
                 as
                 you
                 shall
                 take
                 
                 the
                 connexion
                 of
                 the
                 Temple-bone
                 with
                 a
                 Crown-bone
                 and
                 the
                 wedge-bone
                 ,
                 to
                 be
                 one
                 seams
                 as
                 it
                 is
                 most
                 usually
                 accounted
                 ;
                 or
                 else
                 to
                 be
                 two
                 ;
                 for
                 which
                 ,
                 being
                 the
                 connexions
                 of
                 two
                 and
                 two
                 several
                 bones
                 ,
                 there
                 wants
                 not
                 ,
                 as
                 appears
                 ,
                 some
                 fair
                 and
                 probable
                 ground
                 .
                 And
                 thus
                 though
                 I
                 would
                 not
                 charge
                 with
                 falsity
                 ,
                 that
                 whose
                 truth
                 is
                 from
                 real
                 grounds
                 so
                 plainly
                 defensible
                 ,
                 howbeit
                 somewhat
                 varying
                 from
                 the
                 ordinary
                 methods
                 ,
                 and
                 systems
                 of
                 Authors
                 ;
                 yet
                 I
                 judge
                 it
                 not
                 meet
                 we
                 be
                 too
                 forward
                 ,
                 by
                 reason
                 of
                 our
                 more
                 clear
                 insight
                 into
                 things
                 ,
                 in
                 altering
                 ,
                 slighting
                 and
                 rejecting
                 the
                 Terms
                 ,
                 names
                 ,
                 distinctions
                 and
                 methods
                 ,
                 which
                 have
                 ,
                 time
                 out
                 of
                 mind
                 ,
                 been
                 in
                 continual
                 use
                 with
                 learned
                 men
                 .
                 For
                 names
                 ,
                 that
                 were
                 first
                 coyned
                 for
                 distinction
                 and
                 instruction
                 ,
                 if
                 they
                 should
                 be
                 dayly
                 and
                 hourly
                 changed
                 ,
                 would
                 be
                 sure
                 to
                 introduce
                 confusion
                 .
                 And
                 though
                 in
                 Names
                 and
                 numbers
                 the
                 Antients
                 have
                 used
                 somewhat
                 of
                 authority
                 ,
                 they
                 have
                 notwithstanding
                 ,
                 in
                 the
                 same
                 subject
                 ,
                 fairly
                 exercised
                 their
                 reason
                 .
                 So
                 ,
                 with
                 a
                 very
                 little
                 allowance
                 ,
                 this
                 whole
                 doctrine
                 of
                 the
                 sutures
                 ,
                 has
                 been
                 cast
                 by
                 them
                 into
                 a
                 Mould
                 methodical
                 enough
                 :
                 the
                 sum
                 whereof
                 in
                 few
                 may
                 be
                 this
                 .
                 The
                 sutures
                 of
                 the
                 head
                 either
                 concern
                 not
                 ,
                 or
                 else
                 concern
                 the
                 bones
                 of
                 the
                 upper
                 chap
                 :
                 those
                 which
                 concern
                 not
                 the
                 Bones
                 of
                 the
                 upper
                 chap
                 ,
                 are
                 
                 either
                 largely
                 toothed
                 sutures
                 ,
                 Viz.
                 the
                 Crown
                 suture
                 ,
                 the
                 Dart
                 suture
                 ,
                 and
                 the
                 Lambda
                 suture
                 ;
                 which
                 may
                 also
                 receive
                 another
                 ground
                 of
                 distinction
                 from
                 their
                 concerning
                 the
                 two
                 Crown-bones
                 connexion
                 one
                 with
                 the
                 other
                 ,
                 or
                 with
                 the
                 Forehead
                 bones
                 and
                 the
                 Noul-bone
                 :
                 or
                 otherwise
                 ;
                 other
                 sutures
                 ,
                 not
                 concerning
                 the
                 bones
                 of
                 the
                 upper
                 chap
                 ,
                 are
                 not
                 so
                 fairly
                 not
                 deeply
                 ,
                 sometimes
                 scarce
                 at
                 all
                 ,
                 at
                 least
                 not
                 so
                 manifestly
                 dented
                 ,
                 or
                 not
                 in
                 the
                 same
                 manner
                 ;
                 as
                 the
                 Lepidoidal
                 connexting
                 the
                 Temple-bone
                 to
                 the
                 Crown-bone
                 and
                 to
                 the
                 wedge-bone
                 ;
                 the
                 connexion
                 between
                 the
                 Crown-bone
                 and
                 wedge-bone
                 ;
                 the
                 connexion
                 between
                 the
                 Noul-bone
                 and
                 Temple-bone
                 ,
                 commonly
                 called
                 the
                 Additament
                 of
                 the
                 Lambdoidal
                 ;
                 the
                 connexion
                 between
                 the
                 Nouls
                 additament
                 and
                 the
                 wedge-bone
                 ;
                 and
                 the
                 connexion
                 between
                 the
                 Fore-head-bone
                 and
                 the
                 wedge-bone
                 :
                 which
                 may
                 also
                 receive
                 another
                 ground
                 of
                 distinction
                 from
                 their
                 concerning
                 the
                 connexion
                 of
                 the
                 Temple-bone
                 and
                 wedge-bone
                 one
                 with
                 the
                 other
                 and
                 with
                 the
                 Noul-bone
                 ,
                 Crown-bones
                 ,
                 and
                 Forehead-bone
                 .
                 So
                 the
                 sutures
                 ,
                 which
                 concern
                 the
                 bones
                 of
                 the
                 upper
                 chap
                 ,
                 either
                 concern
                 some
                 one
                 ,
                 or
                 more
                 of
                 them
                 in
                 common
                 with
                 some
                 of
                 the
                 aforementioned
                 bones
                 ▪
                 or
                 else
                 they
                 concern
                 ,
                 the
                 bones
                 of
                 the
                 upper
                 chap
                 alone
                 ,
                 as
                 connexed
                 mutually
                 amongst
                 themselves
                 .
                 The
                 sutures
                 which
                 concern
                 
                 some
                 one
                 ,
                 or
                 more
                 of
                 the
                 bones
                 of
                 the
                 upper
                 chap
                 ,
                 in
                 common
                 with
                 some
                 of
                 the
                 aforementioned
                 ,
                 make
                 connexion
                 only
                 with
                 some
                 of
                 these
                 ,
                 Viz.
                 the
                 Temple-bone
                 ,
                 wedge-bone
                 ,
                 or
                 Fore-head
                 bone
                 ;
                 for
                 the
                 Crown-bones
                 and
                 Noul-bone
                 are
                 no
                 where
                 connexed
                 to
                 the
                 bones
                 of
                 the
                 upper
                 chap
                 :
                 And
                 they
                 are
                 particularly
                 these
                 ;
                 Viz.
                 the
                 connexion
                 of
                 the
                 outer
                 processe
                 of
                 the
                 Fore-head
                 bone
                 ,
                 with
                 the
                 inner
                 processe
                 of
                 the
                 first
                 bone
                 of
                 the
                 upper
                 chap
                 :
                 the
                 connexion
                 of
                 the
                 first
                 bone
                 of
                 the
                 upper
                 chap
                 with
                 the
                 wedge
                 bone
                 :
                 the
                 connexion
                 of
                 the
                 inner
                 processe
                 of
                 the
                 Forehead-bone
                 with
                 the
                 fifth
                 ,
                 fourth
                 ,
                 second
                 and
                 third
                 bones
                 of
                 the
                 upper
                 chap
                 ,
                 being
                 in
                 the
                 order
                 recited
                 ,
                 connexed
                 from
                 within
                 outwards
                 :
                 the
                 connexion
                 of
                 the
                 outer
                 ,
                 or
                 yoke
                 processe
                 of
                 the
                 first
                 bone
                 of
                 the
                 upper
                 chap
                 ,
                 with
                 the
                 yoke
                 processe
                 of
                 the
                 Temple
                 bone
                 ,
                 in
                 the
                 middle
                 of
                 the
                 yoke
                 bone
                 :
                 which
                 yoke
                 bone
                 is
                 not
                 a
                 distinct
                 and
                 several
                 bone
                 ,
                 but
                 made
                 up
                 of
                 two
                 processes
                 of
                 two
                 several
                 bones
                 :
                 and
                 the
                 last
                 is
                 the
                 connexion
                 between
                 part
                 of
                 the
                 upper
                 ,
                 part
                 of
                 the
                 dissepiment
                 and
                 the
                 wedge-bone
                 ,
                 Lastly
                 the
                 seams
                 which
                 concern
                 the
                 bones
                 of
                 the
                 upper
                 chap
                 only
                 among
                 themselves
                 are
                 the
                 connexion
                 of
                 the
                 first
                 and
                 fourth
                 bone
                 ,
                 the
                 connexion
                 of
                 the
                 second
                 bone
                 with
                 the
                 third
                 and
                 fourth
                 ,
                 the
                 connexion
                 of
                 the
                 third
                 and
                 fourth
                 bones
                 ,
                 the
                 connexion
                 of
                 the
                 fourth
                 and
                 fifth
                 
                 bones
                 ,
                 the
                 connexion
                 between
                 the
                 two
                 fifth
                 bones
                 ,
                 the
                 connexion
                 of
                 the
                 fourth
                 and
                 sixth
                 bones
                 ,
                 the
                 connexion
                 between
                 the
                 two
                 sixth
                 bones
                 ,
                 the
                 connexion
                 of
                 the
                 other
                 part
                 of
                 the
                 upper
                 part
                 of
                 the
                 dissepiment
                 with
                 the
                 fifth
                 bones
                 ,
                 the
                 connexion
                 of
                 the
                 lower
                 part
                 ,
                 of
                 the
                 dissepiment
                 in
                 the
                 Plough-share
                 bone
                 with
                 the
                 sixth
                 ,
                 
                   i.
                   e.
                
                 the
                 palat
                 bones
                 and
                 some
                 small
                 part
                 of
                 the
                 fourth
                 bones
                 .
                 In
                 all
                 which
                 ,
                 if
                 somewhat
                 at
                 pleasure
                 ,
                 they
                 have
                 used
                 authority
                 in
                 the
                 numeral
                 order
                 of
                 the
                 bones
                 ,
                 and
                 in
                 the
                 account
                 and
                 order
                 of
                 the
                 seams
                 ,
                 it
                 is
                 no
                 more
                 than
                 is
                 freely
                 allowed
                 to
                 each
                 Artist
                 in
                 his
                 Art
                 ,
                 so
                 long
                 as
                 the
                 Authority
                 they
                 use
                 ,
                 neither
                 bears
                 ,
                 nor
                 creats
                 any
                 repugnancy
                 to
                 the
                 subject
                 matter
                 .
                 And
                 so
                 even
                 in
                 abstract
                 Mathematicks
                 ,
                 somethings
                 are
                 determined
                 ,
                 not
                 by
                 necessity
                 ,
                 because
                 they
                 can
                 be
                 no
                 otherwise
                 ,
                 but
                 by
                 long
                 usage
                 and
                 authority
                 ,
                 because
                 they
                 may
                 conveniently
                 be
                 as
                 is
                 determined
                 .
                 So
                 Arithmetical
                 numerations
                 ,
                 which
                 are
                 every
                 where
                 received
                 and
                 taught
                 in
                 Decuple
                 proportion
                 ;
                 with
                 as
                 much
                 facility
                 and
                 Mathematical
                 accurateness
                 might
                 be
                 instituted
                 in
                 undecuple
                 proportion
                 ;
                 and
                 then
                 the
                 usual
                 Probes
                 ,
                 should
                 not
                 be
                 by
                 casting
                 away
                 of
                 nines
                 but
                 of
                 tens
                 ;
                 or
                 in
                 sexage
                 cuple
                 proportion
                 ;
                 and
                 then
                 the
                 Probes
                 should
                 be
                 by
                 casting
                 away
                 of
                 fifty
                 nines
                 :
                 or
                 non
                 cuple
                 proportion
                 ,
                 and
                 then
                 the
                 Probes
                 should
                 be
                 by
                 casting
                 away
                 of
                 eights
                 .
                 And
                 the
                 
                 like
                 instances
                 might
                 be
                 given
                 in
                 Geometry
                 ;
                 and
                 in
                 several
                 ,
                 if
                 not
                 most
                 parts
                 of
                 Concrete
                 Mathematicks
                 .
                 So
                 in
                 ordering
                 the
                 Account
                 of
                 the
                 Brain
                 ventricles
                 ,
                 of
                 the
                 sides
                 of
                 the
                 same
                 bones
                 ,
                 and
                 sometimes
                 of
                 the
                 branchings
                 of
                 sinews
                 ,
                 veins
                 ,
                 Arteries
                 ,
                 other
                 vessels
                 ,
                 and
                 the
                 like
                 ;
                 custome
                 ,
                 and
                 authority
                 ,
                 commonly
                 gives
                 praecedency
                 to
                 the
                 parts
                 ,
                 cavities
                 ,
                 or
                 branchings
                 ,
                 posited
                 on
                 the
                 right
                 side
                 ;
                 though
                 without
                 offence
                 unto
                 Truth
                 ,
                 all
                 Accounts
                 might
                 as
                 exquisitely
                 be
                 compleated
                 in
                 a
                 quite
                 contrary
                 order
                 .
                 But
                 though
                 these
                 things
                 might
                 be
                 ,
                 and
                 as
                 well
                 ,
                 as
                 what
                 is
                 ;
                 yet
                 what
                 is
                 ,
                 being
                 as
                 accurate
                 ,
                 as
                 what
                 otherwise
                 might
                 be
                 ,
                 it
                 is
                 sufficient
                 to
                 know
                 that
                 these
                 things
                 might
                 be
                 otherwise
                 methodized
                 ,
                 without
                 renouncing
                 the
                 Antients
                 method
                 .
                 I
                 have
                 ,
                 you
                 see
                 ,
                 too
                 high
                 an
                 Opinion
                 of
                 the
                 persons
                 concerned
                 ,
                 to
                 imagine
                 their
                 difference
                 could
                 arise
                 from
                 ,
                 some
                 peculiarities
                 in
                 unusual
                 skulls
                 ;
                 in
                 which
                 it
                 is
                 not
                 uncommon
                 to
                 find
                 great
                 difference
                 in
                 the
                 numbers
                 of
                 bones
                 ,
                 seams
                 and
                 holes
                 ,
                 not
                 only
                 under
                 diversity
                 of
                 Age
                 and
                 sex
                 ,
                 which
                 ,
                 is
                 so
                 common
                 ,
                 it
                 is
                 not
                 otherwise
                 to
                 be
                 expected
                 ;
                 but
                 as
                 well
                 when
                 no
                 such
                 causes
                 can
                 be
                 alleadged
                 ,
                 for
                 the
                 diversity
                 :
                 nay
                 ,
                 many
                 times
                 in
                 the
                 main
                 and
                 principal
                 ,
                 commonly
                 called
                 True
                 seams
                 we
                 do
                 not
                 ever
                 find
                 a
                 constant
                 regularity
                 and
                 uniformity
                 .
                 But
                 of
                 those
                 things
                 ,
                 which
                 
                 seldome
                 are
                 ,
                 we
                 are
                 not
                 to
                 maintain
                 Controversy
                 ,
                 as
                 if
                 they
                 were
                 the
                 way
                 of
                 nature
                 .
                 This
                 ,
                 Sir
                 ,
                 I
                 have
                 written
                 not
                 to
                 Inform
                 ,
                 but
                 confirm
                 your
                 judgement
                 ;
                 which
                 I
                 know
                 so
                 well
                 versed
                 in
                 the
                 Syntaxe
                 of
                 this
                 our
                 humane
                 body
                 ,
                 that
                 it
                 cannot
                 dissent
                 from
                 what
                 is
                 .
              
            
             
               
               
                 The
                 Fifth
                 Answer
                 .
              
               
                 
                   That
                   immense
                   volatility
                   may
                   consist
                   with
                   immense
                   ponderousness
                   .
                   That
                   tincturs
                   may
                   be
                   altered
                   by
                   maturation
                   without
                   any
                   addition
                   ,
                   Whether
                   the
                   appearing
                   and
                   motion
                   of
                   comets
                   may
                   be
                   before
                   their
                   appearance
                   predicted
                   .
                   That
                   no
                   such
                   particular
                   predictions
                   can
                   be
                   made
                   concerning
                   the
                   meteors
                   which
                   are
                   in
                   the
                   Atmosphear
                   of
                   the
                   earth
                   ,
                   nor
                   of
                   the
                   first
                   appearances
                   of
                   such
                   Comets
                   as
                   are
                   supposed
                   to
                   have
                   their
                   Original
                   from
                   new
                   amassements
                   of
                   Cometical
                   matter
                   in
                   other
                   Atmospheares
                   ;
                   though
                   after
                   their
                   first
                   appearance
                   ,
                   upon
                   some
                   observations
                   accurately
                   made
                   ,
                   somewhat
                   ,
                   though
                   nothing
                   so
                   peremptorily
                   as
                   in
                   other
                   cases
                   ,
                   may
                   be
                   predicted
                   ,
                   relating
                   to
                   their
                   future
                   motion
                   .
                   That
                   it
                   is
                   not
                   impossible
                   ,
                   but
                   concerning
                   Comets
                   which
                   are
                   permanent
                   bodies
                   ,
                   and
                   not
                   new
                   amassements
                   ,
                   predictions
                   may
                   be
                   made
                   long
                   before
                   ,
                   of
                   their
                   future
                   appearances
                   and
                   motion
                   .
                
              
               
                 
                   SIR
                   .
                
              
               
                 I
                 Take
                 your
                 satisfaction
                 ,
                 upon
                 my
                 last
                 proposed
                 ,
                 clear
                 ,
                 and
                 doubtless
                 Experiments
                 ,
                 now
                 since
                 by
                 you
                 proved
                 and
                 approved
                 ,
                 as
                 a
                 fair
                 acknowledgment
                 of
                 that
                 Truth
                 ,
                 which
                 however
                 to
                 me
                 upon
                 its
                 former
                 evidences
                 needed
                 no
                 farther
                 confirmation
                 :
                 Viz.
                 that
                 so
                 different
                 are
                 the
                 genuine
                 notions
                 and
                 qualities
                 of
                 fixednes
                 and
                 gravity
                 ,
                 that
                 immense
                 volatility
                 may
                 and
                 is
                 ordinarily
                 consistent
                 with
                 an
                 
                 immense
                 specifick
                 ponderousness
                 ;
                 arising
                 not
                 from
                 the
                 moles
                 and
                 quantity
                 of
                 the
                 bodies
                 under
                 consideration
                 ,
                 but
                 rather
                 from
                 their
                 natures
                 and
                 kinds
                 .
                 That
                 what
                 in
                 your
                 first
                 velitation
                 you
                 assumed
                 as
                 absurdly
                 grosse
                 ,
                 inconsistent
                 ,
                 and
                 impossible
                 ;
                 is
                 now
                 ,
                 upon
                 your
                 own
                 acknowledgments
                 ,
                 most
                 easily
                 and
                 obviously
                 demonstrable
                 by
                 experience
                 .
                 No
                 less
                 certainly
                 is
                 to
                 be
                 acknowledged
                 ,
                 in
                 what
                 you
                 propose
                 for
                 the
                 tingeing
                 of
                 metalline
                 bodies
                 ,
                 only
                 by
                 ordering
                 and
                 attending
                 them
                 in
                 the
                 management
                 of
                 their
                 maturations
                 ,
                 without
                 the
                 addition
                 either
                 of
                 body
                 ,
                 or
                 spirit
                 ,
                 as
                 being
                 all
                 the
                 time
                 of
                 this
                 operation
                 under
                 the
                 sure
                 seal
                 .
                 Could
                 I
                 perswade
                 my self
                 it
                 should
                 not
                 be
                 resented
                 as
                 a
                 grand
                 unkindnes
                 ,
                 to
                 be
                 silent
                 in
                 what
                 you
                 call
                 for
                 my
                 thoughts
                 in
                 ,
                 in
                 the
                 close
                 of
                 yours
                 ;
                 Viz.
                 Whether
                 prognosticks
                 at
                 certainty
                 ,
                 such
                 as
                 are
                 of
                 Eclipses
                 ,
                 of
                 Coitions
                 ,
                 oppositions
                 ,
                 motions
                 of
                 other
                 Starrs
                 ,
                 whether
                 ,
                 I
                 say
                 ,
                 such
                 prognosticks
                 and
                 of
                 like
                 certainty
                 ,
                 may
                 not
                 also
                 be
                 made
                 of
                 Comets
                 ,
                 of
                 their
                 Appearances
                 ,
                 common
                 impediments
                 removed
                 ,
                 and
                 of
                 their
                 motions
                 ;
                 I
                 should
                 ,
                 if
                 it
                 might
                 be
                 ,
                 herein
                 ,
                 lay
                 Harpocrates
                 finger
                 upon
                 my
                 lip
                 ,
                 and
                 seal
                 them
                 up
                 into
                 a
                 pertinacions
                 silence
                 .
                 Not
                 but
                 I
                 am
                 desirous
                 to
                 know
                 ,
                 and
                 willing
                 to
                 search
                 after
                 truth
                 :
                 only
                 I
                 fear
                 me
                 ,
                 these
                 are
                 secrets
                 of
                 nature
                 ,
                 by
                 their
                 peculiar
                 mysteriousnes
                 ,
                 sinking
                 themselves
                 so
                 low
                 
                 into
                 the
                 pit
                 of
                 obscurity
                 ,
                 that
                 the
                 stock
                 of
                 observations
                 and
                 disquisitions
                 about
                 them
                 ,
                 which
                 the
                 world
                 as
                 yet
                 hath
                 ,
                 is
                 not
                 able
                 to
                 raise
                 them
                 so
                 high
                 ,
                 and
                 place
                 them
                 so
                 near
                 day
                 ,
                 as
                 to
                 be
                 within
                 humane
                 reach
                 and
                 discovery
                 .
                 What
                 I
                 now
                 offer
                 is
                 a
                 Caesarean
                 birth
                 of
                 the
                 mind
                 ,
                 not
                 brought
                 forth
                 by
                 me
                 ,
                 but
                 cut
                 out
                 of
                 the
                 womb
                 by
                 your
                 importunity
                 :
                 if
                 it
                 want
                 shape
                 ,
                 licking
                 and
                 lineaments
                 ,
                 accept
                 it
                 as
                 an
                 unripe
                 Abort
                 ,
                 and
                 either
                 hatch
                 it
                 to
                 perfection
                 in
                 your
                 Thigh
                 ,
                 or
                 give
                 it
                 a
                 little
                 dust
                 to
                 cover
                 it
                 .
                 I
                 take
                 by
                 way
                 of
                 praesumption
                 ,
                 that
                 in
                 this
                 Quaery
                 and
                 Question
                 you
                 mean
                 not
                 by
                 Comets
                 ,
                 any
                 of
                 those
                 more
                 usual
                 ,
                 less
                 permanent
                 is
                 regularly
                 moved
                 Meteors
                 ,
                 bred
                 ▪
                 gathered
                 ,
                 fired
                 ,
                 and
                 burning
                 in
                 the
                 Atmosphaere
                 of
                 our
                 Earth
                 ;
                 after
                 their
                 Appearance
                 ,
                 there
                 cannot
                 be
                 much
                 certainty
                 of
                 their
                 Motion
                 ,
                 though
                 sometimes
                 there
                 may
                 be
                 conjectures
                 probable
                 enough
                 ,
                 and
                 in
                 the
                 Event
                 by
                 observation
                 justifyed
                 ,
                 when
                 the
                 fuel
                 ,
                 or
                 fovent
                 matter
                 of
                 such
                 Meteors
                 ,
                 is
                 manifestly
                 ,
                 upon
                 what
                 account
                 soever
                 ,
                 known
                 to
                 be
                 only
                 ,
                 or
                 most
                 copiously
                 situate
                 and
                 disposed
                 some
                 one
                 particular
                 way
                 :
                 but
                 of
                 the
                 generation
                 and
                 first
                 appearance
                 of
                 such
                 Meteors
                 ,
                 particularly
                 the
                 moment
                 when
                 ,
                 and
                 the
                 point
                 ,
                 or
                 exact
                 place
                 where
                 ,
                 they
                 shall
                 appear
                 ,
                 much
                 less
                 of
                 certainty
                 ,
                 in
                 such
                 pronosticks
                 is
                 to
                 be
                 expected
                 ;
                 there
                 being
                 so
                 much
                 variety
                 ,
                 contingency
                 ,
                 and
                 
                 uncertainty
                 ,
                 in
                 the
                 causes
                 and
                 meeting
                 together
                 of
                 those
                 causes
                 ,
                 which
                 contribute
                 to
                 their
                 production
                 .
                 And
                 though
                 there
                 are
                 ,
                 and
                 have
                 been
                 many
                 praedictions
                 in
                 general
                 ,
                 and
                 Rules
                 of
                 praedicting
                 Astrologically
                 ,
                 or
                 Physiologically
                 ,
                 from
                 the
                 seasons
                 of
                 the
                 year
                 ,
                 the
                 temperature
                 ,
                 and
                 distemper
                 of
                 seasons
                 &c.
                 
                 Concerning
                 such
                 meteors
                 in
                 general
                 ;
                 yet
                 in
                 a
                 matter
                 so
                 unallyed
                 unto
                 certainty
                 ,
                 replenished
                 with
                 all
                 manner
                 of
                 casualties
                 ,
                 to
                 promote
                 ,
                 or
                 retard
                 such
                 productions
                 ,
                 I
                 have
                 not
                 known
                 any
                 offer
                 peremptory
                 Pronosticks
                 of
                 the
                 kind
                 of
                 the
                 meteor
                 ,
                 its
                 shape
                 ,
                 magnitude
                 ,
                 duration
                 ,
                 motion
                 ,
                 with
                 absolute
                 determinations
                 of
                 its
                 time
                 and
                 places
                 ,
                 at
                 any
                 time
                 beforehand
                 .
                 The
                 quality
                 and
                 condition
                 of
                 the
                 subject
                 matter
                 ,
                 making
                 it
                 as
                 impossible
                 to
                 bolt
                 out
                 scientifical
                 and
                 oracularly
                 certain
                 predictions
                 of
                 such
                 meteors
                 ,
                 as
                 it
                 is
                 ,
                 a
                 year
                 before
                 ,
                 to
                 prove
                 ,
                 or
                 shew
                 that
                 in
                 such
                 an
                 hour
                 ,
                 and
                 in
                 such
                 a
                 quarter
                 ,
                 shall
                 be
                 a
                 Rainbow
                 ,
                 so
                 colour'd
                 ,
                 so
                 continued
                 ,
                 or
                 discontinued
                 and
                 of
                 such
                 limited
                 dimensions
                 :
                 or
                 that
                 ,
                 such
                 a
                 moment
                 ,
                 from
                 such
                 a
                 point
                 of
                 such
                 an
                 Azymuth
                 ,
                 shall
                 a
                 devolant
                 Star
                 spring
                 forth
                 ,
                 which
                 ,
                 in
                 its
                 fall
                 ,
                 shall
                 run
                 obliquely
                 thorow
                 such
                 and
                 such
                 Azymuths
                 ,
                 and
                 expire
                 at
                 such
                 an
                 height
                 above
                 the
                 Horison
                 ;
                 or
                 that
                 such
                 an
                 hour
                 ,
                 in
                 such
                 a
                 Longitude
                 and
                 Latitude
                 ,
                 shall
                 in
                 such
                 Altitudet
                 and
                 positions
                 ,
                 and
                 of
                 such
                 dimensions
                 ,
                 be
                 
                 seen
                 by
                 day
                 four
                 Suns
                 ,
                 or
                 by
                 night
                 three
                 Moons
                 .
                 To
                 lay
                 aside
                 therefore
                 the
                 consideration
                 of
                 these
                 ,
                 as
                 supposed
                 impertinent
                 to
                 our
                 present
                 purpose
                 ;
                 what
                 may
                 judiciously
                 be
                 concluded
                 upon
                 your
                 Questions
                 concerning
                 those
                 other
                 Comets
                 ,
                 which
                 lye
                 without
                 the
                 compasse
                 of
                 this
                 Earths
                 Atmospaere
                 ?
                 If
                 there
                 be
                 not
                 two
                 sorts
                 of
                 such
                 celestial
                 Comets
                 ;
                 there
                 are
                 at
                 least
                 two
                 several
                 and
                 very
                 different
                 Hypotheses
                 and
                 notions
                 under
                 which
                 they
                 are
                 considered
                 by
                 Artists
                 :
                 and
                 Artists
                 of
                 equal
                 worth
                 and
                 fame
                 ,
                 order
                 their
                 reasonings
                 ,
                 some
                 upon
                 the
                 one
                 supposition
                 ,
                 some
                 upon
                 the
                 other
                 :
                 Viz.
                 Some
                 ,
                 as
                 if
                 Comets
                 were
                 new
                 made
                 bodies
                 ,
                 amassed
                 ,
                 and
                 gathered
                 in
                 some
                 of
                 the
                 superior
                 Atmosphaeres
                 ;
                 many
                 of
                 which
                 are
                 ,
                 not
                 without
                 fair
                 reason
                 ,
                 supposed
                 to
                 be
                 in
                 those
                 vastly
                 remote
                 aethereal
                 regions
                 :
                 others
                 ,
                 as
                 if
                 Comets
                 were
                 coaeval
                 to
                 ,
                 and
                 neither
                 less
                 permanent
                 ,
                 nor
                 more
                 new
                 ,
                 then
                 the
                 rest
                 of
                 the
                 Stars
                 ;
                 only
                 seldome
                 seen
                 ,
                 and
                 when
                 seen
                 ,
                 soon
                 passing
                 again
                 out
                 of
                 sight
                 ,
                 by
                 reason
                 of
                 the
                 Line
                 ,
                 upon
                 which
                 their
                 Center
                 is
                 moved
                 :
                 and
                 nothing
                 as
                 yet
                 appears
                 hindering
                 the
                 truth
                 of
                 the
                 possibility
                 and
                 consistency
                 of
                 both
                 these
                 Opinions
                 ;
                 leaving
                 it
                 especially
                 indifferent
                 in
                 the
                 later
                 Hypothesis
                 ,
                 to
                 call
                 such
                 stars
                 at
                 pleasure
                 by
                 the
                 name
                 ,
                 either
                 of
                 Comets
                 ,
                 or
                 New-stars
                 ,
                 or
                 rather
                 seldome
                 appearing
                 Stars
                 .
                 Comets
                 upon
                 the
                 first
                 supposition
                 ,
                 seem
                 not
                 to
                 want
                 some
                 affinity
                 with
                 several
                 ,
                 especially
                 of
                 
                 the
                 more
                 eminent
                 ,
                 meteors
                 of
                 our
                 Atmosphaere
                 :
                 yet
                 allowing
                 a
                 vast
                 difference
                 between
                 them
                 in
                 place
                 ,
                 proportion
                 ,
                 duration
                 ,
                 motion
                 ,
                 and
                 the
                 like
                 circumstantials
                 .
                 And
                 as
                 our
                 Globe
                 ,
                 though
                 in
                 its
                 self
                 large
                 ,
                 is
                 but
                 a
                 minute
                 thing
                 ,
                 compared
                 with
                 many
                 of
                 the
                 caelestial
                 Bodies
                 ;
                 that
                 the
                 vast
                 error
                 of
                 the
                 Earths
                 semidiameter
                 ,
                 by
                 taking
                 the
                 Earth
                 for
                 a
                 small
                 point
                 ,
                 many
                 times
                 in
                 Astronomical
                 ,
                 Gnomonical
                 ,
                 and
                 other
                 operations
                 ,
                 creates
                 not
                 by
                 consequence
                 any
                 perceiveable
                 Errour
                 :
                 so
                 it
                 is
                 not
                 unlike
                 ,
                 but
                 the
                 Atmosphaere
                 of
                 our
                 Globe
                 is
                 vastly
                 disproportionable
                 to
                 the
                 Atmosphaeres
                 of
                 many
                 of
                 those
                 heavenly
                 lights
                 :
                 which
                 may
                 in
                 a
                 sort
                 be
                 gathered
                 also
                 ,
                 from
                 the
                 small
                 power
                 ,
                 which
                 our
                 Atmosphaere
                 has
                 in
                 the
                 pressures
                 which
                 it
                 makes
                 upon
                 moist
                 and
                 liquid
                 bodies
                 :
                 as
                 also
                 from
                 the
                 common
                 vastnes
                 of
                 cometical
                 bodies
                 ,
                 frequently
                 accepted
                 as
                 much
                 larger
                 then
                 all
                 both
                 our
                 globe
                 and
                 its
                 Atmosphaere
                 ,
                 and
                 therefore
                 requiring
                 a
                 proportionately
                 immense
                 place
                 both
                 for
                 their
                 being
                 and
                 motions
                 .
                 And
                 that
                 such
                 huge
                 dimensions
                 are
                 not
                 groundlesly
                 attributed
                 to
                 the
                 cometical
                 bodies
                 ,
                 is
                 rendered
                 undoubted
                 from
                 the
                 glorious
                 and
                 glaring
                 light
                 ,
                 with
                 which
                 eminently
                 above
                 most
                 of
                 other
                 Stars
                 ,
                 they
                 dazle
                 ,
                 to
                 admiration
                 ,
                 the
                 minds
                 and
                 Eyes
                 of
                 mortals
                 at
                 so
                 great
                 distance
                 ,
                 that
                 some
                 times
                 little
                 ,
                 or
                 no
                 Parallax
                 can
                 be
                 observed
                 in
                 them
                 .
                 As
                 to
                 these
                 therefore
                 to
                 solve
                 the
                 Questions
                 ,
                 
                 unto
                 which
                 you
                 have
                 tasked
                 me
                 ;
                 I
                 cannot
                 but
                 judge
                 it
                 improbable
                 ,
                 with
                 any
                 assurance
                 to
                 make
                 praedictions
                 before
                 hand
                 ,
                 that
                 in
                 such
                 limited
                 moments
                 of
                 futurity
                 ,
                 shall
                 appear
                 such
                 Comets
                 ,
                 as
                 are
                 to
                 be
                 at
                 after
                 amassed
                 ,
                 gathered
                 ,
                 formed
                 ,
                 and
                 embodied
                 in
                 the
                 Atmosphaeres
                 ,
                 which
                 are
                 above
                 :
                 if
                 seeming
                 no
                 more
                 wilde
                 to
                 give
                 heed
                 and
                 credit
                 to
                 the
                 superstitious
                 Auguries
                 and
                 soothsayings
                 of
                 the
                 Antients
                 ,
                 then
                 the
                 most
                 subtle
                 reasonings
                 upon
                 which
                 such
                 predictions
                 can
                 be
                 made
                 ,
                 in
                 a
                 subject
                 of
                 this
                 nature
                 .
                 Yet
                 though
                 their
                 first
                 emerging
                 and
                 appearance
                 cannot
                 upon
                 any
                 calculation
                 be
                 foretold
                 ;
                 however
                 at
                 after
                 they
                 have
                 a
                 while
                 appeared
                 ,
                 and
                 some
                 ,
                 though
                 not
                 very
                 many
                 ,
                 curious
                 observations
                 have
                 been
                 made
                 of
                 the
                 changes
                 of
                 place
                 ,
                 and
                 other
                 circumstances
                 ,
                 which
                 they
                 make
                 in
                 their
                 several
                 appearances
                 ,
                 at
                 such
                 and
                 such
                 exalt
                 distances
                 of
                 time
                 ;
                 it
                 is
                 fairly
                 enough
                 probable
                 ,
                 that
                 often
                 ,
                 by
                 an
                 ingenious
                 prognostick
                 ,
                 the
                 course
                 of
                 such
                 a
                 Comet
                 may
                 be
                 nearly
                 traced
                 ,
                 and
                 set
                 down
                 before
                 hand
                 ,
                 as
                 to
                 way
                 and
                 speed
                 ,
                 only
                 with
                 this
                 limitation
                 ,
                 if
                 the
                 substance
                 of
                 the
                 Comet
                 do
                 not
                 first
                 vanish
                 ,
                 or
                 dissolve
                 ,
                 or
                 by
                 reason
                 of
                 its
                 instant
                 dissolution
                 divert
                 from
                 ,
                 hasten
                 ,
                 or
                 retard
                 its
                 course
                 :
                 upon
                 which
                 score
                 of
                 their
                 dissolubleness
                 ,
                 and
                 therefore
                 their
                 ere
                 long
                 to
                 be
                 expected
                 dissolution
                 ,
                 no
                 prognosticks
                 ,
                 upon
                 them
                 and
                 their
                 course
                 ,
                 can
                 be
                 so
                 absolute
                 and
                 
                 peremptory
                 ,
                 as
                 those
                 which
                 may
                 be
                 made
                 concerning
                 such
                 Comets
                 ,
                 as
                 are
                 properly
                 couched
                 under
                 the
                 second
                 supposition
                 .
                 And
                 the
                 grounds
                 why
                 most
                 what
                 in
                 these
                 Comets
                 of
                 the
                 first
                 constitution
                 ,
                 or
                 supposition
                 ,
                 from
                 past
                 observations
                 ,
                 a
                 fair
                 conjecture
                 may
                 be
                 made
                 of
                 their
                 future
                 course
                 notwithstanding
                 their
                 new
                 production
                 and
                 amassement
                 in
                 Atmosphaeres
                 ,
                 are
                 chiefly
                 hereupon
                 founded
                 :
                 that
                 such
                 cometical
                 bodies
                 ,
                 as
                 they
                 are
                 demonstrably
                 vast
                 ,
                 and
                 manifestly
                 ,
                 and
                 by
                 their
                 nature
                 dissoluble
                 ,
                 so
                 their
                 dissolution
                 is
                 not
                 observed
                 very
                 momentaneous
                 ,
                 but
                 most
                 what
                 more
                 ,
                 leisurely
                 by
                 degrees
                 and
                 in
                 time
                 ;
                 in
                 short
                 ,
                 they
                 flame
                 forth
                 of
                 a
                 sudden
                 ,
                 but
                 expire
                 gradually
                 ;
                 so
                 as
                 to
                 be
                 able
                 ,
                 for
                 a
                 while
                 ,
                 to
                 retain
                 the
                 impressions
                 of
                 regular
                 motion
                 ,
                 whether
                 communicated
                 from
                 somewhat
                 in
                 the
                 Center
                 of
                 the
                 same
                 Atmosphaere
                 ,
                 or
                 from
                 the
                 confluent
                 efficacy
                 of
                 several
                 Agents
                 all
                 contributing
                 constantly
                 ,
                 each
                 according
                 to
                 its
                 nature
                 ,
                 to
                 the
                 Comets
                 alone
                 ,
                 or
                 its
                 and
                 its
                 Atmosphaeres
                 motion
                 ,
                 whether
                 by
                 aequilibration
                 ,
                 pulsion
                 ,
                 traction
                 ,
                 pressure
                 ,
                 manifest
                 ,
                 or
                 secret
                 ;
                 or
                 else
                 inwardly
                 wrought
                 into
                 the
                 very
                 substance
                 and
                 nature
                 of
                 the
                 Comet
                 :
                 especially
                 the
                 Comets
                 place
                 being
                 in
                 those
                 lofty
                 aethereal
                 regions
                 ,
                 where
                 are
                 perpetual
                 circumgyrations
                 and
                 every
                 motion
                 that
                 is
                 observable
                 ,
                 the
                 more
                 accurately
                 it
                 is
                 observed
                 ,
                 it
                 is
                 found
                 the
                 more
                 regular
                 .
                 And
                 
                 if
                 it
                 be
                 urged
                 here-against
                 that
                 the
                 motion
                 of
                 Comets
                 seldome
                 ,
                 or
                 never
                 answers
                 the
                 motion
                 of
                 any
                 other
                 Starre
                 ,
                 but
                 commonly
                 has
                 a
                 cross
                 trajecting
                 motion
                 ,
                 whereby
                 it
                 moves
                 asloope
                 ,
                 athwart
                 ,
                 transversely
                 ,
                 or
                 obliquely
                 ,
                 varying
                 from
                 the
                 usual
                 ,
                 formerly
                 observed
                 motions
                 .
                 As
                 this
                 cannot
                 be
                 denyed
                 ,
                 so
                 it
                 doth
                 not
                 at
                 all
                 infringe
                 the
                 possible
                 regularity
                 of
                 such
                 motions
                 ,
                 which
                 are
                 observed
                 to
                 be
                 most
                 what
                 in
                 the
                 same
                 plain
                 ,
                 though
                 oddly
                 posited
                 :
                 and
                 that
                 there
                 are
                 many
                 and
                 ,
                 diverse
                 such
                 undreamt
                 of
                 motions
                 in
                 the
                 Heavens
                 ,
                 constant
                 and
                 regular
                 ,
                 of
                 which
                 save
                 by
                 some
                 such
                 ,
                 or
                 late
                 discoveries
                 ,
                 no
                 hint
                 hath
                 as
                 yet
                 bin
                 made
                 forth
                 unto
                 us
                 ,
                 may
                 in
                 as
                 good
                 reason
                 be
                 allowed
                 ,
                 as
                 that
                 vast
                 number
                 of
                 Stars
                 in
                 the
                 skyes
                 acknowledged
                 ,
                 of
                 which
                 ,
                 without
                 the
                 Telescop
                 ,
                 't
                 is
                 impossible
                 to
                 have
                 any
                 discerning
                 .
                 And
                 if
                 we
                 grant
                 that
                 in
                 respect
                 of
                 its
                 peculiar
                 Atmosphaere
                 ,
                 the
                 motion
                 of
                 a
                 Comet
                 solely
                 considered
                 as
                 in
                 it
                 ,
                 be
                 not
                 so
                 strictly
                 regular
                 ;
                 yet
                 the
                 regular
                 circumgyration
                 of
                 the
                 Atmosphaere
                 ,
                 in
                 which
                 the
                 Comet
                 is
                 carryed
                 ,
                 in
                 respect
                 of
                 the
                 Universe
                 ,
                 cannot
                 but
                 with
                 those
                 that
                 Cant
                 upon
                 that
                 Hypothesis
                 ,
                 enforce
                 the
                 Confession
                 of
                 a
                 great
                 degree
                 of
                 regularity
                 in
                 the
                 Comets
                 motion
                 :
                 and
                 that
                 Comets
                 ,
                 in
                 what
                 part
                 of
                 the
                 Heavens
                 soever
                 appearing
                 ,
                 move
                 in
                 some
                 conformity
                 with
                 other
                 celestial
                 bodyes
                 ,
                 notwithstanding
                 the
                 contrariety
                 or
                 diversity
                 of
                 their
                 proper
                 ,
                 or
                 peculiar
                 motions
                 from
                 the
                 
                 rest
                 ,
                 is
                 ,
                 beyond
                 all
                 dispute
                 ,
                 evident
                 from
                 their
                 dayly
                 moving
                 ,
                 with
                 the
                 rest
                 ,
                 from
                 East
                 to
                 West
                 .
                 The
                 manner
                 of
                 making
                 observations
                 ,
                 calculating
                 to
                 know
                 the
                 place
                 ,
                 the
                 distance
                 ,
                 speed
                 of
                 such
                 a
                 Comet
                 ,
                 and
                 when
                 it
                 is
                 at
                 nearest
                 ,
                 or
                 remotest
                 in
                 its
                 course
                 from
                 us
                 ,
                 is
                 no
                 other
                 then
                 in
                 Comets
                 of
                 the
                 second
                 supposition
                 .
                 And
                 from
                 this
                 whole
                 discourse
                 is
                 manifest
                 ;
                 the
                 Hypotheses
                 ,
                 upon
                 which
                 Calculations
                 are
                 to
                 be
                 made
                 ,
                 in
                 order
                 to
                 such
                 praedictions
                 ,
                 cannot
                 be
                 particularly
                 in
                 these
                 Comets
                 known
                 before
                 hand
                 ,
                 but
                 in
                 several
                 such
                 Comets
                 may
                 be
                 very
                 different
                 ;
                 and
                 are
                 to
                 be
                 found
                 out
                 by
                 curious
                 ,
                 and
                 diligent
                 Observations
                 ;
                 which
                 ,
                 in
                 each
                 respective
                 Comet
                 ,
                 must
                 be
                 accurate
                 ,
                 and
                 at
                 least
                 four
                 in
                 number
                 :
                 and
                 I
                 think
                 predictions
                 pretended
                 from
                 three
                 observations
                 ,
                 are
                 over
                 hasty
                 ,
                 and
                 ,
                 at
                 best
                 ,
                 can
                 but
                 be
                 conjectural
                 ;
                 nay
                 ,
                 in
                 some
                 cases
                 ,
                 from
                 four
                 ;
                 as
                 if
                 you
                 be
                 unsatisfyed
                 ,
                 I
                 shall
                 be
                 ready
                 ,
                 by
                 demonstrative
                 instances
                 ,
                 to
                 remove
                 your
                 doubt
                 .
                 But
                 if
                 of
                 these
                 Comets
                 any
                 should
                 be
                 found
                 ,
                 to
                 wander
                 ,
                 so
                 as
                 the
                 motion
                 of
                 its
                 Center
                 cannot
                 possibly
                 be
                 reduced
                 any
                 thing
                 nigh
                 unto
                 the
                 same
                 plane
                 ,
                 or
                 regularity
                 in
                 change
                 of
                 Planes
                 ,
                 or
                 that
                 the
                 motion
                 of
                 its
                 Center
                 ,
                 in
                 the
                 same
                 ,
                 or
                 several
                 Planes
                 ,
                 describe
                 not
                 a
                 Line
                 regular
                 ,
                 or
                 near
                 unto
                 a
                 regularity
                 ,
                 it
                 is
                 not
                 probably
                 easy
                 to
                 make
                 predictions
                 at
                 any
                 certainty
                 of
                 truth
                 ,
                 of
                 the
                 future
                 
                 course
                 and
                 motion
                 of
                 such
                 Comets
                 .
                 Concerning
                 Comets
                 of
                 the
                 later
                 kind
                 ,
                 or
                 Hypothesis
                 ;
                 possibly
                 it
                 is
                 more
                 easy
                 to
                 answer
                 your
                 Queries
                 in
                 them
                 ,
                 then
                 effectually
                 without
                 a
                 world
                 of
                 time
                 ,
                 sagacity
                 and
                 diligence
                 to
                 accomplish
                 and
                 attain
                 the
                 purport
                 of
                 those
                 Answers
                 .
                 That
                 the
                 motion
                 of
                 such
                 Comets
                 of
                 the
                 second
                 sort
                 ,
                 being
                 not
                 truly
                 new
                 ,
                 but
                 rather
                 seldom
                 appearing
                 ,
                 and
                 soon
                 disappearing
                 Cometical
                 Stars
                 ;
                 that
                 ,
                 I
                 say
                 the
                 motion
                 of
                 such
                 ,
                 is
                 not
                 without
                 some
                 regularity
                 ,
                 though
                 it
                 may
                 not
                 be
                 concluded
                 without
                 long
                 observations
                 and
                 frequent
                 experiences
                 of
                 their
                 motion
                 had
                 ,
                 which
                 in
                 such
                 seldom
                 appearing
                 ,
                 and
                 soon
                 again
                 disappearing
                 Stars
                 is
                 difficult
                 enough
                 to
                 attain
                 to
                 ;
                 yet
                 if
                 we
                 consider
                 the
                 steady
                 ,
                 unbiassed
                 course
                 ,
                 held
                 by
                 most
                 ,
                 if
                 not
                 all
                 the
                 rest
                 of
                 the
                 permanent
                 Stars
                 without
                 deviation
                 ,
                 though
                 in
                 different
                 periods
                 and
                 with
                 many
                 different
                 latitudes
                 and
                 declinations
                 ,
                 and
                 great
                 diversity
                 of
                 motions
                 ,
                 it
                 cannot
                 be
                 less
                 then
                 mightily
                 probable
                 ,
                 nothing
                 in
                 sense
                 ,
                 reason
                 ,
                 or
                 experience
                 appearing
                 as
                 yet
                 able
                 to
                 demonstrate
                 the
                 contrary
                 ,
                 but
                 that
                 a
                 fair
                 ,
                 celestial
                 regularity
                 may
                 rationally
                 be
                 expected
                 in
                 those
                 their
                 perpetual
                 motions
                 ,
                 whether
                 their
                 Centers
                 be
                 carryed
                 in
                 the
                 same
                 Plane
                 ,
                 or
                 in
                 a
                 continual
                 change
                 ,
                 and
                 alterations
                 of
                 Planes
                 .
                 And
                 in
                 diverse
                 respects
                 in
                 some
                 ,
                 if
                 not
                 all
                 of
                 them
                 ,
                 as
                 in
                 most
                 other
                 Stars
                 ,
                 the
                 same
                 Center
                 may
                 be
                 
                 said
                 to
                 move
                 regularly
                 and
                 perpetually
                 in
                 the
                 same
                 Plane
                 ,
                 and
                 also
                 without
                 elenchtical
                 contradiction
                 to
                 move
                 regularly
                 with
                 an
                 incessant
                 change
                 of
                 Planes
                 :
                 so
                 the
                 motion
                 of
                 any
                 Planet
                 ,
                 or
                 other
                 Starr
                 ,
                 whose
                 motion
                 we
                 are
                 versed
                 in
                 ,
                 taken
                 concretely
                 and
                 with
                 complication
                 of
                 that
                 mundane
                 ,
                 or
                 general
                 circumvolution
                 ,
                 which
                 with
                 all
                 the
                 rest
                 it
                 hath
                 from
                 East
                 by
                 South
                 to
                 West
                 ,
                 though
                 it
                 be
                 regular
                 ,
                 yet
                 it
                 is
                 not
                 without
                 a
                 continual
                 shifting
                 of
                 the
                 Planets
                 ,
                 or
                 Stars
                 center
                 from
                 plane
                 to
                 plane
                 :
                 whereas
                 the
                 proper
                 motion
                 of
                 the
                 same
                 Planet
                 ,
                 or
                 Starr
                 considered
                 abstractly
                 and
                 severed
                 from
                 the
                 general
                 circumvolution
                 it
                 hath
                 with
                 the
                 celestial
                 universe
                 ,
                 as
                 it
                 is
                 regular
                 ,
                 so
                 it
                 may
                 keep
                 the
                 center
                 of
                 the
                 planet
                 ,
                 or
                 Star
                 constant
                 in
                 the
                 same
                 plane
                 .
                 And
                 the
                 Course
                 ,
                 appearance
                 ,
                 and
                 returns
                 of
                 such
                 cometical
                 Stars
                 will
                 be
                 of
                 much
                 more
                 easy
                 calculation
                 ,
                 if
                 their
                 motions
                 ,
                 in
                 the
                 regularity
                 ,
                 propriety
                 and
                 peculiarity
                 of
                 them
                 ,
                 be
                 any
                 wayes
                 reducible
                 ,
                 as
                 is
                 abovesaid
                 ,
                 unto
                 the
                 same
                 plane
                 ;
                 of
                 which
                 plane
                 by
                 observations
                 is
                 to
                 be
                 decided
                 ,
                 whether
                 it
                 pass
                 thorough
                 the
                 center
                 of
                 the
                 Aequator
                 and
                 universe
                 ;
                 and
                 if
                 the
                 center
                 of
                 the
                 universe
                 lye
                 out
                 of
                 it
                 ,
                 what
                 is
                 the
                 perpendicular
                 from
                 the
                 Center
                 unto
                 the
                 plane
                 ;
                 and
                 in
                 the
                 plane
                 ,
                 what
                 regular
                 figure
                 ,
                 or
                 journey
                 is
                 described
                 by
                 the
                 motion
                 of
                 the
                 Comets
                 center
                 ,
                 and
                 what
                 are
                 the
                 Diameters
                 or
                 Axis
                 
                 of
                 such
                 figures
                 ,
                 or
                 centers
                 of
                 such
                 motions
                 in
                 that
                 plane
                 :
                 in
                 the
                 easiest
                 ,
                 and
                 plainest
                 of
                 all
                 which
                 cases
                 ,
                 four
                 observations
                 are
                 at
                 least
                 re-requisite
                 :
                 but
                 where
                 the
                 regularity
                 of
                 the
                 Comets
                 motion
                 ,
                 being
                 irreducible
                 to
                 the
                 same
                 plane
                 ,
                 is
                 in
                 a
                 continual
                 shifting
                 from
                 plane
                 to
                 plane
                 ,
                 there
                 many
                 more
                 observations
                 may
                 scarce
                 be
                 enough
                 ,
                 to
                 clear
                 up
                 the
                 regularity
                 of
                 the
                 Comets
                 motion
                 ,
                 sufficiently
                 for
                 easy
                 calculation
                 :
                 and
                 when
                 a
                 sufficient
                 number
                 of
                 exquisite
                 observations
                 have
                 been
                 made
                 ,
                 the
                 calculation
                 ,
                 as
                 it
                 will
                 hardly
                 be
                 so
                 accurate
                 ,
                 so
                 it
                 will
                 scarce
                 be
                 so
                 easy
                 ,
                 as
                 in
                 other
                 cases
                 -
                 The
                 observations
                 ,
                 in
                 all
                 cases
                 and
                 suppositions
                 ,
                 are
                 to
                 make
                 discovery
                 of
                 the
                 exact
                 place
                 of
                 the
                 Comets
                 center
                 at
                 the
                 moment
                 of
                 the
                 observations
                 ,
                 Viz.
                 of
                 its
                 longitude
                 ,
                 latitude
                 and
                 altitude
                 not
                 in
                 respect
                 of
                 Azymuthal
                 arches
                 ,
                 but
                 in
                 right
                 lines
                 drawn
                 between
                 their
                 center
                 and
                 the
                 center
                 of
                 the
                 universe
                 .
                 And
                 the
                 more
                 exquisite
                 the
                 observations
                 are
                 ,
                 they
                 make
                 way
                 for
                 clearer
                 calculations
                 ;
                 upon
                 which
                 may
                 be
                 founded
                 so
                 much
                 the
                 more
                 assured
                 predictions
                 .
                 To
                 answer
                 therefore
                 your
                 Queries
                 in
                 relation
                 to
                 these
                 Comets
                 of
                 the
                 second
                 sort
                 ,
                 or
                 supposition
                 ;
                 I
                 think
                 it
                 not
                 impossible
                 ,
                 but
                 in
                 time
                 predictions
                 may
                 ,
                 it
                 may
                 be
                 many
                 Ages
                 beforehand
                 ,
                 be
                 made
                 of
                 the
                 Appearing
                 ,
                 or
                 rather
                 Re-appearing
                 of
                 such
                 Comets
                 :
                 according
                 as
                 ,
                 in
                 the
                 regularity
                 of
                 their
                 motion
                 ,
                 they
                 
                 come
                 nearer
                 or
                 are
                 removed
                 Farther
                 off
                 from
                 us
                 .
                 So
                 I
                 remember
                 a
                 person
                 of
                 Quality
                 in
                 the
                 North
                 ,
                 with
                 an
                 high
                 measure
                 of
                 Assurance
                 reported
                 to
                 me
                 ,
                 that
                 one
                 of
                 the
                 late
                 Comets
                 appearance
                 and
                 Emerging
                 ,
                 was
                 by
                 a
                 certain
                 Genoese
                 praedicted
                 at
                 least
                 two
                 years
                 before
                 the
                 Emersion
                 and
                 appearing
                 of
                 it
                 :
                 which
                 if
                 so
                 ,
                 is
                 to
                 be
                 referred
                 ,
                 I
                 doubt
                 not
                 ,
                 to
                 the
                 principle
                 and
                 method
                 ,
                 above
                 insisted
                 on
                 .
                 According
                 to
                 which
                 ,
                 after
                 different
                 intervals
                 of
                 years
                 ,
                 several
                 Comets
                 may
                 appear
                 ,
                 one
                 after
                 another
                 ,
                 and
                 sometimes
                 several
                 such
                 Comets
                 may
                 appear
                 all
                 at
                 once
                 :
                 and
                 yet
                 there
                 is
                 no
                 clear
                 necessity
                 of
                 the
                 constant
                 re-appearing
                 of
                 the
                 same
                 Comet
                 ,
                 in
                 the
                 same
                 Horizon
                 ,
                 still
                 at
                 after
                 equal
                 spaces
                 of
                 time
                 .
                 And
                 if
                 the
                 Emerging
                 of
                 such
                 Comets
                 may
                 be
                 praedicted
                 before
                 their
                 Appearance
                 ;
                 upon
                 the
                 same
                 principles
                 ,
                 the
                 line
                 of
                 their
                 motion
                 ,
                 the
                 casting
                 of
                 their
                 Train
                 ,
                 and
                 their
                 future
                 recess
                 ,
                 may
                 as
                 well
                 be
                 praedicted
                 ,
                 whether
                 before
                 ,
                 or
                 after
                 their
                 first
                 appearance
                 :
                 as
                 also
                 when
                 ,
                 and
                 where
                 shall
                 be
                 their
                 Accelerations
                 and
                 retardations
                 in
                 motion
                 ;
                 especially
                 ,
                 considering
                 the
                 advantage
                 hereunto
                 accruing
                 ,
                 from
                 their
                 nearness
                 to
                 ,
                 or
                 distance
                 from
                 the
                 Eye
                 of
                 the
                 spectator
                 :
                 not
                 that
                 this
                 makes
                 inequality
                 ,
                 or
                 inequability
                 to
                 be
                 ,
                 but
                 rather
                 only
                 to
                 appear
                 ,
                 in
                 their
                 motion
                 .
                 Upon
                 the
                 same
                 accounts
                 ,
                 the
                 dayly
                 rising
                 ,
                 setting
                 ,
                 and
                 Southing
                 of
                 Comets
                 may
                 be
                 calculated
                 :
                 
                 as
                 also
                 what
                 parallel
                 ,
                 or
                 spiral
                 it
                 describes
                 :
                 and
                 how
                 much
                 it
                 gains
                 dayly
                 in
                 its
                 proper
                 motion
                 :
                 as
                 also
                 its
                 place
                 ,
                 as
                 to
                 longitude
                 ,
                 latitude
                 ,
                 altitude
                 ,
                 above
                 any
                 limited
                 Horizon
                 ,
                 at
                 any
                 limite
                 moment
                 :
                 what
                 constellations
                 it
                 shall
                 pass
                 thorough
                 :
                 and
                 in
                 what
                 Aspect
                 it
                 shall
                 be
                 posited
                 to
                 any
                 Planet
                 ,
                 or
                 other
                 Star
                 :
                 so
                 of
                 conjunctions
                 ,
                 oppositions
                 ,
                 quadratures
                 ,
                 combustions
                 ,
                 Eclipses
                 ,
                 and
                 many
                 such
                 like
                 affections
                 ,
                 in
                 common
                 incident
                 to
                 Comets
                 with
                 other
                 Stars
                 ,
                 there
                 may
                 as
                 clear
                 predictions
                 be
                 made
                 :
                 as
                 also
                 when
                 it
                 disappears
                 ,
                 not
                 for
                 its
                 remotenesse
                 from
                 us
                 ,
                 but
                 as
                 other
                 Stars
                 dayly
                 ,
                 because
                 of
                 the
                 Suns
                 propinquity
                 to
                 it
                 ,
                 and
                 our
                 view
                 ,
                 the
                 times
                 of
                 its
                 not
                 being
                 seen
                 by
                 reason
                 of
                 the
                 circumfusion
                 of
                 the
                 greater
                 light
                 of
                 the
                 Sun
                 ,
                 being
                 manifestly
                 calculable
                 ;
                 though
                 of
                 other
                 obscurations
                 by
                 clouds
                 ,
                 and
                 such
                 like
                 impediments
                 and
                 interpositions
                 ,
                 no
                 artificial
                 prognostick
                 can
                 be
                 made
                 .
                 So
                 in
                 the
                 first
                 Approaches
                 ,
                 or
                 last
                 recesses
                 of
                 Comets
                 ,
                 by
                 falling
                 into
                 such
                 circumfusions
                 of
                 solar
                 light
                 ,
                 the
                 first
                 appearance
                 of
                 the
                 Comet
                 will
                 be
                 later
                 then
                 otherwise
                 ,
                 and
                 in
                 the
                 other
                 case
                 ,
                 its
                 appearing
                 cease
                 sooner
                 then
                 might
                 be
                 expected
                 :
                 and
                 predictions
                 are
                 accordingly
                 to
                 be
                 ordered
                 .
                 If
                 I
                 have
                 been
                 unusually
                 long
                 beyond
                 the
                 proportion
                 of
                 a
                 Letter
                 ,
                 suppose
                 it
                 to
                 be
                 the
                 long
                 Train
                 peculiar
                 to
                 such
                 Cometical
                 ,
                 full-lighted
                 ,
                 obscure-natured
                 Stars
                 ,
                 as
                 you
                 are
                 now
                 Quaering
                 
                 about
                 .
                 And
                 not
                 to
                 dismisse
                 you
                 without
                 a
                 Task
                 ;
                 it
                 were
                 a
                 noble
                 disquisition
                 clearly
                 to
                 unfold
                 ,
                 what
                 it
                 is
                 in
                 these
                 Cometical
                 Stars
                 ,
                 alone
                 ,
                 or
                 also
                 in
                 their
                 circumfused
                 Atmospheres
                 ,
                 that
                 enables
                 the
                 Sun
                 ,
                 more
                 especially
                 to
                 affix
                 such
                 glorious
                 ,
                 long
                 luminous
                 Tayls
                 unto
                 them
                 more
                 then
                 to
                 other
                 Stars
                 ,
                 whether
                 constant
                 ,
                 or
                 new
                 ,
                 or
                 rarely
                 appearing
                 Stars
                 ;
                 which
                 in
                 whatsoever
                 position
                 to
                 the
                 Sun
                 ,
                 are
                 still
                 seen
                 without
                 any
                 such
                 long
                 ,
                 beamy
                 ,
                 luminous
                 Trains
                 .
                 But
                 I
                 commit
                 this
                 and
                 those
                 other
                 Queries
                 about
                 the
                 salts
                 of
                 the
                 highest
                 volatiles
                 to
                 your
                 serious
                 leisure
                 .
              
            
             
               
               
                 The
                 sixth
                 Answer
                 .
              
               
                 
                   That
                   the
                   motion
                   of
                   the
                   Celestial
                   bodies
                   about
                   their
                   particular
                   axes
                   ,
                   is
                   not
                   a
                   satisfactory
                   proof
                   of
                   their
                   Copornican
                   Systeme
                   .
                   That
                   Comets
                   though
                   supposed
                   to
                   be
                   nearest
                   to
                   the
                   Earth
                   ,
                   when
                   they
                   are
                   in
                   opposition
                   to
                   the
                   Sun
                   ,
                   yet
                   do
                   not
                   thereby
                   prove
                   the
                   Earths
                   motion
                   .
                
              
               
                 I
                 See
                 that
                 neither
                 the
                 currency
                 of
                 Doctrines
                 ,
                 nor
                 the
                 course
                 of
                 time
                 ,
                 are
                 able
                 to
                 beat
                 off
                 from
                 you
                 your
                 old
                 Philosophical
                 genius
                 .
                 You
                 can
                 swallow
                 nothing
                 upon
                 the
                 mere
                 Tradition
                 of
                 others
                 ,
                 nor
                 yet
                 upon
                 their
                 reasons
                 ,
                 till
                 first
                 examined
                 ,
                 concocted
                 ,
                 and
                 digested
                 in
                 your
                 own
                 .
                 Your
                 doubts
                 seem
                 not
                 causeless
                 which
                 you
                 move
                 upon
                 that
                 point
                 ;
                 whether
                 the
                 motion
                 of
                 the
                 Celestial
                 bodies
                 turning
                 themselves
                 about
                 ,
                 upon
                 their
                 own
                 Axes
                 ,
                 may
                 be
                 allowed
                 for
                 a
                 just
                 proof
                 of
                 the
                 Copernican
                 systeme
                 .
                 It
                 is
                 a
                 golden
                 and
                 noble
                 branch
                 of
                 intellectual
                 Justice
                 ,
                 so
                 far
                 as
                 is
                 possible
                 ,
                 mathematically
                 and
                 to
                 a
                 punctillio
                 ,
                 to
                 clear
                 and
                 set
                 forth
                 the
                 bounders
                 of
                 Truth
                 :
                 yet
                 as
                 your
                 modesty
                 delivers
                 your
                 judgement
                 ,
                 I
                 doubt
                 not
                 ,
                 by
                 way
                 of
                 doubt
                 ;
                 so
                 I
                 desire
                 you
                 read
                 my
                 Answer
                 ,
                 as
                 writ
                 only
                 in
                 the
                 same
                 Dialect
                 ;
                 that
                 I
                 ,
                 in
                 like
                 manner
                 ,
                 may
                 not
                 appear
                 a
                 rash
                 Judger
                 ,
                 of
                 what
                 it
                 seems
                 has
                 been
                 concluded
                 by
                 men
                 of
                 great
                 parts
                 and
                 
                 learning
                 .
                 The
                 Artfullness
                 of
                 the
                 Instruments
                 ,
                 and
                 diligence
                 of
                 the
                 Observers
                 that
                 ,
                 from
                 several
                 marks
                 in
                 several
                 Stars
                 ,
                 have
                 noted
                 this
                 their
                 circumvolution
                 ,
                 I
                 love
                 and
                 honour
                 :
                 and
                 am
                 apt
                 to
                 conjecture
                 ,
                 the
                 like
                 motions
                 may
                 in
                 future
                 be
                 discovered
                 in
                 other
                 Stars
                 ,
                 in
                 which
                 they
                 have
                 not
                 as
                 yet
                 been
                 noted
                 :
                 and
                 shall
                 be
                 ready
                 to
                 improve
                 the
                 Observation
                 in
                 all
                 inferences
                 authorized
                 and
                 recommended
                 unto
                 us
                 by
                 the
                 necessity
                 of
                 their
                 consequence
                 .
                 But
                 as
                 things
                 ,
                 at
                 first
                 glance
                 ,
                 commonly
                 offer
                 a
                 fuller
                 raye
                 then
                 they
                 hold
                 in
                 constant
                 ;
                 so
                 I
                 suspect
                 the
                 glimmering
                 probability
                 of
                 the
                 Earths
                 motion
                 according
                 to
                 the
                 Copernican
                 Hypothesis
                 ,
                 upon
                 the
                 first
                 discovery
                 of
                 this
                 circumgyration
                 of
                 Stars
                 about
                 their
                 own
                 Axes
                 ,
                 will
                 ,
                 upon
                 farther
                 consideration
                 and
                 deliberation
                 ,
                 grow
                 still
                 more
                 faint
                 .
                 There
                 being
                 very
                 few
                 Hypotheses
                 ,
                 with
                 which
                 (
                 to
                 give
                 it
                 a
                 short
                 name
                 )
                 this
                 circumaxal
                 motion
                 of
                 the
                 Stars
                 is
                 expressely
                 inconsistent
                 ;
                 it
                 not
                 being
                 impossible
                 ,
                 if
                 all
                 things
                 else
                 would
                 consist
                 and
                 cohaere
                 ,
                 but
                 such
                 a
                 motion
                 might
                 be
                 accomplished
                 in
                 the
                 Hypothesis
                 of
                 Orbes
                 and
                 Sphaeres
                 ,
                 so
                 the
                 Star
                 were
                 but
                 unfixed
                 and
                 set
                 lax
                 and
                 free
                 to
                 move
                 according
                 to
                 its
                 nature
                 in
                 its
                 own
                 orb
                 ,
                 by
                 a
                 motion
                 discharged
                 ,
                 different
                 and
                 distinct
                 from
                 the
                 motion
                 of
                 its
                 orb
                 :
                 So
                 in
                 the
                 old
                 Hypothesis
                 ,
                 while
                 they
                 affixed
                 and
                 bound
                 the
                 Starre
                 unto
                 its
                 proper
                 Epicycle
                 ,
                 yet
                 in
                 its
                 larger
                 Orb
                 they
                 allowed
                 it
                 to
                 be
                 therein
                 
                 carryed
                 sometimes
                 progressively
                 ,
                 sometimes
                 retrogradely
                 ,
                 and
                 sometimes
                 stationarily
                 ,
                 according
                 to
                 the
                 motion
                 of
                 the
                 Epicycle
                 ,
                 or
                 Episphaere
                 in
                 the
                 larger
                 sphaere
                 .
                 And
                 without
                 any
                 sense
                 as
                 yet
                 ,
                 that
                 it
                 deserves
                 a
                 blush
                 ,
                 I
                 willingly
                 profess
                 ,
                 I
                 want
                 the
                 Lyncean
                 eyes
                 and
                 judgment
                 ,
                 from
                 such
                 a
                 circumaxall
                 moving
                 of
                 Stars
                 ,
                 to
                 deduce
                 the
                 Suns
                 ,
                 or
                 Earths
                 central
                 ,
                 either
                 rest
                 ,
                 or
                 motion
                 :
                 especially
                 the
                 vast
                 distance
                 ,
                 at
                 which
                 these
                 bodyes
                 are
                 distant
                 one
                 from
                 another
                 ,
                 being
                 on
                 all
                 hands
                 yielded
                 ;
                 which
                 if
                 it
                 do
                 not
                 necessarily
                 ,
                 altogether
                 amort
                 ,
                 and
                 abolish
                 the
                 powers
                 ,
                 which
                 one
                 of
                 these
                 bodyes
                 may
                 possibly
                 have
                 upon
                 the
                 others
                 ,
                 yet
                 till
                 such
                 their
                 powers
                 over
                 one
                 another
                 in
                 this
                 point
                 of
                 motion
                 be
                 fairly
                 proved
                 ,
                 it
                 leaves
                 it
                 easily
                 manifest
                 ,
                 that
                 any
                 of
                 these
                 bodyes
                 may
                 be
                 moved
                 according
                 to
                 its
                 own
                 peculiar
                 nature
                 ,
                 without
                 laying
                 ,
                 by
                 so
                 long
                 a
                 reach
                 ,
                 laws
                 of
                 necessity
                 upon
                 other
                 Bodyes
                 ,
                 at
                 so
                 great
                 a
                 distance
                 ,
                 for
                 their
                 motion
                 .
                 As
                 for
                 example
                 the
                 bodyes
                 A.
                 and
                 B.
                 either
                 of
                 the
                 same
                 general
                 ,
                 or
                 special
                 nature
                 ;
                 they
                 being
                 severed
                 and
                 severally
                 placed
                 at
                 competent
                 distance
                 one
                 from
                 another
                 ;
                 both
                 ,
                 or
                 either
                 may
                 be
                 at
                 rest
                 ,
                 or
                 in
                 motion
                 ,
                 neither
                 inferring
                 upon
                 the
                 other
                 the
                 analogy
                 either
                 of
                 its
                 rest
                 ,
                 or
                 motion
                 ,
                 or
                 to
                 be
                 suspected
                 so
                 to
                 doe
                 ,
                 till
                 it
                 be
                 pregnantly
                 either
                 so
                 observed
                 ,
                 or
                 proved
                 :
                 Viz.
                 That
                 either
                 has
                 upon
                 the
                 other
                 ,
                 even
                 at
                 distance
                 ,
                 
                 such
                 a
                 strong
                 and
                 efficacions
                 power·
                 And
                 if
                 the
                 stress
                 of
                 the
                 ratiocination
                 hang
                 upon
                 this
                 ;
                 that
                 otherwise
                 there
                 would
                 not
                 be
                 a
                 conformity
                 between
                 the
                 motions
                 of
                 the
                 Earth
                 ,
                 the
                 Sun
                 ,
                 Saturn
                 ,
                 Jupiter
                 ,
                 &c.
                 to
                 my
                 apprehension
                 ,
                 nothing
                 is
                 more
                 easy
                 ,
                 then
                 not
                 only
                 to
                 turn
                 the
                 edge
                 ,
                 but
                 break
                 the
                 back
                 and
                 overturn
                 the
                 foundation
                 of
                 such
                 an
                 Argument
                 .
                 For
                 first
                 the
                 motions
                 ,
                 or
                 unmovednes
                 of
                 these
                 bodyes
                 ,
                 is
                 not
                 to
                 be
                 made
                 conform
                 unto
                 our
                 retired
                 thoughts
                 ,
                 and
                 speculations
                 in
                 a
                 close
                 study
                 ,
                 or
                 to
                 the
                 motions
                 ,
                 which
                 we
                 have
                 observed
                 ,
                 and
                 justly
                 concluded
                 certain
                 in
                 other
                 noble
                 Bodyes
                 in
                 the
                 world
                 ;
                 but
                 they
                 are
                 rather
                 by
                 all
                 ingenions
                 Arts
                 and
                 instruments
                 ,
                 to
                 be
                 each
                 by
                 its
                 self
                 particularly
                 observed
                 ,
                 how
                 in
                 Truth
                 they
                 are
                 ,
                 and
                 so
                 to
                 be
                 accepted
                 and
                 acknowledged
                 in
                 their
                 appropriate
                 sciences
                 .
                 And
                 what
                 greater
                 necessity
                 is
                 there
                 of
                 making
                 the
                 Sun
                 ,
                 or
                 the
                 Earth
                 in
                 rest
                 and
                 motion
                 to
                 be
                 conform
                 to
                 Saturn
                 ,
                 Jupiter
                 ,
                 &c.
                 then
                 to
                 be
                 conform
                 to
                 them
                 in
                 shape
                 ,
                 magnitude
                 ,
                 respective
                 distances
                 ,
                 which
                 on
                 no
                 hand
                 is
                 ascerted
                 ,
                 but
                 for
                 the
                 impossibility
                 ,
                 and
                 demonstrablenes
                 thereof
                 by
                 observation
                 ,
                 rather
                 the
                 contrary
                 ?
                 And
                 why
                 should
                 there
                 not
                 be
                 as
                 great
                 a
                 necessity
                 for
                 them
                 to
                 be
                 conform
                 in
                 rest
                 and
                 motion
                 ,
                 to
                 the
                 Stars
                 passing
                 generally
                 under
                 the
                 notion
                 and
                 title
                 of
                 Fixed
                 ,
                 which
                 has
                 never
                 yet
                 been
                 offered
                 ,
                 because
                 it
                 is
                 cl●●●
                 it
                 cannot
                 be
                 defended
                 ?
                 And
                 
                 what
                 less
                 necessity
                 is
                 there
                 ,
                 why
                 they
                 should
                 not
                 conform
                 in
                 the
                 celerity
                 and
                 nimbleness
                 of
                 their
                 motion
                 ,
                 as
                 well
                 as
                 the
                 form
                 ,
                 kind
                 and
                 manner
                 of
                 it
                 ?
                 And
                 what
                 greater
                 necessity
                 for
                 conformity
                 of
                 motion
                 in
                 Stars
                 ,
                 then
                 in
                 Animals
                 ,
                 in
                 which
                 some
                 creep
                 ,
                 some
                 fly
                 ,
                 some
                 go
                 upon
                 feet
                 ,
                 some
                 swim
                 ;
                 neither
                 in
                 their
                 supposed
                 and
                 called
                 Elements
                 can
                 any
                 such
                 necessity
                 of
                 the
                 same
                 kind
                 of
                 motion
                 be
                 observed
                 ,
                 though
                 in
                 the
                 same
                 Element
                 .
                 Upon
                 the
                 same
                 way
                 ,
                 method
                 and
                 force
                 of
                 reasoning
                 ,
                 when
                 Jacob
                 is
                 at
                 the
                 Dancing
                 ,
                 or
                 Fencing
                 Schools
                 in
                 
                   Oxford
                   ,
                   Joga
                
                 must
                 be
                 Galliarding
                 or
                 Brandishing
                 it
                 at
                 the
                 like
                 Schools
                 in
                 Salamanca
                 .
                 Every
                 stone
                 has
                 not
                 in
                 it
                 the
                 mysteryes
                 of
                 the
                 Magnet
                 ,
                 and
                 several
                 Jewels
                 have
                 their
                 several
                 splendours
                 ,
                 virtues
                 and
                 properties
                 :
                 so
                 several
                 Stars
                 considering
                 their
                 several
                 situations
                 ,
                 shapes
                 ,
                 magnitudes
                 ,
                 distances
                 and
                 diversities
                 of
                 nature
                 ,
                 cannot
                 but
                 in
                 rest
                 and
                 motion
                 each
                 have
                 something
                 proper
                 and
                 peculiar
                 :
                 at
                 least
                 may
                 not
                 without
                 cause
                 be
                 suspected
                 to
                 have
                 otherwise
                 ,
                 till
                 it
                 be
                 possible
                 by
                 some
                 medium
                 to
                 evince
                 the
                 contrary
                 .
                 And
                 as
                 the
                 Copernican
                 Systeme
                 receives
                 not
                 much
                 confirmation
                 from
                 the
                 praecedent
                 circumaxal
                 motion
                 of
                 Stars
                 ;
                 so
                 that
                 Comets
                 are
                 nearest
                 the
                 Earth
                 when
                 opposite
                 to
                 the
                 Sun
                 ;
                 with
                 what
                 greater
                 strength
                 doth
                 it
                 ,
                 as
                 others
                 urge
                 ,
                 prove
                 the
                 Earths
                 motion
                 ?
                 wherein
                 is
                 requisite
                 that
                 the
                 Hypothesis
                 it self
                 ,
                 be
                 
                 confirmed
                 by
                 frequent
                 Observation
                 in
                 several
                 Comets
                 ,
                 before
                 it
                 be
                 allowed
                 as
                 general
                 ;
                 that
                 as
                 by
                 way
                 of
                 Induction
                 may
                 be
                 inferred
                 ,
                 that
                 every
                 Comet
                 when
                 it
                 is
                 nearest
                 to
                 the
                 Earth
                 ,
                 is
                 then
                 opposite
                 to
                 the
                 Sun
                 ,
                 and
                 when
                 this
                 is
                 allowed
                 and
                 fairly
                 proved
                 ,
                 there
                 is
                 no
                 necessity
                 by
                 which
                 it
                 follows
                 ,
                 or
                 can
                 be
                 inferred
                 from
                 hence
                 ,
                 that
                 the
                 Earth
                 hath
                 its
                 annual
                 Revolutions
                 about
                 the
                 Sun
                 :
                 all
                 that
                 is
                 in
                 the
                 allowed
                 ,
                 and
                 supposedly
                 proved
                 Hypothesis
                 ,
                 being
                 many
                 ways
                 accomplishable
                 ,
                 and
                 as
                 easily
                 and
                 clearly
                 ,
                 the
                 Sun
                 moving
                 about
                 the
                 Earth
                 ,
                 as
                 the
                 Earth
                 moving
                 about
                 the
                 Sun
                 ,
                 in
                 which
                 I
                 doubt
                 not
                 your
                 judgment
                 .
              
            
          
        
      
       
         
           
             
             
               The
               Second
               SECTION
               of
               PHILOSOPHICAL
               ANSWERS
               .
               CONTEINING
               QUERIES
               Relating
               to
               the
               Angle
               of
               Contact
               .
            
             
               LONDON
               ,
               Printed
               by
               
                 T.
                 L.
              
               for
               
                 Nathaniel
                 Brooke
              
               ,
               at
               the
               Angel
               in
               Cornhil
               near
               the
               Royal
               Exchange
               .
               1670.
               
            
          
           
             
             
             
               To
               the
               right
               reverend
               Father
               in
               God
               ,
               Seth
               Lord
               Bishop
               of
               Salisbury
               .
            
             
               To
               the
               right
               reverend
               Father
               in
               God
               Lord
               Bishop
               of
               Chester
               .
            
             
               
                 My
                 Lords
                 .
              
            
             
               ALthough
               I
               know
               ,
               the
               raw
               ,
               untrained
               Reader
               ,
               upon
               this
               last
               Answer
               above
               the
               rest
               ,
               will
               be
               sure
               to
               pronounce
               his
               ,
               
                 Quis
                 leget
                 haec
              
               ?
               and
               to
               bestow
               upon
               it
               ,
               no
               other
               fate
               then
               
                 thuris
                 &
                 scrombrorum
              
               ;
               judging
               all
               things
               impertinent
               ,
               that
               lye
               not
               in
               the
               dirty
               troden
               way
               ;
               or
               cold
               ,
               which
               his
               torpid
               dullness
               is
               unable
               ,
               at
               least
               too
               slothfull
               to
               strike
               any
               fire
               out
               of
               ;
               and
               every
               thing
               to
               be
               destitute
               of
               a
               kernel
               ,
               which
               is
               too
               hard
               for
               his
               delicate
               Gum
               to
               crack
               the
               shell
               of
               :
               however
               ,
               I
               fear
               not
               the
               rendering
               my self
               faulty
               beyond
               Apology
               ,
               in
               laying
               this
               Answer
               ,
               above
               all
               the
               rest
               divulged
               in
               this
               present
               manual
               ,
               at
               your
               Lordships
               feet
               .
               The
               singular
               happines
               of
               both
               your
               Lordships
               in
               your
               studies
               ,
               drawing
               in
               by
               way
               of
               divertisement
               ,
               what
               multitudes
               of
               others
               study
               seriously
               and
               purposely
               for
               ,
               and
               yet
               miss
               of
               ,
               having
               made
               veneration
               to
               wait
               upon
               your
               judgment
               ,
               
               and
               your
               minds
               so
               seemingly
               fruitful
               ,
               in
               multitudes
               of
               as
               new
               as
               ingenious
               inventions
               ;
               hath
               too
               well
               acquainted
               your
               Lordships
               ,
               with
               the
               Scufflings
               which
               have
               been
               in
               former
               ages
               ,
               and
               between
               men
               of
               the
               clearest
               judgments
               ,
               and
               most
               excellent
               parts
               ,
               about
               these
               minute
               angles
               of
               contact
               ;
               being
               ,
               by
               general
               confession
               ,
               small
               beyond
               all
               proportion
               with
               rectilinearies
               ;
               and
               by
               others
               only
               denyed
               to
               be
               minute
               ,
               because
               they
               think
               they
               have
               no
               quantity
               at
               all
               .
               Like
               as
               in
               solar
               Atomes
               ,
               when
               myriades
               of
               them
               are
               before
               us
               ,
               we
               cannot
               be
               peremptory
               in
               asserting
               a
               full
               sight
               of
               any
               one
               of
               them
               .
               And
               however
               I
               may
               have
               missed
               the
               Corinth
               I
               first
               hoysed
               sayl
               for
               ,
               I
               know
               when
               retiring
               from
               your
               higher-region'd
               speculations
               you
               are
               pleased
               to
               make
               unto
               your selves
               a
               Platonick
               divertisement
               ,
               upon
               a
               deigned
               perusall
               of
               this
               scrole
               to
               a
               friend
               ,
               I
               doubt
               not
               but
               your
               Lordships
               will
               justify
               the
               essay
               ,
               and
               allow
               that
               by
               such
               sallyes
               ,
               a
               firm
               comprehension
               may
               at
               last
               be
               attained
               of
               the
               mercurial
               niceness
               of
               this
               question
               ,
               which
               hath
               so
               oft
               eluded
               the
               grasp
               of
               those
               wits
               ,
               who
               have
               formerly
               thought
               they
               have
               had
               it
               so
               firm
               and
               sure
               .
               
                 My
                 Lords
              
               ,
               I
               know
               the
               admirable
               quality
               ,
               and
               mighty
               merits
               of
               the
               persons
               that
               think
               and
               have
               thought
               otherwise
               ,
               then
               I
               have
               delivered
               my
               thoughts
               here
               ;
               however
               I
               should
               have
               dealt
               faithlesly
               with
               my
               friend
               ,
               if
               I
               should
               have
               returned
               him
               answer
               contrary
               to
               my
               judgment
               .
               And
               as
               I
               
               have
               been
               morally
               true
               to
               him
               ,
               so
               what
               right
               ,
               or
               injury
               I
               have
               herein
               done
               unto
               Philosophical
               truths
               in
               this
               nice
               ,
               and
               hitherto
               inexplicable
               subtilty
               ,
               none
               can
               better
               decide
               then
               your
               Lordships
               ;
               at
               whose
               feet
               it
               is
               left
               ,
               by
            
             
               
                 Your
                 Lordships
                 Servant
                 .
                 W.
                 M.
                 
              
            
          
        
         
           
             
             
               The
               Seventh
               Answer
               .
            
             
               
                 Concerning
                 angles
                 of
                 contact
                 .
                 That
                 recto-convexe
                 angles
                 of
                 contact
                 are
                 truly
                 angles
                 according
                 to
                 the
                 definition
                 of
                 plane
                 angles
                 .
                 That
                 recto-convexe
                 angles
                 of
                 contact
                 are
                 not
                 destitute
                 of
                 quantity
                 .
                 How
                 many
                 wayes
                 one
                 quantity
                 may
                 be
                 greater
                 then
                 another
                 ,
                 what
                 kind
                 of
                 homogeneity
                 it
                 is
                 that
                 is
                 requisite
                 for
                 proportionalls
                 .
                 That
                 angles
                 have
                 figuration
                 as
                 well
                 as
                 quantity
                 .
                 That
                 in
                 angles
                 is
                 observable
                 a
                 homogeneity
                 or
                 heterogeneity
                 which
                 is
                 Mathematical
                 ,
                 and
                 also
                 a
                 homogeneity
                 or
                 heterogeneity
                 which
                 is
                 not
                 Mathematical
                 ,
                 nor
                 any
                 way
                 concerning
                 their
                 quantity
                 ,
                 but
                 only
                 their
                 figuration
                 ,
                 &c.
                 
                 Of
                 several
                 special
                 properties
                 of
                 magnitudes
                 and
                 angles
                 .
                 How
                 some
                 angles
                 are
                 Constituted
                 by
                 the
                 concrescency
                 and
                 annexion
                 of
                 several
                 parts
                 and
                 angles
                 which
                 are
                 heterogeneal
                 each
                 to
                 other
                 .
                 Whether
                 like
                 angles
                 in
                 unequal
                 circles
                 ,
                 and
                 in
                 loke
                 segments
                 of
                 circles
                 be
                 always
                 Equal
                 ?
                 That
                 all
                 Equal
                 angles
                 are
                 not
                 like
                 .
                 That
                 all
                 like
                 angles
                 are
                 not
                 Equal
                 .
              
            
             
               THough
               I
               cannot
               consent
               unto
               your
               thoughts
               about
               the
               Angle
               of
               Contact
               ,
               I
               should
               be
               angry
               with
               my
               own
               rashness
               ,
               should
               I
               be
               forward
               to
               blame
               your
               late
               speculations
               thereabout
               ;
               seeing
               so
               many
               noble
               and
               sprightly
               wits
               ,
               and
               such
               as
               have
               deserved
               the
               honour
               of
               Europe
               ,
               siding
               with
               you
               .
               Especially
               I
               honour
               your
               thorow-conversation
               in
               the
               learned
               persons
               works
               ,
               whom
               I
               perceive
               you
               so
               largely
               
               to
               have
               gleaned
               after
               :
               you
               could
               not
               have
               plowed
               in
               these
               hard
               grounds
               with
               a
               more
               worthy
               Assistant
               .
               But
               knowing
               it
               is
               not
               Complement
               you
               expect
               in
               Philosophy
               but
               Reason
               ;
               I
               shall
               in
               as
               short
               ,
               as
               the
               matter
               will
               bear
               ,
               and
               as
               plainly
               as
               I
               can
               ,
               summe
               up
               the
               grounds
               ,
               why
               I
               as
               yet
               remain
               so
               farre
               dissatisfyed
               ,
               that
               I
               cannot
               without
               injustice
               to
               my
               own
               judgment
               ,
               desert
               the
               generous
               and
               learned
               dissenters
               from
               your
               Opinion
               :
               or
               cease
               to
               averre
               Angles
               of
               Contact
               to
               be
               true
               quantities
               ,
               equal
               ;
               or
               unequal
               one
               to
               another
               ,
               though
               the
               highest
               multiplex
               of
               the
               greatest
               citradiametral
               Angle
               of
               Contact
               ,
               be
               never
               able
               to
               exhaust
               ,
               or
               equalize
               the
               least
               right-lined
               part
               of
               the
               most
               acute
               right-lined
               Angle
               .
            
             
               And
               that
               we
               may
               not
               mistake
               each
               other
               about
               the
               State
               of
               the
               Controversy
               ;
               and
               to
               be
               shorter
               ,
               and
               more
               clear
               in
               the
               insuing
               discourse
               ;
               we
               shall
               first
               set
               down
               the
               definitions
               of
               certain
               termes
               ,
               about
               which
               I
               suppose
               there
               will
               be
               no
               difference
               between
               us
               .
               As.
               
            
             
               1
               st
               .
               (
               Our
               discourse
               extending
               no
               further
               then
               to
               plain
               Angles
               )
               that
               by
               a
               Right-lined
               angle
               ,
               we
               understand
               a
               plane
               angle
               contained
               under
               two
               concurring
               sides
               ,
               or
               lines
               ,
               which
               are
               two
               right-lines
               not
               posited
               directly
               the
               one
               in
               the
               production
               of
               the
               other
               ,
               nor
               parallel
               the
               one
               to
               the
               other
               .
               As
               in
               fig.
               1.
               the
               Angle
               B
               
               AC
               is
               a
               right-lined
               angle
               contained
               under
               the
               two
               
               right
               lines
               BA
               and
               CA
               ;
               BA
               and
               CA
               not
               being
               so
               posited
               at
               A
               as
               to
               make
               one
               right
               line
               :
               nor
               can
               they
               be
               parallel
               because
               of
               their
               rectitude
               and
               concurrence
               .
            
             
               2
               ly
               .
               That
               by
               a
               crooked-lined
               Angle
               ,
               we
               understand
               a
               plane
               Angle
               ,
               contained
               under
               two
               concurring
               sides
               ,
               or
               lines
               ,
               which
               are
               two
               crooked
               lines
               ;
               as
               in
               the
               fig.
               2.3
               4.
               the
               Angle
               B
               
               AC
               is
               a
               crooked
               lined
               angle
               contained
               under
               the
               two
               Crooked
               lines
               B
               A
               and
               C
               A.
               
            
             
               3
               ly
               .
               That
               by
               a
               crooked
               lined
               angle
               of
               concaves
               ,
               or
               a
               Concavo-concave
               angle
               ,
               we
               understand
               a
               plane
               angle
               contained
               under
               two
               concurring
               crooked
               lines
               ;
               both
               which
               crooked
               lines
               obvert
               at
               the
               Angle
               first
               of
               all
               inwards
               ,
               the
               concave
               side
               of
               their
               Curvature
               ,
               towards
               the
               angle
               under
               then
               contained
               .
               So
               in
               fig.
               4.
               the
               angle
               B
               
               AC
               is
               a
               crooked-lined
               angle
               of
               Concaves
               ,
               or
               a
               Concavo-concave
               angle
               ,
               contained
               under
               the
               two
               crooked
               lines
               BA
               and
               C
               A
               both
               which
               obvert
               the
               Concave
               side
               of
               their
               curvature
               towards
               the
               angle
               under
               then
               contained
               .
            
             
               4
               ly
               .
               That
               by
               a
               crooked-lined
               angle
               of
               Convexes
               ,
               or
               a
               Convexo-convexe
               angle
               ,
               we
               understand
               a
               plane
               angle
               ,
               contained
               under
               two
               concurring
               crooked-lines
               ;
               both
               which
               crooked-lines
               obvert
               at
               the
               angle
               first
               of
               all
               inwards
               ,
               the
               Convexe
               sides
               of
               their
               Curvature
               towards
               the
               angle
               ,
               under
               them
               contained
               .
               So
               in
               fig.
               2.
               the
               angle
               BAC
               .
               is
               a
               crooked-lined
               
               angle
               of
               convexes
               ,
               or
               a
               Convexo-convexe
               angle
               ,
               contained
               under
               the
               two
               crooked-lines
               BA
               and
               CA
               ,
               both
               which
               obvert
               the
               convexe
               sides
               of
               their
               curvature
               ,
               first
               of
               all
               inwards
               ,
               towards
               the
               Angle
               under
               them
               contained
               .
            
             
               5
               ly
               .
               That
               by
               a
               concavo-convexe
               angle
               we
               understand
               a
               plane
               angle
               ,
               contained
               under
               two
               concurring
               crooked
               lines
               ;
               the
               one
               of
               which
               ,
               at
               the
               angle
               first
               of
               all
               ,
               obverts
               the
               convexe
               side
               of
               its
               curvature
               inwards
               towards
               the
               contained
               angle
               ,
               the
               other
               the
               concave
               side
               of
               its
               curvature
               .
               So
               in
               fig.
               the
               3
               d.
               the
               angle
               BAC
               is
               a
               concave-convexe
               ,
               or
               a
               convexo-concave
               crooked-lined
               angle
               ,
               contained
               under
               the
               two
               crooked
               lines
               BA
               and
               CA
               ,
               whereof
               the
               one
               BA
               obverts
               the
               convexe
               side
               of
               its
               curvature
               towards
               the
               contained
               angle
               ,
               but
               the
               other
               CA
               obverts
               the
               concave
               side
               of
               its
               curvature
               towards
               the
               same
               contained
               angle
               .
            
             
               6
               ly
               .
               That
               by
               a
               mixed
               lined
               angle
               ,
               we
               understand
               a
               plane
               angle
               ,
               contained
               under
               two
               concurring
               lines
               ;
               whereof
               the
               one
               is
               a
               right
               line
               ,
               the
               other
               a
               crooked
               line
               .
               So
               in
               fig.
               5
               th
               .
               6
               th
               .
               the
               angle
               BAC
               is
               a
               mixed-lined
               angle
               ,
               contained
               under
               the
               right
               line
               CA.
               and
               the
               crooked
               line
               B
               A.
               
            
             
               7
               ly
               .
               That
               by
               a
               Recto-concave
               angle
               ,
               we
               understand
               a
               plane
               angle
               ,
               contained
               under
               two
               concurring
               lines
               ;
               whereof
               the
               one
               is
               a
               right
               line
               ,
               the
               other
               a
               crooked
               line
               ,
               having
               the
               concave
               
               side
               of
               its
               curvature
               obverted
               towards
               the
               contained
               angle
               ,
               so
               in
               fig.
               5.
               
               The
               angle
               B.A.C.
               is
               drawn
               a
               recto-concave
               angle
               ,
               being
               contained
               under
               the
               right
               line
               C.A.
               and
               the
               crooked
               line
               B.
               obverting
               the
               concave
               side
               of
               its
               curvature
               towards
               the
               contained
               angle
               .
            
             
               8
               ly
               .
               That
               by
               a
               recto-convexe
               angle
               ,
               we
               understand
               a
               plane
               angle
               ,
               contained
               under
               two
               concurring
               lines
               ;
               whereof
               the
               one
               is
               a
               right
               line
               ,
               the
               other
               a
               crooked
               line
               ,
               having
               the
               convexe
               side
               of
               its
               curvature
               obverted
               towards
               the
               angle
               contained
               ,
               so
               in
               fig.
               6
               th
               .
               the
               angle
               BAC
               .
               is
               drawn
               a
               recto-convexe
               angle
               ,
               being
               contained
               under
               the
               right
               line
               CA.
               and
               the
               crooked
               line
               BA
               .
               obverting
               the
               convexe
               side
               of
               its
               curvature
               towards
               the
               contained
               angle
               .
            
             
               9
               ly
               .
               That
               by
               an
               Angle
               of
               contact
               we
               understand
               a
               plane
               angle
               ,
               contained
               under
               two
               concurring
               lines
               ;
               which
               two
               lines
               are
               neither
               one
               and
               the
               same
               line
               produced
               ,
               nor
               yet
               in
               their
               present
               positure
               and
               inclination
               can
               possibly
               cut
               one
               another
               ,
               how
               farr
               soever
               either
               ,
               or
               both
               of
               them
               be
               produced
               .
               So
               in
               fig.
               7
               th
               .
               and
               fig.
               8
               th
               .
               the
               angle
               B
               
               AC
               .
               being
               an
               angle
               of
               contact
               ,
               if
               in
               either
               figure
               the
               crookd-line
               BA
               ,
               be
               produced
               beyond
               the
               point
               of
               contact
               A
               ,
               unto
               D
               ,
               and
               in
               fig
               ,
               7
               th
               .
               the
               crooked-line
               CA
               ,
               in
               figure
               8
               th
               .
               the
               right
               line
               CA.
               be
               produced
               beyond
               the
               point
               of
               contact
               A
               unto
               E
               ;
               in
               neither
               the
               line
               C
               
               AE
               can
               fal
               into
               the
               production
               of
               the
               line
               BAD
               ,
               nor
               in
               either
               
               shall
               the
               line
               C
               
               AE
               cut
               the
               line
               B
               
               AD.
               And
            
             
               10
               ly
               .
               From
               the
               figuration
               ,
               positure
               and
               properties
               of
               the
               sides
               of
               such
               angles
               of
               contact
               ;
               that
               by
               a
               recto-convexe
               angle
               of
               contact
               ,
               we
               understand
               a
               plane
               angle
               being
               recto-convexe
               ,
               as
               above
               defined
               ,
               and
               being
               also
               an
               angle
               of
               contact
               .
               So
               in
               fig.
               8
               th
               .
               if
               in
               the
               recto-convexe
               angle
               BAC
               ,
               the
               right
               line
               C
               A
               touch
               and
               cut
               not
               the
               convexe
               arch
               B
               A
               the
               angle
               BAC
               is
               a
               recto-convexe
               angle
               of
               contact
               .
            
             
               11
               ly
               .
               That
               ,
               by
               a
               recto-concave
               angle
               of
               contact
               we
               understand
               a
               plane
               angle
               of
               contact
               ,
               which
               is
               also
               recto-concave
               ,
               as
               above
               defined
               .
               As
               in
               fig.
               the
               8
               th
               .
               if
               in
               the
               recto-concave
               angle
               EAB
               the
               right
               line
               EA
               touch
               and
               cut
               not
               the
               concave
               arch
               BA
               ,
               the
               angle
               EAB
               is
               a
               recto-concave
               angle
               of
               contact
               .
            
             
               12.
               
               That
               by
               a
               concavo-convexe
               angle
               of
               contact
               we
               understand
               a
               plane
               angle
               of
               Contact
               ,
               being
               also
               concavo-convexe
               ,
               as
               above
               defined
               .
               So
               in
               fig.
               12
               th
               .
               if
               R
               
               AH
               being
               a
               concavo-convexe
               angle
               ,
               the
               arch
               RA
               touch
               and
               cut
               not
               the
               arch
               H
               A
               then
               the
               angle
               R
               
               AH
               is
               a
               concavo-convexe
               angle
               of
               Contact
               .
            
             
               13.
               
               That
               by
               a
               Citra-diametral-concavo-convexe
               angle
               of
               Contact
               ,
               we
               understand
               a
               plane
               concavo-convexe
               angle
               of
               Contact
               ,
               both
               whose
               conteining
               Arches
               lie
               on
               the
               same
               side
               of
               the
               right
               line
               tangent
               ,
               which
               toucheth
               both
               the
               arches
               in
               the
               same
               point
               of
               Contact
               ,
               As
               in
               fig.
               
               12
               th
               .
               the
               concavo-convexe
               angle
               of
               Contact
               H
               
               AR
               ,
               having
               both
               Arches
               H
               A
               and
               R
               A
               lying
               on
               the
               same
               side
               of
               the
               right
               line
               tangent
               B
               A
               ,
               which
               toucheth
               both
               the
               Arches
               in
               the
               same
               point
               A
               ,
               the
               concavo-convexe
               angle
               of
               Contact
               H
               
               AR
               is
               citradiametral
               .
            
             
               14.
               
               That
               by
               an
               ultra-diametral
               concavo-convexe
               angle
               of
               Contact
               ,
               we
               understand
               a
               plane
               angle
               of
               contact
               being
               concavo-convexe
               ,
               and
               having
               the
               one
               arch
               on
               the
               one
               side
               of
               the
               diameter
               passing
               thorow
               the
               Contact
               point
               ,
               and
               beyond
               the
               right
               line
               tangent
               which
               toucheth
               the
               Arches
               in
               the
               same
               point
               ;
               but
               having
               the
               other
               arch
               on
               the
               other
               side
               of
               that
               diameter
               ,
               and
               on
               the
               other
               side
               of
               the
               right
               line
               tangent
               .
               So
               in
               fig.
               12
               th
               .
               the
               two
               Circles
               F
               
               AH
               and
               
               ADK
               touching
               in
               the
               point
               A
               ,
               and
               AK
               being
               their
               common
               diameter
               and
               AB
               being
               a
               right
               line
               tangent
               to
               both
               ,
               the
               concavo-convexe
               angle
               of
               Contact
               DAH
               is
               ultra-diametral
               .
            
             
               15.
               
               That
               by
               a
               mixed
               crooked
               lined
               angle
               ,
               we
               understand
               a
               plane
               angle
               contain'd
               under
               two
               concurring
               crooked
               lines
               ,
               of
               such
               several
               curvatures
               that
               it
               is
               impossible
               for
               them
               to
               be
               coapted
               from
               the
               angular
               point
               the
               one
               unto
               the
               other
               ,
               or
               the
               one
               ,
               or
               its
               production
               to
               be
               coapted
               to
               the
               other
               ,
               or
               its
               production
               ,
               at
               and
               from
               the
               angular
               point
               ,
               by
               any
               circumduction
               whatsoever
               ,
               so
               in
               fig.
               9.10.11
               .
               the
               angle
               
               B
               
               AC
               ,
               being
               a
               mixed
               crooked
               lined
               angle
               ,
               AB
               as
               being
               of
               a
               different
               curvature
               from
               AC
               ,
               can
               no
               way
               possibly
               be
               coapted
               by
               any
               circumduction
               to
               AC
               from
               the
               angular
               point
               A
               :
               and
               if
               B
               A
               be
               produced
               to
               D
               and
               CA
               to
               E
               the
               crooked-line
               AB
               ,
               because
               of
               its
               different
               curvature
               can
               neither
               be
               coapted
               to
               the
               crooked
               line
               AC
               nor
               to
               its
               production
               
               AE
               :
               nor
               the
               crooked
               line
               
               AC
               to
               the
               crooked
               line
               
               AB
               ,
               nor
               to
               its
               production
               
               AD
               ,
               by
               any
               circumduction
               upon
               the
               angular
               point
               whatsoever
               .
            
             
               16.
               
               That
               by
               the
               isoclitical
               sides
               of
               an
               Angle
               ,
               we
               understand
               two
               such
               lines
               ,
               whether
               right
               ,
               or
               crooked
               ,
               containing
               a
               plane
               angle
               ,
               as
               according
               to
               their
               present
               site
               and
               positure
               ,
               without
               inversion
               when
               crooked
               ,
               moving
               by
               the
               sides
               nearer
               one
               to
               another
               ,
               will
               by
               such
               motion
               at
               last
               come
               to
               be
               coincident
               with
               and
               coapted
               exactly
               the
               one
               unto
               the
               other
               :
               and
               that
               by
               an
               isoclitical
               angle
               we
               understand
               a
               plane
               angle
               ,
               contained
               under
               such
               isoclitical
               sides
               .
               So
               every
               right
               lined
               angle
               is
               manifestly
               isoclitical
               ;
               and
               its
               sides
               are
               isoclitical
               ,
               the
               sides
               by
               such
               motion
               on
               the
               angular
               point
               one
               towards
               another
               being
               manifestly
               coincidible
               and
               coaptable
               So
               also
               in
               fig.
               3d.
               if
               in
               the
               concavo-convexe
               crooked
               lined
               angle
               BAC
               .
               the
               crooked-lined
               side
               B
               A
               by
               moving
               upon
               the
               angular
               point
               A
               towards
               the
               other
               crooked
               lined
               side
               CA
               ,
               will
               at
               last
               coapt
               and
               become
               coincident
               with
               the
               crooked
               
               lined
               side
               C
               A
               then
               are
               B
               A
               and
               C
               A
               isoclitical
               sides
               and
               the
               angle
               B
               
               AC
               anisoclitical
               angle
               .
            
             
               17.
               
               That
               by
               the
               anisoclitical
               sides
               of
               an
               angle
               we
               understand
               two
               such
               lines
               ,
               whether
               the
               one
               right
               &
               the
               other
               crooked
               ,
               or
               else
               both
               crooked
               ,
               and
               whether
               both
               of
               the
               same
               or
               different
               curvatures
               ,
               containing
               a
               plane
               angle
               ,
               so
               as
               according
               to
               their
               present
               site
               and
               positure
               ,
               without
               inversion
               of
               either
               when
               crooked
               ,
               by
               moving
               the
               sides
               nearer
               one
               another
               ,
               by
               the
               continuance
               of
               such
               motion
               they
               can
               never
               be
               brought
               to
               be
               coincident
               with
               ,
               and
               coapted
               exactly
               the
               one
               unto
               the
               other
               and
               that
               by
               an
               anisoclitical
               angle
               ,
               we
               understand
               a
               plane
               angle
               contained
               under
               such
               anisoclitical
               sides
               .
               So
               all
               crooked
               lined
               angles
               of
               concaves
               ,
               though
               having
               sides
               of
               like
               curvature
               ,
               all
               crooked
               lined
               angles
               of
               convexes
               ,
               though
               having
               sides
               of
               like
               curvature
               ,
               all
               concavo-convexe
               angles
               ,
               whose
               sides
               are
               of
               different
               curvatures
               ,
               all
               mixed
               lined
               angles
               ,
               whether
               recto-convexes
               ,
               or
               recto-concaves
               ,
               all
               mixed
               crooked
               lined
               angles
               whatsoever
               ,
               are
               all
               of
               them
               manifestly
               anisoclitical
               angles
               ,
               and
               their
               sides
               anisoclitical
               .
               So
               in
               fig.
               4
               th
               .
               though
               AB
               and
               AC
               .
               be
               supposed
               to
               be
               of
               uniform
               answering
               and
               equal
               curvatures
               ,
               and
               likewise
               in
               fig.
               2
               d.
               though
               AB
               and
               AC
               again
               supposed
               to
               be
               of
               uniform
               ,
               answering
               and
               equal
               curvatures
               ,
               and
               in
               fig.
               3
               d.
               fig
               ,
               5
               ,
               6
               ,
               9
               ,
               10
               ,
               11.
               supposing
               
               AB
               and
               
               AC
               not
               to
               be
               of
               
               uniform
               ,
               answering
               and
               equal
               curvatures
               ,
               by
               moving
               the
               sides
               
               AB
               &
               AC
               in
               that
               their
               present
               fire
               &
               positure
               upon
               the
               angular
               point
               A
               ,
               without
               the
               inversion
               of
               either
               ,
               when
               both
               are
               crooked
               ,
               to
               bring
               the
               sides
               nearer
               the
               one
               to
               the
               other
               ,
               it
               is
               manifest
               that
               by
               the
               continuance
               of
               such
               motion
               they
               can
               never
               be
               brought
               to
               a
               coincidence
               with
               ,
               and
               to
               be
               coapted
               exactly
               the
               one
               unto
               the
               other
               ,
               and
               such
               also
               are
               all
               angles
               of
               Contact
               whatsoever
               ,
               as
               in
               fig.
               7
               ,
               8
               ,
               11.
               
               
               AB
               &
               
               AC
               are
               as
               uncoincidable
               and
               uncoaptable
               as
               in
               the
               former
               ,
               except
               only
               ultradiametral
               ,
               concavo-convexe
               angles
               of
               Contact
               ,
               which
               are
               of
               equal
               and
               answering
               curvatures
               .
            
             
               18.
               
               That
               by
               an
               Angle
               of
               curvature
               ,
               or
               coincidence
               we
               understand
               a
               plane
               angle
               ,
               contained
               by
               any
               two
               parts
               of
               a
               crooked
               line
               at
               the
               point
               of
               their
               concurrences
               any
               where
               to
               be
               imagined
               ,
               or
               taken
               in
               crooked-lines
               .
               So
               in
               the
               7
               th
               .
               fig.
               let
               BAD
               be
               the
               circumference
               of
               a
               Circle
               ,
               Parabola
               ,
               Hyperbola
               ,
               Ellipsis
               &c.
               
               At
               the
               mean
               point
               A
               ,
               the
               sides
               BA
               and
               
               AD
               contain
               an
               Angle
               of
               Curvature
               ,
               or
               coincidence
               .
            
             
               19.
               
               That
               by
               autoclitical
               curvature
               ,
               and
               so
               by
               an
               autoclitical
               crooked
               line
               ,
               we
               understand
               such
               a
               crooked
               line
               ,
               as
               passing
               from
               the
               angular
               point
               of
               a
               right
               lined
               angle
               ,
               between
               the
               two
               sides
               ,
               by
               the
               inclination
               of
               its
               curvature
               ,
               
               keepeth
               the
               convexe
               side
               of
               its
               curvature
               ,
               constantly
               obverted
               to
               one
               of
               the
               sides
               ,
               and
               the
               concave
               side
               of
               its
               curvature
               constantly
               obverted
               to
               the
               other
               .
               As
               in
               fig.
               17.
               in
               the
               right
               lined
               angle
               BAC
               ,
               the
               crooked
               line
               
               AEF
               keeping
               its
               convexity
               constantly
               obverted
               to
               the
               side
               AB
               ,
               and
               its
               concavity
               to
               the
               side
               AC
               ,
               or
               so
               much
               of
               it
               as
               is
               intercepted
               between
               the
               intersection
               at
               F
               and
               the
               angular
               point
               at
               A
               the
               crooked
               line
               
               AEF
               is
               autoclitical
               ,
               
                 i.
                 e.
              
               the
               convexity
               is
               all
               on
               one
               side
               ,
               and
               the
               concavity
               all
               on
               the
               other
               .
            
             
               20.
               
               That
               by
               antanaclitical
               curvature
               and
               so
               by
               an
               antanaclitical
               curve
               line
               ,
               we
               understand
               such
               a
               crooked
               line
               ,
               as
               passing
               from
               the
               angular
               point
               of
               a
               right-lined
               angle
               ,
               between
               the
               two
               sides
               ,
               by
               the
               inclination
               of
               its
               curvature
               ,
               hath
               the
               convexe
               part
               of
               its
               curvature
               ,
               sometimes
               towards
               the
               one
               side
               of
               the
               right
               lined
               angle
               ,
               and
               sometimes
               towards
               the
               other
               side
               of
               the
               right
               lined
               angle
               ,
               
                 i.
                 e.
              
               the
               convexeness
               and
               the
               concaveness
               are
               not
               constantly
               on
               the
               same
               several
               sides
               .
               As
               in
               fig.
               17.
               in
               the
               right
               lined
               angle
               BAC
               ,
               the
               crooked
               line
               AGH
               obverting
               the
               convexity
               at
               G
               towards
               AB
               ,
               and
               at
               H
               towards
               AC
               is
               antanaclitical
               .
            
             
               These
               definitions
               praemised
               ,
               to
               give
               now
               the
               true
               State
               of
               the
               controversy
               ;
               let
               there
               be
               ,
               as
               in
               fig.
               12.
               two
               equall
               Circles
               AHH
               and
               ADK
               
               touching
               in
               the
               point
               A
               :
               and
               let
               AG
               be
               the
               Semidiameter
               of
               the
               circle
               ADK
               :
               and
               AB
               be
               a
               right
               line
               tangent
               touching
               both
               the
               Circles
               in
               the
               point
               A
               :
               and
               let
               
               AEL
               be
               a
               greater
               circle
               then
               either
               ,
               touching
               both
               the
               former
               ,
               and
               also
               the
               right
               line
               tangent
               in
               the
               point
               A
               :
               and
               from
               the
               point
               A
               draw
               the
               right
               line
               AC
               at
               pleasure
               ,
               cutting
               the
               circle
               
               ADK
               in
               the
               point
               D
               ,
               and
               the
               circle
               AEL
               in
               the
               point
               E
               :
               now
               therefore
               whereas
               you
               say
               that
               the
               Recto-convexe
               angles
               of
               Contact
               BAE
               and
               B
               
               AF
               are
               not
               unequal
               ,
               and
               that
               neither
               of
               them
               is
               quantitative
               ;
               and
               that
               the
               crooked-lines
               E
               A
               ,
               DA
               ,
               HA
               are
               coincident
               ,
               Sc.
               so
               as
               to
               make
               no
               Angle
               with
               the
               right
               line
               
               AB
               ,
               or
               one
               with
               another
               ;
               and
               that
               the
               right-lined
               right
               angle
               B
               
               AG
               is
               equal
               to
               the
               mixt-lined
               ,
               
                 i.
                 e.
              
               Recto-concave
               angles
               of
               the
               Semicircles
               E
               
               AG
               and
               D
               
               AG
               severally
               ,
               and
               that
               those
               angles
               of
               the
               Semicircles
               E
               
               AG
               and
               D
               
               AG
               are
               equal
               the
               one
               unto
               the
               other
               ;
               and
               that
               the
               mixed
               lined
               Recto-convexe
               angle
               F
               
               AG
               is
               severally
               equal
               to
               all
               ,
               or
               any
               of
               the
               former
               ;
               and
               that
               there
               is
               no
               heterogeneity
               amongst
               plane
               angles
               ,
               but
               that
               they
               are
               all
               of
               them
               of
               the
               same
               sort
               ,
               and
               homogeneal
               and
               undevideable
               into
               parts
               specifically
               different
               ,
               distinct
               and
               heterogeneal
               in
               respect
               of
               one
               another
               ,
               and
               the
               whole
               ;
               and
               that
               to
               any
               mixt
               ,
               or
               crooked
               lined
               angle
               whatsoever
               that
               is
               quantitative
               it
               is
               not
               impossible
               to
               give
               
               an
               equal
               right
               lined
               angle
               :
               I
               acknowledge
               for
               all
               these
               things
               you
               have
               disputed
               very
               subtilly
               ;
               yet
               I
               must
               with
               a
               clear
               and
               free
               judgment
               own
               and
               declare
               a
               dissent
               from
               you
               in
               them
               all
               ,
               for
               the
               reasons
               to
               be
               alleadged
               in
               the
               insuing
               Discourse
               .
            
             
               To
               clear
               all
               which
               ,
               nothing
               can
               be
               of
               higher
               consequence
               in
               this
               Question
               ,
               then
               truly
               to
               understand
               the
               nature
               of
               an
               Angle
               ,
               what
               an
               Angle
               is
               ,
               what
               is
               an
               Angle
               ,
               and
               what
               is
               not
               .
               And
               to
               exclude
               the
               consideration
               of
               Angles
               herein
               unconcerned
               ;
               they
               are
               plane
               angles
               
                 i.
                 e.
              
               such
               as
               are
               contained
               by
               lines
               ,
               which
               lye
               wholly
               ,
               and
               both
               in
               the
               same
               plane
               ,
               the
               disquisition
               of
               whose
               nature
               we
               are
               now
               about
               .
               And
               such
               is
               the
               affinity
               ,
               which
               the
               inclination
               of
               one
               line
               hath
               to
               another
               in
               the
               same
               plane
               ,
               with
               the
               nature
               of
               an
               Angle
               ,
               that
               without
               it
               a
               plane
               angle
               cannot
               be
               defined
               ,
               or
               conceived
               .
               For
               where
               there
               is
               rectitude
               and
               voidness
               of
               inclination
               ,
               as
               in
               a
               right
               line
               and
               its
               production
               ,
               there
               never
               was
               justly
               suspected
               to
               be
               any
               thing
               latent
               of
               an
               angular
               nature
               ;
               nor
               between
               parallels
               ,
               for
               want
               of
               mutual
               inclination
               .
               But
               yet
               the
               inclination
               of
               line
               upon
               line
               in
               the
               same
               plane
               ,
               is
               not
               sufficient
               to
               make
               up
               the
               nature
               of
               an
               Angle
               :
               for
               in
               the
               same
               plane
               one
               line
               may
               have
               inclination
               to
               another
               ,
               and
               yet
               they
               never
               meet
               ,
               nor
               have
               in
               their
               infinite
               regular
               production
               any
               possibility
               of
               ever
               meeting
               ;
               as
               
               the
               circumference
               of
               an
               Ellipsis
               ,
               or
               circle
               to
               a
               right
               line
               lying
               wholly
               without
               them
               ,
               without
               either
               Section
               ,
               or
               Contact
               ;
               or
               the
               Asymptotes
               in
               conjugate
               Sections
               ,
               which
               though
               ever
               making
               a
               closer
               appropinquation
               to
               the
               circumferences
               of
               the
               conjugate
               figures
               ,
               yet
               infinitely
               produced
               ,
               never
               attain
               a
               concurrency
               .
               And
               though
               by
               possibility
               the
               inclined
               lines
               might
               meet
               ,
               yet
               if
               they
               do
               not
               ,
               an
               Angle
               is
               not
               constituted
               :
               only
               there
               is
               possibility
               of
               an
               Angle
               when
               being
               produced
               they
               shall
               meet
               .
               So
               as
               to
               make
               up
               the
               nature
               of
               an
               Angle
               two
               lines
               must
               be
               inclined
               one
               to
               another
               ,
               and
               also
               concurre
               ,
               or
               meet
               to
               contain
               on
               their
               parts
               a
               certain
               space
               ,
               or
               part
               of
               the
               plane
               between
               their
               productions
               from
               their
               point
               of
               concurrence
               ,
               or
               their
               angular
               point
               .
               And
               this
               being
               the
               general
               ,
               and
               proper
               nature
               of
               a
               plane
               angle
               ;
               as
               it
               is
               manifest
               ,
               that
               in
               a
               right
               line
               there
               is
               no
               Angle
               ,
               so
               it
               is
               dubious
               what
               is
               to
               be
               judged
               of
               those
               curve
               ,
               or
               crooked
               lines
               ,
               whose
               curvature
               is
               either
               equall
               ,
               uniform
               ,
               or
               regular
               :
               Sc.
               whether
               in
               such
               lines
               there
               be
               not
               at
               every
               mean
               point
               an
               angularness
               ,
               the
               lines
               still
               lying
               in
               the
               same
               positure
               .
               It
               is
               clear
               in
               a
               right
               line
               no
               mean
               point
               can
               be
               taken
               ,
               at
               which
               the
               parts
               of
               the
               right
               line
               in
               their
               present
               position
               can
               be
               said
               to
               have
               any
               inclination
               one
               to
               another
               :
               that
               though
               the
               parts
               concurre
               ,
               yet
               they
               want
               inclination
               :
               but
               in
               the
               circumferences
               of
               circles
               ,
               
               Ellipses
               ,
               Hyperbola's
               ,
               Parabola's
               and
               such
               like
               curve
               ,
               or
               crooked
               lines
               ,
               being
               of
               equal
               uniform
               ,
               or
               regular
               curvature
               ,
               no
               point
               can
               be
               assigned
               at
               which
               the
               parts
               in
               their
               present
               position
               have
               not
               a
               special
               inclination
               one
               to
               another
               .
               So
               as
               ,
               as
               before
               ,
               inclination
               of
               lines
               and
               concurrency
               making
               up
               the
               nature
               of
               an
               angle
               ;
               it
               seems
               not
               reasonable
               to
               deny
               angularity
               at
               any
               point
               of
               such
               crooked
               lines
               ,
               however
               their
               curvature
               be
               either
               uniform
               ,
               equal
               ,
               or
               regular
               .
               But
               I
               know
               it
               will
               be
               said
               that
               such
               crooked
               lines
               of
               equal
               ,
               uniform
               &
               regular
               curvature
               are
               but
               one
               line
               ,
               and
               therefore
               ,
               by
               the
               definition
               of
               an
               angle
               ,
               cannot
               contain
               an
               angle
               ;
               which
               requires
               two
               lines
               to
               its
               constitution
               .
               To
               which
               I
               answer
               ,
               that
               in
               like
               manner
               ,
               two
               concurring
               right
               lines
               may
               be
               taken
               for
               one
               line
               continued
               though
               not
               in
               its
               rectitude
               ,
               and
               then
               the
               consideration
               of
               angularity
               between
               them
               is
               excluded
               :
               however
               by
               reason
               of
               the
               inflexion
               and
               inclination
               at
               the
               point
               of
               incurvation
               ,
               there
               is
               an
               aptitude
               in
               that
               one
               produced
               line
               ,
               there
               to
               fall
               ,
               and
               distinguish
               it self
               into
               two
               ,
               with
               inclination
               of
               one
               unto
               the
               other
               ,
               and
               so
               to
               offer
               the
               constitutive
               nature
               of
               an
               angle
               :
               and
               so
               it
               is
               at
               every
               point
               of
               such
               crooked
               lines
               ,
               whose
               curvature
               is
               equal
               ,
               or
               uniform
               and
               regular
               ;
               being
               one
               ,
               or
               more
               lines
               ,
               according
               as
               by
               our
               conceptions
               they
               are
               continued
               ,
               or
               distinguished
               
               And
               as
               a
               continued
               rectitude
               ,
               such
               as
               is
               in
               right
               lines
               ,
               is
               most
               inconsistent
               with
               the
               nature
               of
               an
               angle
               ,
               so
               what
               should
               be
               judged
               more
               accepting
               of
               the
               nature
               of
               angularity
               ,
               then
               curvature
               is
               ,
               being
               thereunto
               contrariously
               opposite
               ,
               and
               by
               all
               confest
               most
               what
               to
               be
               so
               ,
               saving
               in
               the
               aforesaid
               cases
               when
               curvature
               is
               equal
               ,
               uniform
               ,
               or
               regular
               .
               But
               why
               should
               equality
               ,
               uniformity
               ,
               or
               regularity
               of
               curvatures
               ,
               be
               so
               urged
               in
               the
               concern
               of
               angularity
               ,
               by
               none
               of
               which
               is
               angularity
               either
               promoted
               ,
               or
               hindered
               ?
               For
               there
               cannot
               be
               greater
               equality
               ,
               uniformity
               ,
               or
               regularity
               ,
               then
               is
               to
               be
               found
               in
               Rectitude
               as
               well
               as
               angles
               ,
               in
               right
               lines
               as
               well
               as
               Circles
               ;
               the
               things
               constitutive
               of
               the
               nature
               of
               an
               angle
               ,
               being
               things
               quite
               different
               from
               them
               ,
               Viz.
               inflexion
               or
               inclination
               and
               concurrence
               ;
               which
               are
               indifferently
               found
               in
               all
               curvatures
               equal
               and
               unequal
               ,
               uniform
               ,
               or
               not
               uniform
               ,
               regular
               ,
               or
               irregular
               :
               and
               besides
               are
               inseparable
               from
               them
               :
               curvature
               and
               a
               concurrent
               inflexion
               ,
               or
               angularity
               being
               but
               as
               two
               notions
               of
               the
               same
               thing
               ,
               considered
               as
               under
               several
               respects
               :
               Viz.
               curvature
               is
               the
               affection
               of
               a
               line
               considered
               as
               one
               :
               the
               same
               being
               angularity
               and
               inclination
               ,
               when
               from
               any
               point
               of
               the
               curvature
               the
               same
               line
               is
               considered
               as
               two
               :
               so
               two
               concurrent
               right
               lines
               are
               said
               to
               be
               one
               crooked
               line
               :
               but
               
               when
               a
               right-lined
               angle
               is
               said
               to
               be
               contained
               by
               them
               ,
               they
               are
               then
               considered
               as
               two
               from
               the
               point
               of
               their
               incurvation
               ,
               or
               inclination
               :
               and
               equality
               ,
               and
               uniformity
               ,
               and
               regularity
               of
               curvature
               implying
               equality
               ,
               uniformity
               and
               regularity
               of
               inflexion
               ,
               or
               inclination
               ,
               it
               is
               so
               far
               from
               concluding
               against
               angularity
               ,
               that
               it
               inferrs
               it
               with
               an
               additament
               ,
               Viz.
               of
               equal
               angularity
               ,
               uniform
               angularity
               ,
               and
               regular
               angularity
               ;
               as
               might
               be
               at
               large
               declared
               in
               the
               special
               properties
               of
               several
               crooked
               lined
               figures
               .
               And
               in
               several
               uniform
               ,
               and
               regular
               crooked
               lined
               figures
               ,
               there
               are
               some
               special
               points
               ,
               offering
               even
               to
               view
               and
               sense
               a
               clear
               specimen
               of
               a
               more
               then
               ordinary
               angularity
               ,
               without
               any
               such
               loud
               calling
               for
               the
               strong
               operations
               of
               the
               mind
               ;
               as
               the
               vertical
               points
               in
               conjugate
               figures
               ,
               in
               parabola's
               ,
               and
               the
               extream
               points
               of
               either
               axes
               ,
               in
               ellipses
               ,
               and
               the
               like
               ,
               And
               in
               the
               definition
               of
               a
               plane
               angle
               ,
               all
               the
               inclination
               ,
               which
               is
               required
               in
               the
               concurring
               sides
               ,
               is
               only
               ,
               that
               their
               inclination
               be
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               
                 i.
                 e.
              
               so
               as
               the
               sides
               make
               not
               one
               and
               the
               same
               right
               line
               ,
               as
               it
               is
               generally
               understood
               :
               however
               so
               far
               are
               we
               from
               imposing
               upon
               any
               against
               their
               judgements
               ,
               this
               special
               sort
               of
               angles
               ,
               that
               we
               readily
               acknowledge
               ,
               if
               the
               words
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               in
               the
               definition
               of
               a
               plane
               angle
               ,
               be
               forced
               from
               the
               commonly
               received
               
               sense
               ,
               of
               the
               sides
               not
               lying
               in
               the
               same
               right-line
               ,
               to
               signify
               the
               non-coincidency
               of
               the
               production
               of
               the
               one
               side
               with
               the
               other
               ,
               then
               according
               to
               that
               gloss
               upon
               the
               definition
               of
               plane
               angles
               ,
               this
               whole
               sort
               of
               angles
               is
               to
               be
               rejected
               ;
               wherein
               every
               one
               is
               freely
               left
               to
               his
               own
               judgement
               ;
               the
               difference
               being
               a
               question
               and
               quarrel
               about
               words
               ,
               more
               then
               matter
               ,
               and
               not
               concerning
               the
               present
               controversy
               .
            
             
               However
               from
               the
               whole
               it
               out
               of
               Controversy
               appears
               ,
               that
               recto-concave
               angles
               of
               Contact
               are
               true
               angles
               ,
               contained
               under
               two
               inclined
               sides
               ,
               concurring
               :
               also
               that
               ultradiametral
               concavo-convexe
               angles
               of
               Contact
               ,
               whether
               less
               equal
               ,
               or
               greater
               then
               two
               right
               right-lined
               angles
               are
               true
               angles
               ,
               for
               the
               sides
               are
               inclined
               and
               concurr
               :
               of
               their
               concurrence
               can
               be
               no
               doubt
               :
               and
               that
               they
               are
               inclined
               must
               of
               necessity
               be
               yielded
               ,
               seeing
               they
               neither
               lye
               in
               rectitude
               ,
               nor
               which
               to
               some
               might
               be
               a
               causeless
               scruple
               ,
               the
               one
               in
               the
               production
               of
               the
               other
               :
               so
               as
               their
               angularity
               is
               clear
               beyond
               all
               doubt
               :
               the
               inclination
               understood
               in
               the
               definition
               of
               an
               angle
               being
               generally
               any
               positure
               of
               one
               line
               with
               regard
               to
               another
               in
               the
               same
               plane
               ,
               so
               as
               both
               neither
               fall
               in
               one
               right
               line
               ,
               nor
               be
               situated
               parallel
               one
               to
               another
               ,
               nor
               under
               such
               a
               manner
               of
               oblique
               extension
               ,
               as
               may
               
               render
               their
               concurrence
               impossible
               :
               it
               is
               not
               only
               the
               oblique
               positure
               of
               one
               line
               in
               reference
               to
               another
               ,
               as
               contradistinguished
               from
               perpendicularness
               in
               acute
               and
               obsuse
               inclinations
               :
               but
               such
               is
               the
               comprehensiveness
               of
               its
               sense
               ,
               in
               the
               present
               acceptation
               ,
               that
               every
               perpendicularness
               it self
               is
               taken
               for
               an
               inclination
               .
            
             
               So
               likewise
               it
               cannot
               reasonably
               be
               denyed
               but
               those
               special
               angles
               of
               Contact
               ,
               which
               are
               the
               chief
               subject
               of
               this
               present
               controversy
               ,
               I
               mean
               recto-convexe
               angles
               of
               Contact
               ,
               are
               truly
               angles
               ;
               except
               either
               the
               inclination
               of
               their
               sides
               ,
               or
               their
               concurrence
               could
               be
               called
               in
               question
               ;
               nothing
               else
               being
               requisite
               unto
               the
               nature
               ,
               or
               comprized
               in
               the
               definition
               of
               an
               angle
               :
               and
               the
               like
               is
               to
               be
               judged
               of
               all
               other
               concavo-convexe
               and
               convexo-convexe
               angles
               of
               Contact
               .
               But
               you
               said
               that
               in
               the
               recto-convexe
               angle
               of
               Contact
               the
               right-line
               tangent
               and
               circumference
               make
               no
               angle
               ,
               because
               the
               tangent
               is
               not
               inclined
               to
               the
               circumference
               ,
               but
               coincident
               with
               it
               .
               What
               mystery
               of
               reason
               ,
               or
               force
               of
               Argument
               should
               be
               in
               this
               deduction
               ?
               if
               you
               say
               it
               is
               coincidence
               ,
               you
               mean
               by
               non-inclination
               ;
               I
               readily
               yield
               ,
               where
               two
               lines
               ,
               become
               coincident
               ,
               their
               former
               angle
               is
               thereby
               extinct
               ;
               as
               thereby
               they
               come
               under
               the
               consideration
               but
               of
               one
               line
               :
               as
               when
               two
               right-lines
               ,
               or
               
               two
               isoclitical
               crooked
               lines
               ,
               are
               ,
               one
               of
               each
               of
               them
               ,
               so
               moved
               about
               the
               angular
               point
               ,
               till
               the
               two
               lines
               become
               one
               .
               But
               where
               is
               any
               such
               coincidency
               between
               right-line
               tangents
               and
               circumferences
               ?
               or
               what
               possibility
               is
               there
               of
               any
               such
               coincidency
               :
               a
               crooked
               line
               and
               a
               right-line
               may
               no
               doubt
               be
               commensurate
               ,
               or
               proportionable
               in
               length
               ,
               but
               in
               position
               it
               is
               impossible
               :
               and
               if
               we
               imagine
               the
               tangent
               bowed
               to
               such
               a
               coincidence
               ,
               then
               it
               is
               not
               any
               longer
               a
               tangent
               ,
               or
               right-line
               ,
               but
               a
               circumference
               .
               And
               though
               as
               you
               urge
               ,
               a
               right-line
               circumduced
               about
               any
               middle
               point
               in
               the
               side
               of
               a
               regular
               polygone
               ,
               at
               last
               becomes
               coincident
               with
               the
               side
               ,
               and
               looses
               all
               inclination
               and
               angularity
               with
               the
               side
               ;
               what
               doth
               that
               concern
               ,
               or
               how
               doth
               that
               prove
               the
               non-inclination
               of
               the
               tangent
               to
               the
               circumference
               ;
               your selves
               sometimes
               in
               every
               point
               ,
               save
               only
               in
               the
               point
               of
               Contact
               acknowledging
               an
               inclination
               ;
               and
               as
               is
               else
               where
               hinted
               ,
               in
               the
               point
               of
               Contact
               alone
               ,
               and
               abstractedly
               ,
               the
               inclination
               it self
               ,
               which
               is
               the
               habitude
               of
               the
               concurring
               inclined
               sides
               ,
               is
               not
               to
               be
               sought
               ,
               but
               only
               the
               particular
               termination
               of
               their
               inclination
               there
               .
               But
               you
               will
               say
               ,
               it
               is
               non-secancy
               which
               is
               meant
               by
               this
               coincidency
               :
               and
               what
               I
               pray
               is
               that
               more
               then
               the
               Lateran
               bells
               to
               the
               concern
               ,
               or
               constitution
               
               of
               angles
               ?
               Are
               there
               not
               many
               regular
               curve
               lines
               produced
               ,
               some
               infinitely
               without
               Section
               ,
               in
               which
               especially
               the
               circumferences
               of
               circles
               ,
               you
               are
               pleased
               sometimes
               every
               where
               to
               think
               you
               have
               cause
               to
               imagine
               an
               angle
               .
               It
               were
               meet
               to
               know
               the
               meaning
               of
               such
               odly
               connext
               terms
               ,
               before
               reasonings
               upon
               them
               be
               regarded
               in
               questions
               of
               weight
               .
            
             
               That
               the
               Contact
               of
               the
               angular
               sides
               ,
               is
               as
               different
               from
               coincidency
               ,
               as
               from
               secancy
               ,
               is
               most
               unquestionably
               apparent
               in
               angles
               of
               such
               sides
               ,
               as
               are
               capable
               of
               all
               three
               ,
               Viz.
               Contact
               ,
               secancy
               ,
               and
               coincidence
               .
               For
               example
               in
               fig.
               16.
               let
               there
               be
               two
               Arches
               of
               equal
               Circles
               DAF
               and
               BAC
               .
               touching
               one
               another
               in
               the
               point
               A.
               If
               the
               arch
               DAF
               be
               circumduced
               about
               the
               point
               A
               as
               an
               unmoveable
               center
               ;
               at
               after
               an
               infinite
               succession
               of
               secancies
               ,
               at
               last
               all
               will
               terminate
               in
               a
               manifest
               coincidency
               ,
               and
               the
               arch
               AF
               be
               coincident
               with
               the
               arch
               
               AB
               ,
               and
               the
               arch
               AD
               be
               coincident
               with
               the
               arch
               AC
               .
               So
               as
               secancy
               ,
               Contact
               ,
               and
               coincidency
               are
               distinguishable
               one
               from
               another
               with
               as
               much
               ease
               and
               cleareness
               ,
               as
               an
               odde
               number
               from
               an
               even
               .
               But
               if
               it
               be
               urged
               ,
               that
               you
               assert
               not
               a
               coincidency
               between
               the
               arch
               .
               D
               
               AF
               and
               the
               arch
               BAC
               ,
               but
               only
               that
               GH
               being
               a
               right
               line
               and
               touching
               the
               arch
               B
               
               AC
               in
               the
               point
               A
               ,
               that
               I
               say
               the
               right
               line
               
               GH
               and
               the
               arch
               BAC
               are
               coincident
               .
               The
               vanity
               of
               this
               may
               be
               evinced
               ;
               in
               that
               by
               the
               same
               reason
               it
               followes
               ,
               that
               the
               right
               line
               GH
               must
               be
               coincident
               with
               the
               arch
               DAF
               ,
               and
               so
               the
               arch
               DAF
               coincident
               with
               the
               arch
               B
               
               AC
               ;
               the
               contrary
               of
               which
               is
               above-shewn
               and
               confessed
               :
               and
               besides
               hereupon
               should
               two
               arches
               B
               
               AC
               and
               DAF
               being
               convexo-convexely
               posited
               ,
               and
               the
               right
               line
               tangent
               GH
               be
               all
               coincident
               ;
               which
               I
               leave
               for
               others
               to
               say
               rather
               then
               my self
               .
            
             
               When
               a
               right
               line
               tangent
               and
               many
               crooked
               lines
               of
               different
               curvatures
               ,
               all
               touch
               together
               in
               one
               and
               the
               same
               point
               ,
               as
               in
               fig.
               13
               ;
               you
               say
               ,
               though
               without
               the
               angular
               point
               of
               Contact
               the
               sides
               are
               variously
               divaricated
               one
               from
               another
               ,
               yet
               in
               the
               point
               of
               Contact
               ,
               they
               have
               not
               several
               inclinations
               ,
               for
               you
               say
               they
               have
               no
               inclination
               at
               all
               .
               The
               truth
               is
               the
               angular
               point
               of
               its
               self
               is
               not
               capable
               of
               inclinations
               ,
               nor
               for
               the
               indivisibility
               of
               its
               nature
               can
               by
               any
               possibility
               comprehend
               them
               :
               yet
               that
               lines
               concurring
               in
               one
               only
               point
               ,
               and
               presently
               after
               receding
               each
               from
               the
               other
               ,
               should
               not
               be
               inclined
               each
               to
               other
               in
               or
               at
               the
               point
               of
               their
               concurrency
               ,
               whether
               it
               be
               by
               Contact
               ,
               or
               Section
               ,
               though
               in
               the
               case
               of
               Contacts
               the
               inclination
               be
               less
               ,
               then
               can
               be
               expressed
               by
               the
               inclination
               of
               any
               right-line
               upon
               a
               right-line
               ,
               is
               absolutely
               unconceiveable
               ,
               there
               being
               
               no
               lineary
               coincidence
               ,
               but
               only
               of
               one
               point
               between
               them
               .
               For
               as
               else
               where
               ,
               in
               the
               same
               plane
               ,
               neither
               of
               point
               to
               point
               ,
               nor
               of
               point
               to
               line
               ,
               nor
               of
               point
               to
               plane
               can
               be
               any
               inclination
               :
               but
               in
               the
               present
               case
               of
               plane
               angles
               ,
               inclination
               must
               be
               of
               lines
               ,
               and
               may
               be
               of
               them
               in
               the
               very
               point
               of
               concurrence
               ;
               or
               else
               from
               the
               point
               of
               concurrence
               they
               would
               not
               part
               several
               wayes
               :
               for
               it
               is
               their
               diverse
               inclination
               at
               that
               point
               ,
               which
               makes
               their
               departure
               one
               from
               another
               when
               they
               depart
               from
               thence
               .
               And
               as
               even
               in
               the
               angular
               point
               of
               right-lined
               angles
               the
               lines
               have
               the
               same
               inclinations
               as
               else
               where
               ,
               so
               in
               all
               other
               angles
               ,
               save
               only
               such
               angles
               of
               Contact
               ,
               as
               are
               less
               then
               the
               least
               right-lined
               angles
               ,
               by
               drawing
               right-lined
               tangents
               to
               the
               arch
               ,
               or
               arches
               at
               the
               angular
               point
               ,
               is
               shewn
               in
               right-lines
               ,
               either
               the
               very
               inclination
               of
               the
               sides
               ,
               in
               ,
               or
               rather
               at
               the
               angular
               point
               ,
               or
               else
               the
               least
               right-lined
               inclination
               ,
               which
               is
               greater
               ,
               or
               the
               greatest
               right-lined
               inclination
               ,
               which
               is
               less
               :
               for
               though
               they
               may
               differ
               much
               in
               their
               distance
               and
               divarication
               one
               from
               another
               ,
               without
               the
               point
               of
               their
               concurrence
               ;
               in
               the
               point
               of
               their
               concurrence
               ,
               without
               much
               absurdity
               ,
               they
               may
               be
               said
               to
               be
               equally
               distant
               ,
               
                 i.
                 e.
              
               not
               at
               all
               distant
               there
               one
               from
               another
               ,
               for
               there
               they
               are
               not
               indeed
               at
               all
               distant
               any
               of
               them
               from
               the
               rest
               :
               yet
               it
               doth
               
               not
               hence
               any
               way
               follow
               ,
               that
               in
               like
               manner
               it
               may
               be
               said
               of
               them
               without
               absurdity
               ,
               that
               in
               the
               point
               of
               concurrence
               they
               have
               equal
               inclinations
               ,
               
                 i.
                 e.
              
               no
               inclinations
               one
               to
               another
               :
               for
               though
               in
               the
               point
               of
               concurrence
               it
               is
               truth
               ,
               they
               have
               no
               distance
               ,
               yet
               it
               no
               wayes
               as
               may
               appear
               to
               those
               that
               will
               consider
               equally
               follows
               ,
               that
               they
               should
               there
               have
               no
               inclination
               .
            
             
               Besides
               ,
               that
               the
               urging
               of
               the
               coincidency
               of
               the
               sides
               in
               recto-convexe
               angles
               of
               Contact
               is
               most
               directly
               opposite
               to
               the
               nature
               and
               properties
               of
               the
               special
               lines
               under
               which
               such
               angles
               are
               contained
               .
               For
               ,
               t
               is
               the
               special
               propertie
               of
               some
               lines
               that
               they
               can
               touch
               ,
               but
               they
               can
               no
               way
               be-coincident
               ;
               as
               Arches
               of
               unequal
               and
               unlike
               curvature
               :
               and
               a
               right-line
               and
               a
               crooked
               line
               :
               some
               can
               be
               coincident
               ,
               and
               can
               no
               way
               touch
               ,
               as
               right
               lines
               :
               some
               can
               both
               ;
               as
               Arches
               of
               equal
               and
               answering
               curvature
               ,
               which
               set
               concavo-convexely
               ,
               and
               citradiametrally
               cannot
               touch
               ,
               but
               will
               fall
               into
               a
               coincidence
               :
               but
               posited
               convexo-convexely
               ,
               cannot
               be
               coincident
               ,
               but
               may
               construct
               an
               angle
               of
               Contact
               .
            
             
               So
               as
               the
               chimaera
               of
               the
               coincidency
               of
               the
               sides
               in
               recto-convexe
               angles
               of
               Contact
               ,
               if
               persisted
               in
               ,
               is
               worth
               laughing
               at
               ,
               and
               like
               his
               Philosophy
               who
               when
               every
               one
               was
               at
               his
               high
               Lavolta's
               ,
               denyed
               the
               possibility
               of
               
               motion
               in
               the
               world
               :
               But
               to
               justify
               the
               non-inclination
               of
               the
               sides
               against
               the
               eye
               and
               reason
               ,
               this
               horridly
               distorted
               Monster
               of
               their
               coincidency
               was
               introduced
               .
               Indeed
               if
               they
               be
               coincident
               they
               make
               no
               angle
               :
               But
               it
               will
               cramp
               the
               understanding
               of
               an
               Oedipus
               to
               declare
               how
               either
               a
               right-line
               ,
               or
               a
               crooked-line
               ,
               touching
               another
               crooked-line
               in
               one
               only
               point
               and
               no
               more
               ,
               should
               ever
               be
               conceived
               ,
               notwithstanding
               to
               be
               coincident
               with
               the
               production
               of
               the
               other
               crooked-line
               ,
               whether
               the
               tangencie
               of
               the
               crooked-lines
               be
               concavo-convexe
               ,
               or
               convexo-convexe
               ,
               
                 i.
                 e.
              
               the
               one
               within
               the
               other
               ,
               or
               else
               the
               one
               turned
               away
               from
               the
               other
               .
               But
               you
               'll
               say
               ,
               you
               assert
               coincidency
               only
               in
               the
               point
               of
               Contact
               .
               I
               answer
               ,
               that
               's
               frivolous
               ,
               not
               to
               say
               ridiculous
               and
               impertinent
               ;
               for
               coincidency
               in
               the
               present
               question
               of
               angles
               is
               taken
               as
               opposed
               to
               inclination
               ,
               which
               is
               an
               affection
               and
               propriety
               of
               the
               concurring
               sides
               of
               the
               angle
               ,
               not
               only
               of
               the
               angular
               point
               taken
               by
               it self
               abstractedly
               .
               As
               inclination
               cannot
               be
               in
               a
               point
               ,
               though
               it
               may
               be
               at
               a
               point
               ,
               so
               a
               point
               cannot
               be
               said
               to
               be
               inclined
               unto
               a
               line
               ,
               especially
               it self
               being
               in
               the
               same
               line
               :
               it
               may
               be
               said
               to
               hold
               such
               and
               such
               a
               distance
               from
               the
               line
               ,
               when
               it
               is
               without
               it
               ,
               but
               not
               to
               be
               inclined
               unto
               it
               .
               And
               if
               the
               being
               of
               the
               point
               
               of
               Contact
               ,
               generally
               as
               a
               point
               of
               the
               one
               side
               ,
               in
               the
               tangent
               line
               ,
               as
               in
               the
               other
               side
               ,
               whether
               right
               ,
               or
               crooked
               ,
               make
               a
               coincidency
               destroying
               inclination
               ;
               then
               all
               inclinations
               and
               angles
               whatsoever
               are
               destroy'd
               ;
               and
               every
               where
               will
               be
               a
               coincidency
               ,
               for
               that
               is
               common
               to
               every
               angle
               to
               have
               the
               angular
               point
               still
               common
               to
               both
               sides
               :
               and
               the
               secant
               angles
               might
               as
               well
               be
               said
               coincident
               ,
               as
               the
               tangent
               angles
               ,
               For
               what
               you
               say
               ,
               that
               it
               is
               tangency
               as
               opposed
               to
               secancy
               ,
               that
               you
               mean
               by
               coincidency
               ;
               I
               answer
               the
               glosse
               is
               improper
               and
               besides
               the
               anvil
               and
               tangency
               undenied
               ,
               but
               in
               this
               case
               impertinently
               by
               you
               alleadged
               ,
               till
               it
               be
               proved
               that
               tangency
               in
               one
               only
               point
               ,
               and
               no
               more
               ,
               doth
               quite
               annul
               and
               destroy
               the
               inclination
               of
               the
               lines
               ,
               though
               on
               both
               sides
               of
               the
               Contact
               clearly
               receding
               the
               one
               from
               the
               other
               :
               till
               which
               be
               done
               happiness
               to
               my
               friend
               and
               no
               longer
               .
            
             
               I
               might
               adde
               that
               it
               is
               Touching
               which
               is
               only
               mentioned
               in
               the
               definition
               of
               plane
               angles
               ;
               but
               I
               shall
               dispatch
               all
               by
               setting
               the
               case
               before
               you
               in
               this
               Diagramme
               in
               fig.
               12.
               if
               the
               right
               line
               BA
               touch
               the
               arch
               LA
               in
               the
               point
               A
               and
               the
               right-line
               LB
               be
               so
               drawn
               as
               to
               touch
               the
               arch
               LA
               in
               the
               point
               L
               ,
               here
               now
               is
               a
               plane
               on
               every
               side
               bounded
               by
               two
               right-lines
               ,
               Viz.
               BL
               and
               BA
               and
               one
               crooked-line
               ,
               
               Viz.
               the
               arch
               LA.
               That
               LBA
               is
               an
               angle
               will
               not
               be
               doubted
               :
               and
               because
               the
               three
               lines
               perfectly
               bound
               in
               and
               limit
               a
               plane
               on
               all
               sides
               ;
               the
               Arch
               LA
               can
               neither
               be
               coincident
               with
               the
               right
               line
               LB
               ,
               nor
               with
               the
               right-line
               AB
               ,
               and
               in
               the
               angle
               LAB
               ,
               the
               lines
               LA
               and
               BA
               ;
               and
               in
               the
               angle
               ALB
               ,
               the
               lines
               LA
               and
               BL
               concurring
               without
               coincidency
               ,
               and
               without
               lying
               in
               the
               same
               right-line
               ,
               or
               the
               one
               ,
               so
               much
               as
               ,
               in
               the
               production
               of
               the
               other
               ;
               inclination
               ,
               and
               so
               the
               true
               nature
               of
               an
               angle
               ,
               cannot
               be
               denyed
               them
               ;
               according
               to
               the
               most
               severe
               limitations
               and
               hardest
               glosses
               ,
               that
               can
               with
               any
               reason
               be
               deduced
               from
               the
               definition
               of
               a
               plane
               angle
               .
               And
               to
               make
               all
               clear
               let
               a
               Paper
               ,
               or
               other
               plane
               be
               cut
               in
               the
               form
               of
               the
               mixed-lined
               Triangle
               ALB
               and
               the
               wildness
               of
               questioning
               the
               angularity
               of
               the
               two
               recto-convexe
               angles
               of
               contact
               LAB
               and
               BLA
               will
               be
               clear
               to
               all
               persons
               both
               rude
               and
               learned
               .
               I
               take
               it
               therefore
               for
               granted
               ;
               that
               ,
               all
               suspicion
               of
               coincidency
               and
               non-inclination
               ,
               in
               whatsoever
               pertinent
               sense
               ,
               of
               Contact-angle
               sides
               ,
               being
               evicted
               and
               sent
               of
               the
               scene
               ,
               all
               recto-convexe
               angles
               of
               Contact
               are
               truly
               angles
               .
            
             
               To
               passe
               now
               unto
               another
               of
               your
               Thesis's
               in
               which
               you
               peremptorily
               conclude
               Recto-convexe
               angles
               of
               Contact
               to
               be
               devoyd
               of
               all
               
               quantitativeness
               ;
               when
               I
               urge
               their
               quantitativeness
               ,
               I
               mean
               not
               that
               they
               can
               at
               pleasure
               be
               devided
               into
               parts
               in
               any
               given
               and
               limited
               proportion
               ,
               or
               by
               a
               mathematical
               homogeneity
               holding
               any
               proportion
               with
               the
               Angle
               of
               Contact
               devided
               ;
               only
               that
               from
               the
               angular
               point
               between
               the
               sides
               of
               the
               least
               Recto-convexe
               angle
               of
               Contact
               infinite
               other
               lines
               may
               be
               drawn
               ,
               dividing
               the
               angle
               ,
               though
               heterogeneally
               .
               And
               certainly
               its
               being
               an
               angle
               of
               Contact
               cannot
               in
               the
               least
               prejudice
               its
               quantitativeness
               :
               for
               it
               is
               most
               apparent
               that
               a
               convexo-convexe
               angle
               of
               Contact
               contained
               under
               Arches
               of
               equal
               curvature
               ,
               may
               be
               divided
               into
               two
               mixed-lined
               ,
               
                 i.
                 e.
              
               recto-convexe
               angles
               of
               Contact
               ,
               which
               hold
               proportion
               of
               equality
               one
               with
               another
               ,
               and
               each
               of
               them
               is
               in
               subduple
               proportion
               to
               the
               convexo-convexe
               angle
               of
               Contact
               ,
               which
               was
               divided
               :
               and
               infinite
               numbers
               of
               Contact-angles
               of
               several
               sorts
               may
               be
               adjoyned
               one
               unto
               another
               ,
               distinct
               in
               their
               situation
               ,
               without
               drowning
               and
               extinguishing
               one
               another
               and
               each
               lying
               without
               the
               other
               :
               which
               is
               not
               true
               of
               indivisibles
               when
               they
               are
               adjoyned
               to
               ,
               and
               touch
               one
               another
               .
               Besides
               the
               recto-concave
               angle
               of
               Contact
               is
               greater
               then
               the
               greatest
               right-lined
               angle
               ,
               at
               the
               angular
               point
               of
               Contact
               having
               a
               manifest
               inclination
               of
               the
               sides
               ;
               for
               concurring
               ,
               they
               neither
               are
               one
               
               right-line
               ,
               nor
               one
               crooked-line
               .
               Moreover
               ultra-diametral
               concavo-convexe
               Angles
               of
               Contact
               may
               be
               equal
               to
               two
               right
               right-lined
               angles
               ,
               or
               greater
               sometimes
               :
               As
               in
               fig.
               12.
               if
               the
               two
               Circles
               FAH
               and
               ADK
               be
               equal
               and
               touch
               in
               the
               point
               A
               ,
               and
               AK
               be
               diameter
               :
               and
               AR
               be
               another
               Arch
               falling
               beyond
               the
               diameter
               
               AK
               and
               beyond
               the
               right
               line
               tangent
               AB
               :
               it
               is
               manifest
               that
               the
               concavo-convexe
               ultra-diametral
               angle
               of
               Contact
               DAH
               is
               as
               to
               the
               recesse
               of
               the
               sides
               equal
               to
               two
               right-lined
               angles
               :
               for
               the
               recto-concave
               angle
               of
               Contact
               BAH
               is
               equal
               to
               two
               right
               right-lined
               angles
               ,
               deducting
               the
               recto-convexe
               angle
               of
               Contact
               HAS
               :
               and
               the
               recto-convexe
               angle
               of
               Contact
               BAD
               is
               equal
               to
               the
               recto-convexe
               angle
               of
               contact
               H
               
               AS
               ;
               therefore
               the
               ultradiametral
               concavo-convexe
               angle
               of
               Contact
               D
               
               AH
               is
               equal
               to
               two
               right
               right-lined
               angles
               :
               and
               therefore
               the
               ultra-diametral
               concavo-convexe
               angle
               of
               Contact
               D
               
               AR
               is
               greater
               then
               two
               right
               right-lined
               angles
               .
            
             
               And
               then
               ,
               what
               greater
               Monster
               is
               discoverable
               in
               the
               doctrine
               of
               the
               quantitativeness
               of
               the
               recto-convexe
               angle
               of
               Contact
               ?
               it
               is
               demonstrated
               that
               between
               the
               right-line
               tangent
               and
               the
               Arch
               which
               it
               toucheth
               ,
               no
               right-line
               can
               passe
               :
               
                 i.
                 e.
              
               the
               recto-convexe
               angle
               of
               Contact
               is
               less
               then
               the
               least
               right-lined
               
               angle
               :
               but
               why
               should
               hence
               be
               inferred
               that
               the
               recto-convexe
               angle
               of
               Contact
               hath
               no
               true
               quantity
               ?
               you
               will
               say
               ,
               because
               a
               right-lined
               angle
               is
               infinitely
               divisible
               into
               less
               and
               less
               parts
               ,
               and
               therefore
               must
               at
               last
               be
               less
               then
               the
               recto-convexe
               angle
               of
               Contact
               ,
               if
               the
               recto-convexe
               angle
               of
               Contact
               be
               a
               true
               angle
               ,
               having
               truth
               of
               quantity
               .
               I
               answer
               ,
               after
               all
               possible
               divisions
               of
               a
               right-lined
               angle
               into
               parts
               of
               its
               owne
               kind
               ,
               I
               mean
               such
               as
               are
               made
               by
               right
               lines
               ,
               the
               least
               part
               is
               still
               a
               right-lined
               angle
               ;
               then
               which
               the
               recto-convexe
               angle
               of
               Contact
               is
               most
               fairly
               demonstrated
               to
               be
               ever
               less
               :
               
                 i.
                 e.
              
               the
               inclination
               of
               its
               sides
               is
               eve●
               less
               :
               and
               the
               convexe
               arch
               will
               still
               at
               the
               angle
               fall
               within
               :
               so
               as
               truly
               from
               thence
               may
               be
               inferred
               ,
               that
               the
               recto-convexe
               angle
               of
               Contact
               can
               never
               be
               either
               a
               right-lined
               angle
               ,
               or
               equal
               to
               it
               ,
               or
               greater
               than
               a
               right-lin'd
               angle
               :
               but
               that
               therefore
               it
               is
               no
               angle
               ,
               and
               hath
               no
               true
               quantity
               at
               all
               ,
               because
               it
               hath
               not
               the
               quantity
               of
               a
               right-lined
               angle
               ,
               is
               a
               wild
               and
               perverse
               inference
               ,
               and
               else-where
               disproved
               .
            
             
               And
               why
               should
               the
               quantitativeness
               of
               the
               recto-convexe
               angle
               of
               Contact
               be
               called
               in
               question
               ,
               because
               it
               is
               demonstrated
               less
               then
               the
               least
               right-lined
               angle
               ?
               If
               between
               the
               right-lined
               tangent
               and
               the
               arch
               a
               right-line
               could
               be
               drawn
               ,
               then
               you
               would
               confesse
               the
               quantitativeness
               of
               it
               as
               undeniable
               :
               and
               why
               
               doth
               not
               the
               passing
               of
               a
               thousand
               crooked-lines
               from
               the
               angular
               point
               ,
               between
               the
               sides
               of
               the
               recto-convexe
               angle
               of
               Contact
               ,
               as
               well
               prove
               its
               quantity
               and
               divisibility
               ,
               as
               the
               passing
               of
               one
               right-line
               between
               them
               could
               :
               there
               being
               equal
               force
               of
               proof
               from
               the
               one
               ,
               as
               from
               the
               other
               ?
               And
               when
               a
               right-line
               &
               two
               equal
               Circles
               ,
               all
               three
               touch
               in
               the
               same
               point
               ;
               there
               are
               two
               equal
               recto-convexe
               angles
               of
               Contact
               adjacent
               the
               one
               continuedly
               to
               the
               other
               ,
               and
               situate
               the
               one
               without
               the
               other
               :
               which
               in
               indivisibles
               is
               impossible
               .
               And
               when
               the
               question
               is
               of
               the
               quantity
               of
               angles
               ,
               what
               is
               it
               we
               enquire
               ,
               but
               only
               ,
               what
               is
               the
               inclination
               of
               the
               sides
               ,
               especially
               at
               the
               angular
               point
               ?
               And
               in
               recto-convexe
               angles
               of
               Contact
               ,
               the
               answer
               is
               ;
               there
               is
               no
               inclination
               at
               all
               as
               of
               a
               right-line
               to
               a
               right-line
               ,
               but
               only
               as
               of
               a
               crooked-line
               to
               a
               right-line
               :
               that
               it
               were
               wildness
               to
               say
               ;
               because
               in
               recto-convexe
               angles
               of
               Contact
               ,
               there
               cannot
               be
               the
               inclination
               of
               a
               right-line
               to
               a
               right-line
               ,
               that
               therefore
               the
               sides
               meeting
               and
               parting
               one
               from
               another
               ,
               and
               not
               lying
               both
               in
               a
               right-line
               ,
               do
               make
               no
               inclination
               one
               to
               another
               .
            
             
               And
               seeing
               the
               convexo-convexe
               angle
               of
               Contact
               contained
               under
               two
               Arches
               of
               equal
               curvature
               is
               dividable
               into
               two
               equal
               parts
               by
               a
               common
               right-line
               tangent
               to
               them
               both
               ;
               
               angle
               of
               Contact
               ,
               and
               of
               both
               the
               recto-convexe
               angles
               of
               Contact
               ,
               clearly
               appears
               ;
               though
               the
               quantitativenes
               of
               every
               one
               of
               them
               be
               demonstrable
               to
               be
               less
               then
               the
               quantity
               of
               any
               the
               least
               right-lined
               angle
               whatsoever
               :
               because
               right-lines
               cannot
               contain
               a
               less
               angle
               then
               is
               agreable
               to
               the
               inclinations
               they
               possibly
               can
               have
               one
               to
               another
               :
               whereas
               in
               crooked
               lines
               generally
               ,
               these
               are
               the
               constant
               properties
               of
               curvature
               ;
               though
               the
               angles
               of
               coincidence
               ,
               or
               curvature
               ,
               as
               we
               for
               method
               ,
               brevity
               and
               distinction
               sake
               named
               them
               ,
               may
               be
               very
               unequal
               one
               to
               another
               ,
               according
               to
               the
               degrees
               of
               the
               inflexion
               ,
               inclination
               and
               curvature
               ;
               yet
               constantly
               the
               angle
               on
               the
               concave
               side
               ,
               from
               any
               point
               of
               the
               curvature
               manifestly
               if
               it
               be
               equal
               ,
               uniform
               and
               regular
               ,
               is
               still
               greater
               then
               any
               the
               greatest
               acute
               ,
               right
               ,
               or
               obtuse
               right-lined
               angle
               :
               and
               the
               angle
               between
               the
               convexe
               arch
               and
               a
               right-line
               tangent
               ,
               at
               the
               same
               point
               ,
               is
               constantly
               less
               then
               any
               right-lined
               angle
               :
               and
               yet
               either
               may
               be
               made
               still
               infinitely
               less
               ,
               or
               greater
               ,
               the
               recto-convexe
               angle
               of
               Contact
               still
               remaining
               less
               then
               the
               least
               right-lined
               angle
               ,
               and
               the
               angle
               of
               curvature
               ,
               or
               coincidence
               greater
               still
               then
               the
               greatest
               right-lined
               angle
               :
               which
               is
               as
               much
               as
               to
               say
               ,
               that
               in
               curvatures
               the
               difference
               between
               the
               angle
               of
               curvature
               and
               two
               right
               right
               lined
               angles
               ,
               cannot
               be
               a
               
               right-lined
               angle
               ;
               as
               in
               truth
               it
               cannot
               ,
               nor
               is
               in
               reason
               so
               to
               be
               expected
               ,
               but
               of
               necessity
               it
               must
               be
               a
               mixed-lined
               ,
               Sc.
               a
               recto-convexe
               angle
               :
               and
               is
               the
               recto-convexe
               angle
               of
               Contact
               at
               the
               same
               point
               .
               And
               that
               a
               recto-convexe
               angle
               of
               Contact
               ,
               by
               no
               multiplicity
               can
               equal
               ,
               or
               exceed
               a
               right-lined
               angle
               ;
               doth
               not
               disprove
               either
               its
               angular
               nature
               ,
               or
               its
               quantitativeness
               ,
               both
               which
               are
               otherwise
               cleared
               ,
               but
               it
               is
               rather
               a
               confirmation
               of
               the
               heterogeneal
               difference
               ,
               which
               is
               between
               the
               angles
               of
               the
               one
               sort
               and
               the
               other
               .
               And
               in
               my
               judgement
               there
               needed
               no
               greater
               argument
               of
               the
               quantitativeness
               of
               recto-convexe
               angles
               of
               Contact
               ,
               then
               the
               absurdity
               following
               upon
               the
               contrary
               doctrine
               ;
               that
               the
               angle
               of
               a
               semi-Circle
               ,
               and
               a
               right
               right-lined
               angle
               are
               equal
               ,
               Viz.
               The
               whole
               and
               a
               part
               ,
               the
               one
               being
               a
               mixed-lined
               and
               the
               other
               a
               right-lined
               angle
               ;
               and
               in
               the
               indeavour
               of
               coaptation
               ,
               and
               being
               coapted
               on
               one
               side
               ,
               the
               other
               side
               all
               the
               way
               falls
               within
               ,
               or
               without
               the
               other
               ,
               so
               as
               both
               the
               sides
               of
               the
               one
               angle
               are
               impossible
               to
               be
               coapted
               to
               both
               the
               sides
               of
               the
               other
               ,
               but
               will
               both
               still
               lye
               within
               both
               the
               sides
               of
               the
               other
               .
               And
               the
               angles
               of
               semi-Circles
               must
               either
               be
               confessed
               unequal
               in
               unequal
               Circles
               ,
               or
               the
               curvature
               of
               unequal
               circumferences
               ,
               be
               manifestly
               against
               the
               truth
               asserted
               to
               be
               equal
               .
               And
               if
               still
               they
               be
               
               averred
               to
               be
               equal
               ,
               its
               desired
               their
               equality
               should
               be
               demonstrated
               ,
               and
               the
               way
               of
               admeasuring
               their
               equality
               shown
               .
            
             
               But
               you
               will
               say
               ,
               if
               recto-convexe
               angles
               of
               Contact
               be
               quantitative
               ,
               why
               can
               they
               not
               exhaust
               any
               other
               angle
               whatsoever
               contained
               under
               the
               very
               same
               sides
               ;
               for
               possibly
               you
               will
               urge
               ,
               that
               we
               should
               not
               question
               the
               homogeneity
               between
               such
               angles
               ?
               To
               this
               I
               answer
               ,
               without
               examining
               what
               homogeneity
               may
               in
               other
               respects
               be
               between
               them
               ;
               that
               a
               quantitative
               and
               mathematical
               homogeneity
               can
               with
               no
               reason
               between
               them
               be
               imagined
               :
               because
               the
               difference
               which
               is
               between
               them
               is
               a
               right-lined
               angle
               ,
               to
               which
               all
               angles
               of
               Contact
               whatsoever
               are
               heterogeneal
               ;
               and
               your self
               will
               not
               assert
               any
               mathematical
               homogeneity
               ,
               or
               at
               least
               proportionablenes
               which
               as
               to
               this
               purpose
               is
               all
               one
               between
               any
               recto-convexe
               angle
               of
               Contact
               and
               a
               right-lined
               angle
               .
               As
               in
               fig.
               18.
               let
               
               KAB
               be
               a
               recto-convexe
               angle
               of
               Contact
               ,
               and
               
               KAF
               another
               recto-convexe
               angle
               under
               the
               very
               same
               sides
               :
               and
               let
               
               AD
               be
               a
               right-line
               tangent
               upon
               the
               arch
               AF
               :
               therefore
               the
               recto-convexe
               angles
               of
               Contact
               D
               
               AF
               and
               
               KAB
               being
               equall
               ,
               the
               right-lined
               angle
               
               KAD
               is
               the
               difference
               of
               the
               recto-convexe
               angle
               of
               Contact
               
               KAB
               and
               the
               other
               recto-convexe
               angle
               
               KAF
               under
               the
               divarication
               of
               the
               very
               same
               sides
               .
               So
               as
               it
               is
               
               impossible
               to
               divide
               this
               ,
               or
               any
               other
               angle
               whatsoever
               ,
               which
               is
               not
               isoclitical
               ,
               to
               divide
               ,
               I
               say
               ,
               all
               of
               it
               into
               any
               numbers
               at
               pleasure
               given
               of
               parts
               ,
               which
               shall
               be
               homogeneal
               all
               of
               them
               one
               unto
               another
               :
               for
               how
               many
               soever
               be
               homogeneal
               ,
               the
               angle
               of
               Contact
               ,
               or
               that
               which
               is
               taken
               out
               of
               it
               ,
               or
               that
               unto
               which
               it
               adhaeres
               ,
               will
               have
               and
               make
               heterogeneity
               .
            
             
               And
               if
               you
               say
               ,
               how
               can
               recto-convexe
               angles
               of
               Contact
               be
               said
               to
               be
               parts
               of
               such
               concrete
               and
               composite
               angles
               ,
               if
               unable
               by
               any
               multiplicity
               to
               exhaust
               the
               composite
               angle
               ?
               omitting
               the
               answer
               that
               parts
               are
               sometimes
               essential
               and
               of
               the
               definition
               ,
               and
               yet
               by
               no
               multiplicity
               can
               equalize
               the
               whole
               ;
               as
               four
               angles
               in
               the
               definition
               of
               a
               Tetragone
               ,
               and
               a
               foot
               line
               in
               the
               definition
               of
               a
               foot
               Cube
               :
               I
               answer
               chiefly
               ,
               that
               where
               the
               integral
               is
               heterogeneal
               as
               here
               ,
               and
               made
               up
               ,
               and
               properly
               and
               naturally
               resoluble
               into
               several
               heterogeneal
               parts
               ,
               and
               connot
               be
               divided
               into
               any
               numbers
               at
               pleasure
               of
               parts
               all
               homogeneal
               ,
               there
               some
               parts
               may
               never
               be
               able
               by
               any
               multiplicity
               to
               equalize
               the
               whole
               ,
               or
               some
               other
               heterogeneal
               parts
               .
               And
               elss-where
               that
               under
               the
               same
               coaptable
               sides
               may
               be
               angles
               different
               in
               their
               ultimate
               kind
               in
               some
               further
               respects
               ,
               though
               not
               without
               proportionableness
               in
               this
               instance
               ,
               is
               most
               apparent
               among
               right-lined
               angles
               ,
               by
               comparing
               a
               right
               right-lined
               angle
               ,
               
               and
               an
               acute
               right-lined
               angle
               ,
               and
               an
               obtuse
               right-lined
               angle
               together
               ;
               which
               all
               receive
               their
               specifick
               differences
               from
               the
               specifick
               differences
               of
               their
               inclinations
               :
               in
               right
               right-lined
               angles
               the
               inclination
               being
               no
               more
               one
               way
               then
               the
               other
               ,
               no
               more
               from
               the
               angle
               side
               ,
               then
               to
               the
               angle
               side
               ,
               Sc.
               perpendicular
               :
               in
               acute
               angles
               the
               inclination
               being
               to
               the
               angle
               side
               :
               and
               in
               obtuse
               angles
               the
               inclination
               being
               more
               especially
               from
               the
               angle
               side
               .
            
             
               And
               yet
               though
               we
               defend
               the
               quantitativeness
               of
               recto-convexe
               angles
               of
               contact
               we
               are
               equally
               obliged
               to
               assert
               their
               improportionableness
               to
               right-lined
               angles
               :
               nor
               will
               there
               be
               any
               difficulty
               in
               answering
               that
               suggession
               you
               cast
               ,
               though
               in
               anothers
               name
               ;
               that
               a
               recto-convexe
               angle
               of
               Contact
               is
               in
               proportion
               less
               then
               a
               right
               lined
               angle
               ,
               as
               being
               both
               homogeneal
               ;
               and
               that
               by
               the
               multiplicity
               of
               the
               recto-convexe
               angle
               of
               Contact
               ,
               an
               angle
               may
               be
               made
               equal
               to
               a
               right
               lined
               angle
               ,
               or
               greater
               ,
               only
               by
               changing
               its
               kind
               ,
               Sc.
               into
               a
               right-lined
               angle
               :
               Viz.
               in
               the
               same
               manner
               ,
               as
               an
               acute
               right-lined
               angle
               ,
               being
               less
               then
               a
               right
               ,
               or
               any
               obtuse
               right-lined
               angle
               ,
               by
               its
               multiplicity
               may
               with
               change
               of
               its
               sort
               and
               kind
               become
               equal
               ,
               or
               greater
               then
               a
               right
               ,
               or
               any
               given
               obtuse
               right-lined
               angle
               .
               To
               which
               may
               be
               answered
               ,
               omitting
               what
               kind
               of
               distinction
               
               it
               is
               which
               is
               between
               acute
               ,
               right
               and
               obtuse
               right-lined
               angles
               ,
               as
               not
               pertinent
               to
               the
               present
               controversy
               ;
               it
               would
               be
               well
               done
               to
               shew
               what
               multiplex
               of
               any
               recto-convexe
               angle
               of
               Contact
               ,
               is
               equal
               to
               what
               right-lined
               angle
               ,
               that
               so
               a
               right-lined
               angle
               might
               ,
               upon
               that
               proportion
               ,
               be
               formed
               equal
               to
               ,
               or
               less
               then
               the
               recto-convexe
               angle
               of
               Contact
               ;
               contrary
               to
               what
               has
               been
               clearly
               demonstrated
               ,
               and
               is
               generally
               by
               all
               consented
               to
               in
               Geometry
               .
               Though
               acute
               and
               right
               right-lined
               angles
               are
               less
               then
               any
               obtuse
               whatsoever
               that
               are
               right-lined
               ;
               however
               they
               hold
               proportion
               one
               with
               another
               :
               but
               recto-convexe
               angles
               of
               Contact
               cannot
               be
               demonstrated
               to
               hold
               any
               proportion
               with
               any
               right-lined
               angle
               ,
               but
               clearly
               the
               contrary
               .
               And
               as
               by
               the
               divarications
               of
               the
               sides
               of
               an
               acute
               right-lined
               angle
               is
               made
               a
               Genesis
               of
               a
               right
               and
               infinite
               obtuse
               right-lined
               angle
               :
               there
               cannot
               so
               by
               the
               divarication
               of
               the
               sides
               of
               a
               recto-convexe
               angle
               of
               Contact
               be
               effected
               any
               Genesis
               of
               right-lined
               angles
               ,
               but
               only
               of
               recto-convexe
               mixt-lined
               angles
               ,
               whose
               sides
               concurre
               by
               way
               of
               section
               ;
               and
               between
               which
               and
               the
               recto-convexe
               angle
               of
               Contact
               is
               no
               proportion
               ,
               as
               there
               is
               between
               the
               divaricated
               acute
               right-lined
               angle
               and
               the
               other
               right-lined
               angles
               created
               from
               that
               divaricacation
               .
               The
               reason
               of
               which
               is
               clear
               ,
               for
               that
               the
               recto-convexe
               angle
               of
               Contact
               being
               
               demonstrated
               of
               it self
               to
               be
               less
               then
               any
               right-lined
               angle
               whatsoever
               ,
               by
               the
               divarication
               of
               the
               sides
               of
               it
               ,
               there
               are
               continually
               greater
               and
               greater
               right-lined
               angles
               added
               to
               it
               ,
               so
               creating
               an
               improportionality
               between
               the
               one
               ,
               and
               the
               other
               .
            
             
               The
               comparison
               made
               between
               Cyphers
               and
               angles
               of
               Contact
               ,
               to
               draw
               the
               one
               ,
               as
               well
               as
               the
               other
               into
               the
               notion
               of
               nullities
               ,
               is
               unhappy
               enough
               :
               for
               single
               and
               several
               Cyphers
               are
               not
               greater
               ,
               or
               less
               in
               power
               one
               then
               another
               (
               though
               they
               may
               make
               other
               figures
               to
               be
               so
               )
               as
               the
               angles
               of
               Contact
               are
               ,
               and
               may
               be
               made
               larger
               ,
               or
               lesser
               ,
               and
               many
               of
               them
               one
               without
               or
               within
               another
               ,
               contiguously
               and
               continuously
               conjoyned
               together
               ,
               with
               enlargement
               ,
               or
               diminution
               of
               their
               angularity
               ;
               which
               is
               impossible
               in
               indivisibles
               ,
               and
               unappliable
               to
               ,
               and
               unintelligible
               of
               ,
               mere
               nullities
               .
            
             
               That
               your
               Lemma
               is
               without
               exception
               ,
               and
               without
               proof
               might
               have
               been
               admitted
               :
               Viz.
               That
               two
               quantities
               by
               the
               ordinate
               application
               ,
               or
               motion
               of
               a
               line
               ,
               or
               plane
               ,
               increasing
               ,
               or
               decreasing
               proportionably
               ,
               whether
               by
               a
               proportionality
               in
               the
               same
               ,
               or
               different
               powers
               ,
               when
               the
               ordinate
               application
               ,
               or
               motion
               attains
               the
               end
               ,
               and
               bounding
               term
               of
               the
               one
               quantity
               ,
               it
               at
               the
               same
               moment
               reacheth
               &
               hath
               attained
               the
               
               bounding
               term
               of
               the
               other
               ,
               and
               when
               it
               hath
               passed
               the
               one
               it
               hath
               passed
               the
               other
               .
               But
               the
               objection
               against
               the
               quantitativeness
               of
               the
               recto-convexe
               angle
               of
               Contact
               ,
               which
               you
               would
               hereupon
               found
               ,
               hath
               no
               reason
               to
               expect
               the
               like
               allowance
               .
               You
               say
               ,
               the
               right-lined
               right
               angle
               at
               the
               point
               of
               Contact
               contained
               under
               the
               right-line
               tangent
               ,
               and
               the
               diameter
               of
               a
               Circle
               ,
               equally
               with
               the
               circumference
               intercepted
               ,
               increaseth
               ,
               or
               decreaseth
               by
               the
               motion
               of
               the
               diameter
               upon
               the
               point
               of
               Contact
               ,
               as
               a
               Center
               ;
               which
               is
               true
               and
               acknowledged
               :
               and
               whereas
               you
               say
               ,
               that
               therefore
               when
               the
               diameter
               leaves
               nothing
               at
               all
               of
               the
               circumference
               in
               its
               circumvolution
               about
               ,
               un-run
               over
               ,
               but
               attains
               the
               last
               bounder
               and
               termination
               of
               it
               ,
               in
               the
               point
               of
               Contact
               ,
               then
               as
               the
               circumference
               is
               quite
               exhausted
               and
               vanished
               ,
               so
               is
               the
               angle
               too
               :
               this
               also
               we
               acknowledge
               to
               be
               undeniable
               .
               But
               whereas
               when
               the
               diameter
               is
               come
               so
               near
               the
               right-line
               tangent
               in
               its
               circumvolution
               upon
               the
               angular
               point
               of
               Contact
               ,
               as
               to
               intercept
               nothing
               at
               all
               of
               the
               circumference
               between
               them
               ,
               you
               then
               imagine
               still
               an
               angle
               remaining
               ,
               which
               you
               say
               ,
               is
               either
               the
               recto-convexe
               angle
               of
               Contact
               ,
               or
               not
               less
               then
               it
               ;
               you
               herein
               forget
               the
               force
               of
               your
               own
               manifestly
               true
               lemma
               ,
               which
               you
               took
               so
               much
               pains
               to
               prove
               ;
               except
               against
               Geometrical
               
               demonstration
               you
               could
               discover
               a
               possibility
               of
               dividing
               a
               recto-convexe
               angle
               of
               Contact
               by
               a
               right-line
               :
               for
               it
               is
               out
               of
               doubt
               and
               in
               Geometry
               ,
               as
               above
               demonstrated
               ,
               that
               when
               the
               diameter
               in
               such
               circumvolution
               intercepts
               nothing
               of
               the
               circumference
               between
               its
               self
               and
               the
               right-line
               tangent
               ,
               the
               diameter
               is
               then
               coincident
               ,
               and
               the
               same
               right-line
               with
               the
               right-line
               tangent
               ;
               and
               of
               the
               former
               angle
               therefore
               ,
               hath
               lest
               nothing
               at
               all
               ,
               because
               of
               the
               coincidency
               of
               the
               two
               right-lines
               ,
               whose
               parts
               can
               in
               that
               positure
               have
               no
               inclination
               one
               unto
               another
               :
               and
               therefore
               there
               is
               not
               so
               much
               as
               the
               least
               angle
               of
               Contact
               ,
               or
               any
               other
               angle
               whatsoever
               ,
               left
               after
               this
               circumvolution
               :
               so
               as
               the
               whole
               Argumentation
               is
               a
               long
               arrow
               out
               of
               a
               strong
               Bow
               ,
               but
               quite
               besides
               the
               mark
               .
            
             
               It
               is
               a
               seeming
               weighty
               objection
               that
               which
               is
               urged
               out
               of
               opticks
               ,
               and
               the
               usage
               in
               that
               science
               ,
               to
               demonstrate
               in
               conical
               figures
               ,
               the
               angles
               of
               incidence
               and
               reflexion
               to
               be
               equal
               ,
               only
               with
               respect
               to
               the
               right-line
               tangent
               ,
               touching
               the
               figure
               in
               the
               point
               of
               incidence
               and
               reflexion
               ,
               without
               special
               respect
               to
               the
               curvature
               in
               the
               conical
               section
               .
               But
               hereto
               without
               wrong
               ,
               either
               to
               truth
               ,
               nature
               ,
               or
               that
               noble
               science
               ,
               may
               be
               upon
               good
               grounds
               answered
               .
               1.
               
               That
               opticks
               is
               not
               pure
               Geometry
               ,
               and
               obstructed
               stereometry
               ,
               and
               mathematicks
               ;
               
               wherein
               quantities
               ,
               mensurations
               ,
               and
               proportions
               are
               considered
               merely
               as
               in
               themselves
               without
               relation
               to
               matter
               ,
               and
               the
               uses
               to
               which
               in
               other
               faculties
               they
               are
               applicable
               :
               but
               in
               opticks
               ,
               is
               an
               improvement
               of
               what
               in
               nature
               may
               be
               observed
               about
               luminous
               and
               visual
               beams
               ,
               and
               luminous
               mediums
               and
               objects
               ,
               by
               mathematical
               demonstration
               and
               assistance
               .
               Now
               2
               ly
               .
               Nature
               doth
               not
               tye
               it self
               in
               its
               wayes
               so
               strictly
               to
               an
               indivisible
               ,
               absolute
               ,
               vertical
               punctilio
               ,
               that
               if
               it
               cannot
               reach
               ,
               without
               impediments
               ,
               that
               mathematical
               exquisitenes
               ,
               it
               will
               not
               act
               at
               all
               .
               If
               heavy
               things
               on
               the
               clive
               side
               of
               an
               hill
               cannot
               descend
               in
               a
               direct
               line
               towards
               the
               Center
               ,
               however
               they
               will
               seek
               it
               in
               the
               nearest
               oblique
               line
               possibly
               they
               can
               ,
               so
               long
               as
               the
               motion
               brings
               them
               nearer
               :
               all
               animals
               are
               not
               formed
               in
               the
               same
               mould
               ,
               or
               with
               equal
               strength
               of
               Body
               ,
               or
               vigour
               of
               spirit
               .
               So
               as
               in
               these
               optical
               instances
               ,
               the
               brave
               Authors
               sell
               not
               in
               the
               least
               short
               of
               their
               task
               ,
               when
               they
               had
               demonstrated
               the
               most
               equal
               right-lined
               angles
               which
               nature
               could
               cast
               it self
               into
               ,
               in
               the
               alledged
               cases
               ,
               Viz.
               angles
               of
               incidence
               and
               reflexion
               equal
               ,
               if
               compared
               with
               a
               right-line
               tangent
               at
               their
               point
               .
               For
               as
               elsewhere
               ,
               they
               are
               but
               very
               few
               points
               in
               most
               conical
               sections
               ,
               where
               ,
               by
               a
               right-line
               ,
               their
               angularity
               can
               be
               divided
               into
               equal
               parts
               :
               nature
               therefore
               
               when
               it
               cannot
               attain
               its
               prime
               and
               general
               design
               ,
               which
               is
               a
               perfection
               and
               absoluteness
               in
               its
               work
               ,
               it
               doth
               not
               therefore
               slugge
               and
               do
               nothing
               ,
               but
               seeks
               to
               approach
               its
               first
               design
               as
               near
               as
               possible
               :
               and
               that
               is
               in
               making
               with
               right-lined
               beams
               equal
               angles
               respectively
               to
               the
               right-line
               tangents
               at
               the
               points
               concerned
               ;
               which
               angles
               so
               formed
               at
               those
               points
               with
               respect
               unto
               the
               right-line
               tangents
               ,
               are
               as
               is
               elsewhere
               shewn
               ,
               either
               the
               least
               of
               the
               greater
               right-lined
               angles
               ,
               or
               the
               greatest
               of
               the
               less
               right-lined
               angles
               :
               that
               in
               such
               curvatures
               with
               great
               judgement
               the
               quality
               of
               the
               angles
               of
               incidence
               and
               reflexion
               ,
               in
               beams
               passing
               by
               right-lines
               ,
               as
               affected
               for
               their
               directness
               and
               shortness
               ,
               and
               as
               near
               as
               possible
               endeavoured
               by
               nature
               ,
               is
               in
               demonstration
               referred
               to
               examination
               at
               the
               right-line
               tangents
               of
               the
               same
               points
               ;
               an
               absolute
               equality
               by
               right-lines
               to
               be
               made
               ,
               being
               most
               what
               impossible
               ,
               and
               that
               demonstrably
               in
               such
               curvatures
               :
               so
               as
               causeless
               was
               the
               exception
               which
               was
               made
               against
               the
               demonstrations
               of
               the
               noble
               Perspectivists
               :
               nor
               stood
               ,
               either
               they
               ,
               or
               nature
               in
               need
               of
               that
               improper
               ,
               lame
               solution
               and
               help
               ,
               by
               making
               recto-convexe
               angles
               of
               Contact
               to
               be
               neither
               angles
               ,
               nor
               quantitative
               .
            
             
               The
               truth
               of
               the
               angularity
               and
               quantitaveness
               
               of
               recto-convexe
               angles
               of
               Contact
               thus
               asserted
               ;
               however
               ,
               the
               rest
               of
               the
               things
               in
               this
               controversy
               cannot
               be
               determined
               without
               a
               clear
               understanding
               of
               what
               homogenealness
               it
               is
               that
               is
               requisite
               for
               proportionals
               ,
               and
               which
               is
               mentioned
               in
               the
               definition
               of
               proportion
               .
               To
               bring
               our selves
               unto
               the
               right
               understanding
               of
               which
               ,
               know
               there
               are
               three
               wayes
               whereby
               one
               thing
               may
               be
               said
               to
               be
               greater
               then
               another
               .
            
             
               1
               st
               .
               Improportionably
               ,
               and
               by
               the
               whole
               kind
               ;
               as
               in
               heterogeneals
               :
               in
               which
               sometime
               the
               least
               of
               the
               one
               is
               still
               ,
               beyond
               all
               proportion
               ,
               greater
               then
               the
               greatest
               of
               the
               other
               :
               as
               the
               least
               surface
               is
               greater
               then
               any
               line
               ;
               for
               a
               longer
               line
               may
               be
               drawn
               in
               the
               least
               surface
               ,
               then
               any
               the
               longest
               line
               ,
               that
               can
               possibly
               be
               given
               :
               and
               so
               the
               least
               body
               is
               improportionably
               ,
               and
               heterogeneally
               ,
               and
               by
               the
               whole
               kind
               greater
               then
               any
               surface
               :
               for
               by
               the
               dividing
               ,
               or
               altering
               the
               figuration
               of
               the
               least
               body
               ,
               it
               may
               be
               made
               ,
               by
               its
               perimetry
               ,
               to
               exceed
               the
               quantity
               of
               the
               greatest
               surface
               ,
               that
               possibly
               can
               be
               given
               :
               so
               a
               point
               ,
               or
               a
               circle
               of
               a
               foot
               diameter
               ,
               is
               said
               to
               be
               less
               then
               the
               whole
               world
               ;
               which
               is
               to
               be
               understood
               ,
               as
               by
               the
               whole
               kind
               ,
               and
               without
               proportionableness
               .
               And
               so
               I
               doubt
               not
               ,
               but
               it
               will
               be
               confessed
               ,
               that
               the
               least
               recto-convexe
               angle
               of
               Contact
               ,
               is
               greater
               then
               the
               greatest
               acute
               ,
               
               right
               or
               obtuse
               right-lined
               angle
               :
               and
               the
               least
               right-lined
               angle
               ,
               then
               the
               greatest
               recto-convexe
               angle
               of
               Contact
               .
            
             
               2
               ly
               .
               One
               thing
               is
               greater
               then
               another
               infinitely
               ,
               or
               ,
               if
               you
               please
               ,
               indefinitely
               :
               so
               an
               infinitely
               ,
               or
               indefinitely
               inlarged
               line
               ,
               is
               longer
               then
               any
               given
               line
               ,
               without
               limitation
               ,
               and
               so
               consequenely
               without
               proportion
               .
               Or
               ,
            
             
               3
               ly
               .
               One
               thing
               is
               greater
               then
               another
               ,
               rateably
               and
               according
               to
               the
               proportion
               ,
               which
               they
               hold
               at
               the
               same
               standard
               ,
               whereby
               they
               are
               after
               the
               same
               manner
               measured
               ,
               in
               the
               same
               kind
               in
               which
               they
               are
               compared
               .
               So
               one
               lenght
               ,
               or
               line
               is
               longer
               then
               another
               ,
               according
               to
               such
               a
               proportion
               of
               length
               when
               both
               are
               measured
               as
               lengths
               at
               the
               same
               standard
               upon
               an
               indefinite
               line
               ;
               and
               this
               ,
               whether
               the
               lengths
               ,
               or
               lines
               be
               ,
               one
               ,
               both
               ,
               all
               ,
               neither
               ,
               or
               none
               of
               them
               directly
               ,
               or
               indirectly
               posited
               ,
               in
               right
               ,
               or
               crooked
               lines
               ;
               and
               whether
               the
               one
               be
               a
               depth
               ,
               and
               the
               other
               a
               height
               ,
               or
               another
               a
               breadth
               ,
               or
               a
               fourth
               a
               periphery
               .
               So
               one
               number
               is
               greater
               then
               another
               number
               ,
               according
               to
               such
               a
               proportion
               :
               and
               ,
               though
               it
               is
               well
               known
               to
               be
               determined
               in
               Philosophy
               ,
               that
               numbers
               are
               of
               different
               kinds
               ,
               yet
               for
               the
               proportion
               they
               hold
               one
               to
               another
               in
               their
               common
               nature
               ,
               there
               cannot
               be
               denied
               unto
               them
               the
               truth
               of
               
               a
               mathematical
               and
               analogical
               homogenealness
               .
            
             
               And
               as
               things
               are
               said
               to
               be
               several
               wayes
               one
               greater
               then
               another
               ;
               so
               homogeneity
               wants
               not
               its
               several
               acceptations
               .
               What
               homogeneity
               is
               ,
               is
               not
               at
               all
               any
               where
               expressely
               defined
               in
               the
               mathematicks
               ;
               but
               we
               are
               left
               at
               large
               ,
               rationally
               to
               collect
               ,
               what
               is
               by
               that
               Term
               in
               those
               faculties
               to
               be
               understood
               .
               The
               most
               usual
               acceptation
               of
               the
               word
               homogeneity
               in
               Philosophy
               is
               to
               compare
               any
               divisible
               being
               with
               the
               parts
               into
               which
               it
               may
               be
               divided
               :
               those
               things
               being
               said
               to
               be
               homogeneal
               ,
               which
               cannot
               be
               separatingly
               divided
               into
               parts
               of
               any
               other
               name
               ,
               or
               nature
               then
               the
               whole
               is
               :
               as
               the
               least
               separable
               part
               of
               water
               is
               said
               to
               be
               water
               ;
               of
               wine
               ,
               wine
               :
               and
               the
               least
               separable
               part
               of
               a
               line
               ,
               a
               line
               :
               and
               those
               things
               are
               said
               heterogeneal
               ,
               which
               by
               possibility
               may
               be
               separatingly
               divided
               otherwise
               ;
               Viz.
               into
               parts
               of
               different
               name
               and
               nature
               from
               the
               whole
               .
               All
               which
               ,
               as
               appears
               ,
               hath
               its
               dependance
               upon
               the
               similar
               ,
               or
               dissimilar
               natures
               of
               the
               whole
               and
               parts
               .
               So
               all
               solids
               ,
               surfaces
               ,
               lines
               ,
               and
               plane
               angles
               may
               be
               said
               to
               be
               homogeneal
               ;
               for
               by
               dividing
               and
               separating
               them
               part
               from
               part
               ,
               every
               part
               of
               the
               solid
               is
               a
               solid
               ,
               of
               the
               surface
               a
               surface
               ,
               of
               the
               line
               a
               line
               ,
               and
               of
               the
               plane
               angle
               a
               plane
               angle
               :
               and
               by
               taking
               number
               in
               a
               large
               sense
               ,
               every
               
               part
               of
               a
               number
               may
               be
               said
               to
               be
               a
               number
               .
               But
               then
               if
               you
               descend
               lower
               in
               numbers
               and
               angles
               ,
               even
               and
               odde
               numbers
               ,
               &c.
               and
               right-lined
               and
               mixed-lined
               and
               crooked-lined
               angles
               will
               scarce
               be
               able
               to
               defend
               their
               homogeneity
               at
               this
               touch
               :
               for
               Even
               numbers
               may
               be
               divided
               into
               parts
               that
               are
               odde
               :
               and
               a
               right-lined
               plane
               angle
               may
               be
               severed
               into
               parts
               whereof
               the
               one
               may
               be
               a
               recto-concave
               angle
               of
               a
               semi-Circle
               ,
               the
               other
               a
               recto-convexe
               angle
               of
               Contact
               ;
               of
               which
               last
               mentioned
               severed
               part
               you
               say
               ,
               it
               is
               no
               angle
               ,
               and
               therefore
               it
               must
               be
               heterogeneal
               :
               and
               we
               assert
               it
               heterogeneal
               ,
               because
               it
               is
               improportionable
               ,
               and
               its
               quantity
               not
               mensurable
               after
               the
               same
               manner
               ,
               that
               the
               quantity
               of
               right-lined
               angles
               is
               measured
               .
            
             
               And
               with
               reference
               to
               the
               same
               acceptation
               ,
               two
               distinct
               beings
               ,
               being
               compared
               together
               ,
               as
               to
               homogeneity
               and
               heterogeneity
               ,
               they
               are
               then
               said
               to
               be
               homogeneal
               ,
               when
               both
               having
               the
               same
               name
               and
               nature
               ,
               the
               one
               may
               be
               the
               severed
               part
               of
               the
               other
               ,
               or
               both
               together
               the
               severed
               parts
               of
               an
               integral
               of
               the
               same
               both
               name
               and
               nature
               .
               So
               all
               solids
               are
               homogeneal
               :
               all
               surfaces
               :
               all
               lines
               :
               all
               plane
               angles
               and
               all
               numbers
               whatsoever
               .
            
             
               But
               sometimes
               also
               in
               an
               abusive
               acceptation
               ,
               the
               agreeing
               ,
               or
               disagreeing
               of
               things
               in
               some
               general
               ,
               or
               special
               kind
               and
               nature
               ,
               
               passeth
               for
               a
               certain
               kind
               of
               homogeneity
               or
               heterogeneity
               among
               them
               :
               so
               bodies
               and
               surfaces
               and
               lines
               in
               respect
               of
               their
               common
               agreeing
               in
               quantity
               ,
               continuity
               ,
               longitude
               ,
               latitude
               and
               profundity
               ,
               are
               partly
               homogeneal
               ,
               partly
               heterogeneal
               :
               so
               numbers
               in
               respect
               of
               their
               general
               quantitativeness
               ,
               parity
               ,
               imparity
               ,
               symmetry
               and
               a
               symmetry
               &c.
               
               Are
               partly
               homogeneal
               ,
               partly
               heterogeneal
               ;
               and
               so
               solid
               ,
               sphaerical
               ,
               and
               plane
               angles
               ,
               and
               all
               their
               several
               kinds
               are
               partly
               homogeneal
               ,
               and
               partly
               heterogeneal
               .
            
             
               So
               as
               from
               any
               of
               the
               former
               acceptations
               it
               is
               no
               way
               possible
               to
               determine
               what
               is
               that
               mathematical
               homogeneity
               which
               is
               every
               where
               hinted
               unto
               us
               in
               the
               definitions
               of
               proportion
               ;
               for
               the
               bolting
               forth
               of
               which
               we
               shall
               be
               constrained
               of
               necessity
               to
               betake
               our selves
               to
               another
               course
               ,
               in
               the
               quest
               of
               which
               ,
               it
               is
               not
               likely
               but
               we
               shall
               meet
               with
               some
               abstruseness
               and
               difficulty
               .
            
             
               To
               make
               therefore
               an
               Essay
               ;
               proportion
               being
               the
               habitude
               of
               the
               compared
               magnitudes
               according
               to
               quantity
               ,
               it
               is
               an
               homogeneity
               in
               quantity
               which
               is
               only
               herein
               required
               ;
               not
               an
               homogeneity
               in
               substance
               ,
               quality
               ,
               site
               ,
               or
               other
               kinds
               and
               manners
               of
               being
               :
               for
               heterogeneity
               in
               any
               of
               those
               other
               respects
               ,
               as
               substance
               ,
               quality
               ,
               site
               ,
               or
               other
               kinds
               of
               being
               ,
               doth
               not
               hinder
               ,
               but
               they
               may
               
               notwithstanding
               be
               one
               proportionable
               to
               another
               ,
               if
               any
               quantitative
               homogeneity
               be
               to
               be
               found
               amongst
               them
               .
               So
               the
               heterogeneity
               of
               figure
               and
               figuration
               ,
               which
               is
               between
               a
               Circle
               ,
               pentagone
               ,
               square
               ,
               triangle
               ,
               surface
               of
               a
               pyramide
               ,
               &c.
               
               Hinders
               not
               ,
               but
               they
               may
               be
               compared
               and
               hold
               proportion
               in
               respect
               of
               their
               Area's
               and
               superficial
               contents
               ;
               wherein
               they
               retain
               a
               quantitative
               homogeneity
               :
               and
               those
               of
               them
               which
               are
               meerly
               bounded
               planes
               ,
               also
               have
               a
               homogeneity
               in
               respect
               of
               their
               perimetries
               ,
               or
               the
               like
               quantities
               which
               they
               hold
               in
               common
               .
               So
               crookednes
               and
               rectitude
               are
               different
               kinds
               of
               positure
               ,
               and
               make
               an
               heterogeneity
               in
               situation
               ;
               yet
               a
               right
               line
               and
               a
               crooked-line
               ,
               hold
               still
               a
               quantitative
               homogeneity
               ,
               in
               respect
               of
               their
               longitude
               and
               extension
               .
               So
               the
               different
               positures
               of
               the
               surfaces
               in
               a
               sphaere
               ,
               and
               in
               a
               cone
               ,
               and
               in
               a
               plane
               ,
               hindereth
               not
               their
               analogy
               in
               Area
               and
               quantity
               .
            
             
               But
               further
               to
               pursue
               this
               diquisition
               ;
               though
               it
               be
               homogeneity
               in
               quantity
               ,
               which
               is
               herein
               understood
               yet
               it
               is
               not
               the
               being
               quantities
               ,
               or
               magnitudes
               in
               general
               that
               doth
               make
               up
               ,
               or
               can
               compleat
               the
               homogeneity
               hinted
               in
               the
               definition
               of
               proportion
               :
               for
               then
               lines
               ,
               and
               bodies
               ,
               and
               numbers
               ,
               and
               surfaces
               ,
               being
               all
               of
               them
               magnitudes
               and
               quantities
               ,
               there
               should
               be
               a
               proportionablenes
               between
               lines
               
               and
               bodies
               ,
               bodies
               and
               surfaces
               ,
               surfaces
               and
               numbers
               ;
               which
               in
               such
               a
               general-homogeneity
               it
               were
               vanity
               to
               look
               for
               .
               In
               like
               manner
               ,
               though
               plane
               ,
               sphaerical
               and
               solid
               angles
               be
               all
               angles
               ,
               yet
               I
               think
               there
               are
               very
               few
               ,
               that
               because
               of
               that
               their
               general
               homogeneity
               will
               expect
               a
               proportionablenes
               amongst
               them
               .
            
             
               Besides
               as
               little
               necessity
               is
               there
               of
               restraining
               this
               homogeneity
               of
               magnitudes
               ,
               unto
               their
               very
               last
               ,
               and
               in
               every
               respect
               ,
               ultimate
               kinds
               :
               except
               you
               will
               understand
               it
               to
               be
               in
               respect
               of
               some
               indefinite
               ,
               most
               pure
               abstracted
               quantity
               ,
               in
               which
               the
               compared
               magnitudes
               are
               to
               be
               each
               of
               them
               measured
               .
               For
               though
               some
               referr
               all
               numbers
               unto
               quantity
               ,
               distinguishing
               them
               into
               several
               kinds
               ,
               not
               only
               from
               their
               parities
               and
               imparities
               ,
               with
               all
               their
               variations
               thereupon
               ,
               but
               making
               every
               number
               to
               be
               of
               a
               different
               kind
               ,
               receiving
               its
               specification
               from
               the
               last
               unite
               ;
               as
               some
               also
               referr
               unto
               quantity
               all
               lines
               ,
               which
               they
               distinguish
               into
               several
               kinds
               ,
               &
               the
               like
               specifick
               distinctions
               are
               made
               by
               others
               of
               surfaces
               and
               bodies
               :
               yet
               between
               proportionals
               ,
               for
               their
               homogeneity
               ,
               is
               not
               ,
               we
               confesse
               ,
               in
               this
               sense
               required
               ,
               that
               both
               magnitudes
               be
               of
               the
               very
               same
               ultimate
               kind
               :
               for
               hyperbolar
               ,
               parabolar
               ,
               elliptical
               ,
               circular
               arches
               ,
               and
               right-lines
               are
               homogeneal
               in
               length
               ,
               extension
               ,
               and
               as
               lines
               ,
               so
               holding
               analogy
               ,
               though
               in
               respect
               of
               rectitude
               and
               
               curvature
               ,
               and
               their
               several
               kinds
               of
               curvature
               they
               be
               heterogeneal
               .
               So
               all
               numbers
               ,
               even
               ,
               odde
               ,
               commensurable
               ,
               or
               incommensurable
               ,
               however
               ,
               as
               above
               is
               said
               ,
               heterogeneal
               ,
               yet
               as
               multitudes
               and
               numbers
               in
               general
               ,
               and
               the
               accounts
               how
               oft
               an
               unite
               and
               its
               parts
               is
               posited
               ,
               or
               how
               far
               in
               account
               is
               proceeded
               ,
               whether
               retrogradely
               or
               progressively
               from
               an
               unite
               ;
               so
               ,
               all
               numbers
               are
               homogeneal
               ,
               holding
               mutual
               analogy
               one
               to
               another
               .
               So
               notwithstanding
               the
               distinction
               between
               acute
               ,
               right
               ,
               and
               obtuse
               ,
               right-lined
               angles
               ,
               all
               truth
               and
               exquisitenes
               of
               proportion
               ,
               as
               the
               measure
               of
               their
               quantitative
               relation
               ,
               is
               most
               apparent
               amongst
               them
               .
            
             
               Which
               again
               doth
               further
               shew
               unto
               us
               what
               and
               how
               great
               is
               the
               difficulty
               of
               limiting
               the
               homogeneity
               expressed
               in
               the
               definition
               of
               proportion
               ;
               so
               as
               neither
               to
               set
               it
               uselesly
               too
               high
               ,
               nor
               sink
               it
               lower
               then
               is
               necessary
               for
               proportionality
               .
               For
               right
               right-lined
               angles
               are
               proportionate
               ,
               one
               to
               another
               ,
               but
               there
               can
               be
               amongst
               them
               no
               proportion
               of
               inequality
               :
               and
               no
               given
               acute
               right-lined
               angle
               can
               have
               so
               small
               a
               proportion
               to
               any
               acute
               right-lined
               angle
               ,
               as
               it
               may
               have
               and
               hath
               to
               a
               right
               right-lined
               angle
               ,
               or
               to
               any
               obtuse
               right-lined
               angle
               :
               and
               though
               all
               even
               numbers
               hold
               analogy
               and
               proportion
               one
               to
               another
               ,
               yet
               even
               numbers
               are
               not
               capable
               of
               all
               diversities
               of
               proportion
               ;
               as
               in
               uneven
               
               numbers
               ,
               duple
               and
               subduple
               proportions
               are
               impossible
               :
               nay
               numbers
               in
               general
               ,
               I
               mean
               unbroken
               and
               integral
               numbers
               ,
               are
               not
               capable
               of
               all
               kinds
               of
               analogy
               ;
               as
               particularly
               not
               of
               such
               proportions
               as
               are
               asymmetral
               .
            
             
               But
               not
               to
               be
               nice
               in
               my
               thoughts
               to
               my
               friend
               ,
               I
               never
               in
               this
               point
               and
               question
               understood
               any
               other
               thing
               to
               be
               meant
               by
               homogeneity
               in
               the
               definition
               of
               proportion
               ,
               then
               a
               mensurablenes
               of
               the
               quantities
               of
               two
               ,
               or
               more
               magnitudes
               in
               the
               same
               indefinite
               quantity
               for
               kind
               ,
               as
               the
               measure
               of
               their
               quantity
               and
               of
               the
               quantity
               of
               all
               their
               homogeneals
               ,
               the
               mensuration
               therein
               still
               being
               according
               to
               that
               same
               kind
               of
               quantity
               of
               which
               the
               indefinite
               quantity
               is
               .
               And
               so
               proportion
               is
               the
               rate
               and
               habitude
               ,
               which
               the
               rateable
               magnitudes
               hold
               mutually
               one
               to
               the
               other
               in
               respect
               of
               the
               same
               way
               of
               measuring
               their
               quantities
               ,
               or
               in
               respect
               of
               the
               same
               kind
               of
               indefinite
               quantity
               in
               which
               their
               quantities
               are
               measured
               .
               And
               upon
               this
               gloss
               as
               the
               true
               and
               genuine
               meaning
               of
               this
               mathematical
               homogeneity
               ,
               I
               ever
               understood
               that
               postulate
               to
               be
               founded
               in
               which
               is
               required
               and
               granted
               ,
               so
               to
               multiply
               any
               given
               quantity
               ,
               as
               to
               exceed
               any
               other
               given
               quantity
               whatsoever
               of
               the
               same
               kind
               .
               For
               if
               that
               mensurablenes
               in
               the
               same
               indefinite
               quantity
               ,
               as
               a
               measure
               ,
               and
               according
               to
               that
               same
               kind
               of
               quantity
               of
               
               which
               the
               indefinite
               measure
               is
               ,
               were
               not
               the
               very
               thing
               designed
               by
               mathematical
               homogeneity
               ,
               the
               matter
               of
               the
               Postulat
               were
               not
               fit
               to
               be
               granted
               without
               proof
               :
               for
               it
               is
               because
               they
               are
               measured
               in
               the
               same
               indefinite
               quantity
               for
               kind
               ,
               and
               according
               to
               the
               same
               kind
               of
               quantity
               ,
               
                 i.
                 e.
              
               they
               have
               the
               very
               same
               way
               of
               measuring
               in
               the
               same
               indefinite
               quantity
               (
               which
               is
               their
               homogeneity
               ,
               by
               necessity
               of
               consequence
               creating
               a
               proportionablenes
               between
               them
               )
               that
               the
               less
               by
               multiplying
               may
               be
               made
               greater
               then
               the
               greater
               ,
               and
               the
               greater
               by
               a
               continual
               cutting
               off
               still
               more
               then
               half
               may
               be
               made
               less
               then
               the
               less
               .
            
             
               And
               though
               hereby
               homogeneity
               and
               proportionablenes
               be
               not
               made
               to
               be
               the
               very
               same
               thing
               ,
               however
               in
               the
               mathematicks
               ,
               where
               the
               physical
               natures
               of
               things
               are
               not
               inquired
               into
               ,
               the
               one
               by
               a
               necessary
               consequence
               doth
               immediately
               flow
               from
               and
               is
               annexed
               unto
               the
               other
               ;
               and
               because
               of
               their
               necessary
               connexion
               ,
               in
               usual
               speech
               and
               acceptation
               ,
               the
               one
               may
               be
               allowed
               to
               be
               taken
               for
               the
               other
               .
               And
               when
               in
               the
               definition
               of
               proportion
               ,
               proportion
               is
               said
               to
               be
               the
               mutual
               habitude
               of
               magnitudes
               of
               the
               same
               kind
               according
               to
               quantity
               ,
               or
               if
               you
               please
               multiplicity
               ;
               the
               meaning
               is
               no
               other
               ,
               but
               that
               proportion
               is
               the
               mutual
               habitude
               of
               magnitudes
               ,
               which
               have
               their
               mensuration
               after
               the
               same
               
               manner
               ,
               according
               to
               quantity
               ,
               or
               multiplicity
               ;
               taking
               the
               word
               multiplicity
               in
               a
               large
               sense
               :
               
                 i.
                 e.
              
               according
               to
               the
               quantity
               and
               multiplicity
               ,
               which
               they
               have
               each
               to
               other
               in
               the
               same
               indefinite
               quantity
               and
               measure
               ,
               upon
               which
               they
               are
               in
               the
               same
               manner
               and
               according
               to
               the
               same
               kind
               of
               quantity
               measured
               .
               However
               in
               natural
               Philosophy
               for
               very
               weighty
               reasons
               homogeneity
               and
               proportionablenes
               are
               to
               be
               acknowledged
               of
               very
               distant
               and
               different
               natures
               .
               So
               I
               presume
               in
               Mathematicks
               it
               would
               be
               taken
               for
               a
               solaecisme
               to
               say
               a
               body
               and
               a
               line
               were
               homogeneals
               and
               of
               the
               same
               kind
               ,
               because
               all
               separable
               parts
               of
               each
               agree
               in
               their
               being
               all
               of
               them
               continuous
               quantities
               ;
               though
               in
               the
               Physiological
               school
               that
               they
               do
               concenter
               and
               meet
               in
               the
               same
               general
               nature
               is
               not
               deniable
               ;
               and
               so
               they
               may
               carry
               a
               seeming
               shadow
               of
               homogeneity
               ,
               so
               far
               as
               homogeneity
               may
               be
               abusively
               wrested
               to
               denote
               any
               such
               common
               agreement
               in
               a
               general
               notion
               and
               nature
               .
               So
               it
               would
               be
               a
               solaecisme
               in
               Mathematicks
               to
               say
               ,
               that
               a
               solid
               angle
               ,
               a
               sphaerical
               angle
               and
               a
               plane
               angle
               were
               all
               homogeneal
               ,
               because
               they
               are
               all
               angles
               and
               every
               separable
               part
               of
               each
               is
               an
               angle
               :
               but
               to
               pro●e
               Mathematical
               homogeneity
               ,
               the
               mensurablenes
               of
               the
               quantity
               of
               the
               compared
               magnitudes
               in
               the
               same
               indefinite
               quantity
               ,
               or
               measure
               for
               
               kind
               ,
               and
               according
               to
               the
               same
               kind
               of
               quantity
               with
               the
               indefinite
               measure
               is
               to
               be
               made
               out
               :
               for
               that
               all
               are
               quantities
               ,
               or
               all
               angles
               ,
               makes
               them
               not
               in
               the
               mathematical
               school
               homogeneal
               ;
               except
               by
               reason
               of
               this
               mensurability
               of
               the
               quantity
               of
               both
               in
               the
               same
               indefinite
               quantity
               ,
               or
               measure
               ,
               according
               to
               the
               same
               kind
               of
               quantity
               ,
               the
               less
               by
               multiplying
               can
               be
               made
               greater
               then
               the
               greater
               ,
               and
               the
               greater
               by
               dividing
               less
               then
               the
               less
               .
               And
               indeed
               this
               is
               the
               true
               homogeneity
               ,
               not
               denoting
               a
               general
               conveniency
               in
               their
               natures
               in
               respect
               of
               some
               abstracted
               notion
               ,
               but
               rather
               a
               special
               identity
               by
               reason
               of
               their
               mensurablenes
               in
               the
               same
               substrate
               kind
               of
               quantity
               and
               measure
               ,
               only
               with
               difference
               ,
               or
               proportionablenes
               of
               magnitude
               between
               both
               the
               wholes
               ,
               and
               all
               the
               least
               ,
               or
               greatest
               ,
               proper
               ,
               
                 i.
                 e.
              
               homogeneal
               parts
               of
               each
               :
               as
               if
               one
               be
               a
               line
               ,
               so
               is
               the
               other
               ,
               and
               the
               greatest
               and
               least
               proper
               ,
               
                 i.
                 e.
              
               homogeneal
               or
               homometral
               parts
               of
               each
               are
               lines
               and
               proportionable
               to
               either
               .
               For
               ,
               if
               besides
               the
               mensurablenes
               of
               the
               quantity
               of
               the
               compared
               magnitudes
               in
               the
               same
               indefinite
               quantity
               ,
               or
               measure
               ,
               be
               not
               also
               added
               that
               condition
               ,
               that
               in
               that
               indefinite
               quantity
               ,
               or
               measure
               they
               are
               also
               mensurable
               according
               to
               the
               same
               quantity
               for
               kind
               ,
               of
               which
               the
               indefinite
               measure
               is
               ,
               and
               so
               consequently
               proportionable
               one
               to
               another
               ;
               
               by
               nothing
               will
               it
               yet
               be
               determinable
               ,
               whether
               all
               angles
               be
               not
               homogeneal
               :
               for
               in
               every
               angle
               ,
               though
               of
               several
               kinds
               ,
               every
               part
               of
               each
               angle
               is
               an
               angle
               :
               nor
               will
               it
               be
               determinable
               ,
               whether
               all
               continuous
               quantities
               be
               not
               homogeneal
               ;
               for
               every
               part
               even
               of
               heterogeneal
               continuous
               quantities
               is
               a
               continuous
               quantity
               :
               nor
               will
               it
               be
               determinable
               whether
               all
               numbers
               be
               not
               homogeneal
               ;
               for
               that
               all
               numbers
               are
               of
               the
               same
               kind
               will
               be
               found
               a
               doctrine
               of
               very
               hard
               digestion
               any
               where
               save
               in
               the
               mathematick
               school
               :
               yet
               according
               to
               this
               explication
               of
               mathematical
               homogeneity
               ,
               notwithstanding
               the
               diversity
               of
               their
               kinds
               in
               other
               philosophical
               considerations
               ,
               they
               have
               in
               them
               a
               clear
               mathematical
               homogeneity
               :
               and
               even
               an
               unite
               ,
               which
               in
               other
               parts
               of
               Philosophy
               is
               not
               passable
               for
               a
               number
               ,
               will
               fall
               also
               within
               the
               verge
               of
               the
               same
               homogeneity
               ;
               as
               will
               also
               all
               the
               parts
               of
               an
               unite
               ,
               whether
               commensurable
               ,
               or
               incommensurable
               .
               And
               this
               explication
               of
               mathematical
               homogeneity
               will
               be
               allowed
               its
               due
               right
               and
               justification
               more
               easily
               ,
               by
               those
               who
               note
               how
               the
               main
               matter
               and
               design
               of
               mathematical
               definitions
               is
               but
               exegetical
               to
               clear
               up
               what
               is
               meant
               by
               the
               terms
               in
               those
               sciences
               used
               :
               for
               what
               other
               occasion
               could
               there
               be
               in
               the
               Mathematicks
               to
               intermeddle
               with
               homogeneity
               ,
               but
               to
               explain
               
               the
               noble
               points
               of
               proportion
               and
               proportionality
               .
            
             
               And
               yet
               though
               in
               mathematicks
               there
               be
               such
               a
               grand
               affinity
               between
               the
               proportionablenes
               and
               homogeneity
               of
               magnitudes
               and
               in
               common
               use
               and
               spee●h
               the
               one
               may
               be
               put
               for
               the
               other
               ,
               yet
               as
               above
               the
               notions
               are
               easily
               distinguishable
               by
               the
               understanding
               :
               Viz.
               two
               ,
               or
               more
               magnitudes
               are
               said
               homogeneal
               chiefly
               in
               respect
               of
               the
               same
               way
               of
               measuring
               them
               ,
               or
               in
               respect
               of
               the
               same
               kind
               of
               indefinite
               quantity
               in
               which
               they
               are
               both
               mensurable
               :
               but
               they
               are
               said
               proportionable
               in
               respect
               of
               the
               mutual
               habitude
               and
               quantitative
               relation
               which
               is
               between
               themselves
               upon
               such
               their
               mensuration
               ,
               in
               the
               same
               way
               ,
               or
               according
               to
               the
               same
               kind
               of
               quantity
               .
               So
               all
               finite
               lines
               are
               homogeneal
               ,
               as
               mensurable
               in
               the
               same
               indefinite
               line
               ;
               but
               that
               one
               finite
               line
               is
               double
               to
               another
               is
               the
               habitude
               of
               the
               one
               to
               the
               other
               declared
               upon
               that
               mensuration
               .
            
             
               That
               upon
               the
               whole
               matter
               mathematical
               homogeneity
               doth
               not
               insist
               only
               upon
               identity
               in
               kind
               at
               large
               ,
               or
               restrain
               unto
               identity
               in
               kind
               at
               all
               points
               ,
               and
               in
               every
               respect
               and
               consideration
               ;
               but
               it
               is
               identity
               in
               quantity
               and
               therein
               particularly
               in
               the
               manner
               of
               the
               mensuration
               of
               their
               quantities
               ,
               in
               which
               the
               homogeneity
               of
               magnitudes
               is
               chiefly
               
               lodged
               :
               and
               that
               is
               that
               which
               I
               ever
               understood
               in
               those
               words
               (
               according
               to
               quantity
               ,
               or
               if
               you
               please
               ,
               rather
               quotuplicity
               )
               which
               are
               in
               the
               definition
               of
               propo●tion
               ,
               viz.
               quantity
               and
               quotuplicity
               in
               the
               same
               way
               of
               measuring
               .
            
             
               And
               if
               you
               urge
               here
               ,
               according
               to
               this
               account
               I
               must
               conclude
               all
               incommensurables
               to
               be
               heterogeneal
               ;
               I
               answer
               ,
               not
               in
               the
               least
               :
               for
               though
               they
               may
               have
               no
               common
               measure
               ,
               which
               can
               by
               possibility
               exactly
               measure
               both
               or
               all
               ,
               yet
               there
               may
               be
               a
               common
               indefinite
               measure
               ,
               in
               which
               each
               may
               exactly
               measure
               forth
               its
               own
               quantity
               .
               As
               for
               example
               in
               fig.
               24.
               
               Let
               A
               and
               B
               be
               incommensurable
               lines
               ,
               and
               DC
               .
               an
               indefinite
               line
               ,
               beginning
               at
               D
               ,
               and
               on
               the
               part
               of
               C
               infinitely
               produced
               :
               from
               the
               point
               D
               in
               the
               line
               DC
               take
               the
               line
               DE
               equal
               to
               the
               line
               A
               :
               and
               also
               in
               the
               line
               DC
               take
               the
               line
               DF
               equal
               to
               the
               line
               B.
               here
               in
               the
               line
               DC
               as
               a
               common
               indefinite
               measure
               of
               their
               homogeneity
               ,
               the
               two
               lines
               A
               and
               B
               ,
               though
               incommensurable
               ,
               have
               measured
               forth
               themselves
               by
               the
               lines
               DE
               and
               DF.
               So
               in
               fig.
               1.
               upon
               A
               the
               angular
               point
               of
               the
               right-lined
               angle
               BAC
               ,
               as
               Center
               ,
               draw
               the
               Arch
               dfe
               cutting
               the
               line
               AB
               in
               the
               point
               d
               ,
               and
               the
               line
               AC
               in
               the
               point
               f
               ,
               and
               the
               right-line
               Ae
               in
               the
               point
               e
               ,
               so
               as
               the
               Arch
               df
               be
               incommensurable
               to
               the
               
               Arch
               fe
               .
               Here
               the
               two
               right-lined
               angles
               d
               
               Af
               and
               f
               
               Ae
               are
               incommensurable
               ,
               yet
               have
               a
               common
               way
               of
               measuring
               their
               quantity
               and
               proportion
               ,
               Viz.
               by
               Arches
               of
               Circles
               drawn
               upon
               the
               angular
               point
               ,
               as
               center
               ,
               intercepted
               respectively
               between
               the
               sides
               :
               as
               by
               the
               Arch
               dfe
               ,
               the
               one
               being
               in
               that
               Arch
               measured
               by
               the
               Arch
               ,
               df
               ,
               the
               other
               by
               the
               Arch
               fe
               .
               So
               as
               by
               mathematical
               homogeneity
               is
               understood
               an
               homometricalnes
               ,
               or
               autometricalnes
               with
               the
               necessary
               consequent
               of
               a
               rateablenes
               therein
               ,
               without
               any
               necessity
               of
               symmetricalnes
               between
               themselves
               at
               all
               .
               For
               still
               ,
               if
               in
               homogeneity
               besides
               homometricalnes
               in
               the
               same
               indefinite
               quantity
               be
               not
               also
               included
               ,
               that
               the
               mensuration
               of
               both
               be
               according
               to
               the
               same
               kind
               of
               quantity
               ,
               of
               which
               the
               indefinite
               quantity
               ,
               in
               which
               they
               are
               measured
               ,
               is
               ;
               a
               point
               and
               a
               line
               may
               measure
               themselves
               in
               an
               indefinite
               line
               ,
               and
               a
               line
               and
               a
               surface
               may
               measure
               themselves
               in
               an
               indefinite
               surface
               ,
               and
               a
               surface
               and
               a
               solid
               may
               measure
               themselves
               in
               an
               indefinite
               solid
               .
               But
               true
               mathematical
               homogeneity
               is
               when
               two
               ,
               or
               more
               quantities
               being
               mensurable
               in
               the
               same
               indefinite
               quantity
               ,
               or
               measure
               ,
               and
               according
               to
               the
               kind
               of
               the
               indefinite
               quantity
               are
               by
               consequence
               rateably
               ,
               
                 i.
                 e.
              
               proportionably
               and
               homometrically
               equal
               ,
               or
               the
               one
               bigger
               then
               
               the
               other
               ,
               not
               the
               one
               infinitely
               bigger
               than
               the
               other
               ,
               as
               an
               infinite
               line
               is
               longer
               then
               a
               finite
               line
               ,
               for
               between
               such
               it
               is
               acknowledged
               there
               is
               no
               proportion
               nor
               mathematical
               homogeneity
               ;
               nor
               the
               one
               bigger
               then
               the
               other
               by
               the
               whole
               kind
               ,
               as
               every
               solid
               is
               bigger
               then
               any
               surface
               ,
               and
               every
               right-lined
               angle
               then
               any
               recto-convexe
               angle
               of
               Contact
               :
               and
               proportion
               is
               the
               rate
               ,
               or
               quantity
               of
               their
               mutual
               habitude
               in
               that
               their
               homogeneity
               ,
               or
               homometricalnes
               :
               
                 i.
                 e.
              
               proportion
               is
               the
               rate
               ,
               quantity
               ,
               or
               account
               of
               their
               proportionality
               .
               And
               applying
               our
               minds
               rightly
               to
               conceive
               of
               homogeneity
               as
               mathematical
               ;
               of
               necessity
               such
               ,
               and
               no
               other
               can
               be
               the
               notion
               of
               it
               ,
               thereby
               making
               two
               quantities
               to
               be
               rateably
               ,
               
                 i.
                 e.
              
               proportionably
               equal
               ,
               or
               the
               one
               bigger
               then
               the
               other
               ,
               and
               according
               to
               their
               capacity
               and
               possibility
               in
               any
               rate
               and
               proportion
               so
               to
               be
               constituted
               and
               set
               out
               .
               For
               the
               genuine
               well
               known
               notion
               of
               homogeneity
               in
               general
               ,
               what
               is
               it
               but
               that
               all
               and
               each
               of
               the
               proper
               ,
               
                 i.
                 e.
              
               homogeneal
               parts
               ,
               and
               the
               whole
               ,
               fall
               under
               the
               same
               Denomination
               and
               nature
               ,
               as
               if
               one
               be
               a
               line
               ,
               all
               the
               rest
               to
               be
               lines
               ,
               if
               one
               be
               water
               ,
               or
               stone
               ,
               or
               oyle
               ,
               &c.
               all
               the
               rest
               of
               the
               parts
               and
               the
               whole
               to
               be
               so
               also
               ?
               And
               in
               the
               mathematicks
               what
               is
               said
               properly
               and
               homogeneally
               to
               be
               a
               part
               of
               any
               magnitude
               ,
               but
               only
               such
               lesser
               magnitudes
               separable
               from
               it
               as
               are
               able
               to
               measure
               
               out
               ,
               
                 i.
                 e.
              
               by
               their
               multiple
               to
               exhaust
               the
               first
               magnitude
               ?
               For
               a
               surface
               is
               not
               said
               to
               be
               an
               homogeneally
               proper
               part
               of
               a
               solid
               :
               nor
               a
               line
               of
               a
               surface
               .
               And
               by
               such
               measuring
               forth
               of
               the
               integral
               magnitude
               by
               its
               homogeneally
               proper
               ,
               
                 i.
                 e.
              
               exhausting
               parts
               ,
               is
               not
               understood
               the
               symmetry
               of
               the
               parts
               ,
               and
               whole
               ,
               for
               the
               whole
               and
               its
               true
               homogeneal
               parts
               ,
               may
               be
               incommensurable
               ;
               but
               that
               which
               is
               understood
               is
               the
               true
               mathematical
               homogeneity
               of
               the
               whole
               and
               the
               parts
               in
               respect
               of
               their
               common
               way
               of
               measuring
               ,
               and
               the
               proportionality
               ,
               which
               is
               thereupon
               lodged
               between
               them
               So
               as
               by
               laying
               these
               undeniable
               remarks
               together
               ,
               mathematical
               homogeneity
               in
               respect
               of
               the
               same
               integral
               magnitude
               ,
               is
               that
               every
               part
               thereof
               being
               proper
               ,
               
                 i.
                 e.
              
               mensurable
               in
               the
               same
               indefinite
               measure
               according
               to
               the
               special
               kind
               of
               its
               quantity
               ,
               and
               therefore
               able
               to
               exhaust
               the
               whole
               ,
               be
               all
               of
               them
               only
               in
               respect
               of
               their
               quantitativenes
               ,
               not
               their
               figuration
               ,
               or
               other
               respects
               ,
               of
               the
               same
               nature
               and
               denomination
               ,
               and
               any
               proportion
               whatsoever
               (
               according
               to
               the
               capacity
               of
               such
               kind
               of
               quantities
               )
               possible
               to
               be
               constituted
               amongst
               them
               .
               And
               by
               consequence
               several
               magnitudes
               are
               then
               said
               to
               be
               mathematically
               homogeneal
               ,
               when
               being
               the
               one
               able
               to
               exhaust
               the
               other
               ,
               &
               so
               in
               proportion
               one
               unto
               another
               ,
               both
               and
               all
               the
               proper
               parts
               of
               each
               ,
               
               all
               of
               them
               ,
               as
               being
               of
               the
               same
               nature
               fal●
               under
               the
               same
               quantitative
               denomination
               .
            
             
               And
               here
               it
               is
               worth
               our
               noting
               ,
               how
               some
               magnitudes
               are
               so
               homogeneal
               ,
               that
               they
               cannot
               be
               separatingly
               divided
               into
               parts
               ,
               or
               magnitudes
               which
               are
               heterogeneal
               ;
               but
               all
               the
               parts
               into
               which
               they
               are
               separatingly
               divided
               will
               still
               be
               homogeneal
               ,
               both
               one
               with
               another
               and
               with
               the
               whole
               :
               So
               a
               line
               can
               no
               way
               be
               separatingly
               divided
               into
               parts
               which
               are
               magnitudes
               ,
               but
               each
               is
               a
               line
               homogeneal
               to
               the
               whole
               and
               to
               all
               parts
               whatsoever
               of
               whatsoever
               other
               line
               :
               neither
               can
               a
               body
               and
               surface
               be
               separatingly
               divided
               ,
               but
               into
               bodies
               and
               surfaces
               ;
               each
               of
               whose
               parts
               are
               still
               able
               to
               exhaust
               the
               whole
               and
               are
               thereunto
               proportionable
               .
               But
               such
               an
               absolute
               homogeneity
               is
               not
               in
               all
               other
               magnitudes
               ,
               especially
               in
               heterogeneally
               concrete
               magnitudes
               ,
               but
               that
               the
               whole
               may
               be
               divideable
               into
               parts
               separable
               one
               from
               another
               ,
               which
               yet
               are
               heterogeneal
               :
               as
               in
               the
               present
               case
               of
               angles
               a
               right
               right-lined
               angle
               is
               divideable
               into
               a
               recto-convexe
               angle
               of
               Contact
               ,
               and
               the
               recto-concave
               angle
               of
               the
               semi
               Circle
               ;
               which
               are
               separable
               one
               from
               another
               ,
               and
               therefore
               truly
               parts
               ;
               yet
               neither
               of
               them
               is
               homogeneal
               to
               the
               whole
               ,
               nor
               one
               of
               them
               unto
               the
               other
               :
               for
               neither
               can
               the
               whole
               right
               right-lined
               angle
               ,
               nor
               the
               recto-concave
               angle
               
               of
               the
               semi
               Circle
               ever
               be
               exhausted
               by
               any
               number
               whatsoever
               of
               such
               heterogeneal
               parts
               ,
               as
               is
               the
               recto-convexe
               angle
               of
               Contact
               :
               nor
               ever
               any
               equality
               ,
               or
               other
               proportion
               can
               possibly
               be
               shewn
               between
               the
               right
               right-lined
               angle
               ,
               and
               the
               recto-concave
               angle
               of
               the
               semi
               Circle
               ,
               because
               there
               is
               no
               way
               possible
               in
               which
               their
               quantities
               can
               be
               proportionably
               mensurable
               .
               For
               not
               without
               very
               good
               reason
               unto
               all
               magnitudes
               are
               to
               be
               allowed
               their
               special
               properties
               ;
               as
               to
               all
               positures
               and
               figuration
               ,
               theirs
               .
               To
               angles
               these
               things
               are
               peculiar
               ,
               being
               otherwise
               in
               other
               magnitudes
               :
               Viz.
               in
               angles
               which
               are
               truly
               and
               on
               all
               hands
               confessedly
               homogeneal
               ,
               you
               cannot
               to
               any
               given
               angle
               ,
               set
               forth
               another
               of
               the
               same
               kind
               in
               any
               given
               proportion
               at
               pleasure
               :
               for
               every
               right-lined
               angle
               by
               a
               necessity
               of
               nature
               ,
               must
               be
               less
               then
               two
               right
               right-lined
               angles
               :
               and
               in
               a
               plane
               all
               the
               angularity
               at
               any
               point
               cannot
               exceed
               what
               the
               circumjacent
               space
               ,
               or
               plane
               is
               capable
               of
               ;
               which
               is
               only
               four
               right
               right-lined
               angles
               .
               That
               as
               number
               cannot
               be
               infinitely
               divided
               without
               fraction
               ;
               so
               angularity
               cannot
               at
               pleasure
               at
               the
               same
               point
               ,
               in
               the
               same
               plane
               ,
               be
               inlarged
               :
               whereas
               some
               other
               quantities
               have
               both
               infinite
               divisibility
               and
               infinite
               multiplicability
               .
            
             
               So
               another
               property
               of
               the
               magnitude
               of
               angles
               
               is
               ,
               that
               it
               may
               not
               only
               in
               notion
               and
               speculation
               ,
               but
               in
               truth
               and
               severingly
               ,
               be
               divided
               into
               parts
               either
               able
               ,
               or
               unable
               to
               exhaust
               the
               whole
               :
               as
               when
               a
               right
               right-lined
               angle
               is
               divided
               into
               the
               recto-concave
               angle
               of
               a
               semi
               Circle
               ,
               and
               a
               recto-convexe
               angle
               of
               Contact
               :
               you
               may
               sever
               them
               the
               one
               from
               the
               other
               :
               and
               angularity
               is
               equally
               ,
               if
               not
               much
               more
               apparent
               in
               the
               recto-convexe
               angle
               of
               Contact
               ,
               then
               in
               the
               recto-concave
               angle
               of
               the
               semi
               Circle
               ;
               yet
               the
               one
               of
               them
               is
               demonstrated
               and
               confessed
               unable
               ever
               to
               exhaust
               the
               right
               right-lined
               angle
               ,
               the
               other
               not
               .
            
             
               A
               further
               property
               of
               the
               magnitude
               of
               angles
               is
               ,
               that
               sometimes
               the
               same
               part
               ,
               which
               hath
               already
               been
               severed
               from
               it
               ,
               cannot
               exactly
               and
               immediately
               again
               by
               its
               equal
               be
               severed
               from
               it
               on
               the
               same
               side
               ,
               though
               the
               remaining
               angle
               be
               by
               the
               whole
               kind
               greater
               .
               So
               after
               a
               recto-convexe
               angle
               of
               Contact
               is
               taken
               out
               of
               a
               right
               right-lined
               angle
               ,
               there
               cannot
               again
               immediately
               on
               the
               same
               side
               ,
               be
               severed
               from
               the
               remaining
               angle
               another
               angle
               equal
               to
               the
               recto-convexe
               angle
               of
               Contact
               ,
               which
               was
               before
               severed
               from
               it
               .
               If
               it
               can
               ,
               let
               it
               be
               performed
               .
            
             
               Also
               the
               divisibility
               which
               is
               in
               the
               magnitude
               of
               all
               Angles
               ,
               though
               boundles
               and
               infinite
               ,
               in
               some
               however
               ,
               leaves
               the
               dividing
               
               of
               the
               Angle
               into
               two
               equal
               parts
               ,
               impossible
               :
               as
               notwithstanding
               the
               perpetual
               divisibility
               of
               lines
               ,
               the
               side
               and
               diameter
               of
               a
               square
               are
               left
               incommensurable
               .
               So
               some
               other
               angles
               may
               be
               divided
               into
               two
               equal
               parts
               ,
               but
               it
               is
               impossible
               to
               divide
               them
               into
               three
               equal
               parts
               :
               as
               convexo-convexe
               angles
               of
               Contact
               ,
               with
               infinite
               other
               convexo-convexe
               angles
               and
               concavo-concave
               angles
               being
               contained
               under
               equal
               ,
               uniform
               and
               answerable
               Arches
               .
            
             
               To
               consigne
               this
               point
               ,
               the
               principal
               thing
               we
               have
               laboured
               herein
               to
               dilucidate
               ,
               &
               as
               we
               doubt
               not
               have
               effected
               is
               ,
               that
               mathematical
               homogenealness
               is
               not
               an
               homogeneity
               of
               all
               the
               parts
               whatsoever
               ,
               that
               are
               in
               the
               magnitudes
               ,
               which
               are
               homogeneal
               in
               respect
               of
               some
               special
               way
               of
               measuring
               their
               quantities
               ;
               or
               an
               undivideableness
               of
               such
               homogeneal
               magnitudes
               into
               parts
               otherwise
               heterogeneal
               according
               to
               which
               acceptation
               the
               word
               is
               chiefly
               taken
               in
               other
               parts
               of
               Philosophy
               :
               for
               there
               is
               no
               right-lined
               angle
               whatsoever
               ,
               nor
               any
               other
               angle
               whatsoever
               ,
               but
               as
               is
               up
               and
               down
               herein
               shewn
               may
               be
               separatingly
               divided
               into
               heterogeneal
               parts
               :
               but
               mathematical
               homogeneity
               is
               homogeneity
               in
               the
               way
               of
               measuring
               the
               quantity
               of
               the
               compared
               magnitudes
               ,
               Sc.
               in
               the
               same
               indefinite
               measure
               and
               quantity
               ,
               and
               according
               to
               the
               kind
               of
               the
               
               indefinite
               measure
               ,
               and
               which
               thereupon
               follows
               ,
               a
               proportionality
               between
               them
               ,
               in
               respect
               of
               their
               common
               way
               of
               measuring
               :
               and
               of
               this
               mathematical
               homogeneity
               ,
               fair
               foot-steppings
               are
               to
               be
               found
               every
               where
               in
               the
               deducing
               of
               those
               demonstrations
               which
               concern
               proportions
               and
               proportionals
               .
               That
               ,
               such
               magnitudes
               as
               have
               no
               common
               way
               of
               measuring
               their
               quantity
               ,
               as
               weights
               and
               measures
               ,
               are
               heterogeneal
               :
               or
               if
               they
               have
               a
               common
               way
               of
               measuring
               in
               which
               they
               may
               measure
               themselves
               ,
               but
               therein
               do
               not
               measure
               themselves
               according
               to
               the
               same
               kind
               of
               quantity
               with
               the
               indefinite
               measure
               ,
               and
               so
               want
               proportionality
               ,
               yet
               notwithstanding
               they
               are
               heterogeneal
               :
               as
               all
               recto-convexe
               angles
               of
               Contact
               ,
               all
               recto-concave
               angles
               of
               semi
               Circles
               ,
               all
               recto-convexe
               angles
               of
               semi
               Circles
               ,
               all
               acute
               ,
               or
               right
               right-lined
               angles
               ,
               these
               may
               all
               measure
               themselves
               ,
               and
               in
               what
               order
               their
               sides
               fall
               within
               ,
               or
               without
               in
               any
               obtuse
               right-lined
               angle
               whatsoever
               :
               yet
               because
               this
               their
               homometry
               is
               only
               of
               the
               situation
               ,
               or
               order
               ,
               in
               which
               the
               sides
               part
               from
               the
               angular
               point
               ,
               but
               not
               of
               their
               quantity
               in
               an
               indefinite
               measure
               and
               according
               to
               the
               denomination
               of
               the
               same
               quantitative
               measure
               ,
               so
               as
               to
               lodge
               a
               proportionality
               between
               the
               magnitudes
               so
               compared
               together
               in
               their
               common
               way
               of
               measuring
               ;
               they
               are
               not
               ,
               nor
               
               can
               thereby
               be
               vindicated
               from
               their
               otherwise
               innate
               mathematical
               heterogeneity
               :
               which
               concerning
               some
               of
               them
               is
               confessed
               on
               all
               hands
               and
               is
               without
               the
               verge
               of
               the
               controversy
               .
               And
               as
               follows
               ,
               angles
               are
               of
               a
               concrete
               nature
               ,
               having
               in
               them
               something
               quantitative
               and
               something
               not
               quantitative
               :
               whereas
               that
               which
               is
               to
               be
               the
               indefinite
               measure
               of
               homogeneal
               quantities
               is
               to
               be
               considered
               abstractly
               as
               quantity
               without
               heterogeneal
               concretion
               :
               so
               it
               is
               the
               circumference
               of
               a
               Circle
               that
               measures
               all
               right-lined
               angles
               .
               And
               when
               all
               plane
               angles
               are
               said
               to
               be
               homogeneal
               ,
               it
               is
               not
               in
               respect
               of
               a
               common
               indefinite
               quantity
               by
               which
               they
               are
               all
               measured
               ,
               which
               the
               recto-convexe
               angles
               of
               Contact
               doe
               sufficiently
               evince
               ,
               but
               as
               is
               manifest
               it
               is
               only
               because
               of
               the
               position
               and
               situation
               of
               the
               sides
               in
               the
               same
               plane
               ;
               which
               homogeneity
               is
               of
               no
               concern
               unto
               quantity
               ,
               nor
               by
               any
               necessity
               can
               thereupon
               infer
               the
               consequent
               of
               proportionableness
               .
            
             
               But
               to
               proceed
               ;
               as
               is
               said
               ,
               besides
               the
               former
               mathematical
               and
               quantitative
               homogeneity
               and
               heterogeneity
               there
               is
               also
               an
               extramathematical
               and
               extra-quantitative
               homogenealnes
               and
               heterogenealnes
               in
               angles
               every
               where
               observable
               in
               their
               shapes
               ,
               figures
               ,
               positure
               of
               their
               sides
               ,
               such
               like
               schematismes
               and
               other
               respects
               .
               In
               general
               as
               is
               above
               hinted
               ,
               every
               part
               of
               a
               
               plane
               angle
               is
               a
               plane
               angle
               ,
               even
               the
               recto-convexe
               angle
               of
               Contact
               ,
               however
               you
               deny
               it
               to
               be
               an
               angle
               and
               quantitative
               :
               but
               then
               this
               is
               not
               a
               mathematical
               homogeneity
               ,
               but
               only
               in
               respect
               of
               a
               certain
               figuration
               ,
               in
               respect
               of
               the
               positure
               and
               situation
               of
               the
               surface
               in
               which
               those
               angles
               are
               ;
               shewing
               how
               all
               plane
               angles
               from
               the
               greatest
               to
               the
               least
               ,
               agree
               in
               that
               particular
               of
               their
               general
               figuration
               ,
               Viz.
               of
               having
               their
               containing
               sides
               to
               lye
               still
               in
               the
               same
               plane
               ;
               whereby
               they
               distinguish
               themselves
               from
               all
               other
               superficial
               angles
               ,
               which
               are
               heterepipedal
               ,
               whose
               containing
               lines
               ,
               or
               sides
               lye
               in
               several
               planes
               :
               such
               as
               are
               all
               sorts
               of
               sphaerical
               ,
               cylindrical
               and
               conical
               surface-angles
               .
               But
               if
               ever
               a
               mathematical
               and
               quantitative
               homogeneity
               be
               proved
               among
               all
               plane
               angles
               ,
               you
               that
               know
               that
               it
               is
               not
               my
               use
               to
               start
               from
               my
               word
               ,
               shall
               hereby
               rest
               assured
               ,
               upon
               the
               first
               summons
               I
               will
               give
               up
               this
               cause
               .
            
             
               And
               we
               are
               not
               to
               think
               strange
               ,
               that
               a
               figuration
               is
               asserted
               to
               be
               in
               angles
               ;
               for
               if
               we
               seriously
               consider
               ,
               wee
               shall
               find
               there
               is
               shape
               and
               figure
               in
               angles
               ,
               as
               well
               as
               quantity
               ;
               as
               lines
               ,
               and
               surfaces
               ,
               and
               bodies
               have
               their
               figurations
               ,
               the
               positure
               of
               their
               parts
               ,
               their
               shapes
               and
               forms
               ,
               as
               well
               as
               their
               quantities
               and
               magnitudes
               :
               in
               each
               ,
               their
               figuration
               being
               manifest
               ;
               Viz.
               in
               lines
               ,
               in
               respect
               of
               
               their
               lineary
               positure
               :
               in
               surfaces
               ,
               in
               respect
               of
               their
               superficial
               positure
               :
               in
               bodies
               in
               respect
               of
               their
               solid
               positure
               :
               and
               in
               the
               casting
               of
               each
               of
               their
               schematismes
               quantity
               is
               involved
               ,
               as
               length
               ,
               breadth
               ,
               depth
               ,
               Viz.
               their
               quantities
               ,
               and
               the
               quantities
               which
               are
               compounded
               of
               several
               ,
               or
               all
               of
               them
               together
               .
            
             
               And
               here
               by
               the
               figuration
               which
               we
               assert
               in
               angles
               ,
               we
               cannot
               be
               thought
               to
               mean
               that
               any
               right-lined
               figure
               can
               be
               compleated
               ,
               perfectly
               to
               bound
               up
               a
               plane
               on
               all
               sides
               ,
               by
               one
               angle
               ;
               it
               being
               beyond
               the
               power
               of
               two
               :
               and
               three
               ,
               being
               the
               least
               number
               of
               angles
               ,
               requisite
               so
               to
               constitute
               and
               perfectly
               limite
               out
               a
               right-lined
               plane
               figure
               .
               And
               though
               some
               plane
               figures
               are
               perfected
               ,
               and
               perfectly
               bounded
               ,
               without
               any
               such
               angles
               as
               are
               contained
               by
               sides
               concurring
               by
               way
               either
               of
               section
               ,
               or
               Contact
               ,
               as
               namly
               all
               Circles
               and
               Ellipses
               ;
               yet
               the
               angularity
               of
               curve
               coincidence
               is
               every
               where
               found
               ,
               or
               at
               pleasure
               assignable
               in
               the
               bounders
               of
               such
               figures
               .
               But
               our
               meaning
               is
               ,
               a
               plane
               angle
               ,
               though
               most
               what
               it
               do
               not
               by
               the
               continuation
               and
               production
               of
               its
               sides
               ,
               perfectly
               bound
               in
               and
               limit
               out
               a
               certain
               plane
               and
               space
               on
               evry
               side
               ,
               however
               being
               the
               mutual
               habitude
               of
               concurring
               lines
               ,
               it
               gives
               an
               imperfect
               figuration
               to
               the
               plane
               and
               space
               on
               its
               part
               .
               And
               as
               a
               bounded
               plane
               cannot
               be
               
               without
               some
               kind
               of
               plane
               figure
               ,
               so
               a
               limited
               angle
               ever
               implies
               in
               it
               an
               imperfect
               figuration
               of
               some
               sort
               ,
               or
               other
               .
               For
               figuration
               is
               the
               consectary
               of
               material
               finiteness
               and
               limitation
               in
               the
               position
               of
               lengths
               ,
               breadths
               ,
               depths
               ,
               surfaces
               and
               solidities
               ;
               that
               every
               angle
               having
               its
               limits
               and
               bounds
               cannot
               be
               thereof
               destitute
               .
               And
               if
               the
               name
               of
               figures
               be
               so
               frequently
               given
               to
               hyperbola's
               ,
               parabola's
               ,
               and
               the
               like
               ,
               which
               neither
               do
               ,
               nor
               ever
               can
               by
               any
               possible
               production
               perfectly
               bound
               in
               their
               planes
               ;
               what
               reason
               is
               there
               then
               why
               angles
               should
               be
               denied
               an
               imperfect
               interest
               in
               the
               name
               .
               Besides
               as
               a
               plane
               in
               its
               own
               general
               nature
               at
               large
               doth
               not
               denote
               any
               special
               plane
               figure
               ;
               but
               the
               rise
               of
               figures
               ,
               I
               mean
               plane
               figures
               ,
               is
               from
               the
               bounding
               of
               the
               plane
               :
               so
               it
               is
               in
               angles
               ,
               as
               they
               by
               the
               mutual
               habitude
               of
               their
               concurring
               sides
               give
               imperfect
               limits
               and
               bounds
               unto
               the
               space
               and
               plane
               ,
               so
               they
               therein
               make
               an
               imperfect
               figuration
               .
               That
               in
               angles
               something
               of
               form
               and
               figure
               is
               to
               be
               noted
               as
               well
               as
               magnitude
               .
               And
               one
               line
               cannot
               concurre
               with
               and
               be
               inclined
               upon
               another
               ,
               but
               an
               imperfect
               figuration
               will
               arise
               from
               that
               their
               mutual
               inclination
               .
               And
               the
               same
               two
               angles
               may
               have
               the
               inclination
               ,
               
                 i.
                 e.
              
               the
               recesse
               of
               their
               respective
               sides
               one
               from
               another
               equal
               ,
               though
               there
               be
               no
               analogy
               between
               the
               figurations
               
               of
               the
               angles
               ,
               or
               the
               shapes
               in
               the
               which
               the
               sides
               are
               inclined
               in
               the
               one
               and
               in
               the
               other
               .
               For
               by
               reason
               that
               in
               angles
               form
               and
               figure
               are
               to
               be
               observed
               as
               well
               as
               quantity
               ;
               crooked-lined
               and
               right-lined
               angles
               may
               be
               equal
               in
               some
               particular
               quantity
               ,
               yet
               other-wise
               not
               of
               the
               same
               kind
               :
               they
               having
               equality
               in
               some
               magnitude
               ,
               but
               being
               distinct
               in
               the
               manner
               of
               their
               forming
               ,
               figuration
               and
               constitution
               :
               as
               equality
               may
               be
               between
               a
               square
               and
               a
               triangle
               ,
               though
               figures
               altogether
               different
               in
               kind
               .
            
             
               And
               in
               respect
               of
               such
               their
               figurations
               ,
               plane
               angles
               receive
               distinction
               ,
               either
               from
               the
               diverse
               manners
               in
               which
               their
               containing
               sides
               do
               concurre
               ,
               or
               else
               from
               the
               diverse
               natures
               ,
               and
               figurations
               of
               the
               lines
               under
               which
               they
               are
               contained
               ,
               or
               ,
               which
               is
               tantamount
               from
               the
               diversity
               of
               the
               inclinations
               and
               inflexions
               ,
               or
               rather
               inclinablenesses
               and
               imflexiblenesses
               by
               which
               they
               are
               inclined
               each
               to
               other
               ,
               or
               from
               several
               of
               these
               grounds
               of
               distinction
               taken
               together
               .
            
             
               Plane
               angles
               from
               the
               different
               manner
               of
               their
               sides
               concurring
               ,
               may
               aptly
               be
               thus
               distinguished
               ,
               Viz.
               into
               angles
               ,
               whose
               sides
               concurre
               by
               way
               of
               section
               ;
               or
               else
               that
               have
               their
               sides
               concurring
               only
               by
               way
               of
               touch
               in
               some
               single
               singular
               point
               without
               mutual
               section
               ;
               or
               else
               their
               concurrence
               is
               in
               curvature
               ,
               where
               ,
               
               after
               the
               meeting
               of
               their
               sides
               in
               the
               angular
               point
               ,
               the
               sides
               do
               not
               in
               their
               productions
               depart
               one
               from
               another
               ,
               neither
               by
               way
               of
               touch
               ,
               nor
               section
               ;
               but
               become
               ,
               the
               production
               of
               the
               one
               side
               coincident
               with
               the
               other
               side
               ;
               so
               as
               this
               kind
               of
               angles
               may
               aptly
               be
               called
               angles
               of
               coincidency
               ,
               or
               angles
               of
               curvature
               :
               and
               in
               these
               lies
               the
               genuine
               Ratio
               and
               true
               account
               of
               the
               curvature
               .
            
             
               From
               the
               diverse
               figuration
               of
               the
               lines
               ,
               under
               which
               a
               plane
               angle
               is
               contained
               ,
               very
               many
               differences
               of
               angles
               may
               arise
               ,
               according
               to
               the
               various
               distinctions
               ,
               of
               which
               lines
               themselves
               are
               capable
               ,
               I
               mean
               such
               lines
               ,
               as
               fall
               not
               without
               the
               capacity
               and
               comprehensivenes
               of
               the
               same
               plane
               :
               as
               that
               some
               are
               helicoidal
               ,
               some
               parabolar
               ,
               some
               elliptitical
               ,
               &c.
               
               But
               as
               of
               lines
               ,
               so
               hence
               of
               angles
               ,
               the
               chief
               and
               primary
               distinctions
               are
               especially
               these
               ,
               Viz.
               that
               plane
               angles
               are
               either
               right-lined
               angles
               ,
               contained
               under
               two
               inclined
               right-lines
               ;
               or
               not
               right-lined
               angles
               .
            
             
               Not
               right-lined
               angles
               ,
               are
               either
               mixed
               lined
               angles
               ;
               contained
               under
               one
               right-line
               ,
               and
               one
               crooked-line
               :
               or
               crooked-lined
               angles
               contained
               under
               two
               crooked-lined
               sides
               .
               And
               from
               the
               several
               kinds
               of
               special
               ,
               or
               ordinary
               curvatures
               ,
               as
               Circular
               ,
               elliptical
               ,
               hyperbolar
               ,
               &c.
               
               The
               mixt-lined
               ,
               and
               crooked-lined
               angles
               ,
               are
               capable
               of
               many
               farther
               and
               
               more
               particular
               distinctions
               ;
               but
               especially
               from
               the
               site
               of
               the
               convexeness
               ,
               or
               concaveness
               of
               the
               lines
               to
               ,
               or
               from
               the
               angle
               side
               :
               though
               all
               such
               secondary
               distinctions
               rising
               from
               these
               two
               last
               mentioned
               heads
               ,
               are
               as
               properly
               and
               pertinently
               referable
               to
               the
               other
               ground
               of
               distinguishing
               plane
               angles
               ,
               taken
               from
               the
               differences
               which
               may
               be
               in
               the
               inclinations
               of
               the
               one
               containing
               side
               to
               the
               other
               .
               For
               a
               vast
               difference
               is
               in
               the
               inclination
               of
               a
               crooked-line
               ,
               by
               obverting
               the
               concave
               ,
               or
               convexe
               side
               to
               any
               other
               line
               .
               So
               the
               constituting
               a
               circular
               ,
               or
               elliptical
               arch
               &c.
               
               For
               one
               side
               ,
               makes
               a
               vast
               difference
               in
               the
               inclinations
               ,
               because
               of
               the
               difference
               in
               their
               curvatures
               .
            
             
               Also
               another
               principal
               distinction
               of
               angles
               from
               their
               sides
               ,
               may
               be
               into
               angles
               ,
               whose
               sides
               are
               coaptable
               ,
               and
               by
               possibility
               may
               be
               coincident
               one
               with
               another
               :
               or
               else
               such
               as
               have
               between
               them
               no
               possibility
               of
               coaptation
               and
               coincidence
               .
               Of
               the
               former
               sort
               are
               all
               right-lined
               angles
               ,
               and
               all
               concavo-convexe
               angles
               contained
               by
               Arches
               of
               equal
               homogeneal
               ,
               uniform
               ,
               regular
               ,
               or
               answering
               curvatures
               :
               of
               the
               later
               sort
               are
               all
               other
               ;
               whether
               mixt
               lined
               angles
               ,
               or
               crooked-lined
               angles
               ;
               whether
               they
               be
               mixed
               crooked-lined
               angles
               ,
               or
               unmixt
               crooked-lined
               angles
               .
               And
               consequently
               thereupon
               ,
               besides
               the
               numerous
               distinctions
               of
               angles
               
               in
               respect
               of
               their
               different
               inclinations
               ,
               such
               as
               above
               mentioned
               ;
               one
               is
               more
               eminently
               material
               above
               the
               rest
               ,
               that
               the
               inclination
               of
               the
               sides
               ,
               is
               sometimes
               with
               an
               equability
               all
               along
               their
               production
               ;
               though
               imagined
               to
               be
               infinitely
               extended
               ,
               in
               such
               lines
               as
               by
               possibility
               may
               with
               reason
               be
               imagined
               so
               to
               be
               :
               and
               sometimes
               there
               is
               nothing
               of
               equability
               to
               be
               found
               in
               the
               inclination
               of
               the
               several
               parts
               of
               each
               side
               to
               the
               other
               ;
               though
               it
               may
               be
               one
               of
               the
               sides
               be
               a
               right-line
               ,
               or
               an
               Arch
               of
               most
               equal
               ,
               uniform
               ,
               regular
               and
               homogeneal
               curvature
               .
               And
               this
               equability
               and
               inequability
               of
               the
               inclination
               of
               the
               sides
               ,
               strangely
               alters
               the
               properties
               of
               angles
               .
               As
               in
               right-lined
               angles
               ,
               for
               the
               equability
               of
               the
               inclination
               of
               the
               sides
               ,
               no
               parts
               of
               the
               one
               side
               are
               more
               inclined
               then
               the
               rest
               unto
               the
               other
               side
               :
               and
               so
               in
               concavo-convexe
               angles
               of
               equal
               curvatures
               ,
               no
               parts
               of
               the
               one
               Arch
               are
               more
               inclined
               then
               the
               rest
               unto
               the
               other
               ;
               but
               the
               one
               Arch
               is
               all
               along
               inclined
               unto
               the
               other
               ,
               as
               at
               the
               angular
               point
               ;
               and
               the
               inclination
               ,
               which
               the
               one
               bears
               unto
               the
               other
               at
               the
               angular
               point
               ,
               is
               obviously
               expressable
               as
               to
               the
               quantity
               of
               the
               recesses
               ,
               which
               they
               make
               one
               from
               another
               ,
               by
               the
               inclination
               of
               a
               right-line
               to
               a
               right-line
               ,
               except
               when
               the
               inclination
               of
               the
               Arches
               is
               equal
               to
               ,
               or
               greater
               then
               two
               right
               right-lined
               
               angles
               .
               And
               in
               such
               crooked-lined
               angles
               ,
               whose
               sides
               have
               equability
               of
               inclination
               ,
               the
               points
               ,
               which
               from
               the
               angular
               point
               are
               at
               equal
               distances
               along
               the
               Arches
               ,
               are
               also
               absolutely
               at
               equal
               distance
               from
               the
               angular
               point
               ,
               along
               the
               chords
               :
               and
               right-lined
               tangents
               at
               any
               two
               such
               homologal
               points
               ,
               where
               ever
               taken
               ,
               alwayes
               meet
               and
               contain
               a
               right-lined
               angle
               ,
               equal
               as
               to
               the
               quantities
               of
               the
               recesses
               of
               the
               sides
               ,
               to
               the
               crooked-lined
               angle
               contained
               by
               the
               two
               Arches
               ;
               as
               is
               obvious
               to
               demonstrate
               ,
               especially
               in
               circular
               Arches
               .
               And
               the
               right-lined
               angle
               contained
               under
               the
               two
               right-lined
               tangents
               touching
               at
               the
               two
               homologal
               and
               answering
               points
               ,
               which
               is
               equal
               to
               the
               isoclitical
               crooked-lined
               angle
               ,
               if
               the
               two
               right-lined
               tangents
               occurre
               on
               that
               side
               of
               the
               right-line
               connecting
               the
               homologal
               points
               on
               which
               the
               isoclitical
               angle
               falls
               ,
               then
               it
               is
               the
               angle
               contained
               by
               the
               two
               right-lined
               tangents
               into
               whose
               space
               part
               of
               the
               space
               comprized
               between
               the
               two
               Arches
               at
               first
               falls
               ,
               which
               is
               equal
               to
               the
               crooked-lined
               isoclitical
               angle
               :
               but
               if
               they
               occurre
               on
               the
               other
               side
               of
               the
               right-line
               connecting
               the
               two
               homologal
               points
               ,
               
                 i.
                 e.
              
               aversely
               from
               the
               crooked-line
               isoclitical
               angle
               ,
               then
               it
               is
               the
               complement
               of
               such
               an
               angle
               ,
               which
               is
               equal
               to
               the
               crooked-line
               isoclitical
               angle
               :
               but
               if
               the
               two
               right-lined
               tangents
               occurre
               in
               one
               of
               the
               homologal
               points
               ,
               
               the
               angles
               either
               way
               contained
               under
               the
               two
               right
               line
               tangents
               are
               equal
               ,
               viz.
               right
               right-lined
               angles
               ;
               either
               of
               them
               making
               forth
               what
               is
               herein
               asserted
               .
               As
               in
               fig
               :
               19.
               under
               the
               two
               circular
               isoclitical
               arches
               bda
               and
               acn
               let
               there
               be
               constituted
               the
               isoclitical
               angle
               bac
               ;
               and
               let
               the
               right-line
               ag
               touch
               the
               Arch
               acn
               in
               the
               point
               a
               :
               and
               let
               the
               right-line
               af
               touch
               the
               Arch
               adb
               in
               the
               point
               a
               :
               so
               making
               the
               right-lined
               angle
               fag
               equal
               to
               the
               isoclitical
               concavo-convexe
               angle
               bac
               .
               Then
               take
               in
               the
               Arch
               adb
               any
               point
               at
               pleasure
               ,
               Viz.
               the
               point
               d
               :
               and
               draw
               the
               chord
               ad
               .
               Then
               in
               the
               Arch
               acn
               take
               the
               Arch
               ac
               subtended
               by
               the
               chord
               ac
               equal
               to
               the
               chord
               ad
               .
               Therefore
               because
               of
               the
               isocliticalness
               of
               the
               circular
               Arches
               the
               two
               points
               d
               and
               c
               are
               two
               homologal
               ,
               
                 i.
                 e.
              
               answering
               points
               the
               one
               in
               the
               one
               Arch
               ,
               the
               other
               in
               the
               other
               ,
               Viz.
               the
               point
               d
               in
               the
               Arch
               adb
               and
               the
               point
               c
               in
               the
               Arch
               acn
               .
               Then
               draw
               the
               right-line
               dc
               connecting
               the
               two
               homologal
               points
               d
               and
               c.
               Also
               draw
               the
               right-line
               de
               touching
               the
               arch
               adb
               in
               the
               point
               d
               ;
               and
               the
               right-line
               ce
               touching
               the
               Arch
               acn
               in
               the
               point
               c.
               And
               let
               de
               and
               ce
               the
               two
               right-line
               tangents
               be
               produced
               till
               they
               meet
               in
               the
               point
               e
               ;
               which
               in
               this
               figure
               is
               on
               that
               side
               of
               the
               right-line
               dc
               on
               which
               the
               concavo-convexe
               angle
               cab
               lyeth
               .
               I
               say
               therefore
               
               that
               the
               right-lined
               angle
               dec
               contained
               under
               the
               two
               right-lined
               tangents
               de
               and
               ce
               touching
               the
               Arches
               respectively
               at
               the
               homologal
               points
               d
               and
               c
               is
               equal
               to
               the
               right-lined
               angle
               fag
               ,
               contained
               under
               the
               two
               right-line
               tangents
               fa
               and
               ga
               touching
               the
               Arches
               respectively
               at
               a
               the
               angular
               point
               of
               the
               isoclitical
               concavo-convexe
               angle
               .
               For
               the
               right-lined
               tangent
               fa
               cutting
               de
               the
               other
               right-lined
               tangent
               of
               the
               same
               Arch
               adb
               in
               the
               point
               h
               ;
               and
               the
               right-lined
               tangent
               de
               of
               the
               Arch
               adb
               cutting
               the
               chord
               ac
               in
               the
               point
               k
               ;
               upon
               this
               construction
               the
               right-line
               da
               is
               equal
               to
               the
               right-line
               ac
               ;
               and
               the
               right-line
               tangent
               dh
               is
               equal
               to
               the
               right-line
               tangent
               ha
               :
               therefore
               the
               right-lined
               angle
               adh
               is
               equal
               to
               the
               right-lined
               angle
               dah
               and
               so
               to
               the
               right-lined
               angles
               ace
               and
               cag
               severally
               .
               And
               therefore
               the
               right-lined
               angle
               ahe
               being
               equal
               to
               the
               two
               right-lined
               angles
               hda
               and
               dah
               taken
               together
               ,
               and
               the
               right-lined
               angle
               hda
               being
               equal
               to
               the
               right-lined
               angle
               cag
               ;
               the
               right-lined
               angle
               ahe
               is
               equal
               to
               the
               two
               right-lined
               angles
               cag
               and
               dah
               taken
               together
               .
               Therefore
               that
               which
               maketh
               each
               equal
               to
               two
               right
               right-lined
               angles
               ;
               the
               two
               right-lined
               angles
               hka
               and
               hak
               taken
               together
               are
               equal
               to
               the
               two
               right-lined
               angles
               hak
               and
               dal
               taken
               together
               .
               Therefore
               the
               right-lined
               angle
               
               hka
               is
               equal
               to
               the
               right-lined
               angle
               dal
               .
               Therefore
               the
               right-lined
               angle
               cke
               is
               equal
               to
               the
               right-lined
               angle
               dal
               .
               And
               therefore
               that
               which
               makes
               either
               equal
               to
               two
               right
               right-lined
               angles
               ,
               the
               two
               right-lined
               angles
               kce
               and
               kec
               together
               taken
               are
               equal
               to
               the
               right-lined
               angle
               dag
               ,
               which
               is
               equal
               to
               the
               two
               right-lined
               angles
               dac
               and
               cag
               taken
               together
               :
               and
               the
               right-lined
               angle
               cag
               is
               equal
               to
               the
               right-lined
               angle
               kce
               :
               therefore
               the
               right-lined
               angle
               kec
               is
               equal
               to
               the
               right-lined
               angle
               dac
               :
               therefore
               because
               the
               right-lined
               angles
               dah
               and
               cag
               are
               equal
               ;
               also
               the
               right-lined
               angle
               kec
               S
               c.
               dec
               is
               equal
               to
               the
               right-lined
               angle
               hag
               S
               c.
               fag
               ,
               which
               was
               to
               be
               demonstrated
               .
            
             
               But
               if
               the
               two
               right-lined
               tangents
               de
               and
               ce
               as
               in
               fig.
               20.
               do
               not
               occurre
               towards
               the
               concavo-convexe
               angle
               bac
               ,
               but
               on
               the
               other
               side
               of
               the
               right-line
               dc
               in
               the
               point
               e
               ;
               then
               is
               the
               right
               lined
               angle
               dec
               contained
               under
               the
               two
               right
               line
               tangents
               de
               and
               ce
               touching
               at
               the
               homologal
               points
               d
               and
               c
               ,
               not
               equal
               to
               the
               concavo-convexe
               isoclitical
               angle
               bac
               ,
               or
               the
               right
               lined
               angle
               ,
               equal
               unto
               it
               ,
               fag
               ,
               but
               to
               its
               complement
               unto
               two
               right
               right-lined
               angles
               ,
               Viz.
               unto
               the
               right
               lined
               angle
               fal
               ;
               the
               right
               line
               la
               being
               the
               production
               of
               the
               right
               line
               ga
               .
               For
               as
               before
               by
               construction
               ,
               the
               right
               line
               chords
               da
               and
               ca
               to
               the
               
               homologal
               points
               d
               and
               c
               are
               equal
               :
               and
               the
               right
               line
               fa
               cutting
               the
               right
               line
               de
               produced
               in
               the
               point
               h
               ,
               the
               right
               lines
               dh
               and
               ah
               being
               two
               right
               lines
               tangents
               of
               the
               same
               Circle
               adb
               ,
               occurring
               ,
               are
               equal
               .
               And
               let
               the
               right
               line
               ac
               produced
               ,
               occurre
               with
               the
               right
               line
               de
               produced
               ,
               in
               the
               point
               k.
               As
               appears
               ;
               the
               right
               lined
               angles
               adh
               ,
               dah
               and
               cag
               ,
               as
               before
               ,
               are
               equal
               ;
               and
               the
               right
               lined
               angle
               ahe
               is
               equal
               to
               two
               right
               right
               lined
               angles
               ,
               all
               but
               the
               two
               right
               lined
               angles
               hda
               and
               had
               that
               is
               ,
               all
               but
               the
               two
               right
               lined
               angles
               cag
               and
               had
               .
               Therefore
               the
               right
               lined
               angle
               ahe
               being
               equal
               to
               the
               two
               right
               lined
               angles
               hka
               and
               kah
               ;
               the
               two
               right
               lined
               angles
               hka
               and
               kah
               are
               equal
               to
               two
               right
               right
               lined
               angles
               all
               but
               the
               two
               right
               lined
               angles
               cag
               and
               had
               .
               Therefore
               two
               right
               right-lined
               angles
               are
               equal
               to
               the
               four
               right
               lined
               angles
               cag
               and
               had
               and
               hka
               and
               kah
               .
               Therefore
               out
               of
               equals
               taking
               equals
               ,
               the
               right
               lined
               angle
               hka
               ,
               which
               is
               the
               right
               lined
               angle
               cke
               is
               equal
               to
               the
               right
               lined
               angle
               dal
               .
               Therefore
               what
               on
               either
               side
               remains
               to
               make
               up
               two
               right
               right
               lined
               angles
               on
               either
               part
               ;
               the
               two
               right
               lined
               angles
               kce
               and
               kec
               are
               together
               equal
               to
               the
               right
               lined
               angle
               dag
               ,
               which
               is
               equal
               to
               the
               two
               right
               lined
               angles
               dac
               and
               cag
               taken
               together
               .
               And
               producing
               the
               right
               line
               ec
               till
               it
               cut
               the
               right
               line
               ag
               in
               the
               
               point
               g
               ;
               the
               right
               lined
               angles
               cag
               and
               acg
               and
               kce
               are
               equal
               .
               Therefore
               the
               right
               lined
               angle
               kec
               is
               equal
               to
               the
               right
               lined
               angle
               dac
               .
               And
               because
               the
               right-lined
               angle
               dah
               and
               cag
               are
               equal
               ,
               therefore
               adding
               the
               common
               angle
               fac
               ,
               the
               two
               right
               lined
               angles
               dac
               and
               hag
               are
               equal
               .
               And
               therefore
               the
               right
               lined
               angle
               kec
               is
               equal
               to
               the
               right
               lined
               angle
               hag
               .
               And
               therefore
               their
               complements
               unto
               two
               right
               right
               lined
               angles
               ,
               Viz.
               the
               two
               right
               lined
               angles
               dec
               and
               hal
               are
               equal
               ;
               which
               was
               to
               be
               demonstrated
               .
            
             
               So
               in
               fig.
               22.
               if
               the
               right
               line
               tangent
               de
               passe
               directly
               unto
               the
               other
               homologal
               point
               c
               ,
               as
               it
               doth
               when
               the
               isoclitical
               concavo-convexe
               angle
               is
               equal
               to
               a
               right
               right
               lined
               angle
               ,
               and
               the
               homologal
               points
               d
               and
               c
               are
               taken
               at
               quadrantal
               or
               other
               distances
               from
               the
               angular
               point
               ;
               then
               most
               manifestly
               the
               right-lined
               angles
               dcg
               and
               fag
               are
               equal
               ,
               being
               under
               the
               two
               and
               two
               respective
               right
               lined
               tangents
               ;
               and
               between
               the
               angles
               and
               their
               complements
               unto
               two
               right
               right
               lined
               angles
               is
               no
               difference
               ,
               as
               appears
               by
               what
               is
               in
               the
               former
               demonstrations
               .
               So
               in
               fig.
               23.
               if
               the
               homologal
               points
               d
               and
               c
               be
               so
               taken
               ,
               that
               the
               chords
               da
               and
               ca
               be
               the
               diameters
               :
               then
               producing
               the
               right
               line
               tangent
               ce
               till
               it
               occurre
               with
               the
               other
               right
               line
               tangent
               de
               in
               the
               point
               e
               ,
               and
               with
               the
               other
               right-line
               tangent
               
               fa
               in
               the
               point
               f
               ;
               the
               right
               lined
               angles
               cfa
               and
               fag
               are
               equal
               .
               And
               in
               the
               trapezium
               def
               a
               the
               two
               right
               lined
               angles
               eda
               and
               daf
               are
               each
               of
               them
               a
               right
               right-lined
               angle
               ,
               therefore
               the
               right-lined
               angle
               dec
               is
               the
               complement
               of
               the
               right
               lined
               angle
               efa
               unto
               two
               right
               right-lined
               angles
               ,
               
                 i.
                 e.
              
               it
               is
               the
               complement
               of
               the
               angle
               fag
               unto
               two
               right
               right
               lined
               angles
               ;
               which
               was
               to
               be
               demonstrated
               .
               And
               like
               demonstrations
               may
               be
               formed
               upon
               every
               other
               case
               .
               But
               where
               the
               isoclitical
               arches
               are
               not
               circular
               the
               demonstrations
               must
               vary
               according
               to
               the
               propriety
               of
               every
               several
               curvature
               ,
               notwithstanding
               its
               equability
               and
               isocliticalness
               .
            
             
               But
               to
               return
               ,
               in
               angles
               contained
               by
               sides
               whose
               inclination
               each
               to
               other
               is
               without
               any
               such
               equability
               ,
               (
               except
               only
               in
               unmixt
               crooked-lined
               anisoclitical
               angles
               )
               the
               account
               of
               the
               homologal
               points
               along
               the
               chords
               and
               along
               the
               arches
               is
               still
               different
               :
               but
               in
               none
               of
               them
               do
               the
               right
               line
               tangents
               from
               the
               two
               and
               two
               homologal
               points
               ,
               still
               in
               their
               meeting
               make
               the
               two
               same
               angles
               which
               are
               made
               by
               the
               two
               right
               lined
               tangents
               at
               the
               angular
               point
               ;
               nor
               can
               any
               right
               line
               angle
               expresse
               the
               inclination
               ,
               which
               the
               sides
               have
               each
               to
               other
               at
               the
               angular
               point
               .
            
             
               Hence
               is
               manifest
               how
               the
               same
               angle
               may
               from
               several
               of
               the
               grounds
               of
               distinction
               here
               
               proposed
               be
               referable
               to
               several
               heads
               ,
               or
               kinds
               .
               So
               right
               lined
               angles
               as
               isoclitical
               ,
               have
               alwayes
               equability
               of
               inclination
               ;
               and
               the
               concurrence
               of
               their
               sides
               is
               alwayes
               by
               way
               of
               section
               and
               cannot
               be
               by
               Contact
               ,
               or
               coincidency
               .
               So
               mixed
               lined
               angles
               being
               anisoclitical
               have
               always
               inequability
               in
               the
               inclination
               of
               their
               sides
               ;
               and
               their
               concurrence
               may
               be
               either
               by
               Contact
               ,
               or
               Section
               ,
               but
               never
               by
               coincidence
               .
               And
               in
               crooked
               lined
               angles
               ,
               their
               sides
               may
               have
               either
               equability
               ,
               or
               inequability
               of
               inclination
               :
               and
               accordingly
               the
               concurrence
               of
               the
               sides
               may
               be
               by
               section
               ,
               or
               Contact
               ;
               and
               with
               ,
               or
               without
               possibility
               of
               coincidence
               Sc.
               Isoclitical
               crooked-lines
               ,
               but
               they
               must
               be
               posited
               convexo-convexely
               ,
               or
               anisoclitical
               crooked-lines
               posited
               whether
               convexo-convexely
               ,
               or
               concavo-convexely
               ,
               or
               a
               right-line
               and
               a
               crooked
               line
               ,
               any
               two
               of
               these
               may
               touch
               without
               cutting
               ,
               and
               so
               the
               angular
               sides
               have
               inclination
               tangent
               and
               not
               secant
               ,
               So
               isoclitical
               concavo-convexe
               angles
               may
               have
               sides
               circumducted
               to
               coincidence
               :
               but
               the
               same
               sides
               posited
               concavo-concavely
               ,
               or
               convexo-convexely
               become
               anisoclitical
               in
               the
               circumduction
               ,
               one
               in
               respect
               of
               the
               other
               ;
               yet
               either
               is
               isoclitical
               sometimes
               and
               in
               some
               cases
               with
               the
               production
               of
               the
               other
               .
            
             
               From
               these
               things
               though
               distinguishing
               angles
               into
               their
               several
               kinds
               ,
               only
               with
               respect
               
               to
               the
               diversity
               of
               their
               figuration
               ,
               may
               however
               more
               abundantly
               appear
               how
               unmanageable
               a
               task
               they
               take
               upon
               themselves
               ,
               who
               to
               exclude
               recto-convexe
               angles
               of
               Contact
               from
               being
               angles
               ,
               and
               from
               quantitaveness
               ,
               would
               force
               all
               plane
               angles
               to
               be
               of
               the
               same
               kind
               ,
               allowing
               no
               specifick
               difference
               possible
               among
               them
               .
               Not
               here
               to
               pursue
               what
               other
               diversities
               in
               kind
               may
               be
               observed
               among
               angles
               ;
               how
               can
               the
               inclination
               of
               a
               crooked-line
               upon
               a
               right-line
               differ
               less
               then
               in
               kind
               from
               the
               inclination
               of
               a
               right-line
               upon
               a
               right-line
               ?
               For
               as
               a
               right-line
               and
               a
               crooked-line
               agree
               as
               lenghts
               and
               in
               lineariness
               and
               are
               therein
               mathematically
               homogeneal
               ,
               but
               as
               right
               and
               crooked
               differ
               in
               kind
               and
               have
               therein
               heterogeneal
               figuration
               ;
               the
               crookedness
               of
               the
               crooked-line
               having
               no
               analogy
               to
               the
               rectitude
               of
               the
               right-line
               ,
               nor
               the
               rectitude
               of
               the
               right
               line
               any
               proportionableness
               unto
               the
               curvature
               of
               the
               crooked-line
               ;
               so
               the
               inclination
               of
               a
               crooked
               line
               upon
               a
               right
               line
               ,
               and
               of
               a
               right
               line
               upon
               a
               right
               line
               they
               agree
               in
               the
               common
               nature
               of
               inclination
               ,
               and
               the
               one
               may
               be
               greater
               and
               the
               other
               less
               as
               the
               respective
               sides
               fall
               within
               ,
               or
               without
               :
               but
               this
               relation
               of
               greater
               ,
               or
               less
               is
               without
               any
               proportionableness
               ,
               and
               only
               by
               the
               whole
               kind
               ,
               depending
               upon
               the
               passing
               of
               the
               respective
               sides
               ,
               the
               one
               within
               ,
               or
               
               without
               the
               other
               .
               For
               these
               two
               inclinations
               differ
               so
               far
               in
               kind
               that
               neither
               the
               curvature
               of
               the
               mixt
               lined
               inclination
               hath
               any
               thing
               in
               it
               conform
               ,
               or
               proportionable
               to
               the
               rectitude
               of
               the
               right-lined
               inclination
               ,
               nor
               the
               rectitude
               of
               the
               right-lined
               inclination
               to
               the
               curvature
               of
               the
               mixt
               lined
               inclination
               .
               In
               a
               word
               ,
               so
               different
               is
               the
               inclinableness
               of
               a
               crooked
               line
               upon
               a
               right
               line
               ,
               from
               the
               inclinableness
               of
               a
               right
               line
               upon
               a
               right
               line
               ,
               that
               it
               is
               impossible
               for
               the
               one
               ever
               to
               be
               either
               equal
               ,
               or
               any
               way
               determinately
               proportionable
               unto
               the
               other
               :
               because
               the
               coaptation
               of
               a
               right
               line
               as
               a
               right
               line
               ,
               to
               a
               crooked-line
               ,
               as
               a
               crooked
               Iine
               is
               against
               the
               properties
               of
               their
               figurations
               ,
               kinds
               and
               natures
               .
               And
               for
               what
               reason
               should
               there
               be
               lesser
               difference
               between
               a
               crooked-lined
               inclination
               and
               a
               right
               lined
               inclination
               ,
               then
               there
               is
               between
               a
               crooked-line
               and
               a
               right
               line
               ?
               Yet
               all
               this
               their
               distinctness
               concerns
               only
               an
               heterogeneity
               in
               their
               figuration
               and
               not
               at
               all
               ,
               or
               not
               primarily
               their
               quantities
               .
            
             
               The
               argument
               ,
               if
               they
               be
               angles
               ,
               or
               plane
               angles
               ,
               they
               are
               homogeneal
               and
               of
               the
               same
               kind
               ,
               is
               of
               no
               more
               force
               then
               this
               consequence
               ;
               if
               they
               be
               quantities
               ,
               or
               continuous
               magnitudes
               ,
               they
               are
               homogeneal
               and
               of
               the
               same
               kind
               .
               And
               they
               that
               deny
               all
               heterogeneity
               in
               angles
               ,
               because
               they
               are
               all
               angles
               ;
               will
               find
               
               it
               an
               hard
               task
               upon
               the
               same
               ground
               to
               maintain
               an
               analogous
               homogeneity
               ,
               or
               any
               other
               considerable
               homogeneity
               ,
               between
               right
               lined
               angles
               and
               sphaerical
               angles
               ,
               or
               any
               other
               angles
               ,
               made
               by
               planes
               cutting
               the
               heterepipedal
               surfaces
               of
               solids
               ,
               and
               especially
               solid
               angles
               ,
               of
               what
               sort
               soever
               .
               And
               to
               yield
               that
               all
               plane
               angles
               are
               homogeneal
               ;
               for
               it
               is
               true
               :
               and
               the
               most
               absolute
               ,
               proper
               and
               genuine
               homogeneity
               is
               among
               plane
               angles
               :
               
                 i.
                 e.
              
               no
               part
               of
               a
               plane
               angle
               can
               be
               any
               other
               then
               a
               plane
               angle
               ,
               how
               great
               ,
               or
               little
               soever
               ,
               and
               whether
               proportionable
               ,
               or
               improportionable
               one
               to
               another
               ,
               
                 i.
                 e.
              
               whether
               mathematically
               homogeneal
               ,
               or
               heterogeneal
               :
               yet
               if
               we
               seriously
               consider
               what
               is
               this
               homogeneity
               which
               is
               among
               plane
               angles
               ,
               that
               all
               their
               parts
               are
               plane
               angles
               ,
               it
               is
               not
               as
               is
               said
               any
               quantitative
               ,
               or
               mathematical
               homogeneity
               ,
               the
               contrary
               of
               which
               is
               plentifully
               demonstrated
               in
               Geometry
               to
               be
               possible
               ;
               nor
               any
               such
               homogeneity
               in
               respect
               of
               the
               manner
               of
               their
               positure
               and
               figuration
               ,
               as
               to
               exclude
               all
               farther
               distinguishableness
               in
               respect
               of
               figuration
               ;
               but
               only
               denotes
               that
               in
               every
               plane
               angle
               ,
               and
               in
               every
               part
               of
               every
               plane
               angle
               ,
               the
               sides
               lye
               still
               in
               the
               same
               plane
               :
               which
               homogeneity
               ,
               as
               is
               plain
               ,
               excludes
               neither
               heterogeneity
               ,
               in
               respect
               of
               figuration
               ,
               nor
               in
               respect
               of
               proportion
               and
               identity
               in
               the
               way
               of
               measuring
               their
               quantities
               .
            
             
             
               To
               the
               objection
               that
               in
               fig.
               12.
               the
               recto-convexe
               angle
               of
               Contact
               BAF
               can
               be
               added
               to
               the
               right
               right-lined
               angle
               BAG
               ,
               so
               making
               the
               outer
               angle
               of
               a
               semi
               Circle
               F
               
               AG
               ;
               or
               taken
               out
               of
               it
               ,
               Sc.
               the
               recto-convexe
               angle
               of
               Contact
               BAD
               out
               of
               the
               same
               right
               right-lined
               angle
               BAG
               ,
               so
               making
               the
               inner
               angle
               of
               a
               semi
               Circle
               GAD
               :
               and
               that
               therefore
               the
               recto-convexe
               angles
               of
               Contact
               F
               
               AB
               and
               B
               
               AD
               and
               the
               right
               right-lined
               angle
               BAG
               ,
               and
               the
               two
               angles
               of
               the
               semi
               Circle
               GAD
               ,
               and
               FAG
               ,
               are
               all
               of
               them
               homogeneal
               and
               of
               the
               same
               kind
               ;
               I
               answer
               .
               First
               what
               need
               is
               there
               of
               such
               endeavours
               ,
               for
               you
               to
               prove
               their
               homogeneity
               ,
               it
               being
               Geometrically
               demonstrated
               and
               confessed
               that
               there
               is
               no
               analogy
               ,
               or
               proportion
               between
               them
               ;
               I
               mean
               between
               the
               recto-convexe
               angles
               of
               Contact
               and
               either
               the
               right
               right
               lined
               angle
               ,
               or
               either
               of
               the
               angles
               of
               the
               semi
               Circles
               ?
               And
               according
               to
               your
               opinion
               that
               which
               is
               added
               or
               taken
               out
               is
               said
               to
               be
               nothing
               .
               But
               especially
               its
               thought
               strange
               ,
               why
               there
               should
               be
               such
               doubting
               ,
               that
               heterogeneals
               can
               be
               added
               and
               laid
               up
               together
               as
               into
               one
               repository
               ;
               it
               being
               with
               as
               easy
               connexion
               performable
               ,
               as
               is
               usual
               in
               the
               addition
               of
               incommensurables
               and
               specious
               quantities
               of
               which
               it
               is
               not
               known
               ,
               whether
               they
               be
               homogeneal
               ,
               or
               heterogeneal
               .
               And
               out
               of
               an
               heterogeneal
               sum
               ,
               as
               a
               store-house
               ,
               
               why
               cannot
               some
               of
               the
               heterogeneals
               be
               subducted
               ,
               the
               rest
               remaining
               ?
               and
               what
               is
               more
               usual
               then
               the
               adding
               of
               heterogeneal
               figures
               one
               unto
               another
               ?
               and
               subducting
               out
               of
               a
               given
               figure
               some
               other
               figure
               ,
               which
               is
               quite
               heterogeneal
               to
               the
               first
               given
               figure
               ?
               So
               to
               adde
               together
               numbers
               ,
               and
               measures
               ,
               and
               weights
               ?
               the
               sum
               of
               which
               may
               be
               divided
               ,
               multiplyed
               ,
               increased
               ,
               or
               lessened
               ,
               notwithstanding
               its
               heterogeneity
               .
               As
               supposing
               A
               B
               ,
               C
               ,
               D
               ,
               all
               heterogeneal
               ,
               as
               is
               usual
               in
               analyticks
               ,
               the
               half
               ,
               or
               third
               part
               ,
               or
               any
               proportionable
               part
               of
               this
               heterogeneal
               sum
               may
               be
               given
               :
               and
               any
               one
               of
               the
               heterogeneal
               magnitudes
               subducted
               ,
               the
               rest
               remaining
               :
               or
               a
               fifth
               heterogeneal
               magnitude
               added
               to
               the
               former
               sum
               :
               or
               any
               Algorythme
               ,
               ever
               speciously
               ,
               sometimes
               compleatly
               and
               absolutely
               thereupon
               performed
               .
            
             
               Besides
               upon
               geometrical
               demonstration
               and
               your
               own
               confession
               ,
               all
               recto-convexe
               ,
               convexo-convexe
               and
               citradiametral
               concavo-convexe
               angles
               of
               Contact
               must
               necessarily
               by
               your
               own
               principles
               be
               allowed
               to
               be
               absolutely
               heterogeneal
               to
               all
               right-lined
               angles
               whatsoever
               ;
               your self
               acknowledging
               that
               neither
               in
               equality
               ,
               nor
               in
               any
               kind
               of
               multiplicity
               ,
               or
               submultiplicity
               is
               any
               proportionableness
               possible
               amongst
               them
               .
               And
               where
               between
               angles
               a
               mathematical
               homogeneity
               is
               confessed
               
               and
               allowed
               ,
               yet
               heterogeneity
               in
               respect
               of
               their
               Schematismes
               and
               figurations
               is
               undeniable
               .
            
             
               The
               things
               therefore
               constituting
               and
               distinguishing
               plane
               angles
               in
               respect
               of
               their
               figuration
               are
               ,
               as
               above
               ,
               their
               sides
               ,
               their
               inclinations
               ,
               or
               rather
               inclineablenesses
               ,
               and
               their
               concurrence
               .
               That
               when
               two
               angles
               have
               all
               these
               in
               the
               same
               respective
               kinds
               ,
               the
               angles
               are
               upon
               good
               reason
               in
               this
               sense
               concluded
               to
               be
               homogeneal
               :
               but
               when
               between
               two
               angles
               is
               an
               heterogeneity
               in
               any
               of
               these
               things
               ,
               which
               are
               of
               the
               essence
               and
               constitution
               of
               an
               angle
               ,
               those
               angles
               may
               justly
               be
               judged
               in
               this
               sense
               to
               be
               heterogeneal
               .
               And
               that
               such
               a
               specifick
               heterogeneity
               may
               be
               in
               each
               of
               these
               ,
               may
               easily
               be
               declared
               as
               above
               .
               As
               first
               in
               lines
               which
               are
               the
               containing
               sides
               ,
               how
               easy
               is
               it
               to
               discover
               such
               an
               heterogeneity
               ?
               For
               though
               a
               right
               line
               ,
               and
               a
               crooked-line
               agree
               undeniably
               in
               the
               general
               nature
               of
               a
               line
               ,
               and
               of
               length
               ,
               and
               of
               extension
               ,
               yet
               the
               rectitude
               of
               the
               one
               and
               the
               curvature
               of
               the
               other
               ,
               are
               several
               kinds
               of
               positure
               ,
               into
               which
               the
               length
               of
               the
               one
               ,
               and
               of
               the
               other
               is
               disposed
               :
               that
               except
               ,
               in
               contrarieties
               ,
               we
               can
               see
               nothing
               but
               homogeneity
               ,
               such
               an
               heterogeneity
               must
               needs
               be
               acknowledged
               between
               them
               .
               And
               whereas
               homogeneity
               ,
               as
               to
               sides
               ,
               inclination
               and
               concurrence
               ,
               is
               required
               to
               the
               homogeneal
               
               figuration
               of
               angles
               ;
               the
               heterogeneity
               of
               the
               sides
               hinders
               the
               possibility
               of
               ever
               making
               them
               out
               to
               be
               such
               ;
               or
               that
               by
               any
               altering
               their
               divarication
               ,
               keeping
               their
               present
               properties
               they
               can
               be
               coaptable
               .
               And
               that
               angles
               contained
               by
               heterogeneal
               sides
               may
               be
               equal
               ,
               proves
               only
               the
               equality
               of
               the
               inclinations
               in
               either
               ,
               but
               not
               the
               homogeneity
               of
               the
               figuration
               of
               the
               angles
               ,
               or
               inclinations
               ;
               as
               the
               equality
               between
               a
               square
               and
               triangle
               in
               respect
               of
               their
               equal
               perimeters
               ,
               area's
               ,
               heights
               ,
               bases
               ,
               proves
               not
               in
               the
               least
               the
               homogeneity
               of
               their
               figures
               .
               And
               as
               right-lines
               and
               crooked-lines
               are
               heterogeneal
               ,
               as
               above
               ,
               not
               possibly
               to
               be
               coapted
               ,
               with
               the
               precedent
               limitations
               ;
               So
               also
               are
               all
               curve
               lines
               ,
               whose
               curvature
               is
               unequal
               and
               unlike
               ,
               nay
               though
               it
               may
               be
               they
               be
               but
               several
               parts
               of
               the
               same
               line
               ,
               or
               though
               the
               curvature
               of
               both
               ,
               be
               every
               way
               ,
               and
               every
               where
               equal
               and
               like
               ,
               yet
               if
               the
               convexe
               and
               concave
               parts
               of
               the
               one
               be
               not
               alike
               posited
               as
               in
               the
               other
               ,
               there
               will
               be
               a
               manifest
               heterogeneity
               in
               them
               ,
               and
               an
               impossibility
               of
               coapting
               them
               ,
               observing
               the
               limitations
               as
               above
               .
            
             
               And
               why
               doth
               the
               heterogeneity
               in
               the
               sides
               make
               heterogeneity
               in
               the
               angles
               ,
               but
               because
               thereby
               is
               founded
               an
               heterogeneity
               in
               the
               inclinations
               ,
               or
               rather
               in
               the
               inclinablenesses
               of
               the
               one
               side
               to
               the
               other
               ?
               For
               here
               ,
               it
               is
               not
               the
               several
               degrees
               of
               the
               same
               kind
               of
               inclination
               
               that
               is
               intended
               ;
               for
               then
               all
               unequal
               right-lined
               angles
               should
               be
               altogether
               heterogeneal
               one
               to
               another
               :
               but
               it
               is
               a
               more
               then
               gradual
               ,
               a
               specifick
               distinctness
               in
               their
               inclinablenesses
               ,
               which
               we
               are
               now
               discovering
               to
               make
               the
               figuration
               of
               the
               angles
               more
               fairly
               and
               fully
               heterogeneal
               .
               And
               as
               inclination
               is
               the
               habitude
               of
               line
               to
               line
               ,
               not
               being
               posited
               in
               the
               same
               right
               line
               ,
               nor
               parallel
               ,
               for
               even
               perpendiculars
               are
               in
               this
               sense
               here
               said
               to
               be
               inclined
               ;
               so
               ,
               as
               above
               ,
               from
               the
               heterogeneity
               of
               the
               lines
               will
               arise
               an
               heterogeneity
               of
               inclinations
               :
               and
               indeed
               for
               no
               other
               reasons
               do
               heterogeneal
               lines
               make
               heterogeneal
               angles
               ,
               but
               because
               their
               inclinations
               are
               necessarily
               heterogeneal
               .
               And
               ,
               as
               above
               ,
               heterogeneal
               inclinations
               being
               respectively
               equal
               ,
               as
               in
               some
               right
               lined
               and
               crooked
               lined
               angles
               ,
               this
               doth
               not
               in
               the
               least
               annul
               the
               heterogeneity
               of
               their
               inclinations
               ;
               as
               a
               right
               line
               and
               a
               crooked
               line
               may
               be
               equal
               ,
               yet
               as
               to
               the
               positure
               of
               their
               extension
               they
               are
               heterogeneal
               .
            
             
               And
               as
               heterogeneity
               of
               sides
               ,
               or
               inclinations
               makes
               heterogeneity
               of
               angles
               ,
               so
               likewise
               doth
               any
               heterogenealnes
               in
               the
               other
               point
               requisite
               to
               the
               nature
               of
               an
               angle
               ;
               which
               is
               the
               manner
               of
               the
               sides
               concurrence
               .
               And
               there
               are
               only
               three
               wayes
               in
               the
               concurrence
               of
               the
               sides
               of
               angles
               ,
               according
               to
               which
               they
               can
               be
               heterogeneal
               one
               to
               another
               .
               For
               either
               the
               
               production
               of
               the
               one
               concurring
               side
               becomes
               coincident
               with
               the
               other
               concurring
               side
               ,
               or
               else
               it
               departs
               from
               it
               on
               the
               same
               side
               on
               which
               it
               did
               occurre
               ;
               or
               else
               it
               departs
               from
               it
               on
               the
               contrary
               side
               to
               that
               on
               which
               it
               did
               occurre
               :
               all
               which
               are
               clearly
               not
               several
               degrees
               of
               the
               same
               manner
               of
               concurrence
               ,
               but
               several
               kinds
               of
               concurrence
               :
               Viz.
               the
               one
               by
               way
               of
               Contact
               ,
               the
               other
               by
               way
               of
               Section
               ,
               and
               a
               third
               by
               way
               of
               curvature
               ,
               or
               coincidence
               .
               That
               as
               these
               are
               diversified
               in
               angles
               ,
               I
               mean
               from
               kind
               to
               kind
               ,
               not
               from
               degree
               to
               degree
               ,
               so
               there
               is
               thereby
               lodged
               in
               their
               figurations
               an
               heterogeneity
               ,
               though
               in
               some
               mathematical
               respects
               ,
               neither
               sides
               ,
               nor
               inclinations
               can
               sometimes
               be
               denyed
               to
               be
               however
               homogeneal
               .
               So
               particularly
               angles
               of
               Contact
               in
               respect
               of
               their
               figuration
               must
               necessarily
               be
               acknowledged
               clear
               of
               another
               kind
               ,
               then
               all
               other
               angles
               :
               because
               the
               inclination
               of
               their
               sides
               is
               tangent
               ,
               concurring
               only
               in
               a
               punctual
               touch
               ,
               whereas
               the
               inclination
               of
               the
               sides
               of
               all
               other
               angles
               is
               secant
               ,
               and
               at
               the
               point
               of
               their
               concurrence
               by
               reason
               of
               their
               inclination
               they
               cut
               one
               another
               ,
               or
               else
               they
               are
               coincident
               ;
               then
               which
               ,
               what
               can
               make
               a
               more
               material
               difference
               in
               the
               inclination
               of
               the
               sides
               ?
               And
               as
               more
               especially
               relating
               to
               that
               so
               much
               urged
               analogy
               between
               right-lined
               angles
               and
               angles
               of
               Contact
               ;
               the
               inclination
               
               of
               the
               conteining
               sides
               in
               every
               angle
               of
               Contact
               is
               such
               as
               is
               impossible
               to
               be
               between
               the
               sides
               of
               any
               right-lined
               angle
               :
               for
               the
               sides
               of
               no
               right-lined
               angle
               can
               touch
               without
               cutting
               .
               And
               what
               more
               manifest
               and
               material
               difference
               can
               be
               in
               the
               inclinations
               made
               upon
               ,
               or
               unto
               a
               right-line
               ,
               then
               if
               in
               the
               one
               case
               a
               right-line
               be
               inclined
               unto
               it
               ,
               and
               in
               the
               other
               a
               circumference
               ,
               or
               other
               crooked-line
               ?
               Yet
               further
               to
               clear
               that
               differency
               of
               kind
               ,
               which
               is
               between
               angles
               of
               Contact
               and
               other
               angles
               ;
               I
               think
               on
               all
               sides
               it
               will
               be
               judged
               unreasonable
               to
               make
               those
               angles
               of
               the
               same
               kind
               ;
               which
               have
               neither
               one
               common
               way
               of
               measuring
               ,
               nor
               are
               coaptable
               ,
               nor
               any
               way
               proportionable
               one
               to
               another
               ,
               nor
               can
               any
               way
               by
               the
               contraction
               ,
               or
               dilatation
               of
               their
               sides
               be
               made
               equal
               one
               to
               another
               ;
               and
               this
               we
               shall
               find
               to
               be
               the
               condition
               of
               many
               angles
               one
               in
               relation
               to
               another
               .
               However
               mis-understand
               me
               not
               ,
               as
               if
               I
               made
               any
               commensurability
               a
               full
               mark
               of
               a
               full
               homogeneity
               :
               for
               as
               before
               crooked-lined
               and
               right-lined
               angles
               may
               be
               equal
               and
               of
               different
               kinds
               ;
               having
               their
               inclinations
               different
               in
               the
               kinds
               of
               their
               figurations
               ,
               though
               equal
               in
               the
               recesses
               of
               the
               sides
               .
            
             
               And
               thus
               having
               at
               large
               deduced
               the
               grand
               difference
               which
               is
               between
               the
               mathematical
               heterogeneity
               of
               angles
               ,
               and
               their
               heterogeneity
               
               in
               respect
               of
               their
               figurations
               ,
               it
               will
               now
               be
               easy
               for
               us
               to
               extricate
               our selves
               out
               of
               all
               the
               difficulties
               with
               which
               former
               Disquisitions
               upon
               this
               subject
               have
               been
               involved
               .
            
             
               As
               first
               what
               is
               to
               be
               understood
               by
               the
               equality
               ,
               which
               is
               asserted
               to
               be
               between
               right-lined
               ,
               and
               isoclitical
               concavo-convexe
               angles
               .
               For
               it
               is
               out
               of
               controversy
               and
               on
               all
               hands
               yielded
               ,
               that
               to
               any
               right-lined
               angle
               given
               may
               be
               given
               also
               a
               concavo-convexe
               isoclitical
               angle
               equal
               ;
               and
               that
               also
               ,
               in
               a
               thousand
               varieties
               ;
               as
               is
               most
               manifest
               ,
               in
               the
               circumferences
               of
               any
               two
               and
               two
               equal
               Circles
               ,
               or
               any
               two
               and
               two
               equal
               Arches
               .
               And
               so
               in
               a
               converse
               manner
               ;
               to
               any
               isoclitical
               concavo-convexe
               angle
               given
               ,
               whose
               sides
               make
               their
               recesse
               one
               from
               the
               other
               by
               an
               Arch
               less
               then
               a
               semi
               Circle
               ,
               may
               be
               given
               an
               equal
               right-lined
               angle
               :
               although
               in
               the
               infinite
               number
               of
               right
               lined
               angles
               ,
               it
               is
               impossible
               to
               find
               any
               more
               then
               one
               right-lined
               angle
               equal
               to
               the
               given
               concavo-convexe
               angle
               ;
               because
               in
               rectitude
               there
               can
               be
               no
               diversity
               ,
               as
               there
               may
               and
               is
               in
               curvatures
               .
               Now
               in
               the
               above
               recited
               cases
               ,
               why
               is
               equality
               between
               such
               different
               angles
               asserted
               possible
               ?
               and
               what
               is
               meant
               by
               their
               equality
               ?
               and
               whence
               ,
               and
               how
               is
               the
               equality
               of
               them
               to
               be
               demonstrated
               ?
               Of
               necessity
               it
               must
               be
               founded
               upon
               some
               special
               method
               of
               measuring
               angles
               ,
               or
               of
               somewhat
               
               which
               is
               in
               some
               ,
               if
               not
               in
               all
               angles
               ;
               of
               which
               in
               common
               both
               these
               different
               sorts
               of
               angles
               are
               naturally
               and
               indifferently
               capable
               .
               And
               to
               be
               short
               ,
               particular
               and
               plain
               ;
               all
               the
               mysteriousness
               of
               this
               their
               equality
               is
               founded
               upon
               this
               :
               that
               these
               two
               sorts
               of
               angles
               ,
               right
               lined
               angles
               ,
               and
               concavo-convexe
               angles
               of
               equal
               arches
               ,
               they
               both
               have
               in
               common
               one
               special
               property
               ,
               of
               which
               all
               other
               sorts
               of
               plane
               angles
               whatsoever
               are
               destitute
               .
               Viz.
               that
               each
               in
               their
               kind
               are
               isoclitical
               angles
               ,
               and
               the
               sides
               in
               each
               are
               isoclitical
               ,
               and
               in
               each
               angle
               the
               one
               side
               by
               the
               adduction
               ,
               contraction
               ,
               and
               drawing
               together
               of
               the
               sides
               will
               be
               coincident
               and
               coapted
               unto
               the
               other
               .
               And
               as
               the
               coincidence
               of
               right
               lines
               the
               one
               upon
               the
               other
               makes
               a
               right
               lined
               angle
               of
               Contact
               impossible
               ,
               so
               the
               coincidence
               of
               isoclitical
               crooked-lines
               the
               one
               upon
               the
               other
               makes
               an
               isoclitical
               angle
               of
               Contact
               impossible
               ,
               except
               only
               in
               an
               ultradiametral
               positure
               .
               And
               as
               the
               mensuration
               of
               right
               lined
               angles
               is
               by
               the
               Arches
               of
               Circles
               drawn
               upon
               the
               angular
               point
               intercepted
               between
               the
               two
               isoclitical
               sides
               ,
               to
               shew
               how
               far
               they
               are
               departed
               from
               their
               coincidence
               ;
               so
               in
               isoclitical
               crooked
               lined
               angles
               ,
               by
               the
               same
               way
               of
               mensuration
               an
               account
               may
               be
               taken
               of
               the
               departure
               ,
               which
               each
               isoclitical
               side
               hath
               made
               from
               the
               other
               since
               their
               coincidence
               :
               and
               this
               is
               the
               point
               in
               
               which
               their
               equality
               consists
               and
               is
               accounted
               ,
               and
               which
               founds
               the
               mathematical
               homogeneity
               which
               is
               between
               them
               .
               To
               instance
               in
               the
               case
               which
               is
               most
               manifest
               ;
               in
               fig.
               18.
               from
               the
               angular
               point
               A
               ,
               let
               the
               two
               arches
               
               ABC
               .
               and
               AFH
               of
               equall
               circles
               constitute
               and
               contain
               the
               isoclitical
               concavo-convexe
               angle
               CAF
               ,
               and
               let
               the
               arches
               ABC
               and
               
               AFH
               be
               equall
               :
               then
               thorow
               the
               points
               C
               and
               H
               draw
               the
               two
               right-lines
               AD
               and
               
               AG.
               According
               to
               what
               is
               above
               delivered
               ;
               it
               is
               on
               all
               hands
               agreed
               ,
               that
               the
               isoclitical
               concavo-convexe
               angle
               C
               
               AH
               is
               equal
               to
               the
               isoclitical
               right-lined
               angle
               CAH
               :
               as
               is
               copiously
               demonstrable
               from
               the
               equall
               arches
               of
               Circles
               drawn
               upon
               the
               angular
               point
               as
               center
               ,
               cutting
               all
               the
               four
               lines
               :
               viz.
               the
               Arches
               comprized
               between
               the
               two
               isoclitical
               crooked-lines
               ,
               are
               still
               equall
               to
               the
               respective
               arches
               comprized
               between
               the
               two
               isoclitical
               right-lines
               .
               For
               example
               in
               the
               chord
               
               AH
               take
               any
               where
               at
               pleasure
               the
               point
               I
               and
               from
               the
               center
               A
               draw
               the
               arch
               IB
               cutting
               the
               arch
               AFH
               in
               the
               point
               F
               and
               the
               right-line
               AEC
               in
               the
               point
               E
               and
               the
               arch
               ABC
               in
               the
               point
               B.
               The
               arch
               BF
               between
               the
               two
               isoclitical
               arches
               
               ABC
               and
               
               AFH
               is
               still
               equall
               to
               the
               arch
               EI
               intercepted
               between
               the
               two
               right-lines
               DA
               and
               GA.
               For
               the
               arches
               ABC
               and
               
               AFH
               being
               equall
               in
               equall
               circles
               ,
               the
               right-lines
               AC
               and
               
               AH
               are
               equall
               :
               and
               also
               
               AE
               semidiameter
               is
               equall
               to
               
               AI
               semidiameter
               :
               and
               by
               the
               converse
               of
               the
               same
               ratiocination
               
               AB
               arch
               is
               equall
               to
               
               AF
               arch
               :
               so
               as
               in
               short
               by
               superposition
               ,
               or
               adaptation
               the
               arch
               BE
               will
               appear
               to
               be
               equal
               to
               the
               arch
               FI
               :
               and
               therefore
               adding
               the
               common
               arch
               EF
               ;
               the
               arch
               BF
               intercepted
               between
               the
               two
               isoclitical
               crooked-lines
               
               ABC
               and
               
               AFH
               is
               equall
               to
               the
               respective
               arch
               EI
               intercepted
               between
               the
               two
               isoclitical
               right-lines
               :
               and
               this
               wheresoever
               the
               point
               I
               be
               taken
               in
               the
               right-line
               
               AH
               .
               So
               as
               by
               this
               common
               way
               of
               mensuration
               ,
               common
               to
               both
               these
               sorts
               of
               angles
               ,
               by
               reason
               of
               the
               isocliticalness
               ,
               and
               the
               coaptability
               and
               coincidibleness
               of
               the
               sides
               in
               each
               ,
               the
               one
               being
               an
               isoclitical
               concavo
               convexe
               angle
               is
               copiously
               demonstrated
               to
               be
               equal
               to
               the
               other
               being
               a
               right-lined
               angle
               .
               But
               now
               after
               what
               manner
               are
               we
               to
               understand
               this
               equality
               asserted
               between
               such
               right-lined
               and
               isoclitical
               concavo-convexe
               angles
               ?
               It
               is
               not
               an
               every
               way
               absolute
               equality
               which
               is
               between
               the
               angles
               ,
               such
               as
               is
               between
               two
               equall
               squares
               ,
               or
               two
               like
               and
               equall
               triangles
               ,
               or
               any
               two
               regular
               and
               equall
               figures
               of
               the
               same
               kind
               or
               ,
               to
               come
               nearer
               to
               the
               matter
               ,
               it
               is
               not
               such
               as
               is
               between
               two
               equall
               right-lined
               angles
               ,
               or
               between
               two
               equall
               ,
               isoclitical
               ,
               concavo-convexe
               angles
               ,
               all
               whose
               four
               sides
               are
               all
               of
               them
               isoclitical
               each
               in
               respect
               of
               all
               the
               rest
               :
               
               but
               as
               things
               that
               are
               like
               each
               other
               ,
               are
               like
               only
               in
               some
               things
               ,
               and
               unlike
               it
               may
               be
               in
               many
               others
               ;
               such
               is
               the
               equality
               between
               any
               two
               such
               angles
               ;
               Viz.
               only
               a
               respective
               equality
               ,
               such
               as
               is
               possible
               among
               heterogeneals
               ,
               and
               inferring
               a
               necessity
               of
               some
               other
               respective
               inequalities
               .
               And
               such
               an
               equality
               may
               be
               between
               two
               mere
               heterogeneals
               ;
               they
               may
               be
               of
               equal
               length
               ,
               and
               different
               breadth
               ,
               or
               weight
               :
               so
               a
               Triangle
               ,
               and
               a
               square
               and
               a
               Circle
               may
               be
               all
               equal
               ,
               either
               in
               perimetry
               ,
               or
               surface
               ,
               but
               not
               in
               both
               ;
               so
               a
               right-lined
               angle
               ,
               and
               an
               isoclitical
               concavo-convexe
               angle
               may
               be
               equal
               in
               respect
               of
               the
               recesses
               which
               the
               isoclitical
               sides
               make
               each
               from
               other
               and
               from
               their
               coincidence
               and
               coaptation
               ,
               but
               in
               other
               respects
               they
               want
               not
               their
               manifest
               inequalities
               and
               heterogenealness
               .
               As
               a
               solid
               to
               a
               solid
               may
               have
               equal
               proportion
               that
               a
               line
               to
               a
               line
               ,
               yet
               solids
               and
               lines
               are
               heterogeneal
               :
               so
               a
               crooked-line
               from
               a
               crooked-line
               may
               make
               equal
               recesses
               ,
               as
               a
               right-line
               from
               a
               right-line
               ,
               and
               yet
               in
               many
               other
               things
               much
               heterogenealness
               may
               be
               in
               the
               angles
               which
               they
               constitute
               .
               You
               will
               say
               wherein
               ?
               I
               answer
               in
               the
               rectitude
               and
               curvature
               of
               the
               containing
               sides
               .
               And
               in
               these
               different
               respects
               two
               isoclitical
               concavo-convexe
               angles
               may
               be
               both
               equal
               and
               unequal
               the
               one
               unto
               the
               other
               ,
               Viz.
               equal
               in
               the
               recesses
               of
               the
               sides
               ,
               but
               unequal
               
               in
               the
               curvatures
               of
               the
               sides
               :
               in
               the
               same
               manner
               as
               two
               figures
               may
               be
               equal
               in
               their
               perimetry
               ,
               or
               superficial
               ,
               or
               solid
               content
               ,
               and
               yet
               be
               figures
               of
               different
               kinds
               under
               diverse
               inequalities
               :
               as
               the
               one
               a
               Rhombus
               ,
               the
               other
               a
               square
               ,
               the
               one
               a
               Cylinder
               ,
               the
               other
               a
               Dode●aedron
               .
               So
               a
               thousand
               concavo-convexe
               isoclitical
               angles
               may
               be
               equal
               in
               respect
               of
               the
               recesses
               of
               the
               sides
               ,
               yet
               each
               of
               a
               several
               kind
               :
               as
               a
               thousand
               figures
               different
               in
               kind
               may
               be
               equal
               in
               perimetry
               ,
               height
               ,
               base
               ,
               superficial
               ,
               or
               solid
               content
               .
               But
               you
               'll
               say
               ,
               what
               is
               the
               rectitude
               ,
               or
               curvature
               of
               the
               containing
               sides
               to
               the
               nature
               of
               angularity
               ?
               I
               answer
               ,
               they
               are
               of
               essential
               concern
               to
               the
               limiting
               and
               determining
               the
               nature
               of
               angles
               :
               angularity
               being
               the
               habitude
               of
               concurring
               lines
               each
               in
               respect
               of
               the
               other
               ,
               as
               to
               their
               concurrence
               and
               inclination
               .
               And
               though
               the
               inclination
               of
               isoclitical
               crooked
               lines
               may
               be
               equal
               to
               the
               inclination
               of
               right-lines
               one
               upon
               another
               ,
               in
               respect
               of
               the
               equal
               recesses
               and
               departures
               which
               the
               isoclitical
               lines
               make
               each
               from
               the
               other
               ;
               yet
               there
               still
               remains
               a
               vast
               inequality
               ,
               dissimilitude
               ,
               and
               unanalogableness
               between
               the
               angles
               and
               their
               inclinations
               ,
               in
               respect
               of
               that
               little
               of
               figuration
               ,
               without
               which
               neither
               can
               an
               angle
               be
               constituted
               ,
               nor
               an
               inclination
               made
               :
               in
               a
               word
               the
               sides
               may
               make
               equal
               recesses
               ,
               yet
               be
               unequal
               
               in
               their
               curvature
               ,
               and
               unlike
               in
               their
               figuration
               :
               and
               neither
               by
               imagination
               ,
               nor
               circumduction
               ,
               nor
               any
               other
               operation
               can
               the
               one
               possibly
               be
               reduced
               ,
               or
               coapted
               to
               the
               other
               ,
               without
               setting
               the
               homologal
               points
               at
               improper
               and
               undue
               distances
               and
               positures
               one
               from
               another
               ;
               which
               shews
               a
               specifical
               difference
               between
               the
               two
               inclinablenesses
               of
               the
               one
               and
               the
               other
               :
               besides
               that
               a
               right-lined
               angle
               can
               continue
               its
               inclination
               between
               the
               sides
               infinitely
               ,
               but
               many
               isoclitical
               concavo-convexe
               angles
               thereunto
               equal
               by
               the
               necessity
               of
               their
               curvature
               must
               terminate
               within
               a
               very
               little
               space
               :
               circumferences
               and
               several
               other
               arches
               ,
               not
               being
               possible
               to
               be
               produced
               beyond
               their
               integrity
               ;
               so
               as
               some
               three
               given
               angles
               constituting
               a
               given
               triangle
               as
               to
               its
               angles
               ,
               cannot
               ,
               in
               like
               manner
               ,
               constitute
               a
               triangle
               of
               any
               given
               magnitude
               ;
               which
               is
               otherwise
               in
               right-lined
               angles
               .
               And
               that
               the
               equality
               between
               isoclitical
               concavo-convexe
               and
               right-lined
               angles
               is
               not
               so
               absolute
               as
               to
               make
               them
               every
               way
               alike
               ,
               equal
               ,
               and
               of
               the
               same
               kind
               ,
               may
               appear
               especially
               in
               this
               ,
               which
               is
               elsewhere
               demonstrated
               ;
               Viz.
               that
               an
               ultradiametral
               concavo-convexe
               angle
               of
               Contact
               ,
               being
               isoclitical
               ,
               is
               alwayes
               equal
               to
               two
               right
               right-lined
               angles
               ,
               which
               no
               one
               right
               lined-angle
               can
               be
               :
               and
               if
               it
               be
               anisoclitical
               of
               the
               larger
               ,
               it
               is
               ever
               greater
               then
               two
               right
               
               right-lined
               angles
               can
               be
               ,
               which
               is
               impossible
               also
               for
               one
               right-lined
               angle
               to
               be
               .
            
             
               And
               the
               difference
               between
               mathematical
               heterogeneity
               and
               the
               heterogeneity
               of
               angular
               figurations
               being
               as
               above
               discovered
               ;
               the
               nature
               as
               well
               of
               anisoclitical
               crooked-lined
               and
               mixt-lined
               secant
               angles
               will
               as
               clearly
               appear
               :
               Viz.
               that
               comparing
               them
               with
               right-lined
               angles
               they
               are
               compound
               and
               concrete
               angles
               constituted
               of
               right-lined
               angles
               and
               angles
               of
               Contact
               ;
               which
               are
               demonstrated
               every
               way
               heterogeneal
               :
               and
               such
               anisoclitical
               secant
               angles
               cannot
               be
               divided
               into
               any
               number
               at
               pleasure
               of
               parts
               homogeneal
               either
               mathematically
               ,
               or
               in
               respect
               of
               their
               figuration
               ,
               but
               of
               necessity
               some
               of
               them
               must
               be
               both
               wayes
               heterogeneal
               .
               This
               is
               manifest
               ,
               because
               the
               fluxe
               ,
               or
               circumduction
               of
               angles
               of
               Contact
               or
               of
               one
               of
               their
               conteining
               sides
               addes
               only
               a
               right-lined
               angle
               to
               them
               :
               after
               the
               same
               manner
               as
               the
               four
               right
               right-lined
               angles
               ,
               which
               compleat
               the
               space
               in
               any
               plane
               ,
               about
               any
               given
               point
               ,
               may
               be
               exhausted
               by
               the
               circumduction
               of
               a
               crooked
               ,
               concave
               ,
               or
               convexe
               autoclitical
               ,
               or
               antanaclitical
               line
               ,
               as
               well
               as
               by
               the
               circumduction
               of
               a
               right-line
               .
               And
               that
               this
               ties
               not
               both
               angles
               to
               be
               of
               the
               same
               kind
               ,
               may
               easily
               appear
               from
               the
               heterogeneity
               between
               lines
               and
               their
               fluxes
               ,
               which
               are
               superficial
               ;
               or
               surfaces
               and
               their
               fluxes
               ,
               
               which
               are
               solid
               .
               No
               wonder
               therefore
               if
               by
               the
               fluxe
               of
               an
               angle
               of
               contact
               ,
               or
               of
               one
               of
               its
               sides
               ,
               be
               created
               another
               kind
               of
               angle
               ,
               holding
               no
               analogy
               with
               the
               former
               :
               the
               heterogeneity
               and
               improportionableness
               of
               right-lined
               angles
               and
               angles
               of
               Contact
               having
               been
               demonstrated
               .
            
             
               So
               upon
               the
               same
               ground
               we
               may
               be
               assisted
               to
               look
               into
               the
               special
               properties
               ,
               considerable
               in
               the
               several
               kinds
               of
               angles
               peculiar
               unto
               some
               and
               incommunicable
               unto
               others
               ,
               for
               example
               .
            
             
               In
               right
               lined
               angles
               ,
               neither
               the
               greatest
               possible
               angle
               ,
               nor
               the
               least
               possible
               angle
               can
               be
               given
               ;
               though
               all
               usually
               said
               to
               be
               within
               the
               compasse
               but
               of
               one
               kind
               .
               But
               to
               pursue
               the
               difference
               which
               is
               in
               angles
               ;
               angles
               of
               Contact
               ,
               except
               such
               as
               are
               contained
               under
               lines
               of
               the
               same
               rectitude
               and
               curvature
               are
               every
               one
               of
               a
               several
               kind
               ,
               either
               mathematically
               ,
               or
               in
               respect
               of
               their
               figuration
               ,
               or
               both
               .
               And
               except
               convexo-convexe
               angles
               of
               Contact
               of
               equal
               arches
               ,
               which
               may
               be
               divided
               into
               two
               equal
               angles
               by
               common
               right-lined
               tangents
               ;
               all
               other
               angles
               of
               Contact
               are
               ,
               every
               one
               ,
               both
               the
               greatest
               ,
               and
               least
               possible
               ,
               of
               their
               special
               kind
               :
               and
               every
               angle
               of
               Contact
               contained
               under
               the
               convexe
               side
               of
               its
               arch
               ,
               or
               arches
               ,
               is
               the
               least
               possible
               under
               those
               sides
               ;
               which
               I
               suppose
               was
               the
               speculation
               
               unhappily
               missed
               by
               those
               learned
               men
               ,
               who
               would
               have
               imposed
               upon
               the
               world
               ,
               upon
               that
               their
               mistake
               ,
               the
               dream
               of
               the
               coincidence
               of
               the
               sides
               in
               such
               angles
               .
            
             
               Again
               a
               right-line
               may
               be
               drawn
               dividing
               a
               convexo-convexe
               angle
               of
               Contact
               ,
               whether
               it
               divide
               it
               equally
               ,
               or
               unequally
               ,
               but
               a
               right-line
               cannot
               be
               drawn
               dividing
               either
               of
               the
               recto-convexe
               angles
               of
               Contact
               into
               which
               the
               former
               was
               so
               divided
               ;
               whether
               the
               two
               recto-convexe
               angles
               of
               Contact
               be
               of
               the
               same
               ,
               or
               different
               kinds
               :
               as
               is
               up
               and
               down
               demonstrated
               in
               Geometry
               .
            
             
               In
               citradiametral
               concavo-convexe
               angles
               of
               Contact
               ,
               either
               the
               Arches
               are
               of
               unequal
               curvature
               ,
               or
               which
               is
               tantamount
               ,
               though
               they
               be
               of
               equal
               curvature
               ,
               yet
               they
               touch
               not
               at
               homologal
               and
               answering
               points
               ,
               being
               not
               all
               over
               of
               equal
               curvature
               ;
               which
               makes
               them
               notwithstanding
               the
               respective
               but
               not
               answering
               equality
               of
               their
               curvature
               to
               be
               anisoclitical
               .
               And
               between
               their
               Arches
               containing
               the
               angle
               of
               Contact
               ,
               a
               right-line
               cannot
               be
               drawn
               ;
               but
               infinite
               crooked-lines
               in
               number
               may
               be
               drawn
               ,
               bearing
               in
               like
               manner
               their
               convexity
               towards
               the
               concave
               which
               is
               inward
               ,
               and
               their
               concavity
               towards
               the
               convexe
               of
               the
               other
               side
               ,
               which
               is
               also
               inward
               .
            
             
               Concavo-convexe
               angles
               concurring
               by
               way
               of
               section
               ,
               and
               having
               two
               right-line
               tangents
               
               drawn
               upon
               the
               Arches
               at
               the
               angular
               point
               are
               equal
               unto
               the
               right-lined
               angles
               contained
               under
               those
               right-lined
               tangents
               ;
               adding
               respectively
               to
               each
               right-lined
               angle
               one
               of
               the
               recto-convexe
               angles
               of
               Contact
               ,
               and
               subducting
               out
               of
               it
               the
               other
               recto-convexe
               angle
               of
               Contact
               :
               and
               when
               those
               two
               recto-convexe
               angles
               of
               Contact
               are
               equal
               ,
               as
               they
               are
               ,
               when
               the
               sides
               are
               isoclitical
               ,
               then
               the
               concavo-convexe
               angle
               is
               exactly
               equal
               to
               the
               right-lined
               angle
               :
               but
               when
               the
               two
               recto-convexe
               angles
               of
               Contact
               are
               unequal
               ,
               as
               they
               are
               when
               the
               sides
               of
               the
               concavo-convexe
               angle
               are
               anisoclitical
               then
               the
               concavo-convexe
               angle
               and
               the
               right-lined
               angle
               are
               unequal
               .
            
             
               Concavo-concave
               angles
               concurring
               by
               way
               of
               section
               (
               as
               all
               such
               ever
               do
               ;
               or
               by
               coincidence
               ,
               and
               then
               one
               right-lined
               tangent
               gives
               the
               analysme
               of
               them
               ,
               shewing
               the
               two
               recto-convexe
               angles
               of
               Contact
               ,
               by
               which
               the
               angle
               of
               coincidence
               is
               less
               then
               two
               right
               right-lined
               angles
               )
               are
               by
               two
               right-lined
               tangents
               at
               the
               angular
               point
               reduced
               into
               the
               right-lined
               angle
               ;
               which
               is
               the
               least
               of
               those
               right-lined
               angles
               that
               are
               greater
               then
               it
               ,
               exceeding
               it
               only
               by
               two
               recto-convexe
               angles
               of
               Contact
               to
               be
               taken
               out
               of
               it
               .
            
             
               Convexo-convexe
               angles
               concurring
               by
               way
               of
               section
               are
               by
               two
               right-lined
               tangents
               at
               the
               angular
               point
               reduced
               into
               a
               right-lined
               
               angle
               ;
               unto
               which
               to
               make
               it
               equal
               to
               the
               convexo-convexe
               angle
               ,
               are
               to
               be
               added
               two
               recto-convexe
               angles
               of
               Contact
               .
               And
               the
               right-lined
               angle
               is
               the
               greatest
               of
               all
               the
               right-lined
               angles
               that
               are
               less
               then
               the
               convexo-convexe
               angle
               .
            
             
               Recto-concave
               angles
               concurring
               by
               way
               of
               Contact
               are
               the
               greatest
               angles
               possible
               under
               those
               two
               sides
               .
            
             
               Recto-convexe
               angles
               concurring
               by
               way
               of
               Contact
               are
               the
               least
               angles
               possible
               under
               those
               two
               sides
               .
            
             
               If
               we
               compare
               recto-concave
               angles
               of
               Contact
               with
               right
               lined
               angles
               ,
               they
               are
               less
               then
               two
               right
               right-lined
               angles
               by
               one
               only
               recto-convexe
               angle
               of
               Contact
               .
               And
               the
               least
               recto-concave
               angle
               of
               Contact
               is
               greater
               then
               the
               greatest
               right-lined
               angle
               whatsoever
               .
            
             
               Recto-concave
               angles
               concurring
               by
               way
               of
               section
               compared
               with
               right-lined
               angles
               which
               are
               constituted
               ,
               
                 i.
                 e.
              
               compleated
               by
               the
               right-lined
               tangents
               drawn
               upon
               the
               Arches
               at
               the
               angular
               points
               ;
               are
               less
               than
               such
               respective
               right-lined
               angles
               by
               a
               recto-convexe
               angle
               of
               Contact
               .
            
             
               Recto-convexe
               angles
               concurring
               by
               way
               of
               section
               compared
               with
               right-lined
               angles
               which
               are
               constituted
               ,
               
                 i.
                 e.
              
               compleated
               by
               right-line
               tangents
               drawn
               upon
               the
               Arches
               at
               the
               angular
               points
               ;
               are
               greater
               then
               such
               right-lined
               
               angles
               by
               a
               recto-convexe
               angle
               of
               Contact
               .
            
             
               Every
               angle
               of
               curvature
               ,
               or
               coincidence
               ,
               having
               a
               right-line
               tangent
               drawn
               upon
               the
               angular
               point
               ,
               appears
               to
               be
               less
               then
               two
               right
               right-lined
               angles
               by
               two
               recto-convexe
               angles
               of
               Contact
               .
            
             
               The
               inclination
               of
               the
               sides
               without
               the
               angular
               point
               at
               any
               two
               respective
               ,
               or
               other
               points
               ,
               the
               one
               taken
               in
               the
               one
               side
               ,
               the
               other
               in
               the
               other
               ,
               is
               very
               nearly
               shewn
               ,
               and
               as
               nearly
               as
               is
               possible
               in
               right-lines
               ,
               by
               the
               right-line
               tangents
               of
               those
               respective
               points
               :
               but
               in
               mixed
               lined
               and
               mixed
               crooked-lined
               angles
               by
               several
               wayes
               of
               accounting
               ,
               several
               points
               are
               made
               to
               answer
               one
               another
               ,
               as
               by
               accounting
               by
               distance
               from
               the
               angular
               point
               ,
               or
               by
               accounting
               by
               equalness
               of
               lines
               along
               the
               sides
               &c.
               
            
             
               Mixed
               lined
               angles
               of
               Contact
               ,
               when
               they
               can
               be
               ,
               and
               are
               ,
               divided
               by
               a
               right-line
               ,
               the
               parts
               are
               heterogeneal
               and
               unequal
               :
               and
               one
               of
               the
               unequal
               parts
               is
               a
               right-lined
               angle
               .
            
             
               Every
               recto-convexe
               ,
               and
               convexo-convexe
               ,
               or
               citradiametral
               concavo-convexe
               angle
               of
               Contact
               is
               the
               least
               possible
               under
               those
               sides
               .
            
             
               Rectilineary
               mensurableness
               in
               mixt
               lined
               ,
               crooked-lined
               angles
               concurring
               by
               way
               of
               section
               ,
               begins
               from
               the
               recto-convexe
               angle
               
               of
               Contact
               :
               as
               in
               right-lined
               angles
               from
               coincidence
               .
            
             
               A
               convexo-convexe
               angle
               of
               Contact
               ,
               in
               respect
               of
               dividableness
               by
               right-lines
               is
               an
               angle
               made
               up
               only
               of
               heterogeneal
               parts
               ,
               when
               it
               is
               a
               mixt-crooked-lined
               angle
               :
               but
               when
               it
               is
               an
               unmixt
               crooked-lined
               angle
               ,
               it
               hath
               some
               parts
               which
               are
               homogeneal
               ,
               Viz.
               two
               equal
               recto-convexe
               angles
               of
               contact
               ,
               which
               are
               therein
               added
               the
               one
               unto
               the
               other
               .
               And
               those
               two
               equal
               recto-convexe
               angles
               of
               contact
               ,
               as
               they
               are
               homogeneal
               ,
               I
               mean
               of
               the
               same
               kind
               one
               with
               another
               ,
               both
               mathematically
               and
               in
               respect
               of
               their
               figuration
               ;
               so
               mathematically
               they
               are
               homogeneal
               and
               of
               the
               same
               kind
               with
               the
               convexo-convexe
               angle
               which
               was
               divided
               ;
               but
               in
               respect
               of
               it
               ,
               as
               to
               their
               figuration
               ,
               they
               are
               heterogeneal
               and
               of
               another
               kind
               .
            
             
               The
               most
               simple
               angle
               may
               be
               divided
               into
               heterogeneal
               parts
               :
               
                 i.
                 e.
              
               the
               inclinableness
               of
               the
               one
               side
               to
               the
               dividing
               line
               both
               in
               respect
               of
               figuration
               and
               proportion
               may
               be
               specifically
               different
               from
               the
               inclinableness
               of
               the
               other
               side
               to
               the
               same
               dividing
               line
               :
               as
               a
               pentagone
               may
               be
               divided
               into
               a
               tetragone
               and
               a
               triangle
               ,
               so
               a
               recto-convexe
               angle
               of
               Contact
               may
               be
               divided
               into
               two
               parts
               heterogeneal
               the
               one
               to
               the
               other
               ,
               and
               to
               the
               first
               angle
               of
               Contact
               ,
               both
               in
               respect
               of
               figuration
               and
               proportion
               :
               
               viz.
               into
               a
               new
               recto-convexe
               angle
               of
               Contact
               ,
               and
               a
               concavo-convexe
               angle
               of
               Contact
               .
               Therefore
               no
               angle
               can
               be
               said
               homogeneal
               in
               that
               sense
               ,
               as
               if
               it
               could
               not
               be
               divided
               into
               parts
               heterogeneal
               ;
               whether
               you
               please
               to
               understand
               it
               ,
               in
               respect
               of
               mathematical
               homogeneity
               ,
               or
               positure
               ,
               and
               figuration
               ,
               or
               what
               respect
               soever
               else
               that
               limits
               and
               distinguisheth
               plane
               angles
               one
               from
               another
               .
            
             
               And
               to
               give
               a
               brief
               and
               general
               account
               of
               the
               comparative
               admensuration
               of
               angles
               ,
               as
               not
               being
               right-lined
               ,
               yet
               by
               way
               of
               comparative
               admeasurement
               ,
               they
               may
               in
               respect
               of
               their
               rectilineary
               parts
               be
               reduced
               and
               referred
               to
               those
               that
               are
               right-lined
               ;
               the
               containing
               sides
               not
               being
               right-lines
               ,
               at
               the
               angular
               point
               draw
               right-line
               tangents
               touching
               the
               arch
               ,
               or
               arches
               in
               the
               angular
               point
               ;
               and
               the
               right-lined
               angle
               contained
               by
               those
               right-lined
               tangents
               will
               be
               ,
               as
               to
               the
               recesses
               of
               the
               sides
               at
               the
               angular
               point
               ,
               either
               equall
               unto
               the
               first
               proposed
               angle
               ,
               or
               the
               least
               right-lined
               angle
               greater
               then
               it
               ,
               or
               the
               greatest
               right-lined
               angle
               lesser
               then
               it
               :
               or
               if
               two
               right-lined
               tangents
               cannot
               be
               thus
               placed
               at
               the
               angular
               point
               ,
               either
               the
               first
               proposed
               angle
               was
               a
               mixt-lined
               angle
               of
               contact
               ,
               said
               ,
               if
               a
               recto
               convexe
               to
               be
               less
               ,
               if
               a
               recto-concave
               to
               be
               greater
               then
               any
               right-lined
               angle
               ;
               or
               else
               it
               is
               a
               crooked-lined
               angle
               of
               contact
               ,
               which
               if
               convexo-convexe
               ,
               or
               concavo-convexe
               
               and
               citradiametral
               is
               less
               then
               any
               right-lined
               angle
               ,
               but
               if
               concavo-convexe
               and
               ultradiametral
               ,
               is
               greater
               then
               any
               right-lined
               angle
               ,
               nay
               sometimes
               equall
               to
               ,
               or
               greater
               then
               two
               right
               right-lined
               angles
               :
               or
               else
               it
               is
               an
               angle
               of
               coincidence
               ,
               or
               curvature
               .
               All
               which
               is
               to
               be
               understood
               to
               shew
               the
               inclination
               of
               the
               sides
               at
               the
               angular
               point
               ,
               as
               the
               chief
               for
               use
               in
               Geometry
               ,
               but
               not
               necessarily
               else-where
               .
               So
               crooked-lined
               ,
               or
               mixed
               lined
               angles
               are
               compared
               with
               right-lined
               angles
               by
               drawing
               at
               the
               angular
               point
               right-lines
               touching
               the
               Arches
               there
               ,
               and
               comparing
               the
               crooked
               or
               mixed
               lined
               angle
               with
               the
               right-lined
               angle
               so
               constituted
               ,
               respectively
               adding
               ,
               or
               subducting
               the
               recto-convexe
               angles
               of
               Contact
               hereby
               created
               :
               and
               this
               ,
               whether
               the
               Arches
               be
               isoclitical
               ,
               or
               anisoclitical
               ,
               or
               however
               posited
               .
               So
               all
               crooked
               ,
               or
               mixed-lined
               angles
               concurring
               by
               way
               of
               section
               may
               have
               a
               right-lined
               angle
               given
               ,
               which
               if
               it
               fall
               short
               of
               equality
               is
               either
               the
               least
               of
               the
               right-lined
               angles
               that
               are
               greater
               ,
               or
               the
               greatest
               of
               the
               right-lined
               angles
               that
               are
               less
               then
               the
               first
               crooked-lined
               ,
               or
               mixed
               lined
               angle
               .
               And
               so
               an
               analysme
               may
               be
               made
               of
               the
               greater
               angle
               into
               its
               heterogeneal
               parts
               ;
               and
               the
               crooked
               lined
               ,
               or
               mixed-lined
               angle
               may
               be
               reduced
               unto
               ,
               or
               compared
               with
               right-lined
               angles
               ,
               only
               with
               the
               addition
               ,
               or
               substraction
               of
               recto-convexe
               
               angles
               of
               Contact
               ,
               being
               angles
               less
               than
               the
               least
               right-lined
               angle
               whatsoever
               .
            
             
               All
               angles
               have
               their
               inclinations
               compounded
               of
               the
               inclinations
               of
               the
               interjacent
               lines
               each
               to
               other
               in
               order
               :
               and
               of
               the
               inclinations
               of
               the
               sides
               to
               the
               lines
               next
               adjacent
               to
               them
               ;
               which
               composite
               inclination
               may
               be
               heterogeneal
               ,
               as
               well
               as
               homogeneal
               in
               respect
               of
               the
               inclinations
               of
               which
               it
               is
               ,
               or
               may
               be
               compounded
               .
            
             
               Equally
               arched
               convexo-convexe
               ,
               or
               concavo-concave
               angles
               ,
               may
               by
               a
               right-line
               be
               divided
               into
               equal
               parts
               mathematically
               homogeneal
               ,
               but
               heterogeneal
               in
               respect
               of
               their
               figuration
               :
               but
               such
               angles
               cannot
               be
               divided
               into
               any
               more
               ,
               or
               any
               other
               equal
               parts
               ,
               for
               the
               reason
               immediately
               to
               be
               subjoyned
               .
            
             
               Heterogeneals
               taken
               together
               in
               several
               concretes
               proportionably
               ,
               S
               c.
               each
               respectively
               in
               the
               same
               proportion
               ,
               they
               hold
               exact
               proportion
               ,
               concrete
               to
               concrete
               :
               as
               double
               cube
               and
               double
               line
               ,
               are
               double
               ,
               to
               single
               cube
               and
               single
               line
               ,
               Viz.
               the
               concrete
               to
               the
               concrete
               :
               but
               set
               them
               out
               of
               the
               same
               respective
               proportions
               and
               the
               concretes
               are
               no
               way
               proportionable
               ,
               or
               in
               analogy
               ,
               concretely
               to
               be
               compared
               :
               as
               double
               cube
               and
               treble
               line
               ,
               are
               in
               no
               proportion
               ,
               to
               single
               cube
               ,
               and
               single
               line
               .
               So
               double
               number
               ,
               and
               double
               weight
               ,
               and
               double
               measure
               ,
               the
               whole
               concrete
               ,
               is
               
               double
               ,
               to
               single
               number
               ,
               single
               weight
               and
               single
               measure
               :
               but
               setting
               them
               out
               of
               the
               same
               respective
               proportions
               ;
               double
               number
               ,
               and
               double
               weight
               and
               treble
               measure
               ,
               being
               alltogether
               concretely
               taken
               ,
               are
               mathematically
               heterogeneal
               and
               improportionable
               to
               single
               number
               ,
               single
               weight
               and
               single
               measure
               ;
               being
               in
               like
               manner
               concretely
               taken
               :
               because
               the
               heterogeneals
               in
               the
               one
               concrete
               ●old
               not
               the
               same
               respective
               proportions
               to
               the
               answering
               heterogeneals
               in
               the
               other
               .
               So
               convexo-convexe
               ,
               or
               concavo-concave
               equally
               arched
               angles
               being
               secant
               ,
               hold
               proportion
               when
               divided
               equally
               ,
               as
               they
               may
               ,
               by
               right-lines
               :
               but
               they
               are
               merely
               heterogeneal
               and
               without
               proportion
               ,
               when
               divided
               by
               a
               right-line
               unequally
               .
               The
               ground
               of
               which
               is
               the
               heterogeneity
               of
               the
               parts
               ,
               of
               which
               such
               concrete
               angles
               are
               made
               up
               when
               compared
               with
               angularity
               constituted
               by
               right-lines
               ;
               which
               heterogeneal
               parts
               ,
               when
               the
               angle
               is
               divided
               equally
               in
               two
               by
               a
               right-line
               ,
               are
               in
               the
               concretes
               ,
               each
               respectively
               in
               the
               same
               proportion
               ;
               so
               making
               the
               concretes
               ,
               though
               of
               heterogeneal
               parts
               ,
               to
               be
               mathematically
               homogeneal
               and
               proportionable
               one
               unto
               the
               other
               :
               but
               when
               the
               angle
               is
               unequally
               divided
               ;
               in
               the
               two
               concretes
               the
               heterogeneal
               parts
               ,
               of
               which
               they
               are
               made
               up
               ,
               are
               not
               respectively
               in
               the
               same
               proportion
               ;
               for
               the
               recto-convexe
               
               angles
               of
               Contact
               ,
               in
               the
               concrete
               angle
               ,
               are
               divided
               equally
               ,
               and
               the
               right-lined
               angle
               ,
               which
               is
               in
               the
               concrete
               angle
               is
               divided
               unequally
               :
               so
               making
               the
               parts
               of
               the
               divided
               angle
               mathematically
               heterogeneal
               and
               improportionable
               ;
               because
               the
               compounding
               heterogeneal
               parts
               are
               not
               respectively
               according
               to
               the
               same
               proportion
               divided
               .
            
             
               The
               least
               possible
               angle
               under
               any
               two
               given
               incoincidible
               lines
               is
               the
               least
               angle
               of
               Contact
               which
               is
               possible
               under
               them
               .
            
             
               Recto-convexe
               ,
               and
               citradiametral
               concavo-convexe
               angles
               of
               Contact
               as
               they
               are
               the
               least
               angles
               possible
               under
               their
               sides
               ;
               so
               they
               are
               indivisible
               into
               parts
               holding
               all
               each
               to
               other
               any
               thing
               of
               mathematical
               homogeneity
               ,
               or
               proportionableness
               :
               Like
               unites
               they
               may
               be
               multiplyed
               to
               any
               proportion
               as
               whole
               numbers
               ,
               and
               separately
               set
               :
               but
               cannot
               at
               pleasure
               be
               divided
               ,
               nor
               at
               all
               into
               parts
               which
               are
               all
               of
               them
               mathematically
               homogeneal
               :
               nor
               can
               any
               number
               of
               them
               be
               at
               pleasure
               adjoyned
               one
               to
               another
               .
               So
               to
               any
               right-lined
               angle
               another
               may
               be
               imagined
               in
               any
               proportion
               ,
               but
               sometimes
               it
               must
               be
               the
               composition
               of
               seseveral
               angles
               ,
               and
               more
               then
               can
               stand
               at
               the
               space
               circumjacent
               about
               any
               one
               point
               .
               It
               not
               being
               possible
               for
               above
               four
               right
               right-lined
               angles
               to
               stand
               about
               the
               same
               point
               .
               As
               to
               a
               given
               point
               ,
               other
               points
               may
               be
               given
               in
               
               any
               proportion
               ,
               as
               whole
               numbers
               ;
               but
               they
               must
               not
               then
               be
               adjoyned
               one
               to
               another
               :
               but
               properly
               by
               the
               dividing
               of
               a
               point
               ,
               farther
               proportions
               are
               not
               to
               be
               expected
               .
               So
               in
               such
               angles
               ,
               though
               heterogeneal
               divisions
               be
               infinitely
               possible
               ;
               homogeneal
               ,
               I
               mean
               mathematically
               ,
               being
               impossible
               ,
               are
               not
               ,
               as
               above
               ,
               to
               be
               expected
               .
            
             
               Two
               unequal
               anisoclitical
               angles
               contained
               under
               the
               same
               two
               anisoclitical
               sides
               ,
               though
               in
               respect
               of
               their
               figuration
               ,
               both
               are
               formed
               upon
               the
               same
               inclinableness
               of
               the
               sides
               one
               to
               another
               ,
               yet
               they
               are
               mathematically
               heterogeneal
               :
               because
               being
               two
               concrete
               angles
               made
               up
               of
               heterogeneals
               ,
               viz.
               of
               the
               angle
               ,
               or
               angles
               of
               Contact
               and
               right-lined
               angles
               ;
               in
               the
               two
               concrete
               angles
               ,
               the
               angles
               of
               Contact
               are
               in
               the
               proportion
               of
               equality
               ,
               and
               the
               right-lined
               angles
               in
               the
               proportion
               of
               inequality
               :
               so
               as
               the
               concrete
               angles
               can
               have
               no
               proportion
               the
               one
               to
               the
               other
               .
            
             
               The
               half
               of
               the
               inclinableness
               of
               any
               arch
               upon
               its
               self
               ,
               
                 i.
                 e.
              
               upon
               another
               arch
               like
               and
               equal
               ,
               is
               still
               comprehended
               and
               contained
               under
               the
               inclinableness
               of
               a
               right-line
               upon
               the
               same
               arch
               :
               that
               though
               their
               figurations
               are
               ever
               heterogeneal
               ,
               and
               their
               inclinations
               can
               never
               be
               equal
               ,
               yet
               they
               may
               be
               mathematically
               homogeneal
               :
               as
               different
               numbers
               are
               alwayes
               unequal
               though
               ever
               proportionable
               .
            
             
             
               To
               no
               anisoclitical
               angle
               ,
               whether
               of
               Contact
               ,
               or
               concurring
               by
               way
               of
               section
               can
               any
               right-lined
               angle
               be
               made
               equal
               .
               And
               generally
               between
               isoclitical
               and
               anisoclitical
               angles
               equality
               is
               impossible
               .
               And
               when
               anisoclitical
               angles
               are
               compared
               with
               those
               that
               are
               isoclitical
               as
               greater
               ,
               or
               lesser
               ;
               it
               is
               not
               to
               denote
               in
               both
               any
               mathematical
               homogeneity
               ,
               and
               that
               so
               by
               possibility
               they
               are
               reducible
               to
               a
               true
               ,
               compleat
               and
               analogous
               equality
               ;
               but
               the
               intent
               is
               only
               to
               declare
               ,
               whether
               of
               them
               hath
               ,
               or
               can
               have
               the
               containing
               side
               ,
               or
               sides
               falling
               within
               ,
               or
               without
               the
               other
               ;
               which
               is
               only
               from
               their
               inequality
               by
               the
               whole
               kind
               .
               So
               though
               anisoclitical
               angles
               may
               be
               greater
               ,
               or
               less
               then
               a
               right-lined
               angle
               ,
               yet
               between
               the
               one
               and
               the
               other
               there
               is
               neither
               common
               way
               of
               measuring
               ,
               nor
               any
               proportion
               ,
               though
               both
               of
               them
               be
               quantitative
               plane
               angles
               .
               There
               is
               no
               common
               way
               of
               measuring
               them
               ,
               because
               of
               the
               anisocliticalness
               ,
               and
               if
               there
               could
               be
               any
               proportion
               between
               them
               ,
               then
               might
               a
               right-lined
               angle
               be
               given
               equal
               to
               such
               anisoclitical
               angles
               ;
               the
               contrary
               of
               which
               is
               otherwise
               clearly
               demonstrable
               .
               So
               as
               a
               thousand
               angles
               quantitative
               by
               conf●ssion
               and
               having
               right-lined
               angles
               lesser
               and
               greater
               then
               themselves
               ,
               yet
               can
               have
               no
               right-lined
               angle
               equal
               to
               them
               .
            
             
               A
               right-lined
               angle
               cannot
               have
               its
               inclination
               
               at
               the
               angular
               point
               ,
               much
               less
               all
               along
               the
               sides
               divided
               equally
               by
               a
               crooked-line
               ,
               whether
               autoclitical
               ,
               or
               antanaclitical
               ,
               or
               of
               what
               curvature
               soever
               :
               though
               any
               such
               crooked-line
               may
               from
               any
               angle
               divide
               equally
               the
               plane
               bounded
               within
               a
               right-lined
               triangle
               :
               for
               if
               in
               fig.
               17.
               
               BAC
               be
               a
               right-lined
               angle
               :
               I
               say
               ,
               no
               crooked-line
               &c.
               can
               divide
               it
               at
               the
               angular
               point
               equally
               .
               If
               it
               be
               possible
               let
               it
               be
               divided
               into
               equal
               parts
               by
               the
               arch
               AGE
               whether
               autoclitical
               ,
               as
               when
               it
               is
               conceived
               to
               be
               part
               of
               the
               arch
               AGEF
               ,
               or
               antanaclitical
               ,
               as
               when
               it
               is
               conceived
               to
               be
               part
               of
               the
               arch
               AGEHD
               .
               Then
               let
               the
               right-line
               AD
               divide
               equally
               the
               right-lined
               angle
               BAC
               .
               It
               is
               manifest
               ,
               if
               the
               crooked-line
               
               AG
               never
               occurre
               with
               the
               right-line
               AD
               ,
               that
               then
               it
               doth
               not
               divide
               the
               angle
               B
               
               AC
               equally
               ;
               for
               the
               right
               and
               crooked-lines
               cannot
               be
               coapted
               .
               Let
               therefore
               ,
               if
               they
               do
               occurre
               ,
               the
               place
               of
               their
               first
               meeting
               ,
               or
               occurrence
               be
               at
               E.
               Therefore
               the
               arch
               AGE
               falleth
               within
               the
               angle
               BAD
               and
               therefore
               divides
               the
               whole
               angle
               BAC
               unequally
               ,
               whether
               it
               be
               autoclitical
               ,
               or
               antanaclitical
               .
               And
               by
               the
               same
               demonstration
               appears
               that
               it
               is
               as
               impossible
               for
               it
               to
               divide
               the
               inclination
               all
               along
               the
               sides
               equally
               .
               And
               though
               a
               crooked-line
               may
               divide
               any
               right-lined
               figure
               ,
               because
               of
               its
               perfect
               bounds
               ,
               from
               any
               angle
               
               into
               equal
               parts
               ;
               yet
               this
               no
               way
               evinceth
               any
               possibility
               of
               dividing
               a
               right-lined
               angle
               by
               a
               crooked
               line
               into
               equal
               parts
               :
               and
               though
               from
               any
               point
               of
               such
               a
               dividing
               crooked
               line
               ,
               lines
               might
               be
               drawn
               making
               up
               a
               figure
               ,
               whose
               parts
               divided
               by
               the
               crooked
               line
               from
               the
               angle
               are
               equal
               ;
               yet
               this
               is
               no
               more
               then
               is
               performable
               upon
               any
               point
               of
               any
               crooked
               line
               drawn
               between
               the
               sides
               of
               a
               right-lined
               angle
               at
               randome
               ,
               whether
               the
               parts
               of
               the
               angle
               be
               homogeneal
               ,
               or
               heterogeneal
               ,
               equal
               ,
               or
               unequal
               .
            
             
               Difference
               of
               curvature
               by
               a
               perpetual
               necessity
               infers
               difference
               of
               inclination
               ,
               whether
               the
               curve
               line
               be
               inclined
               upon
               a
               right-line
               ,
               or
               upon
               a
               curve
               line
               ;
               for
               still
               the
               one
               of
               the
               curve
               lines
               will
               fall
               within
               ,
               or
               without
               the
               other
               .
               So
               let
               inclination
               upon
               a
               right-line
               be
               in
               a
               recto-concave
               angle
               of
               Contact
               ,
               recto-convexe
               angle
               of
               Contact
               ,
               recto-concave
               ,
               or
               recto-convexe
               secant
               angles
               ;
               by
               no
               right-line
               can
               any
               of
               those
               inclinations
               be
               made
               upon
               the
               first
               right-line
               ,
               nor
               by
               any
               other
               crooked-line
               :
               but
               still
               the
               lines
               will
               fall
               either
               within
               ,
               or
               without
               ▪
               And
               all
               angles
               of
               Contact
               under
               lines
               of
               different
               curvature
               and
               rectitude
               ,
               as
               they
               are
               of
               ●
               thousand
               Mathematically
               and
               extra-quantitatively
               different
               kinds
               one
               from
               another
               ,
               so
               they
               are
               〈◊〉
               manifestly
               distinct
               in
               kind
               from
               all
               right-lined
               angl●●
               whatsoever
               .
               
               And
               as
               every
               number
               is
               a
               different
               kind
               ,
               in
               like
               manner
               as
               to
               positure
               and
               figuration
               is
               every
               line
               a
               distinct
               kind
               differing
               from
               all
               other
               not
               agreeing
               with
               it
               in
               rectitude
               and
               curvature
               ;
               whether
               the
               curvatures
               be
               homogeneal
               ,
               
                 i.
                 e.
              
               every
               where
               equal
               ,
               as
               in
               Circles
               ,
               or
               heterogeneal
               ,
               
                 i.
                 e.
              
               unequal
               in
               the
               several
               parts
               ,
               as
               in
               ellipses
               ,
               hyperbolas
               ,
               parabolas
               ,
               &c.
               
               And
               accordingly
               judgment
               is
               to
               be
               made
               of
               the
               sides
               under
               which
               angles
               are
               contain'd
               ,
               and
               the
               angles
               contain'd
               under
               them
               :
               alter
               the
               kind
               of
               either
               side
               ,
               and
               the
               kind
               of
               the
               angle
               is
               chang'd
               ;
               ever
               ,
               in
               respect
               of
               figuration
               ;
               and
               most
               what
               ,
               mathematically
               ;
               because
               the
               former
               inclinablenes
               of
               the
               sides
               is
               taken
               away
               ,
               and
               a
               new
               kind
               of
               inclinablenes
               introduced
               between
               them
               .
               Hence
               appears
               ;
               no
               two
               recto-concaves
               ,
               no
               two
               recto-convexes
               ,
               no
               two
               concavo-convexes
               ,
               no
               two
               convexo-convexes
               ,
               being
               all
               angles
               of
               Contact
               can
               be
               equal
               ,
               except
               their
               sides
               have
               the
               very
               same
               rectitude
               and
               curvature
               .
               In
               general
               ,
               equality
               is
               not
               to
               be
               asserted
               between
               angles
               ,
               except
               either
               for
               the
               mensurablenes
               of
               both
               in
               some
               common
               way
               of
               measuring
               ,
               or
               at
               least
               because
               they
               can
               be
               so
               cast
               into
               a
               coaptation
               ,
               as
               that
               either
               shall
               contain
               all
               the
               quantity
               which
               is
               in
               the
               other
               .
               And
               as
               he
               erred
               in
               squaring
               the
               superficial
               content
               of
               a
               circle
               ,
               that
               in
               his
               quadrature
               left
               out
               a
               small
               lunular
               figure
               ;
               so
               neither
               can
               he
               be
               
               justifyed
               to
               have
               given
               one
               anisoclitical
               angle
               equal
               to
               another
               ,
               who
               ,
               as
               he
               must
               of
               necessity
               ,
               takes
               in
               ,
               or
               leaves
               out
               an
               angle
               ,
               or
               some
               angles
               of
               contact
               which
               are
               in
               the
               one
               and
               not
               in
               the
               other
               .
            
             
               Particularly
               no
               mixed
               lined
               ,
               nor
               mixed
               crooked
               lined
               angles
               whatsoever
               ,
               whether
               secant
               ,
               or
               of
               contact
               ,
               can
               by
               any
               possibility
               ,
               either
               by
               right-line
               ,
               or
               crooked
               line
               be
               divided
               into
               equal
               parts
               ,
               or
               angles
               ,
               whether
               of
               the
               same
               ,
               or
               of
               different
               ,
               kinds
               .
               And
               in
               few
               ,
               those
               angles
               ,
               which
               cannot
               be
               divided
               into
               two
               equal
               parts
               cannot
               be
               divided
               into
               three
               ,
               fower
               ,
               five
               ,
               or
               any
               other
               number
               of
               equal
               parts
               .
            
             
               The
               comparative
               admensurement
               ,
               as
               above
               ,
               of
               right-lined
               and
               not
               right-lined
               angles
               is
               as
               well
               of
               their
               heterogeneal
               inequality
               ,
               when
               heterogeneal
               ,
               as
               of
               their
               homogeneal
               equality
               and
               proportionablenes
               when
               homogeneal
               .
            
             
               All
               not
               right-lined
               angles
               comparatively
               admeasured
               ,
               to
               right-lined
               angles
               ,
               only
               with
               the
               addition
               ,
               or
               only
               with
               the
               subduction
               of
               any
               angle
               ,
               or
               angles
               of
               contact
               ,
               are
               heterogeneal
               to
               all
               right-lined
               angles
               ,
               as
               also
               ,
               if
               when
               one
               angle
               of
               contact
               is
               to
               be
               added
               ,
               and
               another
               subducted
               ,
               the
               two
               angles
               of
               contact
               be
               unequal
               .
               And
               as
               is
               manifest
               ,
               of
               several
               not
               right-lined
               angles
               ,
               that
               is
               still
               the
               greater
               ,
               whose
               comparative
               and
               relative
               admeasurement
               
               is
               made
               to
               the
               greater
               right-lined
               angle
               :
               and
               of
               those
               ,
               whose
               comparative
               admeasurement
               is
               unto
               equal
               right
               lined
               angles
               ,
               they
               are
               greater
               ,
               or
               equal
               ,
               or
               less
               according
               to
               the
               equality
               ,
               or
               inequality
               ,
               and
               comparing
               together
               of
               their
               angles
               of
               Contact
               ▪
               And
               of
               concavo-concave
               angles
               the
               right-lined
               angle
               cannot
               be
               given
               ,
               which
               is
               the
               greatest
               of
               those
               right-lined
               angles
               that
               are
               less
               then
               the
               concavo-concave
               angle
               :
               nor
               in
               convexo-convexe
               angles
               can
               be
               given
               a
               right-lined
               angle
               ,
               which
               is
               the
               least
               of
               those
               right-lined
               angles
               which
               are
               greater
               then
               the
               convexo-convexe
               angle
               :
               nor
               in
               recto-concaves
               the
               greatest
               of
               the
               less
               :
               nor
               in
               recto-convexes
               the
               least
               of
               the
               greater
               :
               whereas
               in
               concavo-convexe
               angles
               ,
               the
               right-lined
               angle
               to
               which
               the
               comparative
               admeasurement
               is
               made
               ,
               may
               according
               to
               the
               case
               be
               either
               equal
               ,
               as
               when
               the
               sides
               are
               isoclitical
               ;
               or
               the
               greatest
               right-lined
               angle
               that
               is
               less
               ,
               as
               when
               the
               concave
               side
               is
               of
               less
               curvature
               ;
               or
               the
               least
               of
               the
               greater
               ,
               as
               when
               the
               concave
               side
               is
               of
               greater
               curvature
               .
            
             
               From
               these
               things
               may
               appear
               how
               that
               objection
               is
               to
               be
               answered
               ,
               in
               which
               you
               urge
               that
               there
               is
               a
               proportion
               between
               mixed
               lined
               angles
               ,
               whose
               sides
               are
               seca●●
               ,
               and
               right-lined
               angles
               ,
               because
               the
               one
               by
               its
               multiple
               may
               exceed
               the
               other
               ;
               and
               that
               therefore
               an
               homogeneity
               ,
               mathematically
               to
               be
               understood
               ,
               
               is
               to
               be
               acknowledged
               amongst
               them
               .
               To
               this
               is
               answered
               that
               all
               anisoclitical
               angles
               whatsoever
               having
               their
               sides
               concurring
               by
               way
               of
               section
               when
               compared
               unto
               right-lined
               angles
               are
               heterogeneal
               compositions
               of
               angularity
               ,
               
                 i.
                 e.
              
               cannot
               be
               divided
               into
               any
               number
               at
               pleasure
               of
               parts
               all
               equal
               ;
               but
               as
               is
               manifest
               ,
               they
               in
               that
               their
               relative
               nature
               are
               concrete
               and
               composite
               angles
               ,
               formed
               of
               right-lined
               angles
               by
               adding
               to
               them
               ,
               or
               subducting
               from
               them
               mixed
               lined
               angles
               of
               Contact
               :
               so
               as
               the
               right-lined
               part
               of
               the
               secant
               anisoclitical
               angle
               ,
               or
               any
               the
               least
               part
               of
               it
               hath
               true
               proportion
               with
               all
               right-lined
               angles
               ;
               and
               the
               whole
               being
               heterogeneal
               ,
               yet
               hath
               a
               semblance
               of
               proportion
               with
               right-lined
               angles
               ,
               because
               of
               those
               parts
               which
               it
               hath
               which
               are
               homogeneal
               with
               them
               ;
               but
               between
               the
               other
               heterogeneal
               part
               which
               is
               the
               mixt-lined
               angle
               of
               Contact
               ,
               and
               right-lined
               angles
               ,
               neither
               is
               ,
               nor
               can
               be
               any
               proportion
               :
               and
               it
               is
               because
               of
               this
               part
               that
               it
               is
               not
               a
               true
               ,
               but
               a
               seeming
               proportion
               only
               ,
               which
               is
               between
               the
               whole
               concrete
               heterogeneal
               ,
               composite
               anisoclitical
               angle
               ,
               and
               right-lined
               angles
               :
               for
               if
               it
               were
               a
               true
               proportion
               and
               proportionablenes
               which
               were
               between
               them
               ,
               it
               should
               then
               be
               possible
               to
               give
               a
               right-lined
               angle
               equal
               to
               such
               an
               anisoclitical
               secant
               angle
               ;
               which
               when
               done
               ,
               I
               have
               no
               more
               to
               
               say
               ,
               being
               well
               assured
               I
               am
               able
               to
               demonstrate
               the
               contrary
               upon
               whatsoever
               right-lined
               angle
               shall
               be
               offered
               under
               that
               notion
               .
               And
               whereas
               it
               is
               urged
               that
               a
               right-lined
               angle
               is
               ,
               and
               may
               be
               equal
               to
               a
               mixed
               lined
               angle
               ;
               all
               their
               difference
               being
               only
               in
               this
               ,
               that
               the
               sides
               of
               the
               one
               are
               more
               spread
               and
               divaricated
               then
               the
               other
               ;
               what
               is
               this
               else
               but
               to
               say
               ,
               that
               they
               are
               equal
               ,
               saying
               that
               the
               one
               is
               bigger
               then
               the
               other
               .
               For
               angles
               of
               Contact
               can
               divide
               angularity
               ,
               or
               space
               equally
               ,
               or
               unequally
               into
               more
               ,
               or
               fewer
               parts
               :
               and
               they
               are
               not
               indivisible
               ,
               because
               every
               one
               contains
               innumerably
               more
               in
               it
               :
               and
               by
               the
               definition
               of
               an
               angle
               ,
               lines
               have
               a
               sufficient
               inclination
               to
               constitute
               an
               angle
               ,
               if
               in
               the
               same
               plane
               ,
               they
               lye
               not
               both
               in
               the
               same
               right-line
               .
               And
               to
               all
               other
               cases
               and
               objections
               of
               the
               like
               nature
               grounded
               upon
               them
               athematical
               heterogeneity
               of
               the
               parts
               ,
               of
               which
               such
               special
               angles
               are
               constituted
               in
               respect
               of
               right-lined
               angles
               ,
               like
               answers
               may
               be
               addressed
               .
            
             
               Though
               a
               right-lined
               angle
               cannot
               be
               divided
               into
               equal
               parts
               of
               the
               same
               kind
               mathematically
               ,
               or
               in
               shape
               by
               a
               crooked-line
               ,
               nor
               a
               mixed
               ,
               or
               crooked-line
               angle
               by
               a
               right-line
               into
               parts
               of
               the
               same
               kind
               in
               respect
               of
               figuration
               ;
               yet
               some
               crooked
               lined
               angles
               may
               be
               divided
               into
               parts
               exactly
               equal
               of
               the
               same
               kind
               one
               with
               another
               ,
               and
               with
               the
               whole
               
               mathematically
               by
               a
               right-line
               .
               For
               it
               is
               apparent
               ,
               there
               may
               be
               between
               magnitudes
               a
               sufficient
               homogeneity
               for
               proportionablenes
               without
               excluding
               all
               further
               discriminableness
               between
               them
               :
               as
               between
               equal
               arched
               convexo-convexes
               ,
               or
               concavo-concaves
               and
               their
               parts
               when
               they
               are
               equally
               divided
               .
               So
               between
               lines
               of
               all
               kinds
               ,
               and
               numbers
               of
               all
               kinds
               ,
               is
               proportionality
               .
               Yet
               still
               their
               proportionality
               is
               in
               respect
               of
               somewhat
               which
               is
               homogeneal
               in
               them
               .
               Viz.
               that
               they
               are
               all
               resoluble
               into
               parts
               that
               are
               homogeneal
               ,
               or
               parts
               into
               which
               the
               rest
               are
               homogeneally
               resoluble
               ,
               or
               after
               the
               same
               manner
               mensurable
               ,
               as
               equal
               arched
               convexo-convexe
               angles
               into
               two
               equal
               recto-convexes
               :
               yet
               notwithstanding
               the
               proportionablenes
               between
               the
               convexo-convexes
               and
               the
               recto-convexes
               ;
               such
               an
               heterogeneity
               is
               in
               their
               figuration
               ,
               that
               by
               no
               divarication
               of
               sides
               can
               they
               ever
               be
               made
               equal
               .
               And
               though
               the
               equal
               arched
               convexo-convexes
               and
               concavo
               concaves
               may
               be
               divided
               by
               a
               right-line
               into
               two
               equal
               parts
               ;
               by
               no
               lines
               whatsoever
               can
               they
               be
               divided
               into
               any
               more
               equal
               parts
               nor
               their
               recto-convexes
               by
               any
               line
               whatsoever
               into
               two
               equal
               parts
               ;
               such
               parts
               of
               such
               angles
               being
               more
               impossible
               to
               be
               given
               in
               Geometry
               ,
               then
               the
               square
               roots
               of
               unsquare
               numbers
               ,
               or
               cubicke
               roots
               of
               uncubical
               numbers
               are
               in
               Arithmeticke
               .
               As
               
               any
               two
               points
               ,
               or
               any
               two
               indivisibles
               may
               have
               some
               few
               proportions
               amongst
               them
               ,
               but
               no
               more
               :
               the
               latitude
               of
               proportionablenes
               being
               limited
               by
               the
               special
               natures
               and
               kinds
               of
               things
               :
               so
               between
               special
               kinds
               of
               numbers
               cannot
               be
               all
               proportions
               :
               nor
               between
               special
               lines
               all
               angles
               ,
               or
               all
               proportions
               in
               all
               angles
               .
            
             
               Hence
               also
               we
               may
               clear
               the
               possibility
               of
               that
               speculation
               of
               the
               learned
               persons
               by
               you
               named
               ,
               that
               in
               heterogeneals
               there
               may
               be
               a
               passing
               from
               greater
               to
               less
               ,
               and
               from
               less
               to
               greater
               in
               a
               continuous
               manner
               without
               ,
               ever
               passing
               thorow
               equality
               ;
               however
               you
               are
               pleased
               to
               bestow
               upon
               it
               a
               contemptuous
               smile
               ,
               if
               not
               an
               hiss
               :
               Viz.
               that
               we
               should
               assert
               that
               mixed
               lined
               ,
               or
               crooked
               lined
               angles
               can
               pass
               by
               the
               divarication
               of
               the
               same
               sides
               from
               being
               greater
               to
               be
               less
               then
               given
               angles
               ,
               or
               contrarily
               by
               their
               contraction
               ,
               and
               yet
               in
               that
               transit
               ,
               never
               be
               equal
               .
               What
               is
               more
               obvious
               then
               to
               give
               instances
               of
               anisoclitical
               angles
               ,
               less
               then
               a
               given
               isoclitical
               angle
               ,
               and
               by
               divaricating
               and
               distending
               the
               sides
               of
               the
               anisoclitical
               angle
               ,
               it
               may
               be
               made
               to
               exceed
               the
               first
               given
               isoclitical
               angle
               ;
               yet
               in
               all
               the
               way
               they
               could
               never
               be
               equal
               the
               one
               to
               the
               other
               ;
               the
               one
               being
               isoclitical
               ,
               the
               other
               anisoclitical
               :
               coaptation
               and
               the
               passing
               of
               the
               sides
               from
               the
               angular
               point
               every
               where
               prove
               
               their
               inequality
               :
               and
               let
               those
               that
               assert
               their
               equality
               any
               way
               shew
               and
               admeasure
               it
               .
               And
               the
               sober
               understanding
               of
               those
               sayings
               ,
               that
               such
               angles
               passe
               from
               less
               to
               greater
               without
               ever
               being
               equal
               ,
               is
               not
               to
               assert
               any
               homogeneity
               ,
               or
               proportion
               of
               any
               kind
               between
               them
               ,
               or
               common
               way
               of
               measuring
               their
               quantity
               in
               and
               according
               to
               the
               same
               indefinite
               quantity
               ,
               but
               only
               to
               shew
               how
               the
               sides
               may
               pass
               within
               and
               without
               each
               other
               ;
               but
               because
               of
               their
               anisocliticalnes
               and
               difference
               in
               respect
               of
               rectitude
               and
               curvatures
               ,
               they
               can
               never
               be
               brought
               to
               be
               coincident
               .
               And
               why
               should
               this
               appear
               so
               monstrous
               ,
               that
               transitions
               should
               be
               ,
               in
               the
               sense
               abovesaid
               ,
               made
               from
               greater
               to
               less
               ,
               without
               passing
               thorow
               equality
               ?
               seeing
               it
               is
               most
               manifest
               that
               though
               a
               crooked-line
               circumducted
               about
               the
               angular
               point
               of
               a
               right-lined
               angle
               makes
               infinite
               divisions
               of
               the
               right-lined
               angle
               ;
               yet
               they
               are
               ever
               unequal
               ,
               and
               never
               by
               possibility
               can
               be
               equal
               :
               though
               notwithstanding
               ,
               the
               same
               crooked
               line
               may
               divide
               a
               crooked
               lined
               isoclitical
               angle
               equal
               to
               the
               given
               right-lined
               angle
               into
               equal
               parts
               ;
               which
               besides
               plentifully
               shews
               the
               heterogeneity
               which
               is
               between
               equal
               right-lined
               and
               crooked-lined
               isoclitical
               angles
               ,
               notwithstanding
               their
               equality
               :
               yet
               by
               making
               up
               the
               right-lined
               angle
               into
               a
               compleately
               bounded
               figure
               ,
               the
               crooked-line
               
               may
               divide
               the
               plane
               of
               it
               into
               equal
               isoepipedal
               and
               isorrhopical
               parts
               :
               as
               is
               more
               easy
               to
               demonstrate
               then
               that
               there
               should
               be
               any
               need
               to
               set
               it
               down
               .
               And
               it
               is
               not
               to
               be
               stranged
               at
               ,
               that
               we
               assert
               a
               crooked-line
               can
               divide
               a
               right-lined
               triangle
               into
               equal
               parts
               ,
               but
               not
               any
               of
               its
               right-lined
               angles
               :
               for
               the
               figuration
               of
               the
               one
               is
               compleat
               ,
               and
               the
               production
               of
               the
               crooked-line
               ,
               as
               well
               as
               of
               the
               lines
               containing
               the
               right-lined
               angle
               ,
               in
               the
               triangle
               are
               limited
               ;
               all
               which
               are
               quite
               otherwise
               in
               mere
               angles
               ,
               being
               in
               many
               kinds
               of
               angles
               and
               inclinations
               very
               alterable
               by
               the
               production
               of
               the
               lines
               .
               And
               if
               you
               will
               pertinaciously
               say
               that
               a
               crooked-line
               may
               divide
               equally
               a
               right-lined
               angle
               ,
               shew
               their
               equality
               ,
               and
               your
               way
               of
               admeasuring
               the
               equality
               of
               the
               parts
               :
               both
               coaptation
               of
               sides
               ,
               in
               whose
               habitude
               the
               nature
               of
               angles
               chiefly
               consists
               ,
               and
               also
               the
               way
               of
               measuring
               by
               interjected
               arches
               demonstrating
               and
               declaring
               the
               contrary
               :
               Nay
               sometimes
               in
               quantities
               among
               which
               there
               is
               true
               proportion
               ,
               the
               special
               differences
               of
               their
               kinds
               may
               be
               the
               authour
               of
               little
               less
               .
               So
               commensurable
               quantities
               ,
               being
               in
               their
               commensurability
               infinitely
               divisible
               ,
               may
               be
               continually
               increased
               ,
               or
               lessened
               by
               quantities
               vastly
               ,
               less
               then
               any
               given
               quantity
               ;
               yet
               as
               they
               pass
               from
               less
               to
               greater
               never
               can
               hit
               equality
               with
               any
               the
               like
               
               quantities
               of
               the
               same
               general
               kind
               ,
               being
               incommensurable
               .
               So
               in
               the
               present
               question
               ,
               heterogeneity
               of
               sides
               makes
               an
               impossibility
               to
               coapt
               ,
               though
               by
               straitning
               and
               divaricating
               they
               may
               fall
               within
               ,
               or
               without
               ,
               and
               so
               be
               less
               ,
               or
               greater
               ,
               but
               being
               impossible
               to
               coapt
               ,
               and
               incapable
               of
               any
               other
               common
               way
               of
               measuring
               ,
               equality
               cannot
               be
               conceived
               in
               them
               ;
               in
               which
               is
               praesupposed
               a
               common
               way
               of
               measuring
               to
               declare
               and
               prove
               their
               asserted
               proportion
               and
               equality
               .
               And
               in
               this
               whole
               matter
               what
               is
               more
               said
               then
               that
               the
               cutting
               inclination
               of
               a
               right-line
               upon
               a
               right-line
               ,
               and
               of
               a
               right-line
               upon
               a
               crooked
               ,
               or
               of
               a
               crooked-line
               upon
               two
               crooked-lines
               being
               all
               of
               different
               curvature
               ,
               may
               any
               of
               them
               be
               greater
               ,
               or
               less
               then
               one
               another
               heterogeneally
               ,
               but
               can
               never
               be
               equal
               ;
               as
               to
               those
               that
               will
               consider
               ,
               is
               most
               clearly
               manifest
               and
               necessary
               .
               And
               this
               is
               no
               more
               strange
               ,
               then
               that
               odde
               numbers
               may
               be
               less
               ,
               or
               greater
               then
               any
               even
               number
               ,
               at
               least
               above
               two
               ,
               yet
               by
               their
               constitutive
               nature
               ,
               they
               can
               never
               be
               equal
               :
               and
               for
               the
               same
               reason
               magnitudes
               commensurable
               and
               incommensurable
               may
               be
               greater
               ,
               and
               less
               then
               one
               another
               ,
               and
               by
               a
               less
               quantity
               then
               any
               quantity
               that
               can
               be
               given
               ;
               yet
               ,
               for
               the
               specialty
               of
               their
               natures
               ,
               they
               can
               never
               be
               equal
               ,
               remaining
               so
               distinct
               in
               their
               ultimate
               kind
               :
               and
               however
               they
               have
               
               many
               general
               things
               in
               which
               they
               agree
               ,
               yet
               they
               are
               not
               accountable
               specially
               ,
               but
               by
               several
               wayes
               and
               algorythmes
               .
               And
               as
               it
               is
               the
               binding
               up
               of
               numbers
               to
               the
               specifick
               properties
               of
               Evenness
               ,
               or
               oddness
               ,
               and
               of
               magnitudes
               to
               commensurability
               ,
               or
               incommensurability
               ,
               that
               makes
               equality
               between
               their
               kinds
               impossible
               ;
               so
               it
               is
               the
               incoaptablenes
               and
               want
               and
               impossibility
               of
               a
               common
               way
               of
               measuring
               ,
               which
               is
               between
               the
               two
               and
               two
               sides
               of
               such
               heterogeneally
               unequal
               angles
               ,
               keeping
               all
               along
               the
               properties
               of
               their
               inclination
               ,
               that
               renders
               equality
               between
               their
               kinds
               impossible
               ;
               though
               either
               may
               sometimes
               truly
               and
               manifestly
               be
               greater
               or
               less
               then
               the
               other
               ,
               however
               without
               proportion
               .
               For
               in
               heterogeneals
               may
               be
               a
               two
               fold
               inequality
               ,
               either
               an
               inequality
               according
               to
               proportion
               ,
               or
               an
               inequality
               without
               proportion
               :
               and
               though
               they
               be
               heterogeneals
               ,
               if
               it
               be
               a
               true
               limited
               ,
               determinate
               proportion
               of
               inequality
               which
               is
               between
               them
               ,
               as
               between
               an
               odde
               number
               and
               an
               even
               ,
               a
               commensurable
               magnitude
               and
               an
               incommensurable
               ,
               they
               are
               then
               considered
               as
               some
               way
               ,
               or
               according
               to
               something
               which
               is
               ,
               homogeneal
               ,
               not
               heterogeneal
               ,
               in
               both
               :
               but
               if
               their
               inequality
               ,
               though
               real
               and
               apparent
               be
               without
               any
               true
               and
               homogeneally
               determinable
               proportionablenes
               ,
               as
               between
               right-lined
               and
               
               anisoclitical
               angles
               ,
               then
               the
               consideration
               had
               of
               them
               is
               in
               their
               heterogeneity
               .
               So
               to
               the
               objection
               ,
               that
               right-lined
               angles
               ,
               and
               recto-convexe
               angles
               of
               Contact
               have
               proportion
               of
               greater
               and
               less
               ,
               and
               are
               therefore
               homogeneal
               ,
               if
               any
               quantitativenes
               be
               to
               be
               asserted
               in
               the
               recto-convexe
               angles
               of
               Contact
               ;
               and
               so
               of
               the
               rest
               :
               I
               answer
               ,
               there
               is
               not
               a
               proportionable
               inequality
               between
               right-lined
               angles
               and
               recto-convexe
               angles
               of
               Contact
               ,
               and
               the
               like
               ;
               but
               an
               inequality
               ,
               which
               is
               improportionable
               :
               as
               when
               the
               Earth
               is
               said
               to
               be
               intruth
               greter
               then
               a
               point
               ;
               though
               in
               many
               particular
               hypotheses
               contrarily
               conceived
               ,
               for
               the
               better
               observing
               and
               accounting
               several
               phaenomena
               ,
               and
               the
               better
               accommodating
               of
               instruments
               for
               the
               making
               of
               observations
               .
               So
               in
               the
               genesis
               of
               quantities
               and
               figures
               ,
               when
               one
               quantity
               hath
               its
               genesis
               by
               the
               fluxe
               ,
               or
               motion
               of
               another
               ,
               the
               quantity
               formed
               by
               that
               motion
               is
               greater
               then
               the
               quantity
               moved
               ,
               though
               still
               without
               any
               proportion
               :
               so
               by
               the
               divaricating
               of
               the
               sides
               ,
               by
               the
               motion
               of
               one
               of
               the
               sides
               ,
               of
               a
               recto-convexe
               angle
               of
               Contact
               ,
               is
               a
               genesis
               of
               recto-convexe
               angles
               of
               section
               ;
               and
               the
               recto-convexe
               angles
               of
               section
               are
               still
               greater
               then
               the
               recto-convexe
               angle
               of
               Contact
               ,
               or
               it
               s
               respectively
               moved
               side
               :
               and
               all
               still
               without
               any
               proportion
               .
               And
               inequality
               without
               proportionality
               being
               so
               usually
               asserted
               ,
               
               and
               familiar
               between
               mere
               heterogeneals
               ,
               doth
               easily
               acquit
               it self
               from
               the
               reproach
               you
               charge
               it
               withal
               ,
               of
               being
               a
               contradiction
               in
               the
               very
               terms
               ;
               for
               finite
               and
               infinite
               never
               were
               denyed
               to
               be
               unequal
               ,
               yet
               never
               can
               be
               made
               out
               to
               be
               mathematically
               homogeneal
               and
               proportionable
               .
               For
               though
               all
               proportion
               is
               either
               of
               equality
               ,
               or
               inequality
               ;
               that
               hinders
               not
               but
               an
               inequality
               in
               heterogeneals
               may
               be
               admitted
               ,
               without
               an
               asserting
               of
               proportion
               between
               them
               .
               So
               as
               mathematical
               homogeneity
               is
               not
               proved
               by
               inequality
               ,
               till
               proportionablenes
               be
               as
               well
               proved
               as
               inequality
               ▪
               Of
               the
               same
               leaven
               is
               that
               strange
               kind
               of
               reasoning
               you
               use
               ;
               to
               any
               right-lined
               angle
               a
               crooked-lined
               angle
               may
               be
               made
               equal
               (
               we
               confess
               a
               thousand
               several
               crooked-lined
               angles
               may
               be
               made
               equal
               to
               any
               one
               right-lined
               )
               angle
               )
               but
               you
               from
               thence
               inferre
               that
               therefore
               all
               crooked
               lined
               angles
               and
               right-lined
               angles
               are
               of
               the
               same
               kind
               ;
               without
               adding
               as
               well
               ,
               that
               to
               any
               crooked-lined
               ,
               or
               mixed
               lined
               angle
               ,
               a
               right-lined
               angle
               also
               might
               be
               made
               equal
               ;
               which
               can
               never
               be
               done
               ;
               the
               contrary
               thereof
               being
               confessedly
               demonstrated
               in
               Geometry
               :
               or
               if
               any
               be
               so
               opinionated
               that
               it
               is
               either
               easy
               ,
               or
               feasible
               ,
               let
               them
               give
               a
               right-lined
               angle
               equal
               to
               an
               anisoclitical
               angle
               ,
               whose
               sides
               concurre
               by
               way
               of
               section
               or
               to
               an
               ultradiametral
               convexo-concave
               angle
               
               of
               Contact
               :
               and
               trye
               whether
               the
               equality
               of
               the
               two
               angles
               be
               not
               disproveable
               .
               In
               the
               same
               manner
               inequality
               without
               proportion
               ,
               is
               asserted
               between
               both
               the
               recto-concave
               and
               recto-convexe
               angles
               of
               the
               semi-circle
               ,
               
                 i.
                 e.
              
               the
               inner
               and
               outer
               angles
               of
               the
               semi-circle
               ,
               and
               a
               right
               ,
               or
               any
               other
               right-lined
               angle
               ▪
               But
               to
               return
               ,
               from
               the
               concrete
               and
               composite
               nature
               of
               not
               right-lined
               angles
               ,
               when
               compared
               with
               right-lined
               angles
               ;
               the
               enigmaticalnes
               of
               the
               proposition
               ,
               that
               magnitudes
               of
               one
               sort
               may
               be
               greater
               ,
               and
               less
               then
               a
               given
               magnitude
               of
               another
               ,
               but
               never
               equal
               ,
               becomes
               most
               clear
               and
               doubtles
               :
               as
               infinite
               solids
               may
               be
               given
               greater
               ,
               or
               lesser
               then
               the
               heterogeneal
               concrete
               magnitude
               of
               a
               cube
               foot
               and
               a
               foot-line
               ;
               but
               never
               can
               any
               solid
               be
               given
               thereunto
               equal
               :
               so
               likewise
               infinite
               heterogeneal
               concrete
               magnitudes
               consisting
               each
               of
               solids
               with
               the
               accrescency
               ,
               or
               annexion
               of
               a
               foot-line
               ,
               may
               be
               given
               greater
               ,
               or
               lesser
               then
               a
               cube
               elle
               ;
               but
               in
               such
               comparisons
               ,
               never
               can
               the
               concretes
               of
               the
               heterogeneal
               magnitudes
               be
               equal
               ,
               or
               in
               proportion
               to
               any
               solitary
               ,
               single
               ,
               one
               ,
               of
               the
               heterogeneals
               .
            
             
               In
               like
               manner
               it
               manifestly
               appears
               what
               is
               to
               be
               thought
               of
               that
               often
               pretended
               equality
               between
               the
               two
               angles
               of
               a
               semi-circle
               and
               a
               right
               right-lined
               angle
               :
               for
               by
               coaptation
               and
               
               the
               goings
               forth
               of
               the
               lines
               from
               the
               angular
               point
               it
               appears
               otherwise
               .
               And
               how
               will
               they
               that
               assert
               their
               equality
               demonstrate
               it
               ?
               and
               by
               what
               way
               will
               they
               admeasure
               it
               ?
               for
               interjacent
               and
               intercepted
               arches
               of
               Circles
               drawn
               upon
               the
               angular
               point
               as
               center
               ,
               can
               in
               these
               angles
               contribute
               nothing
               to
               the
               admeasuring
               of
               them
               :
               and
               coaptation
               makes
               against
               their
               equality
               :
               and
               is
               so
               farre
               from
               making
               a
               right
               right-lined
               angle
               to
               be
               the
               constant
               standard
               of
               all
               angles
               of
               semi-circles
               ,
               that
               it
               manifestly
               shews
               the
               angles
               of
               greater
               semi-circles
               to
               be
               greater
               and
               of
               less
               ,
               lesser
               .
               The
               objection
               ,
               that
               unequal
               circles
               ,
               semi
               circles
               ,
               and
               segments
               of
               equal
               degrees
               ,
               cannot
               be
               judged
               like
               and
               homologal
               figures
               ,
               except
               all
               their
               respective
               angles
               be
               equal
               ,
               as
               well
               as
               the
               sides
               homologal
               :
               this
               may
               be
               easily
               answered
               ;
               Viz.
               that
               it
               is
               like
               genesis
               that
               makes
               like
               figures
               ,
               which
               in
               all
               right-lined
               figures
               makes
               the
               answering
               angles
               equal
               ,
               and
               in
               all
               figures
               the
               sides
               to
               be
               homologal
               ,
               and
               the
               difference
               of
               the
               homologal
               angles
               to
               be
               less
               then
               the
               least
               right-lined
               angle
               .
            
             
               In
               Circles
               the
               same
               genesis
               ,
               by
               the
               circumduction
               of
               the
               semi-diameter
               about
               the
               Center
               ,
               which
               makes
               the
               figures
               like
               ,
               and
               the
               sides
               homologal
               ,
               makes
               in
               unequal
               circles
               the
               curvatures
               ,
               angles
               of
               coincidence
               ,
               the
               angles
               of
               the
               semi
               circles
               ,
               and
               of
               homologal
               segments
               ,
               
               necessarily
               unequal
               .
               Or
               ,
               then
               figures
               〈◊〉
               be
               judged
               like
               when
               in
               a
               concentrick
               〈…〉
               the
               perimeter
               ,
               sides
               and
               lines
               of
               the
               〈…〉
               proportionable
               to
               those
               in
               the
               other
               ,
               〈…〉
               or
               may
               be
               placed
               parallel
               ,
               or
               〈…〉
               those
               in
               the
               one
               ,
               to
               those
               in
               the
               other
               ▪
               ,
               〈…〉
               which
               in
               right-lined
               figures
               ,
               t
               is
               true
               ,
               th●●●
               alwayes
               follows
               equality
               between
               the
               answering
               angles
               ,
               but
               not
               so
               in
               other
               lined
               figures
               .
               And
               because
               equality
               of
               angles
               in
               like
               right-lined
               figures
               is
               so
               much
               urged
               ;
               the
               difference
               of
               the
               case
               of
               angles
               in
               like
               right-lined
               and
               like
               crooked-lined
               figures
               may
               plainly
               appear
               in
               another
               remarke
               ;
               Viz.
               that
               in
               like
               right-lined
               figures
               ,
               the
               sides
               of
               the
               answering
               angles
               may
               from
               the
               answering
               angular
               point
               be
               exactly
               coapted
               one
               to
               another
               ;
               which
               in
               unequal
               Circles
               ,
               though
               never
               so
               like
               figures
               ,
               is
               most
               apparently
               impossible
               :
               that
               it
               cannot
               but
               be
               unreasonable
               to
               expect
               as
               absolute
               a
               conformity
               between
               the
               answering
               angles
               in
               like
               crooked
               lined
               figures
               ,
               as
               there
               is
               in
               like
               right-lined
               figures
               ;
               being
               so
               manifestly
               against
               both
               the
               eye
               and
               demonstration
               .
               And
               therefore
               like
               segments
               of
               Circles
               ,
               are
               not
               defined
               by
               equality
               of
               angles
               contained
               between
               their
               arches
               and
               their
               chords
               ;
               but
               by
               the
               equality
               of
               the
               angles
               contained
               in
               them
               ;
               the
               one
               being
               certain
               ,
               constant
               in
               all
               ,
               and
               demonstrable
               ;
               the
               other
               in
               most
               cases
               ,
               not
               only
               doubtful
               ,
               but
               
               impossible
               .
               So
               the
               objection
               ,
               from
               the
               asserted
               equality
               of
               alternate
               angles
               ,
               made
               by
               a
               right-line
               cutting
               parallel
               circumferences
               ;
               is
               readily
               answered
               by
               denying
               the
               truth
               of
               what
               is
               presumed
               in
               the
               objection
               ,
               as
               never
               by
               any
               demonstrated
               ,
               or
               ever
               possibly
               demonstrable
               ,
               Viz.
               that
               the
               alternate
               and
               vertically
               opposite
               angles
               ,
               made
               by
               a
               right-line
               cutting
               parallel
               circumferences
               are
               equal
               .
               And
               the
               tyranny
               of
               forcing
               lines
               out
               of
               their
               natures
               and
               special
               properties
               ,
               may
               appear
               in
               that
               very
               instance
               of
               compelling
               the
               parallelisme
               of
               curve
               lines
               to
               answer
               the
               consectaries
               and
               idioms
               of
               the
               parallelisme
               of
               right-lines
               :
               in
               which
               ,
               to
               omit
               the
               alleadged
               instance
               ,
               as
               by
               you
               unproved
               ,
               and
               for
               good
               reasons
               by
               us
               to
               be
               denyed
               ,
               a
               right-line
               tangent
               of
               the
               lesser
               concentrick
               Circle
               cuts
               the
               circumference
               of
               the
               greater
               ,
               and
               infinite
               right-lines
               cutting
               the
               greater
               ,
               neither
               cut
               ,
               nor
               touch
               the
               lesser
               ,
               which
               is
               repugnant
               to
               the
               nature
               of
               parallelisme
               in
               right
               lines
               .
               That
               that
               which
               is
               so
               much
               contended
               for
               ,
               that
               a
               crooked-line
               and
               a
               right-line
               are
               homogeneal
               as
               to
               length
               and
               their
               general
               lineariness
               ,
               was
               ,
               or
               ought
               never
               to
               be
               denied
               ,
               there
               being
               all
               possibility
               of
               equality
               ,
               and
               truly
               proportionable
               inequality
               between
               them
               ,
               what
               kind
               of
               curvature
               soever
               the
               crooked-lines
               bear
               :
               but
               that
               they
               are
               homogeneal
               as
               to
               the
               positure
               of
               their
               longitude
               ;
               the
               site
               and
               
               manner
               of
               their
               extension
               ,
               hath
               unto
               me
               been
               alwayes
               unconceivable
               :
               whence
               the
               truths
               on
               both
               hands
               clearly
               follow
               ,
               Viz.
               that
               an
               arch
               and
               a
               right-line
               may
               be
               equal
               and
               hold
               alwayes
               a
               true
               limited
               ,
               exact
               proportion
               one
               to
               another
               ,
               but
               the
               arch
               and
               it
               s
               chorde
               never
               can
               be
               equal
               ,
               
                 i.
                 e.
              
               there
               never
               can
               be
               equality
               between
               a
               right-line
               and
               a
               crooked-line
               ,
               both
               posited
               between
               the
               same
               two
               terminateing
               points
               ,
               nor
               any
               analogy
               between
               the
               rectitude
               of
               the
               one
               and
               the
               curvature
               of
               the
               other
               .
               And
               the
               seeking
               to
               prove
               the
               equality
               of
               angles
               contain'd
               under
               homologal
               lines
               ,
               the
               one
               curve
               ,
               the
               other
               a
               right-line
               from
               your
               usual
               fancy
               of
               a
               regular
               polygone
               of
               infinite
               angles
               in
               every
               circle
               ,
               is
               too
               wild
               to
               be
               perswasive
               :
               for
               though
               at
               any
               mean
               point
               in
               curve
               lines
               ,
               the
               two
               parts
               of
               the
               curve-line
               may
               be
               conceived
               specially
               to
               meet
               as
               several
               parts
               and
               lines
               ,
               and
               so
               to
               have
               inclination
               the
               one
               to
               the
               other
               ,
               and
               so
               to
               constitute
               an
               angle
               ;
               which
               we
               call
               the
               angle
               of
               curvature
               and
               coincidence
               ,
               not
               reasonably
               to
               be
               denyed
               by
               those
               ,
               with
               whom
               it
               is
               so
               ordinary
               to
               make
               such
               suppositions
               ,
               and
               especially
               such
               as
               can
               so
               usually
               against
               possibility
               imagine
               angles
               in
               a
               right-line
               remaining
               a
               right-line
               :
               yet
               that
               angles
               should
               be
               without
               sides
               ,
               and
               a
               perimeter
               of
               any
               figure
               conceived
               at
               once
               to
               be
               all
               angular
               points
               and
               no
               lineary
               sides
               ,
               clearly
               
               ●●●●stes
               the
               perimeter
               of
               the
               nature
               of
               a
               line
               :
               〈◊〉
               to
               me
               it
               seems
               far
               from
               the
               nature
               of
               a
               ●●●ular
               figure
               ,
               that
               hath
               nothing
               but
               points
               ●●stead
               of
               lines
               to
               bound
               it
               :
               but
               which
               is
               most
               material
               ;
               that
               a
               number
               actually
               infinite
               should
               be
               so
               easily
               given
               ,
               is
               hard
               to
               allow
               :
               and
               that
               indivisibles
               ,
               as
               points
               ,
               should
               be
               so
               adjacent
               〈◊〉
               to
               another
               ,
               one
               without
               another
               ,
               with●●●
               coincidence
               ,
               identity
               and
               unity
               ,
               is
               new
               ●●●●osophy
               ,
               and
               not
               easily
               capable
               of
               any
               in
               〈◊〉
               defense
               .
               Therefore
               that
               argumentation
               〈◊〉
               ,
               that
               such
               a
               regular
               polygone
               of
               in
               〈◊〉
               whether
               sides
               or
               angles
               is
               either
               a
               circle
               ,
               〈◊〉
               inscribable
               in
               a
               circle
               ,
               is
               too
               vain
               :
               for
               it
               can
               be
               neither
               ,
               being
               nothing
               ,
               because
               there
               neither
               is
               ,
               nor
               can
               be
               any
               such
               thing
               :
               for
               if
               any
               such
               were
               allowed
               they
               must
               of
               necessity
               have
               equal
               and
               infinite
               perimeters
               ;
               which
               is
               too
               gross
               to
               be
               admitted
               in
               it self
               ,
               and
               besides
               renders
               the
               whole
               matter
               unapplyable
               to
               Circles
               ,
               which
               are
               acknowledged
               ,
               to
               be
               some
               less
               then
               others
               .
               So
               as
               all
               discourses
               of
               a
               regular
               polygone
               of
               infinite
               angles
               ,
               are
               discourses
               not
               only
               of
               a
               non-entity
               ,
               but
               an
               absolute
               impossibility
               ,
               which
               renders
               all
               suppositions
               thereof
               unjustifiable
               .
               And
               of
               the
               same
               fineness
               are
               those
               sayings
               ,
               that
               the
               magnitude
               of
               an
               angle
               is
               not
               to
               be
               judged
               of
               from
               the
               divarication
               which
               the
               sides
               have
               without
               the
               angular
               point
               ,
               or
               point
               of
               concurrence
               ,
               but
               from
               the
               divarication
               
               which
               they
               have
               in
               the
               point
               of
               concurrence
               ;
               as
               if
               in
               an
               indivisible
               point
               they
               could
               have
               any
               divarication
               at
               all
               .
               But
               as
               if
               it
               were
               resolved
               that
               even
               this
               should
               be
               transcended
               in
               monstrosity
               ,
               for
               the
               justifying
               of
               the
               equality
               of
               mixed
               lined
               angles
               contain'd
               by
               homologal
               sides
               in
               unequal
               Circles
               ,
               by
               an
               instance
               from
               the
               coapting
               of
               unequal
               hexagones
               to
               the
               same
               line
               ,
               as
               a
               common
               side
               in
               them
               all
               ,
               divided
               equally
               by
               a
               perpendicular
               passing
               thorow
               the
               centers
               of
               all
               ,
               a
               right-lined
               angle
               is
               strangely
               constituted
               either
               of
               three
               right-lines
               concurring
               ,
               but
               not
               in
               the
               same
               point
               ,
               or
               of
               two
               lines
               without
               any
               concurrence
               ,
               or
               else
               the
               instance
               must
               be
               void
               of
               all
               pertinency
               to
               the
               question
               .
               So
               to
               all
               those
               objections
               seemingly
               founded
               upon
               that
               proposition
               ,
               or
               postulate
               ,
               that
               what
               is
               less
               then
               any
               positive
               quantity
               whatsoever
               is
               not
               any
               quantity
               at
               all
               ,
               is
               justly
               answered
               ;
               that
               the
               proposition
               or
               postulate
               is
               most
               true
               and
               reasonable
               ,
               and
               cannot
               by
               any
               of
               sound
               mind
               be
               denyed
               ,
               or
               doubted
               :
               but
               no
               force
               of
               objection
               could
               be
               made
               out
               of
               that
               ,
               if
               other
               things
               of
               a
               less
               veritable
               nature
               had
               not
               been
               taken
               in
               ;
               as
               in
               most
               of
               them
               the
               fancyed
               possibility
               of
               a
               regular
               polygone
               of
               infinite
               angles
               ;
               and
               frequently
               that
               a
               Circle
               is
               that
               regular
               polygone
               .
            
             
               But
               besides
               ,
               though
               what
               is
               less
               then
               any
               positive
               quantity
               whatsoever
               be
               not
               any
               quantity
               
               at
               all
               ;
               yet
               this
               hinders
               not
               ,
               but
               quantityes
               may
               be
               mathematically
               heterogeneal
               and
               improportionable
               one
               to
               another
               :
               so
               every
               surface
               is
               less
               then
               any
               solid
               :
               and
               angles
               of
               Contact
               are
               not
               less
               then
               any
               quantity
               whatsoever
               ,
               for
               there
               is
               in
               the
               least
               of
               them
               an
               endles
               ,
               unexhausted
               divisibility
               ;
               which
               how
               it
               can
               consist
               with
               a
               nonquantitativenes
               ,
               let
               those
               that
               have
               a
               mind
               to
               be
               serious
               solemnely
               consider
               .
               To
               the
               objection
               that
               would
               prove
               ,
               neither
               semi-circumference
               to
               contain
               an
               angle
               with
               the
               right-lined
               tangent
               of
               it
               in
               its
               extreem
               point
               ,
               because
               the
               two
               semi
               circumferences
               contain
               no
               angle
               at
               that
               point
               ,
               but
               are
               one
               regularly
               continued
               line
               ,
               and
               the
               circumference
               and
               right-lined
               tangent
               are
               lines
               coincident
               ,
               at
               least
               as
               to
               the
               point
               of
               Contact
               ;
               manifest
               and
               reasonable
               answers
               cannot
               be
               to
               seek
               out
               of
               what
               hath
               already
               been
               said
               .
               For
               first
               what
               hinders
               the
               reasonable
               conceiving
               of
               angularity
               at
               any
               point
               of
               a
               curve-line
               ,
               where
               is
               both
               concurrence
               ,
               inclination
               and
               divisibility
               ,
               more
               then
               the
               notion
               of
               divisibility
               at
               any
               mean
               point
               of
               a
               right-line
               ?
               And
               not
               to
               doubt
               but
               a
               curve
               line
               may
               be
               conceived
               reasonably
               as
               one
               continued
               line
               ,
               as
               well
               as
               two
               ,
               or
               more
               ,
               inclined
               and
               concurring
               right-lines
               ;
               yet
               that
               the
               right-line
               tangent
               and
               curve-line
               which
               it
               toucheth
               ,
               should
               be
               said
               to
               be
               coincident
               lines
               ,
               in
               such
               sense
               as
               to
               exclude
               angularity
               ;
               
               or
               that
               any
               two
               lines
               can
               be
               so
               coincident
               in
               one
               only
               point
               ,
               as
               to
               exclude
               angularity
               ,
               and
               the
               inflexion
               of
               one
               to
               ,
               or
               from
               the
               other
               ,
               except
               both
               lye
               in
               one
               and
               the
               same
               right-line
               ;
               hath
               ,
               as
               elsewhere
               ,
               been
               plainly
               and
               abundantly
               answered
               to
               .
               To
               the
               objection
               that
               the
               Area
               of
               a
               Circle
               is
               equal
               to
               a
               rect-angle
               under
               the
               semidiameter
               and
               semi
               circumference
               ;
               and
               that
               therefore
               the
               semidiameter
               in
               a
               Circle
               is
               perpendicular
               to
               the
               circumference
               in
               a
               Circle
               and
               makes
               at
               the
               circumference
               four
               equal
               right-angles
               ;
               is
               answered
               ,
               that
               the
               whole
               objection
               is
               a
               manifest
               paralogisme
               .
               For
               it
               is
               not
               denyed
               but
               in
               the
               right-lined
               rect-angle
               under
               the
               semidiameter
               ,
               and
               a
               right-line
               equal
               to
               the
               semi
               circumference
               ,
               is
               presumed
               ,
               and
               by
               the
               definition
               of
               a
               rect-angle
               inferred
               ,
               that
               the
               angle
               under
               those
               two
               right-lines
               is
               a
               right
               right-lined
               angle
               :
               besides
               it
               is
               not
               denyed
               ,
               but
               the
               diameter
               falls
               perpendicularly
               in
               the
               circle
               upon
               the
               circumference
               :
               and
               that
               the
               four
               angles
               made
               by
               the
               falling
               of
               the
               semidiameter
               upon
               the
               circumference
               differ
               from
               one
               another
               less
               than
               the
               least
               right-lined
               angle
               :
               however
               that
               cannot
               force
               the
               falling
               of
               the
               semidiameter
               perpendicularly
               upon
               the
               circumference
               into
               the
               properties
               of
               perpendicularness
               between
               right
               lines
               ,
               which
               still
               divides
               the
               space
               at
               the
               angular
               point
               into
               four
               angles
               always
               ,
               every
               way
               alike
               ,
               and
               equal
               ;
               which
               in
               right
               lines
               perpendiculars
               
               upon
               curve
               lines
               in
               the
               same
               plane
               ,
               is
               impossible
               to
               be
               ,
               and
               therefore
               impossible
               ever
               to
               be
               demonstrated
               .
            
             
               It
               will
               not
               be
               unuseful
               here
               to
               enquire
               wherein
               the
               likeness
               and
               unlikeness
               of
               angles
               doth
               consist
               ,
               and
               whether
               there
               be
               any
               such
               thing
               as
               likeness
               and
               unlikeness
               in
               angles
               ,
               or
               whether
               the
               likeness
               ,
               or
               unlikeness
               of
               figures
               be
               only
               in
               the
               similitude
               ,
               or
               dissimilitude
               of
               the
               sides
               .
               And
               that
               by
               a
               circumspect
               consideration
               of
               the
               nature
               of
               mathematical
               similitude
               in
               other
               cases
               ,
               we
               may
               be
               the
               better
               guided
               into
               the
               true
               and
               most
               rational
               notion
               of
               similitude
               in
               angles
               ,
               let
               us
               remember
               what
               hath
               already
               been
               judged
               in
               this
               point
               ,
               and
               what
               is
               herein
               confessed
               on
               all
               hands
               .
               First
               ,
               in
               right
               lined
               figures
               ,
               those
               figures
               are
               judged
               like
               ,
               whose
               answering
               angles
               are
               equal
               ,
               and
               the
               answering
               sides
               and
               other
               lines
               proportionable
               ;
               and
               if
               they
               be
               equal
               ,
               they
               may
               be
               coapted
               ,
               homologal
               side
               to
               homologal
               side
               ,
               and
               answering
               angle
               to
               answering
               angle
               ;
               or
               whether
               they
               be
               equal
               ,
               or
               unequal
               ,
               all
               the
               sides
               and
               other
               answering
               lines
               of
               the
               one
               may
               be
               set
               ,
               as
               from
               the
               same
               center
               ,
               each
               at
               parallelisme
               ,
               or
               coincidency
               with
               the
               answering
               sides
               ,
               or
               lines
               of
               the
               other
               ,
               so
               as
               in
               like
               right
               lined
               figures
               is
               proportionablenes
               ▪
               in
               the
               answering
               sides
               ,
               equality
               of
               the
               answering
               angles
               ,
               coaptability
               of
               all
               the
               answering
               sides
               into
               either
               coincidence
               ,
               or
               equidistance
               ,
               and
               a
               proportionate
               
               distance
               of
               the
               answering
               angles
               ,
               each
               from
               the
               other
               .
               But
               now
               the
               similitude
               of
               figuration
               which
               is
               in
               circles
               founded
               upon
               the
               like
               genesis
               of
               all
               circles
               is
               in
               the
               equidistance
               ,
               or
               coincidence
               of
               their
               circumferences
               ,
               when
               the
               center
               of
               the
               one
               is
               coapted
               to
               the
               center
               of
               the
               other
               ;
               and
               that
               equal
               angles
               from
               the
               center
               intercept
               proportionable
               parts
               of
               the
               circumferences
               ,
               and
               that
               proportionable
               parts
               of
               the
               circumferences
               are
               connected
               by
               proportionable
               chords
               ,
               and
               contain
               and
               sustain
               equal
               right
               lined
               angles
               .
               And
               the
               like
               speculations
               might
               be
               pursued
               in
               other
               figures
               both
               plane
               and
               solid
               .
               In
               a
               rational
               application
               of
               which
               to
               the
               disquisition
               of
               angles
               ,
               it
               may
               be
               first
               enquired
               ,
               whether
               there
               be
               any
               such
               thing
               as
               similitude
               &
               dissimilitude
               to
               be
               own'd
               or
               observ'd
               among
               angles
               ;
               and
               if
               so
               ,
               how
               that
               similitude
               is
               to
               be
               understood
               ;
               and
               whether
               it
               be
               inconsistent
               with
               inequality
               in
               the
               answering
               angles
               .
               To
               clear
               all
               which
               we
               must
               know
               ,
               that
               in
               unequal
               but
               like
               right
               lined
               figures
               ,
               the
               homologal
               angles
               are
               always
               equal
               ,
               being
               contained
               in
               both
               figures
               under
               right
               lines
               :
               but
               in
               unequal
               ,
               and
               like
               mixed
               lined
               and
               crooked
               lined
               figures
               ,
               the
               homologal
               mixed
               lined
               ,
               or
               crooked
               lined
               angles
               ,
               neither
               are
               ,
               nor
               can
               be
               equal
               ;
               only
               their
               difference
               is
               ever
               less
               than
               the
               least
               right
               lined
               angle
               ;
               and
               their
               similitude
               hath
               never
               rationally
               yet
               by
               any
               been
               questioned
               but
               
               with
               good
               reason
               according
               to
               the
               following
               gloss
               is
               to
               be
               justified
               .
               The
               more
               clearly
               to
               demonstrate
               all
               which
               in
               Fig.
               21.
               upon
               the
               common
               center
               A.
               draw
               two
               unequal
               Circles
               ,
               Viz.
               HEG.
               the
               lesser
               ,
               and
               BCD
               .
               the
               greater
               .
               Then
               from
               any
               point
               B.
               in
               the
               greater
               circle
               BCD
               .
               draw
               the
               right
               line
               BEAFK
               .
               thorough
               the
               common
               center
               A.
               cutting
               the
               circumference
               of
               the
               lesser
               circle
               HEG.
               in
               the
               point
               E.
               then
               take
               AF.
               equal
               to
               BE.
               and
               upon
               the
               center
               F.
               and
               semidiameter
               FE
               .
               draw
               another
               circle
               ECKD
               .
               equal
               to
               the
               greater
               circle
               CBD
               .
               Here
               on
               all
               hands
               is
               agreed
               ,
               that
               the
               lesser
               circle
               HEG.
               and
               the
               greater
               circle
               CBD
               .
               are
               like
               figures
               ,
               and
               that
               therefore
               the
               two
               mixed
               lined
               recto-concave
               angles
               ABD
               .
               and
               AEG
               .
               are
               like
               angles
               .
               And
               by
               the
               construction
               it
               is
               apparent
               that
               the
               two
               recto-concave
               angles
               ABD
               .
               and
               AED
               .
               are
               equal
               :
               and
               that
               the
               two
               recto-concave
               angles
               AEG
               .
               and
               AED
               .
               are
               unequal
               :
               and
               that
               the
               recto-concave
               angle
               AED
               .
               is
               greater
               than
               the
               recto-concave
               angle
               AEG
               .
               and
               in
               all
               like
               cases
               it
               is
               always
               so
               ;
               however
               the
               difference
               of
               the
               two
               angles
               must
               necessarily
               be
               less
               than
               any
               right
               lined
               angle
               ,
               because
               all
               such
               citradiametral
               concavo-convexe
               angles
               of
               contact
               ,
               as
               GED
               .
               are
               always
               less
               than
               any
               right
               lined
               angle
               ;
               as
               is
               consequent
               to
               what
               hath
               been
               demonstrated
               in
               Geometry
               ,
               which
               was
               to
               be
               shewn
               .
               Whence
               we
               may
               clearly
               observe
               
               that
               similitude
               of
               Figures
               lies
               chiefly
               in
               the
               proportionality
               and
               like
               positure
               of
               homologal
               sides
               ,
               in
               respect
               of
               parallelisme
               and
               coincidence
               ,
               without
               imposing
               any
               other
               necessity
               for
               the
               equality
               of
               answering
               angles
               ,
               then
               as
               it
               may
               consist
               with
               the
               proportionating
               and
               like
               positing
               of
               the
               homologal
               sides
               and
               lines
               .
               And
               such
               inequality
               of
               the
               answering
               angles
               ,
               as
               is
               requisite
               to
               the
               proportionating
               and
               alike
               positing
               of
               the
               homologal
               sides
               and
               lines
               in
               like
               and
               unequal
               mixed
               ,
               or
               crooked
               lined
               figures
               ,
               is
               so
               far
               from
               being
               inconsistent
               with
               their
               figurative
               similitude
               ,
               that
               they
               cannot
               without
               it
               ,
               under
               inequality
               keep
               similitude
               in
               their
               figuration
               .
               And
               though
               the
               inequality
               ,
               which
               is
               between
               angle
               and
               angle
               be
               less
               then
               that
               which
               is
               ,
               or
               may
               be
               ,
               between
               the
               Homologal
               sides
               and
               lines
               ;
               yet
               the
               inequality
               of
               the
               angles
               is
               more
               different
               ,
               being
               an
               inequality
               without
               proportion
               ,
               whereas
               the
               inequality
               of
               the
               homologal
               sides
               and
               lines
               is
               ever
               according
               to
               proportion
               .
               Upon
               the
               whole
               ,
               it
               is
               not
               equality
               that
               generally
               makes
               angles
               to
               be
               like
               ,
               for
               a
               right
               lined
               ,
               and
               an
               Isoclitical
               concavo
               Convexe
               may
               be
               equal
               angles
               ,
               but
               never
               can
               be
               like
               ,
               nor
               were
               ever
               suspected
               to
               be
               so
               :
               but
               that
               which
               makes
               angles
               to
               be
               like
               is
               rather
               their
               being
               contained
               under
               homologal
               sides
               ,
               posited
               so
               as
               to
               construct
               a
               like
               &
               homologal
               figuration
               .
               And
               this
               whole
               matter
               depends
               upon
               what
               I
               before
               hinted
               ,
               Viz.
               the
               figuration
               of
               
               lines
               and
               angles
               ;
               Sc.
               rectitude
               ,
               being
               one
               single
               ,
               simple
               ,
               figuration
               of
               lines
               incapable
               of
               any
               variety
               ,
               like
               angles
               under
               right
               lines
               are
               always
               equal
               ,
               and
               never
               can
               be
               unequal
               :
               but
               〈◊〉
               ,
               being
               infinitely
               variable
               ,
               those
               〈…〉
               are
               said
               to
               be
               like
               ,
               
                 i.
                 e.
              
               homologal
               ,
               ●hose
               construction
               is
               like
               ,
               so
               as
               in
               like
               figure●
               upon
               a
               common
               center
               ,
               to
               set
               homologal
               sides
               and
               lines
               proportionably
               equidistant
               ,
               or
               coincident
               ;
               as
               circumferences
               of
               like
               though
               unequal
               circles
               ,
               ellipses
               ,
               &c.
               
               And
               so
               under
               a
               thousand
               inequalities
               such
               mixed
               and
               crooked
               lined
               angles
               may
               be
               like
               :
               as
               in
               Fig.
               21.
               the
               recto-concave
               angles
               AEG
               .
               and
               AED
               .
               being
               unequal
               ,
               are
               both
               like
               to
               the
               angle
               
               ABD
               .
               and
               so
               is
               every
               angle
               how
               different
               soever
               ,
               if
               contained
               under
               a
               diameter
               and
               a
               circumference
               .
               And
               indeed
               the
               figuration
               of
               angles
               ,
               being
               incompleat
               ,
               and
               the
               length
               of
               their
               sides
               undetermined
               ,
               neither
               parallelisme
               ,
               nor
               coincidency
               ,
               nor
               proportionality
               ,
               nor
               homologal
               positure
               ,
               can
               ,
               when
               they
               are
               unequal
               ,
               be
               conceived
               in
               their
               sides
               ,
               without
               special
               relation
               to
               some
               compleat
               Figure
               and
               its
               Center
               :
               so
               the
               recto-concave
               angles
               
               AEG
               .
               and
               
               ABD
               in
               Fig.
               21.
               are
               like
               ,
               as
               conceived
               to
               be
               each
               contained
               respectively
               under
               a
               diameter
               and
               a
               circumference
               ,
               and
               so
               upon
               a
               common
               center
               positable
               into
               parallelisme
               ,
               coincidence
               and
               proportionableness
               ,
               and
               all
               possible
               likeness
               and
               homologalness
               
               of
               figuration
               .
               In
               right-lined
               angles
               ,
               where
               homologal
               and
               like
               angles
               are
               always
               equal
               ▪
               for
               the
               same
               reason
               every
               angle
               equal
               to
               a
               right-lined
               angle
               ,
               is
               not
               presently
               a
               like
               angle
               :
               〈◊〉
               a
               thousand
               equal
               angles
               ,
               are
               all
               ,
               ever
               ,
               and
               to
               all
               purposes
               ,
               unlike
               ;
               as
               two
               equal
               isoclitical
               right
               lined
               and
               crooked
               lined
               angles
               ,
               because
               they
               can
               never
               be
               coapted
               to
               be
               answering
               angles
               ,
               in
               like
               Figures
               ,
               or
               to
               set
               their
               containing
               sides
               homologally
               ,
               and
               in
               parallelisme
               ,
               or
               coincidence
               .
               That
               equality
               of
               answering
               angles
               is
               not
               so
               of
               the
               essence
               of
               like
               Figures
               ,
               as
               proportionality
               of
               sides
               ,
               and
               answering
               lines
               ,
               with
               their
               parallelisme
               ,
               or
               coincidency
               :
               only
               from
               the
               propriety
               of
               like
               plane
               Figures
               ,
               follows
               an
               equality
               in
               all
               like
               right-lined
               angles
               ;
               and
               in
               like
               curve
               lined
               Figures
               ,
               that
               their
               inequality
               is
               ever
               less
               than
               the
               least
               right-lined
               angle
               .
               Hence
               therefore
               appears
               that
               from
               the
               similitude
               which
               is
               in
               unequal
               circles
               ,
               the
               equality
               between
               angles
               of
               semi-circles
               &
               right
               right-lined
               angles
               is
               not
               effectually
               proved
               .
               And
               notwithstanding
               any
               thing
               in
               those
               arguments
               tendred
               and
               proved
               ,
               every
               recto-concave
               angle
               contained
               under
               a
               concave
               arch
               of
               a
               circle
               and
               a
               right-line
               ,
               which
               is
               perpendicular
               to
               the
               right
               line
               tangent
               of
               the
               arch
               at
               the
               angular
               point
               ,
               is
               greater
               than
               any
               right-lined
               acute
               angle
               ,
               and
               less
               than
               a
               right
               right-lined
               angle
               :
               and
               the
               recto-convexe
               angle
               contained
               
               under
               the
               convexe
               arch
               and
               the
               right
               lined
               tangent
               is
               less
               than
               any
               right
               lined
               angle
               whatsoever
               :
               and
               the
               other
               recto-convexe
               angle
               contained
               under
               the
               convexe
               arch
               and
               the
               right
               line
               ,
               which
               is
               perpendicular
               to
               the
               right
               line
               tangent
               at
               the
               angular
               point
               ,
               is
               greater
               than
               a
               right
               right-lined
               angle
               ,
               and
               less
               than
               any
               obtuse
               right-lined
               angle
               whatsoever
               .
               And
               whereas
               you
               object
               that
               if
               ,
               as
               in
               Fig.
               12.
               the
               right
               line
               KG
               A.
               be
               the
               diameter
               of
               the
               circle
               K
               
               AD.
               and
               
               AB
               the
               right
               line
               tangent
               ,
               then
               KAB
               is
               a
               right
               right
               lined
               angle
               :
               and
               the
               recto-convexe
               angle
               of
               contact
               DAB
               is
               no
               part
               of
               the
               right
               right
               lined
               angle
               KAB
               .
               that
               therefore
               the
               angle
               of
               the
               semicitcle
               KAD
               is
               still
               equal
               to
               a
               right
               right
               lined
               angle
               ,
               because
               what
               is
               taken
               out
               of
               it
               was
               no
               part
               of
               it
               .
               I
               answer
               ,
               the
               recto-convexe
               angle
               of
               contact
               DAB
               is
               indeed
               no
               proportionable
               part
               of
               the
               right
               right-lined
               angle
               KAB
               ,
               but
               yet
               it
               is
               truly
               a
               part
               ,
               though
               improportionable
               ,
               and
               so
               mathematically
               heterogeneal
               :
               for
               if
               it
               had
               been
               no
               part
               at
               all
               ,
               and
               nothing
               ,
               then
               the
               angle
               of
               the
               semicircle
               KAD
               .
               (
               nothing
               being
               taken
               out
               of
               the
               right
               right-lined
               angle
               KAB
               ,
               but
               the
               recto-convexe
               angle
               of
               contact
               D
               
               AB
               ,
               which
               is
               said
               by
               you
               to
               be
               nothing
               ,
               and
               no
               angle
               )
               it
               should
               ●●ill
               remain
               a
               right
               right-lined
               angle
               ;
               which
               is
               not
               by
               any
               asserted
               ,
               the
               contrary
               being
               so
               manifest
               :
               besides
               that
               the
               separability
               of
               the
               recto-convexe
               
               angle
               D
               
               AB
               from
               the
               recto-concave
               angle
               KAD
               makes
               clear
               and
               certain
               ,
               the
               truth
               of
               its
               being
               a
               part
               of
               the
               right-lined
               angle
               KAB
               .
            
             
               And
               likewise
               from
               what
               hath
               been
               before
               declared
               in
               our
               opening
               the
               nature
               of
               a
               plane
               angle
               may
               clearly
               appear
               ,
               that
               we
               are
               not
               to
               understand
               that
               a
               plane
               angle
               is
               meerely
               the
               angular
               point
               ,
               or
               meerely
               in
               the
               angular
               point
               ,
               as
               contradistinguished
               from
               the
               containing
               sides
               ,
               though
               it
               there
               terminate
               ,
               or
               thence
               have
               its
               rise
               ;
               but
               angles
               are
               in
               the
               habitude
               of
               the
               concurring
               ,
               containing
               and
               inclined
               sides
               :
               Viz.
               the
               habitude
               which
               they
               hold
               each
               to
               other
               all
               along
               their
               tendency
               unto
               the
               angular
               point
               ,
               or
               their
               rise
               from
               thence
               ,
               if
               we
               would
               have
               the
               full
               notion
               ,
               inclination
               and
               figuration
               of
               an
               angle
               .
               For
               there
               is
               often
               a
               great
               inequality
               and
               vast
               imparity
               between
               the
               inclination
               ,
               sometimes
               of
               one
               part
               of
               the
               containing
               side
               to
               the
               other
               containing
               side
               ,
               and
               the
               inclination
               thereunto
               ,
               of
               other
               parts
               of
               the
               same
               first
               containing
               side
               ;
               as
               may
               appear
               in
               all
               mixed
               lined
               ,
               mixed
               crooked-lined
               and
               all
               other
               anisoclitical
               angles
               .
               And
               the
               nature
               of
               an
               angle
               consisting
               in
               inclination
               as
               well
               as
               in
               concurrence
               ,
               though
               concurrence
               may
               be
               and
               is
               in
               a
               point
               ,
               and
               inclination
               at
               a
               point
               ,
               yet
               inclination
               must
               be
               in
               the
               lines
               and
               of
               the
               lines
               ,
               and
               cannot
               be
               in
               a
               point
               separately
               .
               And
               methinks
               the
               nature
               of
               an
               angle
               ,
               and
               its
               inclination
               ,
               is
               
               scarcely
               so
               fully
               held
               forth
               ,
               when
               the
               inclination
               of
               the
               two
               lineary
               sides
               containing
               it
               ,
               as
               if
               the
               sides
               were
               not
               therein
               concerned
               ,
               is
               ordered
               to
               be
               observed
               only
               in
               the
               angular
               point
               ,
               and
               not
               out
               of
               it
               :
               because
               ,
               as
               you
               say
               ,
               though
               you
               urge
               it
               to
               the
               contrary
               ,
               many
               times
               out
               of
               the
               angular
               point
               in
               the
               containing
               sides
               ,
               no
               two
               points
               can
               be
               shewen
               in
               the
               one
               side
               ,
               where
               it
               hath
               the
               same
               inclination
               unto
               the
               other
               .
               Certainly
               in
               an
               indivisible
               ,
               such
               as
               is
               the
               angular
               point
               ,
               if
               abstractly
               considered
               ,
               it
               were
               vain
               to
               expect
               and
               impossible
               to
               observe
               any
               inclination
               :
               and
               no
               doubt
               as
               the
               magnitudes
               inclined
               are
               without
               the
               angular
               point
               ,
               so
               is
               also
               the
               inclination
               :
               though
               as
               they
               terminate
               in
               the
               angular
               point
               ,
               so
               doth
               the
               inclination
               .
               So
               the
               angle
               of
               a
               semi
               circle
               is
               not
               the
               common
               terme
               of
               the
               diameter
               and
               the
               semi
               circumference
               ,
               excluding
               the
               diameter
               and
               the
               semicir-cumference
               for
               then
               in
               an
               abstracted
               point
               it
               should
               be
               possible
               to
               observe
               an
               inclination
               ;
               and
               a
               point
               being
               indivisible
               should
               be
               inclined
               unto
               its
               self
               ;
               which
               is
               not
               convenient
               to
               assert
               :
               but
               rather
               the
               angle
               of
               a
               semi
               circle
               is
               the
               inclination
               of
               the
               semi
               circumference
               to
               the
               diameter
               ,
               terminated
               in
               the
               angular
               point
               ,
               which
               is
               common
               to
               both
               .
               And
               whereas
               it
               is
               said
               that
               out
               of
               the
               angular
               point
               ,
               no
               two
               points
               can
               be
               shewen
               in
               the
               diameter
               ,
               at
               which
               the
               diameter
               is
               equally
               inclined
               to
               the
               semi
               circumference
               ,
               nor
               in
               the
               semi
               circumference
               where
               it
               
               is
               equally
               inclined
               to
               the
               diameter
               ;
               all
               is
               allowed
               and
               averred
               as
               glosseably
               true
               ,
               and
               this
               is
               that
               which
               makes
               the
               great
               difference
               between
               anisoclitical
               and
               isoclitical
               angles
               ,
               and
               renders
               it
               so
               impossible
               to
               give
               an
               anisoclitical
               angle
               equal
               to
               an
               isoclitical
               angle
               .
               For
               in
               isoclitical
               angles
               the
               inclination
               of
               the
               sides
               ,
               the
               one
               unto
               the
               other
               ,
               is
               at
               all
               points
               the
               same
               without
               any
               variation
               ;
               as
               every
               where
               appears
               by
               the
               interjected
               arches
               of
               circles
               drawn
               upon
               the
               angular
               point
               as
               Center
               :
               but
               in
               anisoclitical
               angles
               ,
               at
               every
               several
               point
               ,
               the
               one
               containing
               side
               hath
               several
               and
               different
               inclinations
               to
               the
               other
               containing
               side
               :
               which
               is
               the
               cause
               that
               isoclitical
               angles
               may
               possibly
               and
               easily
               be
               given
               ,
               sometimes
               greater
               ,
               and
               sometimes
               lesser
               then
               anisoclitical
               angles
               ,
               but
               never
               equal
               :
               because
               in
               the
               one
               the
               inclination
               of
               the
               containing
               sides
               still
               varieth
               ,
               in
               the
               other
               not
               at
               all
               .
               And
               if
               the
               whole
               nature
               of
               an
               angle
               lye
               in
               the
               angular
               point
               ,
               without
               extending
               the
               habitude
               of
               it
               farther
               into
               the
               production
               and
               figuration
               of
               the
               containing
               sides
               ;
               it
               will
               not
               only
               be
               necessary
               for
               us
               to
               yield
               unto
               you
               that
               recto-convexe
               angles
               of
               contact
               are
               not
               quantitative
               ;
               but
               besides
               both
               you
               and
               we
               ,
               contrary
               to
               what
               we
               have
               alwayes
               hitherto
               judged
               ,
               shall
               be
               constrained
               to
               acknowledge
               that
               there
               is
               no
               quantitativenes
               neither
               in
               crooked-lined
               ,
               nor
               right-lined
               ,
               nor
               any
               other
               
               angles
               whatsoewer
               ,
               whether
               superficial
               ,
               in
               one
               ,
               or
               several
               plaines
               ,
               or
               solid
               .
               And
               can
               any
               thing
               be
               more
               horrid
               then
               to
               say
               ,
               the
               quantity
               of
               angles
               is
               not
               to
               be
               measured
               by
               the
               divarication
               of
               the
               sides
               at
               the
               angular
               point
               ,
               but
               by
               their
               divarication
               in
               the
               angular
               point
               ,
               where
               they
               have
               none
               at
               all
               ?
               But
               yet
               though
               it
               is
               thus
               evident
               ,
               that
               the
               inclination
               of
               the
               sides
               at
               the
               angular
               point
               ,
               may
               and
               frequently
               is
               much
               less
               or
               greater
               then
               the
               inclination
               of
               the
               same
               sides
               at
               other
               points
               ;
               which
               ,
               as
               is
               above
               hinted
               ,
               is
               not
               to
               be
               left
               out
               in
               the
               full
               ,
               genuine
               and
               clear
               consideration
               of
               the
               nature
               of
               angles
               ,
               their
               kinds
               ,
               figurations
               and
               quantities
               :
               however
               the
               inclination
               of
               the
               sides
               at
               the
               angular
               point
               ,
               is
               that
               which
               is
               most
               usually
               enquired
               after
               ,
               and
               most
               useful
               to
               be
               searched
               and
               observed
               in
               Geometry
               ;
               for
               the
               discoveries
               which
               are
               from
               thence
               made
               of
               lines
               ,
               how
               they
               fall
               coincidently
               ,
               or
               within
               ,
               or
               without
               others
               .
               To
               the
               objection
               that
               in
               fig.
               12.
               the
               right-lined
               tangent
               AB
               and
               the
               arch
               AL
               make
               all
               one
               and
               the
               same
               equal
               inclination
               to
               the
               right-line
               secant
               AC
               in
               the
               common
               angular
               point
               A
               ;
               and
               that
               therefore
               the
               right-lined
               angle
               BAC
               under
               the
               right-lined
               secant
               AC
               and
               the
               right-line
               tangent
               AB
               is
               equal
               to
               the
               mixed-lined
               recto-concave
               angle
               EAL
               under
               the
               right-line
               secant
               AC
               and
               the
               arch
               AL
               ;
               I
               answer
               as
               before
               ,
               inclination
               is
               not
               in
               the
               angular
               point
               abstractly
               
               considered
               without
               regard
               to
               the
               sides
               passing
               out
               of
               it
               ,
               but
               inclination
               is
               the
               relative
               situation
               which
               the
               concurring
               sides
               have
               at
               the
               angular
               point
               :
               at
               least
               that
               is
               their
               inclination
               there
               :
               for
               a
               point
               to
               a
               line
               can
               have
               no
               inclination
               :
               it
               may
               have
               distance
               from
               the
               line
               ,
               but
               cannot
               be
               inclined
               unto
               it
               ,
               because
               of
               its
               indivisibility
               .
               And
               having
               already
               shewn
               the
               inclination
               of
               side
               to
               side
               to
               be
               of
               the
               essence
               ,
               notion
               and
               nature
               of
               an
               angle
               ;
               a
               little
               may
               be
               reply
               enough
               to
               all
               those
               hypersceptical
               objections
               which
               are
               sounded
               upon
               the
               imagination
               of
               an
               angle
               in
               a
               right-line
               ,
               or
               any
               inclination
               ,
               or
               angularity
               imagined
               between
               a
               right
               line
               and
               a
               point
               ,
               especially
               the
               point
               being
               in
               the
               right-line
               .
               And
               equality
               ,
               or
               inequality
               of
               angles
               is
               not
               ,
               nor
               can
               be
               judged
               of
               by
               the
               so
               abstractly
               considered
               angular
               point
               ;
               in
               which
               a
               thousand
               several
               sides
               of
               several
               and
               unequal
               angles
               may
               meet
               indifferently
               ;
               but
               the
               judgment
               of
               the
               magnitude
               and
               equality
               and
               inequality
               of
               angles
               is
               from
               the
               sides
               ,
               and
               the
               order
               of
               the
               divarications
               ,
               in
               which
               they
               passe
               ,
               especially
               first
               of
               all
               ,
               from
               the
               angular
               point
               .
               Besides
               how
               strangely
               is
               it
               taken
               for
               granted
               ,
               without
               proving
               ,
               th●●
               the
               right-line
               tangent
               AB
               and
               the
               arch
               
               AL
               are
               equally
               in
               the
               angular
               point
               A
               inclined
               unto
               the
               right-line
               secant
               
               AC
               ?
               If
               that
               could
               be
               once
               proved
               ,
               the
               concern
               of
               it
               would
               turn
               the
               scales
               of
               the
               controversy
               :
               but
               demonstration
               
               is
               so
               clear
               to
               the
               contrary
               ,
               that
               as
               without
               proof
               it
               is
               not
               fit
               to
               be
               admitted
               ,
               so
               for
               the
               proof
               of
               it
               ,
               I
               know
               nothing
               can
               be
               produced
               besides
               an
               utter
               despair
               of
               ever
               making
               it
               out
               .
            
             
               For
               if
               the
               congruency
               of
               the
               sides
               terminatively
               in
               the
               angular
               point
               ,
               were
               sufficient
               to
               constitute
               equality
               in
               angles
               ,
               it
               appears
               not
               how
               any
               angles
               meeting
               in
               the
               same
               or
               different
               angular
               points
               could
               be
               unequal
               :
               every
               point
               by
               reason
               of
               its
               indivisibility
               being
               incapable
               of
               inequality
               ,
               as
               well
               as
               inclination
               .
            
             
               And
               if
               all
               such
               angles
               so
               constituted
               by
               the
               falling
               within
               ,
               or
               without
               of
               the
               sides
               ,
               shall
               be
               doubted
               and
               questioned
               whether
               they
               be
               true
               and
               quantitative
               angles
               ,
               and
               whether
               the
               addition
               ,
               or
               subduction
               of
               them
               be
               able
               to
               diversify
               other
               angles
               and
               their
               quantities
               ;
               all
               the
               pains
               of
               the
               Geometricians
               to
               prove
               the
               intracadency
               and
               extracadency
               of
               the
               angular
               sides
               from
               the
               same
               angular
               point
               ,
               were
               vain
               and
               to
               no
               purpose
               ;
               the
               angles
               remaining
               altogether
               the
               same
               and
               equal
               ,
               whether
               such
               angles
               of
               contact
               be
               added
               to
               them
               ,
               or
               taken
               from
               them
               .
               But
               yet
               though
               the
               true
               ,
               〈◊〉
               and
               genuine
               nature
               of
               an
               angle
               consist
               in
               the
               mutual
               habitude
               and
               inclination
               of
               the
               containing
               and
               concurring
               sides
               ,
               it
               is
               not
               ever
               necessary
               to
               consider
               it
               with
               such
               a
               largenes
               in
               Geometry
               .
               The
               inclination
               which
               the
               sides
               bear
               mutually
               each
               
               to
               other
               at
               the
               angular
               point
               is
               out
               of
               doubt
               that
               which
               is
               of
               most
               constant
               necessity
               ,
               highest
               concern
               and
               usefulnes
               in
               all
               angles
               to
               be
               observed
               .
               It
               is
               true
               in
               isoclitical
               angles
               ,
               the
               inclination
               of
               the
               two
               containing
               sides
               being
               every
               where
               the
               same
               and
               equal
               ,
               it
               is
               indifferently
               by
               Geometricians
               taken
               by
               a
               circle
               whose
               center
               is
               in
               the
               angular
               point
               ,
               of
               what
               diameter
               soever
               thereunto
               applyable
               ,
               and
               at
               what
               same
               distance
               soever
               from
               the
               angle
               ,
               or
               at
               what
               same
               longitude
               soever
               from
               thence
               in
               the
               sides
               the
               points
               be
               ,
               at
               which
               their
               inclination
               is
               observed
               by
               intercepted
               arches
               :
               and
               the
               same
               two
               points
               ,
               terminating
               the
               intercepted
               arches
               at
               which
               their
               mutual
               inclination
               is
               observed
               ,
               constantly
               offer
               themselves
               together
               ,
               whether
               you
               take
               points
               at
               equal
               distance
               from
               the
               angular
               point
               ,
               or
               intercepting
               in
               the
               sides
               equal
               longitudes
               between
               them
               and
               the
               same
               angular
               point
               But
               in
               anisoclitical
               angles
               ;
               the
               inclination
               of
               the
               anisoclitical
               sides
               varying
               still
               in
               the
               continuity
               of
               their
               production
               ,
               if
               we
               use
               the
               former
               method
               of
               measuring
               the
               inclination
               of
               the
               sides
               by
               intercepted
               arches
               of
               a
               circle
               drawn
               upon
               the
               angular
               point
               as
               center
               ;
               in
               them
               therefore
               Geometricians
               concern
               themselves
               little
               further
               then
               to
               observe
               the
               mutual
               inclination
               of
               the
               sides
               at
               the
               very
               point
               of
               their
               angle
               :
               and
               not
               at
               any
               other
               points
               in
               the
               anisoclitical
               sides
               ,
               save
               only
               the
               point
               of
               their
               
               concurrence
               ;
               because
               of
               the
               constant
               variation
               of
               their
               inclination
               ,
               both
               in
               respect
               of
               such
               ,
               and
               every
               other
               method
               and
               way
               of
               measuring
               ,
               according
               to
               the
               continuity
               of
               their
               production
               .
               And
               by
               arches
               of
               circle
               drawn
               upon
               the
               angular
               point
               as
               center
               ,
               it
               is
               impossible
               to
               measure
               the
               inclination
               of
               the
               anisoclitical
               lines
               at
               the
               point
               of
               their
               concurrence
               ;
               the
               only
               way
               therefore
               which
               remains
               unto
               Geometricians
               to
               measure
               such
               anisoclitical
               angles
               ,
               
                 i.
                 e
              
               ,
               the
               inclination
               of
               their
               sides
               at
               the
               point
               of
               their
               concurrence
               ,
               is
               by
               observing
               the
               lines
               in
               what
               order
               they
               depart
               from
               the
               angular
               point
               :
               Viz.
               which
               line
               falls
               within
               ,
               which
               without
               ,
               and
               which
               is
               coincident
               with
               that
               unto
               which
               it
               is
               compared
               .
               So
               in
               fig.
               12.
               if
               AHF
               and
               ADK
               be
               equal
               circles
               touching
               in
               the
               point
               A
               :
               and
               G
               the
               center
               ,
               and
               
               AGK
               the
               diameter
               of
               the
               circle
               ADK
               :
               and
               AEL
               the
               arch
               of
               a
               greater
               circle
               touching
               both
               the
               former
               circles
               in
               the
               same
               point
               A
               and
               with
               its
               concave
               side
               at
               A
               respecting
               the
               center
               G
               ,
               and
               with
               its
               convexe
               side
               at
               A
               respecting
               the
               circle
               HAF
               :
               also
               if
               AB
               be
               a
               right-line
               tangent
               touching
               all
               the
               three
               former
               circles
               in
               the
               point
               A
               ,
               and
               the
               right-line
               ADEC
               cutt
               the
               circle
               
               ADK
               in
               the
               point
               ●
               ,
               and
               the
               circle
               AEL
               in
               the
               point
               E
               :
               here
               the
               Geometrician
               demonstrates
               the
               angle
               K
               AB
               to
               be
               a
               right
               right
               lined
               angle
               ,
               and
               that
               the
               arch
               
               AFH
               passeth
               out
               of
               the
               angular
               point
               A
               without
               and
               
               beyond
               the
               right-line
               tangent
               AB
               ,
               and
               without
               the
               right
               right-lined
               angle
               K
               AB
               :
               and
               that
               the
               arch
               AEL
               passeth
               from
               the
               angular
               point
               A
               within
               ,
               or
               on
               this
               side
               of
               the
               right-line
               tangent
               AB
               ,
               and
               within
               the
               right
               right-lined
               angle
               KAB
               ,
               but
               without
               the
               arch
               ADK
               ,
               and
               without
               the
               angle
               of
               the
               semi
               circle
               KAD
               :
               and
               that
               the
               right-line
               ADEC
               passeth
               from
               the
               angular
               point
               A
               within
               the
               arch
               
               ADK
               ,
               and
               within
               the
               angle
               of
               the
               semi
               circle
               K
               
               AD.
               And
               this
               is
               all
               which
               is
               intended
               in
               Geometry
               when
               recto-convexe
               and
               such
               like
               angles
               of
               contact
               are
               said
               to
               be
               less
               then
               the
               least
               right-lined
               angle
               :
               not
               that
               there
               is
               any
               proportion
               between
               any
               angles
               of
               contact
               and
               right-lined
               angles
               ;
               for
               there
               is
               none
               :
               but
               that
               the
               sides
               of
               any
               such
               angle
               of
               contact
               are
               coaptable
               ,
               both
               of
               them
               within
               any
               the
               least
               right-lined
               angle
               .
               In
               like
               manner
               when
               one
               angle
               of
               contact
               is
               said
               to
               be
               greater
               or
               lesser
               then
               another
               ;
               it
               is
               not
               the
               intendment
               of
               the
               Geometrician
               to
               assert
               any
               proportionablenes
               between
               them
               ,
               but
               only
               to
               set
               down
               in
               what
               order
               the
               inclined
               lines
               pass
               ,
               each
               in
               respect
               of
               the
               rest
               ,
               from
               the
               angular
               point
               ,
               or
               point
               of
               concurrence
               :
               and
               how
               each
               is
               conseque●●ly
               more
               ,
               or
               less
               inclined
               in
               that
               respect
               ,
               to
               any
               one
               of
               the
               rest
               ,
               though
               by
               the
               whole
               kind
               and
               without
               proportion
               ,
               and
               without
               a
               common
               way
               of
               measuring
               their
               quantities
               in
               some
               indefinite
               quantity
               ,
               according
               to
               
               the
               kind
               of
               the
               measuring
               ,
               indefinite
               quantity
               ▪
               the
               semblablenes
               of
               saying
               ,
               one
               angle
               of
               contact
               is
               greater
               ,
               or
               less
               then
               another
               being
               only
               in
               this
               ,
               that
               when
               one
               side
               is
               coapted
               ,
               the
               other
               falls
               within
               ,
               or
               without
               ;
               or
               when
               neither
               sides
               can
               possibly
               be
               coapted
               ,
               both
               fall
               within
               ,
               or
               both
               fall
               without
               ,
               so
               as
               the
               one
               is
               really
               greater
               ,
               or
               less
               then
               the
               other
               ,
               not
               in
               proportion
               ,
               but
               by
               the
               whole
               kind
               and
               heterogeneally
               ,
               as
               a
               two
               foot
               right-line
               is
               lesser
               then
               a
               four
               foot
               square
               ,
               and
               a
               two
               foot
               right-line
               cannot
               be
               contained
               in
               an
               inch
               square
               ,
               though
               in
               an
               inch
               square
               may
               be
               drawn
               a
               line
               ten
               thousand
               times
               longer
               ,
               only
               out
               of
               rectitude
               .
               And
               to
               prove
               such
               an
               heterogeneal
               inequality
               ,
               whether
               of
               the
               greater
               ,
               or
               of
               the
               less
               between
               angles
               ,
               is
               that
               which
               is
               frequently
               sufficient
               for
               many
               Geometrical
               purposes
               .
            
             
               Only
               two
               things
               now
               remain
               of
               all
               your
               objections
               and
               scrupulous
               quaeries
               .
               Viz.
               
            
             
               First
               ,
               why
               in
               some
               quantities
               ,
               between
               the
               homogeneals
               ,
               to
               a
               given
               quantity
               you
               cannot
               give
               another
               in
               what
               proportion
               you
               please
               greater
               or
               less
               ,
               The
               answer
               is
               clear
               out
               of
               the
               foregoing
               discourse
               :
               because
               some
               quantities
               have
               but
               a
               limited
               extensibility
               :
               and
               other
               quantities
               have
               but
               a
               limited
               divisibility
               .
               All
               proportions
               at
               pleasure
               with
               respect
               to
               any
               given
               quantity
               ,
               are
               only
               assigneable
               in
               those
               homogeneals
               whose
               quantityes
               have
               both
               an
               unlimited
               
               extensibility
               and
               an
               unlimited
               divisibility
               ;
               either
               of
               which
               failing
               ,
               though
               they
               cannot
               be
               ,
               save
               in
               a
               limited
               proportion
               one
               to
               another
               ,
               and
               to
               every
               given
               homogeneal
               ,
               yet
               they
               cannot
               be
               in
               any
               proportion
               ,
               at
               pleasure
               ,
               to
               any
               given
               homogeneal
               .
            
             
               And
               2ly
               .
               for
               your
               other
               Quaery
               ;
               whether
               this
               controversy
               about
               the
               recto-convexe
               angle
               of
               contact
               concern
               abstract
               Mathematicks
               ,
               whose
               glory
               used
               to
               be
               that
               it
               was
               devoid
               of
               Controversy
               ,
               locking
               forth
               and
               keeping
               out
               all
               unpeaceful
               bickerings
               with
               the
               diamond-key
               of
               doubtless
               demonstration
               ;
               or
               whether
               it
               concern
               concrete
               Mathematicks
               ,
               which
               for
               its
               concretion
               unto
               matter
               cannot
               so
               well
               free
               its
               self
               from
               the
               intanglements
               of
               doubts
               and
               disputes
               ?
               and
               if
               it
               concern
               the
               more
               abstract
               part
               of
               the
               Mathematicks
               ;
               what
               was
               the
               first
               slip
               that
               drew
               so
               unworthy
               a
               disparagement
               upon
               that
               most
               noble
               piece
               of
               learning
               ?
               In
               the
               preceding
               discourse
               you
               may
               observe
               ,
               I
               have
               been
               free
               and
               clear
               in
               my
               judgement
               concerning
               this
               ;
               that
               it
               is
               a
               controversy
               of
               the
               purer
               and
               more
               noble
               part
               of
               the
               Mathematicks
               :
               and
               therefore
               of
               the
               higher
               concern
               ,
               that
               it
               be
               brought
               to
               a
               fair
               decision
               and
               irrefragable
               diremption
               :
               and
               that
               the
               first
               spark
               from
               whence
               all
               this
               heat
               afterward
               arose
               ,
               was
               at
               first
               struck
               out
               of
               the
               dubiousnes
               and
               aequivocation
               of
               the
               word
               homogeneity
               ,
               mentioned
               in
               certain
               Mathematical
               
               definitions
               and
               Postulates
               ;
               but
               of
               it self
               no
               where
               expressely
               defined
               in
               the
               Mathematicks
               ;
               except
               we
               shall
               take
               one
               of
               the
               Pustulates
               ,
               for
               its
               definition
               cast
               into
               the
               form
               of
               a
               Postulate
               ;
               or
               as
               cryptically
               implying
               that
               definition
               and
               the
               postulate
               to
               be
               thereupon
               immediately
               formed
               :
               especially
               when
               they
               compared
               homogeneity
               in
               a
               mistaken
               sense
               ,
               with
               some
               mathematical
               conclusions
               ,
               which
               they
               had
               observed
               to
               be
               fairly
               demonstrated
               .
            
             
               Pardon
               this
               my
               zeal
               on
               the
               behalf
               of
               the
               old
               learned
               Romane
               professour
               .
               And
               if
               these
               papers
               seem
               long
               ,
               remember
               yours
               were
               not
               short
               ;
               and
               the
               question
               hath
               long
               troubled
               the
               world
               :
               and
               in
               matters
               that
               are
               new
               ,
               and
               scarce
               yet
               well
               understood
               ,
               to
               inculcate
               once
               and
               again
               the
               same
               things
               ,
               is
               not
               only
               justify'd
               as
               allowable
               ,
               but
               judged
               expedient
               and
               in
               such
               cases
               necessary
               .
               If
               you
               please
               ,
               let
               the
               whole
               passe
               for
               a
               fuse
               Comment
               upon
               this
               ;
               that
               in
               recto-convexe
               angles
               of
               contact
               is
               the
               inclination
               required
               for
               an
               angle
               by
               the
               definition
               of
               an
               angle
               ;
               and
               therefore
               they
               are
               angles
               and
               their
               sides
               not
               coincident
               :
               and
               they
               are
               angular
               parts
               of
               acknowledged
               angles
               separable
               from
               the
               remaining
               angular
               parts
               ;
               and
               there
               is
               an
               infinite
               divisibility
               in
               them
               ,
               so
               as
               they
               must
               be
               quantitative
               :
               and
               yet
               it
               is
               demonstrated
               that
               they
               are
               improportionable
               to
               all
               right-lined
               angles
               ;
               therefore
               they
               are
               heterogeneal
               :
               which
               
               inference
               is
               the
               more
               necessary
               ,
               when
               upon
               examination
               we
               find
               nothing
               so
               convenably
               to
               set
               forth
               unto
               us
               the
               nature
               of
               mathematical
               homogeneity
               ,
               as
               the
               homometricalnes
               and
               the
               thence
               arising
               proportionablenes
               of
               the
               magnitudes
               ,
               and
               of
               mathematical
               heterogeneity
               as
               their
               heterometricalnes
               and
               improportionablenes
               .