Elements of philosophy the first section, concerning body / written in Latine by Thomas Hobbes of Malmesbury ; and now translated into English ; to which are added Six lessons to the professors of mathematicks of the Institution of Sr. Henry Savile, in the University of Oxford.
         De corpore. English
         Hobbes, Thomas, 1588-1679.
      
       
         
           1656
        
      
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             Elements of philosophy the first section, concerning body / written in Latine by Thomas Hobbes of Malmesbury ; and now translated into English ; to which are added Six lessons to the professors of mathematicks of the Institution of Sr. Henry Savile, in the University of Oxford.
             De corpore. English
             Hobbes, Thomas, 1588-1679.
          
           [14], 394 p., 13 leaves of plates (l folded)
           
             Printed by R. & W. Leybourn for Andrew Crooke ...,
             London :
             1656.
          
           
             A translation, and to some extent adaptation, of "Elementorum philosophiae. Sectio prima. De corpore ..." (see "The translator to the reader", prelim. p. [3]).
             "Six lessons ..." is a reply to John Wallis's and Seth Ward's attacks on De corpore ...
             "Six lessons ..." has been filmed separately and may be found as Wing H2260 at reel 1360:1.
             Reproduction of original in the University of Illinois (Urbana-Champaign Campus). Library.
          
        
      
    
     
       
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         eng
      
       
         
           Philosophy -- Early works to 1800.
        
      
    
     
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           ELEMENTS
           OF
           PHILOSOPHY
           ,
           THE
           
             FIRST
             SECTION
          
           ,
           CONCERNING
           BODY
           .
           Written
           in
           Latine
           by
           THOMAS
           HOBBES
           OF
           MALMESBURY
           .
           And
           now
           translated
           into
           ENGLISH
           .
           To
           which
           are
           added
           Six
           Lessons
           to
           the
           Professors
           of
           Mathematicks
           of
           the
           Institution
           of
           
             S
             
             r.
             HENRY
             SAVILE
          
           ,
           in
           the
           University
           of
           OXFORD
           .
        
         
           LONDON
           ,
           Printed
           by
           
             R.
             &
             W.
             Leybourn
          
           ,
           for
           
             Andrew
             Crocke
          
           ,
           at
           the
           Green
           Dragon
           in
           Pauls
           Church-yard
           ,
           1656.
           
        
      
       
         
         
         
           THE
           TRANSLATOR
           To
           the
           Reader
           .
        
         
           IF
           when
           I
           had
           finished
           my
           Translation
           of
           this
           first
           Section
           of
           the
           Elements
           of
           Philosophy
           ,
           I
           had
           presently
           committed
           the
           same
           to
           the
           Press
           ,
           it
           might
           have
           come
           to
           your
           hands
           sooner
           then
           now
           it
           doth
           .
           But
           as
           I
           undertook
           it
           with
           much
           diffidence
           of
           my
           own
           ability
           to
           perform
           it
           well
           ;
           so
           I
           thought
           fit
           before
           I
           published
           it
           ,
           to
           pray
           Mr.
           Hobbes
           to
           view
           ,
           correct
           and
           order
           it
           according
           to
           his
           own
           minde
           and
           pleasure
           .
           Wherefore
           ,
           though
           you
           find
           some
           places
           enlarged
           ,
           others
           altered
           ,
           and
           two
           Chapters
           (
           the
           18th
           and
           20th
           )
           almost
           wholly
           changed
           ,
           you
           may
           nevertheless
           remain
           assured
           ,
           that
           as
           now
           I
           present
           it
           to
           you
           ,
           
           it
           doth
           not
           at
           all
           vary
           from
           the
           Authours
           own
           sense
           and
           meaning
           .
           As
           for
           the
           six
           Lessons
           to
           the
           Savilian
           Professors
           at
           Oxford
           ,
           they
           are
           not
           of
           my
           translation
           ,
           but
           were
           written
           ,
           as
           here
           you
           have
           them
           in
           English
           ,
           by
           Mr.
           Hobbes
           himself
           ;
           and
           are
           joyned
           to
           this
           Book
           ,
           because
           they
           are
           chiefly
           in
           defence
           of
           the
           same
           .
        
      
       
         
         
           The
           Authors
           Epistle
           Dedicatory
           ,
           TO
           THE
           
             RIGHT
             HONORABLE
          
           ,
           My
           most
           Honored
           LORD
           ,
           WILLIAM
           Earl
           of
           Devonshire
           .
        
         
           THis
           first
           Section
           of
           the
           Elements
           of
           Philosophy
           ,
           the
           Monument
           of
           my
           Service
           ,
           &
           your
           Lordships
           bounty
           ,
           though
           (
           after
           the
           third
           Section
           published
           )
           long
           deferred
           ,
           yet
           at
           last
           finished
           ,
           I
           now
           present
           (
           my
           most
           Excellent
           Lord
           )
           and
           dedicate
           to
           your
           Lordship
           .
           A
           little
           Book
           ▪
           but
           full
           ;
           and
           great
           enough
           ,
           if
           men
           count
           well
           for
           great
           ;
           and
           to
           an
           attentive
           Reader
           versed
           in
           the
           Demonstrations
           of
           Mathematicians
           that
           is
           ,
           to
           your
           Lordship
           〈◊〉
           &
           easy
           to
           understand
           ;
           and
           almost
           new
           througho●●
           ▪
           without
           any
           offensive
           Novelty
           .
           I
           know
           that
           that
           part
           of
           Philosophy
           wherein
           are
           considered
           Lines
           and
           Figures
           ,
           has
           been
           delivered
           to
           us
           notably
           improved
           by
           the
           Ancients
           ;
           
           and
           withall
           a
           most
           perfect
           pattern
           of
           the
           Logique
           by
           which
           they
           were
           enabled
           to
           finde
           out
           and
           demonstrate
           such
           excellent
           Theoremes
           as
           they
           have
           done
           .
           I
           know
           also
           that
           the
           Hypothesis
           of
           the
           Earths
           Diurnal
           Motion
           was
           the
           invention
           of
           the
           Ancients
           ;
           but
           that
           both
           it
           ,
           and
           Astronomy
           (
           that
           is
           ,
           Coelestial
           Physiques
           )
           springing
           up
           together
           with
           it
           ,
           were
           by
           succeeding
           Philosophers
           strangled
           with
           the
           snares
           of
           Words
           .
           And
           therefore
           the
           beginning
           of
           Astronomy
           (
           except
           Observations
           )
           I
           think
           is
           not
           to
           be
           derived
           from
           farther
           time
           then
           from
           
             Nicolaus
             Copernicus
          
           ;
           who
           in
           the
           Age
           next
           preceding
           the
           present
           ,
           revived
           the
           opinion
           of
           
             Pythagoras
             ,
             Aristarchus
             &
             Philolaus
          
           .
           After
           him
           ,
           the
           Doctrine
           of
           the
           Motion
           of
           the
           Earth
           being
           now
           received
           ,
           &
           a
           difficult
           Question
           thereupon
           arising
           concerning
           the
           Descent
           of
           Heavy
           Bodies
           ,
           Galilaeus
           in
           our
           time
           striving
           with
           that
           difficulty
           ,
           was
           the
           first
           that
           opened
           to
           us
           the
           〈◊〉
           Natural
           Philosophy
           Universal
           ,
           which
           is
           the
           knowledge
           of
           the
           Nature
           of
           Motion
           .
           So
           that
           neither
           can
           the
           Age
           of
           Natural
           Philosophy
           be
           reckoned
           higher
           then
           to
           him
           .
           Lastly
           ,
           the
           Science
           of
           
             Mans
             Body
          
           ,
           the
           most
           profitable
           part
           of
           Natural
           
           Science
           ,
           was
           first
           discovered
           with
           admirable
           sagacity
           by
           our
           Countryman
           Doctor
           Harvey
           ,
           principal
           Physician
           to
           King
           James
           and
           King
           Charles
           ,
           in
           his
           Books
           of
           the
           
             Motion
             of
             the
             Blood
          
           and
           of
           the
           
             Generation
             of
             Living
             Creatures
          
           ;
           who
           is
           the
           onely
           man
           I
           know
           ,
           that
           conquering
           envy
           ,
           hath
           established
           a
           new
           Doctrine
           in
           his
           life
           time
           .
           Before
           these
           ,
           there
           was
           nothing
           certain
           in
           Natural
           Philosophy
           but
           every
           mans
           Experiments
           to
           himself
           ,
           and
           the
           Natural
           Histories
           ,
           if
           they
           may
           be
           called
           certain
           ,
           that
           are
           no
           certainer
           then
           Civil
           Histories
           ▪
           But
           since
           these
           ,
           Astronomy
           &
           Natural
           Philosophy
           in
           general
           have
           for
           so
           little
           time
           been
           extraordinarily
           advanced
           by
           
             Joannes
             Keplerus
             ,
             Petrus
             Gassendus
             ,
             &
             Marinus
             Mersennus
          
           ;
           &
           the
           Science
           of
           Humane
           Bodies
           in
           special
           by
           the
           wit
           &
           industry
           of
           Physicians
           (
           the
           onely
           true
           Natural
           Philosophers
           )
           especially
           of
           our
           most
           Learned
           Men
           of
           the
           Colledge
           of
           Physicians
           in
           London
           .
           Natural
           Philosophy
           is
           therefore
           but
           young
           ;
           but
           Civil
           Philosophy
           yet
           much
           younger
           ,
           as
           being
           no
           older
           (
           I
           say
           it
           provoked
           ,
           &
           that
           my
           Detractors
           may
           know
           how
           little
           they
           have
           wrought
           upon
           me
           )
           then
           my
           own
           Book
           
             de
             Cive
          
           .
           But
           what
           ?
           Were
           there
           no
           Philosophers
           Natural
           nor
           Civil
           among
           the
           
           ancient
           Greeks
           ?
           There
           were
           men
           so
           called
           ;
           Witness
           Lucian
           ,
           by
           whom
           they
           are
           derided
           ;
           Witness
           divers
           Cities
           ,
           from
           which
           they
           have
           been
           often
           by
           publique
           Edicts
           banished
           .
           But
           it
           follows
           not
           that
           there
           was
           Philosophy
           .
           There
           walked
           in
           old
           Greece
           a
           certain
           Phantasme
           ,
           for
           superficial
           gravity
           (
           though
           full
           within
           of
           fraud
           &
           filth
           )
           a
           little
           like
           Philosophy
           ;
           which
           unwary
           men
           thinking
           to
           be
           it
           ,
           adhered
           to
           the
           Professors
           of
           it
           ,
           some
           to
           one
           ,
           some
           to
           another
           (
           though
           they
           disagreed
           among
           themselves
           )
           and
           with
           great
           Salary
           put
           their
           children
           to
           them
           to
           be
           taught
           ,
           in
           stead
           of
           Wisdome
           ,
           nothing
           but
           to
           dispute
           ;
           &
           neglecting
           the
           Laws
           ,
           to
           determine
           every
           Question
           ▪
           according
           to
           their
           own
           fancies
           .
           The
           first
           Doctors
           of
           the
           Church
           next
           the
           Apostles
           ,
           born
           in
           those
           times
           ,
           whilest
           they
           endeavored
           to
           defend
           the
           Christian
           Faith
           against
           the
           Gentiles
           by
           Natural
           Reason
           ,
           began
           also
           to
           make
           use
           of
           Philosophy
           ,
           &
           with
           the
           Decrees
           of
           Holy
           Scripture
           to
           mingle
           the
           Sentences
           of
           Heathen
           Philosophers
           ;
           &
           first
           some
           harmless
           ones
           of
           Plato
           ;
           but
           afterwards
           also
           many
           foolish
           &
           false
           ones
           out
           of
           the
           Physicks
           &
           Metaphysicks
           of
           Aristotle
           ;
           &
           bringing
           in
           the
           Enemies
           betrayed
           unto
           them
           the
           Cittadel
           of
           Christianity
           From
           that
           time
           ,
           in
           stead
           of
           the
           Worship
           of
           God
           ,
           there
           entred
           a
           thing
           called
           School-Divinity
           ,
           
           walking
           on
           one
           foot
           firmly
           ,
           which
           is
           the
           Holy
           Scripture
           ,
           but
           halted
           on
           the
           other
           rotten
           foot
           ,
           which
           the
           Apostle
           Paul
           called
           Vain
           ,
           &
           might
           have
           called
           
             Pernicious
             Philosophy
          
           ;
           for
           it
           hath
           raised
           an
           infinite
           number
           of
           Controversies
           in
           the
           Christian
           World
           concerning
           Religion
           ,
           &
           from
           those
           Controversies
           Wars
           .
           It
           is
           like
           that
           Empusa
           in
           the
           Athenian
           Comick
           Poet
           ,
           which
           was
           taken
           in
           Athens
           for
           a
           Ghost
           that
           changed
           shapes
           ,
           having
           one
           brazenleg
           ,
           but
           the
           other
           was
           the
           leg
           of
           an
           Ass
           ,
           &
           was
           sent
           (
           as
           was
           believed
           )
           by
           Hecate
           ,
           as
           a
           signe
           of
           some
           approaching
           evil
           fortune
           .
           Against
           this
           Empusa
           I
           think
           there
           cannot
           be
           invented
           a
           better
           Exorcisme
           ,
           then
           to
           distinguish
           between
           the
           Rules
           of
           Religion
           ,
           that
           is
           ,
           the
           Rules
           of
           Honoring
           God
           ,
           which
           we
           have
           from
           the
           Laws
           ,
           and
           the
           Rules
           of
           Philosophy
           ,
           that
           is
           ,
           the
           Opinions
           of
           private
           men
           ;
           &
           to
           yeild
           what
           is
           due
           to
           Religion
           to
           the
           Holy
           Scripture
           ,
           and
           what
           is
           due
           to
           Philosophy
           to
           Natural
           Reason
           .
           And
           this
           I
           shall
           do
           ,
           if
           I
           but
           handle
           the
           Elements
           of
           Philosophy
           truly
           &
           clearly
           ▪
           as
           I
           endevour
           to
           do
           .
           Therefore
           having
           in
           the
           ●d
           Section
           w
           ch
           I
           have
           published
           &
           dedicated
           to
           your
           Lordship
           long
           since
           ▪
           reduced
           all
           Power
           Ecclesiastical
           and
           Civil
           by
           strong
           Argu●en●●
           of
           Reason
           ,
           without
           repugnance
           to
           Gods
           Word
           ,
           to
           one
           and
           the
           same
           Soveraign
           Authority
           ;
           I
           intend
           now
           ,
           
           by
           putting
           into
           a
           clear
           Method
           the
           true
           Foundations
           of
           Natural
           Philosophy
           ,
           to
           fright
           and
           drive
           away
           this
           Metaphysical
           Empusa
           ;
           not
           by
           skirmish
           ,
           but
           by
           letting
           in
           the
           light
           upon
           her
           .
           For
           I
           am
           confident
           (
           if
           any
           confidence
           of
           a
           Writing
           can
           proceed
           from
           the
           Writers
           fear
           ,
           circumspection
           &
           diffidence
           )
           that
           in
           the
           three
           former
           parts
           of
           this
           Book
           ,
           all
           that
           I
           have
           said
           is
           sufficiently
           demonstrated
           from
           Definitions
           ;
           &
           all
           in
           the
           fourth
           part
           ,
           from
           Suppositions
           not
           absurd
           .
           But
           if
           there
           appear
           to
           your
           Lordship
           any
           thing
           less
           fully
           demonstrated
           then
           to
           satisfie
           every
           Reader
           ,
           the
           cause
           was
           this
           ,
           that
           I
           professed
           to
           write
           not
           all
           to
           all
           ,
           but
           some
           things
           to
           Geometricians
           onely
           .
           But
           that
           your
           Lordship
           will
           be
           satisfied
           J
           cannot
           doubt
           .
        
         
           There
           remains
           the
           second
           Section
           ,
           which
           is
           concerning
           Man.
           That
           part
           thereof
           where
           J
           handle
           the
           Optiques
           ,
           contayning
           six
           Chapters
           ,
           together
           with
           the
           Tables
           of
           the
           Figures
           belonging
           to
           them
           ,
           I
           have
           already
           written
           &
           engravenlying
           by
           me
           above
           these
           six
           years
           .
           The
           rest
           shall
           ,
           as
           soon
           as
           J
           can
           ,
           be
           added
           to
           it
           ;
           though
           by
           the
           contumelies
           &
           petty
           injuries
           of
           some
           unskilful
           men
           ,
           I
           know
           already
           by
           experience
           how
           much
           greater
           thanks
           will
           be
           due
           ,
           then
           payed
           me
           ,
           for
           telling
           Men
           the
           truth
           of
           what
           
           Men
           are
           .
           But
           the
           burthen
           I
           have
           taken
           on
           me
           I
           mean
           to
           carry
           through
           ;
           not
           striving
           to
           appease
           ,
           but
           rather
           to
           revenge
           my self
           of
           Envy
           ,
           by
           encreasing
           it
           .
           For
           it
           contents
           me
           that
           I
           have
           your
           Lordships
           favour
           ;
           which
           ,
           (
           being
           all
           you
           require
           )
           J
           acknowledge
           ;
           and
           for
           which
           ,
           with
           my
           prayers
           to
           Almighty
           God
           for
           your
           Lordships
           safety
           ,
           J
           shall
           (
           to
           my
           power
           )
           be
           always
           thankefull
           .
        
         
           
             London
             ,
             
               April
               23
               ,
               1655.
               
            
          
           
             YOUR
             LORDSHIPS
             most
             humble
             Servant
             Thomas
             Hobbes
             .
          
        
      
       
         
         
           The
           Authors
           Epistle
           
             To
             the
             Reader
          
           .
        
         
           THink
           not
           (
           courteous
           Reader
           )
           that
           the
           Philosophy
           the
           Elements
           whereof
           I
           am
           going
           to
           set
           in
           order
           ,
           is
           that
           which
           makes
           Philosophers
           Stones
           ,
           nor
           that
           which
           is
           found
           in
           the
           Metaphsique
           Codes
           .
           But
           that
           it
           is
           the
           Natural
           Reason
           of
           Man
           busily
           flying
           up
           and
           down
           among
           the
           Creatures
           ,
           &
           bringing
           back
           a
           true
           report
           of
           their
           Order
           ,
           Causes
           &
           Effects
           .
           Philosophy
           therefore
           ,
           the
           Childe
           of
           the
           World
           and
           your
           own
           Mind
           ,
           is
           within
           your self
           ;
           perhaps
           not
           fashioned
           yet
           ,
           but
           like
           the
           World
           its
           Father
           ,
           as
           it
           was
           in
           the
           beginning
           ,
           a
           thing
           confused
           .
           Do
           therefore
           as
           the
           Statuaries
           do
           ,
           who
           by
           hewing
           off
           that
           which
           is
           superfluous
           ,
           do
           not
           make
           but
           find
           the
           Image
           .
           Or
           imitate
           the
           Creation
           .
           If
           you
           will
           be
           a
           Philosopher
           in
           good
           earnest
           ,
           let
           your
           Reason
           move
           upon
           the
           Deep
           of
           your
           own
           Cogitations
           and
           Experience
           .
           Those
           things
           that
           lie
           in
           Confusion
           must
           be
           set
           asunder
           ,
           distinguished
           ,
           and
           every
           one
           stampt
           with
           its
           own
           name
           set
           in
           order
           ;
           that
           is
           to
           say
           ,
           your
           Method
           must
           resemble
           that
           of
           the
           Creation
           .
           The
           order
           of
           the
           Creation
           was
           ,
           
             Light
             ,
             Distinction
             of
             Day
             and
             Night
             ,
          
           the
           Firmament
           ,
           the
           
             Luminaries
             ,
             Sensible
             Creatures
             ,
             Man
          
           ;
           and
           after
           the
           Creation
           ,
           the
           Commandement
           .
           Therefore
           the
           order
           of
           Contemplation
           will
           be
           ,
           
             Reason
             ,
             Definition
             ,
             Space
             ,
             the
             Starres
             ,
             Sensible
             Quality
             ,
             Man
          
           ;
           and
           after
           Man
           is
           grown
           up
           ,
           
             Subjection
             to
             Command
          
           .
           In
           the
           first
           part
           of
           this
           Section
           which
           is
           entitled
           Logique
           ,
           I
           set
           up
           the
           light
           of
           Reason
           .
           In
           the
           Second
           (
           which
           hath
           for
           title
           the
           Grounds
           of
           Philosophy
           )
           I
           distinguish
           the
           most
           common
           Notions
           by
           accurate
           definition
           ,
           for
           the
           avoiding
           of
           confusion
           and
           obscurity
           .
           The
           third
           part
           concerns
           the
           Expansion
           of
           Space
           ,
           that
           is
           ,
           Geometry
           .
           The
           fourth
           contains
           the
           Motion
           of
           the
           Starres
           ,
           together
           with
           the
           doctrine
           of
           Sensible
           Qualities
           .
        
         
         
           In
           the
           second
           Section
           (
           if
           it
           please
           God
           )
           shall
           be
           handled
           Man.
           In
           the
           third
           Section
           the
           doctrine
           of
           Subjection
           is
           handled
           already
           .
           This
           is
           the
           Method
           I
           followed
           ;
           and
           if
           it
           like
           you
           you
           may
           use
           the
           same
           ;
           for
           I
           do
           but
           propound
           ,
           not
           commend
           to
           you
           any
           thing
           of
           mine
           .
           But
           whatsoever
           shall
           be
           the
           Method
           you
           will
           like
           ,
           I
           would
           very
           fain
           commend
           Philosophy
           to
           you
           ,
           that
           is
           to
           say
           ,
           the
           study
           of
           Wisdome
           ,
           for
           want
           of
           which
           we
           have
           all
           suffered
           much
           dammage
           lately
           .
           For
           even
           they
           that
           study
           Wealth
           ,
           do
           it
           out
           of
           love
           to
           Wisdome
           ;
           for
           their
           Treasures
           serve
           them
           but
           for
           a
           Looking-glass
           ,
           wherin
           to
           behold
           and
           contemplate
           their
           owne
           Wisdome
           .
           Nor
           do
           they
           that
           love
           to
           be
           employed
           in
           publike
           business
           ,
           aime
           at
           any
           thing
           but
           place
           wherein
           to
           shew
           their
           Wisdome
           .
           Neither
           do
           Voluptuous
           men
           neglect
           Philosophy
           ,
           but
           onely
           because
           they
           know
           not
           how
           great
           a
           pleasure
           it
           is
           to
           the
           Mind
           of
           Man
           to
           be
           ravished
           in
           the
           vigorous
           and
           perpetual
           embraces
           of
           the
           most
           beauteous
           World.
           Lastly
           ,
           though
           for
           nothing
           else
           ,
           yet
           (
           because
           the
           Mind
           of
           Man
           is
           no
           less
           impatient
           of
           Empty
           Time
           ,
           then
           Nature
           is
           of
           Empty
           Place
           )
           to
           the
           end
           you
           be
           not
           ▪
           forced
           for
           want
           of
           what
           to
           do
           ,
           to
           be
           troublesome
           to
           men
           that
           have
           business
           ,
           or
           take
           hurt
           by
           falling
           into
           idle
           Company
           ,
           but
           have
           somewhat
           of
           your
           own
           wherewith
           to
           fill
           up
           your
           time
           ,
           I
           recommend
           unto
           you
           the
           Study
           of
           Philosophy
           .
           Farewell
           .
        
         
           
             T.
             H.
             
          
        
      
       
         
         
           The
           Titles
           of
           the
           CHAPTERS
           .
        
         
           
             The
             first
             Part
             ,
             or
             Logique
             .
          
           
             CHAP.
             1
             Of
             Philosophy
             .
          
           
             CHAP.
             2
             Of
             Names
             .
          
           
             CHAP.
             3
             Of
             Proportion
             .
          
           
             CHAP.
             4
             Of
             Syllogisme
             .
          
           
             CHAP.
             5
             Of
             Erring
             ,
             Falsity
             and
             Captions
             .
          
           
             CHAP.
             6
             Of
             Method
             .
          
        
         
           
             The
             Second
             Part
             ,
             or
             The
             first
             Grounds
             of
             Philosophy
             .
          
           
             CHAP.
             7
             Of
             Place
             and
             Time.
             
          
           
             CHAP.
             8
             Of
             Body
             and
             Accident
             .
          
           
             CHAP.
             9
             Of
             Cause
             and
             Effect
             .
          
           
             CHAP.
             10
             Of
             Power
             and
             Act.
             
          
           
             CHAP.
             11
             Of
             Identity
             and
             Difference
             .
          
           
             CHAP.
             12
             Of
             Quantity
             .
          
           
             CHAP.
             13
             Of
             Analogisme
             ,
             or
             the
             Same
             Proportion
             .
          
           
             CHAP.
             14
             Of
             Straight
             and
             Crocked
             ,
             Angle
             and
             Figure
             .
          
        
         
           
             The
             third
             Part
             ,
             Of
             the
             Proportions
             of
             Motions
             and
             Magnitudes
             .
          
           
             CHAP.
             15
             Of
             the
             Nature
             ,
             Properties
             ,
             and
             divers
             considerations
             ,
             of
             Motion
             and
             Endeavour
             .
          
           
             CHAP.
             16
             Of
             Motion
             Accelerated
             and
             Uniform
             ,
             and
             of
             Motion
             by
             Concourse
             .
          
           
             CHAP.
             17
             Of
             Figures
             Deficient
             .
          
           
             CHAP.
             18
             Of
             the
             Equation
             of
             Straight
             Lines
             ,
             which
             the
             Crooked
             Lines
             of
             Parabolas
             ,
             and
             other
             Figures
             made
             in
             imitation
             of
             Parabolas
             .
          
           
             CHAP.
             19
             Of
             Angles
             of
             Incidence
             and
             Reflexion
             ,
             equal
             by
             supposition
             .
          
           
             CHAP.
             20
             Of
             the
             Dimension
             of
             a
             Circle
             ,
             and
             the
             Division
             of
             Arches
             or
             Angles
             .
          
           
             CHAP.
             21
             Of
             Circular
             Motion
             .
          
           
             CHAP.
             22
             Of
             other
             Variety
             of
             Motions
             .
          
           
             CHAP.
             23
             Of
             the
             Center
             of
             Equiponderation
             of
             Bodies
             pressing
             downwards
             in
             straight
             parallel
             lines
             .
          
           
             CHAP.
             24
             Of
             Refraction
             and
             Reflexion
             .
          
        
         
           
             The
             fourth
             Part
             ,
             of
             Physiques
             ,
             or
             the
             Phaenomena
             of
             Nature
             .
          
           
             CHAP.
             25
             Of
             Sense
             and
             Animall
             Motion
             .
          
           
             CHAP.
             26
             Of
             the
             World
             and
             of
             the
             Starres
             .
          
           
             CHAP.
             27
             Of
             Light
             ,
             Heat
             ,
             and
             of
             Colours
             .
          
           
             CHAP.
             28
             Of
             Cold
             ,
             Wind
             ,
             Hard
             ,
             Ice
             ,
             Restitution
             of
             Bodies
             bent
             ,
             Diaphanous
             ,
             Lightning
             and
             Thunder
             ,
             and
             of
             the
             Heads
             of
             Rivers
             .
          
           
             CHAP.
             29
             Of
             Sound
             ,
             Odour
             ,
             Savour
             ,
             and
             Touch.
             
          
           
             CHAP.
             30
             Of
             Gravity
             .
          
        
      
    
     
       
         
         
           COMPUTATION
           OR
           LOGIQUE
           .
        
         
           
             
               CHAP.
               I.
            
             Of
             Philosophy
             .
          
           
             
               
                 1
                 ▪
                 The
                 Introduction
                 .
              
               
                 2
                 The
                 Definition
                 of
                 Philosophy
                 explained
                 .
              
               
                 3
                 Ratiocination
                 of
                 the
                 Mind
                 .
              
               
                 4
                 Properties
                 what
                 they
                 are
                 .
              
               
                 5
                 How
                 Properties
                 are
                 known
                 by
                 Generation
                 ,
                 &
                 contrarily
                 .
              
               
                 6
                 The
                 Scope
                 of
                 Philosophy
                 .
              
               
                 7
                 The
                 Utility
                 of
                 it
                 .
              
               
                 8
                 The
                 Subject
                 .
              
               
                 9
                 The
                 Parts
                 of
                 it
                 .
              
               
                 10
                 The
                 Epilogue
                 .
              
            
          
           
             PHILOSOPHY
             seems
             to
             me
             to
             be
             amongst
             men
             now
             ,
             in
             the
             same
             manner
             as
             Corn
             and
             Wine
             are
             said
             to
             have
             been
             in
             the
             world
             in
             ancient
             time
             .
             For
             from
             the
             beginning
             there
             were
             Vines
             and
             Ears
             of
             Corn
             growing
             here
             and
             there
             in
             the
             fields
             ;
             but
             no
             care
             was
             taken
             for
             the
             planting
             and
             sowing
             of
             them
             .
             Men
             lived
             therefore
             upon
             Akorns
             ;
             or
             if
             any
             were
             so
             bold
             as
             to
             venture
             upon
             the
             eating
             of
             those
             unknown
             and
             doubtfull
             fruits
             ,
             they
             did
             it
             with
             danger
             of
             their
             health
             .
             In
             like
             manner
             ,
             every
             man
             brought
             Philosophy
             ,
             that
             is
             ,
             Naturall
             Reason
             ,
             into
             the
             world
             with
             him
             ;
             for
             all
             men
             can
             reason
             to
             some
             degree
             ,
             and
             concerning
             some
             things
             :
             but
             where
             there
             is
             need
             of
             a
             long
             series
             of
             Reasons
             ,
             there
             most
             men
             wander
             out
             of
             the
             way
             ,
             and
             fall
             into
             Error
             for
             want
             of
             Method
             ,
             as
             it
             were
             for
             want
             of
             sowing
             and
             planting
             ,
             that
             is
             ,
             of
             improving
             their
             Reason
             .
             And
             from
             hence
             it
             comes
             to
             passe
             ,
             that
             they
             who
             content
             themselves
             with
             daily
             experience
             ,
             which
             may
             be
             likened
             to
             feeding
             upon
             Akorns
             ,
             and
             either
             reject
             ,
             or
             not
             much
             regard
             Philosophy
             ,
             
             are
             commonly
             esteemed
             ,
             and
             are
             indeed
             ,
             men
             of
             sounder
             judgement
             ,
             then
             those
             ,
             who
             from
             opinions
             ,
             though
             not
             vulgar
             ,
             yet
             full
             of
             uncertainty
             ,
             and
             carelesly
             received
             ,
             do
             nothing
             but
             dispute
             and
             wrangle
             ,
             like
             men
             that
             are
             not
             well
             in
             their
             wits
             .
             I
             confesse
             indeed
             ,
             that
             that
             part
             of
             Philophy
             by
             which
             Magnitudes
             and
             Figures
             are
             computed
             ,
             is
             highly
             improved
             .
             But
             because
             I
             have
             not
             observed
             the
             like
             advancement
             in
             the
             other
             parts
             of
             it
             ,
             my
             purpose
             is
             ,
             as
             far
             forth
             as
             I
             am
             able
             ,
             to
             lay
             open
             the
             few
             and
             first
             Elements
             of
             Philosophy
             in
             generall
             ,
             as
             so
             many
             Seeds
             ,
             from
             which
             pure
             and
             true
             Philosophy
             may
             hereafter
             spring
             up
             by
             little
             and
             little
             .
          
           
             I
             am
             not
             ignorant
             how
             hard
             a
             thing
             it
             is
             to
             weed
             out
             of
             mens
             mindes
             such
             inveterate
             opinions
             as
             have
             taken
             root
             there
             ,
             and
             been
             cōfirmed
             in
             them
             by
             the
             authority
             of
             most
             eloquent
             Writers
             ;
             especially
             ,
             seeing
             true
             (
             that
             is
             accurate
             )
             Philosophy
             ,
             professedly
             rejects
             not
             only
             the
             paint
             and
             false
             colours
             of
             Language
             ,
             but
             even
             the
             very
             ornaments
             and
             graces
             of
             the
             same
             ;
             and
             the
             first
             Grounds
             of
             all
             Science
             ,
             are
             not
             only
             not
             beautifull
             ,
             but
             poore
             ,
             aride
             ,
             and
             in
             appearance
             deformed
             .
             Neverthelesse
             ,
             there
             being
             certainly
             some
             men
             ,
             though
             but
             few
             ,
             who
             are
             delighted
             with
             Truth
             and
             strength
             of
             Reason
             in
             all
             things
             ,
             I
             thought
             I
             might
             do
             well
             to
             take
             this
             pains
             for
             the
             sake
             even
             of
             those
             few
             .
             I
             proceed
             therefore
             to
             the
             matter
             ,
             and
             take
             my
             beginning
             from
             the
             very
             Definition
             of
             Philosophy
             ,
             which
             is
             this
             .
          
           
             2
             PHILOSOPHY
             
               is
               such
               knowledge
               of
               Effects
               or
               Appearances
               ,
               as
               we
               acquire
               by
               true
               Ratiocination
               from
               the
               knowledge
               we
               have
               first
               of
               their
               Causes
               or
               Generation
               :
               And
               again
               ,
               of
               such
               Causes
               or
               Generations
               as
               may
               be
               from
               knowing
               first
               their
               Effects
               .
            
          
           
             For
             the
             better
             understanding
             of
             which
             Definition
             ,
             we
             must
             consider
             ;
             first
             ,
             that
             although
             Sense
             and
             Memory
             of
             things
             ,
             which
             are
             common
             to
             Man
             and
             all
             living
             Creatures
             ,
             be
             Knowledge
             ,
             yet
             because
             they
             are
             given
             us
             immediately
             by
             Nature
             ,
             and
             not
             gotten
             by
             Ratiocination
             ,
             they
             are
             not
             Philosophy
             .
          
           
             Secondly
             ,
             Seeing
             Experience
             is
             nothing
             but
             Memory
             ;
             and
             Prudence
             ,
             or
             Prospect
             into
             the
             future
             time
             ,
             nothing
             but
             Expectation
             of
             such
             things
             as
             we
             have
             already
             had
             experience
             of
             ,
             Prudence
             also
             is
             not
             to
             be
             esteemed
             Philosophy
             .
          
           
             By
             RATIOCINATION
             ,
             I
             mean
             Computation
             .
             Now
             to
             compute
             ,
             
             is
             either
             to
             collect
             the
             sum
             of
             many
             things
             that
             are
             added
             together
             ,
             or
             to
             know
             what
             remains
             when
             one
             thing
             is
             taken
             out
             of
             another
             .
             Ratiocination
             therefore
             is
             the
             same
             with
             Addition
             and
             Substraction
             ;
             and
             if
             any
             man
             adde
             Multiplication
             and
             Division
             ,
             I
             will
             not
             be
             against
             it
             ,
             seeing
             Multiplication
             is
             nothing
             but
             Addition
             of
             equals
             one
             to
             another
             ,
             and
             Division
             nothing
             but
             a
             Substraction
             of
             equals
             one
             from
             another
             ,
             as
             often
             as
             is
             possible
             .
             So
             that
             all
             Ratiocination
             is
             comprehended
             in
             these
             two
             operations
             of
             the
             minde
             ,
             Addition
             and
             Substraction
             .
          
           
             3
             But
             how
             by
             the
             Ratiocination
             of
             our
             Minde
             ,
             we
             Adde
             and
             Substract
             in
             our
             silent
             thoughts
             ,
             without
             the
             use
             of
             words
             ,
             it
             will
             be
             necessary
             for
             me
             to
             make
             intelligible
             by
             an
             example
             or
             two
             .
             If
             therefore
             a
             man
             see
             something
             a
             far
             off
             and
             obscurely
             ,
             although
             no
             appellation
             had
             yet
             been
             given
             to
             any
             thing
             ,
             he
             will
             notwithstanding
             have
             the
             same
             Idea
             of
             that
             thing
             ,
             for
             which
             now
             by
             imposing
             a
             name
             on
             it
             ,
             we
             call
             it
             Body
             .
             Again
             ,
             when
             by
             comming
             neerer
             ,
             he
             sees
             the
             same
             thing
             thus
             and
             thus
             ,
             now
             in
             one
             place
             and
             now
             in
             another
             ,
             he
             will
             have
             a
             new
             Idea
             thereof
             ,
             namely
             that
             ,
             for
             which
             we
             now
             call
             such
             a
             thing
             Animated
             .
             Thirdly
             ,
             when
             standing
             neerer
             he
             perceives
             the
             figure
             ,
             hears
             the
             voice
             ,
             and
             sees
             other
             things
             ,
             which
             are
             signes
             of
             a
             Rationall
             minde
             ,
             he
             has
             a
             third
             Idea
             ,
             though
             it
             have
             yet
             no
             appellation
             ,
             namely
             ,
             that
             for
             which
             we
             now
             call
             any
             thing
             Rationall
             .
             Lastly
             ,
             when
             by
             looking
             fully
             and
             distinctly
             upon
             it
             he
             conceaves
             all
             that
             he
             has
             seen
             as
             one
             thing
             ,
             the
             Idea
             he
             has
             now
             ,
             is
             compounded
             of
             his
             former
             Ideas
             ,
             which
             are
             put
             together
             in
             the
             Minde
             ,
             in
             the
             same
             order
             ,
             in
             which
             these
             three
             single
             names
             
               Body
               ,
               Animated
               ,
               Rationall
            
             ,
             are
             in
             speech
             compounded
             into
             this
             one
             name
             Body-Animated-Rationall
             ,
             or
             Man.
             In
             like
             manner
             ,
             of
             the
             severall
             conceptions
             of
             
               four
               sides
               ,
               equality
               of
               sides
               ,
               and
               right
               angles
               ,
            
             is
             compounded
             the
             conception
             of
             a
             Square
             .
             For
             the
             mind
             may
             conceive
             a
             figure
             of
             foure
             sides
             without
             any
             conception
             of
             their
             equality
             ;
             &
             of
             that
             equality
             without
             conceiving
             a
             right
             angle
             ;
             and
             may
             joyne
             together
             all
             these
             single
             conceptions
             into
             one
             conception
             or
             one
             Idea
             of
             a
             Square
             .
             And
             thus
             we
             see
             how
             the
             Conceptions
             of
             the
             mind
             are
             compounded
             .
             Again
             ,
             whosoever
             sees
             a
             man
             standing
             neer
             him
             ,
             conceives
             the
             whole
             Idea
             of
             that
             man
             ;
             and
             if
             as
             he
             goes
             away
             he
             follow
             him
             with
             his
             eyes
             onely
             ,
             he
             will
             lose
             the
             Idea
             of
             those
             things
             which
             
             were
             signes
             of
             his
             being
             Rationall
             ,
             whilest
             neverthelesse
             the
             Idea
             of
             a
             Body-Animated
             remaines
             still
             before
             his
             eies
             ;
             so
             that
             the
             Idea
             of
             Rationall
             is
             substracted
             from
             the
             whole
             Idea
             of
             Man
             ,
             that
             is
             to
             say
             of
             Body-Animated-Rationall
             ,
             and
             there
             remaines
             that
             of
             Body-Animated
             ;
             &
             a
             while
             after
             at
             a
             greater
             distance
             the
             Idea
             of
             Animated
             will
             be
             lost
             ,
             &
             that
             of
             Body
             only
             will
             remain
             ;
             so
             that
             at
             last
             ,
             when
             nothing
             at
             all
             can
             be
             seen
             ,
             the
             whole
             Idea
             will
             vanish
             out
             of
             sight
             .
             By
             which
             examples
             ,
             I
             thinke
             it
             is
             manifest
             enough
             ,
             what
             is
             the
             internall
             Ratiocination
             of
             the
             Mind
             ,
             without
             words
             .
          
           
             We
             must
             not
             therefore
             thinke
             that
             Computation
             ,
             that
             is
             ,
             Ratiocination
             ,
             has
             place
             onely
             in
             numbers
             ;
             as
             if
             man
             were
             distinguished
             from
             other
             living
             Creatures
             (
             which
             is
             said
             to
             have
             been
             the
             opinion
             of
             Pythagoras
             )
             by
             nothing
             but
             the
             faculty
             of
             numbring
             ;
             for
             
               Magnitude
               ,
               Body
               ,
               Motion
               ,
               Time
               ,
               Degrees
               of
               Quality
               ,
               Action
               ,
               Conception
               ,
               Proportion
               ,
               Speech
               and
               Names
            
             (
             in
             which
             all
             the
             kinds
             of
             Philosophy
             consist
             )
             are
             capable
             of
             Addition
             and
             Substraction
             .
             Now
             such
             things
             as
             we
             adde
             or
             substract
             ,
             that
             is
             ,
             which
             we
             put
             into
             an
             account
             ,
             we
             are
             said
             to
             consider
             ,
             in
             Greek
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             ;
             in
             which
             language
             also
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             signifies
             to
             
               Compute
               ,
               Reason
            
             or
             Reckon
             .
          
           
             4
             But
             Effects
             and
             the
             Appearances
             of
             things
             to
             sense
             ,
             are
             faculties
             or
             Powers
             of
             Bodies
             ,
             which
             make
             us
             distinguish
             them
             from
             one
             another
             ;
             that
             is
             to
             say
             ,
             conceive
             one
             Body
             to
             be
             equall
             or
             unequall
             ,
             like
             or
             unlike
             to
             another
             Body
             ;
             as
             in
             the
             example
             above
             ,
             when
             by
             coming
             neer
             enough
             to
             any
             Body
             ,
             we
             perceive
             the
             Motion
             and
             Going
             of
             the
             same
             ,
             we
             distinguish
             it
             thereby
             from
             a
             Tree
             ,
             a
             Column
             and
             other
             fixed
             Bodies
             ;
             and
             so
             that
             motion
             ,
             or
             going
             is
             the
             Property
             thereof
             ,
             as
             being
             proper
             to
             living
             creatures
             ,
             and
             a
             faculty
             by
             which
             they
             make
             us
             distinguish
             them
             from
             other
             Bodies
             .
          
           
             5
             How
             the
             knowledge
             of
             any
             Effect
             may
             be
             gotten
             from
             the
             knowledge
             of
             the
             Generation
             thereof
             ,
             may
             easily
             be
             understood
             by
             the
             example
             of
             a
             Circle
             :
             For
             if
             there
             be
             set
             before
             us
             a
             plain
             figure
             having
             as
             neer
             as
             may
             be
             the
             figure
             of
             a
             Circle
             ,
             we
             cannot
             possibly
             perceive
             by
             sense
             whether
             it
             be
             a
             true
             Circle
             or
             no
             ;
             then
             which
             neverthelesse
             nothing
             is
             more
             easie
             to
             be
             known
             ,
             to
             him
             that
             knowes
             first
             the
             Generation
             of
             the
             propounded
             figure
             .
             For
             let
             it
             be
             known
             that
             the
             figure
             was
             made
             by
             the
             circumduction
             of
             
             a
             Body
             whereof
             one
             end
             remained
             unmoved
             ,
             and
             we
             may
             reason
             thus
             ;
             a
             Body
             carried
             about
             ,
             retaining
             alwayes
             the
             same
             length
             ,
             applies
             it selfe
             first
             to
             one
             Radius
             ,
             then
             to
             another
             ,
             to
             a
             third
             ,
             a
             fourth
             ,
             and
             successively
             to
             all
             ;
             and
             therefore
             the
             same
             length
             ,
             from
             the
             same
             point
             ,
             toucheth
             the
             circumference
             in
             every
             part
             thereof
             ;
             which
             is
             as
             much
             to
             say
             as
             all
             the
             Radii
             are
             equal
             .
             We
             know
             therefore
             that
             from
             such
             generation
             proceeds
             a
             figure
             ,
             from
             whose
             one
             middle
             point
             all
             the
             extreame
             points
             are
             reached
             unto
             by
             equal
             Radii
             .
             And
             in
             like
             manner
             ,
             by
             knowing
             first
             what
             figure
             is
             set
             before
             us
             ,
             we
             may
             come
             by
             Ratiocination
             to
             some
             Generation
             of
             the
             same
             ,
             though
             perhaps
             not
             that
             by
             which
             it
             was
             made
             ,
             yet
             that
             by
             w
             ch
             it
             might
             have
             been
             made
             ;
             for
             he
             that
             knows
             that
             a
             Circle
             has
             the
             property
             above
             declared
             ,
             will
             easily
             know
             whether
             a
             Body
             carried
             about
             ,
             as
             is
             said
             ,
             will
             generate
             a
             Circle
             or
             no.
             
          
           
             6
             The
             End
             or
             Scope
             of
             Philosophy
             ,
             is
             ,
             that
             we
             may
             make
             use
             to
             our
             benefit
             of
             effects
             formerly
             seen
             ;
             or
             that
             by
             applicatiō
             of
             Bodies
             to
             one
             another
             ;
             we
             may
             produce
             the
             like
             effects
             of
             those
             we
             conceive
             in
             our
             minde
             ,
             as
             far
             forth
             as
             matter
             ,
             strength
             &
             industry
             will
             permit
             ,
             for
             the
             commodity
             of
             humane
             life
             .
             For
             he
             inward
             glory
             and
             triumph
             of
             mind
             that
             a
             man
             may
             have
             ,
             for
             the
             mastering
             of
             some
             difficult
             and
             doutfull
             matter
             ,
             or
             for
             the
             discovery
             of
             some
             hidden
             truth
             ,
             is
             not
             worth
             so
             much
             paines
             as
             the
             study
             of
             Philosophy
             requires
             ;
             nor
             need
             any
             man
             care
             much
             to
             teach
             another
             what
             he
             knowes
             himselfe
             ,
             if
             he
             think
             that
             will
             be
             the
             onely
             benefit
             of
             his
             labour
             .
             The
             end
             of
             Knowledge
             is
             Power
             ;
             and
             the
             use
             of
             Theoremes
             (
             which
             among
             Geometricians
             serve
             for
             the
             finding
             out
             of
             Properties
             )
             is
             for
             the
             construction
             of
             Problemes
             ;
             and
             lastly
             ,
             the
             scope
             of
             all
             speculation
             is
             the
             performing
             of
             some
             action
             ,
             or
             thing
             to
             be
             done
             .
          
           
             7
             But
             what
             the
             Utility
             of
             Philosophy
             is
             ,
             especially
             of
             Natural
             Philosophy
             and
             Geometry
             ,
             will
             be
             best
             understood
             by
             reckoning
             up
             the
             chief
             commodities
             of
             which
             mankind
             is
             capable
             ;
             and
             by
             comparing
             the
             manner
             of
             life
             of
             such
             as
             enjoy
             them
             ,
             with
             that
             of
             others
             which
             want
             the
             same
             .
             Now
             the
             greatest
             commodities
             of
             mankind
             are
             the
             Arts
             ,
             namely
             of
             measuring
             Matter
             and
             Motion
             ;
             of
             moving
             ponderous
             Bodies
             ;
             of
             Architecture
             ;
             of
             Navigation
             ;
             of
             making
             instruments
             for
             all
             uses
             ;
             of
             calculating
             the
             Coelestiall
             Motions
             ,
             the
             Aspects
             of
             the
             Stars
             ,
             and
             the
             parts
             of
             Time
             ;
             of
             Geography
             ,
             
             &c.
             
             By
             which
             Sciences
             ,
             how
             great
             benefits
             men
             receive
             ,
             is
             more
             easily
             understood
             then
             expressed
             .
             These
             benefits
             are
             enjoyed
             by
             almost
             all
             the
             people
             of
             Europe
             ,
             by
             most
             of
             those
             of
             Asia
             ,
             and
             by
             some
             of
             Africa
             ;
             but
             the
             Americans
             ,
             and
             they
             that
             live
             neer
             the
             Poles
             do
             totally
             want
             them
             .
             But
             why
             ?
             Have
             they
             sharper
             wits
             then
             these
             ?
             Have
             not
             all
             men
             one
             kinde
             of
             soule
             ,
             and
             the
             same
             faculties
             of
             mind
             ?
             What
             then
             makes
             this
             difference
             ,
             except
             Philosophy
             ?
             Philosophy
             therefore
             is
             the
             cause
             of
             all
             these
             benefits
             .
             But
             the
             Utility
             of
             Morall
             and
             Civil
             Philosophy
             is
             to
             be
             estimated
             not
             so
             much
             by
             the
             commodities
             we
             have
             by
             knowing
             these
             Sciences
             ,
             as
             by
             the
             calamities
             we
             receive
             from
             not
             knowing
             them
             .
             Now
             all
             such
             calamities
             as
             may
             be
             avoided
             by
             humane
             industry
             arise
             from
             warre
             ,
             but
             chiefly
             from
             Civil
             warre
             ;
             for
             from
             this
             proceed
             Slaughter
             ,
             Solitude
             ,
             and
             the
             want
             of
             all
             things
             .
             But
             the
             cause
             of
             warre
             is
             not
             that
             men
             are
             willing
             to
             have
             it
             ;
             for
             the
             Will
             has
             nothing
             for
             Object
             but
             Good
             ,
             at
             least
             that
             which
             seemeth
             good
             .
             Nor
             is
             it
             from
             this
             ,
             that
             men
             know
             not
             that
             the
             effects
             of
             war
             are
             evil
             ;
             for
             who
             is
             there
             that
             thinks
             not
             poverty
             and
             losse
             of
             life
             to
             be
             great
             evils
             ?
             The
             cause
             therefore
             of
             Civill
             warre
             ,
             is
             that
             men
             know
             not
             the
             causes
             neither
             of
             Warre
             nor
             Peace
             ,
             there
             being
             but
             few
             in
             the
             world
             that
             have
             learned
             those
             duties
             which
             unite
             and
             keep
             men
             in
             peace
             ,
             that
             is
             to
             say
             ,
             that
             have
             learned
             the
             rules
             of
             civill
             life
             sufficiently
             .
             Now
             the
             knowledge
             of
             these
             rules
             is
             Morall
             Philosophy
             .
             But
             why
             have
             they
             not
             learned
             them
             ,
             unlesse
             for
             this
             reason
             that
             none
             hitherto
             have
             taught
             them
             in
             a
             clear
             and
             exact
             method
             ?
             For
             what
             shall
             we
             say
             ?
             Could
             the
             ancient
             Masters
             of
             Greece
             ,
             Egypt
             ,
             Rome
             ,
             and
             others
             perswade
             the
             unskillfull
             multitude
             to
             their
             innumerable
             opinions
             concerning
             the
             nature
             of
             their
             Gods
             ,
             which
             they
             themselves
             knew
             not
             whether
             they
             were
             true
             or
             false
             ,
             and
             which
             were
             indeed
             manifestly
             false
             &
             absurd
             ;
             &
             could
             they
             not
             perswade
             the
             same
             multitude
             to
             civill
             duty
             ,
             if
             they
             themselves
             had
             understood
             it
             ?
             Or
             shall
             those
             few
             writings
             of
             Geometricians
             which
             are
             extant
             ,
             be
             thought
             sufficient
             for
             the
             taking
             away
             of
             all
             controversy
             in
             the
             matters
             they
             treat
             of
             ,
             and
             shall
             those
             innumerable
             and
             huge
             Volumes
             of
             Ethicks
             be
             thought
             unsufficient
             ,
             if
             what
             they
             teach
             had
             been
             certain
             and
             well
             demonstrated
             ?
             What
             then
             can
             be
             imagined
             to
             be
             the
             cause
             that
             the
             writings
             of
             
             those
             men
             have
             increased
             science
             ,
             and
             the
             writings
             of
             these
             have
             increased
             nothing
             but
             words
             ,
             saving
             that
             the
             former
             were
             written
             by
             men
             that
             knew
             ,
             and
             the
             later
             by
             such
             as
             knew
             not
             the
             doctrine
             they
             taught
             onely
             for
             ostentation
             of
             their
             wit
             and
             eloquence
             ?
             Neverthelesse
             ,
             I
             deny
             not
             but
             the
             reading
             of
             some
             such
             books
             is
             very
             delightfull
             ;
             for
             they
             are
             most
             eloquently
             written
             ,
             and
             containe
             many
             cleer
             ,
             wholsome
             and
             choice
             sentences
             ;
             which
             yet
             are
             not
             universally
             true
             ,
             though
             by
             them
             universally
             pronounced
             .
             From
             whence
             it
             comes
             to
             passe
             ,
             that
             the
             circumstances
             of
             times
             ,
             places
             and
             persons
             being
             changed
             ,
             they
             are
             no
             lesse
             frequently
             made
             use
             of
             to
             confirme
             wicked
             men
             in
             their
             purposes
             ,
             then
             to
             make
             them
             understand
             the
             precepts
             of
             Civill
             duties
             .
             Now
             that
             which
             is
             chiefly
             wanting
             in
             them
             ,
             is
             a
             true
             and
             certaine
             rule
             of
             our
             actions
             ,
             by
             which
             we
             might
             know
             whether
             that
             we
             undertake
             be
             just
             or
             unjust
             .
             For
             it
             is
             to
             no
             purpose
             to
             be
             bidden
             in
             every
             thing
             to
             do
             Right
             ,
             before
             there
             be
             a
             certain
             Rule
             and
             measure
             of
             Right
             established
             ;
             which
             no
             man
             hitherto
             hath
             established
             .
             Seeing
             therefore
             from
             the
             not
             knowing
             of
             Civill
             duties
             ,
             that
             is
             ,
             from
             the
             want
             of
             Morall
             science
             proceed
             Civill
             warres
             ,
             and
             the
             greatest
             calamities
             of
             mankind
             ,
             we
             may
             very
             well
             attribute
             to
             such
             science
             the
             production
             of
             the
             contrary
             commodities
             .
             And
             thus
             much
             is
             sufficient
             ,
             to
             say
             nothing
             of
             the
             prayses
             and
             other
             contentment
             proceeding
             from
             Philosophy
             ,
             to
             let
             you
             see
             the
             Utility
             of
             the
             same
             in
             every
             kinde
             thereof
             .
          
           
             8
             The
             Subject
             of
             Philosophy
             ,
             or
             the
             matter
             it
             treats
             of
             ,
             is
             every
             Body
             of
             which
             we
             can
             conceive
             any
             generation
             ,
             and
             which
             we
             may
             by
             any
             consideration
             thereof
             compare
             with
             other
             Bodies
             ;
             or
             which
             is
             capable
             of
             composition
             and
             resolution
             ;
             that
             is
             to
             say
             ,
             every
             Body
             ,
             of
             whose
             Generation
             or
             Properties
             we
             can
             have
             any
             knowledge
             ..
             And
             this
             may
             be
             deduced
             from
             the
             Definition
             of
             Philosophy
             ,
             whose
             profession
             it
             is
             to
             search
             out
             the
             Properties
             of
             Bodies
             from
             their
             Generation
             ,
             or
             their
             Generation
             from
             their
             Properties
             ;
             and
             therefore
             where
             there
             is
             no
             Generation
             nor
             Property
             ,
             there
             is
             no
             Philosophy
             .
             Therefore
             it
             excludes
             Theology
             ,
             I
             meane
             the
             doctrine
             of
             God
             ,
             Eternal
             ,
             Ingenerable
             ,
             Incomprehensible
             ,
             and
             in
             whom
             there
             is
             nothing
             neither
             to
             divide
             nor
             compound
             ,
             nor
             any
             Generation
             to
             be
             conceived
             .
          
           
             It
             excludes
             the
             doctrine
             of
             Angels
             ;
             and
             all
             such
             things
             as
             are
             
             thought
             to
             be
             neither
             Bodies
             ,
             nor
             properties
             of
             Bodies
             ;
             there
             being
             in
             them
             no
             place
             neither
             for
             composition
             ,
             nor
             division
             ,
             nor
             any
             capacity
             of
             more
             and
             lesse
             ;
             that
             is
             to
             say
             ,
             no
             place
             for
             Ratiocination
             .
          
           
             It
             excludes
             History
             ,
             as
             well
             Naturall
             as
             Politicall
             ,
             though
             most
             usefull
             (
             nay
             necessary
             )
             to
             Philosophy
             ;
             because
             such
             Knowledge
             is
             but
             Experience
             ,
             or
             Authority
             ,
             and
             not
             Ratiocination
             .
          
           
             It
             excludes
             all
             such
             Knowledge
             as
             is
             acquired
             by
             Divine
             Inspiration
             ,
             or
             Revelation
             ,
             as
             not
             derived
             to
             us
             by
             Reason
             ,
             but
             by
             Divine
             grace
             in
             an
             instant
             ,
             and
             as
             it
             were
             by
             some
             sense
             supernaturall
             .
          
           
             It
             excludes
             ,
             not
             onely
             all
             Doctrines
             which
             are
             false
             ,
             but
             such
             also
             as
             are
             not
             well
             grounded
             ;
             for
             whatsoever
             we
             know
             by
             right
             Ratiocination
             ,
             can
             neither
             be
             false
             nor
             doubtfull
             ;
             and
             therefore
             Astrology
             ,
             as
             it
             is
             now
             held
             forth
             ,
             and
             all
             such
             Divinations
             rather
             then
             sciences
             ,
             are
             excluded
             .
          
           
             Lastly
             ,
             the
             doctrine
             of
             
               Gods
               Worship
            
             is
             excluded
             from
             Philosophy
             ,
             as
             being
             not
             to
             be
             known
             by
             naturall
             reason
             ,
             but
             by
             the
             authority
             of
             the
             Church
             ;
             and
             as
             being
             the
             object
             of
             Faith
             ,
             and
             not
             of
             Knowledge
             .
          
           
             9
             The
             principall
             parts
             of
             Philosophy
             are
             two
             .
             For
             two
             chief
             kinds
             of
             Bodies
             ,
             and
             very
             different
             from
             one
             another
             ,
             offer
             themselves
             to
             such
             as
             search
             after
             their
             Generation
             &
             Properties
             ;
             One
             whereof
             being
             the
             worke
             of
             Nature
             ,
             is
             called
             a
             
               Naturall
               Body
            
             ;
             the
             other
             is
             called
             a
             Commonwealth
             ,
             and
             is
             made
             by
             the
             wills
             and
             agreement
             of
             men
             .
             And
             from
             these
             spring
             the
             two
             parts
             of
             Philosophy
             called
             Naturall
             and
             Civill
             .
             But
             seeing
             that
             for
             the
             knowledge
             of
             the
             Properties
             of
             a
             Common-wealth
             ,
             it
             is
             necessary
             first
             to
             know
             the
             Dispositions
             ,
             Affections
             and
             Manners
             of
             men
             ,
             Civill
             Philosophy
             is
             againe
             commonly
             divided
             into
             two
             parts
             ;
             whereof
             one
             which
             treats
             of
             mens
             Dispositions
             and
             Manners
             is
             called
             Ethicks
             ,
             and
             the
             other
             which
             takes
             cognisance
             of
             their
             Civil
             Duties
             is
             caled
             Politicks
             or
             simply
             
               Civill
               Philosophy
            
             .
             In
             the
             first
             place
             therefore
             (
             after
             I
             have
             set
             downe
             such
             Premisses
             as
             appertaine
             to
             the
             nature
             of
             Philosophy
             in
             general
             )
             I
             will
             discourse
             of
             
               Bodies
               Naturall
            
             ;
             in
             the
             second
             ,
             of
             the
             
               Dispositions
               and
               Manners
               of
               men
            
             ;
             and
             in
             the
             third
             ,
             of
             the
             
               Civill
               Duties
               of
               Subjects
            
             .
          
           
             10
             To
             conclude
             ,
             seeing
             there
             may
             be
             many
             who
             will
             not
             like
             
             this
             my
             Definition
             of
             Philosophy
             ,
             and
             will
             say
             that
             from
             the
             liberty
             which
             a
             man
             may
             take
             of
             so
             defining
             as
             seemes
             best
             to
             himselfe
             ,
             he
             may
             conclude
             any
             thing
             from
             any
             thing
             (
             though
             I
             thinke
             it
             no
             hard
             matter
             to
             demonstrate
             ,
             that
             this
             Definition
             of
             mine
             agrees
             with
             the
             sense
             of
             all
             men
             ;
             )
             yet
             lest
             in
             this
             point
             there
             should
             be
             any
             cause
             of
             dispute
             betwixt
             me
             and
             them
             ,
             I
             here
             undertake
             no
             more
             then
             to
             deliver
             the
             Elements
             of
             that
             Science
             ,
             by
             which
             the
             Effects
             of
             any
             thing
             may
             be
             found
             out
             from
             the
             known
             Generation
             of
             the
             same
             ,
             or
             contrarily
             the
             Generation
             from
             the
             Effects
             ;
             to
             the
             end
             that
             they
             who
             search
             after
             other
             Philosophy
             ,
             may
             be
             admonished
             to
             seeke
             it
             from
             other
             Principles
             .
          
        
         
           
           
             
               CHAP.
               II.
            
             Of
             Names
             .
          
           
             
               
                 1
                 The
                 necessity
                 of
                 sensible
                 Moniments
                 or
                 Marks
                 for
                 the
                 help
                 of
                 Memory
                 ,
                 A
                 Marke
                 defined
                 .
              
               
                 2
                 The
                 necessity
                 of
                 Marks
                 for
                 the
                 signification
                 of
                 the
                 conceptions
                 of
                 the
                 Mind
                 .
              
               
                 3
                 Names
                 supply
                 both
                 those
                 necessities
                 .
              
               
                 4
                 The
                 Definition
                 of
                 a
                 Name
                 .
              
               
                 5
                 Names
                 are
                 Signes
                 not
                 of
                 Things
                 ,
                 but
                 of
                 our
                 Cogitations
                 .
              
               
                 6
                 What
                 it
                 is
                 we
                 give
                 Names
                 to
                 .
              
               
                 7
                 Names
                 Positive
                 and
                 Negative
                 .
              
               
                 8
                 Contradictory
                 Names
                 .
              
               
                 9
                 A
                 Common
                 Name
                 .
              
               
                 10
                 Names
                 of
                 the
                 first
                 and
                 second
                 Intention
                 .
              
               
                 11
                 Universall
                 ,
                 Particular
                 ,
                 Individuall
                 ,
                 and
                 Indefinite
                 Names
                 .
              
               
                 12
                 Names
                 Univocall
                 and
                 Equivocal
                 .
              
               
                 13
                 Absolute
                 and
                 Relative
                 Names
                 .
              
               
                 14
                 Simple
                 and
                 Compounded
                 Names
                 .
              
               
                 15
                 A
                 Praedicament
                 described
                 .
              
               
                 16
                 Some
                 things
                 to
                 be
                 noted
                 concerning
                 Praedicaments
                 .
              
            
          
           
             1
             HOw
             unconstant
             and
             fading
             mens
             thoughts
             are
             ,
             and
             how
             much
             the
             recovery
             of
             them
             depends
             upon
             chance
             ,
             there
             is
             none
             but
             knows
             by
             infallible
             experience
             in
             himself
             .
             For
             no
             man
             is
             able
             to
             remember
             Quantities
             without
             sensible
             and
             present
             Measures
             ,
             nor
             Colours
             without
             sensible
             and
             present
             Patterns
             ,
             nor
             Number
             without
             the
             Names
             of
             Numbers
             disposed
             in
             order
             and
             learned
             by
             heart
             .
             So
             that
             whatsoever
             a
             man
             has
             put
             together
             in
             his
             mind
             by
             ratiocination
             without
             such
             helps
             ,
             will
             presently
             slip
             from
             him
             ,
             and
             not
             be
             revocable
             but
             by
             beginning
             his
             ratiocination
             anew
             .
             From
             which
             it
             followes
             ,
             that
             for
             the
             acquiring
             of
             Philosophy
             some
             sensible
             Moniments
             are
             necessary
             ,
             by
             which
             our
             past
             thoughts
             may
             be
             not
             onely
             reduced
             ,
             but
             also
             registred
             every
             one
             in
             its
             own
             order
             .
             These
             Moniments
             I
             call
             MARKS
             ,
             namely
             ,
             sensible
             things
             taken
             at
             pleasure
             ,
             that
             by
             the
             sense
             of
             them
             such
             thoughts
             may
             be
             recalled
             to
             our
             mind
             ,
             as
             are
             like
             those
             thoughts
             for
             which
             we
             [
             took
             them
             .
             ]
          
           
           
             2
             Again
             ,
             though
             some
             one
             man
             ,
             of
             how
             excellent
             a
             wit
             soever
             ,
             should
             spend
             all
             his
             time
             ,
             partly
             in
             reasoning
             and
             partly
             in
             inventing
             Marks
             for
             the
             help
             of
             his
             memory
             ,
             and
             advancing
             himself
             in
             learning
             ;
             who
             sees
             not
             ,
             that
             the
             benefit
             he
             reapes
             to
             himselfe
             will
             not
             be
             much
             ,
             and
             to
             others
             none
             at
             all
             ?
             For
             unlesse
             he
             communicate
             his
             notes
             with
             others
             ,
             his
             science
             will
             perish
             with
             him
             .
             But
             if
             the
             same
             notes
             be
             made
             common
             to
             many
             ,
             and
             so
             one
             mans
             inventions
             be
             taught
             to
             others
             ,
             sciences
             will
             thereby
             be
             encreased
             to
             the
             generall
             good
             of
             mankind
             .
             It
             is
             therefore
             necessary
             for
             the
             acquiring
             of
             Philosophy
             that
             there
             be
             certain
             Signes
             ,
             by
             which
             what
             one
             man
             finds
             out
             may
             be
             manifested
             and
             made
             known
             to
             others
             .
             Now
             those
             things
             we
             call
             SIGNES
             ,
             are
             the
             
               Antecedents
               of
               their
               Consequents
               ,
               and
               the
               Consequents
               of
               their
               Antecedents
               ,
               as
               often
               as
               we
               observe
               them
               to
               go
               before
               or
               follow
               after
               in
               the
               same
               manner
               .
            
             For
             example
             ,
             a
             thick
             Cloud
             is
             a
             Signe
             of
             Rain
             to
             follow
             ;
             and
             Rain
             a
             Signe
             ,
             that
             a
             Cloud
             has
             gone
             before
             ,
             for
             this
             reason
             onely
             ,
             that
             we
             seldom
             see
             Clouds
             without
             the
             Consequence
             of
             Rain
             ,
             nor
             Rain
             at
             any
             time
             but
             when
             a
             Cloud
             has
             gone
             before
             .
             And
             of
             Signs
             some
             are
             Naturall
             ,
             whereof
             I
             have
             already
             given
             an
             example
             ;
             others
             are
             Arbitrary
             ,
             namely
             ,
             those
             we
             make
             choice
             of
             at
             our
             own
             pleasure
             ;
             as
             a
             bush
             hung
             up
             ,
             signifies
             that
             Wine
             is
             to
             be
             sold
             there
             ;
             a
             stone
             set
             in
             the
             ground
             ,
             signifies
             the
             bound
             of
             a
             field
             ;
             and
             words
             so
             and
             so
             connected
             ,
             signifie
             the
             Cogitations
             and
             Motions
             of
             our
             Minde
             .
             The
             difference
             therefore
             betwixt
             Markes
             and
             Signes
             is
             this
             ,
             that
             we
             make
             those
             for
             our
             own
             use
             ,
             but
             these
             for
             the
             use
             of
             others
             .
          
           
             3
             Words
             so
             connected
             ,
             as
             that
             they
             become
             signes
             of
             our
             Thoughts
             ,
             are
             called
             SPEECH
             ,
             of
             which
             every
             part
             is
             a
             Name
             .
             But
             seeing
             (
             as
             is
             said
             )
             both
             Markes
             and
             Signes
             are
             necessary
             for
             the
             acquiring
             of
             Philosophy
             ,
             (
             Marks
             by
             which
             we
             may
             remember
             our
             own
             thoughts
             ,
             and
             Signes
             by
             which
             we
             may
             make
             our
             thoughts
             known
             to
             others
             ,
             )
             Names
             do
             both
             these
             offices
             ;
             but
             they
             serve
             for
             Marks
             before
             they
             be
             used
             as
             Signes
             .
             For
             though
             a
             man
             were
             alone
             in
             the
             world
             ,
             they
             would
             be
             usefull
             to
             him
             in
             helping
             him
             to
             remember
             ;
             but
             to
             teach
             others
             ,
             (
             unlesse
             there
             were
             some
             others
             to
             be
             taught
             ,
             )
             of
             no
             use
             at
             all
             .
             Again
             ,
             Names
             ▪
             though
             standing
             singly
             by
             themselves
             ,
             are
             Marks
             ,
             because
             they
             serve
             to
             recall
             our
             own
             thoughts
             to
             mind
             ;
             but
             they
             cannot
             be
             Signes
             ,
             otherwise
             then
             by
             being
             disposed
             and
             ordered
             in
             Speech
             ,
             
             as
             parts
             of
             the
             same
             .
             For
             example
             ,
             a
             man
             may
             begin
             with
             a
             word
             ,
             whereby
             the
             hearer
             may
             frame
             an
             Idea
             of
             something
             in
             his
             mind
             ,
             which
             neverthelesse
             he
             cannot
             conceive
             to
             be
             the
             Idea
             which
             was
             in
             the
             mind
             of
             him
             that
             spake
             ,
             but
             that
             he
             would
             say
             something
             which
             began
             with
             that
             word
             ,
             though
             perhaps
             not
             as
             by
             it selfe
             ,
             but
             as
             part
             of
             another
             word
             .
             So
             that
             the
             nature
             of
             a
             name
             consists
             principally
             in
             this
             ,
             that
             it
             is
             a
             Mark
             ta
             ken
             for
             Memories
             sake
             ;
             but
             it
             serves
             also
             by
             accident
             ,
             to
             signifie
             and
             make
             known
             to
             others
             what
             we
             remember
             our selves
             ;
             and
             therefore
             I
             will
             define
             it
             thus
             :
          
           
             4
             A
             NAME
             
               is
               a
               Word
               taken
               at
               pleasure
               to
               serve
               for
               a
               Mark
               ,
               which
               may
               raise
               in
               our
               Mind
               a
               thought
               like
               to
               some
               thought
               we
               had
               before
               ,
               and
               which
               being
               pronounced
               to
               others
               ,
               may
               be
               to
               them
               a
               Sign
               of
               what
               thought
               the
               speaker
               had
               or
               had
               not
               before
               in
               his
               mind
               .
            
             And
             it
             is
             for
             brevities
             sake
             that
             I
             suppose
             the
             Originall
             of
             Names
             to
             be
             Arbitrary
             ,
             judging
             it
             a
             thing
             that
             maybe
             assumed
             as
             unquestionable
             .
             For
             cōsidering
             that
             new
             Names
             are
             daily
             made
             ,
             and
             old
             ones
             laid
             aside
             ;
             that
             diverse
             Nations
             use
             different
             Names
             ,
             and
             how
             impossible
             it
             is
             either
             to
             observe
             similitude
             ,
             or
             make
             any
             comparison
             betwixt
             a
             Name
             and
             a
             Thing
             ,
             how
             can
             any
             man
             imagine
             that
             the
             Names
             of
             Things
             were
             imposed
             from
             their
             natures
             ?
             For
             though
             some
             Names
             of
             living
             creatures
             and
             other
             things
             ,
             which
             our
             first
             Parents
             used
             ,
             were
             taught
             by
             God
             himselfe
             ;
             yet
             they
             were
             by
             him
             arbitrarily
             imposed
             ,
             and
             afterwards
             both
             at
             the
             Tower
             of
             Babel
             ,
             and
             since
             in
             processe
             of
             time
             ,
             growing
             every
             where
             out
             of
             use
             ,
             are
             quite
             forgotten
             ,
             and
             in
             their
             roomes
             have
             succeeded
             others
             ,
             invented
             and
             received
             by
             men
             at
             pleasure
             .
             Moreover
             ,
             whatsoever
             the
             common
             use
             of
             words
             be
             ,
             yet
             Philosophers
             ,
             who
             were
             to
             teach
             their
             knowledge
             to
             others
             ,
             had
             alwayes
             the
             liberty
             ,
             and
             sometimes
             they
             both
             had
             and
             will
             have
             a
             necessity
             ,
             of
             taking
             to
             themselves
             such
             Names
             as
             they
             please
             for
             the
             signifying
             of
             their
             meaning
             ,
             if
             they
             would
             have
             it
             understood
             .
             Nor
             had
             Mathematicians
             need
             to
             aske
             leave
             of
             any
             but
             themselves
             to
             name
             the
             figures
             they
             invented
             
               Parabolas
               ,
               Hyperboles
               ,
               Cissoeides
               ,
               Quadratrices
               ,
               &c.
            
             or
             to
             call
             one
             Magnitude
             A
             ,
             another
             B.
             
          
           
             5
             But
             seeing
             Names
             ordered
             in
             speech
             (
             as
             is
             defined
             )
             are
             signes
             of
             our
             Conceptions
             ,
             it
             is
             manifest
             they
             are
             not
             signes
             of
             the
             Things
             themselves
             ;
             for
             that
             the
             sound
             of
             this
             word
             Stone
             should
             be
             the
             
             signe
             of
             a
             Stone
             ,
             cannot
             be
             understood
             in
             any
             sense
             but
             this
             ,
             that
             he
             that
             heares
             it
             ,
             collects
             that
             he
             that
             pronouunces
             it
             thinkes
             of
             a
             Stone
             .
             And
             therefore
             that
             disputation
             ,
             whether
             Names
             signifie
             the
             Matter
             or
             Form
             ,
             or
             something
             compounded
             of
             both
             ,
             and
             other
             like
             subtleties
             of
             the
             Metaphysicks
             ,
             is
             kept
             up
             by
             erring
             men
             ,
             and
             such
             as
             understand
             not
             the
             words
             they
             dispute
             about
             .
          
           
             6
             Nor
             indeed
             is
             it
             at
             all
             necessary
             that
             every
             Name
             should
             be
             the
             Name
             of
             some
             Thing
             .
             For
             as
             these
             ,
             a
             Man
             ,
             a
             Tree
             ,
             a
             Stone
             ,
             are
             the
             Names
             of
             the
             Things
             themselves
             ;
             so
             the
             Images
             of
             a
             Man
             ,
             of
             a
             Tree
             and
             of
             a
             stone
             ,
             which
             are
             represented
             to
             men
             sleeping
             ,
             have
             their
             Names
             also
             ,
             though
             they
             be
             not
             Things
             ,
             but
             onely
             fictions
             and
             Phantasmes
             of
             things
             .
             For
             we
             can
             remember
             these
             ;
             and
             therefore
             it
             is
             no
             lesse
             necessary
             that
             they
             have
             Names
             to
             mark
             and
             signifie
             them
             ,
             then
             the
             Things
             themselves
             .
             Also
             this
             word
             Future
             is
             a
             Name
             ,
             but
             no
             future
             thing
             has
             yet
             any
             being
             ,
             nor
             do
             we
             know
             whether
             that
             which
             we
             call
             Future
             ,
             shall
             ever
             have
             a
             being
             or
             no.
             Neverthelesse
             ,
             seeing
             we
             use
             in
             our
             mind
             to
             knit
             together
             things
             Past
             with
             those
             that
             are
             Present
             ,
             the
             Name
             Future
             serves
             to
             signifie
             such
             knitting
             together
             .
             Moreover
             ,
             that
             which
             neither
             is
             ,
             nor
             has
             been
             ,
             nor
             ever
             shall
             or
             ever
             can
             be
             ,
             has
             a
             name
             ,
             namely
             ,
             
               That
               which
               neither
               is
               ,
               nor
               has
               been
               ,
               &c.
               
            
             Or
             more
             briefly
             this
             ,
             Impossible
             .
             To
             conclude
             ,
             this
             word
             Nothing
             ,
             is
             a
             name
             ,
             which
             yet
             cannot
             be
             the
             name
             of
             any
             thing
             .
             For
             when
             (
             for
             example
             )
             we
             substract
             2
             and
             3
             from
             5
             ,
             and
             so
             nothing
             remaining
             we
             would
             call
             that
             substraction
             to
             mind
             ,
             this
             speech
             
               Nothing
               remains
            
             ,
             and
             in
             it
             the
             word
             Nothing
             is
             not
             unusefull
             .
             And
             for
             the
             same
             reason
             we
             say
             truly
             
               Lesse
               then
               Nothing
            
             remaines
             ,
             when
             we
             substract
             More
             from
             Lesse
             ;
             for
             the
             minde
             feigns
             such
             remaines
             as
             these
             for
             Doctrines
             sake
             ,
             and
             desires
             as
             often
             as
             is
             necessary
             ,
             to
             call
             the
             same
             to
             memory
             .
             But
             seeing
             every
             name
             has
             some
             relation
             to
             that
             which
             is
             named
             ,
             though
             that
             which
             we
             name
             be
             not
             alwaies
             a
             thing
             that
             has
             a
             being
             in
             Nature
             ,
             yet
             it
             is
             lawfull
             for
             Doctrines
             sake
             to
             apply
             the
             word
             Thing
             ,
             to
             whasoever
             we
             name
             ;
             as
             if
             it
             were
             all
             one
             whether
             that
             thing
             be
             truly
             existent
             or
             be
             onely
             feigned
             .
          
           
             7
             The
             first
             distinction
             of
             Names
             ,
             is
             that
             some
             are
             Positive
             ,
             or
             Affirmative
             ;
             others
             Negative
             ,
             which
             are
             also
             called
             Privative
             and
             
               Indefinite
               ▪
               Positive
            
             are
             such
             as
             we
             impose
             for
             the
             likenesse
             ,
             Equality
             
             or
             Identity
             of
             the
             things
             we
             consider
             ;
             Negative
             for
             the
             diversity
             ,
             Unliknesse
             ,
             or
             Inequality
             of
             the
             same
             .
             Examples
             of
             the
             former
             are
             
               a
               Man
               ,
               a
               Philosopher
            
             ;
             for
             
               a
               Man
            
             denotes
             any
             one
             of
             a
             multitude
             of
             men
             ,
             and
             
               a
               Philosopher
            
             any
             one
             of
             many
             Philosophers
             by
             reason
             of
             their
             similitude
             ;
             Also
             Socrates
             is
             a
             Positive
             name
             ,
             because
             it
             signifies
             alwayes
             one
             and
             the
             same
             man.
             Examples
             of
             Negatives
             are
             such
             Positives
             as
             have
             the
             Negative
             particle
             Not
             added
             to
             them
             ,
             as
             
               Not-Man
               ,
               Not-Philosopher
            
             .
             But
             Positives
             were
             before
             Negatives
             ;
             for
             otherwise
             there
             could
             have
             been
             no
             use
             at
             all
             of
             these
             .
             For
             when
             the
             name
             of
             White
             was
             imposed
             upon
             certain
             things
             ,
             and
             afterwards
             upon
             other
             things
             the
             names
             of
             
               Black
               ,
               Blew
               ,
               Transparent
               ,
               &c.
            
             the
             infinite
             dissimilitudes
             of
             these
             with
             White
             could
             not
             be
             comprehended
             in
             any
             one
             Name
             ,
             save
             that
             which
             had
             in
             it
             the
             Negation
             of
             White
             ,
             that
             is
             to
             say
             ,
             the
             Name
             Not-White
             ,
             or
             some
             other
             equivalent
             to
             it
             ,
             in
             which
             the
             word
             White
             is
             repeated
             ,
             such
             as
             
               Unlike
               to
               white
               ,
               &c.
            
             
             And
             by
             these
             
               Negative
               ▪
               names
            
             ,
             we
             take
             notice
             our selves
             ,
             and
             signifie
             to
             others
             what
             we
             have
             not
             thought
             of
             .
          
           
             8
             Positive
             and
             Negative
             names
             are
             Contradictory
             to
             one
             another
             ,
             so
             that
             they
             cannot
             both
             be
             the
             name
             of
             the
             same
             thing
             .
             Besides
             ,
             of
             Contradictory
             names
             ,
             one
             is
             the
             name
             of
             any
             thing
             whatsoever
             ;
             for
             whatsoever
             is
             ,
             is
             either
             Man
             or
             Not-man
             ,
             White
             or
             Not-white
             ,
             and
             so
             of
             the
             rest
             .
             And
             this
             is
             so
             manifest
             ,
             that
             it
             needs
             no
             further
             proofe
             or
             explication
             ;
             for
             they
             that
             say
             
               the
               same
               thing
               cannot
               both
               be
               ,
               and
               not
               be
               ,
            
             speak
             obscurely
             ;
             but
             they
             that
             say
             ,
             
               Whatsoever
               is
               ,
               either
               is
               ,
               or
               is
               not
               ,
            
             speake
             also
             absurdly
             and
             ridiculously
             .
             The
             certainty
             of
             this
             Axiome
             ,
             viz.
             
               Of
               two
               Contradictory
               Names
               ,
               one
               is
               the
               Name
               of
               any
               thing
               whatsoever
               ,
               the
               other
               not
               ,
            
             is
             the
             originall
             and
             foundation
             of
             all
             Ratiocination
             ,
             that
             is
             ,
             of
             all
             Philosophy
             ;
             and
             therefore
             it
             ought
             to
             be
             so
             exactly
             propounded
             ,
             that
             it
             may
             be
             of
             it selfe
             cleare
             and
             perspicuous
             to
             all
             men
             ;
             as
             indeed
             it
             is
             ,
             saving
             to
             such
             ,
             as
             reading
             the
             long
             discourses
             made
             upon
             this
             subject
             by
             the
             Writers
             of
             Metaphysicks
             (
             which
             they
             beleeve
             to
             be
             some
             egregious
             learning
             )
             thinke
             they
             understand
             not
             ,
             when
             they
             do
             .
          
           
             9
             Secondly
             ,
             of
             Names
             ,
             some
             are
             Common
             to
             many
             things
             ,
             as
             a
             
               Man
               ,
               a
               Tree
            
             ;
             others
             Proper
             to
             one
             thing
             ,
             as
             
               he
               that
               writ
               the
               Iliad
               ,
               Homer
               ,
               this
               man
               ,
               that
               man.
            
             And
             a
             Common
             name
             ,
             being
             the
             name
             of
             many
             things
             severally
             taken
             ,
             but
             not
             collectively
             of
             all
             together
             (
             as
             Man
             
             is
             not
             the
             name
             of
             all
             mankind
             ,
             but
             of
             every
             one
             ,
             as
             of
             Peter
             ,
             Iohn
             ,
             and
             the
             rest
             severally
             )
             is
             therefore
             called
             an
             
               Universall
               name
            
             ;
             and
             therefore
             this
             word
             Universall
             is
             never
             the
             name
             of
             any
             thing
             existent
             in
             nature
             ,
             nor
             of
             any
             Idea
             or
             Phantasme
             formed
             in
             the
             mind
             ,
             but
             alwayes
             the
             name
             of
             some
             word
             or
             name
             ;
             so
             that
             when
             
               a
               Living
               creature
               ,
               a
               Stone
               ,
               a
               Spirit
               ,
            
             or
             any
             other
             thing
             is
             said
             to
             be
             Universal
             ,
             it
             is
             not
             to
             be
             understood
             ,
             that
             any
             Man
             ,
             Stone
             ,
             &c.
             ever
             was
             or
             can
             be
             Universall
             ,
             but
             onely
             that
             these
             words
             ,
             
               Living
               creature
               ,
               Stone
               ,
               &c.
            
             are
             
               Universall
               names
            
             ,
             that
             is
             ,
             Names
             common
             to
             many
             things
             ;
             and
             the
             Conceptions
             answering
             them
             in
             our
             minde
             ,
             are
             the
             Images
             and
             Phantasmes
             of
             severall
             Living
             Creatures
             ,
             or
             other
             things
             .
             And
             therfore
             for
             the
             understanding
             of
             the
             extent
             of
             an
             Universal
             name
             ,
             we
             need
             no
             other
             faculty
             but
             that
             of
             our
             imagination
             ,
             by
             which
             we
             remember
             that
             such
             names
             bring
             sometimes
             one
             thing
             ,
             sometimes
             another
             into
             our
             minde
             .
             Also
             of
             Common
             Names
             some
             are
             more
             ,
             some
             lesse
             Common
             .
             
               More
               Common
            
             ,
             is
             that
             which
             is
             the
             name
             of
             more
             things
             ;
             
               Lesse
               Common
            
             ,
             the
             name
             of
             fewer
             things
             .
             As
             Living-Creature
             is
             more
             Common
             then
             Man
             ,
             or
             Horse
             or
             Lion
             ,
             because
             it
             comprehends
             them
             all
             ;
             and
             therefore
             a
             more
             Common
             name
             ,
             in
             respect
             of
             a
             lesse
             Common
             ,
             is
             called
             the
             Genus
             or
             a
             
               Generall
               name
            
             ;
             and
             this
             in
             respect
             of
             that
             ,
             the
             Species
             ,
             or
             a
             
               Speciall
               Name
            
             .
          
           
             10
             And
             from
             hence
             proceeds
             the
             third
             distinction
             of
             Names
             ,
             which
             is
             ,
             that
             some
             are
             called
             names
             of
             the
             First
             ,
             others
             of
             the
             
               Second
               Intention
            
             .
             Of
             the
             
               first
               Intention
            
             are
             the
             names
             of
             Things
             ,
             a
             
               Man
               ,
               Stone
               ,
               &c.
            
             of
             the
             second
             are
             the
             names
             of
             names
             and
             speeches
             ,
             as
             
               Universall
               ,
               Particular
               ,
               Genus
               ,
               Species
               ,
               Syllogisme
               ,
            
             and
             the
             like
             .
             But
             it
             is
             hard
             to
             say
             why
             those
             are
             called
             names
             of
             the
             First
             ,
             and
             these
             of
             the
             
               Second
               Intention
            
             ,
             unlesse
             perhaps
             it
             was
             first
             intended
             by
             us
             to
             give
             names
             to
             those
             things
             which
             are
             of
             daily
             use
             in
             this
             life
             ,
             and
             afterwards
             to
             such
             things
             as
             appertaine
             to
             science
             ,
             that
             is
             ,
             that
             our
             Second
             Intention
             was
             to
             give
             names
             to
             Names
             .
             But
             whatsoever
             the
             cause
             hereof
             may
             be
             ,
             yet
             this
             is
             manifest
             ,
             that
             
               Genus
               ,
               Species
               ,
               Definition
               ,
               &c.
            
             are
             names
             of
             Words
             and
             Names
             onely
             ;
             and
             therefore
             to
             put
             Genus
             and
             Species
             for
             Things
             ,
             and
             Definition
             for
             the
             nature
             of
             any
             thing
             ,
             as
             the
             Writers
             of
             Metaphysicks
             have
             done
             ,
             is
             not
             right
             ,
             seeing
             they
             be
             only
             significations
             of
             what
             we
             thinke
             of
             the
             nature
             of
             Things
             .
          
           
             11
             Fourthly
             ,
             some
             Names
             are
             of
             
               certaine
               ▪
            
             and
             determined
             ,
             others
             
             of
             uncertaine
             and
             undetermined
             signification
             .
             Of
             determined
             and
             certain
             signification
             is
             ,
             first
             ,
             that
             name
             which
             is
             giuen
             to
             any
             one
             thing
             by
             it self
             ,
             and
             is
             called
             an
             
               Individuall
               Name
            
             ;
             as
             
               Homer
               ,
               this
               tree
               ,
               that
               living
               Creature
               ,
               &c.
               
            
             Secondly
             that
             which
             has
             any
             of
             these
             words
             
               All
               ,
               Every
               ,
               Both
               ,
               Either
               ,
            
             or
             the
             like
             added
             to
             it
             ;
             and
             it
             is
             therefore
             called
             an
             Universall
             Name
             ,
             because
             it
             signifies
             every
             one
             of
             those
             things
             to
             which
             it
             is
             Common
             ;
             and
             of
             certaine
             signification
             for
             this
             reason
             ,
             that
             he
             which
             heares
             ,
             conceives
             in
             his
             minde
             the
             same
             thing
             that
             he
             which
             speakes
             would
             have
             him
             conceive
             .
             Of
             Indefinite
             signification
             is
             ,
             first
             ,
             that
             Name
             which
             has
             the
             word
             some
             ,
             or
             the
             like
             added
             to
             it
             ,
             and
             is
             called
             a
             
               Particular
               name
            
             ;
             Secondly
             a
             Common
             Name
             set
             by
             it selfe
             without
             any
             note
             either
             of
             Universality
             or
             Particularity
             ,
             as
             
               Man
               ,
               Stone
            
             ,
             and
             is
             called
             an
             
               Indefinite
               Name
            
             ;
             but
             both
             Particular
             and
             Indefinite
             names
             are
             of
             uncertaine
             signification
             ,
             because
             the
             Hearer
             knowes
             not
             what
             thing
             it
             is
             the
             Speaker
             would
             have
             him
             conceive
             ;
             and
             therefore
             in
             Speech
             ,
             Particular
             and
             Indefinite
             names
             are
             to
             be
             esteemed
             equivalent
             to
             one
             another
             .
             But
             these
             words
             ,
             
               All
               ,
               Every
               ,
               Some
               ,
               &c.
            
             which
             denote
             Universality
             and
             Particularity
             ,
             are
             not
             Names
             ,
             but
             parts
             onely
             of
             Names
             ;
             So
             that
             
               Every
               Man
            
             ,
             and
             
               That
               Man
               which
               the
               Hearer
               conceives
               in
               his
               mind
               ,
            
             are
             all
             one
             ;
             and
             
               Some
               Man
            
             ,
             and
             
               That
               Man
               which
               the
               Speaker
               thought
               of
            
             signifie
             the
             same
             .
             From
             whence
             it
             is
             evident
             ,
             that
             the
             use
             of
             signes
             of
             this
             kind
             ,
             is
             not
             for
             a
             mans
             own
             sake
             ,
             or
             for
             his
             getting
             of
             knowledge
             by
             his
             own
             private
             meditation
             (
             for
             every
             man
             has
             his
             own
             ,
             Thoughts
             sufficiently
             determined
             without
             such
             helpes
             as
             these
             )
             but
             for
             the
             sake
             of
             others
             ;
             that
             is
             ,
             for
             the
             teaching
             and
             signifying
             of
             our
             Conceptions
             to
             others
             ;
             nor
             were
             they
             invented
             onely
             to
             make
             us
             remember
             ,
             but
             to
             make
             us
             able
             to
             discourse
             with
             others
             .
          
           
             12
             Fifthly
             ,
             Names
             are
             usually
             distinguished
             into
             Univocall
             ,
             and
             
               Equivocall
               .
               Univocall
            
             are
             those
             which
             in
             the
             same
             train
             of
             Discourse
             signifie
             alwayes
             the
             same
             thing
             ;
             but
             Equivocall
             those
             which
             meane
             sometimes
             one
             thing
             ,
             and
             sometimes
             another
             .
             Thus
             ,
             the
             Name
             Triangle
             is
             said
             to
             be
             Univocall
             ,
             because
             it
             is
             alwayes
             taken
             in
             the
             same
             sense
             ;
             and
             Parabola
             to
             be
             Equivocall
             ,
             for
             the
             signification
             it
             has
             sometimes
             of
             Allegory
             or
             Similitude
             ,
             and
             sometimes
             of
             a
             certaine
             Geometricall
             figure
             .
             Also
             every
             Metaphor
             is
             by
             profession
             Equivocall
             .
             But
             this
             distinction
             belongs
             not
             so
             much
             to
             Names
             ,
             as
             to
             those
             that
             use
             Names
             ;
             for
             some
             use
             them
             properly
             and
             accurately
             
             for
             the
             finding
             out
             of
             truth
             ;
             others
             draw
             them
             from
             their
             proper
             sense
             ,
             for
             Ornament
             ,
             or
             Deceipt
             .
          
           
             13
             Sixtly
             ,
             of
             Names
             ,
             some
             are
             Absolute
             ,
             others
             
               Relative
               .
               Relative
            
             are
             such
             as
             are
             imposed
             for
             some
             Comparison
             ,
             as
             
               Father
               ,
               Sonne
               ,
               Cause
               ,
               Effect
               ,
               Like
               ,
               Unlike
               ,
               Equal
               ,
               Unequal
               ,
               Master
               ,
               Servant
               ,
               &c.
               
            
             And
             those
             that
             signifie
             no
             Comparison
             at
             all
             are
             
               Absolute
               Names
            
             .
             But
             as
             it
             was
             noted
             above
             ,
             that
             Universality
             is
             to
             be
             attributed
             to
             Words
             &
             Names
             onely
             ,
             and
             not
             to
             Things
             ;
             so
             the
             same
             is
             to
             be
             said
             of
             other
             distinctions
             of
             Names
             ;
             for
             no
             Things
             are
             either
             Univocall
             or
             Equivocall
             ,
             or
             Relative
             or
             Absolute
             .
             There
             is
             also
             another
             distinction
             of
             Names
             into
             Concrete
             and
             Abstract
             ;
             but
             because
             Abstract
             Names
             proceed
             from
             Proposition
             ,
             and
             can
             have
             no
             place
             where
             there
             is
             no
             Affirmation
             ,
             I
             shall
             speake
             of
             them
             hereafter
             .
          
           
             14
             Lastly
             there
             are
             Simple
             and
             
               Compounded
               Names
            
             .
             But
             here
             it
             is
             to
             be
             noted
             ,
             that
             a
             name
             is
             not
             taken
             in
             Philosophy
             ,
             as
             in
             Grammar
             ,
             for
             one
             single
             word
             ,
             but
             for
             any
             number
             of
             words
             put
             together
             to
             signifie
             one
             Thing
             ;
             for
             among
             Philosophers
             
               Sentient
               Animated
               Body
            
             ,
             passes
             but
             for
             one
             Name
             ,
             being
             the
             Name
             of
             every
             living
             Creature
             ;
             which
             yet
             ,
             among
             Grāmarians
             is
             accounted
             three
             Names
             .
             Also
             a
             
               Simple
               Name
            
             is
             not
             here
             distinguished
             from
             a
             
               Compounded
               Name
            
             by
             a
             Preposition
             ,
             as
             in
             Grammar
             .
             But
             I
             call
             a
             
               Simple
               Name
            
             ,
             that
             which
             in
             every
             kind
             is
             the
             most
             Common
             or
             most
             Universall
             ;
             and
             that
             a
             
               Compounded
               Name
            
             ,
             which
             by
             the
             joyning
             of
             another
             Name
             to
             it
             is
             made
             lesse
             Universall
             ,
             and
             signifies
             that
             more
             conceptions
             then
             one
             were
             in
             the
             mind
             ,
             for
             which
             that
             later
             Name
             was
             added
             .
             For
             example
             ,
             in
             the
             conception
             of
             Man
             (
             as
             is
             shewn
             in
             the
             former
             Chap.
             )
             First
             ,
             he
             is
             conceived
             to
             be
             something
             that
             has
             Extension
             ,
             which
             is
             marked
             by
             the
             word
             
               Body
               .
               Body
            
             therefore
             is
             a
             
               Simple
               Name
            
             ,
             being
             put
             for
             that
             first
             single
             Conception
             ;
             Afterwards
             ,
             upon
             the
             sight
             of
             such
             and
             such
             motion
             ,
             another
             Conception
             arises
             for
             which
             he
             is
             called
             an
             
               Animated
               Body
            
             ;
             and
             this
             I
             here
             call
             a
             
               Compounded
               Name
            
             ,
             as
             I
             doe
             also
             the
             name
             Animal
             ,
             which
             is
             equivalent
             to
             an
             
               Animated
               Body
            
             .
             And
             in
             the
             same
             manner
             an
             
               Animated
               Rational
               Body
            
             ,
             as
             also
             a
             Man
             ,
             which
             is
             equivalent
             to
             it
             ,
             is
             a
             more
             Compounded
             Name
             .
             And
             by
             this
             we
             see
             how
             the
             Composition
             of
             Conceptions
             in
             the
             mind
             is
             answerable
             to
             the
             Composition
             of
             Names
             ;
             for
             as
             in
             the
             minde
             one
             Idea
             or
             Phantasme
             succeeds
             to
             another
             ,
             and
             to
             this
             a
             third
             ;
             so
             to
             one
             Name
             is
             added
             another
             and
             another
             successively
             ,
             
             and
             of
             them
             all
             is
             made
             one
             Compounded
             Name
             .
             Neverthelesse
             we
             must
             not
             thinke
             Bodies
             ,
             which
             are
             without
             the
             Minde
             ,
             are
             compounded
             in
             the
             same
             manner
             ,
             namely
             that
             there
             is
             in
             Nature
             a
             Body
             ,
             or
             any
             other
             imaginable
             Thing
             existent
             ,
             which
             at
             the
             first
             has
             no
             Magnitude
             ,
             and
             then
             by
             the
             addition
             of
             Magnitude
             ,
             comes
             to
             have
             Quantity
             ,
             and
             by
             more
             or
             lesse
             Quantity
             to
             have
             Density
             or
             Rarity
             ,
             and
             again
             by
             the
             addition
             of
             Figure
             to
             be
             Figurate
             ,
             and
             after
             this
             by
             the
             injection
             of
             Light
             or
             Colour
             ,
             to
             become
             Lucid
             or
             Coloured
             ;
             though
             such
             has
             been
             the
             Philosophy
             of
             many
             .
          
           
             15
             The
             Writers
             of
             Logique
             have
             endeavoured
             to
             digest
             the
             Names
             of
             all
             the
             kinds
             of
             Things
             into
             certaine
             Scales
             or
             Degrees
             ,
             by
             the
             continual
             subordination
             of
             Names
             lesse
             Common
             ,
             to
             Names
             more
             Common
             .
             In
             the
             Scale
             of
             Bodies
             they
             put
             in
             the
             first
             and
             highest
             place
             Body
             simply
             ,
             and
             in
             the
             next
             place
             under
             it
             lesse
             Common
             Names
             ,
             by
             which
             it
             may
             be
             more
             limited
             and
             determined
             ,
             namely
             Animated
             and
             Inanimated
             ,
             and
             so
             on
             till
             they
             come
             to
             Individualls
             .
             In
             like
             manner
             in
             the
             Scale
             of
             Quantities
             they
             assign
             the
             first
             place
             to
             Quantity
             ,
             and
             the
             next
             to
             
               Line
               ,
               Superficies
            
             ,
             and
             Solid
             ,
             which
             are
             Names
             of
             lesse
             latitude
             ;
             and
             these
             Orders
             or
             Scales
             of
             Names
             they
             usually
             call
             Praedicaments
             and
             Categories
             ;
             And
             of
             this
             Ordination
             not
             onely
             Positive
             but
             Negative
             Names
             also
             are
             capable
             ;
             which
             may
             be
             exemplified
             by
             such
             Formes
             of
             the
             Praedicaments
             as
             follow
             .
          
           
             
               The
               Form
               of
               the
               Praedicament
               of
               Body
               .
            
             
               Not-Body
               ;
               or
               Accident
            
             
               Body
               
                 
                   Not
                   animated
                
                 
                   Animated
                   
                     
                       Not
                       Living-Creature
                    
                     
                       Living-Creature
                       
                         
                           Not
                           Man
                        
                         
                           Man
                           
                             
                               Not
                               Peter
                            
                             
                               Peter
                            
                          
                        
                      
                    
                  
                
              
            
             
               Both
               Accident
               and
               Body
               are
               considered
               
                 
                   Absolutely
                   as
                   ,
                   
                     
                       Quantity
                       ,
                       or
                       so
                       much
                    
                     
                       Quality
                       ,
                       or
                       such
                    
                  
                
                 
                   or
                   Comparatively
                   ,
                   which
                   is
                   called
                   their
                   Relation
                
              
            
          
           
             
             
               The
               Forme
               of
               the
               Praedicament
               of
               Quantity
               .
            
             
               Quātity
               
                 
                   Not
                   Continual
                   ,
                   as
                   Number
                   ,
                
                 
                   Continuall
                   
                     
                       Of
                       it selfe
                       ,
                       as
                       
                         
                           Line
                        
                         
                           Superficies
                        
                         
                           Solid
                        
                      
                    
                     
                       By
                       Accident
                       ,
                       as
                       —
                       
                         
                           Time
                           ,
                           by
                           Line
                        
                         
                           Motion
                           ,
                           by
                           Line
                           and
                           Time
                        
                         
                           Force
                           ,
                           by
                           Motion
                           and
                           Solid
                        
                      
                    
                  
                
              
            
          
           
             Where
             it
             is
             to
             be
             noted
             ,
             that
             
               Line
               ,
               Superficies
            
             and
             Solid
             may
             be
             said
             to
             be
             of
             such
             and
             such
             Quantity
             ,
             that
             is
             ,
             to
             be
             originally
             and
             of
             their
             own
             nature
             capable
             of
             Equality
             and
             Inequality
             ;
             But
             we
             cannot
             say
             there
             is
             either
             Majority
             or
             Minority
             ,
             or
             Equality
             ,
             or
             indeed
             any
             Quantity
             at
             all
             ,
             in
             Time
             ,
             without
             the
             help
             of
             Line
             and
             Motion
             ;
             nor
             in
             Motion
             ,
             without
             Line
             and
             Time
             ;
             nor
             in
             Force
             ,
             otherwise
             then
             by
             Motion
             and
             Solid
             .
          
           
             
               The
               Forme
               of
               the
               Praedicament
               of
               Quality
               .
            
             
               Quality
               
                 
                   Perceptiō
                   by
                   Sense
                   
                     
                       Primary
                       
                         
                           Seeing
                        
                         
                           Hearing
                        
                         
                           Smelling
                        
                         
                           Tasting
                        
                         
                           Touching
                        
                      
                    
                     
                       Secondary
                       
                         
                           Imagination
                        
                         
                           Affection
                           —
                           
                             
                               Pleasant
                            
                             
                               Unpleasant
                            
                          
                        
                      
                    
                  
                
                 
                   Sensible
                   Quality
                   
                     
                       By
                       Seeing
                       ,
                       as
                       Light
                       and
                       Colour
                    
                     
                       By
                       Hearing
                       ,
                       as
                       Sound
                    
                     
                       By
                       Smelling
                       ,
                       as
                       Odors
                    
                     
                       By
                       Tasting
                       ,
                       as
                       Savours
                    
                     
                       By
                       Touching
                       ,
                       as
                       Hardnesse
                       ,
                       Heat
                       ,
                       Cold
                       ,
                       &c.
                       
                    
                  
                
              
            
          
           
             
             
               The
               forme
               of
               the
               Praedicament
               of
               Relation
               .
            
             
               Relation
               of
               
                 
                   Magnitudes
                   ,
                   as
                   Equality
                   and
                   Inequality
                
                 
                   Qualities
                   ,
                   as
                   Likenesse
                   and
                   Unlikenesse
                
                 
                   Order
                   
                     
                       Together
                       
                         
                           In
                           Place
                        
                         
                           In
                           Time
                        
                      
                    
                     
                       Not
                       together
                       
                         
                           In
                           Place
                           
                             
                               Former
                            
                             
                               Later
                            
                          
                        
                         
                           In
                           Time
                           
                             
                               Former
                            
                             
                               Later
                            
                          
                        
                      
                    
                  
                
              
            
          
           
             16
             Concerning
             which
             Praedicaments
             it
             is
             to
             be
             noted
             in
             the
             first
             place
             ,
             That
             as
             the
             division
             is
             made
             in
             the
             first
             Praedicament
             into
             Contradictory
             Names
             ,
             so
             it
             might
             have
             been
             done
             in
             the
             rest
             .
             For
             as
             there
             ,
             Body
             is
             divided
             into
             Animated
             and
             Not-Animated
             ,
             so
             in
             the
             second
             Praedicament
             
               Continuall
               Quantity
            
             may
             be
             divided
             into
             Line
             and
             Not-line
             ,
             and
             again
             ,
             Not-line
             into
             Superficies
             and
             Not-Superficies
             ,
             and
             so
             in
             the
             rest
             ;
             but
             it
             was
             not
             necessary
             .
          
           
             Secondly
             ,
             it
             is
             to
             be
             observed
             ,
             that
             of
             Positive
             Names
             the
             former
             comprehends
             the
             later
             ;
             but
             of
             Negatives
             the
             former
             is
             comprehended
             by
             the
             later
             .
             For
             example
             Living-Creature
             is
             the
             Name
             of
             every
             Man
             ,
             and
             therefore
             it
             comprehends
             the
             Name
             Man
             ;
             but
             on
             the
             contrary
             Not-Man
             is
             the
             Name
             of
             every
             Thing
             which
             is
             Not-Living-Creature
             ,
             and
             therefore
             the
             Name
             Not-Living-Creature
             which
             is
             put
             first
             ,
             is
             comprehended
             by
             the
             later
             Name
             Not-Man
             .
          
           
             Thirdly
             ,
             we
             must
             take
             heed
             we
             do
             not
             thinke
             ,
             that
             as
             Names
             ,
             so
             the
             diversities
             of
             Things
             themselves
             may
             be
             searched
             out
             and
             determined
             by
             such
             Distinctions
             as
             these
             ;
             or
             that
             arguments
             may
             be
             taken
             from
             hence
             (
             as
             some
             have
             done
             ridiculously
             )
             to
             prove
             that
             the
             kinds
             of
             Things
             are
             not
             infinite
             .
          
           
             Fourthly
             ,
             I
             would
             not
             have
             any
             man
             thinke
             I
             deliver
             the
             Forms
             above
             for
             a
             true
             and
             exact
             Ordination
             of
             Names
             ;
             for
             this
             cannot
             be
             performed
             as
             long
             as
             Philosophy
             remains
             imperfect
             ;
             Nor
             that
             by
             placing
             (
             for
             example
             )
             Light
             in
             the
             Praedicament
             of
             Qualities
             ,
             while
             another
             places
             the
             same
             in
             the
             Praedicament
             of
             Bodies
             ,
             I
             pretend
             that
             either
             of
             us
             ought
             for
             this
             to
             be
             drawn
             
             from
             his
             opinion
             ;
             for
             this
             is
             to
             be
             done
             onely
             by
             Arguments
             and
             Ratiocination
             ,
             and
             not
             by
             disposing
             of
             words
             into
             Classes
             .
          
           
             Lastly
             ,
             I
             confesse
             I
             have
             not
             yet
             seen
             any
             great
             use
             of
             the
             Praedicaments
             in
             Philosophy
             .
             I
             beleeve
             Aristotle
             when
             he
             saw
             he
             could
             not
             digest
             the
             Things
             themselves
             into
             such
             Orders
             ,
             might
             neverthelesse
             desire
             out
             of
             his
             owne
             Authority
             to
             reduce
             Words
             to
             such
             Formes
             ,
             as
             I
             have
             done
             ;
             but
             I
             doe
             it
             onely
             for
             this
             end
             ,
             that
             it
             may
             be
             understood
             what
             this
             Ordination
             of
             Words
             is
             ,
             and
             not
             to
             have
             it
             received
             or
             true
             ,
             till
             it
             be
             demonstrated
             by
             good
             reason
             to
             be
             so
             .
          
        
         
           
           
             
               CHAP.
               III.
            
             Of
             Proposition
             .
          
           
             
               
                 1
                 Divers
                 Kinds
                 of
                 Speech
                 .
              
               
                 2
                 Proposition
                 defined
                 .
              
               
                 3
                 Subject
                 ,
                 Praedicate
                 and
                 Copula
                 what
                 they
                 are
                 ,
                 and
                 Abstract
                 and
                 Concrete
                 what
                 .
                 The
                 Use
                 and
                 Abuse
                 of
                 Names
                 Abstract
                 .
              
               
                 5
                 Proposition
                 Universal
                 and
                 Particular
                 .
              
               
                 6
                 Affirmative
                 and
                 Negative
                 .
              
               
                 7
                 True
                 and
                 False
                 .
              
               
                 8
                 True
                 and
                 False
                 belongs
                 to
                 Speech
                 ,
                 and
                 not
                 to
                 Things
                 .
              
               
                 9
                 Proposition
                 Primary
                 ,
                 not
                 Primary
                 ,
                 Definion
                 ,
                 Axiome
                 ,
                 Petition
                 .
              
               
                 10
                 Proposition
                 Necessary
                 and
                 Contingent
                 .
              
               
                 11
                 Categoricall
                 and
                 Hypotheticall
                 .
              
               
                 12
                 The
                 same
                 Proposition
                 diversly
                 pronounced
                 .
              
               
                 13
                 Propositions
                 that
                 may
                 be
                 reduced
                 to
                 the
                 same
                 Categoricall
                 Proposition
                 ,
                 are
                 Equipollent
                 .
              
               
                 14
                 Universal
                 Propositions
                 converted
                 by
                 Contradictory
                 Names
                 ,
                 are
                 Equipollent
                 .
              
               
                 15
                 Negative
                 Propositions
                 are
                 the
                 same
                 ,
                 whether
                 the
                 Negation
                 be
                 before
                 or
                 after
                 the
                 Copula
                 .
              
               
                 16
                 Particular
                 Propositions
                 simply
                 converted
                 ,
                 are
                 Equipollent
                 .
              
               
                 17
                 What
                 are
                 Subaltern
                 ,
                 Contrary
                 ,
                 Subcontrary
                 and
                 Contradictory
                 Propositions
                 .
              
               
                 18
                 Consequence
                 what
                 it
                 is
                 .
              
               
                 19
                 Falsity
                 cannot
                 follow
                 from
                 Truth
                 .
              
               
                 20
                 How
                 one
                 Proposition
                 is
                 the
                 Cause
                 of
                 another
                 .
              
            
          
           
             1_FRom
             the
             Connection
             or
             Contexture
             of
             Names
             arise
             diverse
             kinds
             of
             Speech
             ,
             whereof
             some
             signifie
             the
             Desires
             and
             Affections
             of
             Men
             ;
             such
             are
             first
             Interrogations
             ,
             which
             denote
             the
             desire
             of
             Knowing
             ;
             as
             
               Who
               is
               a
               good
               Man
            
             ?
             In
             which
             speech
             there
             is
             one
             Name
             expressed
             ,
             &
             another
             desired
             and
             expected
             from
             him
             of
             whom
             we
             aske
             the
             same
             .
             Then
             Prayers
             ,
             which
             signifie
             the
             desire
             of
             having
             something
             ;
             
               Promises
               ,
               Threats
               ,
               Wishes
               ,
               Commands
               ,
               Complaints
               ,
            
             and
             other
             significations
             of
             other
             Affections
             .
             Speech
             may
             also
             be
             Absurd
             and
             Insignificant
             ;
             as
             when
             there
             is
             a
             succession
             of
             Words
             ,
             to
             which
             there
             can
             be
             no
             succession
             of
             Thoughts
             in
             the
             mind
             to
             answer
             them
             ;
             and
             this
             happens
             often
             to
             such
             ,
             as
             understanding
             nothing
             in
             some
             subtil
             matter
             ,
             doe
             neverthelesse
             ,
             to
             make
             others
             beleeve
             they
             understand
             ,
             speake
             of
             the
             same
             
             incoherently
             ;
             For
             the
             connection
             of
             incoherent
             Words
             ,
             though
             it
             want
             the
             end
             of
             Speech
             (
             which
             is
             Signification
             )
             yet
             it
             is
             Speech
             ;
             and
             is
             used
             by
             the
             Writers
             of
             Metaphysicks
             almost
             as
             frequently
             as
             Speech
             significative
             .
             In
             Philosophy
             there
             is
             but
             one
             kinde
             of
             Speech
             usefull
             ,
             which
             some
             call
             in
             Latine
             Dictum
             ,
             others
             
               Enuntiatum
               &
               Pronunciatum
            
             ;
             but
             most
             men
             call
             it
             Proposition
             ,
             and
             is
             the
             speech
             of
             those
             that
             Affirm
             or
             Deny
             ,
             and
             expresseth
             Truth
             or
             Falsity
             .
          
           
             2
             A
             PROPOSITION
             
               is
               a
               Speech
               consisting
               of
               two
               Names
               copulated
               by
               which
               he
               that
               speaketh
               signifies
               he
               conceives
               the
               later
               Name
               to
               be
               the
               Name
               of
               the
               same
               thing
               whereof
               the
               former
               is
               the
               Name
               ;
               or
            
             (
             which
             is
             all
             one
             )
             
               that
               the
               former
               Name
               is
               comprehended
               by
               the
               later
               .
            
             For
             example
             ,
             this
             speech
             
               Man
               is
               a
               Living
               Creature
            
             ,
             in
             which
             two
             Names
             are
             copulated
             by
             the
             verb
             Is
             ,
             is
             a
             Proposition
             ,
             for
             this
             reason
             ,
             that
             he
             that
             speakes
             it
             conceives
             both
             
               Living
               Creature
            
             and
             Man
             to
             be
             Names
             of
             the
             same
             thing
             ,
             or
             that
             the
             former
             Name
             Man
             is
             comprehended
             by
             the
             later
             Name
             
               Living
               Creature
            
             .
             Now
             the
             former
             Name
             is
             commonly
             called
             the
             Subject
             ,
             or
             Antecedent
             ,
             or
             the
             
               Contained
               Name
            
             ,
             and
             the
             later
             the
             
               Praedicat
               ,
               Consequent
            
             or
             
               Containing
               Name
            
             .
             The
             signe
             of
             Connection
             amongst
             most
             Nations
             is
             either
             some
             word
             ,
             as
             the
             word
             is
             in
             the
             Proposition
             
               Man
               is
               a
               living
               Creature
            
             ,
             or
             some
             Case
             or
             Termination
             of
             a
             word
             ,
             as
             in
             this
             Proposition
             ,
             
               Man
               walketh
            
             (
             which
             is
             equivalent
             to
             this
             ,
             
               Man
               is
               walking
            
             )
             the
             Termination
             by
             which
             it
             is
             said
             he
             walketh
             ,
             rather
             then
             he
             
               is
               walking
            
             ,
             signifieth
             that
             those
             two
             are
             understood
             to
             be
             copulated
             ,
             or
             to
             be
             Names
             of
             the
             same
             Thing
             .
          
           
             But
             there
             are
             ,
             or
             certainly
             may
             be
             some
             Nations
             that
             have
             no
             word
             which
             answers
             to
             our
             Verbe
             Is
             ,
             who
             neverthelesse
             forme
             Propositions
             by
             the
             position
             onely
             of
             one
             Name
             after
             another
             ,
             as
             if
             instead
             of
             
               Man
               is
               a
               Living
               Creature
            
             ,
             it
             should
             be
             said
             
               Man
               a
               Living
               Creature
            
             ;
             for
             the
             very
             order
             of
             the
             Names
             may
             sufficiently
             shew
             their
             connection
             ;
             and
             they
             are
             as
             apt
             and
             usefull
             in
             Philosophy
             ,
             as
             if
             they
             were
             copulated
             by
             the
             Verbe
             Is.
             
          
           
             3
             Wherefore
             in
             every
             Proposition
             three
             things
             are
             to
             be
             considered
             ,
             viz.
             the
             two
             Names
             ,
             which
             are
             the
             Subject
             and
             the
             Praedicate
             ,
             and
             their
             Copulation
             ;
             both
             which
             Names
             raise
             in
             our
             Minde
             the
             Thought
             of
             one
             and
             the
             same
             Thing
             ;
             but
             the
             Copulation
             makes
             us
             thinke
             of
             the
             Cause
             for
             which
             those
             Names
             were
             imposed
             on
             that
             Thing
             .
             As
             for
             example
             ,
             when
             we
             say
             
               a
               Body
               is
               moveable
            
             ,
             though
             we
             conceive
             the
             same
             thing
             to
             be
             designed
             by
             both
             
             those
             Names
             ,
             yet
             our
             Minde
             rests
             not
             there
             ,
             but
             searches
             further
             what
             it
             is
             
               to
               be
               a
               Body
            
             ,
             or
             
               to
               be
               Moveable
            
             ,
             that
             is
             ,
             wherein
             consists
             the
             difference
             betwixt
             these
             and
             other
             Things
             ,
             for
             which
             these
             are
             so
             called
             ,
             others
             are
             not
             so
             called
             .
             They
             therefore
             that
             seeke
             what
             it
             is
             
               to
               be
            
             any
             thing
             ,
             as
             
               to
               be
               Moveable
               ,
               to
               be
               Hot
               ,
               &c.
            
             seek
             in
             Things
             the
             causes
             of
             their
             Names
             .
          
           
             And
             from
             hence
             arises
             that
             distinction
             of
             Names
             (
             touched
             in
             the
             last
             Chap.
             )
             into
             Concrete
             and
             Abstract
             .
             For
             Concrete
             is
             the
             Name
             of
             any
             thing
             which
             we
             suppose
             to
             have
             a
             being
             ,
             and
             is
             therefore
             called
             the
             Subject
             ,
             in
             Latine
             Suppositum
             ,
             and
             in
             Greek
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             ;
             as
             
               Body
               ,
               Moveable
               ,
               Moved
               ,
               Figurate
               ,
               a
               Cubit
               high
               ,
               Hot
               ,
               Cold
               ,
               Like
               ,
               Equal
               ,
               Appius
               ,
               Lentulus
            
             and
             the
             like
             ;
             and
             Abstract
             is
             that
             which
             in
             any
             Subject
             denotes
             the
             Cause
             of
             the
             Concrete
             Name
             ,
             as
             
               to
               be
               a
               Body
               ,
               to
               be
               Moveable
               ,
               to
               be
               Moved
               ,
               to
               be
               Figurate
               ,
               to
               be
               of
               such
               Quantity
               ,
               to
               be
               Hot
               ,
               to
               be
               Cold
               ,
               to
               be
               Like
               ,
               to
               be
               Equall
               ,
               to
               be
               Appius
               ,
               to
               be
               Lentulus
               ,
               &c.
               
            
             Or
             Names
             equivalent
             to
             these
             ,
             which
             are
             most
             commonly
             called
             Abstract
             Names
             ,
             as
             
               Corporeity
               ,
               Mobility
               ,
               Motion
               ,
               Figure
               ,
               Quantity
               ,
               Heat
               ,
               Cold
               ,
               Likenesse
               ,
               Equality
               ,
            
             and
             (
             as
             Cicero
             has
             it
             )
             Appiety
             and
             Lentulity
             .
             Of
             the
             same
             kind
             also
             are
             Infinitives
             ;
             for
             
               to
               Live
            
             and
             
               to
               Move
            
             are
             the
             same
             with
             Life
             and
             Motion
             ,
             or
             
               to
               be
               Living
            
             ,
             and
             
               to
               be
               Moved
            
             .
             But
             
               Abstract
               Names
            
             denote
             onely
             the
             Causes
             of
             
               Concrete
               Names
            
             ,
             and
             not
             the
             Things
             themselves
             .
             For
             example
             when
             we
             see
             any
             thing
             ,
             or
             conceive
             in
             our
             Minde
             any
             Visible
             thing
             ,
             that
             Thing
             appears
             to
             us
             ,
             or
             is
             conceived
             by
             us
             ,
             not
             in
             one
             Point
             ,
             but
             as
             having
             Parts
             distant
             from
             one
             another
             ,
             that
             is
             ,
             as
             being
             extended
             and
             filling
             some
             space
             .
             Seeing
             therefore
             we
             call
             the
             Thing
             so
             conceived
             Body
             ,
             the
             cause
             of
             that
             name
             is
             ,
             that
             that
             Thing
             is
             extended
             ,
             or
             the
             Extension
             or
             Corporeity
             of
             it
             .
             So
             when
             we
             see
             a
             Thing
             appeare
             sometimes
             here
             ,
             sometimes
             there
             ,
             and
             call
             it
             Moved
             or
             Removed
             ,
             the
             Cause
             of
             that
             Name
             is
             that
             it
             
               is
               Moved
            
             or
             the
             Motion
             of
             the
             same
             .
          
           
             And
             these
             Causes
             of
             Names
             ;
             are
             the
             same
             with
             the
             Causes
             of
             our
             Conceptions
             ,
             namely
             some
             Power
             or
             Action
             ,
             or
             Affection
             of
             the
             Thing
             conceived
             ,
             which
             some
             call
             the
             Manner
             by
             which
             any
             thing
             workes
             upon
             our
             senses
             ,
             but
             by
             most
             men
             they
             are
             called
             Accidents
             ;
             I
             say
             Accidents
             ,
             not
             in
             that
             sense
             in
             which
             Accident
             is
             opposed
             to
             Necessary
             ;
             but
             so
             ,
             as
             being
             neither
             the
             Things
             themselves
             ,
             nor
             parts
             thereof
             ,
             do
             neverthelesse
             accompany
             the
             
             Things
             in
             such
             manner
             ,
             that
             (
             saving
             Extension
             )
             they
             may
             all
             perish
             and
             〈◊〉
             destroyed
             ,
             but
             can
             never
             be
             abstracted
             .
          
           
             4
             There
             is
             also
             this
             difference
             betwixt
             Concrete
             and
             
               Abstract
               Names
            
             ,
             that
             those
             were
             invented
             before
             Propositions
             ,
             but
             these
             after
             ;
             for
             these
             could
             have
             no
             being
             till
             there
             were
             Propositions
             ,
             from
             whose
             Copula
             they
             proceed
             .
             Now
             in
             all
             matters
             that
             concerne
             this
             life
             ,
             but
             chiefly
             in
             Philosophy
             ,
             there
             is
             both
             great
             Use
             and
             great
             Abuse
             of
             
               Abstract
               Names
            
             ;
             and
             the
             use
             consists
             in
             this
             ,
             that
             without
             them
             we
             cannot
             for
             the
             most
             part
             either
             Reason
             ,
             or
             compute
             the
             Properties
             of
             Bodies
             ;
             for
             when
             we
             would
             multiply
             ,
             divide
             ,
             adde
             or
             substract
             Heat
             ,
             Light
             ,
             or
             Motion
             ,
             if
             we
             should
             double
             or
             adde
             them
             together
             by
             Concrete
             Names
             ,
             saying
             (
             for
             example
             )
             Hot
             is
             double
             to
             Hot
             ,
             Light
             double
             to
             Light
             ,
             or
             Moved
             double
             to
             Moved
             ,
             we
             should
             not
             double
             the
             Properties
             ,
             but
             the
             Bodies
             themselves
             that
             are
             Hot
             ,
             Light
             ,
             Moved
             ,
             &c.
             which
             we
             would
             not
             doe
             .
             But
             the
             Abuse
             proceeds
             from
             this
             ,
             that
             some
             men
             seeing
             they
             can
             consider
             ,
             that
             is
             (
             as
             I
             said
             before
             )
             bring
             into
             account
             the
             Increasings
             and
             Decreasings
             of
             Quantity
             ,
             Heat
             and
             other
             Accidents
             ,
             without
             considering
             their
             Bodies
             or
             Subjects
             (
             which
             they
             call
             Abstracting
             ,
             or
             making
             to
             exist
             apart
             by
             themselves
             ,
             )
             they
             speake
             of
             Accidents
             ,
             as
             if
             they
             might
             be
             separated
             from
             all
             Bodies
             .
             And
             from
             hence
             proceed
             the
             grosse
             errors
             of
             writers
             of
             Metaphysicks
             ;
             for
             ,
             because
             they
             can
             consider
             Thought
             without
             the
             consideration
             of
             Body
             ,
             they
             inferre
             there
             is
             no
             need
             of
             a
             ●●inking-Body
             ;
             and
             because
             Quantity
             may
             be
             considered
             without
             considering
             Body
             ,
             they
             thinke
             also
             that
             Quantity
             may
             be
             without
             Body
             ,
             and
             Body
             without
             Quantity
             ;
             and
             that
             a
             Body
             ●as
             Quantity
             by
             the
             addition
             of
             Quantity
             to
             it
             .
             From
             the
             same
             fountaine
             spring
             those
             insignificant
             words
             ,
             
               Abstract
               Substance
               ,
               Separated
               Essence
            
             and
             the
             like
             ;
             as
             also
             that
             confusion
             of
             words
             derived
             from
             the
             Latine
             Verb
             Est
             ,
             as
             
               Essence
               ,
               Essentiality
               ,
               Entity
               ,
               Entitative
            
             ;
             besides
             
               Reality
               ,
               Aliquiddity
               ,
               Quiddity
            
             ,
             &c.
             which
             could
             never
             have
             been
             heard
             of
             among
             such
             Nations
             as
             do
             not
             Copulate
             their
             Names
             by
             the
             Verbe
             Is
             ,
             but
             by
             Adjective
             Verbs
             ,
             as
             Runneth
             ,
             Readeth
             ,
             &c.
             or
             by
             the
             mere
             placing
             of
             one
             Name
             after
             another
             ;
             and
             yet
             seeing
             such
             Nations
             Compute
             and
             Reason
             ,
             it
             is
             evident
             that
             Philosophy
             has
             no
             need
             of
             those
             words
             
               Essence
               ,
               Entity
            
             and
             other
             the
             like
             barbarous
             Termes
             .
          
           
           
             There
             are
             many
             Distinctions
             of
             Propositions
             ,
             whereof
             the
             first
             is
             ,
             that
             some
             are
             Universall
             ,
             others
             Particular
             ,
             others
             Indefinite
             ,
             and
             others
             Singular
             ;
             and
             this
             is
             commonly
             called
             the
             distinction
             of
             Quantity
             .
             An
             
               Universall
               Proposition
            
             is
             that
             whose
             subject
             is
             affected
             with
             the
             sign
             of
             an
             Universall
             Name
             ,
             as
             
               Every
               man
               is
               a
               Living
               Creature
               .
               Particular
               ,
            
             that
             whose
             subject
             is
             affected
             with
             the
             sign
             of
             a
             Particular
             Name
             ,
             as
             
               Some
               Man
               is
               learned
            
             .
             An
             
               Indefinite
               Proposition
            
             has
             for
             its
             Subject
             a
             Common
             Name
             ,
             and
             put
             without
             any
             sing
             ;
             as
             
               Man
               is
               a
               Living
               Creature
               ,
               Man
               is
               Learned
               .
            
             And
             a
             
               Singular
               Proposition
            
             is
             that
             whose
             Subject
             is
             a
             singular
             Name
             ,
             as
             
               Socrates
               is
               a
               Philosopher
               ,
               This
               man
               is
               black
               .
            
          
           
             6
             The
             Second
             Distinction
             is
             into
             Affirmative
             and
             Negative
             ,
             and
             is
             called
             the
             Distinction
             of
             Quality
             .
             An
             
               Affirmative
               Proposition
            
             is
             that
             whose
             Praedicate
             is
             a
             Positive
             Name
             ,
             as
             
               Man
               is
               a
               Living
               Creature
               .
               Negative
               ,
            
             that
             whose
             Praedicate
             is
             a
             Negative
             Name
             ,
             as
             
               Man
               is
               Not-a-stone
            
             .
          
           
             7
             The
             third
             Distinction
             is
             ,
             that
             one
             is
             True
             ,
             another
             False
             .
             A
             
               True
               Proposition
            
             is
             that
             ,
             whose
             Praedicate
             containes
             ,
             or
             comprehends
             its
             Subject
             ,
             or
             whose
             Praedicate
             is
             the
             Name
             of
             every
             thing
             ,
             of
             which
             the
             Subject
             is
             the
             Name
             ;
             as
             
               Man
               is
               a
               Living
               Creature
            
             is
             therefore
             a
             True
             Proposition
             ,
             because
             whatsoever
             is
             called
             Man
             ,
             the
             same
             is
             also
             called
             
               Living
               Creature
            
             ;
             and
             
               Some
               Man
               is
               sick
            
             ,
             is
             True
             ,
             because
             sick
             is
             the
             Name
             of
             
               Some
               Man.
            
             That
             which
             is
             not
             True
             ,
             or
             that
             whose
             Praedicate
             does
             not
             containe
             its
             Subject
             ,
             is
             called
             a
             
               False
               Proposition
            
             ,
             as
             
               Man
               is
               a
               Stone
            
             .
          
           
             Now
             these
             words
             
               True
               ,
               Truth
            
             ,
             and
             
               True
               Proposition
            
             are
             equivalent
             to
             one
             another
             ;
             For
             Truth
             consists
             in
             Speech
             ,
             and
             not
             in
             the
             Things
             spoken
             of
             ;
             and
             though
             True
             be
             sometimes
             opposed
             to
             to
             Apparent
             or
             Feigned
             ,
             yet
             it
             is
             alwayes
             to
             be
             referred
             to
             the
             Truth
             of
             Proposition
             ;
             for
             the
             Image
             of
             a
             Man
             in
             a
             Glasse
             ,
             or
             a
             Ghost
             ,
             is
             therefore
             denyed
             to
             be
             a
             very
             Man
             ,
             because
             this
             Proposition
             ,
             
               A
               Ghost
               is
               a
               man
            
             ,
             is
             not
             True
             ;
             for
             it
             cannot
             be
             denied
             but
             that
             a
             Ghost
             is
             a
             very
             Ghost
             .
             And
             therefore
             Truth
             or
             Verity
             is
             not
             any
             Affection
             of
             the
             Thing
             ,
             but
             of
             the
             Proposition
             concerning
             it
             .
             As
             for
             that
             which
             the
             Writers
             of
             Metaphysicks
             say
             ,
             that
             
               A
               thing
               ,
               One
               thing
            
             ,
             and
             
               a
               Very
               thing
            
             ,
             are
             equivalent
             to
             one
             another
             ,
             it
             is
             but
             trifling
             and
             childish
             ;
             for
             who
             does
             not
             know
             ,
             that
             
               A
               Man
               ,
               One
               Man
            
             ,
             and
             a
             
               Very
               Man
            
             ,
             signifie
             the
             same
             .
          
           
           
             8
             And
             from
             hence
             it
             is
             evident
             ,
             that
             Truth
             and
             Falsity
             have
             no
             place
             but
             amongst
             such
             Living
             Creatures
             as
             use
             Speech
             .
             For
             though
             some
             brute
             Creatures
             ,
             looking
             upon
             the
             Image
             of
             a
             Man
             in
             a
             Glasse
             ,
             may
             be
             affected
             with
             it
             as
             if
             it
             were
             the
             Man
             himselfe
             ,
             and
             for
             this
             reason
             feare
             it
             or
             fawne
             upon
             it
             in
             vain
             ;
             Yet
             they
             doe
             not
             apprehend
             it
             as
             True
             or
             False
             ,
             but
             onely
             as
             Like
             ;
             and
             in
             this
             they
             are
             not
             deceived
             .
             Wherefore
             ,
             as
             men
             owe
             all
             their
             True
             Ratiocination
             to
             the
             right
             understanding
             of
             Speech
             ;
             So
             also
             they
             owe
             their
             Errors
             to
             the
             misunderstanding
             of
             the
             same
             ;
             and
             as
             all
             the
             Ornaments
             of
             Philosophy
             proceed
             onely
             from
             Man
             ,
             so
             from
             Man
             also
             is
             derived
             the
             ugly
             absurdity
             of
             False
             opinions
             .
             For
             Speech
             has
             something
             in
             it
             like
             to
             a
             Spiders
             Web
             (
             as
             it
             was
             said
             of
             old
             of
             Solons
             laws
             ;
             )
             for
             by
             contexture
             of
             words
             tender
             and
             delicate
             Wits
             are
             insnared
             and
             stopt
             ;
             but
             strong
             Wits
             breake
             easily
             through
             them
             .
          
           
             From
             hence
             also
             this
             may
             be
             deduced
             ,
             that
             the
             first
             Truths
             were
             arbitrarily
             made
             by
             those
             that
             first
             of
             all
             imposed
             Names
             upon
             Things
             ,
             or
             received
             them
             from
             the
             imposition
             of
             others
             .
             For
             it
             is
             true
             (
             for
             example
             )
             that
             
               Man
               is
               a
               Living
               Creature
            
             ;
             but
             it
             is
             for
             this
             reason
             ,
             that
             it
             pleased
             men
             to
             impose
             both
             those
             Names
             on
             the
             same
             thing
             .
          
           
             9
             Fourthly
             ,
             Propositions
             are
             distinguished
             into
             Primary
             and
             
               not
               Primary
               .
               Primary
            
             is
             that
             wherein
             the
             Subject
             is
             explicated
             by
             a
             Praedicate
             of
             many
             Names
             ,
             as
             
               Man
               ,
               is
               a
               Body
               Animated
               Rationall
            
             ;
             for
             that
             which
             is
             comprehended
             in
             the
             Name
             Man
             is
             more
             largely
             expressed
             in
             the
             Names
             
               Body
               ,
               Animated
            
             ,
             and
             Rationall
             joyned
             together
             ;
             and
             it
             is
             called
             Primary
             ,
             because
             it
             is
             first
             in
             Ratiocination
             ;
             for
             nothing
             can
             be
             proved
             ,
             without
             understanding
             first
             the
             name
             of
             the
             Thing
             in
             question
             .
             Now
             
               Primary
               Propositions
            
             are
             nothing
             but
             Definitions
             ,
             or
             parts
             of
             Definitions
             ,
             and
             these
             onely
             are
             the
             principles
             of
             Demonstration
             ,
             being
             Truths
             constituted
             arbitrarily
             by
             the
             Inventors
             of
             Speech
             ,
             and
             therefore
             not
             to
             be
             demonstrated
             .
             To
             these
             Propositions
             ,
             some
             have
             added
             others
             ,
             which
             they
             call
             Primary
             and
             Principles
             ;
             namely
             Axiomes
             and
             
               Common
               Notions
            
             ;
             which
             (
             though
             they
             be
             so
             evident
             that
             they
             need
             no
             proofe
             ,
             yet
             )
             because
             they
             may
             be
             proved
             ,
             are
             not
             truly
             Principles
             ;
             &
             the
             lesse
             to
             be
             received
             for
             such
             ,
             in
             regard
             Propositions
             not
             intelligible
             ,
             and
             sometimes
             manifestly
             false
             ,
             are
             thrust
             on
             us
             under
             the
             Name
             of
             Principles
             
             by
             the
             clamour
             of
             Men
             ,
             who
             obtrude
             for
             evident
             to
             others
             all
             that
             they
             themselves
             thinke
             True.
             Also
             certaine
             Petitions
             are
             commonly
             received
             into
             the
             number
             of
             Principles
             ;
             as
             for
             example
             ,
             
               That
               a
               streight
               line
               may
               be
               drawne
               between
               two
               points
               ,
            
             and
             other
             Petitions
             of
             the
             Writers
             of
             Geometry
             ;
             and
             these
             are
             indeed
             the
             Principles
             of
             Art
             or
             Construction
             ,
             but
             not
             of
             Science
             and
             Demonstration
             .
          
           
             10
             Fiftly
             ,
             Propositions
             are
             distinguished
             into
             Necessary
             ,
             that
             is
             ,
             necessarily
             True
             ;
             and
             True
             ,
             but
             not
             necessarily
             ,
             which
             they
             call
             Contingent
             .
             A
             
               Necessary
               Proposition
            
             is
             when
             nothing
             can
             at
             any
             time
             be
             conceived
             or
             feigned
             ,
             whereof
             the
             Subject
             is
             the
             Name
             ,
             but
             the
             Praedicate
             also
             is
             the
             Name
             of
             the
             same
             thing
             ;
             as
             
               Man
               is
               a
               Living-Creature
            
             is
             a
             necessary
             Proposition
             ,
             because
             at
             what
             time
             soever
             we
             suppose
             the
             name
             Man
             agrees
             with
             any
             thing
             ,
             at
             that
             time
             the
             name
             Living-Creature
             also
             agrees
             with
             the
             same
             .
             But
             a
             
               Contingent
               Proposition
            
             is
             that
             which
             at
             one
             time
             may
             be
             true
             ,
             at
             another
             time
             false
             ;
             as
             
               Every
               Crow
               is
               Black
            
             ;
             which
             may
             perhaps
             be
             true
             now
             ,
             but
             false
             hereafter
             .
             Again
             ,
             in
             every
             
               Necessary
               Proposition
            
             ,
             the
             Praedicate
             is
             either
             equivalent
             to
             the
             Subject
             ,
             as
             in
             this
             ,
             
               Man
               is
               a
               Rational
               Living-Creature
            
             ;
             or
             part
             of
             an
             equivalent
             Name
             ,
             as
             in
             this
             ,
             
               Man
               is
               a
               Living-Creature
            
             ,
             for
             the
             Name
             
               Rational
               Living-Creature
            
             ,
             or
             Man
             ,
             is
             compounded
             of
             these
             two
             ,
             Rational
             and
             Living-Creature
             .
             But
             in
             a
             
               Contingent
               Proposition
            
             this
             cannot
             be
             ;
             for
             though
             this
             were
             true
             ,
             
               Every
               Man
               is
               a
               Lyar
            
             ,
             yet
             because
             the
             word
             Lyar
             is
             no
             part
             of
             a
             Compounded
             name
             equivalent
             to
             the
             Name
             Man
             ,
             that
             Proposition
             is
             not
             to
             be
             called
             Necessary
             ,
             but
             Contingent
             ,
             though
             it
             should
             happen
             to
             be
             true
             alwayes
             .
             And
             therefore
             those
             Propositions
             only
             are
             Necessary
             ,
             which
             are
             of
             Sempiternal
             Truth
             ,
             that
             is
             ,
             true
             at
             all
             times
             .
             From
             hence
             also
             it
             is
             manifest
             ,
             that
             Truth
             adheres
             not
             to
             Things
             ,
             but
             to
             Speech
             onely
             ;
             for
             some
             Truths
             are
             eternal
             ;
             for
             it
             will
             be
             eternally
             true
             ,
             
               if
               Man
               ,
               then
               Living-Creature
            
             ;
             but
             that
             any
             Man
             ,
             or
             Living-Creature
             should
             exist
             eternally
             ,
             is
             not
             necessary
             .
          
           
             11
             A
             sixth
             Distinction
             of
             Propositions
             is
             into
             Categorical
             and
             Hypotheticall
             .
             A
             
               Categoricall
               Proposition
            
             is
             that
             which
             is
             simply
             or
             absolutely
             pronounced
             ,
             as
             
               Every
               Man
               is
               a
               Living
               Creature
               ,
               No
               Man
               is
               a
               Tree
            
             ;
             and
             Hypotheticall
             is
             that
             which
             is
             pronounced
             conditionally
             ,
             as
             ,
             
               If
               any
               thing
               be
               a
               Man
               the
               same
               is
               also
               a
               Living
               Creature
               ,
               If
               any
               thing
               be
               a
               Man
               the
               same
               is
               also
               Not-a-stone
               .
            
          
           
             
               A
               Categoricall
               Proposition
            
             ,
             and
             an
             Hypothetical
             answering
             it
             ,
             doe
             
             both
             signifie
             the
             same
             ,
             if
             the
             Propositions
             be
             Necessary
             ;
             but
             not
             if
             they
             be
             Contingent
             .
             For
             example
             ,
             if
             this
             ,
             
               Every
               Man
               is
               a
               Living-Creature
            
             ,
             be
             true
             ,
             this
             also
             will
             be
             true
             ,
             
               If
               any
               thing
               be
               a
               Man
               ,
               the
               same
               is
               also
               a
               Living-Creature
            
             ;
             but
             in
             Contingent
             Propositions
             ,
             though
             this
             be
             true
             ,
             
               Every
               Crow
               is
               Black
            
             ;
             yet
             this
             ,
             
               If
               any
               thing
               be
               a
               Crow
               the
               same
               is
               Black
               ,
            
             is
             false
             .
             But
             an
             
               Hypotheticall
               Proposition
            
             is
             then
             rightly
             said
             to
             be
             true
             ,
             when
             the
             Consequence
             is
             true
             ;
             as
             
               Every
               Man
               is
               a
               Living-Creature
            
             is
             rightly
             said
             to
             be
             a
             true
             Proposition
             ,
             because
             of
             whatsoever
             it
             is
             truly
             said
             
               That
               is
               a
               Man
            
             ,
             it
             cannot
             but
             be
             truly
             said
             also
             
               The
               same
               is
               a
               Living-Creature
            
             .
             And
             therefore
             whensoever
             an
             
               Hypotheticall
               Proposition
            
             is
             true
             ,
             the
             Categoricall
             answering
             it
             is
             not
             only
             true
             ,
             but
             also
             necessary
             ;
             which
             I
             thought
             worth
             the
             noting
             ,
             as
             an
             argument
             ,
             that
             Philosophers
             may
             in
             most
             things
             reason
             more
             solidly
             by
             Hypotheticall
             then
             
               Categoricall
               Propositions
            
             .
          
           
             12
             But
             seeing
             every
             Proposition
             may
             be
             &
             uses
             to
             be
             pronounced
             and
             written
             in
             many
             formes
             ,
             and
             we
             are
             obliged
             to
             speake
             in
             the
             same
             manner
             as
             most
             men
             speake
             ;
             yet
             they
             that
             learne
             Philosophy
             from
             Masters
             ,
             had
             need
             to
             take
             heed
             they
             be
             not
             deceived
             by
             the
             Variety
             of
             expressions
             .
             And
             therefore
             whensoever
             they
             meet
             with
             any
             obscure
             Proposition
             ,
             they
             ought
             to
             reduce
             it
             to
             its
             most
             Simple
             and
             Categoricall
             forme
             ;
             in
             which
             the
             Copulative
             word
             Is
             must
             be
             expressed
             by
             it selfe
             ,
             and
             not
             mingled
             in
             any
             manner
             either
             with
             the
             Subject
             or
             Praedicate
             ,
             both
             which
             must
             be
             separated
             and
             cleerly
             distinguished
             one
             from
             another
             .
             For
             example
             ,
             if
             this
             Proposition
             
               Man
               can
               not
               sinne
            
             ,
             be
             compared
             with
             this
             ,
             
               Man
               cannot
               sinne
            
             ,
             their
             difference
             will
             easily
             appeare
             if
             they
             be
             reduced
             to
             these
             ,
             
               Man
               is
               able
               not
               to
               sinne
               ,
            
             and
             
               Man
               is
               not
               able
               to
               sinne
               ,
            
             where
             the
             Praedicates
             are
             manifestly
             different
             .
             But
             they
             ought
             to
             doe
             this
             silently
             by
             themselves
             ,
             or
             betwixt
             them
             and
             their
             Masters
             onely
             ;
             for
             it
             will
             be
             thought
             both
             ridiculous
             and
             absurd
             ,
             for
             a
             man
             to
             use
             such
             language
             publiquely
             .
             Being
             therefore
             to
             speake
             of
             
               Equipollent
               Propositions
            
             ,
             I
             put
             in
             the
             first
             place
             all
             those
             for
             Equipollent
             ,
             that
             may
             be
             reduced
             p●rely
             to
             one
             and
             the
             same
             Categoricall
             Proposition
             .
          
           
             13
             Secondly
             ,
             that
             which
             is
             Categoricall
             and
             Necessary
             ,
             is
             Equipollent
             to
             its
             Hypotheticall
             Proposition
             ;
             as
             this
             Categorical
             ,
             
               A
               Right-lined
               Triangle
               has
               its
               three
               Angles
               equal
               to
               two
               Right
               angles
               ,
            
             to
             this
             Hypotheticall
             ,
             
               If
               any
               Figure
               be
               a
               Right-lined
               Triangle
               ,
               the
               three
               Angles
               of
               it
               are
               equal
               to
               two
               Right
               Angles
               .
            
          
           
           
             14
             Also
             ,
             any
             two
             Universall
             Propositions
             ,
             of
             which
             the
             Termes
             of
             the
             one
             (
             that
             is
             ,
             the
             Subject
             and
             Praedicate
             )
             are
             Contradictory
             to
             the
             Termes
             of
             the
             other
             ,
             and
             their
             Order
             inverted
             ,
             as
             these
             ,
             
               Every
               man
               is
               a
               Living
               Creature
               ,
            
             and
             
               Every
               thing
               that
               is
               not
               a
               Living
               Creature
               ,
               is
               not
               a
               Man
               ,
            
             are
             Equipollent
             .
             For
             seeing
             
               Every
               Man
               is
               a
               Living
               Creature
            
             is
             a
             true
             Proposition
             ,
             the
             Name
             
               Living
               Creature
            
             containes
             the
             Name
             Man
             ;
             but
             they
             are
             both
             Positive
             Names
             ;
             and
             therefore
             (
             by
             the
             last
             Article
             of
             the
             praecedent
             Chapter
             )
             the
             Negative
             Name
             
               Not
               Living
               Creature
            
             containes
             the
             Negative
             Name
             
               Not
               Man
            
             ;
             Wherefore
             
               Everything
               that
               is
               not
               a
               Living
               Creature
               is
               not
               a
               Man
            
             is
             a
             true
             Proposition
             .
             Likewise
             these
             ,
             
               No
               Man
               is
               a
               Tree
               ,
               No
               Tree
               is
               a
               Man
               ,
            
             are
             Equipollent
             .
             For
             if
             it
             be
             true
             that
             Tree
             is
             not
             the
             Name
             of
             any
             Man
             ,
             then
             no
             one
             thing
             can
             be
             signified
             by
             the
             two
             Names
             Man
             and
             Tree
             ,
             wherefore
             
               No
               Tree
               is
               a
               Man
            
             is
             a
             true
             Proposition
             .
             Also
             to
             this
             ,
             
               Whatsoever
               is
               not
               a
               Living
               Creature
               is
               not
               a
               Man
               ,
            
             where
             both
             the
             Termes
             are
             Negative
             ,
             this
             other
             Proposition
             is
             
               Equipollent
               ,
               Onely
               a
               Living
               Creature
               is
               a
               Man.
               
            
          
           
             15
             Fourthly
             ,
             Negative
             Propositions
             ,
             whether
             the
             particle
             of
             Negation
             be
             set
             after
             the
             Copula
             as
             some
             Nations
             doe
             ,
             or
             before
             it
             ,
             as
             it
             is
             in
             Latine
             and
             Greeke
             ,
             if
             the
             Termes
             be
             the
             same
             ,
             are
             equipollent
             ;
             as
             for
             example
             ,
             
               Man
               is
               not-a-Tree
            
             and
             
               Man
               is
               not-a-Tree
            
             ,
             are
             equipollent
             ,
             though
             Aristotle
             deny
             it
             .
             Also
             these
             ,
             
               Every
               Man
               is
               not
               a
               Tree
               ,
            
             and
             
               No
               Man
               is
               a
               Tree
            
             are
             equipollent
             ,
             and
             that
             so
             manifestly
             ,
             as
             it
             needs
             not
             be
             demonstrated
             .
          
           
             16
             Lastly
             ,
             all
             Particular
             Propositions
             that
             have
             their
             Termes
             inverted
             ,
             as
             these
             ,
             
               Some
               Man
               is
               Blind
               ,
               Some
               Blind
               thing
               is
               a
               Man
               ,
            
             are
             equipollent
             ;
             for
             either
             of
             the
             two
             Names
             ,
             is
             the
             Name
             of
             some
             one
             and
             the
             same
             Man
             ;
             and
             therefore
             in
             which
             soever
             of
             the
             two
             Orders
             they
             be
             connected
             ,
             they
             signifie
             the
             same
             Truth
             .
          
           
             17
             Of
             Propositions
             that
             have
             the
             same
             Termes
             ,
             and
             are
             placed
             in
             the
             same
             Order
             ,
             but
             varied
             either
             by
             Quantity
             or
             Quality
             ,
             some
             are
             called
             Subaltern
             ,
             others
             Contrary
             ,
             others
             Subcontrary
             ,
             and
             others
             Contradictory
             .
          
           
             Subaltern
             ,
             are
             Universal
             and
             Particular
             Propositions
             of
             the
             same
             Quality
             ;
             as
             ,
             
               Every
               Man
               is
               a
               Living
               Creature
               ,
               Some
               Man
               is
               a
               Living
               Creature
            
             ;
             or
             ,
             
               No
               Man
               is
               Wise
               ,
               Some
               Man
               is
               not
               Wise.
            
             Of
             these
             i●
             the
             Universal
             be
             true
             ,
             the
             Particular
             will
             be
             true
             also
             .
          
           
             Contrary
             ,
             are
             Universal
             Propositions
             of
             different
             Quality
             ;
             as
             
             
               Every
               Man
               is
               happy
               ,
               No
               Man
               is
               happy
               .
            
             And
             of
             these
             if
             one
             be
             true
             the
             other
             is
             false
             ;
             also
             they
             may
             both
             be
             false
             ,
             as
             in
             the
             example
             given
             .
          
           
             Subcontrary
             ,
             are
             Particular
             Propositions
             of
             different
             Quality
             ;
             
               as
               Some
               Man
               is
               learned
               ,
               Some
               Man
               is
               not
               learned
            
             ;
             which
             cannot
             be
             both
             false
             ,
             but
             they
             may
             be
             both
             true
             .
          
           
             Contradictory
             are
             those
             that
             differ
             both
             in
             Quantity
             and
             Quality
             ;
             as
             
               Every
               Man
               is
               a
               Living
               Creature
               ,
               Some
               Man
               is
               not
               a
               Living
               Creature
            
             ;
             which
             can
             neither
             be
             both
             true
             ,
             nor
             both
             false
             .
          
           
             18
             A
             Proposition
             is
             said
             to
             follow
             from
             two
             other
             Propositions
             ,
             when
             these
             being
             granted
             to
             be
             true
             ,
             it
             cannot
             be
             denyed
             but
             the
             other
             is
             true
             also
             .
             For
             example
             ,
             let
             these
             two
             Propositions
             ,
             
               Every
               Man
               is
               a
               Living
               Creature
               ,
            
             and
             
               Every
               Living
               Creature
               is
               a
               Body
               ,
            
             be
             supposed
             true
             ,
             that
             is
             ,
             that
             Body
             is
             the
             Name
             of
             
               Every
               Living
               Creature
               ,
               and
               Living
               Creature
               ,
            
             the
             Name
             of
             
               Every
               Man.
            
             Seeing
             therefore
             ,
             if
             these
             be
             understood
             to
             be
             true
             ,
             it
             cannot
             be
             understood
             that
             Body
             is
             not
             the
             name
             of
             
               Every
               man
            
             ,
             that
             is
             ,
             that
             
               Every
               Man
               is
               a
               Body
            
             ,
             is
             false
             ,
             this
             Proposition
             will
             be
             said
             to
             follow
             from
             those
             two
             ,
             or
             to
             be
             necessarily
             inferred
             from
             them
             .
          
           
             19
             That
             a
             true
             Proposition
             may
             follow
             from
             false
             Propositions
             ,
             may
             happen
             sometimes
             ;
             but
             false
             from
             true
             ,
             never
             .
             For
             if
             these
             ,
             
               Every
               Man
               is
               a
               Stone
            
             ,
             and
             
               Every
               Stone
               is
               a
               Living
               Creature
               ,
            
             (
             which
             are
             both
             false
             )
             be
             granted
             to
             be
             true
             ,
             it
             is
             granted
             also
             that
             
               Living
               Creature
            
             is
             the
             name
             of
             
               Every
               Stone
            
             ,
             and
             Stone
             of
             
               Every
               Man
            
             ,
             that
             is
             ,
             that
             
               Living
               Creature
            
             is
             the
             Name
             of
             
               Every
               Man
            
             ;
             that
             is
             to
             say
             ,
             this
             Proposition
             
               Every
               Man
               is
               a
               Living
               Creature
               ,
            
             is
             true
             ,
             as
             it
             is
             indeed
             true
             .
             Wherefore
             a
             true
             Proposition
             may
             sometimes
             follow
             from
             false
             .
             But
             if
             any
             two
             Propositions
             be
             true
             ,
             a
             false
             one
             can
             never
             follow
             from
             them
             .
             For
             if
             true
             follow
             from
             false
             ,
             for
             this
             reason
             onely
             ,
             that
             the
             false
             are
             granted
             to
             be
             true
             ,
             then
             truth
             from
             two
             truths
             granted
             will
             follow
             in
             the
             same
             manner
             .
          
           
             20
             Now
             seeing
             none
             but
             a
             true
             Proposition
             will
             follow
             from
             true
             ,
             and
             that
             the
             understanding
             of
             two
             Propositions
             to
             be
             true
             ,
             is
             the
             cause
             of
             understanding
             that
             also
             to
             be
             true
             which
             is
             deduced
             from
             them
             ;
             the
             two
             Antecedent
             Propositions
             are
             commonly
             called
             the
             Causes
             of
             the
             inferred
             Proposition
             ,
             or
             Conclusion
             .
             And
             from
             hence
             it
             is
             that
             Logicians
             say
             ,
             the
             Premisses
             are
             Causes
             of
             the
             Conclusion
             ;
             which
             may
             passe
             ,
             though
             it
             be
             not
             properly
             spoken
             ;
             for
             
             though
             Understanding
             be
             the
             cause
             of
             Understanding
             ,
             yet
             Speech
             is
             not
             the
             cause
             of
             Speech
             .
             But
             when
             they
             say
             ,
             the
             Cause
             of
             the
             Properties
             of
             any
             thing
             ,
             is
             the
             Thing
             it self
             ,
             they
             speake
             absurdly
             .
             Eor
             example
             ,
             if
             a
             Figure
             be
             propounded
             which
             is
             Triangular
             ;
             Seeing
             every
             Triangle
             has
             all
             its
             angles
             together
             equal
             to
             two
             right
             angles
             ,
             from
             whence
             it
             follows
             that
             all
             the
             angles
             of
             that
             Figure
             are
             equal
             to
             two
             right
             angles
             ,
             they
             say
             for
             this
             reason
             ,
             that
             that
             Figure
             is
             the
             Cause
             of
             that
             Equality
             .
             But
             seeing
             the
             Figure
             does
             not
             it self
             make
             its
             angles
             ,
             and
             therefore
             cannot
             be
             said
             to
             be
             the
             Efficient-Cause
             ,
             they
             call
             it
             the
             Formall-Cause
             ;
             whereas
             in
             deed
             it
             is
             no
             Cause
             at
             all
             ;
             nor
             does
             the
             Property
             of
             any
             Figure
             follow
             the
             Figure
             ,
             but
             has
             its
             Being
             at
             the
             same
             time
             with
             it
             ;
             only
             the
             Knowledge
             of
             the
             Figure
             goes
             before
             the
             Knowledge
             of
             the
             Properties
             ;
             and
             one
             Knowledge
             is
             truly
             the
             Cause
             of
             another
             Knowledge
             ,
             namely
             the
             Efficient-Cause
             .
          
           
             And
             thus
             much
             concerning
             Proposition
             ;
             which
             in
             the
             Progress
             of
             Philosophy
             is
             the
             first
             Step
             ,
             like
             the
             moving
             towards
             of
             one
             Foot.
             By
             the
             due
             addition
             of
             another
             Step
             I
             shall
             proceed
             to
             Syllogisme
             ,
             and
             make
             a
             compleat
             Pace
             .
             Of
             which
             in
             the
             next
             Chapter
             .
          
        
         
           
           
             
               CHAP.
               IV.
            
             Of
             Syllogisme
             .
          
           
             
               
                 1
                 The
                 Definition
                 of
                 Syllogisme
                 .
              
               
                 2
                 In
                 a
                 Syllogisme
                 there
                 are
                 but
                 three
                 Termes
                 .
              
               
                 3
                 Major
                 ,
                 Minor
                 and
                 Middle
                 Term
                 ;
                 also
                 Major
                 and
                 Minor
                 Proposition
                 ,
                 what
                 they
                 are
                 .
              
               
                 4
                 The
                 Middle
                 Terme
                 in
                 every
                 Syllogisme
                 ought
                 to
                 be
                 determined
                 in
                 both
                 the
                 Propositions
                 to
                 one
                 and
                 the
                 same
                 thing
                 .
              
               
                 5
                 From
                 two
                 Particular
                 Propositions
                 nothing
                 can
                 be
                 concluded
                 .
              
               
                 6
                 A
                 Syllogisme
                 is
                 the
                 Collection
                 of
                 two
                 Propositions
                 into
                 one
                 Summe
                 .
              
               
                 7
                 The
                 Figure
                 of
                 a
                 Syllogisme
                 what
                 it
                 is
                 .
              
               
                 8
                 What
                 is
                 in
                 the
                 mind
                 answering
                 to
                 a
                 Syllogisme
                 .
              
               
                 9
                 The
                 first
                 Indirect
                 Figure
                 how
                 it
                 is
                 made
                 .
              
               
                 10
                 The
                 second
                 Indirect
                 Figure
                 how
                 made
                 .
              
               
                 11
                 How
                 the
                 third
                 Indirect
                 Figure
                 is
                 made
              
               
                 12
                 There
                 are
                 many
                 Moods
                 in
                 every
                 Figure
                 ,
                 but
                 most
                 of
                 them
                 Uselesse
                 in
                 Philosophy
                 .
              
               
                 13
                 An
                 Hypotheticall
                 Syllogisme
                 when
                 equipollent
                 to
                 a
                 Categoricall
                 .
              
            
          
           
             1.
             
             A
             Speech
             consisting
             of
             three
             Propositions
             ,
             from
             two
             of
             which
             the
             third
             followes
             ,
             is
             called
             a
             SYLLOGISME
             ;
             and
             that
             which
             followes
             is
             called
             the
             Conclusion
             ;
             the
             other
             two
             Premisses
             .
             For
             example
             this
             Speech
             ,
             
               Every
               man
               is
               a
               Living
               Creature
               ,
               Every
               Living
               Creature
               is
               a
               Body
               ,
            
             therefore
             ,
             
               Every
               Man
               is
               a
               Body
            
             ,
             is
             a
             Syllogisme
             ,
             because
             the
             third
             Proposition
             follows
             from
             the
             two
             first
             ;
             that
             is
             ,
             if
             those
             be
             granted
             to
             be
             true
             ,
             this
             must
             also
             be
             granted
             to
             be
             true
             .
          
           
             2
             From
             two
             Propositions
             which
             have
             not
             one
             Terme
             common
             ,
             no
             Conclusion
             can
             follow
             ;
             and
             therefore
             no
             Syllogisme
             can
             be
             made
             of
             them
             .
             For
             let
             any
             two
             Premisses
             ,
             
               A
               man
               is
               a
               Living
               Creature
               ,
               A
               Tree
               is
               a
               Plant
               ,
            
             be
             both
             of
             them
             true
             ,
             yet
             because
             it
             cannot
             be
             collected
             from
             them
             that
             Plant
             is
             the
             Name
             of
             a
             Man
             ,
             or
             Man
             the
             Name
             of
             a
             Plant
             ,
             it
             is
             not
             necessary
             that
             this
             Conclusion
             ,
             
               A
               Man
               is
               a
               Plant
            
             should
             be
             true
             .
             Corollary
             ,
             Therefore
             in
             the
             Premisses
             of
             a
             Syllogisme
             there
             can
             be
             but
             three
             Termes
             .
          
           
             Besides
             there
             can
             be
             no
             Terme
             in
             the
             Conclusion
             ,
             which
             was
             not
             in
             the
             Premisses
             .
             For
             let
             any
             two
             Premisses
             be
             ,
             
               A
               Man
               is
               a
               Living
               Creature
               ,
               A
               Living
               Creature
               is
               a
               Body
               ,
            
             yet
             if
             any
             other
             Terme
             be
             put
             
             in
             the
             Conclusion
             ,
             as
             
               Man
               is
               two
               footed
            
             ;
             though
             it
             be
             true
             ,
             it
             cannot
             follow
             from
             the
             Premisses
             ,
             because
             from
             them
             it
             cannot
             be
             collected
             ,
             that
             the
             Name
             
               Two
               footed
            
             belongs
             to
             a
             Man
             ;
             and
             therefore
             againe
             ,
             In
             every
             Syllogisme
             there
             can
             be
             but
             three
             Termes
             .
          
           
             3
             Of
             these
             Termes
             ,
             that
             which
             is
             the
             Predicate
             in
             the
             Conclusion
             ,
             is
             commonly
             called
             the
             Major
             ;
             that
             which
             is
             the
             Subject
             in
             the
             Conclusion
             ,
             the
             Minor
             ,
             and
             the
             other
             is
             the
             
               Middle
               Term
            
             ;
             as
             in
             this
             Syllogisme
             ,
             
               A
               Man
               is
               a
               Living
               Creature
               ,
               A
               Living
               Creature
               is
               a
               Body
               ,
            
             therefore
             ,
             
               A
               Man
               is
               a
               Body
               ,
               Body
            
             is
             the
             
               Major
               ,
               Man
            
             the
             Minor
             and
             
               Living
               Creature
            
             the
             
               Middle
               Term.
            
             Also
             of
             the
             Premisses
             ,
             that
             in
             which
             the
             
               Major
               Terme
            
             is
             found
             ,
             is
             called
             the
             
               Major
               Proposition
            
             ,
             and
             that
             which
             has
             the
             
               Minor
               Term
            
             the
             
               Minor
               Proposition
            
             .
          
           
             4
             If
             the
             Middle
             Terme
             be
             not
             in
             both
             the
             Premisses
             determined
             to
             one
             and
             the
             same
             singular
             thing
             ,
             no
             Conclusion
             will
             follow
             ,
             nor
             Syllogisme
             be
             made
             .
             For
             let
             the
             
               Minor
               Terme
            
             be
             Man
             ,
             the
             
               Middle
               Terme
               Living
               Creature
            
             ,
             and
             the
             
               Major
               Term
               Lyon
            
             ;
             and
             let
             the
             Premisses
             be
             
               Man
               is
               a
               Living
               Creature
               ,
               Some
               Living
               Creature
               is
               a
               Lyon
               ,
            
             yet
             it
             will
             not
             follow
             that
             ,
             Every
             or
             
               Any
               Man
               is
               a
               Lyon.
            
             By
             which
             it
             is
             manifest
             ,
             that
             in
             every
             Syllogisme
             ,
             that
             Proposition
             which
             has
             the
             
               Middle
               Terme
            
             for
             its
             Subject
             ,
             ought
             to
             be
             either
             Universal
             or
             Singular
             ,
             but
             not
             Particular
             nor
             Indefinite
             .
             For
             example
             ,
             this
             Syllogism
             ,
             
               Every
               man
               is
               a
               Living
               Creature
               ,
               some
               Living
               Creature
               is
               four-footed
               ,
            
             therefore
             
               some
               Man
               is
               four-footed
            
             ,
             is
             therefore
             faulty
             ,
             because
             the
             
               Middle
               Term
               ,
               Living
               Creature
            
             ,
             is
             in
             the
             first
             of
             the
             Premisses
             determined
             onely
             to
             Man
             ,
             for
             there
             the
             Name
             of
             
               Living
               Creature
            
             is
             given
             to
             
               Man
               onely
            
             ,
             but
             in
             the
             later
             Premisse
             it
             may
             be
             understood
             of
             some
             other
             Living
             Creature
             besides
             Man.
             But
             if
             the
             later
             Premisse
             had
             been
             Universall
             ,
             as
             here
             ,
             
               Every
               Man
               is
               a
               Living
               Creature
               ,
               Every
               Living
               Creature
               is
               a
               Body
               ,
            
             therefore
             
               Every
               Man
               is
               a
               Body
            
             ,
             the
             Syllogisme
             had
             been
             true
             ;
             for
             it
             would
             have
             followed
             that
             Body
             had
             been
             the
             Name
             of
             
               Every
               Living
               Creature
            
             ,
             that
             is
             of
             Man
             ,
             that
             is
             to
             say
             ,
             the
             Conclusion
             
               Every
               Man
               is
               a
               Body
            
             had
             been
             true
             .
             Likewise
             when
             the
             
               Middle
               Term
            
             is
             a
             Singular
             Name
             ,
             a
             Syllogisme
             may
             be
             made
             ,
             I
             say
             a
             true
             Syllogisme
             ,
             though
             uselesse
             in
             Philosophy
             ,
             as
             this
             ,
             
               Some
               Man
               is
               Socrates
               ,
               Socrates
               is
               a
               Philosopher
               ,
            
             therefore
             
               Some
               Man
               is
               a
               Philosopher
            
             ;
             for
             the
             Premisses
             being
             granted
             ,
             the
             Conclusion
             cannot
             be
             denyed
             .
          
           
             5
             And
             therefore
             of
             two
             Premisses
             ,
             in
             both
             which
             the
             Middle
             
             Terme
             is
             Particular
             ,
             a
             Syllogisme
             cannot
             be
             made
             ;
             for
             whether
             the
             
               Middle
               Terme
            
             be
             the
             Subject
             in
             both
             the
             Premisses
             ,
             or
             the
             Predicate
             in
             both
             ,
             or
             the
             Subject
             in
             one
             and
             the
             Predicate
             in
             the
             other
             ,
             it
             will
             not
             be
             necessarily
             determined
             to
             the
             same
             thing
             .
             For
             let
             the
             Premisses
             be
             ,
             
               
                 Some
                 Man
                 is
                 blind
              
               
                 Some
                 Man
                 is
                 learned
              
            
             In
             both
             which
             the
             
               Middle
               Term
            
             is
             the
             Subject
             ,
          
           
             It
             will
             not
             follow
             ,
             that
             Blind
             is
             the
             Name
             of
             any
             
               learned
               Man
            
             ,
             or
             Learned
             the
             Name
             of
             any
             
               Blind
               Man
            
             ,
             seeing
             the
             Name
             Learned
             does
             not
             containe
             the
             Name
             Blind
             ,
             nor
             this
             that
             ;
             and
             therefore
             it
             is
             not
             necessary
             that
             both
             should
             be
             Names
             of
             the
             same
             Man.
             So
             from
             these
             Premisses
             .
             
               
                 Every
                 Man
                 is
                 a
                 Living
                 Creature
              
               
                 Every
                 Horse
                 is
                 a
                 Living
                 Creature
              
            
             In
             both
             which
             the
             
               Middle
               Terme
            
             is
             the
             Predicate
             ,
          
           
             Nothing
             will
             follow
             .
             For
             seeing
             
               Living
               Creature
            
             is
             in
             both
             of
             them
             Indefinite
             ,
             which
             is
             equivalent
             to
             Particular
             ,
             and
             that
             Man
             may
             be
             one
             kind
             of
             
               Living
               Creature
            
             ,
             and
             Horse
             another
             kind
             ,
             it
             is
             not
             necessary
             that
             Man
             should
             the
             be
             Name
             of
             Horse
             ,
             or
             Horse
             of
             Man.
             Or
             if
             the
             Premisses
             be
             ,
             
               
                 Every
                 Man
                 is
                 a
                 Living
                 Creature
              
               
                 Some
                 Living
                 Creature
                 is
                 four-footed
              
            
             In
             one
             of
             which
             the
             
               Middle
               Terme
            
             is
             the
             Subject
             ,
             and
             in
             the
             other
             the
             Predicate
             ,
          
           
             The
             Conclusion
             will
             follow
             ,
             because
             the
             Name
             
               Living
               Creature
            
             being
             not
             determined
             ,
             it
             may
             in
             one
             of
             them
             be
             understood
             of
             Man
             in
             the
             other
             of
             Not-Man
             .
          
           
             6
             Now
             it
             is
             manifest
             from
             what
             has
             been
             said
             ,
             that
             a
             Syllogisme
             is
             nothing
             but
             a
             Collection
             of
             the
             summe
             of
             two
             Propositions
             ,
             joyned
             together
             by
             a
             common
             Term
             ,
             which
             is
             called
             the
             
               Middle
               Terme
            
             .
             And
             as
             Proposition
             is
             the
             Addition
             of
             two
             Names
             ,
             so
             Syllogisme
             is
             the
             adding
             together
             of
             three
             .
          
           
             7
             Syllogismes
             are
             usually
             distinguished
             according
             to
             their
             diversity
             of
             Figures
             ,
             that
             is
             ,
             by
             the
             diverse
             position
             of
             the
             Middle
             Term.
             And
             againe
             in
             Figure
             there
             is
             a
             distinction
             of
             certain
             
             Moods
             ,
             which
             consist
             of
             the
             differences
             of
             Propositions
             in
             
               Quantity
               &
               Quality
            
             .
             The
             first
             Figure
             is
             that
             ,
             in
             which
             the
             Terms
             are
             placed
             one
             after
             another
             according
             to
             their
             latitude
             of
             Signification
             ;
             in
             which
             order
             the
             
               Minor
               Term
            
             is
             first
             ,
             the
             
               Middle
               Term
            
             next
             ,
             and
             the
             Major
             last
             ;
             as
             if
             the
             Minor
             Term
             be
             Man
             ,
             the
             
               Middle
               Term
               Living
               Creature
            
             and
             the
             
               Major
               Term
               Body
            
             ,
             then
             ,
             
               Man
               is
               a
               Living
               Creature
               ,
               is
               a
               Body
               ,
            
             will
             be
             a
             Syllogisme
             in
             the
             first
             Figure
             ;
             in
             which
             ,
             
               Man
               is
               a
               Living
               Creature
            
             ,
             is
             the
             Minor
             Proposition
             ;
             the
             
               Major
               ,
               Living
               Creature
               is
               a
               Body
               ,
            
             and
             the
             Conclusion
             or
             sum
             of
             both
             
               Man
               is
               a
               Body
            
             .
             Now
             this
             Figure
             is
             called
             Direct
             ,
             because
             the
             Termes
             stand
             in
             direct
             Order
             ;
             and
             it
             is
             varied
             by
             Quantity
             and
             Quality
             into
             four
             Moods
             ;
             of
             which
             the
             first
             is
             that
             wherein
             all
             the
             Terms
             are
             Positive
             ,
             and
             the
             Minor
             Term
             Universal
             ,
             as
             
               Every
               Man
               is
               a
               Living
               Creature
               ,
               Every
               Living
               Creature
               is
               a
               Body
            
             ;
             in
             which
             all
             the
             Propositions
             are
             Affirmative
             and
             Universall
             .
             But
             if
             the
             Major
             Term
             be
             a
             Negative
             Name
             ,
             and
             the
             Minor
             an
             Universall
             Name
             ,
             the
             Figure
             will
             be
             in
             the
             second
             Mood
             ,
             as
             ,
             
               Every
               Man
               is
               a
               Living
               Creature
               ,
               Every
               Living
               Creature
               is
               not
               a
               Tree
               ,
            
             in
             which
             the
             Major
             Proposition
             and
             Conclusion
             are
             both
             Universall
             and
             Negative
             .
             To
             these
             two
             are
             commonly
             addded
             two
             more
             ,
             by
             making
             the
             Minor
             Term
             Particular
             .
             Also
             it
             may
             happen
             that
             both
             the
             Major
             and
             Middle
             Termes
             are
             Negative
             Terms
             ,
             and
             then
             there
             arises
             another
             Mood
             ,
             in
             which
             all
             the
             Propositions
             are
             Negative
             ,
             and
             yet
             the
             Syllogisme
             will
             be
             good
             ;
             as
             ,
             if
             the
             Minor
             Term
             be
             Man
             ,
             the
             Middle
             Term
             
               Not
               a
               Stone
            
             ,
             and
             the
             Major
             Terme
             ,
             
               Not
               a
               Flint
            
             ,
             this
             Syllogisme
             ,
             
               No
               Man
               is
               a
               Stone
               ,
               Whatsoever
               is
               not
               a
               Stone
               is
               not
               a
               Flint
               ,
            
             therefore
             
               No
               Man
               is
               a
               Flint
            
             ,
             is
             true
             ,
             though
             it
             consist
             of
             three
             Negatives
             .
             But
             in
             Philosophy
             ,
             the
             Profession
             whereof
             is
             to
             establish
             Universall
             Rules
             concerning
             the
             Properties
             of
             Things
             ,
             seeing
             the
             difference
             betwixt
             Negatives
             and
             Affirmatives
             is
             onely
             this
             ,
             that
             in
             the
             former
             the
             Subject
             is
             affirmed
             by
             a
             Negative
             Name
             ,
             and
             by
             a
             Positive
             in
             the
             later
             ,
             it
             is
             superfluous
             to
             consider
             any
             other
             Mood
             in
             direct
             Figure
             ,
             besides
             that
             ,
             in
             which
             all
             the
             Propositions
             are
             both
             Universal
             and
             Affirmative
             .
          
           
             8.
             
             The
             Thoughts
             in
             the
             mind
             answering
             to
             a
             Direct
             Syllogism
             ,
             proceed
             in
             this
             manner
             ;
             First
             ,
             there
             is
             conceived
             a
             Phantasme
             of
             the
             thing
             named
             ,
             with
             that
             Accident
             or
             Quality
             thereof
             for
             which
             it
             is
             in
             the
             Minor
             Proposition
             called
             by
             that
             name
             which
             
             is
             the
             Subject
             ;
             next
             ,
             the
             Mind
             has
             a
             Phantasme
             of
             the
             same
             thing
             with
             that
             Accident
             or
             Quality
             for
             which
             it
             hath
             the
             name
             that
             in
             the
             same
             Proposition
             is
             the
             Predicate
             ;
             Thirdly
             ,
             the
             Thought
             returns
             of
             the
             same
             thing
             as
             having
             that
             Accident
             in
             it
             ,
             for
             which
             it
             is
             called
             by
             the
             Name
             that
             is
             in
             the
             Predicate
             of
             the
             Major
             Proposition
             ;
             and
             lastly
             ,
             remembring
             that
             all
             those
             are
             the
             Accidents
             of
             one
             and
             the
             same
             thing
             ,
             it
             concludes
             that
             those
             three
             Names
             are
             also
             Names
             of
             one
             and
             the
             same
             thing
             ;
             that
             is
             to
             say
             ,
             the
             Conclusion
             is
             true
             .
             For
             example
             ,
             when
             this
             Syllogisme
             is
             made
             ,
             
               Man
               is
               a
               Living
               Creature
               ,
               A
               Living
               Creature
               is
               a
               Body
               ,
            
             therefore
             
               Man
               is
               a
               Body
            
             ,
             the
             Mind
             conceives
             first
             an
             image
             of
             a
             Man
             speaking
             or
             discoursing
             ,
             and
             remembers
             that
             that
             which
             so
             appears
             ,
             is
             called
             Man
             ;
             then
             it
             has
             the
             image
             of
             the
             same
             Man
             moving
             ,
             and
             remembers
             that
             that
             which
             appeares
             so
             is
             called
             
               Living
               Creature
            
             ;
             thirdly
             ,
             it
             conceives
             an
             image
             of
             the
             same
             Man
             as
             
               filling
               some
               place
            
             or
             space
             ,
             and
             remembers
             that
             what
             appeares
             so
             is
             called
             Body
             ;
             and
             lastly
             ,
             when
             it
             remembers
             ,
             that
             that
             thing
             which
             was
             extended
             ,
             and
             moved
             and
             spake
             ,
             was
             one
             and
             the
             same
             thing
             ,
             it
             concludes
             that
             the
             three
             Names
             
               Man
               ,
               Living
               Creature
            
             ,
             and
             Body
             ,
             are
             Names
             of
             the
             same
             thing
             ,
             and
             that
             therefore
             
               Man
               is
               a
               Living
               Creature
            
             is
             a
             true
             Proposition
             .
             From
             whence
             it
             is
             manifest
             ,
             that
             Living
             Creatures
             that
             have
             not
             the
             use
             of
             Speech
             ,
             have
             no
             Conception
             or
             Thought
             in
             the
             Mind
             ,
             answering
             to
             a
             Syllogisme
             made
             of
             Universall
             Propositions
             ;
             seeing
             it
             is
             necessary
             to
             Thinke
             not
             only
             of
             the
             Thing
             ,
             but
             also
             by
             turnes
             to
             remember
             the
             diverse
             Names
             ,
             which
             for
             diverse
             considerations
             thereof
             are
             applied
             to
             the
             same
             .
          
           
             9
             The
             rest
             of
             the
             Figures
             arise
             either
             from
             the
             Inflexion
             ,
             or
             Inversion
             of
             the
             first
             or
             direct
             Figure
             ;
             which
             is
             done
             by
             changing
             the
             Major
             ,
             or
             Minor
             ,
             or
             both
             the
             Propositions
             into
             converted
             Propositions
             aequipollent
             to
             them
             .
          
           
             From
             whence
             follow
             three
             other
             Figures
             ;
             of
             which
             ,
             two
             are
             Inflected
             ,
             and
             the
             third
             Inverted
             .
             The
             first
             of
             these
             three
             is
             made
             by
             the
             Conversion
             of
             the
             Major
             Proposition
             .
             For
             let
             the
             Minor
             ,
             Middle
             and
             Major
             Terms
             stand
             in
             direct
             order
             ,
             thus
             ,
             
               Man
               is
               a
               Living
               Creature
               ,
               Is
               not
               a
               Stone
               ,
            
             which
             is
             the
             first
             or
             direct
             Figure
             ;
             the
             Inflexion
             will
             be
             by
             converting
             the
             Major
             Proposition
             in
             this
             manner
             ,
          
           
           
             
               Man
               is
               a
               Living
               Creature
            
             ,
          
           
             
               A
               stone
               is
               not
               a
               Living
               Creature
            
             ;
             And
             this
             is
             the
             second
          
           
             Figure
             ,
             or
             the
             first
             of
             the
             Indirect
             Figures
             ;
             in
             which
             the
             Conclusion
             will
             be
             ,
             
               Man
               is
               not
               a
               stone
            
             .
             For
             (
             having
             shewn
             in
             the
             last
             Chap.
             14
             Article
             ,
             that
             Universall
             Propositions
             converted
             by
             contradiction
             of
             the
             Termes
             are
             aequipollent
             ,
             )
             both
             those
             Syllogismes
             conclude
             alike
             ;
             so
             that
             if
             the
             Major
             be
             read
             (
             like
             Hebrew
             )
             backwards
             ,
             thus
             ,
             
               A
               Living
               Creature
               is
               not
               a
               Stone
               ,
            
             it
             will
             be
             direct
             again
             ,
             as
             it
             was
             before
             .
             In
             like
             manner
             this
             Direct
             Syllogisme
             ,
             
               Man
               is
               not
               a
               Tree
               ,
               is
               not
               a
               Pear-tree
               ,
            
             will
             be
             made
             Indirect
             by
             converting
             the
             Major
             Proposition
             (
             by
             contradiction
             of
             the
             Termes
             )
             into
             another
             aequipollent
             to
             it
             ,
             thus
             ,
          
           
             
               Man
               is
               not
               a
               Tree
            
             ,
          
           
             
               A
               Pear-tree
               is
               a
               Tree
            
             ;
             for
             the
             same
             Conclusion
             will
             follow
             ,
             
               Man
               is
               not
               a
               Pear-tree
            
             .
          
           
             But
             for
             the
             Conversion
             of
             the
             Direct
             Figure
             into
             the
             first
             Indirect
             Figure
             ,
             the
             Major
             Terme
             in
             the
             Direct
             Figure
             ought
             to
             be
             Negative
             .
             For
             though
             this
             Direct
             ,
             
               Man
               is
               a
               Living
               Creature
               ,
               is
               a
               Body
            
             be
             made
             Indirect
             ,
             by
             converting
             the
             Major
             Proposition
             ,
             thus
             ,
          
           
             
               Man
               is
               a
               Living
               Creature
            
             .
          
           
             
               Not
               a
               Body
               is
               not
               a
               Living
               Creature
               ,
            
             Therefore
          
           
             
               Every
               Man
               is
               a
               Body
            
             ;
             yet
             this
             Conversion
             appeares
             so
             obscure
             ,
             that
             this
             Mood
             is
             of
             no
             use
             at
             all
             .
             By
             the
             Conversion
             of
             the
             Major
             Proposition
             ,
             it
             is
             manifest
             ,
             that
             in
             this
             Figure
             ,
             the
             Mi
             dle
             Terme
             is
             alwayes
             the
             Predicate
             in
             both
             the
             Premisses
             .
          
           
             10
             The
             second
             Indirect
             Figure
             is
             made
             by
             converting
             the
             Minor
             Proposition
             ,
             so
             as
             that
             the
             Middle
             Term
             is
             the
             Subject
             in
             both
             .
             But
             this
             never
             concludes
             Uniuersally
             ,
             and
             therefore
             is
             of
             no
             use
             in
             Philosophy
             .
             Neverthelesse
             I
             will
             set
             down
             an
             example
             of
             it
             ;
             by
             which
             this
             Direct
             .
          
           
             
               Every
               Man
               is
               a
               Living
               Creature
               ,
            
          
           
             
               Every
               Living
               Creature
               is
               a
               Body
               ,
            
             by
             Conversion
             of
             the
             Minor
             Proposition
             will
             stand
             thus
             ,
          
           
             
               Some
               Living
               Creature
               is
               a
               Man
               ,
            
          
           
             
               Every
               Living
               Creature
               is
               a
               Body
               ,
            
             Therefore
          
           
             
               Some
               Man
               is
               a
               Body
            
             .
          
           
             For
             
               Every
               Man
               is
               a
               Living
               Creature
               ,
            
             cannot
             be
             converted
             into
             this
             ,
             
               Every
               Living
               Creature
               is
               a
               Man
            
             ;
             and
             therefore
             if
             this
             Syllogisme
             
             be
             restored
             to
             its
             Direct
             forme
             ,
             the
             Minor
             Proposition
             will
             be
             
               Some
               Man
               is
               a
               Living
               Creature
               ,
            
             and
             consequently
             the
             Conclusion
             will
             be
             
               Some
               Man
               is
               a
               Body
            
             ,
             seeing
             the
             Minor
             Terme
             Man
             ,
             which
             is
             the
             Subject
             in
             the
             Conclusion
             ,
             is
             a
             Particular
             Name
             .
          
           
             11
             The
             third
             Indirect
             or
             Inverted
             Figure
             ,
             is
             made
             by
             the
             Conversion
             of
             both
             rhe
             Premisses
             .
             For
             Example
             ,
             this
             Direct
             Syllogisme
             ,
          
           
             
               Every
               Man
               is
               a
               Living
               Creature
               ,
            
          
           
             
               Every
               Living
               Creature
               is
               not
               a
               Stone
               ,
            
             Therefore
          
           
             
               Every
               Man
               is
               not
               a
               Stone
            
             ;
             being
             Inverted
             will
             stand
             thus
             ,
          
           
             
               Every
               Stone
               is
               not
               a
               Living
               Creature
            
          
           
             
               Whatsoever
               is
               not
               a
               Living
               Creature
               ,
               is
               not
               a
               Man
               ,
            
             Therefore
          
           
             
               Every
               Stone
               is
               not
               a
               Man.
            
             Which
             Conclusion
             is
             the
             Converse
             of
             the
             Direct
             Conclusion
             ,
             and
             aequipollent
             to
             the
             same
             .
             The
             Figures
             therefore
             of
             Syllogisms
             ,
             if
             they
             be
             numbred
             by
             the
             diverse
             scituation
             of
             the
             Middle
             Terme
             onely
             ,
             are
             but
             three
             ;
             in
             the
             first
             whereof
             the
             Middle
             Term
             has
             the
             Middle
             place
             ;
             in
             the
             second
             ,
             the
             last
             ;
             and
             in
             the
             third
             ,
             the
             first
             place
             .
             But
             if
             they
             be
             numbred
             according
             to
             the
             scituation
             of
             the
             Termes
             simply
             ,
             they
             are
             four
             ;
             for
             the
             first
             may
             be
             distinguished
             againe
             into
             two
             ,
             namely
             into
             Direct
             and
             Inverted
             .
             From
             whence
             it
             is
             evident
             ,
             that
             the
             controversie
             among
             Logicians
             concerning
             the
             fourth
             Figure
             ,
             is
             a
             meer
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             ,
             or
             contention
             about
             the
             Name
             thereof
             ;
             for
             as
             for
             the
             thing
             it selfe
             ,
             it
             is
             plain
             ,
             that
             the
             scituation
             of
             the
             Termes
             (
             not
             considering
             the
             Quantity
             or
             Quality
             by
             which
             the
             Moods
             are
             distinguished
             )
             makes
             four
             differences
             of
             Syllogismes
             ,
             which
             may
             be
             called
             Figures
             ,
             or
             have
             any
             other
             Name
             at
             pleasure
             .
          
           
             12
             In
             every
             one
             of
             these
             Figures
             there
             are
             many
             Moods
             ,
             which
             are
             made
             by
             varying
             the
             Premisses
             according
             to
             all
             the
             differences
             they
             are
             capable
             of
             ,
             by
             Quantity
             and
             Quality
             ;
             as
             namely
             ,
             in
             the
             Direct
             Figure
             there
             are
             six
             Moods
             ;
             in
             the
             first
             Indirect
             Figure
             ,
             four
             ;
             in
             the
             second
             ,
             fourteen
             ;
             and
             in
             the
             third
             ,
             eighteen
             .
             But
             because
             from
             the
             Direct
             Figure
             I
             rejected
             as
             superfluous
             all
             Moods
             besides
             that
             which
             consists
             of
             Universal
             Propositions
             ,
             and
             whose
             Minor
             Proposition
             is
             Affirmative
             ,
             I
             doe
             together
             with
             it
             reject
             the
             Moods
             of
             the
             rest
             of
             the
             Figures
             which
             are
             made
             by
             Conversion
             of
             the
             Premisses
             in
             the
             Direct
             Figure
             .
          
           
           
             13
             As
             it
             was
             shewed
             before
             ,
             that
             in
             Necessary
             Propositions
             a
             Categoricall
             and
             Hypotheticall
             Proposition
             are
             aequipollent
             ;
             so
             likewise
             it
             is
             manifest
             that
             a
             Categoricall
             and
             Hypotheticall
             Syllogisme
             are
             aequivalent
             .
             For
             every
             Categoricall
             Syllogisme
             ,
             as
             this
             ,
          
           
             
               Every
               Man
               is
               a
               Living
               Creature
               ,
            
          
           
             
               Every
               Living
               Creature
               is
               a
               Body
               ,
            
             Therefore
          
           
             
               Every
               Man
               is
               a
               Body
            
             ,
             is
             of
             equall
             force
             with
             this
             Hypotheticall
             Syllogisme
             .
          
           
             
               If
               any
               thing
               be
               a
               Man
               ,
               the
               same
               is
               also
               a
               Living
               Creature
               ,
            
          
           
             
               If
               any
               thing
               be
               a
               Living
               Creature
               ,
               the
               same
               is
               a
               Body
               ,
            
             Therefore
          
           
             
               If
               any
               thing
               be
               a
               Man
               ,
               the
               same
               is
               a
               Body
               .
            
             In
             like
             manner
             ,
             this
             Categorical
             Syllogisme
             in
             an
             Indirect
             Figure
          
           
             
               No
               Stone
               is
               a
               Living
               Creature
               ,
            
          
           
             
               Every
               Man
               is
               a
               Living
               Creature
               ,
            
             Therefore
          
           
             
               No
               Man
               is
               a
               Stone
            
             ,
             or
          
           
             
               No
               Stone
               is
               a
               Man
            
             ,
             is
             aequivalent
             to
             this
             Hypotheticall
             Syllogisme
             .
          
           
             
               If
               any
               thing
               be
               a
               Man
               ,
               the
               same
               is
               a
               Living
               Creature
               ,
            
          
           
             
               If
               any
               thing
               be
               a
               a
               Stone
               ,
               the
               same
               is
               not
               a
               Living
               Creature
               ,
            
             Therefore
          
           
             
               If
               any
               thing
               be
               a
               Stone
               ,
               the
               same
               is
               not
               a
               Man
               ,
            
             or
          
           
             
               If
               any
               thing
               be
               a
               Man
               ,
               the
               same
               is
               not
               a
               Stone
               .
            
          
           
             And
             thus
             much
             seemes
             sufficient
             for
             the
             nature
             of
             Syllogismes
             ;
             (
             for
             the
             doctrine
             of
             Moods
             and
             Figures
             is
             cleerely
             delivered
             by
             others
             that
             have
             written
             largely
             and
             profitably
             of
             the
             same
             .
             )
             Nor
             are
             Precepts
             so
             necessary
             as
             Practice
             for
             the
             attaining
             of
             true
             Ratiocination
             ;
             and
             they
             that
             study
             the
             Demonstrations
             of
             Mathematicians
             ,
             will
             sooner
             learn
             true
             Logick
             ,
             then
             they
             that
             spend
             time
             in
             reading
             the
             Rules
             of
             Syllogi●ing
             which
             Logicians
             have
             made
             ;
             no
             otherwise
             then
             little
             Children
             learn
             to
             goe
             ,
             not
             by
             Precepts
             ,
             but
             by
             exercising
             their
             feet
             .
             This
             therefore
             may
             serve
             for
             the
             first
             Pace
             in
             the
             way
             to
             Philosophy
             .
          
           
             In
             the
             next
             place
             I
             shall
             speak
             of
             the
             Faults
             and
             Errors
             ,
             into
             which
             men
             that
             reason
             unwarily
             ,
             are
             apt
             to
             fall
             ;
             and
             of
             their
             Kinds
             and
             Causes
             .
          
        
         
           
           
             
               CHAP.
               V.
            
             Of
             
               Erring
               ,
               Falsity
               and
               Captions
            
             .
          
           
             
               
                 1.
                 
                 Erring
                 and
                 Falsity
                 how
                 they
                 differ
                 .
                 Error
                 of
                 the
                 Mind
                 by
                 it selfe
                 without
                 the
                 use
                 of
                 Words
                 ,
                 how
                 it
                 happens
                 .
              
               
                 2
                 A
                 sevenfold
                 Incoherency
                 of
                 Names
                 ,
                 every
                 one
                 of
                 which
                 makes
                 allwayes
                 a
                 false
                 Proposition
                 .
              
               
                 3.
                 
                 Examples
                 of
                 the
                 first
                 manner
                 of
                 Incoherency
                 .
              
               
                 4
                 Of
                 the
                 second
                 .
              
               
                 5
                 Of
                 the
                 third
                 .
              
               
                 6
                 Of
                 the
                 fourth
                 .
              
               
                 7
                 Of
                 the
                 fifth
                 .
              
               
                 8
                 Of
                 the
                 sixth
                 .
              
               
                 9
                 Of
                 the
                 seventh
                 .
              
               
                 10
                 Falsity
                 of
                 Propositions
                 detected
                 by
                 resolving
                 the
                 Terms
                 with
                 Definitions
                 continued
                 till
                 they
                 ●ome
                 to
                 Simple
                 Names
                 ,
                 or
                 Names
                 that
                 are
                 the
                 most
                 Generall
                 of
                 their
                 kind
                 .
              
               
                 11
                 Of
                 the
                 fault
                 of
                 a
                 Syllogisme
                 consisting
                 in
                 the
                 Implication
                 of
                 the
                 Termes
                 which
                 the
                 Copula
              
               
                 12
                 Of
                 the
                 fault
                 which
                 consists
                 in
                 Equivocation
                 .
              
               
                 13
                 Sophisticall
                 Captions
                 are
                 oftner
                 faulty
                 in
                 the
                 matter
                 then
                 in
                 the
                 forme
                 of
                 Syllogismes
                 .
              
            
          
           
             1
             MEn
             are
             subject
             to
             Erre
             not
             onely
             in
             Affiming
             and
             Denying
             ;
             but
             also
             in
             Perception
             ,
             and
             in
             silent
             Cogitation
             .
             In
             Affirming
             and
             Denying
             ,
             when
             they
             call
             any
             thing
             by
             a
             Name
             ,
             which
             is
             not
             the
             Name
             thereof
             ;
             as
             if
             from
             seeing
             the
             Sun
             first
             by
             reflection
             in
             Water
             ,
             and
             afterwards
             again
             directly
             in
             the
             Firmament
             ,
             we
             should
             to
             both
             those
             appearances
             give
             the
             Name
             of
             Sunne
             ,
             and
             say
             there
             are
             two
             Sunnes
             ;
             which
             none
             but
             men
             can
             d●e
             ;
             for
             no
             other
             Living
             Creatures
             have
             the
             use
             of
             Names
             .
             This
             kind
             of
             Error
             onely
             deserves
             the
             name
             of
             Falsity
             ,
             as
             arising
             ,
             not
             from
             sense
             ,
             nor
             from
             the
             Things
             themselves
             but
             from
             pronouncing
             rashly
             ;
             for
             Names
             have
             their
             constitution
             ,
             not
             from
             the
             Species
             of
             Things
             ,
             but
             from
             the
             Will
             and
             Consent
             of
             Men.
             And
             hence
             it
             comes
             to
             passe
             ,
             that
             men
             pronounce
             Falsely
             by
             their
             own
             negligence
             ,
             in
             departing
             from
             such
             appellations
             of
             things
             as
             are
             agreed
             upon
             ,
             and
             are
             not
             deceived
             neither
             by
             the
             Things
             ,
             nor
             by
             the
             Sense
             ;
             for
             they
             do
             not
             perceive
             that
             the
             thing
             they
             see
             is
             called
             Sunne
             ,
             but
             they
             give
             it
             that
             Name
             from
             their
             owne
             will
             and
             agreement
             .
             Tacite
             Errors
             ,
             or
             the
             Errors
             of
             
             Sense
             and
             Cogitation
             ,
             are
             made
             ,
             by
             passing
             from
             one
             Imaginatition
             to
             the
             Imagination
             of
             another
             different
             thing
             ;
             or
             by
             feigning
             that
             to
             be
             Past
             ,
             or
             Future
             ,
             which
             never
             was
             ,
             nor
             ever
             shall
             be
             ;
             as
             when
             by
             seeing
             the
             Image
             of
             the
             Sunne
             in
             Water
             ,
             we
             imagine
             the
             Sunne
             it selfe
             to
             be
             there
             ;
             or
             by
             seeing
             swords
             ,
             that
             there
             has
             been
             or
             shall
             be
             fighting
             ,
             because
             it
             uses
             to
             be
             so
             for
             the
             most
             part
             ;
             or
             when
             from
             Promises
             we
             feigne
             the
             mind
             of
             the
             Promiser
             to
             be
             such
             and
             such
             ;
             or
             lastly
             ,
             when
             from
             any
             Signe
             we
             vainly
             imagine
             something
             to
             be
             signified
             ,
             which
             is
             not
             .
             And
             Errors
             of
             this
             sort
             are
             common
             to
             all
             things
             that
             have
             sense
             ;
             and
             yet
             the
             Deception
             proceeds
             neither
             from
             our
             senses
             ,
             nor
             from
             the
             Things
             we
             perceive
             ;
             but
             from
             our selves
             ,
             while
             we
             feigne
             such
             things
             as
             are
             but
             meer
             Images
             ,
             to
             be
             something
             more
             then
             Images
             .
             But
             neither
             Things
             ,
             nor
             Imaginations
             of
             Things
             can
             be
             said
             to
             be
             False
             ,
             seeing
             they
             are
             truly
             what
             they
             are
             ;
             nor
             doe
             they
             as
             Signes
             promise
             any
             thing
             which
             they
             do
             not
             performe
             ;
             for
             they
             indeed
             do
             not
             promise
             at
             all
             ,
             but
             we
             from
             them
             ;
             nor
             doe
             the
             Clouds
             ,
             but
             we
             from
             seeing
             the
             Clouds
             ,
             say
             it
             shall
             rain
             .
             The
             best
             way
             therefore
             to
             free
             our selves
             from
             such
             Errors
             as
             arise
             from
             naturall
             Signes
             ,
             is
             first
             of
             all
             ,
             before
             we
             begin
             to
             reason
             concerning
             such
             conjecturall
             things
             ,
             to
             suppose
             our selves
             ignorant
             ,
             and
             then
             to
             make
             use
             of
             our
             Ratiocination
             ;
             for
             these
             Errors
             proceed
             from
             the
             want
             of
             Ratiocination
             ;
             whereas
             Errors
             which
             consist
             in
             Affirmation
             and
             Negation
             ,
             (
             that
             is
             ,
             the
             Falsity
             of
             Propositions
             )
             proceed
             only
             from
             Reasoning
             amisse
             .
             Of
             these
             therefore
             ,
             as
             repugnant
             to
             Philosophy
             ,
             I
             will
             speake
             principally
             .
          
           
             2
             Errors
             which
             happen
             in
             Reasoning
             ,
             that
             is
             ,
             in
             Syllogizing
             ,
             consist
             either
             in
             the
             Falsity
             of
             the
             Premisses
             ,
             or
             of
             the
             Inference
             .
             In
             the
             first
             of
             these
             cases
             ,
             a
             Syllogisme
             is
             said
             to
             be
             faulty
             in
             the
             Matter
             of
             it
             ;
             and
             in
             the
             second
             case
             ,
             in
             the
             Forme
             .
             I
             will
             first
             consider
             the
             Matter
             ,
             namely
             how
             many
             wayes
             a
             Proposition
             may
             be
             false
             ;
             and
             next
             the
             Forme
             ,
             and
             how
             it
             comes
             to
             pass
             ,
             that
             when
             the
             Premises
             are
             True
             ,
             the
             Inference
             is
             notwithstanding
             False
             .
          
           
             Seeing
             therefore
             that
             Proposition
             onely
             is
             True
             ,
             (
             Chap.
             3.
             
             Art.
             7.
             )
             in
             which
             are
             copulated
             two
             Names
             of
             one
             and
             the
             same
             thing
             ;
             and
             that
             alwayes
             False
             ,
             in
             which
             Names
             of
             different
             things
             are
             copulated
             ;
             look
             how
             many
             wayes
             Names
             of
             different
             things
             
             may
             be
             copulated
             ,
             and
             so
             many
             wayes
             a
             False
             Proposition
             may
             be
             made
             .
          
           
             Now
             all
             things
             to
             which
             we
             give
             Names
             ,
             may
             be
             reduced
             to
             these
             four
             kinds
             ,
             namely
             ,
             
               Bodies
               ,
               Accidents
               ,
               Phantasmes
            
             ,
             and
             Names
             themselves
             ;
             and
             therefore
             in
             every
             true
             Proposition
             it
             is
             necessary
             that
             the
             Names
             copulated
             ,
             be
             both
             of
             them
             Names
             of
             Bodies
             ,
             or
             both
             Names
             of
             Accidents
             ,
             or
             both
             Names
             of
             Phantasmes
             ,
             or
             both
             Names
             of
             Names
             .
             For
             Names
             otherwise
             copulated
             are
             incoherent
             ,
             and
             constitute
             a
             False
             Proposition
             .
             It
             may
             happen
             also
             that
             the
             Name
             of
             a
             Body
             ,
             of
             an
             Accident
             ,
             or
             of
             a
             Phantasme
             may
             be
             copulated
             with
             the
             Name
             of
             a
             Speech
             .
             So
             that
             copulated
             Names
             may
             be
             Incoherent
             seven
             manner
             of
             wayes
             .
          
           
             
               1
               If
               the
               Name
               of
               a
               Body
               be
               copulated
               with
               the
               Name
               of
               an
               Accident
               .
            
             
               2
               If
               the
               Name
               of
               a
               Body
               be
               copulated
               with
               the
               Name
               of
               a
               Phantasme
               .
            
             
               3
               If
               the
               Name
               of
               a
               Body
               be
               copulated
               with
               the
               Name
               of
               a
               Name
               .
            
             
               4
               If
               the
               Name
               of
               an
               Accident
               be
               copulated
               with
               the
               Name
               of
               a
               Phantasme
               .
            
             
               5
               If
               the
               Name
               of
               an
               Accident
               be
               copulated
               with
               the
               Name
               of
               a
               Name
               .
            
             
               6
               If
               the
               Name
               of
               a
               Phantasme
               be
               copulated
               with
               the
               Name
               of
               a
               Name
               .
            
             
               7
               If
               the
               Name
               of
               a
               Body
               ,
               of
               an
               Accident
               or
               of
               a
               Phantasme
               be
               copulated
               with
               the
               Name
               of
               a
               Speech
               .
            
          
           
             Of
             all
             which
             I
             will
             give
             some
             examples
             .
          
           
             3
             After
             the
             first
             of
             these
             wayes
             Propositions
             are
             false
             ,
             when
             Abstract
             Names
             are
             copulated
             with
             Concrete
             Names
             ;
             as
             (
             in
             Latine
             and
             Greek
             )
             
               Esse
               est
               Ens
               ,
               Essenti●
               est
               Ens
               ,
            
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             (
             i.
             )
             
               Quidditas
               est
               Ens
            
             ,
             and
             many
             the
             like
             which
             are
             found
             in
             Aristotles
             Metaphysicks
             .
             Also
             ,
             the
             
               Understanding
               worketh
            
             ,
             the
             
               Understanding
               understandeth
            
             ,
             the
             
               Sight
               seeth
            
             ,
             A
             Body
             is
             Magnitude
             ,
             A
             Body
             is
             Quantity
             ,
             A
             Body
             is
             Extension
             ,
             To
             
               Be
               a
               Man
            
             is
             
               a
               Man
               ,
               Whitenesse
            
             is
             a
             
               White
               thing
               ,
               &c.
            
             which
             is
             as
             if
             one
             should
             say
             The
             Runner
             is
             the
             Running
             ,
             or
             the
             
               Walke
               Walketh
            
             .
             Moreover
             ,
             Essence
             is
             
               separated
               ,
               Substance
            
             is
             Abstracted
             ;
             and
             others
             like
             these
             ,
             or
             derived
             from
             these
             (
             with
             which
             common
             Philosophy
             abounds
             )
             .
             For
             seeing
             no
             Subject
             of
             an
             Accident
             ,
             (
             that
             is
             ,
             No
             Body
             )
             is
             an
             Accident
             ;
             no
             Name
             of
             an
             Accident
             ought
             to
             be
             given
             to
             a
             Body
             ,
             nor
             of
             a
             Body
             to
             an
             Accident
             .
          
           
             4
             False
             in
             the
             second
             Manner
             are
             such
             Propositions
             as
             these
             ,
             
             A
             
               Ghost
               is
               a
               Body
            
             ,
             or
             a
             Spirit
             ,
             that
             is
             ,
             a
             thinne
             Body
             ;
             
               Sensible
               Species
               fly
               up
               and
               down
               in
               the
               Air
               ,
            
             or
             
               are
               moved
               hither
               and
               thither
            
             ,
             which
             is
             proper
             to
             Bodies
             ;
             also
             ,
             A
             
               Shadow
               is
               Moved
            
             ,
             or
             
               is
               a
               Body
               ;
               Light
               is
               Moved
               ,
            
             or
             
               is
               a
               Body
               ;
               Colour
               is
               the
               Ob●ect
               of
               Sight
               ,
               Sound
            
             of
             
               Hearing
               ;
               Space
            
             or
             
               Place
               is
               Extended
            
             ;
             and
             innumerable
             others
             of
             this
             kind
             .
             For
             seeing
             
               Ghosts
               ,
               sensible
               Species
            
             ,
             a
             
               Shadow
               ,
               Light
               ,
               Colour
               ,
               Sound
               ,
               Space
               ,
               &c.
            
             appeare
             to
             us
             no
             lesse
             sleeping
             then
             waking
             ,
             they
             cannot
             be
             things
             without
             us
             ,
             but
             onely
             Phantasmes
             of
             the
             mind
             that
             imagins
             them
             ;
             and
             therefore
             the
             Names
             of
             these
             ,
             copulated
             with
             the
             Names
             of
             Bodies
             ,
             cannot
             constitute
             a
             true
             Proposition
             .
          
           
             5
             False
             Propositions
             of
             the
             third
             kind
             ,
             are
             such
             as
             these
             ;
             
               Genus
               est
               Ens
               ,
               Universale
               est
               Ens
               ,
               Ens
               de
               Ente
               Praedicatur
               .
            
             For
             Genus
             ,
             and
             Universale
             ,
             and
             Praedicare
             are
             Names
             of
             Names
             ,
             and
             not
             of
             Things
             .
             Also
             
               Number
               is
               Infinite
            
             ,
             is
             a
             false
             Proposition
             ;
             for
             no
             number
             can
             be
             Infinite
             ,
             but
             onely
             the
             word
             Number
             is
             then
             called
             an
             Indefinite
             Name
             when
             there
             is
             no
             determined
             Number
             answering
             to
             it
             in
             the
             Mind
             .
          
           
             6
             To
             the
             fourth
             kind
             belong
             such
             false
             Propositions
             as
             these
             ,
             ▪
             
               An
               Object
               is
               of
               such
               Magnitude
               or
               Figure
               as
               appeares
               to
               the
               Beholders
               ;
               Colour
               ,
               Light
               ,
               Sound
               ,
               are
               in
               the
               Object
            
             ;
             and
             the
             like
             .
             For
             the
             same
             Object
             appeares
             sometimes
             greater
             ,
             sometimes
             lesser
             ,
             sometimes
             square
             ,
             sometimes
             round
             according
             to
             the
             diversity
             of
             the
             Distance
             and
             Medium
             ;
             but
             the
             true
             Magnitude
             and
             Figure
             of
             the
             thing
             seen
             is
             allwayes
             one
             and
             the
             same
             ;
             so
             that
             the
             magnitude
             and
             figure
             which
             appeares
             ,
             is
             not
             the
             true
             magnitude
             and
             figure
             of
             the
             Object
             ,
             nor
             any
             thing
             but
             Phantasme
             ;
             and
             therefore
             in
             such
             Propositions
             as
             these
             ,
             the
             Names
             of
             Accidents
             are
             copulated
             with
             the
             Names
             of
             Phantasmes
             .
          
           
             7
             Propositions
             are
             false
             in
             the
             fifth
             manner
             ,
             when
             it
             is
             said
             that
             
               The
               Definition
               is
               the
               Essence
               of
               a
               thing
               ;
               Whitenesse
               ,
            
             or
             some
             other
             Accident
             
               is
               the
               Genus
               ,
               or
               Universal
               .
            
             For
             Definition
             is
             not
             the
             Essence
             of
             any
             thing
             ,
             but
             a
             speech
             signifying
             what
             we
             conceive
             of
             the
             Essence
             thereof
             ;
             and
             so
             also
             not
             Whitenesse
             it selfe
             ,
             but
             the
             word
             Whitenesse
             ,
             is
             a
             Genus
             ,
             or
             an
             
               Universall
               Name
            
             .
          
           
             8
             In
             the
             sixth
             manner
             they
             Erre
             ,
             that
             say
             the
             
               Idea
               of
               any
               thing
               is
               Universal
            
             ;
             as
             if
             there
             could
             be
             in
             the
             Mind
             an
             Image
             of
             
               a
               Man
            
             which
             were
             not
             the
             Image
             of
             
               some
               one
               Man
            
             ,
             but
             of
             
               Man
               simply
            
             ,
             which
             is
             impossible
             ;
             for
             every
             Idea
             is
             one
             ,
             and
             of
             onething
             ;
             but
             they
             are
             
             deceived
             in
             this
             ,
             that
             they
             put
             the
             Name
             of
             the
             thing
             for
             the
             Idea
             thereof
             .
          
           
             9
             They
             erre
             in
             the
             seventh
             manner
             ,
             that
             make
             this
             distinction
             between
             things
             that
             have
             being
             ,
             that
             some
             of
             them
             
               exist
               by
               themselves
            
             ,
             others
             
               by
               Accident
            
             ;
             Namely
             ,
             because
             
               Socrates
               is
               a
               Man
            
             is
             a
             Necessary
             Proposition
             ,
             and
             
               Socrates
               is
               a
               Musician
            
             a
             Contingent
             Proposition
             ,
             therefore
             they
             say
             some
             things
             
               exist
               necessarily
            
             or
             
               by
               themselves
            
             ,
             others
             contingently
             or
             
               by
               Accident
            
             ;
             whereby
             ,
             seeing
             
               Necessary
               ,
               Contingent
               ,
               By
               it selfe
               ,
               By
               Accident
               ,
            
             are
             not
             Names
             of
             Things
             ,
             but
             of
             Propositions
             ,
             they
             that
             say
             
               any
               thing
               that
               has
               being
               ,
               exists
               by
               Accident
               ,
            
             copulate
             the
             Name
             of
             a
             Proposition
             with
             the
             Name
             of
             a
             Thing
             .
             In
             the
             same
             manner
             also
             they
             Erre
             ,
             which
             place
             some
             Ideas
             in
             the
             Understanding
             ,
             others
             in
             the
             Fancy
             ;
             as
             if
             from
             the
             Understanding
             of
             this
             Proposition
             
               Man
               is
               a
               Living
               Creature
            
             ,
             we
             had
             one
             Idea
             or
             Image
             of
             a
             Man
             derived
             from
             sense
             to
             the
             Memory
             ,
             and
             another
             to
             the
             Understanding
             ;
             wherein
             that
             which
             deceives
             them
             is
             this
             ,
             that
             they
             think
             one
             Idea
             should
             be
             answerable
             to
             a
             Name
             ,
             another
             to
             a
             Proposition
             ;
             which
             is
             false
             ;
             for
             Pr●position
             signifies
             onely
             the
             order
             of
             those
             things
             one
             after
             another
             ,
             which
             we
             observe
             in
             the
             same
             Idea
             of
             Man
             ;
             so
             that
             this
             Proposition
             
               Man
               is
               a
               Living
               Creature
            
             ,
             raises
             but
             one
             Idea
             in
             us
             ,
             though
             in
             that
             Idea
             we
             consider
             that
             first
             ,
             for
             which
             he
             is
             called
             Man
             ,
             and
             next
             that
             for
             which
             he
             is
             called
             
               Living
               Creature
            
             .
             The
             Falsities
             of
             Propositions
             in
             all
             these
             several
             manners
             ,
             is
             to
             be
             discovered
             by
             the
             Definitions
             of
             the
             Copulated
             Names
             .
          
           
             10
             But
             when
             Names
             of
             Bodies
             are
             copulated
             with
             Names
             of
             Bodies
             ,
             Names
             of
             Accidents
             with
             Names
             of
             Accidents
             ,
             Names
             of
             Names
             with
             Names
             of
             Names
             ,
             and
             Names
             of
             Phantasmes
             with
             Names
             of
             Phantasmes
             ,
             if
             we
             neverthelesse
             remaine
             still
             doubtfull
             whether
             such
             Propositions
             are
             true
             ;
             we
             ought
             then
             in
             the
             first
             place
             to
             find
             out
             the
             Definition
             of
             both
             those
             Names
             ,
             and
             againe
             the
             Definitions
             of
             such
             Names
             as
             are
             in
             the
             former
             Definition
             ,
             and
             so
             proceed
             by
             a
             continuall
             Resolution
             till
             we
             come
             to
             a
             simple
             Name
             ,
             that
             is
             ,
             to
             the
             most
             Generall
             or
             most
             Universall
             Name
             of
             that
             kind
             ;
             and
             if
             after
             all
             this
             the
             Truth
             or
             Falsity
             thereof
             be
             not
             evident
             ,
             we
             must
             search
             it
             out
             by
             Philosophy
             ,
             and
             Ratiocination
             ,
             beginning
             from
             Definitions
             .
             For
             every
             Proposition
             Universally
             true
             ,
             is
             either
             a
             Definition
             ,
             or
             part
             of
             a
             Definition
             ,
             or
             the
             evidence
             of
             it
             depends
             upon
             Definitions
             .
          
           
           
             11
             That
             fault
             of
             a
             Syllogisme
             which
             lyes
             bid
             in
             the
             Forme
             thereof
             ,
             will
             allwayes
             be
             found
             either
             in
             the
             implication
             of
             the
             Copula
             with
             one
             of
             the
             Termes
             ,
             or
             in
             the
             Aequivocation
             of
             some
             word
             ;
             and
             in
             either
             of
             these
             wayes
             there
             will
             be
             
               four
               Terms
            
             ,
             which
             (
             as
             I
             have
             shewne
             )
             cannot
             stand
             in
             a
             true
             Syllogisme
             .
             Now
             the
             implication
             of
             the
             Copula
             with
             either
             Terme
             ,
             is
             easily
             detected
             by
             reducing
             the
             Propositions
             to
             plain
             and
             cleere
             Praedication
             ;
             as
             (
             for
             example
             )
             if
             any
             man
             should
             argue
             thus
             ,
          
           
             
               The
               Hand
               toucheth
               the
               Pen
            
             ,
          
           
             
               The
               Pen
               toucheth
               the
               Paper
            
             ,
             Therefore
          
           
             
               The
               Hand
               toucheth
               the
               Paper
            
             ;
             the
             Fallacy
             will
             easily
             appear
             by
             reducing
             it
             ,
             thus
             ,
          
           
             
               The
               Hand
               ,
               is
               ,
               touching
               the
               Pen
               ,
            
          
           
             
               The
               Pen
               ,
               is
               ,
               touching
               the
               Paper
               ,
            
             Therefore
          
           
             
               The
               Hand
               ,
               is
               ,
               touching
               the
               Paper
            
             ;
             where
             there
             are
             manifestly
             these
             four
             Termes
             ,
             
               The
               Hand
               ,
               Touching
               the
               Pen
               ,
               The
               Pen
               ,
            
             and
             
               Touching
               the
               Paper
            
             .
             But
             the
             danger
             of
             being
             deceived
             by
             Sophismes
             of
             this
             kind
             ,
             does
             not
             seem
             to
             be
             so
             great
             ,
             as
             that
             I
             need
             insist
             longer
             upon
             them
             .
          
           
             12
             And
             though
             there
             may
             be
             Fallacy
             in
             Aequivocal
             Terms
             ,
             yet
             in
             those
             that
             be
             manifestly
             such
             there
             is
             none
             at
             all
             ;
             nor
             in
             Metaphors
             ;
             for
             they
             professe
             the
             transferring
             of
             Names
             from
             one
             thing
             to
             another
             .
             Neverthelesse
             sometimes
             Aequivocalls
             (
             and
             those
             not
             very
             obscure
             )
             may
             deceive
             ;
             as
             in
             this
             argumentation
             ,
             
               It
               belongs
               to
               Metaphysicks
               ,
               to
               treat
               of
               Principles
               ;
               But
               the
               first
               Principles
               of
               all
               ,
               is
               ,
               that
               the
               same
               thing
               cannot
               both
               exist
               and
               not
               exist
               at
               the
               same
               time
               ;
               and
               therefore
               it
               belongs
               to
               Metaphysicks
               to
               treat
               whether
               the
               same
               thing
               may
               both
               exist
               and
               not
               exist
               at
               the
               same
               time
            
             ;
             where
             the
             Fallacy
             lies
             in
             the
             Aequivocation
             of
             the
             word
             Principle
             ;
             for
             whereas
             Aristotle
             in
             the
             beginning
             of
             his
             Metaphysicks
             sayes
             ,
             that
             
               the
               treating
               of
               Principles
               belongs
               to
               primary
               science
               ,
            
             he
             understands
             by
             
               Principles
               ,
               Causes
            
             of
             things
             ,
             and
             certaine
             Existences
             which
             he
             calls
             Primary
             ;
             but
             where
             he
             sayes
             
               a
               Primary
               Proposition
               is
               a
               Principle
               ,
            
             by
             Principle
             there
             he
             means
             the
             
               beginning
               and
               cause
               of
               Knowledge
            
             ,
             that
             is
             the
             understanding
             of
             words
             ,
             which
             if
             any
             man
             want
             ,
             he
             is
             incapable
             of
             learning
             .
          
           
             13
             But
             the
             Captions
             of
             Sophists
             and
             Scepticks
             ,
             by
             which
             they
             were
             wont
             of
             old
             to
             deride
             and
             oppose
             Truth
             ,
             were
             faulty
             for
             
             the
             most
             part
             ,
             not
             in
             the
             Forme
             ,
             but
             in
             the
             Matter
             of
             Syllogisme
             ;
             and
             they
             deceived
             not
             others
             oftner
             then
             they
             were
             themselves
             deceived
             .
             For
             the
             force
             of
             that
             famous
             argument
             of
             Zeno
             against
             Motion
             ,
             consisted
             in
             this
             Proposition
             ,
             
               Whatsoever
               may
               be
               divided
               into
               parts
               infinite
               in
               number
               ,
               the
               same
               is
               infinite
            
             ;
             which
             he
             without
             doubt
             thought
             to
             be
             true
             ,
             yet
             neverthelesse
             is
             false
             .
             For
             to
             be
             divided
             into
             
               infinite
               parts
            
             ,
             is
             nothing
             else
             but
             to
             be
             divided
             ●●●o
             
               as
               many
               parts
               as
               any
               man
               will.
            
             But
             it
             is
             not
             necessary
             that
             a
             Line
             should
             have
             parts
             infinite
             in
             number
             ,
             or
             be
             infinite
             ,
             because
             I
             can
             divide
             and
             subdivide
             it
             as
             often
             as
             I
             please
             ;
             for
             how
             many
             parts
             soever
             I
             make
             ,
             yet
             their
             number
             is
             finite
             ;
             but
             because
             he
             that
             sayes
             Parts
             ,
             simply
             ,
             without
             adding
             
               how
               many
            
             ,
             does
             not
             limit
             any
             number
             ,
             but
             leaves
             it
             to
             the
             determination
             of
             the
             Hearer
             ,
             therefore
             we
             say
             commonly
             a
             line
             may
             be
             divided
             infinitely
             ;
             which
             cannot
             be
             true
             in
             any
             other
             sense
             .
          
           
             And
             thus
             much
             may
             suffice
             concerning
             Syllogisme
             ,
             which
             is
             as
             it
             were
             the
             first
             Pace
             towards
             Philosophy
             ;
             in
             which
             I
             have
             said
             as
             much
             as
             is
             necessary
             ,
             to
             teach
             any
             man
             from
             whence
             all
             true
             argumentation
             has
             its
             force
             .
             And
             to
             enlarge
             this
             Treatise
             with
             all
             that
             may
             be
             heaped
             together
             ,
             would
             be
             as
             superfluous
             ,
             as
             if
             one
             should
             (
             as
             I
             said
             before
             )
             give
             a
             young
             child
             Precepts
             for
             the
             teaching
             of
             him
             to
             goe
             ;
             for
             the
             Art
             of
             Reasoning
             is
             not
             so
             well
             learned
             by
             Precepts
             as
             by
             Practice
             ,
             and
             by
             the
             reading
             of
             those
             books
             in
             which
             the
             Conclusions
             are
             all
             made
             by
             severe
             Demonstration
             .
             And
             so
             I
             pass
             on
             to
             the
             way
             of
             Philosophy
             ,
             that
             is
             ,
             to
             the
             Method
             of
             Study
             .
          
        
         
           
           
             
               CHAP.
               VI.
            
             Of
             
               Method
               ▪
            
          
           
             
               
                 1
                 Method
                 and
                 Science
                 defined
                 .
              
               
                 2
                 It
                 is
                 more
                 easily
                 known
                 concerning
                 Singular
                 then
                 Universall
                 things
                 ,
                 That
                 they
                 are
                 ;
                 and
                 contrarily
                 ,
                 it
                 i●
                 more
                 easily
                 knowne
                 concerning
                 Universall
                 then
                 Singular
                 things
                 ,
                 Why
                 they
                 are
                 ,
                 or
                 what
                 are
                 their
                 Causes
                 .
              
               
                 3
                 What
                 it
                 is
                 Philosophers
                 seek
                 to
                 know
                 .
              
               
                 4
                 The
                 first
                 Part
                 ,
                 by
                 which
                 Principles
                 are
                 found
                 out
                 is
                 purely
                 Analyticall
                 .
              
               
                 5
                 The
                 highest
                 Causes
                 ,
                 and
                 most
                 Universall
                 in
                 every
                 kind
                 ,
                 are
                 knowne
                 by
                 themselves
                 .
              
               
                 6
                 Method
                 from
                 Principles
                 fonnd
                 out
                 ,
                 tending
                 to
                 Science
                 simply
                 ,
                 what
                 it
                 is
                 .
              
               
                 7
                 That
                 Method
                 of
                 Civill
                 and
                 Naturall
                 Science
                 which
                 proceeds
                 from
                 Sense
                 to
                 Principles
                 ,
                 is
                 Analytical
                 ;
                 and
                 againe
                 that
                 which
                 begins
                 at
                 Principles
                 ,
                 is
                 Syntheticall
                 .
              
               
                 8
                 The
                 Method
                 of
                 searching
                 out
                 ,
                 whether
                 any
                 thing
                 propounded
                 ,
                 be
                 Matter
                 or
                 Accident
                 .
              
               
                 9
                 The
                 Method
                 of
                 seeking
                 whether
                 any
                 Accident
                 be
                 in
                 this
                 ,
                 or
                 in
                 that
                 Subject
                 .
              
               
                 10
                 The
                 Method
                 of
                 searching
                 after
                 the
                 Cause
                 of
                 any
                 Effect
                 propounded
                 .
              
               
                 11
                 Words
                 serve
                 to
                 Invention
                 ,
                 as
                 Markes
                 ;
                 to
                 Demonstration
                 ,
                 as
                 Signes
                 .
              
               
                 12
                 The
                 Method
                 of
                 Demonstration
                 is
                 Syntheticall
                 .
              
               
                 13
                 Definitions
                 onely
                 are
                 Primary
                 and
                 Universal
                 Propositions
                 .
              
               
                 14
                 The
                 Nature
                 and
                 Definition
                 of
                 a
                 Definition
                 .
              
               
                 15
                 The
                 Properties
                 of
                 a
                 Definition
                 .
              
               
                 16
                 The
                 Nature
                 of
                 a
                 Demonstration
                 .
              
               
                 17
                 The
                 Properties
                 of
                 a
                 Demonstration
                 ,
                 and
                 Order
                 of
                 things
                 to
                 be
                 demonstrated
                 .
              
               
                 18
                 The
                 Faults
                 of
                 a
                 Demonstration
                 .
              
               
                 19
                 Why
                 the
                 Analyticall
                 Method
                 of
                 Ge●metricians
                 cannot
                 be
                 treated
                 of
                 in
                 this
                 place
                 .
              
            
          
           
             1
             FOr
             the
             understanding
             of
             Method
             ,
             it
             will
             be
             necessary
             for
             me
             to
             repeat
             the
             definition
             of
             Philosophy
             ,
             delivered
             above
             (
             Chap.
             1.
             
             Art.
             2.
             )
             in
             this
             manner
             ,
             
               Philosophy
               is
               the
               knowledge
               we
               acquire
               by
               true
               Ratiocination
               ,
               of
               Appearances
               ,
               or
               apparent
               Effects
               ,
               from
               the
               knowledge
               we
               have
               of
               some
               possible
               Production
               or
               Generation
               of
               the
               same
               ;
               and
               of
               such
               Production
               as
               has
               been
               or
               may
               be
               ,
               from
               the
               knowledge
               we
               have
               of
               the
               Effects
               .
            
             METHOD
             therefore
             in
             the
             Study
             of
             Philosophy
             ,
             
               is
               the
               shortest
               way
               of
               finding
               out
               Effects
               by
               their
            
             
             
               known
               Causes
               ,
               or
               of
               Causes
               by
               their
               known
               Effects
               .
            
             But
             we
             are
             then
             said
             to
             know
             any
             Effect
             ,
             when
             we
             know
             ,
             
               that
               there
               be
               Causes
               of
               the
               same
               ,
            
             and
             
               in
               what
               Subiect
               those
               Causes
               are
               ,
            
             and
             
               in
               what
               Subiect
               they
               produce
               that
               Effect
               ,
            
             and
             
               in
               what
               Manner
               they
               work
               the
               same
               .
            
             And
             this
             is
             the
             Science
             of
             Causes
             ,
             or
             as
             they
             call
             it
             of
             the
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             .
             All
             other
             Science
             ,
             which
             is
             called
             the
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             ,
             is
             either
             Perception
             by
             Sense
             ,
             or
             the
             Imagination
             ,
             or
             Memory
             remaining
             after
             such
             Perception
             .
          
           
             The
             first
             Beginnings
             therefore
             of
             Knowledge
             ,
             are
             the
             Phantasmes
             of
             Sense
             and
             Imagination
             ;
             and
             that
             there
             be
             such
             Phantasmes
             we
             know
             well
             enough
             by
             Nature
             ;
             but
             to
             know
             why
             they
             be
             ,
             or
             from
             what
             Causes
             they
             proceed
             ,
             is
             the
             work
             of
             Ratiocination
             ;
             which
             consists
             (
             as
             is
             said
             above
             ,
             in
             the
             1.
             
             Chap.
             2.
             
             Art.
             )
             in
             Composition
             ,
             and
             Division
             or
             Resolution
             .
             There
             is
             therefore
             no
             Method
             ,
             by
             which
             we
             find
             out
             the
             Causes
             of
             things
             ,
             but
             is
             either
             Compositive
             ,
             or
             Resolutive
             ,
             or
             
               partly
               Compositive
            
             ,
             and
             
               partly
               Resolutive
            
             .
             And
             the
             Resolutive
             is
             commonly
             called
             Analyticall
             Method
             ,
             as
             the
             Compositive
             is
             called
             Syntheticall
             .
          
           
             2
             It
             is
             common
             to
             all
             sorts
             of
             Method
             ,
             to
             proceed
             from
             known
             things
             to
             unknown
             ;
             and
             this
             is
             manifest
             from
             the
             cited
             Definition
             of
             Philosophy
             .
             But
             in
             Knowledge
             by
             Sense
             ,
             the
             whole
             object
             is
             more
             known
             ,
             then
             any
             part
             thereof
             ;
             as
             when
             we
             see
             a
             Man
             ,
             the
             Conception
             or
             
               whole
               Idea
            
             of
             that
             Man
             is
             first
             or
             more
             known
             ,
             then
             the
             
               particular
               Ideas
            
             of
             his
             being
             
               figurate
               ,
               animate
               ,
               and
               rationall
            
             ;
             that
             is
             ,
             we
             first
             see
             the
             whole
             Man
             ,
             and
             take
             notice
             of
             his
             Being
             ,
             before
             we
             observe
             in
             him
             those
             other
             Particulars
             .
             And
             therefore
             in
             any
             knowledge
             of
             the
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             ,
             or
             that
             any
             thing
             is
             ,
             the
             beginning
             of
             our
             search
             is
             from
             the
             
               whole
               Idea
            
             ;
             and
             contrarily
             ,
             in
             our
             knowledge
             of
             the
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             ,
             or
             of
             the
             Causes
             of
             any
             thing
             ,
             that
             is
             ,
             in
             the
             Sciences
             ,
             we
             have
             more
             knowledge
             of
             the
             Causes
             of
             the
             Parts
             ,
             then
             of
             the
             Whole
             .
             For
             the
             Cause
             of
             the
             Whole
             is
             compounded
             of
             the
             Causes
             of
             the
             Parts
             ;
             but
             it
             is
             necessary
             that
             we
             know
             the
             things
             that
             are
             to
             be
             compounded
             ,
             before
             we
             can
             know
             the
             whole
             Compound
             .
             Now
             by
             Parts
             ,
             I
             do
             not
             here
             mean
             Parts
             of
             the
             thing
             it self
             ,
             but
             Parts
             of
             its
             Nature
             ;
             as
             ,
             by
             the
             Parts
             of
             Man
             I
             do
             not
             understand
             his
             Head
             ,
             his
             Shoulders
             ,
             his
             Arms
             ,
             &c.
             but
             his
             Figure
             ,
             Quantity
             ,
             Motion
             ,
             Sense
             ,
             Reason
             ,
             and
             the
             like
             ;
             which
             Accidents
             being
             compounded
             or
             put
             together
             ,
             constitute
             the
             whole
             Nature
             of
             Man
             ,
             but
             not
             the
             man
             himselfe
             .
             And
             this
             is
             the
             meaning
             of
             that
             
             common
             saying
             ,
             namely
             ,
             that
             some
             things
             are
             more
             knowne
             to
             us
             ,
             others
             more
             known
             to
             Nature
             ;
             for
             I
             do
             not
             thinke
             that
             they
             which
             so
             distinguish
             ,
             mean
             ,
             that
             something
             is
             known
             to
             Nature
             ,
             which
             is
             known
             to
             no
             man
             ;
             and
             therefore
             ,
             by
             those
             things
             that
             are
             more
             known
             to
             Us
             ,
             we
             are
             to
             understand
             ,
             things
             we
             take
             notice
             of
             by
             our
             Senses
             ,
             and
             by
             more
             known
             to
             Nature
             ,
             those
             we
             acquire
             the
             knowledge
             of
             by
             Reason
             ;
             for
             in
             this
             sense
             it
             is
             ,
             that
             the
             Whole
             ,
             that
             is
             ,
             those
             things
             that
             have
             Universal
             Names
             ,
             (
             which
             for
             brevities
             sake
             I
             call
             Universall
             )
             are
             more
             knowne
             to
             us
             then
             the
             Parts
             ,
             that
             is
             ,
             such
             things
             as
             have
             Names
             lesse
             Universal
             ,
             (
             which
             I
             therefore
             call
             Singular
             ;
             )
             and
             the
             Causes
             of
             the
             Parts
             ,
             are
             more
             known
             to
             Nature
             then
             the
             Cause
             of
             the
             Whole
             ;
             that
             is
             ,
             Universalls
             then
             Singulars
             .
          
           
             3
             In
             the
             Study
             of
             Philosophy
             men
             search
             after
             Science
             either
             Simply
             ,
             or
             Indefinitely
             ;
             that
             is
             ,
             to
             know
             as
             much
             as
             they
             can
             ,
             without
             propounding
             to
             themselves
             any
             limited
             question
             ;
             or
             they
             enquire
             into
             the
             Cause
             of
             some
             determined
             Appearance
             ,
             or
             endeavour
             to
             find
             out
             the
             certainty
             of
             something
             in
             question
             ;
             as
             what
             is
             the
             cause
             of
             Light
             ,
             of
             Heat
             ,
             of
             Gravity
             ,
             of
             a
             Figure
             propounded
             ,
             and
             the
             like
             ;
             or
             in
             what
             Subiect
             any
             propounded
             Accident
             is
             inhaerent
             ;
             or
             what
             may
             conduce
             most
             to
             the
             generation
             of
             some
             propounded
             Effect
             from
             many
             Accidents
             ;
             or
             in
             what
             manner
             particular
             Causes
             ought
             to
             be
             compounded
             for
             the
             production
             of
             some
             certaine
             Effect
             .
             Now
             according
             to
             this
             variety
             of
             things
             in
             question
             ,
             sometimes
             the
             
               Analyticall
               Method
            
             is
             to
             be
             used
             ,
             and
             sometimes
             the
             Syntheticall
             .
          
           
             4
             But
             to
             those
             that
             search
             after
             Science
             indefinitely
             ,
             which
             consists
             in
             the
             knowledge
             of
             the
             Causes
             of
             all
             things
             ,
             as
             far
             forth
             as
             it
             may
             be
             attained
             ,
             and
             the
             Causes
             of
             Singular
             things
             are
             compounded
             of
             the
             Causes
             of
             Universall
             or
             Simple
             things
             ,
             it
             is
             necessary
             that
             they
             know
             the
             Causes
             of
             Universall
             things
             ,
             or
             of
             such
             Accidents
             as
             are
             common
             to
             all
             Bodies
             ,
             that
             is
             ,
             to
             all
             Matter
             ,
             before
             they
             can
             know
             the
             Causes
             of
             Singular
             things
             ,
             that
             is
             ,
             of
             those
             Accidents
             by
             which
             one
             thing
             is
             distinguished
             from
             another
             .
             And
             againe
             they
             must
             know
             what
             those
             Universall
             things
             are
             ,
             before
             they
             can
             know
             their
             Causes
             .
             Moreover
             ,
             seeing
             Universall
             things
             are
             contained
             in
             the
             Nature
             of
             Singular
             things
             ,
             the
             knowledge
             of
             them
             is
             to
             be
             acquired
             by
             Reason
             ,
             that
             is
             ,
             by
             Resolution
             .
             
             ▪
             For
             example
             ,
             if
             there
             be
             propounded
             a
             Conception
             or
             Idea
             of
             some
             Singular
             thing
             ,
             as
             of
             a
             Square
             ,
             this
             Square
             is
             to
             be
             resolved
             into
             a
             
               Plain
               ,
               terminated
               with
               a
               certaine
               number
               of
               equall
               and
               straight
               lines
               and
               right
               angles
               .
            
             For
             by
             this
             Resolution
             we
             have
             these
             things
             Universall
             or
             agreeable
             to
             all
             Matter
             ,
             namely
             ,
             
               Line
               ,
               Plain
            
             ,
             (
             which
             containes
             
               Superficies
               ,
               )
               Terminated
               ,
               Angle
               ,
               Straightness
               ,
               Rectitude
               and
               Equality
            
             ;
             and
             if
             we
             can
             find
             out
             the
             Causes
             of
             these
             ,
             we
             may
             compound
             them
             all
             together
             into
             the
             Cause
             of
             a
             Square
             .
             Againe
             ,
             if
             any
             man
             propound
             to
             himselfe
             the
             Conception
             of
             Gold
             ,
             he
             may
             by
             Resolving
             come
             to
             the
             Ideas
             of
             
               Solid
               ,
               Visible
               ,
               Heavy
            
             ,
             (
             that
             is
             ,
             tending
             to
             the
             Center
             of
             the
             Earth
             ,
             or
             downwards
             ,
             )
             and
             many
             other
             more
             Universall
             then
             Gold
             it selfe
             ;
             and
             these
             he
             may
             Resolve
             againe
             ,
             till
             he
             come
             to
             such
             things
             as
             are
             most
             Universall
             .
             And
             in
             this
             manner
             by
             Resolving
             continually
             ,
             we
             may
             come
             to
             know
             what
             those
             things
             are
             ,
             whose
             Causes
             being
             first
             known
             severally
             ,
             and
             afterwards
             compounded
             ,
             bring
             us
             to
             the
             Knowledge
             of
             Singular
             things
             .
             I
             conclude
             therefore
             ,
             that
             the
             Method
             of
             attaining
             to
             the
             Universall
             Knowledge
             of
             Things
             ,
             is
             purely
             Analyticall
             .
          
           
             5
             But
             the
             Causes
             of
             Universall
             things
             (
             of
             those
             at
             least
             that
             have
             any
             Cause
             )
             are
             manifest
             of
             themselues
             ,
             or
             (
             as
             they
             say
             commonly
             )
             known
             to
             Nature
             ;
             so
             that
             they
             need
             no
             Method
             at
             all
             ;
             for
             they
             have
             all
             but
             one
             Universall
             Cause
             ,
             which
             is
             Motion
             .
             For
             the
             variety
             of
             all
             Figures
             arises
             out
             of
             the
             variety
             of
             those
             Motions
             by
             which
             they
             are
             made
             ;
             and
             Motion
             cannot
             be
             understood
             to
             have
             any
             other
             Cause
             besides
             Motion
             ;
             nor
             has
             the
             Variety
             of
             those
             things
             we
             perceive
             by
             Sense
             ,
             as
             of
             
               Colours
               ,
               Sounds
               ,
               Savours
               ,
               &c.
            
             any
             other
             Cause
             then
             Motion
             ,
             residing
             partly
             in
             the
             Objects
             that
             work
             upon
             our
             Senses
             ,
             and
             partly
             in
             our selves
             ,
             in
             such
             manner
             ,
             as
             that
             it
             is
             manifestly
             some
             kind
             of
             Motion
             ,
             though
             we
             cannot
             without
             Ratiocination
             come
             to
             know
             what
             kind
             .
             For
             though
             many
             cannot
             understand
             till
             it
             be
             in
             some
             sort
             demonstrated
             to
             them
             ,
             that
             all
             Mutation
             consists
             in
             Motion
             ;
             yet
             this
             happens
             not
             from
             any
             obscurity
             in
             the
             thing
             it selfe
             ,
             (
             for
             it
             is
             not
             intelligible
             that
             any
             thing
             can
             depart
             either
             from
             Rest
             ,
             or
             from
             the
             Motion
             it
             has
             ,
             except
             by
             Motion
             ;
             )
             but
             either
             by
             having
             their
             Naturall
             Discourse
             corrupted
             with
             former
             Opinions
             received
             from
             their
             Masters
             ,
             or
             else
             for
             this
             ,
             that
             they
             do
             not
             at
             all
             bend
             their
             mind
             to
             the
             enquiring
             out
             of
             Truth
             ,
          
           
           
             6
             By
             the
             Knowledge
             therefore
             of
             Universalls
             ,
             and
             of
             their
             Causes
             (
             which
             are
             the
             first
             Principles
             by
             which
             we
             know
             the
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             of
             things
             ,
             )
             we
             have
             in
             the
             first
             place
             their
             Definitions
             ,
             (
             which
             are
             nothing
             but
             the
             explication
             of
             our
             Simple
             Conceptions
             .
             )
             For
             example
             ,
             he
             that
             has
             a
             true
             Conception
             of
             Place
             ,
             cannot
             be
             ignorant
             of
             this
             Definition
             ,
             
               Place
               is
               that
               space
               which
               is
               possessed
               or
               filled
               adaequately
               by
               some
               Body
            
             ;
             and
             so
             ,
             he
             that
             conceives
             Motion
             aright
             ,
             cannot
             but
             know
             ,
             that
             
               Motion
               is
               the
               privation
               of
               one
               Place
               ,
               and
               the
               acquisition
               of
               another
               .
            
             In
             the
             next
             place
             ,
             we
             have
             their
             Generations
             or
             Descriptions
             ;
             as
             ,
             (
             for
             example
             ,
             )
             that
             
               a
               Line
               is
               made
               by
               the
               Motion
               of
               a
               Point
               ,
               Superficies
               by
               the
               Motion
               of
               a
               Line
               ,
            
             and
             
               one
               Motion
               by
               another
               Motion
               ,
               &c.
               
            
             It
             remains
             ,
             that
             we
             enquire
             ,
             what
             Motion
             begets
             such
             and
             such
             Effects
             ;
             as
             ,
             what
             Motion
             makes
             a
             Straight
             line
             ,
             and
             what
             a
             Circular
             ;
             what
             Motion
             thrusts
             ,
             what
             drawes
             ,
             and
             by
             what
             way
             ;
             what
             makes
             a
             thing
             which
             is
             seen
             or
             heard
             ,
             to
             be
             seen
             or
             heard
             sometimes
             in
             one
             manner
             ,
             sometimes
             in
             another
             .
             Now
             the
             Method
             of
             this
             kind
             of
             Enquiry
             ,
             is
             Compositive
             .
             For
             first
             we
             are
             to
             observe
             what
             Effect
             a
             Body
             moved
             produceth
             ,
             when
             we
             consider
             nothing
             in
             it
             besides
             its
             Motion
             ;
             and
             we
             see
             presently
             that
             this
             makes
             a
             Line
             ,
             or
             length
             ;
             next
             ,
             what
             the
             Motion
             of
             a
             long
             Body
             produces
             ,
             which
             we
             find
             to
             be
             Superficies
             ;
             and
             so
             forwards
             ,
             till
             we
             see
             what
             the
             Effects
             of
             Simple
             Motion
             are
             ;
             and
             then
             in
             like
             manner
             ,
             we
             are
             to
             observe
             what
             proceeds
             from
             the
             Addition
             ,
             Multiplication
             ,
             Substraction
             and
             Division
             of
             these
             Motions
             ,
             and
             what
             Effects
             ,
             what
             Figures
             ,
             and
             what
             Properties
             they
             produce
             ;
             from
             which
             kind
             of
             Contemplation
             sprung
             that
             part
             of
             Philosophy
             which
             is
             called
             Geometry
             .
          
           
             From
             this
             consideration
             of
             what
             is
             produced
             by
             Simple
             Motion
             ,
             we
             are
             to
             passe
             to
             the
             consideration
             of
             what
             Effects
             one
             Body
             moved
             worketh
             upon
             another
             ;
             and
             because
             there
             may
             be
             Motion
             in
             all
             the
             severall
             parts
             of
             a
             Body
             ,
             yet
             so
             as
             that
             the
             whole
             Body
             remain
             still
             in
             the
             same
             place
             ,
             we
             must
             enquire
             ,
             first
             ,
             what
             Motion
             causeth
             such
             and
             such
             Motion
             in
             the
             whole
             ,
             that
             is
             ,
             when
             one
             Body
             invades
             another
             Body
             which
             is
             either
             at
             Rest
             ,
             or
             in
             Motion
             ,
             what
             way
             ,
             and
             with
             what
             swiftnesse
             the
             invaded
             Body
             shall
             move
             ;
             and
             again
             ,
             what
             Motion
             this
             second
             Body
             will
             generate
             in
             a
             third
             ,
             and
             so
             forwards
             .
             From
             which
             Contemplation
             shall
             be
             drawn
             that
             part
             of
             Philosophy
             which
             treats
             of
             Motion
             .
          
           
           
             In
             the
             Third
             place
             we
             must
             proceed
             to
             the
             Enquiry
             of
             such
             Effects
             ,
             as
             are
             made
             by
             the
             Motion
             of
             the
             Parts
             of
             any
             Body
             ,
             as
             ,
             how
             it
             comes
             to
             passe
             ,
             that
             things
             when
             they
             are
             the
             same
             ,
             yet
             seeme
             not
             to
             be
             the
             same
             ,
             but
             changed
             .
             And
             here
             the
             things
             we
             search
             after
             are
             sensible
             Qualities
             ,
             such
             as
             
               Light
               ,
               Colour
               ,
               Transparency
               ,
               Opacity
               ,
               Sound
               ,
               Odour
               ,
               Savour
               ,
               Heat
               ,
               Cold
            
             and
             the
             like
             ;
             which
             because
             they
             cannot
             be
             known
             till
             we
             know
             the
             Causes
             of
             Sense
             it selfe
             ,
             therefore
             the
             consideration
             of
             the
             Causes
             of
             
               Seeing
               ,
               Hearing
               ,
               Smelling
               ,
               Tasting
            
             and
             Touching
             belongs
             to
             this
             third
             place
             ;
             and
             all
             those
             qualities
             and
             Changes
             above
             mentioned
             are
             to
             be
             referred
             to
             the
             fourth
             place
             ;
             which
             two
             considerations
             comprehend
             that
             part
             of
             Philosophy
             which
             is
             called
             Physiques
             .
             And
             in
             these
             four
             parts
             is
             contained
             whatsoever
             in
             Naturall
             Philosophy
             may
             be
             explicated
             by
             Demonstration
             properly
             so
             called
             .
             For
             if
             a
             Cause
             were
             to
             be
             rendred
             of
             Natural
             Appearances
             in
             special
             ,
             as
             ,
             what
             are
             the
             Motions
             and
             Influences
             of
             the
             heavenly
             Bodies
             ,
             and
             of
             their
             parts
             ,
             the
             reason
             hereof
             must
             either
             be
             drawn
             from
             the
             parts
             of
             the
             Sciences
             above
             mentioned
             ,
             or
             no
             reason
             at
             all
             will
             be
             given
             ,
             but
             all
             left
             to
             uncertaine
             conjecture
             .
          
           
             After
             Physiques
             we
             must
             come
             to
             
               Morall
               Philosophy
            
             ;
             in
             which
             we
             are
             to
             consider
             the
             Motions
             of
             the
             Mind
             ,
             namely
             
               Appetite
               ,
               Aversion
               ,
               Love
               ,
               Benevolence
               ,
               Hope
               ,
               Fear
               ,
               Anger
               ,
               Emulation
               ,
               Envy
               ,
               &c.
            
             what
             Causes
             they
             have
             ,
             and
             of
             what
             they
             be
             Causes
             .
             And
             the
             reason
             why
             these
             are
             to
             be
             considered
             after
             Physiques
             ,
             is
             ,
             that
             they
             have
             their
             Causes
             in
             Sense
             and
             Imagination
             ,
             which
             are
             the
             Subject
             of
             Physicall
             Contemplation
             .
             Also
             the
             reason
             why
             all
             these
             Things
             are
             to
             be
             searched
             after
             in
             the
             order
             abovesaid
             ,
             is
             ,
             that
             Physiques
             cannot
             be
             understood
             ,
             except
             we
             know
             first
             what
             Motions
             are
             in
             the
             smallest
             parts
             of
             Bodies
             ;
             nor
             such
             Motion
             of
             Parts
             ,
             till
             we
             know
             what
             it
             is
             that
             makes
             another
             Body
             move
             ;
             nor
             this
             ,
             till
             we
             know
             what
             Simple
             Motion
             will
             effect
             .
             And
             because
             all
             Appearance
             of
             things
             to
             sense
             is
             determined
             ,
             and
             made
             to
             be
             of
             such
             and
             such
             Quality
             and
             Quantity
             by
             Compounded
             Motions
             ,
             every
             one
             of
             which
             has
             a
             certaine
             degree
             of
             Velocity
             ,
             and
             a
             certaine
             and
             determined
             way
             ;
             therefore
             in
             the
             first
             place
             we
             are
             to
             search
             out
             the
             wayes
             of
             Motion
             simply
             ,
             (
             in
             which
             Geometry
             consists
             ;
             )
             next
             the
             wayes
             of
             such
             generated
             Motions
             as
             are
             manifest
             ;
             and
             lastly
             the
             wayes
             of
             internal
             and
             invisible
             Motions
             ,
             (
             which
             is
             the
             
             Enquiry
             of
             Naturall
             Philosophers
             .
             )
             And
             therefore
             they
             that
             study
             Naturall
             Philosophy
             ,
             study
             in
             vaine
             ,
             except
             they
             begin
             at
             Geometry
             ;
             and
             such
             Writers
             or
             Disputers
             thereof
             ,
             as
             are
             ignorant
             of
             Geometry
             ,
             do
             but
             make
             their
             Readers
             and
             Hearers
             lose
             their
             time
             .
          
           
             7
             Civill
             and
             
               Morall
               Philosophy
            
             doe
             not
             so
             adhere
             to
             one
             another
             ,
             but
             that
             they
             may
             be
             severed
             .
             For
             the
             Causes
             of
             the
             Motions
             of
             the
             Mind
             are
             known
             not
             onely
             by
             Ratiocination
             ,
             but
             also
             by
             the
             Experience
             of
             every
             man
             that
             takes
             the
             paines
             to
             observe
             those
             Motions
             within
             himselfe
             .
             And
             therefore
             not
             only
             they
             that
             have
             attained
             the
             knowledge
             of
             the
             Passions
             and
             Perturbations
             of
             the
             Mind
             ,
             by
             the
             
               Syntheticall
               Method
            
             and
             from
             the
             very
             first
             Principles
             of
             Philosophy
             ,
             may
             by
             proceeding
             in
             the
             same
             way
             come
             to
             the
             Causes
             and
             Necessity
             of
             constituting
             Common-wealths
             ,
             and
             to
             get
             the
             Knowledge
             of
             what
             is
             Naturall
             Right
             ,
             and
             what
             are
             Civill
             Duties
             ;
             and
             in
             every
             kind
             of
             Government
             ,
             what
             are
             the
             Rights
             of
             the
             Commonwealth
             ,
             and
             all
             other
             Knowledge
             appertaining
             to
             Civill
             Philosophy
             ,
             for
             this
             reason
             ,
             that
             the
             Principles
             of
             the
             Politiques
             consist
             in
             the
             Knowledge
             of
             the
             Motions
             of
             the
             Mind
             ,
             and
             the
             Knowledge
             of
             these
             Motions
             from
             the
             knowledge
             of
             Sense
             and
             Imagination
             ;
             but
             even
             they
             also
             that
             have
             not
             learned
             the
             first
             part
             of
             Philosophy
             ,
             namely
             Geometry
             and
             Physiques
             ,
             may
             notwithstanding
             attain
             the
             Principles
             of
             Civill
             Philosophy
             ,
             by
             the
             
               Analyticall
               Method
            
             .
             For
             if
             a
             Question
             be
             propounded
             ,
             as
             
               Whether
               such
               an
               Action
               be
               Just
               or
               Uniust
            
             ;
             if
             that
             Uniust
             be
             resolved
             into
             
               Fact
               against
               Law
            
             ,
             and
             that
             notion
             of
             Law
             into
             the
             Command
             of
             him
             or
             them
             that
             have
             
               Coercive
               Power
            
             ;
             and
             that
             Power
             be
             derived
             from
             the
             Wills
             of
             Men
             that
             constitute
             such
             Power
             to
             the
             end
             they
             may
             live
             in
             Peace
             ,
             they
             may
             at
             last
             come
             to
             this
             ,
             that
             the
             Appetites
             of
             Men
             and
             the
             Passions
             of
             their
             Minds
             are
             such
             ,
             that
             unlesse
             they
             be
             restrained
             by
             some
             Power
             ,
             they
             will
             alwayes
             be
             making
             warre
             upon
             one
             another
             ;
             which
             may
             be
             known
             to
             be
             so
             by
             any
             mans
             experience
             ,
             that
             will
             but
             examine
             his
             owne
             Mind
             .
             And
             therefore
             from
             hence
             he
             may
             proceed
             by
             Compounding
             ,
             to
             the
             determination
             of
             the
             Justice
             or
             Injustice
             of
             any
             propounded
             Action
             .
             So
             that
             it
             is
             manifest
             by
             what
             has
             been
             said
             ,
             that
             the
             Method
             of
             Philosophy
             to
             such
             as
             seek
             Science
             simply
             ,
             without
             propounding
             to
             themselves
             the
             Solution
             of
             any
             Particular
             question
             ,
             is
             partly
             Analyticall
             ,
             
             and
             partly
             Syntheticall
             ;
             namely
             ,
             that
             which
             proceeds
             from
             Sense
             to
             the
             invention
             of
             Principles
             ,
             Analyticall
             ;
             and
             the
             rest
             Syntheticall
             .
          
           
             8
             To
             those
             that
             seek
             the
             Cause
             of
             some
             certaine
             and
             pro
             pounded
             Appearance
             or
             Effect
             ,
             it
             happens
             sometimes
             ,
             that
             they
             know
             not
             whether
             the
             thing
             whose
             Cause
             is
             sought
             after
             ,
             be
             Matter
             or
             Body
             ,
             or
             some
             Accident
             of
             a
             Body
             .
             For
             though
             in
             Geometry
             ,
             when
             the
             Cause
             is
             sought
             of
             Magnitude
             ,
             or
             Proportion
             ,
             or
             Figure
             ,
             it
             be
             certainly
             known
             that
             these
             things
             ,
             namely
             Magnitude
             ,
             Proportion
             and
             Figure
             are
             Accidents
             ;
             yet
             in
             Naturall
             Philosophy
             ,
             where
             all
             questions
             are
             concerning
             the
             Causes
             of
             the
             Phantasmes
             of
             sensible
             things
             ,
             it
             is
             not
             so
             easie
             to
             discern
             between
             the
             things
             themselves
             from
             which
             those
             Phantasmes
             proceed
             ,
             and
             the
             Appearances
             of
             those
             things
             to
             the
             sense
             ;
             which
             have
             deceived
             many
             ,
             especially
             when
             the
             Phantasmes
             have
             been
             made
             by
             Light.
             For
             Example
             ,
             a
             Man
             that
             looks
             upon
             the
             Sunne
             ,
             has
             a
             certaine
             shining
             Idea
             of
             ●●e
             Magnitude
             of
             about
             a
             fo●t
             over
             ;
             and
             this
             he
             calls
             the
             Sunne
             ,
             thoug●…e
             know
             the
             Sunne
             to
             be
             truly
             a
             great
             deale
             bigger
             and
             in
             like
             〈…〉
             ,
             the
             Phantasme
             of
             the
             same
             thing
             appears
             sometimes
             ●●und
             ▪
             by
             being
             〈…〉
             a
             ●arre
             off
             ,
             and
             sometimes
             square
             ,
             by
             being
             neerer
             .
             Whereupon
             ●t
             may
             well
             be
             doubted
             whether
             that
             Phantasme
             be
             Ma●…
             ▪
             or
             some
             Body
             Naturall
             ,
             or
             onely
             some
             Accident
             of
             a
             Body
             ;
             in
             the
             examination
             of
             which
             doubt
             we
             may
             use
             this
             Method
             .
             The
             Properties
             of
             Matter
             and
             Accidents
             already
             found
             out
             by
             Us
             by
             the
             
               Syntheticall
               Method
            
             from
             their
             Definitions
             ,
             are
             to
             be
             compared
             with
             the
             Idea
             we
             have
             before
             us
             ;
             and
             if
             it
             agree
             with
             the
             Properties
             of
             Matter
             or
             Body
             ,
             then
             it
             is
             a
             Body
             ;
             otherwise
             it
             is
             an
             Accident
             .
             Seeing
             therefore
             Matter
             cannot
             by
             any
             endeavour
             of
             ours
             be
             either
             Made
             or
             Destroyed
             ,
             or
             Encreased
             ,
             or
             Diminished
             ,
             or
             Moved
             out
             of
             its
             place
             ,
             whereas
             that
             Idea
             Appeares
             ,
             Vanishes
             ,
             is
             Encreased
             ,
             and
             Diminished
             ,
             and
             Moved
             hither
             and
             thither
             at
             pleasure
             ;
             we
             may
             certainly
             conclude
             that
             it
             is
             not
             a
             Body
             ,
             but
             an
             Accident
             onely
             .
             And
             this
             Method
             is
             Syntheticall
             .
          
           
             9
             But
             if
             there
             be
             a
             doubt
             made
             concerning
             the
             Subject
             of
             any
             known
             Accident
             ,
             (
             for
             this
             may
             be
             doubted
             sometimes
             ,
             as
             in
             the
             praecedent
             example
             doubt
             may
             be
             made
             in
             what
             Subject
             that
             Splendor
             and
             apparent
             Magnitude
             of
             the
             Sunne
             is
             )
             then
             our
             enquiry
             
             must
             proceed
             in
             this
             manner
             .
             First
             ,
             Matter
             in
             Generall
             must
             be
             divided
             into
             parts
             ,
             as
             into
             Object
             ,
             Medium
             ,
             and
             the
             Sentient
             it selfe
             ,
             or
             such
             other
             parts
             as
             seem
             most
             conformable
             to
             the
             thing
             propounded
             .
             Next
             ,
             these
             parts
             are
             severally
             to
             be
             examined
             how
             they
             agree
             with
             the
             Definition
             of
             the
             Subject
             ;
             and
             such
             of
             them
             as
             are
             not
             capable
             of
             that
             Accident
             are
             to
             be
             rejected
             .
             For
             example
             ,
             If
             by
             any
             true
             Ratiocination
             the
             Sunne
             be
             found
             to
             be
             greater
             then
             its
             apparent
             Magnitude
             ,
             then
             that
             Magnitude
             is
             not
             in
             the
             Sunne
             ;
             If
             the
             Sunne
             be
             in
             one
             determined
             straight
             line
             ,
             and
             one
             determined
             distance
             ,
             and
             the
             Magnitude
             and
             Splendor
             be
             seen
             in
             more
             lines
             and
             distances
             then
             one
             ,
             as
             it
             is
             in
             Reflection
             or
             Refraction
             ,
             then
             neither
             that
             Splendor
             nor
             apparent
             Magnitude
             are
             in
             the
             Sun
             it self
             ,
             and
             therefore
             the
             Body
             of
             the
             Sun
             cannot
             be
             the
             Subject
             of
             that
             Splendor
             and
             Magnitude
             .
             And
             for
             the
             same
             reasons
             the
             Aire
             and
             other
             parts
             will
             be
             rejected
             ,
             till
             at
             last
             nothing
             remain
             which
             can
             be
             the
             Subject
             of
             that
             Splendor
             and
             Magnitude
             but
             the
             Sentient
             it selfe
             .
             And
             this
             Method
             ,
             in
             regard
             the
             Subject
             is
             divided
             into
             parts
             is
             Analitycall
             ;
             and
             in
             regard
             the
             Properties
             both
             of
             the
             Subject
             and
             Accident
             are
             compared
             with
             the
             Accident
             concerning
             whose
             Subject
             the
             enquiry
             is
             made
             ,
             it
             is
             Syntheticall
             .
          
           
             10
             But
             when
             we
             seek
             after
             the
             Cause
             of
             any
             propounded
             Effect
             ;
             we
             must
             in
             the
             first
             place
             get
             into
             our
             Mind
             an
             exact
             Notion
             or
             Idea
             of
             that
             which
             we
             call
             Cause
             ,
             namely
             ,
             that
             
               A
               Cause
               is
               the
               Summe
               or
               Aggregate
               of
               all
               such
               Accidents
               both
               in
               the
               Agents
               and
               the
               Patient
               ,
               as
               concurre
               to
               the
               producing
               of
               the
               Effect
               propounded
               ;
               all
               which
               existing
               together
               ,
               it
               cannot
               be
               understood
               but
               that
               the
               Effect
               existeth
               with
               them
               ;
               or
               that
               it
               cannot
               possibly
               exist
               if
               any
               one
               of
               them
               be
               absent
               .
            
             This
             being
             known
             ,
             in
             the
             next
             place
             we
             must
             examine
             singly
             every
             Accident
             that
             accompanies
             or
             praecedes
             the
             Effect
             ,
             as
             farre
             forth
             as
             it
             seemes
             to
             conduce
             in
             any
             manner
             to
             the
             production
             of
             the
             same
             ,
             and
             see
             whether
             the
             propounded
             Effect
             may
             be
             conceived
             to
             exist
             ,
             without
             the
             existence
             of
             any
             of
             those
             Accidents
             ;
             and
             by
             this
             meanes
             separate
             such
             Accidents
             as
             do
             not
             concurre
             ,
             from
             such
             as
             concurre
             to
             produce
             the
             said
             Effect
             ;
             which
             being
             done
             ,
             we
             are
             to
             put
             together
             the
             concurring
             Accidents
             ,
             and
             consider
             whether
             we
             can
             possibly
             conceive
             that
             when
             these
             are
             all
             present
             ,
             the
             Effect
             propounded
             will
             not
             follow
             ;
             and
             if
             it
             be
             evident
             that
             
             the
             Effect
             will
             follow
             ,
             then
             that
             Aggregate
             of
             Accidents
             is
             the
             entire
             Cause
             ,
             otherwise
             not
             ;
             but
             we
             must
             still
             search
             out
             and
             put
             together
             other
             Accidents
             .
             For
             example
             ,
             if
             the
             Cause
             of
             Light
             be
             propounded
             to
             be
             sought
             out
             ;
             first
             ,
             we
             examine
             things
             without
             us
             ,
             and
             find
             that
             whensoever
             Light
             appeares
             ,
             there
             is
             some
             principall
             Object
             ,
             as
             it
             were
             the
             fountaine
             of
             Light
             ,
             without
             which
             we
             cannot
             have
             any
             perception
             of
             Light
             ;
             and
             therefore
             the
             concurrence
             of
             that
             Object
             is
             necessary
             to
             the
             generation
             of
             Light.
             Next
             we
             consider
             the
             Medium
             ,
             and
             find
             that
             unlesse
             it
             be
             disposed
             in
             a
             certaine
             manner
             ,
             namely
             ,
             that
             it
             be
             transparent
             ,
             though
             the
             Object
             remain
             the
             same
             ,
             yet
             the
             Effect
             will
             not
             follow
             ;
             and
             therefore
             the
             concurrence
             of
             Transparency
             is
             also
             necessary
             to
             the
             generation
             of
             Light.
             Thirdly
             ,
             we
             observe
             our
             own
             Body
             ,
             and
             find
             that
             by
             the
             indisposition
             of
             the
             Eyes
             ,
             the
             Brain
             ,
             the
             Nerves
             ,
             and
             the
             Heart
             ,
             that
             is
             ,
             by
             Obstructions
             ,
             Stupidity
             and
             Debility
             we
             are
             deprived
             of
             Light
             ,
             so
             that
             a
             fitting
             disposition
             of
             the
             Organs
             to
             receive
             impressions
             from
             without
             is
             likewise
             a
             necessary
             part
             of
             the
             Cause
             of
             Light.
             Again
             ,
             of
             all
             the
             Accidents
             inhaerent
             in
             the
             Object
             ,
             there
             is
             none
             that
             can
             conduce
             to
             the
             effecting
             of
             Light
             ,
             but
             onely
             Action
             ,
             (
             or
             a
             certain
             Motion
             ,
             )
             which
             cannot
             be
             conceived
             to
             be
             wanting
             ,
             whensoever
             the
             Effect
             is
             present
             ;
             for
             ,
             that
             any
             thing
             may
             shine
             ,
             it
             is
             not
             requisite
             that
             it
             be
             of
             such
             or
             such
             ●agnitude
             or
             Figure
             ,
             or
             that
             the
             whole
             Body
             of
             it
             be
             moved
             out
             of
             the
             place
             it
             is
             in
             ,
             (
             unlesse
             it
             may
             perhaps
             be
             said
             ,
             that
             in
             the
             Sun
             or
             other
             Body
             ,
             that
             which
             causeth
             Light
             is
             the
             light
             it
             hath
             in
             it selfe
             ;
             which
             yet
             is
             but
             a
             trifling
             exception
             ,
             seeing
             nothing
             is
             meant
             thereby
             but
             the
             Cause
             of
             Light
             ;
             as
             if
             any
             man
             should
             say
             that
             the
             Cause
             of
             Light
             is
             that
             in
             the
             Sunne
             which
             produceth
             it
             ;
             )
             it
             remaines
             therefore
             that
             the
             Action
             by
             which
             Light
             is
             generated
             ,
             is
             Motion
             only
             in
             the
             parts
             of
             the
             Object
             .
             Which
             being
             understood
             ,
             we
             may
             easily
             conceive
             what
             it
             is
             the
             Medium
             contributes
             ,
             namely
             ,
             the
             continuation
             of
             that
             Motion
             to
             the
             Eye
             ;
             and
             lastly
             what
             the
             Eye
             and
             the
             rest
             of
             the
             Organs
             of
             the
             Sentient
             contribute
             ,
             namely
             ,
             the
             continuation
             of
             the
             same
             Motion
             to
             the
             last
             Organ
             of
             Sense
             ,
             the
             Heart
             .
             And
             in
             this
             manner
             the
             Cause
             of
             Light
             may
             be
             made
             up
             of
             Motion
             continued
             from
             the
             Original
             of
             the
             same
             Motion
             ,
             to
             the
             Original
             of
             Vitall
             Motion
             ,
             Light
             being
             nothing
             but
             the
             alteration
             of
             Vitall
             Motion
             ,
             made
             by
             the
             impression
             upon
             it
             of
             Motion
             continued
             from
             the
             Object
             .
             
             But
             I
             give
             this
             onely
             for
             an
             example
             ,
             for
             I
             shall
             speak
             more
             at
             large
             of
             Light
             ,
             and
             the
             generation
             of
             it
             in
             its
             proper
             place
             .
             In
             the
             mean
             time
             it
             is
             manifest
             ,
             that
             in
             the
             searching
             out
             of
             Causes
             ,
             there
             is
             need
             partly
             of
             the
             Analyticall
             ,
             and
             partly
             of
             the
             
               Syntheticall
               Method
            
             ;
             of
             the
             Analyticall
             ,
             to
             conceive
             how
             circumstances
             conduce
             severally
             to
             the
             production
             of
             Effects
             ;
             and
             of
             the
             Syntheticall
             ,
             for
             the
             adding
             together
             and
             compounding
             of
             what
             they
             can
             effect
             singly
             by
             themselves
             .
             And
             thus
             much
             may
             serve
             for
             the
             Method
             of
             Invention
             .
             It
             remaines
             that
             I
             speake
             of
             the
             Method
             of
             Teaching
             ,
             that
             is
             ,
             of
             Demonstration
             ,
             and
             of
             the
             Meanes
             by
             which
             we
             demonstrate
             .
          
           
             11
             In
             the
             Method
             of
             Invention
             the
             use
             of
             words
             consists
             in
             this
             ,
             that
             they
             may
             serve
             for
             Marks
             ,
             by
             which
             ,
             whatsoever
             we
             have
             found
             out
             may
             be
             recalled
             to
             memory
             ;
             for
             without
             this
             all
             our
             Inventions
             perish
             ,
             nor
             will
             it
             be
             possible
             for
             us
             to
             go
             on
             from
             Principles
             beyond
             a
             Syllogisme
             or
             two
             ,
             by
             reason
             of
             the
             weaknesse
             of
             Memory
             .
             For
             example
             ,
             if
             any
             man
             by
             considering
             a
             Triangle
             set
             before
             him
             ,
             should
             find
             that
             all
             its
             angles
             together
             taken
             are
             equall
             to
             two
             right
             angles
             ,
             and
             that
             by
             thinking
             of
             the
             same
             tacitely
             ,
             without
             any
             use
             of
             words
             either
             understood
             or
             expressed
             ;
             and
             it
             should
             happen
             afterwards
             that
             another
             Triangle
             unlike
             the
             former
             ,
             or
             the
             same
             in
             different
             scituation
             should
             be
             offered
             to
             his
             consideration
             ,
             he
             would
             not
             know
             readily
             whether
             the
             same
             property
             were
             in
             this
             last
             or
             no
             ;
             but
             would
             be
             forced
             as
             often
             as
             a
             different
             Triangle
             were
             brought
             before
             him
             (
             and
             the
             difference
             of
             Triangles
             is
             infinite
             )
             to
             begin
             his
             contemplation
             anew
             ;
             which
             he
             would
             have
             no
             need
             to
             do
             if
             he
             had
             the
             use
             of
             Names
             ;
             for
             every
             Universal
             Name
             denotes
             the
             conceptions
             we
             have
             of
             infinite
             Singular
             things
             ..
             Neverthelesse
             as
             I
             said
             above
             ,
             they
             serve
             as
             Markes
             for
             the
             helpe
             of
             our
             Memory
             ,
             whereby
             we
             register
             to
             our selves
             our
             own
             Inventions
             ;
             but
             not
             as
             Signes
             by
             which
             we
             declare
             the
             same
             to
             others
             ;
             so
             that
             a
             man
             may
             be
             a
             Philosopher
             alone
             by
             himselfe
             without
             any
             Master
             ;
             Adam
             had
             this
             capacity
             ;
             But
             to
             Teach
             ,
             that
             is
             to
             Demonstrate
             ,
             supposes
             two
             at
             the
             least
             ,
             and
             Syllogisticall
             Speech
             .
          
           
             12
             And
             seeing
             Teaching
             is
             nothing
             but
             leading
             the
             Mind
             of
             him
             we
             teach
             ,
             to
             the
             knowledge
             of
             our
             Inventions
             ,
             in
             that
             Track
             by
             which
             we
             attained
             the
             same
             with
             our
             own
             Mind
             ;
             therefore
             
             the
             same
             Method
             that
             served
             for
             our
             Invention
             ,
             will
             serve
             also
             for
             Demonstration
             to
             others
             ,
             saving
             that
             we
             omit
             the
             first
             part
             of
             Method
             which
             proceeded
             from
             the
             Sense
             of
             Things
             to
             Universal
             Principles
             ;
             which
             because
             they
             are
             Principles
             ,
             cannot
             be
             demonstrated
             ;
             and
             seeing
             they
             are
             known
             by
             Nature
             (
             as
             was
             said
             above
             in
             the
             5th
             .
             Article
             )
             they
             need
             no
             Demonstration
             ,
             though
             they
             need
             Explication
             .
             The
             whole
             Method
             therefore
             of
             Demonstration
             is
             Syntheticall
             ,
             consisting
             of
             that
             order
             of
             Speech
             ,
             which
             begins
             from
             Primary
             or
             most
             Universall
             Propositions
             ,
             which
             are
             manifest
             of
             themselves
             ,
             and
             proceeds
             by
             a
             perpetuall
             composition
             of
             Propositions
             into
             Syllogismes
             ,
             till
             at
             last
             the
             Learner
             understand
             the
             truth
             of
             the
             Conclusion
             sought
             after
             .
          
           
             13
             Now
             such
             Principles
             are
             nothing
             but
             Definitions
             ;
             whereof
             there
             are
             two
             sorts
             ;
             one
             ,
             of
             Names
             that
             signifie
             such
             things
             as
             have
             some
             conceiveable
             Cause
             ,
             and
             another
             of
             such
             Names
             as
             signifie
             things
             of
             which
             we
             can
             conceive
             no
             Cause
             at
             all
             .
             Names
             of
             the
             former
             kind
             are
             ,
             Body
             or
             
               Matter
               ,
               Quantity
            
             or
             
               Extension
               ,
               Motion
            
             ,
             and
             whatsoever
             is
             common
             to
             all
             Matter
             .
             Of
             the
             second
             kind
             are
             ,
             
               such
               a
               Body
               ,
               such
               and
               so
               great
               Motion
               ,
               so
               great
               Magnitude
               ,
               such
               Figure
               ,
            
             and
             whatsoever
             we
             can
             distinguish
             one
             Body
             from
             another
             by
             .
             And
             Names
             of
             the
             former
             kind
             are
             well
             enough
             defined
             ,
             when
             by
             Speech
             as
             short
             as
             may
             be
             ,
             we
             raise
             in
             the
             Mind
             of
             the
             Hearer
             perfect
             and
             cleer
             Ideas
             or
             Conceptions
             of
             the
             Things
             named
             ,
             as
             when
             we
             Define
             Motion
             to
             be
             
               the
               leaving
               of
               one
               place
               ,
               and
               the
               acquiring
               of
               another
               continually
            
             ;
             for
             though
             no
             Thing
             Moved
             ,
             nor
             any
             Cause
             of
             Motion
             be
             in
             that
             Definition
             ,
             yet
             at
             the
             hearing
             of
             that
             Speech
             ,
             there
             will
             come
             into
             the
             Mind
             of
             the
             Hearer
             an
             Idea
             of
             Motion
             cl●er
             enough
             .
             But
             Definitions
             of
             things
             which
             may
             be
             understood
             to
             have
             some
             Cause
             ,
             must
             consist
             of
             such
             Names
             as
             expresse
             the
             Cause
             or
             Manner
             of
             their
             Generation
             ,
             as
             when
             we
             Define
             a
             Circle
             to
             be
             a
             Figure
             made
             by
             the
             circumduction
             of
             a
             straight
             line
             in
             a
             plaine
             ,
             &c.
             
             Besides
             Definitions
             ,
             there
             is
             no
             other
             Proposition
             that
             ought
             to
             be
             called
             Primary
             ,
             or
             (
             according
             to
             severe
             truth
             )
             be
             received
             into
             the
             number
             of
             Principles
             .
             For
             those
             Axiomes
             of
             Euclide
             ,
             seeing
             they
             may
             be
             demonstrated
             are
             no
             Principles
             of
             Demonstration
             ,
             though
             they
             have
             by
             the
             consent
             of
             all
             Men
             gotten
             the
             authority
             of
             Principles
             ,
             because
             they
             need
             not
             be
             Demonstrated
             .
             Also
             those
             Petitious
             or
             Postulata
             (
             as
             they
             call
             them
             )
             though
             they
             be
             Principles
             
             yet
             they
             are
             not
             Principles
             of
             Demonstration
             ,
             but
             of
             Construction
             onely
             ;
             that
             is
             ,
             not
             of
             Science
             ,
             but
             of
             Power
             ;
             or
             (
             which
             is
             all
             one
             )
             not
             of
             Theoremes
             ,
             which
             are
             Speculations
             ,
             but
             of
             Problemes
             ,
             which
             belong
             to
             Practice
             ,
             or
             the
             doing
             of
             something
             .
             But
             as
             for
             those
             common
             received
             Opinions
             ,
             
               Nature
               abhorres
               Vanity
               ,
               Nature
               doth
               nothing
               in
               Vaine
               ,
            
             and
             the
             like
             ,
             which
             are
             neither
             evident
             in
             themselves
             ,
             nor
             at
             all
             to
             be
             demonstrated
             ,
             and
             which
             are
             oftner
             false
             then
             true
             ,
             they
             are
             much
             lesse
             to
             be
             ackowledged
             for
             Principles
             .
          
           
             To
             returne
             therefore
             to
             Definitions
             ,
             The
             reason
             why
             I
             say
             that
             the
             Cause
             and
             Generation
             of
             such
             things
             as
             have
             any
             Cause
             or
             Generation
             ,
             ought
             to
             enter
             into
             their
             Definitions
             ,
             is
             this
             .
             The
             End
             of
             Science
             ,
             is
             the
             Demonstration
             of
             the
             Causes
             and
             Generations
             of
             Things
             ;
             which
             if
             they
             be
             not
             in
             the
             Definitions
             ,
             they
             cannot
             be
             found
             in
             the
             Conclusion
             of
             the
             first
             Syllogisme
             that
             is
             made
             from
             those
             Definitions
             ;
             and
             if
             they
             be
             not
             in
             the
             first
             Conclusion
             ,
             they
             will
             not
             be
             found
             in
             any
             further
             Conclusion
             deduced
             from
             that
             ;
             and
             therefore
             by
             proceeding
             in
             this
             manner
             we
             shall
             never
             come
             to
             Science
             ;
             which
             is
             against
             the
             scope
             and
             intention
             of
             Demonstration
             .
          
           
             14
             Now
             seeing
             Definitions
             (
             as
             I
             have
             said
             )
             are
             Principles
             or
             Primary
             Propositions
             ,
             they
             are
             therefore
             Speeches
             ;
             and
             seeing
             they
             are
             used
             for
             the
             raising
             of
             an
             Idea
             of
             some
             Thing
             in
             the
             mind
             of
             the
             Learner
             ,
             whensoever
             that
             Thing
             has
             a
             Name
             ,
             the
             Definition
             of
             it
             can
             be
             nothing
             but
             the
             Explication
             of
             that
             Name
             by
             Speech
             ;
             and
             if
             that
             Name
             be
             given
             it
             for
             some
             compounded
             Conception
             ,
             the
             Definition
             is
             nothing
             but
             a
             Resolution
             of
             that
             Name
             into
             its
             most
             Universall
             parts
             .
             As
             when
             we
             define
             Man
             ,
             saying
             ,
             
               Man
               is
               a
               Body
               Animated
               ,
               Sentient
               ,
               Rationall
               ,
            
             those
             Names
             
               Body
               Animated
               ,
               &c.
            
             are
             parts
             of
             that
             whole
             Name
             Man
             ;
             so
             that
             Definitions
             of
             this
             kind
             alwayes
             consist
             of
             Genus
             and
             Difference
             ;
             the
             former
             Names
             being
             all
             till
             the
             last
             ,
             Generall
             ;
             and
             the
             last
             of
             all
             ,
             Difference
             .
             But
             if
             any
             Name
             be
             the
             most
             Universall
             in
             its
             kind
             ,
             then
             the
             Definition
             of
             it
             cannot
             consist
             of
             Genus
             and
             Difference
             ,
             but
             is
             to
             be
             made
             by
             such
             circumlocution
             as
             best
             explicateth
             the
             force
             of
             that
             Name
             .
             Again
             ,
             it
             is
             possible
             ,
             and
             happens
             often
             that
             the
             Genus
             and
             Difference
             are
             put
             together
             ,
             and
             yet
             make
             no
             Definition
             ;
             as
             these
             Words
             
               a
               Straight
               Line
            
             containe
             
             both
             the
             Genus
             and
             Difference
             ;
             but
             are
             not
             a
             Definition
             ,
             unlesse
             we
             should
             thinke
             a
             Straight
             Line
             may
             be
             thus
             defined
             ,
             
               A
               Straight
               Line
               is
               a
               Straight
               Line
            
             ;
             and
             yet
             if
             there
             were
             added
             another
             Name
             consisting
             of
             different
             Words
             ,
             but
             signifying
             the
             same
             thing
             which
             these
             signifie
             ,
             then
             these
             might
             be
             the
             Definition
             of
             that
             Name
             .
             From
             what
             has
             been
             said
             it
             may
             be
             understood
             how
             a
             Definition
             ought
             to
             be
             defined
             ,
             namely
             ,
             
               That
               it
               is
               a
               Proposition
               ,
               whose
               Praedicate
               Resolves
               the
               Subiect
               ,
               when
               it
               may
               ,
               and
               when
               it
               may
               not
               ,
               it
               exemplifies
               the
               same
               .
            
          
           
             15
             The
             Properties
             of
             a
             Definition
             are
             ,
          
           
             First
             ,
             that
             it
             takes
             away
             Aequivocation
             ,
             as
             also
             all
             that
             multitude
             of
             Distinctions
             ,
             which
             are
             used
             by
             such
             as
             think
             they
             may
             learn
             Philosophy
             by
             Disputation
             .
             For
             the
             Nature
             of
             a
             Definition
             is
             to
             define
             ,
             that
             is
             ,
             to
             determine
             the
             signification
             of
             the
             defined
             Name
             ,
             and
             to
             pare
             from
             it
             all
             other
             Signification
             besides
             what
             is
             contained
             in
             the
             Definition
             it selfe
             ;
             and
             therefore
             one
             Definition
             does
             as
             much
             ,
             as
             all
             the
             Distinctions
             (
             how
             many
             soever
             )
             that
             can
             be
             used
             about
             the
             Name
             defined
             .
          
           
             Secondly
             ,
             That
             it
             gives
             an
             Universall
             Notion
             of
             the
             thing
             defined
             ,
             representing
             a
             certaine
             Universall
             Picture
             thereof
             ,
             not
             to
             the
             Eye
             ,
             but
             to
             the
             Mind
             .
             For
             as
             when
             one
             paints
             a
             Man
             ,
             he
             paints
             the
             image
             of
             some
             Man
             ;
             so
             he
             that
             defines
             the
             Name
             Man
             ,
             makes
             a
             Representation
             of
             some
             Man
             to
             the
             mind
             .
          
           
             Thirdly
             ,
             That
             it
             is
             not
             necessary
             to
             dispute
             whether
             Definitions
             are
             to
             be
             admitted
             or
             no.
             For
             when
             a
             Master
             is
             instructing
             his
             Scholar
             ,
             if
             the
             Scholar
             understand
             all
             the
             parts
             of
             the
             thing
             defined
             ,
             which
             are
             Resolved
             in
             the
             Definition
             ,
             and
             yet
             will
             not
             admit
             of
             the
             Definition
             ,
             there
             needs
             no
             further
             Controversie
             betwixt
             them
             ,
             it
             being
             all
             one
             as
             if
             he
             refused
             to
             be
             taught
             .
             But
             if
             he
             understand
             nothing
             ,
             then
             certainely
             the
             Definition
             is
             faulty
             ;
             for
             the
             nature
             of
             a
             Definition
             consists
             in
             this
             ,
             that
             it
             exhibit
             a
             cleare
             Idea
             of
             the
             thing
             defined
             ;
             and
             Principles
             are
             either
             known
             by
             themselves
             ,
             or
             else
             they
             are
             not
             Principles
             .
          
           
             Fourthly
             ,
             That
             in
             Philosophy
             ,
             Definitions
             are
             before
             defined
             Names
             .
             For
             in
             teaching
             Philosophy
             ,
             the
             first
             beginning
             is
             from
             Definitions
             ,
             and
             all
             progression
             in
             the
             same
             till
             we
             come
             to
             the
             Knowledge
             of
             the
             thing
             compounded
             ,
             is
             Compositive
             .
             Seeing
             therefore
             Definition
             is
             the
             explication
             of
             a
             Compounded
             Name
             
             by
             Resolution
             ,
             and
             the
             progression
             is
             from
             the
             parts
             to
             the
             compound
             ,
             Definitions
             must
             be
             understood
             before
             Compounded
             Names
             ;
             nay
             when
             the
             Names
             of
             the
             parts
             of
             any
             Speech
             be
             explicated
             ,
             is
             it
             not
             necessary
             that
             the
             Definition
             should
             be
             a
             Name
             Compounded
             of
             them
             .
             For
             example
             ,
             when
             these
             Names
             ,
             
               Aequilaterall
               ,
               Quadrilaterall
               ,
               Right-angled
            
             ,
             are
             sufficiently
             understood
             ,
             it
             is
             not
             necessary
             in
             Geometry
             that
             there
             should
             be
             at
             all
             such
             a
             Name
             as
             Square
             ;
             for
             defined
             Names
             are
             received
             in
             Philosophy
             for
             brevities
             sake
             onely
             .
          
           
             Fiftly
             ,
             That
             Compounded
             Names
             which
             are
             defined
             one
             way
             in
             some
             one
             part
             of
             Philosophy
             ,
             may
             in
             another
             part
             of
             the
             same
             be
             otherwise
             defined
             ;
             as
             a
             Parabola
             and
             an
             Hyperbole
             have
             one
             Definition
             in
             Geometry
             ,
             and
             another
             in
             Rhetorique
             ;
             for
             Definitions
             are
             instituted
             and
             serve
             for
             the
             understanding
             of
             the
             Doctrine
             which
             is
             treated
             of
             .
             And
             therefore
             as
             in
             one
             part
             of
             Philosophy
             ,
             a
             Definition
             may
             have
             in
             it
             some
             one
             fit
             Name
             for
             the
             more
             briefe
             explanation
             of
             some
             proposition
             in
             Geometry
             ;
             so
             it
             may
             have
             the
             same
             liberty
             in
             other
             parts
             of
             Philosophy
             ;
             for
             the
             use
             of
             Names
             is
             particular
             (
             even
             where
             many
             agree
             to
             the
             setling
             of
             them
             )
             and
             arbitrary
             .
          
           
             Sixtly
             ,
             That
             no
             Name
             can
             be
             defined
             by
             any
             one
             Word
             ;
             because
             no
             one
             Word
             is
             sufficient
             for
             the
             Resolving
             of
             one
             or
             more
             words
             .
          
           
             Seventhly
             ,
             That
             a
             defined
             Name
             ought
             not
             to
             be
             repeated
             in
             the
             Definition
             .
             For
             a
             defined
             Name
             ,
             is
             the
             whole
             Compound
             ,
             and
             a
             Definition
             is
             the
             Resolution
             of
             that
             Compound
             into
             parts
             ;
             but
             no
             Totall
             can
             be
             part
             of
             it selfe
             .
          
           
             16
             Any
             two
             Definitions
             that
             may
             be
             compounded
             into
             a
             Syllogisme
             ,
             produce
             a
             Conclusion
             ;
             which
             because
             it
             is
             derived
             from
             Principles
             ,
             that
             is
             ,
             from
             Definitions
             ,
             is
             said
             to
             be
             Demonstrated
             ;
             and
             the
             Derivation
             or
             Composition
             it selfe
             is
             called
             a
             Demonstration
             .
             In
             like
             manner
             ,
             if
             a
             Syllogisme
             be
             made
             of
             two
             Propositions
             ,
             whereof
             one
             is
             a
             Definition
             ,
             the
             other
             a
             Demonstrated
             Conclusion
             ,
             or
             neither
             of
             them
             is
             a
             Definition
             ,
             but
             both
             formerly
             demonstrated
             ,
             that
             Syllogisme
             is
             also
             called
             a
             Demonstration
             ,
             and
             so
             successively
             .
             The
             Definition
             therefore
             of
             a
             Demonstration
             is
             this
             ,
             A
             DEMONSTRATION
             
               is
               a
               Syllogism
               or
               Series
               of
               Syllogisms
               derived
               and
               continued
               from
               the
               Definitions
               of
               Names
               ,
               to
               the
               last
               Conclusion
               .
            
             
             And
             from
             hence
             it
             may
             be
             understood
             ,
             that
             all
             true
             Ratiocination
             ,
             which
             taketh
             its
             beginning
             from
             true
             Principles
             ,
             produceth
             Science
             ,
             and
             is
             true
             Demonstration
             .
             For
             as
             for
             the
             Originall
             of
             the
             Name
             ,
             although
             that
             which
             the
             Greeks
             called
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             ,
             and
             the
             Latines
             Demonstratio
             was
             understood
             by
             them
             for
             that
             sort
             onely
             of
             Ratiocination
             ,
             in
             which
             by
             the
             describing
             of
             certaine
             Lines
             and
             Figures
             ,
             they
             placed
             the
             thing
             they
             were
             to
             prove
             ,
             as
             it
             were
             before
             mens
             Eyes
             ,
             which
             is
             properly
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             ,
             or
             to
             shew
             by
             the
             Figure
             ;
             yet
             they
             seem
             to
             have
             done
             it
             for
             this
             reason
             ,
             that
             unlesse
             it
             were
             in
             Geometry
             (
             in
             which
             only
             there
             is
             place
             for
             such
             Figures
             )
             there
             was
             no
             Ratiocination
             certaine
             ,
             and
             ending
             in
             Science
             ,
             their
             Doctrines
             concerning
             all
             other
             things
             being
             nothing
             but
             Controversie
             and
             Clamour
             ;
             which
             neverthelesse
             hapned
             not
             because
             the
             Truth
             to
             which
             they
             pretended
             could
             not
             be
             made
             evident
             without
             Figures
             ,
             but
             because
             they
             wanted
             true
             Principles
             ,
             from
             which
             they
             might
             derive
             their
             Ratiocination
             ;
             and
             therefore
             there
             is
             no
             reason
             but
             that
             if
             true
             Definitions
             were
             praemised
             in
             all
             sorts
             of
             Doctrines
             ,
             the
             Demonstrations
             also
             would
             be
             true
             .
          
           
             17
             It
             is
             proper
             to
             Methodical
             Demonstration
             ,
          
           
             First
             ,
             That
             there
             be
             a
             true
             Succession
             of
             one
             Reason
             to
             another
             ,
             according
             to
             the
             Rules
             of
             Syllogizing
             delivered
             above
             .
          
           
             Secondly
             ,
             That
             the
             Praemisses
             of
             all
             Syllogismes
             be
             demonstrated
             from
             the
             first
             Definitions
             .
          
           
             Thirdly
             ,
             That
             after
             Definitions
             ,
             he
             that
             Teaches
             or
             Demonstrates
             any
             thing
             ,
             proceed
             in
             the
             same
             Method
             by
             which
             he
             found
             it
             out
             ;
             namely
             ,
             that
             in
             the
             first
             place
             those
             things
             be
             demonstrated
             which
             immediately
             succeed
             to
             Universal
             Definitions
             (
             in
             which
             is
             contained
             that
             part
             of
             Philosophy
             which
             is
             called
             
               Philosophia
               Prima
               .
            
             )
             Next
             ,
             those
             things
             which
             may
             be
             demonstrated
             by
             Simple
             Motion
             (
             in
             which
             Geometry
             consists
             .
             )
             After
             Geome
             try
             ,
             such
             things
             as
             may
             be
             taught
             or
             shewed
             by
             manifest
             Action
             ,
             that
             is
             ,
             by
             Thrusting
             from
             ,
             or
             Pulling
             towards
             .
             And
             after
             these
             ,
             the
             Motion
             or
             Mutation
             of
             the
             invisible
             parts
             of
             Things
             ,
             and
             the
             Doctrine
             of
             Sense
             &
             Imagination
             &
             of
             the
             internal
             Passions
             ,
             especially
             those
             of
             Men
             ,
             in
             which
             are
             comprehended
             the
             Grounds
             of
             Civil
             Duties
             ,
             or
             Civil
             Philosophy
             ;
             which
             takes
             up
             the
             last
             place
             .
             And
             that
             this
             Method
             ought
             to
             be
             kept
             in
             all
             sorts
             of
             Philosophy
             ,
             is
             evident
             from
             hence
             ,
             that
             such
             things
             as
             I
             have
             said
             are
             to
             be
             taught
             
             last
             cannot
             be
             demonstrated
             ,
             till
             such
             as
             are
             propounded
             to
             be
             first
             treated
             of
             ,
             be
             fully
             understood
             .
             Of
             which
             Method
             no
             other
             Example
             can
             be
             given
             ,
             but
             that
             Treatise
             of
             the
             Elements
             of
             Philosophy
             ,
             which
             I
             shall
             begin
             in
             the
             next
             Chapter
             ,
             and
             continue
             to
             the
             end
             of
             the
             worke
             .
          
           
             18
             Besides
             those
             Paralogismes
             ,
             whose
             fault
             lies
             either
             in
             the
             Falsity
             of
             the
             Praemisses
             ,
             or
             the
             want
             of
             true
             Composition
             ,
             of
             which
             I
             have
             spoken
             in
             the
             praecedent
             Chapter
             ,
             there
             are
             two
             more
             which
             are
             frequent
             in
             Demonstration
             ;
             one
             whereof
             is
             commonly
             called
             
               Petitio
               Principii
            
             ;
             the
             other
             is
             the
             supposing
             of
             a
             
               False
               Cause
            
             ;
             and
             these
             do
             not
             onely
             deceive
             Unskilfull
             Learners
             but
             sometimes
             Masters
             themselves
             ,
             by
             making
             them
             take
             that
             for
             well
             demonstrated
             which
             is
             not
             demonstrated
             at
             all
             .
             
               Petitio
               Principii
            
             ,
             is
             when
             the
             Conclusion
             to
             be
             proved
             ,
             is
             disguised
             in
             other
             Words
             ,
             and
             put
             for
             the
             Definition
             or
             Principle
             from
             whence
             it
             is
             to
             be
             demonstrated
             ;
             and
             thus
             by
             putting
             for
             the
             Cause
             of
             the
             Thing
             sought
             ,
             either
             the
             Thing
             it selfe
             or
             some
             Effect
             of
             it
             ,
             they
             make
             a
             Circle
             in
             their
             Demonstration
             .
             As
             for
             example
             ,
             He
             that
             would
             Demonstrate
             that
             the
             Earth
             stands
             still
             in
             the
             Center
             of
             the
             World
             ,
             and
             should
             suppose
             the
             Earths
             Gravity
             to
             be
             the
             Cause
             thereof
             ,
             and
             define
             Gravity
             to
             be
             a
             quality
             by
             which
             every
             heavy
             Body
             tends
             towards
             the
             Center
             of
             the
             World
             ,
             would
             lose
             his
             labour
             ;
             for
             the
             question
             is
             ,
             What
             is
             the
             Cause
             of
             that
             quality
             in
             the
             Earth
             ;
             and
             therefore
             he
             that
             supposes
             Gravity
             to
             be
             the
             Cause
             ,
             puts
             the
             Thing
             it selfe
             for
             its
             own
             Cause
             .
          
           
             Of
             a
             
               False
               Cause
            
             I
             find
             this
             example
             in
             a
             certaine
             Treatise
             where
             the
             thing
             to
             be
             demonstrated
             is
             the
             Motion
             of
             the
             Earth
             .
             He
             begins
             therefore
             with
             this
             ,
             that
             seeing
             the
             Earth
             and
             the
             Sun
             are
             not
             alwayes
             in
             the
             same
             scituation
             ,
             it
             must
             needs
             be
             that
             one
             of
             them
             be
             locally
             moved
             ;
             which
             is
             true
             ;
             next
             he
             affirms
             that
             the
             Vapours
             which
             the
             Sun
             raises
             from
             the
             Earth
             and
             Sea
             are
             by
             reason
             of
             this
             Motion
             necessarily
             moved
             ;
             which
             also
             is
             true
             ;
             from
             whence
             he
             infers
             the
             Winds
             are
             made
             ,
             and
             this
             may
             passe
             for
             granted
             ;
             and
             by
             these
             Winds
             he
             sayes
             the
             Waters
             of
             the
             Sea
             are
             moved
             ,
             and
             by
             their
             Motion
             the
             bottome
             of
             the
             Sea
             ,
             as
             if
             it
             were
             beaten
             forwards
             ,
             moves
             round
             ;
             and
             let
             this
             also
             be
             granted
             ;
             wherefore
             he
             concludes
             ,
             the
             Earth
             is
             moved
             ;
             which
             is
             neverthelesse
             is
             a
             Paralogisme
             .
             For
             if
             that
             wind
             were
             
             the
             Cause
             why
             the
             Earth
             was
             from
             the
             beginning
             moved
             round
             ,
             and
             the
             Motion
             either
             of
             the
             Sunne
             or
             the
             Earth
             were
             the
             Cause
             of
             that
             Wind
             ,
             then
             the
             Motion
             of
             the
             Sunne
             or
             the
             Earth
             was
             before
             the
             Wind
             it self
             ;
             and
             if
             the
             Earth
             were
             Moved
             before
             the
             Wind
             was
             made
             ,
             then
             the
             Wind
             could
             not
             be
             the
             cause
             of
             the
             Earths
             revolution
             ;
             but
             if
             the
             Sunne
             were
             Moved
             ,
             and
             the
             Earth
             stand
             still
             ,
             then
             it
             is
             manifest
             the
             Earth
             might
             remain
             Unmoved
             notwithstanding
             that
             Wind
             ;
             and
             therefore
             that
             motion
             was
             not
             made
             by
             the
             Cause
             which
             he
             alledgeth
             .
             But
             Parallogismes
             of
             this
             kind
             are
             very
             frequent
             among
             the
             Writers
             of
             Physiques
             ,
             though
             none
             can
             be
             more
             elaborate
             then
             this
             in
             the
             Example
             given
             .
          
           
             19
             It
             may
             to
             some
             men
             seem
             pertinent
             to
             treat
             in
             this
             place
             of
             that
             Art
             of
             the
             Geometricians
             ,
             which
             they
             call
             Logistica
             ,
             that
             is
             ,
             the
             Art
             ,
             by
             which
             ,
             from
             supposing
             the
             thing
             in
             question
             to
             be
             true
             ,
             they
             proceed
             by
             Ratiocination
             ,
             till
             either
             they
             come
             to
             something
             knowne
             ,
             by
             which
             they
             may
             demonstrate
             the
             truth
             of
             the
             thing
             sought
             for
             ;
             or
             to
             something
             which
             is
             impossible
             ,
             from
             whence
             they
             collect
             that
             to
             be
             false
             which
             they
             supposed
             true
             .
             But
             this
             Art
             cannot
             be
             explicated
             here
             ,
             for
             this
             reason
             ,
             that
             the
             Method
             of
             it
             can
             neither
             be
             practised
             ,
             nor
             understood
             unlesse
             by
             such
             as
             are
             well
             versed
             in
             Geometry
             ;
             and
             among
             Geometricians
             themselves
             ,
             they
             that
             have
             most
             Theoremes
             in
             readiness
             ,
             are
             the
             most
             ready
             in
             the
             use
             of
             this
             Logistica
             ;
             so
             that
             indeed
             it
             is
             not
             a
             distinct
             thing
             from
             Geometry
             it selfe
             ;
             for
             there
             are
             in
             the
             Method
             of
             it
             three
             parts
             ;
             the
             first
             whereof
             consists
             in
             the
             finding
             out
             of
             Equality
             betwixt
             known
             and
             unknown
             things
             ,
             which
             they
             call
             Equation
             ;
             and
             this
             Equation
             cannot
             be
             found
             out
             but
             by
             such
             as
             know
             perfectly
             the
             Nature
             ,
             Properties
             and
             Transpositions
             of
             Proportion
             ,
             as
             also
             the
             Addition
             ,
             Substraction
             ,
             Multiplication
             ,
             and
             Division
             of
             Lines
             and
             Superficies
             ,
             and
             the
             Extraction
             of
             Roots
             ;
             which
             are
             the
             parts
             of
             no
             meane
             Geometrician
             .
             The
             Second
             is
             ,
             when
             an
             Equation
             is
             found
             ,
             to
             be
             able
             to
             judge
             whether
             the
             Truth
             or
             Falsity
             of
             the
             Question
             may
             be
             deduced
             from
             it
             or
             no
             ;
             which
             yet
             requires
             greater
             Knowledge
             .
             And
             the
             third
             is
             ,
             when
             such
             an
             Equation
             is
             found
             ,
             as
             is
             fit
             for
             the
             solution
             of
             the
             Question
             ,
             to
             know
             how
             to
             Resolve
             the
             same
             in
             such
             manner
             ,
             that
             the
             Truth
             or
             Falsity
             may
             thereby
             
             manifestly
             appeare
             ,
             which
             in
             hard
             questions
             cannot
             be
             done
             without
             the
             Knowledge
             of
             the
             Nature
             of
             Crooked-lined
             Figures
             ;
             but
             he
             that
             understands
             readily
             the
             Nature
             and
             Properties
             of
             these
             ,
             is
             a
             Compleat
             Geometrician
             .
             It
             happens
             besides
             ,
             that
             for
             the
             finding
             out
             of
             Equations
             there
             is
             no
             certaine
             Method
             ,
             but
             he
             is
             best
             able
             to
             do
             it
             ,
             that
             has
             the
             best
             Naturall
             Wit.
             
          
        
      
       
         
         
           THE
           FIRST
           GROVNDS
           OF
           PHILOSOPHY
           .
        
         
           
             
               CHAP.
               VII
            
             .
             Of
             
               Place
               and
               Time.
            
             
          
           
             
               
                 1
                 Things
                 that
                 have
                 no
                 existence
                 ,
                 may
                 neverthelesse
                 be
                 under
                 stood
                 and
                 computed
                 .
              
               
                 2
                 What
                 is
                 space
                 .
              
               
                 3
                 Time.
                 
              
               
                 4
                 Part.
                 
              
               
                 5
                 Division
                 .
              
               
                 6
                 One
              
               
                 7
                 Number
                 .
              
               
                 8
                 Composition
                 .
              
               
                 9
                 The
                 Whole
                 .
              
               
                 10
                 Spaces
                 and
                 Times
                 Contiguous
                 ,
                 and
                 Continuall
                 .
              
               
                 11
                 Beginning
                 ,
                 End
                 ,
                 Way
                 ,
                 Finite
                 ,
                 Infinite
                 .
              
               
                 12
                 What
                 is
                 Infinite
                 in
                 Power
                 .
                 Nothing
                 Infinite
                 can
                 be
                 truly
                 said
                 to
                 be
                 either
                 Whole
                 ,
                 or
                 One
                 ;
                 Nor
                 Infinite
                 Spaces
                 or
                 Times
                 ,
                 Many
                 .
              
               
                 13
                 Division
                 proceeds
                 not
                 to
                 the
                 Least
                 .
              
            
          
           
             1
             IN
             the
             Teaching
             of
             Naturall
             Philosophy
             ,
             I
             cannot
             begin
             better
             (
             as
             I
             have
             already
             shewn
             )
             then
             from
             Privation
             ;
             that
             is
             ,
             from
             feigning
             the
             World
             to
             be
             annihilated
             .
             But
             if
             such
             annihilation
             of
             all
             things
             be
             supposed
             ,
             it
             may
             perhaps
             be
             asked
             ,
             what
             would
             remain
             for
             any
             Man
             (
             whom
             onely
             I
             except
             from
             this
             Universal
             annihilation
             of
             things
             )
             to
             consider
             as
             the
             Subject
             of
             Philosophy
             ,
             or
             at
             all
             to
             reason
             upon
             ;
             or
             what
             to
             give
             Names
             unto
             for
             Ratiocinations
             sake
             .
          
           
             I
             say
             therefore
             there
             would
             remain
             to
             that
             Man
             Ideas
             of
             the
             World
             ,
             and
             of
             all
             such
             Bodies
             as
             he
             had
             ,
             before
             their
             annihilation
             ,
             seen
             with
             his
             eies
             ,
             or
             perceived
             by
             any
             other
             Sense
             ;
             that
             is
             to
             say
             ,
             the
             Memory
             and
             Imagination
             of
             Magnitudes
             ,
             Motions
             ,
             Sounds
             ,
             Colours
             ,
             &c.
             as
             also
             of
             their
             order
             &
             parts
             .
             All
             w
             ch
             things
             though
             they
             be
             nothing
             but
             Ideas
             &
             Phantasms
             ▪
             hapning
             internally
             to
             him
             that
             imagineth
             ;
             yet
             they
             will
             appear
             as
             if
             they
             were
             externall
             ,
             and
             not
             at
             all
             depending
             upon
             any
             power
             of
             the
             Mind
             .
             And
             these
             are
             
             the
             things
             to
             which
             he
             would
             give
             Names
             ,
             and
             substract
             them
             from
             ,
             and
             compound
             them
             with
             one
             another
             .
             For
             seeing
             that
             after
             the
             destruction
             of
             all
             other
             things
             ,
             I
             suppose
             Man
             still
             remaining
             ,
             and
             namely
             that
             he
             thinkes
             ,
             imagines
             ,
             and
             remembers
             ,
             there
             can
             be
             nothing
             for
             him
             to
             thinke
             of
             but
             what
             is
             Past
             ;
             Nay
             ,
             if
             we
             do
             but
             observe
             diligently
             what
             it
             is
             we
             doe
             when
             we
             consider
             and
             reason
             ,
             we
             shall
             find
             ,
             that
             though
             all
             things
             be
             still
             remaining
             in
             the
             world
             ,
             yet
             we
             compute
             nothing
             but
             our
             own
             Phantasmes
             .
             For
             when
             we
             calculate
             the
             magnitude
             and
             motions
             of
             Heaven
             or
             Earth
             ,
             we
             doe
             not
             ascend
             into
             Heaven
             that
             we
             may
             divide
             it
             into
             parts
             ,
             or
             measure
             the
             motions
             thereof
             ,
             but
             we
             doe
             it
             sitting
             still
             in
             our
             Closets
             or
             in
             the
             Darke
             .
             Now
             things
             may
             be
             considered
             ,
             that
             is
             ,
             be
             brought
             into
             Account
             ,
             either
             as
             internal
             Accidents
             of
             our
             Mind
             ,
             in
             which
             manner
             we
             consider
             them
             when
             the
             question
             is
             about
             some
             Faculty
             of
             the
             Mind
             ;
             or
             as
             Species
             of
             external
             things
             ,
             not
             as
             really
             existing
             ,
             but
             appearing
             onely
             to
             exist
             ,
             or
             to
             have
             a
             Being
             without
             Us.
             And
             in
             this
             manner
             we
             are
             now
             to
             consider
             them
             .
          
           
             2
             If
             therefore
             we
             remember
             ,
             or
             have
             a
             Phantasme
             of
             any
             thing
             that
             was
             in
             the
             world
             before
             the
             supposed
             annihilation
             of
             the
             same
             ;
             and
             consider
             ,
             not
             that
             the
             thing
             was
             such
             or
             such
             ,
             but
             onely
             that
             it
             had
             a
             Being
             without
             the
             Mind
             ,
             we
             have
             presently
             a
             Conception
             of
             that
             we
             call
             Space
             :
             an
             Imaginary
             Space
             indeed
             ,
             because
             a
             meere
             Phantasme
             ,
             yet
             that
             very
             thing
             which
             all
             men
             call
             so
             .
             For
             no
             man
             calls
             it
             Space
             for
             being
             already
             filled
             ,
             but
             because
             it
             may
             be
             filled
             ;
             nor
             does
             any
             man
             think
             Bodies
             carry
             their
             Places
             away
             with
             them
             ,
             but
             that
             the
             same
             Space
             contains
             sometimes
             one
             ,
             sometimes
             another
             Body
             ;
             which
             could
             not
             be
             if
             Space
             should
             alwayes
             accompany
             the
             Body
             which
             is
             once
             in
             it
             .
             And
             this
             is
             of
             it selfe
             so
             manifest
             ,
             that
             I
             should
             not
             thinke
             it
             needed
             any
             explaining
             at
             all
             ,
             but
             that
             I
             finde
             Space
             to
             be
             falsely
             defined
             by
             certaine
             Philosophers
             ,
             who
             inferre
             from
             thence
             ,
             One
             ,
             that
             the
             world
             is
             Infinite
             ;
             for
             taking
             Space
             to
             be
             the
             Extension
             of
             Bodies
             ,
             and
             thinking
             Extension
             may
             encrease
             continually
             ,
             he
             inferres
             that
             Bodies
             may
             be
             infinitely
             Extended
             ;
             and
             Another
             from
             the
             same
             Definition
             concludes
             rashly
             ,
             that
             it
             is
             impossible
             even
             to
             God
             himselfe
             to
             create
             more
             Worlds
             then
             one
             ;
             for
             if
             another
             World
             were
             to
             be
             created
             ,
             he
             sayes
             ,
             that
             seeing
             there
             is
             nothing
             without
             this
             world
             ,
             and
             therefore
             (
             according
             to
             his
             
             Definition
             )
             no
             Space
             ,
             that
             new
             world
             must
             be
             placed
             in
             nothing
             ;
             but
             in
             nothing
             nothing
             can
             be
             placed
             ;
             which
             he
             affirms
             onely
             ,
             without
             shewing
             any
             reason
             for
             the
             same
             ;
             whereas
             the
             contrary
             is
             the
             truth
             :
             for
             more
             cannot
             be
             put
             into
             a
             Place
             allready
             filled
             ,
             so
             much
             is
             Empty
             Space
             fitter
             then
             that
             which
             is
             Full
             for
             the
             receiving
             of
             new
             Bodies
             .
             Having
             therefore
             spoken
             thus
             much
             for
             these
             mens
             sakes
             ,
             and
             for
             theirs
             that
             assent
             to
             them
             ,
             I
             return
             to
             my
             purpose
             ,
             and
             define
             Space
             thus
             ,
             SPACE
             
               is
               the
               Phantasme
               of
               a
               Thing
               existing
               without
               the
               Mind
               simply
            
             ;
             that
             is
             to
             say
             ,
             that
             Phantasme
             ,
             in
             which
             we
             consider
             no
             other
             Accident
             ,
             but
             onely
             that
             it
             appears
             without
             us
             .
          
           
             3
             As
             a
             Body
             leaves
             a
             Phantasme
             of
             its
             Magnitude
             in
             the
             mind
             ,
             so
             also
             a
             Moved
             Body
             leaves
             a
             Phantasme
             of
             its
             Motion
             ,
             namely
             an
             Idea
             of
             that
             Body
             passing
             out
             of
             one
             Space
             into
             another
             by
             continuall
             succession
             .
             And
             this
             Idea
             or
             Phantasme
             ,
             is
             that
             which
             (
             without
             receding
             much
             from
             the
             common
             opinion
             ,
             or
             from
             Aristotles
             Definition
             )
             I
             call
             Time.
             For
             seeing
             all
             men
             confesse
             a
             Yeare
             to
             be
             Time
             ,
             and
             yet
             do
             not
             think
             a
             Year
             to
             be
             the
             Accident
             or
             Affection
             of
             any
             Body
             ,
             they
             must
             needs
             confesse
             it
             to
             be
             ,
             not
             in
             the
             things
             without
             Us
             ,
             but
             only
             in
             the
             Thought
             of
             the
             Mind
             .
             So
             when
             they
             speake
             of
             the
             Times
             of
             their
             Predecessors
             ,
             they
             do
             not
             think
             after
             their
             Predecessors
             are
             gone
             ,
             that
             their
             Times
             can
             be
             any
             where
             else
             then
             in
             the
             Memory
             of
             those
             that
             remember
             them
             .
             And
             as
             for
             those
             that
             say
             ,
             Dayes
             ,
             Years
             and
             Moneths
             are
             the
             Motions
             of
             the
             Sunne
             and
             Moon
             ,
             seeing
             it
             is
             all
             one
             to
             say
             ,
             Motion
             Past
             and
             Motion
             Destroyed
             ,
             and
             that
             Future
             Motion
             is
             the
             same
             with
             Motion
             which
             
               Is
               not
               yet
               begun
            
             ,
             they
             say
             ,
             that
             which
             they
             do
             not
             meane
             ,
             that
             there
             neither
             is
             ,
             nor
             has
             been
             ,
             nor
             shall
             be
             any
             Time
             :
             for
             of
             whatsoever
             it
             may
             be
             said
             ,
             
               It
               has
               been
            
             or
             
               It
               shall
               be
            
             ,
             of
             the
             same
             also
             it
             might
             have
             been
             said
             heretofore
             ,
             or
             may
             be
             said
             hereafter
             ,
             
               It
               is
            
             .
             What
             then
             can
             Dayes
             ,
             Moneths
             and
             Yeares
             be
             ,
             but
             the
             Names
             of
             such
             Computations
             made
             in
             our
             Mind
             ?
             Time
             therefore
             is
             a
             Phantasme
             ,
             but
             a
             Phantasme
             of
             Motion
             ,
             for
             if
             we
             would
             know
             by
             what
             Moments
             Time
             passes
             away
             ,
             we
             make
             use
             of
             some
             Motion
             or
             other
             ,
             as
             of
             the
             Sun
             ,
             of
             a
             Clock
             ,
             of
             the
             sand
             in
             an
             Hour-glasse
             ,
             or
             we
             mark
             some
             Line
             upon
             which
             we
             imagine
             something
             to
             be
             Moved
             ,
             there
             being
             no
             other
             means
             by
             which
             we
             can
             take
             notice
             of
             any
             Time
             at
             all
             .
             And
             yet
             when
             I
             say
             Time
             is
             a
             Phantasme
             
             of
             Motion
             ,
             I
             doe
             not
             say
             this
             is
             sufficient
             to
             define
             it
             by
             ;
             for
             this
             word
             Time
             comprehends
             the
             notion
             of
             Former
             and
             Later
             ,
             or
             of
             Succession
             in
             the
             motion
             of
             a
             Body
             ,
             in
             as
             much
             as
             it
             is
             first
             Here
             then
             There
             .
             Wherefore
             a
             compleat
             Definition
             of
             Time
             is
             such
             as
             this
             ,
             TIME
             
               is
               the
               Phantasme
               of
               Before
               and
               After
               in
               Motion
            
             ;
             which
             agrees
             with
             this
             Definition
             of
             
               Aristotle
               ,
               Time
               is
               the
               Number
               of
               Motion
               according
               to
               Former
               and
               Later
            
             ;
             for
             that
             Numbring
             is
             an
             act
             of
             the
             Mind
             ;
             and
             therefore
             it
             is
             all
             one
             to
             say
             ,
             
               Time
               is
               the
               Number
               of
               Motion
               according
               to
               Former
               and
               Later
            
             ;
             and
             
               Time
               is
               a
               Phantasme
               of
               Motion
               Numbred
               .
            
             But
             that
             other
             Definition
             ,
             
               Time
               is
               the
               Measure
               of
               Motion
               ,
            
             is
             not
             so
             exact
             ;
             for
             we
             measure
             Time
             by
             Motion
             and
             not
             Motion
             by
             Time.
             
          
           
             4
             One
             Space
             is
             called
             Part
             of
             another
             Space
             ,
             and
             one
             Time
             Part
             of
             another
             Time
             ,
             when
             this
             containes
             that
             and
             something
             besides
             .
             From
             whence
             it
             may
             be
             collected
             ,
             that
             nothing
             can
             rightly
             be
             called
             a
             PART
             ,
             but
             that
             which
             is
             compared
             with
             something
             that
             contains
             it
             .
          
           
             5
             And
             therefore
             to
             
               make
               parts
            
             ,
             or
             to
             Part
             or
             DIVIDE
             Space
             or
             Time
             ,
             is
             nothing
             else
             but
             to
             consider
             One
             and
             Another
             within
             the
             same
             ;
             so
             that
             if
             any
             Man
             Divide
             Space
             or
             Time
             ,
             the
             diverse
             Conceptions
             he
             has
             are
             more
             by
             one
             ,
             then
             the
             Parts
             he
             makes
             ;
             for
             his
             first
             Conception
             is
             of
             that
             which
             is
             to
             be
             divided
             ,
             then
             of
             some
             Part
             of
             it
             ,
             and
             again
             of
             some
             other
             Part
             of
             it
             ,
             and
             so
             forwards
             as
             long
             as
             he
             goes
             on
             in
             Dividing
             .
          
           
             But
             it
             is
             to
             be
             noted
             ,
             that
             here
             by
             Division
             ,
             I
             doe
             not
             mean
             the
             severing
             or
             pulling
             asunder
             of
             one
             Space
             or
             Time
             from
             another
             (
             for
             does
             any
             man
             think
             that
             one
             Hemisphere
             may
             be
             separated
             from
             the
             other
             Hemisphere
             ,
             or
             the
             first
             Hour
             from
             the
             second
             ?
             )
             but
             Diversity
             of
             Consideration
             ;
             so
             that
             Division
             is
             not
             made
             by
             the
             operation
             of
             the
             Hands
             but
             of
             the
             Mind
             .
          
           
             6
             When
             Space
             or
             Time
             is
             considered
             among
             other
             Spaces
             or
             Times
             ,
             it
             is
             said
             to
             be
             ONE
             ,
             namely
             
               One
               of
               them
            
             ;
             for
             except
             One
             Space
             might
             be
             added
             to
             another
             ,
             and
             substracted
             from
             another
             Space
             ,
             and
             so
             of
             Time
             ,
             it
             would
             be
             sufficient
             to
             say
             Space
             or
             Time
             simply
             ,
             and
             superfluous
             to
             say
             One
             Space
             or
             One
             Time
             ,
             if
             it
             could
             not
             be
             conceived
             that
             there
             were
             another
             .
             The
             common
             Definition
             of
             One
             ,
             namely
             ,
             that
             
               One
               is
               that
               which
               is
               Undivided
               ,
            
             is
             obnoxious
             to
             an
             absurd
             Consequence
             ;
             for
             it
             may
             thence
             be
             inferred
             ,
             that
             whatsoever
             is
             Divided
             ,
             is
             many
             things
             ,
             that
             is
             ,
             that
             every
             Divided
             
             thing
             ,
             is
             Divided
             Things
             ,
             which
             is
             Insignificant
             .
          
           
             7
             NUMBER
             is
             .
             One
             and
             One
             ,
             or
             One
             One
             and
             One
             ,
             and
             so
             forwards
             ;
             namely
             One
             and
             One
             make
             the
             Number
             Two
             ,
             and
             One
             One
             and
             One
             ,
             the
             Number
             Three
             ;
             and
             so
             are
             all
             other
             Numbers
             made
             ;
             which
             is
             all
             one
             as
             if
             we
             should
             say
             ,
             
               Number
               is
               Unities
            
             .
          
           
             8
             To
             COMPOUND
             Space
             of
             Spaces
             ,
             or
             Time
             of
             Times
             ,
             is
             first
             to
             Consider
             them
             one
             after
             another
             ,
             and
             then
             altogether
             as
             One
             ;
             as
             if
             one
             should
             reckon
             first
             the
             Head
             ,
             the
             Feet
             ,
             the
             Armes
             and
             the
             Body
             severally
             ,
             and
             then
             for
             the
             account
             of
             them
             all
             together
             put
             Man.
             And
             that
             which
             is
             so
             put
             for
             all
             the
             severalls
             of
             which
             it
             consists
             ,
             is
             called
             the
             WHOLE
             ;
             and
             those
             severalls
             ,
             when
             by
             the
             Division
             of
             the
             Whole
             ,
             they
             come
             again
             to
             be
             considered
             singly
             ,
             are
             parts
             thereof
             ;
             and
             therefore
             the
             Whole
             ,
             and
             
               all
               the
               Parts
               taken
               together
            
             ,
             are
             the
             same
             thing
             .
             And
             as
             I
             noted
             above
             ,
             that
             in
             Division
             it
             is
             not
             necessary
             to
             pull
             the
             Parts
             asunder
             ;
             so
             in
             Composition
             it
             is
             to
             be
             understood
             ,
             that
             for
             the
             making
             up
             of
             a
             whole
             there
             is
             no
             need
             of
             putting
             the
             Parts
             together
             ,
             so
             as
             to
             make
             them
             touch
             one
             another
             ,
             but
             onely
             of
             collecting
             them
             into
             one
             summe
             in
             the
             Mind
             .
             For
             thus
             all
             Men
             being
             considered
             together
             ,
             make
             up
             the
             Whole
             of
             Mankind
             ,
             though
             never
             so
             much
             dispersed
             by
             Time
             and
             Place
             ;
             and
             twelve
             Hours
             ,
             though
             the
             hours
             of
             severall
             dayes
             ,
             may
             be
             Compounded
             into
             one
             Number
             of
             Twelve
             .
          
           
             9
             This
             being
             well
             understood
             ,
             it
             is
             manifest
             ,
             that
             nothing
             can
             rightly
             be
             called
             a
             Whole
             ,
             that
             is
             not
             conceived
             to
             be
             compounded
             of
             Parts
             ,
             and
             that
             it
             may
             be
             divided
             into
             parts
             ;
             so
             that
             if
             we
             deny
             that
             a
             thing
             has
             parts
             ,
             we
             deny
             the
             same
             to
             be
             a
             Whole
             .
             For
             example
             ,
             if
             we
             say
             the
             soul
             can
             have
             no
             Parts
             ,
             we
             affirme
             that
             no
             soul
             can
             be
             a
             Whole
             soul.
             Also
             it
             is
             manifest
             ,
             that
             Nothing
             has
             Parts
             till
             it
             be
             Divided
             ;
             and
             when
             a
             Thing
             is
             Divided
             ,
             the
             Parts
             are
             onely
             so
             many
             as
             the
             Division
             makes
             them
             .
             Againe
             ,
             that
             a
             Part
             of
             a
             Part
             is
             a
             Part
             of
             the
             Whole
             ;
             &
             thus
             any
             Part
             of
             the
             Number
             Four
             ,
             as
             Two
             ,
             is
             a
             Part
             of
             the
             Number
             Eight
             ;
             for
             Four
             is
             made
             of
             Two
             and
             Two
             ;
             but
             Eight
             is
             compounded
             of
             
               Two
               ,
               Two
            
             and
             Four
             ;
             and
             therefore
             Two
             which
             is
             a
             Part
             of
             the
             Part
             Four
             ,
             is
             also
             a
             Part
             of
             the
             whole
             Eight
             .
          
           
             10
             Two
             Spaces
             are
             said
             to
             be
             CONTIGUOUS
             ,
             when
             there
             is
             no
             other
             Space
             betwixt
             them
             .
             But
             two
             Times
             ,
             betwixt
             which
             
             there
             is
             no
             other
             Time
             ,
             are
             called
             IMMEDIATE
             ,
             A
             —
             B
             —
             C
             as
             AB
             ,
             BC.
             And
             any
             two
             Spaces
             as
             well
             as
             Times
             are
             said
             to
             be
             CONTINUALL
             ,
             when
             they
             have
             one
             common
             part
             ,
             A
             —
             B
             —
             C
             —
             D
             as
             AC
             ,
             BD
             ,
             where
             the
             part
             BC
             is
             common
             ;
             and
             more
             Spaces
             and
             Times
             are
             Continual
             ,
             when
             every
             two
             which
             are
             next
             one
             another
             are
             Continual
             .
          
           
             11
             That
             Part
             which
             is
             between
             two
             other
             Parts
             ,
             is
             called
             a
             MEAN
             ;
             &
             that
             which
             is
             not
             between
             two
             other
             parts
             ,
             an
             EXTREME
             .
             And
             of
             Extremes
             ,
             that
             which
             is
             first
             reckoned
             is
             the
             BEGINNING
             ,
             and
             that
             which
             last
             ,
             the
             END
             ;
             and
             all
             the
             Means
             together
             taken
             ,
             are
             the
             WAY
             .
             Also
             
               Extreme
               Parts
            
             and
             Limits
             are
             the
             same
             thing
             .
             And
             from
             hence
             it
             is
             manifest
             ,
             that
             Beginning
             and
             End
             depend
             upon
             the
             order
             in
             which
             we
             number
             them
             ;
             and
             that
             to
             Terminate
             or
             Limit
             Space
             and
             Time
             ,
             is
             the
             same
             thing
             with
             
               imagining
               their
               Beginning
               and
               End
            
             ;
             as
             also
             that
             every
             thing
             is
             FINITE
             or
             INFINITE
             ,
             acording
             as
             we
             imagine
             or
             not
             imagine
             it
             Limited
             or
             Terminated
             every
             way
             ;
             and
             that
             the
             Limits
             of
             any
             Number
             are
             Unities
             ,
             and
             of
             these
             ,
             that
             which
             is
             the
             first
             in
             our
             Numbering
             is
             the
             Beginning
             ,
             and
             that
             which
             we
             number
             last
             ,
             is
             the
             End.
             When
             we
             say
             Number
             is
             Infinite
             ,
             we
             mean
             only
             that
             no
             Number
             is
             expressed
             ;
             for
             when
             we
             speak
             of
             the
             Numbers
             
               Two
               ,
               Three
               ,
               a
               Thousand
               ,
               &c.
            
             they
             are
             always
             Finite
             .
             But
             when
             no
             more
             is
             said
             but
             this
             ,
             
               Number
               is
               Infinite
            
             ,
             it
             is
             to
             be
             understood
             as
             if
             it
             were
             said
             ,
             this
             Name
             Number
             is
             an
             Indefinite
             Name
             .
          
           
             12
             Space
             or
             Time
             is
             said
             to
             be
             
               Finite
               in
               Power
            
             ,
             or
             Terminable
             ,
             when
             there
             may
             be
             assigned
             a
             Number
             of
             finite
             Spaces
             or
             Times
             ,
             as
             of
             Paces
             or
             Hours
             ,
             than
             which
             there
             can
             be
             no
             greater
             Number
             of
             the
             same
             measure
             ,
             in
             that
             Space
             or
             Time
             ;
             and
             
               Infinite
               in
               Power
            
             is
             that
             Space
             or
             Time
             ,
             in
             which
             a
             greater
             Number
             of
             the
             said
             Paces
             or
             Hours
             may
             be
             assigned
             ,
             than
             any
             Number
             that
             can
             be
             given
             .
             But
             we
             must
             note
             ,
             that
             although
             in
             that
             Space
             or
             Time
             which
             is
             Infinite
             in
             Power
             ,
             there
             may
             be
             numbered
             more
             Paces
             or
             Hours
             then
             any
             number
             that
             can
             be
             assigned
             ,
             yet
             their
             number
             will
             alwayes
             be
             Finite
             ;
             for
             every
             Number
             is
             Finite
             .
             And
             therefore
             his
             Ratiocination
             was
             not
             good
             ,
             that
             undertaking
             to
             prove
             the
             World
             to
             be
             Finite
             ,
             reasoned
             thus
             ;
             
               If
               the
               world
               be
               Infinite
               ,
               then
               there
               may
               be
               taken
               in
               it
               some
               Part
               which
               is
               distant
               from
               us
               an
               Infinite
               number
               of
               Paces
               :
               But
               no
               such
               Part
               can
               be
               taken
               ;
               wherefore
               the
            
             
             
               world
               is
               not
               infinite
            
             ;
             because
             that
             Consequence
             of
             the
             Major
             Proposition
             is
             false
             ;
             for
             in
             an
             Infinite
             space
             ,
             whatsoever
             we
             take
             ,
             or
             design
             in
             our
             Mind
             ,
             the
             distance
             of
             the
             same
             from
             us
             is
             a
             Finite
             space
             ;
             for
             in
             the
             very
             designing
             of
             the
             place
             thereof
             ,
             we
             put
             an
             End
             to
             that
             space
             ,
             of
             whch
             we
             our selves
             are
             the
             Beginning
             ,
             and
             whatsoever
             any
             man
             with
             his
             Mind
             cuts
             off
             both
             wayes
             from
             Infinite
             ,
             he
             determines
             the
             same
             ,
             that
             is
             ,
             he
             makes
             it
             Finite
             .
          
           
             Of
             Infinite
             Space
             or
             Time
             ,
             it
             cannot
             be
             said
             that
             it
             is
             a
             Whole
             ,
             or
             One
             ;
             not
             a
             Whole
             ,
             because
             not
             compounded
             of
             Parts
             ;
             for
             seeing
             Parts
             ,
             how
             many
             soever
             they
             be
             ,
             are
             severally
             Finite
             ,
             they
             will
             also
             when
             they
             are
             all
             put
             together
             make
             a
             whole
             Finite
             ;
             Nor
             One
             ,
             because
             nothing
             can
             be
             said
             to
             be
             One
             ,
             except
             there
             be
             Another
             to
             compare
             it
             with
             ;
             but
             it
             cannot
             be
             conceived
             that
             there
             are
             two
             Spaces
             ,
             or
             two
             Times
             Infinite
             .
             Lastly
             ,
             when
             we
             make
             question
             whether
             the
             World
             be
             Finite
             or
             Infinite
             ,
             we
             have
             nothing
             in
             our
             Minde
             answering
             to
             the
             Name
             World
             ;
             for
             whatsoever
             we
             Imagine
             ,
             is
             therefore
             Finite
             ,
             though
             our
             Computation
             reach
             the
             fixed
             Stars
             ,
             or
             the
             ninth
             or
             tenth
             ,
             nay
             ,
             the
             thousanth
             Sphere
             .
             The
             meaning
             of
             the
             Question
             is
             this
             onely
             ,
             whether
             God
             has
             actually
             made
             so
             great
             an
             Addition
             of
             Body
             to
             Body
             ,
             as
             we
             are
             able
             to
             make
             of
             Space
             to
             Space
             .
          
           
             13
             And
             therefore
             that
             which
             is
             commonly
             said
             ,
             that
             Space
             and
             Time
             may
             be
             divided
             Infinitely
             ,
             is
             not
             to
             be
             so
             understood
             ,
             as
             if
             there
             might
             be
             any
             Infinite
             or
             Eternal
             Division
             ;
             but
             rather
             to
             be
             taken
             in
             this
             sense
             ,
             
               Whatsover
               is
               Divided
               ,
               is
               divided
               into
               such
               Parts
               as
               may
               again
               be
               Divided
            
             ;
             or
             thus
             ,
             
               The
               Least
               Divisible
               thing
               is
               not
               to
               be
               given
            
             ;
             or
             as
             Geometricians
             have
             it
             ,
             
               No
               Quantity
               is
               so
               small
               ,
               but
               a
               Less
               may
               be
               taken
            
             ;
             which
             may
             easily
             be
             demonstrated
             in
             this
             manner
             .
             Let
             any
             Space
             or
             Time
             (
             that
             which
             was
             thought
             to
             be
             the
             Least
             Divisible
             )
             be
             divided
             into
             two
             equal
             Parts
             A
             and
             B.
             I
             say
             either
             of
             them
             ,
             as
             A
             ,
             may
             be
             divided
             again
             .
             For
             suppose
             the
             Part
             A
             to
             be
             contiguous
             to
             the
             Part
             B
             of
             one
             side
             ,
             and
             of
             the
             other
             side
             to
             some
             other
             Space
             equal
             to
             B.
             This
             whole
             Space
             therefore
             (
             being
             greater
             then
             the
             Space
             given
             )
             is
             divisible
             .
             Wherefore
             if
             it
             be
             divided
             into
             two
             equal
             Parts
             ,
             the
             Part
             in
             the
             middle
             ,
             which
             is
             A
             ,
             will
             be
             also
             divided
             into
             two
             equal
             Parts
             ;
             and
             therefore
             A
             was
             Divisible
             .
          
        
         
           
           
             
               CHAP.
               VIII
            
             .
             Of
             Body
             and
             Accident
             .
          
           
             
               
                 1
                 Body
                 defined
                 .
              
               
                 2
                 Accident
                 defined
                 .
              
               
                 3
                 How
                 an
                 Accident
                 may
                 be
                 understood
                 to
                 be
                 in
                 its
                 subject
                 .
              
               
                 4
                 Magnitude
                 ,
                 what
                 it
                 is
                 .
              
               
                 5
                 Place
                 what
                 it
                 is
                 ,
                 and
                 that
                 it
                 is
                 Immoveable
                 .
              
               
                 6
                 What
                 is
                 Full
                 and
                 Empty
                 .
              
               
                 7
                 Here
                 ,
                 There
                 ,
                 Somewhere
                 ,
                 what
                 they
                 signifie
                 .
              
               
                 8
                 Many
                 Bodies
                 cannot
                 be
                 in
                 One
                 place
                 ,
                 nor
                 One
                 Body
                 in
                 Many
                 places
                 .
              
               
                 9
                 Contiguous
                 and
                 Continual
                 what
                 they
                 are
                 .
              
               
                 10
                 The
                 definition
                 of
                 Motion
                 .
                 No
                 Motion
                 intelligible
                 but
                 with
                 Time.
                 
              
               
                 11
                 What
                 it
                 is
                 to
                 be
                 at
                 Rest
                 ,
                 to
                 have
                 been
                 Moved
                 ,
                 and
                 to
                 be
                 Moved
                 .
                 No
                 Motion
                 to
                 be
                 conceived
                 ,
                 without
                 the
                 conception
                 of
                 Past
                 and
                 Future
                 .
              
               
                 12
                 A
                 Point
                 ,
                 a
                 Line
                 ,
                 Superficies
                 and
                 Solid
                 ,
                 what
                 they
                 are
                 .
              
               
                 13
                 Equal
                 ,
                 Greater
                 and
                 Lesse
                 in
                 Bodies
                 and
                 Magnitudes
                 ,
                 what
                 they
                 are
                 .
              
               
                 14
                 One
                 and
                 the
                 same
                 Body
                 has
                 alwayes
                 one
                 and
                 the
                 the
                 same
                 Magnitude
                 .
              
               
                 15
                 Velocity
                 what
                 it
                 is
                 .
              
               
                 16
                 Equal
                 ,
                 Greater
                 and
                 Lesse
                 in
                 Times
                 what
                 they
                 are
                 .
              
               
                 17
                 Equal
                 ,
                 Greater
                 and
                 Lesse
                 in
                 Velocity
                 ,
                 what
                 .
              
               
                 18
                 Equal
                 ,
                 Greater
                 and
                 Lesse
                 in
                 Motion
                 ,
                 what
                 .
              
               
                 19
                 That
                 which
                 is
                 at
                 Rest
                 will
                 alwayes
                 be
                 at
                 Rest
                 except
                 it
                 be
                 Moved
                 by
                 some
                 external
                 thing
                 ;
                 and
                 that
                 which
                 is
                 Moved
                 will
                 alwayes
                 be
                 Moved
                 ,
                 unless
                 it
                 be
                 hindered
                 by
                 some
                 external
                 thing
                 .
              
               
                 20
                 Accidents
                 are
                 Generated
                 and
                 Destroyed
                 ,
                 but
                 Bodies
                 not
                 so
                 .
              
               
                 21
                 An
                 Accident
                 cannot
                 depart
                 from
                 its
                 Subject
                 .
              
               
                 22
                 Nor
                 be
                 Moved
                 .
              
               
                 23
                 Essence
                 ,
                 Form
                 ,
                 and
                 Matter
                 ,
                 what
                 they
                 are
                 .
              
               
                 24
                 First
                 Matter
                 ,
                 what
                 .
              
               
                 25
                 That
                 the
                 whole
                 is
                 greater
                 then
                 any
                 Part
                 thereof
                 ,
                 why
                 demonstrated
                 .
              
            
          
           
             1
             HAving
             understood
             what
             Imaginary
             Space
             is
             ,
             in
             which
             we
             supposed
             nothing
             remaining
             without
             us
             ,
             but
             all
             those
             things
             to
             be
             destroyed
             that
             by
             existing
             heretofore
             left
             Images
             of
             themselves
             in
             our
             Minds
             ;
             let
             us
             now
             suppose
             some
             one
             of
             those
             things
             to
             be
             placed
             again
             in
             the
             World
             ,
             or
             created
             anew
             .
             It
             is
             necessary
             therefore
             that
             this
             new
             created
             or
             replaced
             thing
             do
             not
             onely
             fill
             some
             part
             of
             the
             Space
             above-mentioned
             ,
             or
             be
             coincident
             and
             coextended
             with
             it
             ,
             but
             also
             ,
             that
             it
             have
             no
             dependance
             upon
             our
             thought
             .
             And
             this
             is
             that
             which
             for
             the
             
             Extension
             of
             it
             we
             commonly
             call
             Body
             ;
             and
             because
             it
             depends
             not
             upon
             our
             Thought
             ,
             we
             say
             is
             
               a
               thing
               subsisting
               of
               itself
            
             ;
             as
             also
             existing
             ,
             because
             without
             Us
             ;
             and
             lastly
             ,
             it
             is
             called
             the
             Subject
             ,
             because
             it
             is
             so
             placed
             in
             and
             subjected
             to
             Imaginary
             Space
             ,
             that
             it
             may
             be
             understood
             by
             Reason
             ,
             as
             well
             as
             perceived
             by
             Sense
             .
             The
             Definition
             therefore
             of
             Body
             may
             be
             this
             ,
             A
             BODY
             
               is
               that
               which
               having
               no
               dependance
               upon
               our
               Thought
               is
               coincident
               or
               coextended
               with
               some
               part
               of
               Space
               .
            
          
           
             2
             But
             what
             an
             Accident
             is
             ,
             cannot
             so
             easily
             be
             explained
             by
             any
             Definition
             ,
             as
             by
             Examples
             .
             Let
             us
             imagine
             therefore
             that
             a
             Body
             fills
             any
             Space
             ,
             or
             is
             coextended
             with
             it
             ,
             that
             Coextention
             is
             not
             the
             coextended
             Body
             ;
             And
             in
             like
             manner
             ,
             let
             us
             imagine
             that
             the
             same
             Body
             is
             removed
             out
             of
             its
             place
             ,
             that
             Removing
             is
             not
             the
             removed
             Body
             ;
             Or
             let
             us
             think
             the
             same
             not
             removed
             ,
             that
             notremoving
             or
             Rest
             ,
             is
             not
             the
             resting
             Body
             .
             What
             then
             are
             these
             things
             ?
             They
             are
             Accidents
             of
             that
             Body
             .
             But
             the
             thing
             in
             question
             is
             
               What
               is
               an
               Accident
            
             ;
             which
             is
             an
             Enquiry
             after
             that
             which
             we
             know
             already
             ,
             and
             not
             that
             which
             we
             should
             enquire
             after
             .
             For
             who
             does
             not
             alwayes
             and
             in
             the
             same
             manner
             understand
             him
             that
             sayes
             any
             thing
             is
             Extended
             ,
             or
             Moved
             ,
             or
             not
             Moved
             ?
             But
             most
             men
             will
             have
             it
             be
             said
             that
             
               an
               Accident
               is
               something
            
             ,
             namely
             some
             part
             of
             a
             natural
             thing
             ,
             when
             indeed
             it
             is
             no
             part
             of
             the
             same
             .
             To
             satisfie
             these
             men
             ,
             as
             well
             as
             may
             be
             ,
             they
             answer
             best
             that
             define
             an
             Accident
             to
             be
             
               the
               Manner
               by
               which
               any
               Body
               is
               conceived
            
             ;
             which
             is
             all
             one
             as
             if
             they
             should
             say
             ,
             
               An
               Accident
               is
               that
               faculty
               of
               any
               Body
               by
               which
               it
               works
               in
               us
               a
               Conception
               of
               itself
               .
            
             Which
             Definition
             though
             it
             be
             not
             an
             Answer
             to
             the
             Question
             propounded
             ,
             yet
             it
             is
             an
             Answer
             to
             that
             Question
             which
             should
             have
             been
             propounded
             ,
             namely
             ,
             
               whence
               does
               it
               happen
               that
               one
               part
               of
               any
               Body
               appears
               here
               ,
               another
               part
               there
               ?
            
             For
             this
             is
             well
             answered
             thus
             ,
             
               It
               happens
               from
               the
               Extension
               of
               that
               Body
               .
            
             Or
             ,
             
               How
               comes
               it
               to
               pass
               that
               the
               whole
               Body
               by
               succession
               is
               seen
               now
               here
               now
               there
               ?
            
             and
             the
             answer
             will
             be
             ,
             
               By
               reason
               of
               its
               Motion
            
             .
             Or
             lastly
             ,
             
               Whence
               is
               it
               that
               any
               Body
               possesseth
               the
               same
               space
               for
               sometime
               ?
            
             And
             the
             answer
             will
             be
             ,
             
               because
               it
               is
               not
               moved
            
             .
             For
             if
             concerning
             the
             Name
             of
             a
             Body
             ,
             that
             is
             ,
             concerning
             a
             Concrete
             Name
             ,
             it
             be
             asked
             ,
             
               what
               is
               it
            
             ?
             the
             answer
             must
             be
             made
             by
             Definition
             ;
             for
             the
             Question
             is
             concerning
             the
             signification
             of
             the
             Name
             .
             But
             if
             it
             be
             asked
             concerning
             an
             Abstract
             Name
             ,
             
               what
               is
               it
            
             ?
             the
             Cause
             
             is
             demanded
             why
             a
             thing
             appears
             so
             or
             so
             .
             As
             if
             it
             be
             asked
             ,
             
               what
               is
               Hard
            
             ?
             The
             Answer
             will
             be
             ,
             Hard
             is
             that
             ,
             whereof
             no
             Part
             gives
             place
             ,
             but
             when
             the
             Whole
             gives
             place
             .
             But
             if
             it
             be
             demanded
             ,
             
               what
               is
               Hardness
            
             ?
             A
             Cause
             must
             be
             shewn
             why
             a
             Part
             does
             not
             give
             place
             except
             the
             Whole
             give
             place
             .
             Wherefore
             I
             define
             an
             ACCIDENT
             to
             be
             
               the
               Ma●ner
               of
               our
               conception
               of
               Body
               .
            
          
           
             3
             When
             an
             Accident
             is
             said
             
               to
               be
               in
               a
               Body
            
             ,
             it
             is
             not
             so
             to
             be
             understood
             ,
             as
             if
             any
             thing
             were
             conteined
             in
             that
             Body
             ;
             as
             if
             ,
             for
             example
             ,
             Redness
             were
             in
             Blood
             ,
             in
             the
             same
             manner
             ,
             as
             Blood
             is
             in
             a
             bloody
             cloth
             ,
             that
             is
             ,
             as
             a
             Part
             in
             the
             Whole
             ;
             for
             so
             an
             Accident
             would
             be
             a
             Body
             also
             .
             But
             as
             Magnitude
             ,
             or
             Rest
             ,
             or
             Motion
             ,
             is
             in
             that
             which
             is
             Great
             ,
             or
             which
             Resteth
             ,
             or
             which
             is
             Moved
             (
             which
             how
             it
             is
             to
             be
             understood
             ,
             every
             man
             understands
             )
             so
             also
             it
             is
             to
             be
             understood
             that
             every
             other
             Accident
             
               is
               in
            
             its
             Subject
             .
             And
             this
             also
             is
             explicated
             by
             Aristotle
             no
             otherwise
             then
             negatively
             ,
             namely
             ,
             that
             
               An
               Accident
               is
               in
               its
               Subject
               ,
               not
               as
               any
               part
               thereof
               ,
               but
               so
               as
               that
               it
               may
               be
               away
               ,
               the
               Subject
               still
               remaining
            
             ;
             which
             is
             right
             ,
             saving
             that
             there
             are
             certain
             Accidents
             which
             can
             never
             perish
             except
             the
             Body
             perish
             also
             ;
             for
             no
             Body
             can
             be
             conceived
             to
             be
             without
             Extension
             ,
             or
             without
             Figure
             .
             All
             other
             Accidents
             ,
             which
             are
             not
             common
             to
             all
             Bodies
             ,
             but
             peculiar
             to
             some
             onely
             ,
             as
             
               To
               be
               at
               Rest
               ,
               to
               be
               Moved
               ,
               Colour
               ,
               Hardness
               ,
            
             and
             the
             like
             ,
             do
             perish
             continually
             ,
             and
             are
             succeeded
             by
             others
             ;
             yet
             so
             ,
             as
             that
             the
             Body
             never
             perisheth
             .
             And
             as
             for
             the
             opinion
             that
             some
             may
             have
             ,
             that
             all
             other
             Accidents
             are
             not
             in
             their
             Bodies
             in
             the
             same
             manner
             that
             Extension
             ,
             Motion
             ,
             Rest
             ,
             or
             Figure
             are
             in
             the
             same
             ;
             for
             example
             ,
             that
             Colour
             ,
             Heat
             ,
             Odour
             ,
             Vertue
             ,
             Vice
             and
             the
             like
             ,
             are
             otherwise
             in
             them
             ,
             and
             (
             as
             they
             say
             )
             inherent
             ;
             I
             desire
             they
             would
             suspend
             their
             judgement
             for
             the
             present
             ,
             and
             expect
             a
             little
             ,
             till
             it
             be
             found
             out
             by
             Ratiocination
             ,
             whether
             these
             very
             Accidents
             are
             not
             also
             certain
             Motions
             ,
             either
             of
             the
             Mind
             of
             the
             perceiver
             ;
             or
             of
             the
             Bodies
             themselves
             which
             are
             perceived
             ;
             for
             in
             the
             search
             of
             this
             ,
             a
             great
             part
             of
             Naturall
             Philosophy
             consists
             .
          
           
             4
             The
             Extension
             of
             a
             Body
             ,
             is
             the
             same
             thing
             with
             the
             MAGNITUDE
             of
             it
             ,
             or
             that
             which
             some
             call
             
               Real
               Space
            
             .
             But
             this
             Magnitude
             does
             not
             depend
             upon
             our
             Cogitation
             ,
             as
             Imaginary
             Space
             doth
             ;
             for
             this
             is
             an
             Effect
             of
             our
             Imagination
             ,
             but
             Magnitude
             is
             the
             Cause
             of
             it
             ;
             this
             is
             an
             Accident
             of
             the
             Mind
             ,
             that
             of
             a
             Body
             existing
             out
             of
             the
             Mind
             .
          
           
           
             5
             That
             Space
             (
             by
             which
             word
             I
             here
             understand
             Imaginary
             Space
             )
             which
             is
             coincident
             with
             the
             Magnitude
             of
             any
             Body
             ,
             is
             called
             the
             PLACE
             of
             that
             Body
             ;
             and
             the
             Body
             it self
             is
             that
             which
             we
             call
             the
             
               Thing
               Placed
            
             .
             Now
             Place
             ,
             and
             the
             Magnitude
             of
             the
             Thing
             Placed
             differ
             :
             First
             in
             this
             ,
             that
             a
             Body
             keeps
             alwayes
             the
             same
             Magnitude
             both
             when
             it
             is
             at
             Rest
             ,
             and
             when
             it
             is
             Moved
             ;
             but
             when
             it
             is
             Moved
             ,
             it
             does
             not
             keep
             the
             same
             Place
             .
             Secondly
             ,
             in
             this
             ,
             that
             Place
             is
             a
             Phantasme
             of
             any
             Body
             of
             such
             and
             such
             Quantity
             and
             Figure
             ;
             but
             Magnitude
             is
             the
             peculiar
             Accident
             of
             every
             Body
             ;
             for
             one
             Body
             may
             at
             several
             times
             have
             several
             Places
             ,
             but
             has
             always
             one
             and
             the
             same
             Magnitude
             .
             Thirdly
             ,
             in
             this
             ,
             that
             Place
             is
             nothing
             out
             of
             the
             Mind
             ,
             nor
             Magnitude
             any
             thing
             within
             it
             .
             And
             lastly
             ,
             Place
             is
             feigned
             Extension
             but
             Magnitude
             true
             Extension
             ,
             and
             a
             Placed
             Body
             is
             not
             Extension
             ,
             but
             a
             Thing
             Extended
             .
             Besides
             ,
             
               Place
               is
               Immoveable
            
             ;
             for
             seeing
             that
             which
             is
             Moved
             ,
             is
             understood
             to
             be
             carried
             from
             Place
             to
             Place
             ,
             if
             Place
             were
             Moved
             ,
             it
             would
             also
             be
             carried
             from
             Place
             to
             Place
             ,
             so
             that
             one
             Place
             must
             have
             another
             Place
             ,
             and
             that
             Place
             another
             Place
             ,
             and
             so
             on
             infinitely
             ,
             which
             is
             ridiculous
             .
             And
             as
             for
             those
             ,
             that
             by
             making
             Place
             to
             be
             of
             the
             same
             Nature
             with
             
               Real
               Space
            
             ,
             would
             from
             thence
             maintain
             it
             to
             be
             Immoveable
             ,
             they
             also
             make
             Place
             (
             though
             they
             do
             not
             perceive
             they
             make
             it
             so
             )
             to
             be
             a
             meer
             Phantasme
             .
             For
             whilest
             One
             affirms
             that
             Place
             is
             therefore
             said
             to
             be
             Immoveable
             ,
             because
             Space
             in
             general
             is
             considered
             there
             ;
             if
             he
             had
             remembred
             that
             nothing
             is
             General
             or
             Universal
             besides
             Names
             or
             Signes
             ,
             he
             would
             easily
             have
             seen
             that
             that
             Space
             which
             he
             sayes
             is
             considered
             in
             general
             ,
             is
             nothing
             but
             a
             Phantasme
             in
             the
             Mind
             or
             the
             Memory
             ,
             of
             a
             Body
             of
             such
             Magnitude
             and
             such
             Figure
             .
             And
             whilest
             another
             sayes
             ,
             Real
             Space
             is
             made
             Immoveable
             by
             the
             Understanding
             ;
             as
             when
             under
             the
             Superficies
             of
             running
             water
             ,
             we
             imagine
             other
             and
             other
             water
             to
             come
             by
             continual
             succession
             ,
             that
             Superficies
             fixed
             there
             by
             the
             Understanding
             is
             the
             
               Immoveable
               Place
            
             of
             the
             River
             ,
             what
             else
             does
             he
             make
             it
             to
             be
             but
             a
             Phantasm
             ,
             though
             he
             doe
             it
             obscurely
             ,
             and
             in
             perplexed
             words
             ?
             Lastly
             ,
             the
             nature
             of
             Place
             does
             not
             consist
             in
             the
             
               Superficies
               of
               the
               Ambient
            
             ,
             but
             in
             
               Solid
               Space
            
             ;
             for
             the
             whole
             Placed
             Body
             is
             coextended
             with
             its
             whole
             Place
             ,
             and
             every
             part
             of
             it
             with
             every
             answering
             part
             of
             the
             same
             Place
             ;
             but
             seeing
             every
             
             Placed
             Body
             is
             a
             Solid
             thing
             ,
             it
             cannot
             be
             understood
             to
             be
             coextended
             with
             Superficies
             .
             Besides
             ,
             how
             can
             any
             whole
             Body
             be
             Moved
             ,
             unless
             all
             its
             parts
             be
             moved
             together
             with
             it
             ?
             Or
             how
             can
             the
             internall
             Parts
             of
             it
             be
             Moved
             ,
             but
             by
             leaving
             their
             Place
             ?
             But
             the
             internal
             Parts
             of
             a
             body
             cannot
             leave
             the
             Superficies
             of
             an
             external
             part
             contiguous
             to
             it
             ;
             and
             therefore
             it
             followes
             ,
             that
             if
             Place
             be
             the
             Superficies
             of
             the
             Ambient
             ,
             then
             the
             parts
             of
             a
             Body
             Moved
             ,
             that
             is
             Bodies
             moved
             ,
             are
             not
             Moved
             .
          
           
             6
             Space
             (
             or
             Place
             )
             that
             is
             possessed
             by
             a
             Body
             ,
             is
             called
             FULL
             ,
             and
             that
             which
             is
             not
             so
             possessed
             is
             called
             EMPTY
             .
          
           
             7
             
               Here
               ,
               There
               ,
               In
               the
               Country
               ,
               In
               the
               City
               ,
            
             and
             other
             the
             like
             Names
             by
             which
             answer
             is
             made
             to
             the
             question
             
               Where
               is
               it
            
             ,
             are
             not
             properly
             Names
             of
             Place
             ,
             nor
             doe
             they
             of
             themselves
             bring
             into
             the
             mind
             the
             Place
             that
             is
             sought
             ;
             for
             Here
             and
             There
             signifie
             nothing
             ,
             unlesse
             the
             thing
             be
             shewn
             at
             the
             same
             time
             with
             the
             finger
             or
             something
             else
             ,
             but
             when
             the
             Eye
             of
             him
             that
             seeks
             ,
             is
             by
             pointing
             ,
             or
             some
             other
             signe
             directed
             to
             the
             thing
             sought
             ,
             the
             Place
             of
             it
             is
             not
             hereby
             defined
             by
             him
             that
             answers
             ,
             but
             found
             out
             by
             him
             that
             askes
             the
             question
             .
             Now
             such
             Shewings
             as
             are
             made
             by
             words
             onely
             ,
             as
             when
             we
             say
             ,
             
               In
               the
               Countrey
            
             ,
             or
             
               In
               the
               City
            
             ,
             are
             some
             of
             greater
             latitude
             then
             others
             ,
             as
             when
             we
             say
             
               In
               the
               Countrey
               ,
               In
               the
               City
               ,
               In
               such
               a
               Street
               ,
               In
               a
               House
               ,
               In
               the
               Chamber
               ,
               In
               Bed
               ,
            
             &c.
             
             For
             these
             do
             by
             little
             and
             little
             direct
             the
             Seeker
             neerer
             to
             the
             proper
             Place
             ;
             &
             yet
             they
             do
             not
             determine
             the
             same
             ,
             but
             onely
             restrain
             it
             to
             a
             lesser
             Space
             ,
             &
             signifie
             no
             more
             then
             that
             the
             Place
             of
             the
             Thing
             is
             within
             a
             certain
             Space
             designed
             by
             those
             Words
             ,
             as
             a
             Part
             is
             in
             the
             Whole
             .
             And
             all
             such
             Names
             (
             by
             which
             answer
             is
             made
             to
             the
             question
             Where
             )
             have
             for
             their
             highest
             Genus
             the
             Name
             Somewhere
             .
             From
             whence
             it
             may
             be
             understood
             ,
             that
             whatsoever
             is
             Somewhere
             ,
             is
             in
             some
             Place
             properly
             so
             called
             ,
             which
             Place
             is
             part
             of
             that
             greater
             Space
             that
             is
             signified
             by
             some
             of
             these
             Names
             ,
             
               In
               the
               Countrey
               ,
               In
               the
               City
               ,
            
             or
             the
             like
             .
          
           
             8
             A
             Body
             ,
             and
             the
             Magnitude
             ,
             and
             the
             Place
             thereof
             ,
             are
             divided
             by
             one
             and
             the
             same
             act
             of
             the
             Mind
             ;
             for
             ,
             to
             divide
             an
             Extended
             Body
             ,
             and
             the
             Extension
             thereof
             ,
             and
             the
             Idea
             of
             that
             Extension
             ,
             which
             is
             Place
             ,
             is
             the
             same
             with
             dividing
             any
             one
             of
             them
             ;
             because
             they
             are
             coincident
             ,
             and
             it
             cannot
             be
             done
             but
             by
             the
             Mind
             ,
             that
             is
             by
             the
             Division
             of
             Space
             .
             From
             whence
             it
             is
             manifest
             ,
             that
             
             neither
             Two
             Bodies
             can
             be
             together
             in
             the
             same
             Place
             ,
             nor
             One
             Body
             be
             in
             Two
             Places
             at
             the
             same
             Time.
             Not
             Two
             Bodies
             in
             the
             same
             Place
             ;
             because
             when
             a
             Body
             that
             fills
             its
             whole
             Place
             is
             divided
             into
             Two
             ,
             the
             Place
             it self
             is
             divided
             into
             Two
             also
             ,
             so
             that
             there
             will
             be
             Two
             Places
             ,
             Nor
             One
             Body
             in
             Two
             Places
             ;
             for
             ,
             the
             Place
             that
             a
             Body
             fills
             being
             divided
             into
             Two
             ,
             the
             Placed
             Body
             will
             be
             also
             divided
             into
             Two
             ,
             (
             for
             ,
             as
             I
             said
             ,
             a
             Place
             and
             the
             Body
             that
             fills
             that
             Place
             are
             divided
             both
             together
             )
             and
             so
             there
             will
             be
             two
             Bodies
             .
          
           
             9
             Two
             Bodies
             are
             said
             to
             be
             Contiguous
             to
             one
             another
             ,
             and
             Continual
             ,
             in
             the
             same
             manner
             as
             Spaces
             are
             ;
             namely
             ,
             
               those
               are
               Contiguous
               ,
               between
               which
               there
               is
               no
               Space
               .
            
             Now
             by
             Space
             I
             understand
             here
             as
             formerly
             an
             Idea
             or
             Phantasme
             of
             a
             Body
             .
             Wherefore
             ,
             though
             between
             two
             Bodies
             there
             be
             put
             no
             other
             Body
             ,
             and
             consequently
             no
             Magnitude
             ,
             or
             (
             as
             they
             call
             it
             )
             Real
             Space
             ,
             yet
             if
             another
             Body
             may
             be
             put
             between
             them
             ,
             that
             is
             ,
             if
             there
             intercede
             any
             imagined
             Space
             which
             may
             receive
             another
             Body
             ,
             then
             those
             Bodies
             are
             not
             Contiguous
             .
             And
             this
             is
             so
             easie
             to
             be
             understood
             ,
             that
             I
             should
             wonder
             at
             some
             men
             ,
             who
             being
             otherwise
             skilful
             enough
             in
             Philosophy
             ,
             are
             of
             a
             different
             opinion
             ,
             but
             that
             I
             finde
             that
             most
             of
             those
             that
             affect
             Metaphysical
             subtilties
             ,
             wander
             from
             Truth
             ,
             as
             if
             they
             were
             led
             out
             of
             their
             way
             by
             an
             
               Ignis
               Fatuus
            
             .
             For
             can
             any
             man
             that
             has
             his
             natural
             Senses
             ,
             think
             that
             two
             Bodies
             must
             therefore
             necessarily
             Touch
             one
             another
             ,
             because
             no
             other
             Body
             is
             between
             them
             ?
             Or
             that
             there
             can
             be
             no
             Vacuum
             ,
             because
             Vacuum
             is
             nothing
             ,
             or
             as
             they
             call
             it
             ,
             
               Non
               Ens
            
             ?
             Which
             is
             as
             childish
             ,
             as
             if
             one
             should
             reason
             thus
             ;
             No
             Man
             can
             Fast
             ,
             because
             to
             Fast
             is
             to
             eat
             Nothing
             ;
             but
             Nothing
             cannot
             be
             eaten
             .
             
               Continual
               ,
               are
               any
               two
               Bodies
               that
               have
               a
               common
               part
               ;
               and
               more
               then
               two
               are
               Continual
               ,
               when
               every
               two
               that
               are
               next
               to
               one
               another
               ,
               are
               continual
               .
            
          
           
           
             11
             
               That
               is
               said
               to
               be
               at
               Rest
               ,
               which
               during
               anytime
               is
               in
               one
               place
               ;
               and
               that
               .
               to
               be
               Moved
               ,
               or
               to
               have
               been
               Moved
               ,
               which
               whether
               it
               be
               now
               at
               Rest
               ,
               or
               Moved
               ,
               was
               formerly
               in
               another
               Place
               then
               that
               which
               it
               is
               now
               in
               .
            
             From
             which
             Definitions
             it
             may
             be
             inferred
             ,
             First
             ,
             that
             
               Whatsoever
               is
               Moved
               ,
               has
               been
               Moved
            
             ;
             for
             if
             it
             be
             still
             in
             the
             same
             Place
             in
             which
             it
             was
             formerly
             ,
             it
             is
             at
             Rest
             ,
             that
             is
             ,
             it
             is
             not
             Moved
             ,
             by
             the
             Definition
             of
             Rest
             ;
             but
             if
             it
             be
             in
             another
             Place
             ,
             it
             has
             been
             Moved
             ,
             by
             the
             Definition
             of
             Moved
             .
             Secondly
             ,
             that
             
               what
               is
               Moved
               ,
               will
               yet
               be
               Moved
            
             ;
             for
             that
             which
             is
             Moved
             ,
             leaveth
             the
             Place
             where
             it
             is
             ,
             and
             therefore
             will
             be
             in
             another
             Place
             ,
             and
             consequently
             will
             be
             moved
             still
             .
             Thirdly
             ,
             that
             
               whatsoever
               is
               Moved
               ,
               is
               not
               in
               One
               place
               during
               any
               time
               ,
               how
               little
               soever
               that
               time
               be
            
             ;
             for
             by
             the
             Definition
             of
             Rest
             ,
             that
             which
             is
             in
             one
             Place
             during
             any
             time
             ,
             is
             at
             Rest.
             
          
           
             There
             is
             a
             certain
             Sophisme
             against
             Motion
             ,
             which
             seems
             to
             spring
             from
             the
             not
             understanding
             of
             this
             last
             Proposition
             .
             For
             they
             say
             ,
             that
             ,
             
               If
               any
               Body
               be
               Moved
               ,
               it
               is
               Moved
               either
               in
               the
               Place
               where
               it
               is
               ,
               or
               in
               the
               Place
               where
               it
               is
               not
               ;
               both
               which
               are
               false
               ;
               and
               therefore
               nothing
               is
               Moved
               .
            
             But
             the
             falsity
             lies
             in
             the
             Major
             Proposition
             ;
             for
             that
             which
             is
             Moved
             ,
             is
             neither
             Moved
             in
             the
             Place
             where
             it
             is
             ,
             nor
             in
             the
             Place
             where
             it
             is
             not
             ;
             but
             from
             the
             Place
             where
             it
             is
             ,
             to
             the
             Place
             where
             it
             is
             not
             .
             Indeed
             it
             cannot
             be
             denied
             but
             that
             whatsoever
             is
             Moved
             ,
             is
             Moved
             somewhere
             ,
             that
             is
             ,
             within
             some
             Space
             ;
             but
             then
             the
             Place
             of
             that
             Body
             is
             not
             that
             whole
             Space
             ,
             but
             a
             part
             of
             it
             ,
             as
             is
             said
             above
             in
             the
             seventh
             Article
             .
             From
             what
             is
             above
             demonstrated
             ,
             namely
             ,
             that
             whatsoever
             is
             Moved
             ,
             has
             also
             
             been
             Moved
             ,
             and
             will
             be
             Moved
             ,
             this
             also
             may
             be
             collected
             ,
             that
             there
             can
             be
             no
             conception
             of
             Motion
             ,
             without
             conceiving
             Past
             and
             Future
             time
             .
          
           
             12
             Though
             there
             be
             no
             Body
             which
             has
             not
             some
             Magnitude
             ,
             yet
             if
             when
             any
             Body
             is
             moved
             ,
             the
             Magnitude
             of
             it
             be
             not
             at
             all
             considered
             ,
             the
             way
             it
             makes
             it
             called
             a
             LINE
             ,
             or
             one
             single
             Dimension
             ;
             &
             the
             Space
             through
             which
             it
             passeth
             ,
             is
             called
             LENGTH
             ;
             and
             the
             Body
             it self
             ,
             a
             POINT
             ;
             in
             which
             sense
             the
             Earth
             is
             called
             a
             Point
             ,
             and
             the
             Way
             of
             its
             yearly
             Revolution
             ,
             the
             
               Ecliptick
               Line
            
             .
             But
             if
             a
             Body
             which
             is
             Moved
             ,
             be
             considered
             as
             long
             ,
             and
             be
             supposed
             to
             be
             so
             Moved
             ,
             as
             that
             all
             the
             several
             parts
             of
             it
             be
             understood
             to
             make
             several
             Lines
             ,
             then
             the
             Way
             of
             every
             part
             of
             that
             Body
             is
             called
             BREADTH
             ,
             and
             the
             Space
             which
             is
             made
             is
             called
             SUPERFICIES
             ,
             consisting
             of
             two
             Dimensions
             ,
             one
             whereof
             to
             every
             several
             part
             of
             the
             other
             is
             applyed
             whole
             .
             Again
             ,
             if
             a
             Body
             be
             considered
             as
             having
             Superficies
             ,
             and
             be
             understood
             to
             be
             so
             Moved
             ,
             that
             all
             the
             several
             parts
             of
             it
             describe
             several
             Lines
             ,
             then
             the
             Way
             of
             every
             part
             of
             that
             Body
             is
             called
             THICKNESS
             ,
             or
             DEPTH
             ,
             and
             the
             Space
             which
             is
             Made
             is
             called
             SOLID
             ,
             consisting
             of
             three
             Dimensions
             ,
             any
             two
             whereof
             are
             applyed
             whole
             to
             every
             several
             part
             of
             the
             third
             .
          
           
             But
             if
             a
             Body
             be
             considered
             as
             Solid
             ,
             then
             it
             is
             not
             possible
             that
             all
             the
             several
             parts
             of
             it
             should
             describe
             several
             lines
             ;
             for
             what
             way
             soever
             it
             be
             Moved
             ,
             the
             way
             of
             the
             following
             part
             will
             fall
             into
             the
             way
             of
             the
             part
             before
             it
             ,
             so
             that
             the
             same
             Solid
             will
             still
             be
             made
             which
             the
             formost
             Superficies
             would
             have
             made
             by
             it self
             .
             And
             therefore
             there
             can
             be
             no
             other
             Dimension
             in
             any
             Body
             ,
             as
             it
             is
             a
             Body
             ,
             then
             the
             three
             which
             I
             have
             now
             described
             ;
             though
             as
             it
             shall
             be
             shewed
             hereafter
             ,
             Velocity
             ,
             which
             is
             Motion
             according
             to
             Length
             ,
             may
             be
             being
             applyed
             to
             all
             the
             parts
             of
             a
             Solid
             ,
             make
             a
             Magnitude
             of
             Motion
             consisting
             of
             four
             Dimensions
             ;
             as
             the
             goodness
             of
             Gold
             computed
             in
             all
             the
             parts
             of
             it
             makes
             the
             Price
             and
             Value
             thereof
             .
          
           
             13
             Bodies
             (
             how
             many
             soever
             they
             be
             )
             that
             can
             fill
             every
             one
             the
             place
             of
             every
             one
             ,
             are
             said
             to
             be
             EQUAL
             every
             one
             to
             every
             other
             .
             Now
             one
             Body
             may
             fill
             the
             same
             Place
             which
             another
             Body
             filleth
             ,
             though
             it
             be
             not
             of
             the
             same
             Figure
             with
             that
             other
             Body
             ,
             if
             so
             be
             that
             it
             may
             be
             understood
             to
             be
             reducible
             to
             the
             same
             
             Figure
             ,
             either
             by
             Flexion
             or
             Transposition
             of
             the
             parts
             .
             And
             
               One
               Body
               is
            
             GREATER
             
               then
               another
               Body
               ,
               when
               a
               part
               of
               that
               is
               equal
               to
               all
               this
               ;
               and
            
             LESSE
             ,
             
               when
               all
               that
               is
               equal
               to
               a
               part
               of
               this
               .
            
             Also
             Magnitudes
             are
             Equal
             ,
             or
             Greater
             ,
             or
             Lesser
             then
             one
             another
             for
             the
             same
             consideration
             ,
             namely
             ,
             when
             the
             Bodies
             of
             which
             they
             are
             the
             Magnitudes
             ,
             are
             either
             Equal
             or
             Greater
             or
             Lesse
             ,
             &c.
             
          
           
             14
             One
             and
             the
             same
             Body
             ,
             is
             alwayes
             of
             one
             and
             the
             same
             Magnitude
             .
             For
             seeing
             a
             Body
             and
             the
             Magnitude
             and
             Place
             thereof
             cannot
             be
             comprehended
             in
             the
             Minde
             ,
             otherwise
             then
             as
             they
             are
             Coincident
             ,
             if
             any
             Body
             be
             understood
             to
             be
             at
             Rest
             ,
             that
             is
             ,
             to
             remain
             in
             the
             same
             Place
             during
             some
             time
             ,
             and
             the
             Magnitude
             thereof
             be
             in
             one
             part
             of
             that
             time
             Greater
             ,
             and
             in
             another
             part
             Lesse
             ,
             that
             Bodies
             Place
             ,
             which
             is
             one
             and
             the
             same
             ,
             will
             be
             coincident
             sometimes
             with
             Greater
             ,
             sometimes
             with
             Lesse
             Magnitude
             ,
             that
             is
             ,
             the
             same
             Place
             will
             be
             greater
             and
             lesse
             then
             it self
             ,
             which
             is
             impossible
             .
             But
             there
             would
             be
             no
             need
             at
             all
             of
             Demonstrating
             a
             thing
             that
             is
             in
             it self
             so
             manifest
             ,
             if
             there
             were
             not
             some
             ,
             whose
             opinion
             concerning
             Bodies
             and
             their
             Magnitudes
             is
             ,
             that
             a
             Body
             may
             exist
             separated
             from
             its
             Magnitude
             ,
             and
             have
             greater
             or
             lesse
             Magnitude
             bestowed
             upon
             it
             ,
             making
             use
             of
             this
             Principle
             for
             the
             explication
             of
             the
             nature
             of
             Rarum
             and
             Densum
             .
          
           
             15
             Motion
             ,
             in
             as
             much
             as
             a
             certain
             Length
             may
             in
             a
             certain
             Time
             be
             transmitted
             by
             it
             ,
             is
             called
             VELOCITY
             or
             Swiftness
             :
             &c.
             
             For
             though
             Swift
             be
             very
             often
             understood
             with
             relation
             to
             Slower
             or
             
               less
               Swift
            
             ,
             as
             Great
             is
             in
             respect
             of
             Less
             ,
             yet
             nevertheless
             ,
             as
             Magnitude
             is
             by
             Philosophers
             taken
             absolutely
             for
             Extension
             ,
             so
             also
             Velocity
             or
             Swiftness
             may
             be
             put
             absolutely
             for
             Motion
             accord
             ing
             to
             Length
             .
          
           
             16
             Many
             Motions
             are
             said
             to
             be
             made
             in
             Equal
             Times
             ,
             when
             every
             one
             of
             them
             begins
             and
             ends
             together
             with
             some
             other
             Motion
             ,
             or
             if
             it
             had
             begun
             together
             ,
             would
             also
             have
             ended
             together
             with
             the
             same
             .
             For
             Time
             which
             is
             a
             Phantasme
             of
             Mo
             tion
             ,
             cannot
             be
             reckoned
             but
             by
             some
             exposed
             Motion
             ;
             as
             in
             Dials
             by
             the
             Motion
             of
             the
             Sun
             or
             of
             the
             Hand
             ;
             and
             if
             two
             or
             more
             Motions
             begin
             and
             end
             with
             this
             Motion
             ,
             they
             are
             said
             to
             be
             made
             in
             equal
             times
             ;
             from
             whence
             also
             it
             is
             easie
             to
             understand
             what
             it
             is
             to
             be
             moved
             in
             Greater
             or
             Longer
             time
             ,
             &
             in
             lesse
             time
             or
             not
             so
             long
             ;
             namely
             ,
             that
             that
             is
             longer
             Moved
             ,
             which
             beginning
             with
             another
             ,
             
             ends
             later
             ;
             or
             ending
             together
             ,
             began
             sooner
             .
          
           
             17
             Motions
             are
             said
             to
             be
             Equally
             Swift
             ,
             when
             Equal
             lengths
             are
             transmitted
             in
             Equal
             times
             ;
             and
             Greater
             Swiftness
             is
             that
             ,
             wherein
             Greater
             length
             is
             passed
             in
             Equal
             time
             ,
             or
             Equal
             length
             in
             less
             time
             .
             Also
             that
             Swiftness
             by
             which
             Equal
             lengths
             are
             passed
             in
             Equal
             parts
             of
             time
             ,
             is
             called
             Uniform
             Swiftness
             or
             Motion
             ;
             and
             of
             Motions
             
               not
               Uniform
            
             ,
             such
             as
             become
             Swifter
             or
             Slower
             by
             equal
             Increasings
             or
             Decreasings
             in
             equal
             parts
             of
             time
             ,
             are
             said
             to
             be
             Accelerated
             or
             
               Retarded
               Uniformly
            
             .
          
           
             18
             But
             Motion
             is
             said
             to
             be
             Greater
             ,
             Lesse
             ,
             and
             Equal
             ,
             not
             onely
             in
             regard
             of
             the
             Length
             which
             is
             transmitted
             in
             a
             certain
             time
             ,
             that
             is
             ,
             in
             regard
             of
             Swiftness
             onely
             ,
             but
             of
             Swiftness
             applyed
             to
             every
             smallest
             particle
             of
             Magnitude
             ;
             For
             when
             any
             Body
             is
             Moved
             ,
             every
             part
             of
             it
             is
             also
             Moved
             ;
             and
             supposing
             the
             parts
             to
             be
             halves
             ,
             the
             Motions
             of
             those
             halves
             have
             their
             Swiftness
             equal
             to
             one
             another
             ,
             and
             severally
             equal
             to
             that
             of
             the
             Whole
             ;
             but
             the
             Motion
             of
             the
             Whole
             is
             equal
             to
             those
             two
             Motions
             ,
             either
             of
             which
             is
             of
             equal
             Swiftness
             with
             it
             ;
             and
             therefore
             it
             is
             one
             thing
             for
             two
             Motions
             to
             be
             Equal
             to
             one
             another
             ,
             &
             another
             thing
             for
             them
             to
             be
             Equally
             Swift
             .
             And
             this
             is
             manifest
             in
             two
             Horses
             that
             draw
             abreast
             ,
             where
             the
             Motion
             of
             both
             the
             Horses
             together
             is
             of
             Equal
             Swiftness
             with
             the
             Motion
             of
             either
             of
             them
             singly
             ;
             but
             the
             Motion
             of
             both
             is
             Greater
             then
             the
             Motion
             of
             one
             of
             them
             ,
             namely
             Double
             .
             Wherefore
             
               Motions
               are
               said
               to
               be
               simply
               Equal
               to
               one
               another
               when
               the
               Swiftness
               of
               one
               computed
               in
               every
               part
               of
               its
               Magnitude
               ,
               is
               Equal
               to
               the
               Swiftness
               of
               the
               other
               cōputed
               also
               in
               every
               part
               of
               its
               Magnitude
               :
               &
               Greater
               then
               one
               another
               ,
               when
               the
               Swiftness
               of
               one
               computed
               as
               above
               ,
               is
               Greater
               then
               the
               Swiftness
               of
               the
               other
               so
               computed
               ;
               and
               Lesse
               ,
               when
               Lesse
               .
            
             Besides
             ,
             the
             Magnitude
             of
             Motion
             computed
             in
             this
             manner
             is
             that
             which
             is
             commonly
             called
             FORCE
             .
          
           
             19
             
               Whatsoever
               is
               at
               Rest
               ,
               will
               alwayes
               be
               at
               Rest
               ,
               unless
               there
               be
               some
               other
               Body
               besides
               it
               ,
               which
               by
               endeavouring
               to
               get
               into
               its
               Place
               by
               motion
               ,
               suffers
               it
               no
               longer
               to
               remain
               at
               Rest.
            
             For
             suppose
             that
             some
             Finite
             Body
             exist
             ,
             and
             be
             at
             Rest
             ,
             and
             that
             all
             Space
             besides
             be
             Empty
             ;
             if
             now
             this
             Body
             begin
             to
             be
             Moved
             ,
             it
             will
             certainly
             be
             Moved
             some
             way
             ;
             Seeing
             therefore
             there
             was
             nothing
             in
             that
             Body
             which
             did
             not
             dispose
             it
             to
             Rest
             ,
             the
             reason
             why
             it
             is
             Moved
             this
             way
             is
             in
             something
             out
             of
             it
             ;
             and
             in
             like
             manner
             ,
             if
             it
             had
             been
             Moved
             any
             
             other
             way
             ,
             the
             reason
             of
             Motion
             that
             way
             had
             also
             been
             in
             something
             out
             of
             it
             ;
             but
             seeing
             it
             was
             supposed
             that
             Nothing
             is
             out
             of
             it
             ,
             the
             reason
             of
             its
             Motion
             one
             way
             would
             be
             the
             same
             with
             the
             reason
             of
             its
             Motion
             every
             other
             way
             ;
             wherefore
             it
             would
             be
             Moved
             alike
             all
             wa●es
             at
             once
             ;
             which
             is
             impossible
             .
          
           
             In
             like
             manner
             ,
             
               Whatsoever
               is
               Moved
               ,
               will
               alwayes
               be
               Moved
               ,
               except
               there
               be
               some
               other
               Body
               besides
               it
               ,
               which
               causeth
               it
               to
               Rest.
            
             For
             if
             we
             suppose
             Nothing
             to
             be
             without
             it
             ,
             there
             will
             be
             no
             reason
             why
             it
             should
             Rest
             now
             ,
             rather
             then
             at
             another
             time
             ;
             wherefore
             its
             Motion
             would
             cease
             in
             every
             particle
             of
             time
             alike
             ;
             which
             is
             not
             intelligible
             .
          
           
             When
             we
             say
             a
             Living
             Creature
             ,
             a
             Tree
             ,
             or
             any
             other
             specified
             Body
             is
             Generated
             ,
             or
             Destroyed
             ,
             it
             is
             not
             to
             be
             so
             understood
             as
             if
             there
             were
             made
             a
             Body
             of
             that
             which
             is
             not-Body
             ,
             or
             not
             a
             Body
             of
             a
             Body
             ,
             but
             of
             a
             Living
             Creature
             not
             a
             Living
             Creature
             ,
             of
             a
             Tree
             not
             a
             Tree
             ,
             &c.
             that
             is
             ,
             that
             those
             Accidents
             for
             which
             we
             call
             one
             thing
             a
             Living
             Creature
             ,
             another
             thing
             a
             Tree
             ,
             and
             another
             by
             some
             other
             Name
             ,
             are
             Generated
             and
             Destroyed
             ;
             and
             that
             therefore
             the
             same
             Names
             are
             not
             to
             be
             given
             to
             them
             now
             ,
             which
             were
             given
             them
             before
             .
             But
             that
             Magnitude
             for
             which
             we
             give
             to
             any
             thing
             the
             Name
             of
             Body
             is
             neither
             Generated
             nor
             Destroyed
             .
             For
             though
             we
             may
             feign
             in
             our
             Mind
             that
             a
             Point
             may
             swell
             to
             a
             huge
             bulk
             ,
             and
             that
             this
             may
             again
             contract
             it selfe
             to
             a
             Point
             ;
             that
             is
             ,
             though
             we
             may
             imagine
             something
             to
             arise
             where
             before
             was
             Nothing
             ,
             and
             Nothing
             to
             be
             there
             where
             before
             was
             something
             ,
             yet
             we
             cannot
             comprehend
             in
             our
             Minde
             how
             this
             may
             possibly
             be
             done
             in
             Nature
             .
             And
             therefore
             Philosophers
             ,
             who
             tye
             themselves
             to
             Naturall
             Reason
             ,
             Suppose
             that
             a
             Body
             can
             neither
             be
             Generated
             nor
             Destroyed
             ,
             but
             onely
             that
             it
             may
             appear
             otherwise
             then
             it
             did
             to
             Us
             ,
             that
             is
             under
             different
             Species
             ,
             and
             consequently
             be
             called
             by
             other
             and
             other
             Names
             ;
             so
             that
             that
             which
             is
             now
             called
             Man
             ,
             may
             at
             another
             time
             have
             the
             Name
             of
             Not-Man
             ;
             but
             that
             which
             is
             once
             called
             Body
             ,
             can
             never
             be
             called
             Not-Body
             .
             But
             it
             is
             manifest
             ,
             that
             all
             other
             Accidents
             besides
             Magnitude
             or
             Extension
             may
             be
             Generated
             and
             Destroyed
             ;
             as
             when
             a
             White
             thing
             is
             made
             Black
             ,
             the
             Whiteness
             that
             was
             in
             it
             Perisheth
             ,
             and
             the
             Blackness
             that
             was
             not
             in
             it
             is
             now
             Generated
             ;
             and
             therefore
             Bodies
             ,
             and
             the
             Accidents
             under
             which
             they
             appear
             diversly
             ,
             have
             
             this
             difference
             ,
             that
             Bodies
             are
             Things
             ,
             and
             not
             Generated
             ;
             Accidents
             are
             Generated
             ,
             and
             not
             Things
             .
          
           
             21
             And
             therefore
             when
             any
             thing
             appears
             otherwise
             then
             it
             did
             ,
             by
             reason
             of
             other
             and
             other
             Accidents
             ,
             it
             is
             not
             to
             be
             thought
             that
             an
             Accident
             goes
             out
             of
             one
             Subject
             into
             another
             (
             for
             they
             are
             not
             ,
             as
             I
             said
             above
             ,
             in
             their
             Subjects
             as
             a
             Part
             in
             the
             Whole
             ,
             or
             as
             a
             Conteined
             thing
             in
             that
             which
             Conteins
             it
             ,
             or
             as
             a
             Master
             of
             a
             Family
             in
             his
             House
             ,
             )
             but
             that
             one
             Accident
             Perisheth
             ,
             and
             another
             is
             Generated
             .
             For
             example
             ,
             when
             the
             Hand
             being
             Moved
             ,
             Moves
             the
             Pen
             ,
             Motion
             does
             not
             go
             out
             of
             the
             Hand
             into
             the
             Pen
             ,
             for
             so
             the
             Writing
             might
             be
             continued
             though
             the
             Hand
             stood
             still
             ,
             but
             a
             new
             Motion
             is
             Generated
             in
             the
             Pen
             ,
             and
             is
             the
             Pens
             Motion
             .
          
           
             22
             And
             therefore
             also
             it
             is
             improper
             to
             say
             an
             Accident
             is
             Moved
             ;
             as
             when
             in
             stead
             of
             saying
             ,
             
               Figure
               is
               an
               Accident
               of
               a
               Body
               carried
               away
               ,
            
             we
             say
             ,
             
               A
               Body
               carries
               away
               its
               Figure
               .
            
          
           
             23
             Now
             that
             Accident
             for
             which
             we
             give
             a
             certain
             Name
             to
             any
             Body
             ,
             or
             the
             Accident
             which
             denominates
             its
             Subject
             ,
             is
             commonly
             called
             the
             ESSENCE
             thereof
             ;
             as
             Rationality
             is
             the
             Essence
             of
             a
             Man
             ,
             Whiteness
             ;
             of
             any
             White
             Thing
             and
             Extension
             the
             Essence
             of
             a
             Body
             .
             And
             the
             same
             Essence
             in
             as
             much
             as
             it
             is
             Generated
             ,
             is
             called
             the
             FORM.
             Again
             ,
             a
             Body
             ,
             in
             respect
             of
             any
             accident
             is
             called
             the
             SUBJECT
             ,
             and
             in
             respect
             of
             the
             Form
             it
             is
             called
             the
             MATTER
             .
          
           
             Also
             ,
             the
             Production
             or
             Perishing
             of
             any
             Accident
             ,
             makes
             its
             Subject
             be
             said
             
               to
               be
               Changed
            
             ;
             onely
             the
             Production
             or
             Perishing
             of
             Form
             ,
             make
             it
             be
             said
             it
             is
             Generated
             or
             Destroyed
             ;
             but
             in
             all
             Generation
             and
             Mutation
             ,
             the
             name
             of
             Matter
             still
             remains
             .
             For
             a
             Table
             made
             of
             Wood
             ,
             is
             not
             onely
             Wooden
             ,
             but
             Wood
             ;
             and
             a
             Statue
             of
             Brass
             is
             Brass
             as
             well
             as
             Brazen
             ;
             though
             Aristotle
             in
             his
             Metaphysiques
             say
             ,
             that
             whatsoever
             is
             made
             of
             any
             thing
             ought
             not
             to
             be
             called
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             ,
             but
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             ,
             as
             that
             which
             is
             made
             of
             Wood
             not
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             ,
             but
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             ,
             that
             is
             ,
             not
             Wood
             ,
             but
             Wooden
             .
          
           
             24
             And
             as
             for
             that
             Matter
             which
             is
             common
             to
             all
             things
             ,
             and
             which
             Philosophers
             following
             Aristotle
             ,
             usually
             call
             
               Materia
               Prima
            
             ,
             that
             is
             ,
             
               First
               Matter
            
             ,
             it
             is
             not
             any
             Body
             distinct
             from
             all
             other
             Bodies
             ,
             nor
             is
             it
             one
             of
             them
             .
             What
             then
             is
             it
             ?
             A
             mere
             Name
             ;
             yet
             a
             Name
             which
             is
             not
             of
             vain
             Use
             ;
             for
             it
             signifies
             a
             conception
             of
             
             Body
             without
             the
             consideration
             of
             any
             Form
             or
             other
             Accident
             except
             onely
             Magnitude
             or
             Extension
             ,
             &
             aptness
             to
             receive
             Form
             &
             other
             Accidents
             ;
             so
             that
             whensoever
             we
             have
             use
             of
             the
             Name
             
               Body
               in
               general
            
             ,
             if
             we
             use
             that
             of
             
               Materia
               Prima
            
             ,
             we
             do
             well
             .
             For
             as
             ,
             when
             a
             Man
             not
             knowing
             which
             was
             first
             ,
             Water
             or
             Ice
             ,
             would
             finde
             out
             which
             of
             the
             two
             were
             the
             Matter
             of
             both
             ,
             he
             would
             be
             fain
             to
             suppose
             some
             third
             Matter
             which
             were
             neither
             of
             these
             two
             ;
             so
             he
             that
             would
             finde
             out
             what
             is
             the
             Matter
             of
             all
             things
             ,
             ought
             to
             suppose
             such
             as
             is
             not
             the
             Matter
             of
             any
             thing
             that
             exists
             .
             Wherefore
             
               Materia
               Prima
            
             is
             no
             Thing
             ;
             and
             therefore
             they
             do
             not
             attribute
             to
             it
             either
             Form
             or
             any
             other
             Accident
             besides
             Quantity
             ;
             whereas
             all
             singular
             things
             have
             their
             Forms
             and
             Accidents
             certain
             .
          
           
             
               Materia
               Prima
            
             therefore
             is
             Body
             in
             general
             ,
             that
             is
             Body
             considered
             Universally
             ,
             not
             as
             having
             neither
             Form
             nor
             any
             Accident
             ,
             but
             in
             which
             no
             Form
             nor
             any
             other
             Accident
             but
             Quantity
             are
             at
             all
             considered
             ,
             that
             is
             ,
             they
             are
             not
             drawn
             into
             Argumentation
             .
          
           
             25
             From
             what
             has
             been
             said
             ,
             those
             Axiomes
             may
             be
             demonstrated
             which
             are
             assumed
             by
             Euclide
             in
             the
             beginning
             of
             his
             first
             Element
             about
             the
             Equality
             and
             Inequality
             of
             Magnitudes
             ;
             of
             (
             which
             omitting
             the
             rest
             )
             I
             will
             here
             demonstrate
             onely
             this
             one
             ,
             
               The
               Whole
               is
               greater
               then
               any
               Part
               thereof
            
             ;
             to
             the
             end
             that
             the
             Reader
             may
             know
             that
             those
             Axioms
             are
             not
             indemonstrable
             ,
             &
             therefore
             not
             Principles
             of
             Demonstration
             ;
             and
             from
             hence
             learn
             to
             be
             wary
             how
             he
             admits
             any
             thing
             for
             a
             Principle
             ,
             which
             is
             not
             at
             least
             as
             evident
             as
             these
             are
             .
             Greater
             is
             defined
             to
             be
             that
             ,
             whose
             Part
             is
             Equal
             to
             the
             Whole
             of
             another
             .
             Now
             if
             we
             suppose
             any
             Whole
             to
             be
             A
             ,
             and
             a
             Part
             of
             it
             to
             be
             B
             ;
             seeing
             the
             Whole
             B
             is
             Equal
             to
             it self
             ,
             and
             the
             same
             B
             is
             a
             Part
             of
             A
             ;
             therefore
             a
             Part
             of
             A
             will
             be
             Equal
             to
             the
             Whole
             B.
             Wherefore
             by
             the
             Definition
             above
             ,
             A
             is
             Greater
             then
             B
             ,
             which
             was
             to
             be
             proved
             .
          
        
         
           
           
             
               CHAP.
               IX
            
             .
             Of
             Cause
             and
             Effect
             .
          
           
             
               
                 1
                 Action
                 and
                 Passion
                 what
                 they
                 are
                 .
              
               
                 2
                 Action
                 and
                 Passion
                 Mediate
                 and
                 Immediate
                 .
              
               
                 3
                 Cause
                 simply
                 taken
                 .
                 Cause
                 without
                 which
                 no
                 Effect
                 follows
                 ,
                 or
                 Cause
                 Necessary
                 by
                 Supposition
                 .
              
               
                 4
                 Cause
                 Efficient
                 and
                 Material
                 .
              
               
                 5
                 An
                 Entire
                 Cause
                 is
                 alwayes
                 sufficient
                 to
                 produce
                 its
                 Effect
                 .
                 At
                 the
                 same
                 instant
                 that
                 the
                 Cause
                 is
                 Entire
                 ,
                 the
                 Effect
                 is
                 produced
                 .
                 Every
                 Effect
                 has
                 a
                 Necessary
                 Cause
                 .
              
               
                 6
                 The
                 Generation
                 of
                 Effects
                 is
                 Continual
                 .
                 What
                 is
                 the
                 Beginning
                 in
                 Causation
                 .
              
               
                 7
                 No
                 Cause
                 of
                 Motion
                 but
                 in
                 a
                 Body
                 Contiguous
                 and
                 Moved
                 .
              
               
                 8
                 The
                 same
                 Agents
                 and
                 Patients
                 if
                 alike
                 disposed
                 ,
                 produce
                 like
                 Effects
                 ,
                 though
                 at
                 different
                 times
                 .
              
               
                 9
                 All
                 Mutation
                 is
                 Motion
                 .
              
               
                 10
                 Contingent
                 Accidents
                 what
                 they
                 are
                 .
              
            
          
           
             1
             A
             Body
             is
             said
             to
             Work
             upon
             or
             Act
             ,
             that
             is
             to
             say
             ,
             Do
             some
             thing
             to
             another
             Body
             ,
             when
             it
             either
             generates
             or
             destroys
             some
             Accident
             in
             it
             ;
             and
             the
             Body
             in
             which
             an
             Accident
             is
             generated
             or
             destroyed
             is
             said
             to
             Suffer
             ,
             that
             is
             ,
             to
             have
             something
             Done
             to
             it
             by
             another
             Body
             ,
             As
             when
             one
             Body
             by
             putting
             forwards
             another
             Body
             generates
             Motion
             in
             it
             ,
             it
             is
             called
             the
             AGENT
             ;
             and
             the
             Body
             in
             which
             Motion
             is
             so
             generated
             ,
             is
             called
             the
             PATIENT
             ;
             so
             Fire
             that
             warms
             the
             Hand
             is
             the
             Agent
             ,
             and
             the
             Hand
             which
             is
             warmed
             is
             the
             Patient
             .
             That
             Accident
             which
             is
             generated
             in
             the
             Patient
             is
             called
             the
             EFFECT
             .
          
           
             2
             When
             an
             Agent
             and
             Patient
             are
             Contiguous
             to
             one
             another
             ,
             their
             Action
             and
             Reason
             are
             then
             said
             to
             be
             Immediate
             ,
             otherwise
             Mediate
             ;
             and
             when
             another
             Body
             lying
             betwixt
             the
             Agent
             and
             Patient
             is
             Contiguous
             to
             them
             both
             ,
             it
             is
             then
             it self
             both
             an
             Agent
             and
             a
             Patient
             ,
             an
             Agent
             in
             respect
             of
             the
             Body
             next
             after
             it
             ,
             upon
             which
             it
             Works
             ,
             and
             a
             Patient
             in
             respect
             of
             the
             Body
             next
             before
             it
             ,
             from
             which
             it
             suffers
             .
             Also
             if
             many
             Bodies
             be
             so
             ordered
             that
             every
             two
             which
             are
             next
             to
             one
             another
             be
             contiguous
             ,
             then
             all
             those
             that
             are
             betwixt
             the
             first
             and
             the
             last
             are
             both
             Agents
             and
             Patients
             ,
             and
             the
             first
             is
             an
             Agent
             onely
             ,
             and
             the
             last
             a
             Patient
             onely
             .
          
           
           
             3
             An
             Agent
             is
             understood
             to
             produce
             its
             determined
             or
             certain
             Effect
             in
             the
             Patient
             ,
             according
             to
             some
             certain
             Accident
             ,
             or
             Accidents
             ,
             with
             which
             both
             it
             and
             the
             Patient
             are
             affected
             ;
             that
             is
             to
             say
             ,
             the
             Agent
             hath
             its
             Effect
             precisely
             such
             ,
             not
             because
             it
             is
             a
             Body
             ,
             but
             because
             such
             a
             Body
             ,
             or
             so
             Moved
             ;
             For
             otherwise
             all
             Agents
             ,
             seeing
             they
             are
             all
             Bodies
             alike
             ,
             would
             produce
             like
             Effects
             in
             all
             Patients
             ;
             and
             therefore
             the
             Fire
             (
             for
             example
             )
             does
             not
             warm
             ,
             because
             it
             is
             a
             Body
             ,
             but
             because
             it
             is
             Hot
             ;
             nor
             does
             one
             Body
             put
             forward
             another
             Body
             because
             it
             is
             a
             Body
             ,
             but
             because
             it
             is
             moved
             into
             the
             place
             of
             that
             other
             Body
             .
             The
             Cause
             therefore
             of
             all
             Effects
             consists
             in
             certain
             Accidents
             both
             in
             the
             Agents
             and
             in
             the
             Patient
             ;
             which
             when
             they
             are
             all
             present
             ,
             the
             Effect
             is
             produced
             ;
             but
             if
             any
             one
             of
             them
             be
             wanting
             it
             is
             not
             produced
             ;
             and
             that
             Accident
             either
             of
             the
             Agent
             or
             Patient
             ,
             without
             which
             the
             Effect
             cannot
             be
             produced
             ,
             is
             called
             
               Causa
               sine
               qua
               non
            
             ,
             or
             
               Cause
               Necessary
               by
               Supposition
            
             ,
             as
             also
             the
             
               Cause
               Requisite
               for
               the
               Production
               of
               the
               Effect
               .
            
             But
             a
             CAUSE
             .
             simply
             ,
             or
             
               An
               Entire
               Cause
               ,
               is
               the
               Aggregate
               of
               all
               the
               Accidents
               both
               of
               the
               Agents
               how
               many
               soever
               they
               be
               ,
               and
               of
               the
               Patient
               ,
               put
               together
               ;
               which
               when
               they
               are
               all
               supposed
               to
               be
               present
               ,
               it
               cannot
               be
               understood
               but
               that
               the
               Effect
               is
               produced
               at
               the
               same
               instant
               ;
               and
               if
               any
               one
               of
               them
               be
               wanting
               ,
               it
               cannot
               be
               understood
               but
               that
               the
               Effect
               is
               not
               produced
               .
            
          
           
             4
             The
             Aggregate
             of
             Accidents
             in
             the
             Agent
             or
             Agents
             ,
             requisite
             for
             the
             production
             of
             the
             Effect
             ,
             the
             Effect
             being
             produced
             ,
             is
             called
             the
             
               Efficient
               Cause
            
             thereof
             ;
             and
             the
             Aggregate
             of
             Accidents
             in
             the
             Patient
             ,
             the
             Effect
             being
             produced
             ,
             is
             usually
             called
             the
             
               Material
               Cause
            
             ;
             I
             say
             the
             Effect
             being
             produced
             ;
             for
             where
             there
             is
             no
             Effect
             ,
             there
             can
             be
             no
             Cause
             ;
             for
             nothing
             can
             be
             called
             a
             Cause
             where
             there
             is
             nothing
             that
             can
             be
             called
             an
             Effect
             .
             But
             the
             Efficient
             and
             Material
             Causes
             ,
             are
             both
             but
             Partial
             Causes
             ,
             or
             Parts
             of
             that
             Cause
             which
             in
             the
             next
             precedent
             article
             I
             called
             an
             Entire
             Cause
             .
             And
             from
             hence
             it
             is
             manifest
             ,
             that
             the
             Effect
             we
             expect
             ,
             though
             the
             Agents
             be
             not
             defective
             on
             their
             part
             ,
             may
             nevertheless
             be
             frustrated
             by
             a
             defect
             in
             the
             Patient
             ;
             and
             when
             the
             Patient
             is
             sufficient
             ,
             by
             a
             defect
             in
             the
             Agents
             .
          
           
             5
             An
             Entire
             Cause
             is
             alwayes
             sufficient
             for
             the
             production
             of
             its
             Effect
             ,
             if
             the
             Effect
             be
             at
             all
             possible
             .
             For
             let
             any
             Effect
             whatsoever
             be
             propounded
             to
             be
             produced
             ;
             if
             the
             
             same
             be
             produced
             ,
             it
             is
             manifest
             that
             the
             Cause
             which
             produced
             it
             was
             a
             sufficient
             Cause
             ;
             but
             if
             it
             be
             not
             produced
             ,
             and
             yet
             be
             possible
             ,
             it
             is
             evident
             that
             something
             was
             wanting
             either
             in
             some
             Agent
             ,
             or
             in
             the
             Patient
             ,
             without
             which
             it
             could
             not
             be
             produced
             ;
             that
             is
             ,
             that
             some
             Accident
             was
             wanting
             which
             was
             requisite
             for
             its
             Production
             ;
             and
             therefore
             that
             Cause
             was
             not
             Entire
             ,
             which
             is
             contrary
             to
             what
             was
             supposed
             .
          
           
             It
             follows
             also
             from
             hence
             ,
             that
             in
             whatsoever
             instant
             the
             Cause
             is
             Entire
             ,
             in
             the
             same
             instant
             the
             Effect
             is
             produced
             .
             For
             if
             it
             be
             not
             produced
             ,
             something
             is
             still
             wanting
             ,
             which
             is
             requisite
             for
             the
             production
             of
             it
             ;
             and
             therefore
             the
             Cause
             was
             not
             Entire
             ,
             as
             was
             supposed
             .
          
           
             And
             seeing
             a
             Necessary
             Cause
             is
             defined
             to
             be
             that
             ,
             which
             being
             supposed
             ,
             the
             Effect
             cannot
             but
             follow
             ;
             this
             also
             may
             be
             collected
             ,
             that
             whatsoever
             Effect
             is
             produced
             at
             any
             time
             ,
             the
             same
             is
             produced
             by
             a
             Necessary
             Cause
             .
             For
             whatsoever
             is
             produced
             ,
             in
             as
             much
             as
             it
             is
             produced
             ,
             had
             an
             Entire
             Cause
             ,
             that
             is
             ,
             had
             all
             those
             things
             ,
             which
             being
             supposed
             ,
             it
             cannot
             be
             understood
             but
             that
             the
             Effect
             follows
             ;
             that
             is
             ,
             it
             had
             a
             Necessary
             Cause
             .
             And
             in
             the
             same
             manner
             it
             may
             be
             shewn
             ,
             that
             whatsoever
             Effects
             are
             hereafter
             to
             be
             produced
             ,
             shall
             have
             a
             Necessary
             Cause
             ;
             so
             that
             all
             the
             Effects
             ,
             that
             have
             been
             or
             shall
             be
             produced
             ,
             have
             their
             Ne
             cessity
             in
             things
             antecedent
             .
          
           
             6
             And
             from
             this
             ,
             that
             whensoever
             the
             Cause
             is
             Entire
             ,
             the
             Effect
             is
             produced
             in
             the
             same
             instant
             ,
             it
             is
             manifest
             ,
             that
             Causation
             and
             the
             Production
             of
             Effects
             consist
             in
             a
             certain
             continual
             Progress
             ;
             so
             that
             as
             there
             is
             a
             continual
             Mutation
             in
             the
             Agent
             or
             Agents
             by
             the
             working
             of
             other
             Agents
             upon
             them
             ,
             so
             also
             the
             Patient
             upon
             which
             they
             work
             is
             continually
             altered
             and
             changed
             .
             For
             example
             ,
             as
             the
             Heat
             of
             the
             Fire
             encreases
             more
             and
             more
             ,
             so
             also
             the
             Effects
             thereof
             ,
             namely
             the
             Heat
             of
             such
             Bodies
             as
             are
             next
             to
             it
             ,
             &
             again
             of
             such
             other
             Bodies
             as
             are
             next
             to
             them
             ,
             encreases
             more
             &
             more
             accordingly
             ;
             which
             is
             already
             no
             litle
             argument
             that
             all
             Mutation
             consists
             in
             Motion
             onely
             ;
             the
             truth
             whereof
             shall
             be
             further
             demonstrated
             in
             the
             ninth
             Article
             .
             But
             in
             this
             Progress
             of
             Causation
             ,
             that
             is
             ,
             of
             Action
             and
             Passion
             ,
             if
             any
             man
             comprehend
             in
             his
             imagination
             a
             part
             thereof
             ,
             and
             divide
             the
             same
             into
             parts
             ,
             the
             first
             part
             or
             Beginning
             of
             it
             cannot
             be
             considered
             otherwise
             then
             
             as
             Action
             or
             Cause
             ;
             for
             if
             it
             should
             be
             considered
             as
             Effect
             or
             Passion
             ,
             then
             it
             would
             be
             necessary
             to
             consider
             something
             before
             it
             for
             its
             Cause
             or
             Action
             ;
             which
             cannot
             be
             ;
             for
             nothing
             can
             be
             before
             the
             Beginning
             .
             And
             in
             like
             manner
             ,
             the
             last
             part
             is
             considered
             onely
             as
             Effect
             ;
             for
             it
             cannot
             be
             called
             Cause
             if
             nothing
             follow
             it
             ;
             but
             after
             the
             last
             nothing
             follows
             .
             And
             from
             hence
             it
             is
             ,
             that
             in
             all
             Action
             the
             Beginning
             and
             Cause
             are
             taken
             for
             the
             same
             thing
             .
             But
             every
             one
             of
             the
             intermediate
             parts
             are
             both
             Action
             and
             Passion
             ,
             and
             Cause
             and
             Effect
             ,
             according
             as
             they
             are
             compared
             with
             the
             antecedent
             or
             subsequent
             part
             .
          
           
             7
             There
             can
             be
             no
             Cause
             of
             Motion
             ,
             except
             in
             a
             Body
             Contiguous
             ,
             and
             Moved
             .
             For
             let
             there
             be
             any
             two
             Bodies
             which
             are
             not
             contiguous
             ,
             and
             betwixt
             which
             the
             intermediate
             Space
             is
             empty
             ,
             or
             if
             filled
             ,
             filled
             with
             another
             Body
             which
             is
             at
             Rest
             ;
             and
             let
             one
             of
             the
             propounded
             Bodies
             be
             supposed
             to
             be
             at
             Rest
             ,
             I
             say
             it
             shall
             always
             be
             at
             Rest.
             For
             if
             it
             shall
             be
             Moved
             ,
             the
             Cause
             of
             that
             Motion
             (
             by
             the
             8th
             .
             Chapter
             19th
             .
             Article
             )
             will
             be
             in
             some
             external
             Body
             ;
             and
             therefore
             if
             between
             it
             and
             that
             external
             Body
             there
             be
             nothing
             but
             empty
             Space
             ,
             then
             whatsoever
             the
             disposition
             be
             of
             that
             external
             Body
             ,
             or
             of
             the
             Patient
             it self
             ,
             yet
             if
             it
             be
             supposed
             to
             be
             now
             at
             Rest
             ,
             we
             may
             conceive
             it
             wil
             continue
             so
             til
             it
             be
             touched
             by
             some
             other
             Body
             ;
             but
             seeing
             Cause
             (
             by
             the
             Definition
             )
             is
             the
             Aggregate
             of
             all
             such
             Accidents
             ,
             which
             being
             supposed
             to
             be
             present
             it
             cannot
             be
             conceived
             but
             that
             the
             Effect
             will
             follow
             ,
             those
             Accidents
             which
             are
             either
             in
             external
             Bodies
             ,
             or
             in
             the
             Patient
             it self
             ,
             cannot
             be
             the
             Cause
             of
             future
             Motion
             ;
             and
             in
             like
             manner
             ,
             seeing
             we
             may
             conceive
             ,
             that
             whatsoever
             is
             at
             Rest
             ,
             will
             still
             be
             at
             Rest
             ,
             though
             it
             be
             touched
             by
             some
             other
             Body
             ,
             except
             that
             other
             Body
             be
             moved
             ,
             therefore
             in
             a
             contiguous
             Body
             which
             is
             at
             Rest
             ,
             there
             can
             be
             no
             Cause
             of
             Motion
             .
             Wherefore
             there
             is
             no
             Cause
             of
             Motion
             in
             any
             Body
             ,
             except
             it
             be
             Contiguous
             and
             Moved
             .
          
           
             The
             same
             reason
             may
             serve
             to
             prove
             ,
             that
             whatsoever
             is
             Moved
             ,
             will
             alwayes
             be
             Moved
             on
             in
             the
             same
             way
             and
             with
             the
             same
             Velocity
             ,
             except
             it
             be
             hindered
             by
             some
             other
             Contiguous
             and
             Moved
             Body
             ;
             and
             consequently
             that
             no
             Bodies
             either
             when
             they
             are
             at
             Rest
             ,
             or
             when
             there
             is
             an
             interposition
             of
             Vacuum
             ,
             can
             generate
             or
             ●●tinguish
             or
             lesson
             Motion
             in
             other
             Bodies
             .
             There
             is
             one
             that
             has
             
             written
             ,
             that
             things
             Moved
             are
             more
             resisted
             by
             things
             at
             Rest
             ,
             then
             by
             things
             contrarily
             Moved
             ,
             for
             this
             reason
             ,
             that
             he
             conceived
             Motion
             not
             to
             be
             so
             contrary
             to
             Motion
             as
             Rest.
             That
             which
             deceived
             him
             was
             ,
             that
             the
             words
             Rest
             and
             Motion
             are
             but
             contradictory
             Names
             ;
             whereas
             Motion
             indeed
             is
             not
             resisted
             by
             Rest
             ,
             but
             by
             contrary
             Motion
             .
          
           
             8
             But
             if
             a
             Body
             work
             upon
             another
             Body
             at
             one
             time
             ,
             and
             afterwards
             the
             same
             Body
             work
             upon
             the
             same
             Body
             at
             another
             time
             so
             ,
             that
             both
             the
             Agent
             and
             Patient
             ,
             and
             all
             their
             parts
             ,
             be
             in
             all
             things
             as
             they
             were
             ;
             and
             there
             be
             no
             difference
             except
             onely
             in
             time
             ,
             that
             is
             ,
             that
             one
             Action
             be
             former
             the
             other
             later
             in
             time
             ;
             it
             is
             manifest
             of
             it self
             ,
             that
             the
             Effects
             will
             be
             Equal
             and
             Like
             ;
             as
             not
             differing
             in
             any
             thing
             besides
             time
             .
             And
             as
             Effects
             themselves
             proceed
             from
             their
             Causes
             ;
             so
             the
             diversity
             of
             them
             depends
             upon
             the
             diversity
             of
             their
             Causes
             also
             .
          
           
             9
             This
             being
             true
             ,
             it
             is
             necessary
             that
             Mutation
             can
             be
             nothing
             else
             ,
             but
             Motion
             of
             the
             Parts
             of
             that
             Body
             which
             is
             Changed
             .
             For
             First
             ,
             we
             do
             not
             say
             any
             thing
             is
             Changed
             ,
             but
             that
             which
             appears
             to
             our
             Senses
             otherwise
             then
             it
             appeared
             formerly
             .
             Secondly
             ,
             both
             those
             Appearances
             are
             Effects
             produced
             in
             the
             Sentient
             ;
             &
             therefore
             if
             they
             be
             differēt
             ,
             it
             is
             necessary
             (
             by
             the
             preceding
             article
             )
             that
             either
             some
             part
             of
             the
             Agent
             which
             was
             formerly
             at
             Rest
             ,
             is
             now
             Moved
             ,
             and
             so
             the
             Mutation
             consists
             in
             this
             Motion
             ;
             or
             some
             part
             which
             was
             formerly
             Moved
             ,
             is
             now
             otherwise
             Moved
             ,
             and
             so
             also
             the
             Mutation
             consists
             in
             this
             new
             Motion
             ;
             or
             which
             being
             formerly
             Moved
             ,
             is
             now
             at
             Rest
             ,
             which
             (
             as
             I
             have
             shewn
             above
             )
             cannot
             come
             to
             pass
             without
             Motion
             ,
             and
             so
             again
             Mutation
             is
             Motion
             ;
             or
             ,
             lastly
             ,
             it
             happens
             in
             some
             of
             these
             manners
             to
             the
             Patient
             or
             some
             of
             its
             parts
             ;
             so
             that
             Mutation
             ,
             howsoever
             it
             be
             made
             ,
             will
             consist
             in
             the
             Motion
             of
             the
             parts
             either
             of
             the
             Body
             which
             is
             perceived
             ,
             or
             of
             the
             Sentient
             Body
             ,
             or
             of
             both
             .
             Mutation
             therefore
             is
             Motion
             ,
             (
             namely
             of
             the
             parts
             either
             of
             the
             Agent
             or
             of
             the
             Patient
             ;
             )
             which
             was
             to
             be
             demonstrated
             .
             And
             to
             this
             it
             is
             consequent
             ,
             that
             Rest
             cannot
             be
             the
             Cause
             of
             any
             thing
             ;
             nor
             can
             any
             Action
             proceed
             from
             it
             ,
             seeing
             neither
             Motion
             nor
             Mutation
             can
             be
             caused
             by
             it
             .
          
           
           
             10
             Accidents
             ,
             in
             respect
             of
             other
             Accidents
             which
             precede
             them
             ,
             or
             are
             before
             them
             in
             time
             ,
             &
             upon
             which
             they
             do
             not
             depend
             as
             upon
             their
             Causes
             ,
             are
             called
             Contingent
             Accidents
             ;
             I
             say
             in
             respect
             of
             those
             Accidents
             by
             which
             they
             are
             not
             generated
             ;
             for
             in
             respect
             of
             their
             Causes
             all
             things
             come
             to
             pass
             with
             equal
             necessity
             ,
             for
             otherwise
             ,
             they
             would
             have
             no
             Causes
             at
             all
             ;
             which
             of
             things
             generated
             is
             not
             intelligible
             .
          
        
         
           
           
             
               CHAP.
               X.
            
             Of
             Power
             and
             Act.
             
          
           
             
               
                 1
                 Power
                 and
                 Cause
                 are
                 the
                 same
                 thing
                 .
              
               
                 2
                 An
                 Act
                 is
                 produced
                 at
                 the
                 same
                 instant
                 in
                 which
                 the
                 Power
                 is
                 Plenary
                 .
              
               
                 3
                 Active
                 and
                 Passive
                 Power
                 are
                 parts
                 onely
                 of
                 Plenary
                 Power
                 .
              
               
                 4
                 An
                 Act
                 when
                 said
                 to
                 be
                 Possible
                 .
              
               
                 5
                 An
                 Act
                 Necessary
                 and
                 Contingent
                 ,
                 what
                 .
              
               
                 6
                 Active
                 Power
                 consists
                 in
                 Motion
                 .
              
               
                 7
                 Cause
                 Formal
                 and
                 Final
                 ,
                 what
                 they
                 are
                 .
              
            
          
           
             1_COrrespondent
             to
             Cause
             and
             Effect
             are
             POWER
             and
             ACT
             ;
             Nay
             ,
             those
             and
             these
             are
             the
             same
             things
             ,
             though
             for
             divers
             considerations
             they
             have
             divers
             names
             .
             Forwhensoever
             any
             Agent
             has
             all
             those
             Accidents
             which
             are
             necessarily
             requisite
             for
             the
             production
             of
             some
             Effect
             in
             the
             Patient
             ,
             then
             we
             say
             that
             Agent
             has
             Power
             to
             produce
             that
             Effect
             ,
             if
             it
             be
             applyed
             to
             a
             Patient
             .
             But
             (
             as
             I
             have
             shewn
             in
             the
             precedent
             Chapter
             ,
             )
             those
             Accidents
             constitute
             the
             Efficient
             Cause
             ;
             and
             therefore
             the
             same
             Accidents
             which
             constitute
             the
             Efficient
             Cause
             ,
             constitute
             also
             the
             Power
             of
             the
             Agent
             .
             Wherefore
             the
             
               Power
               of
               the
               Agent
            
             ,
             and
             the
             
               Efficient
               Cause
            
             are
             the
             same
             thing
             .
             But
             they
             are
             considered
             with
             this
             difference
             ,
             that
             Cause
             is
             so
             called
             in
             respect
             of
             the
             Effect
             already
             produced
             ,
             and
             Power
             in
             respect
             of
             the
             same
             Effect
             to
             be
             produced
             hereafter
             ,
             so
             that
             Cause
             respects
             the
             Past
             ,
             Power
             the
             Future
             time
             .
             Also
             the
             
               Power
               of
               the
               Agent
            
             ,
             is
             that
             which
             is
             commonly
             called
             
               Active
               Power
            
             .
          
           
             In
             like
             manner
             ,
             whensoever
             any
             Patient
             has
             all
             those
             Accidents
             which
             it
             is
             requisite
             it
             should
             have
             for
             the
             production
             of
             some
             Effect
             in
             it
             ,
             we
             say
             it
             is
             in
             the
             Power
             of
             that
             Patient
             to
             produce
             that
             Effect
             ,
             if
             it
             be
             applyed
             to
             a
             fitting
             Agent
             .
             But
             those
             Accidents
             (
             as
             is
             defined
             in
             the
             precedent
             Chapter
             )
             constitute
             the
             Material
             
             Cause
             ;
             and
             therefore
             the
             
               Power
               of
               the
               Patient
            
             ,
             (
             commonly
             called
             
               Passive
               Power
            
             )
             and
             
               Material
               Cause
            
             are
             the
             same
             thing
             ;
             but
             with
             this
             different
             consideration
             ,
             that
             in
             Cause
             the
             Past
             time
             ,
             and
             in
             Power
             the
             Future
             is
             respected
             .
             Wherefore
             the
             Power
             of
             the
             Agent
             and
             Patient
             together
             ,
             which
             may
             be
             called
             Entire
             or
             
               Plenary
               Power
            
             ,
             is
             the
             same
             thing
             with
             
               Entire
               Cause
            
             ;
             for
             they
             both
             consist
             in
             the
             Sum
             or
             Aggregate
             of
             all
             the
             Accidents
             as
             well
             in
             the
             Agent
             as
             in
             the
             Patient
             ,
             which
             are
             requisite
             for
             the
             production
             of
             the
             Effect
             .
             Lastly
             ,
             as
             the
             Accident
             produced
             is
             in
             respect
             of
             the
             Cause
             called
             an
             Effect
             ;
             so
             in
             respect
             of
             the
             Power
             it
             is
             called
             an
             Act.
             
          
           
             2
             As
             therefore
             the
             Effect
             is
             produced
             in
             the
             same
             instant
             in
             which
             the
             Cause
             is
             Entire
             ;
             so
             also
             every
             Act
             that
             may
             be
             produced
             ,
             is
             produced
             in
             the
             same
             instant
             ,
             in
             which
             the
             Power
             is
             Plenary
             .
             And
             as
             there
             can
             be
             no
             Effect
             ,
             but
             from
             a
             Sufficient
             and
             Necessary
             Cause
             ;
             so
             also
             no
             Act
             can
             be
             produced
             ,
             but
             by
             Sufficient
             Power
             ,
             or
             that
             Power
             by
             which
             it
             could
             not
             but
             be
             produced
             .
          
           
             3
             And
             as
             it
             is
             manifest
             ,
             (
             as
             I
             have
             shewn
             )
             that
             the
             Efficient
             and
             Material
             Causes
             are
             severally
             and
             by
             themselves
             parts
             onely
             of
             an
             Entire
             Cause
             ,
             and
             cannot
             produce
             any
             Effect
             but
             by
             being
             joyned
             together
             ;
             so
             also
             Power
             Active
             and
             Passive
             ,
             are
             parts
             onely
             of
             Plenary
             and
             Entire
             Power
             ;
             nor
             ,
             except
             they
             be
             joyned
             ,
             can
             any
             Act
             proceed
             from
             them
             ;
             and
             therefore
             these
             Powers
             (
             as
             I
             said
             in
             the
             first
             Article
             )
             are
             but
             conditionall
             ,
             namely
             ,
             
               the
               Agent
               has
               Power
               ,
               if
               it
               be
               applyed
               to
               a
               Patient
               ;
               and
               the
               Patient
               has
               Power
               ,
               if
               it
               be
               applyed
               to
               an
               Agent
            
             ;
             otherwise
             neither
             of
             them
             have
             Power
             ,
             nor
             can
             the
             Accidents
             which
             are
             in
             them
             severally
             be
             properly
             called
             Powers
             ;
             nor
             any
             Action
             be
             said
             to
             be
             Possible
             ,
             for
             the
             Power
             of
             the
             Agent
             alone
             ,
             or
             of
             the
             Patient
             alone
             .
          
           
             4
             For
             that
             is
             an
             Impossible
             Act
             ,
             for
             the
             production
             of
             which
             there
             is
             no
             Power
             Plenary
             .
             For
             seeing
             Plenary
             Power
             is
             that
             in
             which
             all
             things
             concurre
             which
             are
             requisite
             for
             the
             production
             of
             an
             Act
             ,
             if
             the
             Power
             shall
             never
             be
             Plenary
             ,
             there
             will
             always
             be
             wanting
             some
             of
             those
             things
             ,
             without
             which
             the
             Act
             cannot
             be
             produced
             ;
             wherefore
             that
             Act
             shall
             never
             be
             produced
             ,
             that
             is
             ,
             that
             Act
             is
             IMPOSSIBLE
             :
             And
             every
             Act
             which
             is
             not
             Impossible
             ,
             is
             POSSIBLE
             .
             Every
             Act
             therefore
             which
             is
             Possible
             shall
             at
             some
             time
             be
             produced
             ;
             for
             if
             it
             shall
             never
             be
             produced
             ,
             then
             those
             things
             shall
             never
             concurre
             which
             are
             requisite
             for
             the
             production
             
             of
             it
             ;
             wherefore
             that
             Act
             is
             Impossible
             by
             the
             Definition
             ;
             which
             is
             contrary
             to
             what
             was
             supposed
             .
          
           
             5
             A
             
               Necessary
               Act
            
             is
             that
             ,
             the
             production
             whereof
             it
             is
             Impossible
             to
             hinder
             ;
             and
             therefore
             every
             Act
             that
             shall
             be
             produced
             ;
             shall
             necessarily
             be
             produced
             ;
             for
             that
             it
             shall
             not
             be
             produced
             is
             Impossible
             ,
             because
             (
             as
             is
             already
             demonstrated
             )
             every
             Possible
             Act
             shall
             at
             some
             time
             be
             produced
             ;
             Nay
             ,
             this
             Proposition
             ,
             
               What
               shall
               be
               ,
               shall
               be
               ,
            
             is
             as
             necessary
             a
             Proposition
             ,
             as
             this
             ,
             
               A
               Man
               is
               a
               Man.
            
             
          
           
             But
             here
             perhaps
             some
             man
             may
             ask
             ,
             whether
             those
             Future
             things
             ,
             which
             are
             commonly
             called
             Contingents
             ,
             are
             Necessary
             .
             I
             say
             therefore
             that
             generally
             all
             Contingents
             ,
             have
             their
             Necessary
             Causes
             ,
             (
             as
             is
             shewn
             in
             the
             preceding
             Chapter
             ,
             )
             but
             are
             called
             Contingents
             in
             respect
             of
             other
             Events
             upon
             which
             they
             do
             not
             depend
             ;
             as
             the
             Rain
             which
             shall
             be
             to
             morrow
             ,
             shall
             be
             Necessarily
             ,
             (
             that
             is
             from
             necessary
             Causes
             ;
             )
             but
             we
             think
             and
             say
             it
             happens
             by
             chance
             ,
             because
             we
             doe
             not
             yet
             perceive
             the
             Causes
             thereof
             ,
             though
             they
             exist
             now
             ;
             for
             men
             commonly
             call
             that
             Casuall
             or
             Contingent
             ,
             whereof
             they
             do
             not
             perceive
             the
             necessary
             Cause
             ;
             and
             in
             the
             same
             manner
             they
             use
             to
             speake
             of
             things
             past
             ,
             when
             not
             knowing
             whether
             a
             thing
             be
             done
             or
             no
             ,
             they
             say
             it
             is
             possible
             it
             never
             was
             done
             .
          
           
             Wherefore
             all
             Propositions
             concerning
             Future
             things
             contingent
             or
             not
             contingent
             ,
             as
             this
             ,
             
               It
               will
               rayne
               to
               morrow
            
             ,
             or
             this
             ,
             
               To
               morrow
               the
               Sun
               will
               rise
               ,
            
             are
             either
             necessarily
             true
             ,
             or
             necessarily
             false
             ;
             but
             we
             call
             them
             Contingent
             because
             we
             doe
             not
             yet
             know
             whether
             they
             be
             true
             or
             false
             ;
             whereas
             their
             Verity
             depends
             not
             upon
             our
             Knowledge
             ,
             but
             upon
             the
             foregoing
             of
             their
             Causes
             .
             But
             there
             are
             some
             who
             though
             they
             confess
             this
             whole
             Proposition
             ,
             
               To
               morrow
               it
               will
               either
               rain
               or
               not
               rain
               ,
            
             to
             be
             true
             ,
             yet
             they
             will
             not
             acknowledge
             the
             parts
             of
             it
             ,
             as
             
               To
               morrow
               it
               will
               rain
            
             ,
             or
             
               To
               morrow
               it
               will
               not
               rain
               ,
            
             to
             be
             either
             of
             them
             true
             by
             it self
             ,
             because
             they
             say
             neither
             this
             nor
             that
             is
             true
             determinately
             .
             But
             what
             is
             this
             
               determinately
               true
            
             ,
             but
             true
             upon
             our
             knowledge
             ,
             or
             evidently
             true
             ?
             and
             therefore
             they
             say
             no
             more
             but
             that
             it
             is
             not
             yet
             known
             whether
             it
             be
             true
             or
             no
             ;
             but
             they
             say
             it
             more
             obscurely
             ,
             and
             darken
             the
             Evidence
             of
             the
             truth
             with
             the
             same
             words
             with
             which
             they
             endevour
             to
             hide
             their
             own
             ignorance
             .
          
           
             6
             In
             the
             9th
             .
             Article
             of
             the
             precedent
             Chapter
             I
             have
             shewn
             ,
             
             that
             the
             Efficient
             Cause
             of
             all
             Motion
             and
             Mutation
             consists
             in
             the
             Motion
             of
             the
             Agent
             or
             Agents
             ;
             And
             in
             the
             first
             Article
             of
             this
             Chapter
             ,
             that
             the
             Power
             of
             the
             Agent
             is
             the
             same
             thing
             with
             the
             Efficient
             Cause
             .
             From
             whence
             it
             may
             be
             understood
             ,
             that
             all
             Active
             Power
             consists
             in
             Motion
             also
             ;
             and
             that
             Power
             is
             not
             a
             certain
             Accident
             which
             differs
             from
             all
             Acts
             ,
             but
             is
             indeed
             an
             Act
             ,
             namely
             Motion
             ,
             which
             is
             therefore
             called
             Power
             ,
             because
             another
             Act
             shall
             be
             produced
             by
             it
             afterwards
             .
             For
             example
             ,
             if
             of
             three
             Bodies
             the
             first
             put
             forwards
             the
             second
             ,
             and
             this
             the
             third
             ,
             the
             Motion
             of
             the
             second
             in
             respect
             of
             the
             first
             which
             produceth
             it
             ,
             is
             the
             Act
             of
             the
             second
             Body
             ,
             but
             in
             respect
             of
             the
             third
             it
             is
             the
             Active
             Power
             of
             the
             same
             second
             Body
             .
          
           
             7
             The
             Writers
             of
             Metaphysiques
             reckon
             up
             two
             other
             Causes
             besides
             the
             Efficient
             and
             Material
             ,
             namely
             the
             ESSENCE
             ,
             which
             some
             call
             the
             
               Formal
               Cause
            
             ;
             and
             the
             End
             ,
             or
             
               Final
               Cause
            
             ;
             both
             which
             are
             nevertheless
             Efficient
             Causes
             .
             For
             when
             it
             is
             said
             ,
             the
             Essence
             of
             a
             thing
             is
             the
             Cause
             thereof
             ,
             
               as
               to
               be
               Rational
               ,
               is
               the
               Cause
               of
               Man
               ,
            
             it
             is
             not
             intelligible
             ;
             for
             it
             is
             all
             one
             as
             if
             it
             were
             said
             ,
             
               To
               be
               a
               Man
               is
               the
               Cause
               of
               Man
               ,
            
             which
             is
             not
             well
             said
             .
             And
             yet
             the
             knowledge
             of
             the
             Essence
             of
             any
             thing
             ,
             is
             the
             Cause
             of
             the
             knowledge
             of
             the
             thing
             it selfe
             ;
             for
             if
             I
             first
             know
             that
             a
             thing
             is
             Rational
             ,
             I
             know
             from
             thence
             that
             the
             same
             is
             Man
             ;
             but
             this
             is
             no
             other
             then
             an
             Efficient
             Cause
             .
             A
             
               Final
               Cause
            
             has
             no
             place
             but
             in
             such
             things
             as
             have
             Sense
             and
             Will
             ;
             and
             this
             also
             I
             shall
             prove
             hereafter
             to
             be
             an
             Efficient
             Cause
             .
          
        
         
           
           
             
               CHAP.
               XI
            
             .
             Of
             Identity
             and
             Difference
             .
          
           
             
               
                 1
                 What
                 it
                 is
                 for
                 one
                 thing
                 to
                 Differ
                 from
                 another
                 .
              
               
                 2
                 To
                 Differ
                 in
                 Number
                 ,
                 Magnitude
                 ,
                 Species
                 and
                 Genus
                 ,
                 what
                 .
              
               
                 3
                 What
                 is
                 Relation
                 ,
                 Proportion
                 ,
                 and
                 Relatives
                 .
              
               
                 4
                 Proportionals
                 what
                 .
              
               
                 5
                 The
                 Proportion
                 of
                 Magnitudes
                 to
                 one
                 another
                 ,
                 wherein
                 it
                 consists
                 .
              
               
                 6
                 Relation
                 is
                 no
                 new
                 Accident
                 ,
                 but
                 one
                 of
                 those
                 that
                 were
                 in
                 the
                 Relative
                 before
                 the
                 Relation
                 or
                 Comparison
                 was
                 made
                 .
                 Also
                 the
                 Causes
                 of
                 Accidents
                 in
                 the
                 Correlatives
                 are
                 the
                 Cause
                 of
                 Relation
                 .
              
               
                 7
                 Of
                 the
                 Beginning
                 of
                 Individuation
                 .
              
            
          
           
             1_HItherto
             I
             have
             spoken
             of
             Body
             simply
             ,
             and
             of
             Accidents
             common
             to
             all
             Bodies
             as
             
               Magnitude
               ,
               Motion
               ,
               Rest
               ,
               Action
               ,
               Passion
               ,
               Power
               ,
               Possible
               ,
               &c.
               
            
             And
             I
             should
             now
             descend
             to
             those
             Accidents
             by
             which
             one
             Body
             is
             distinguished
             from
             ano●●er
             ,
             but
             that
             it
             is
             first
             to
             be
             declared
             what
             it
             is
             to
             be
             Distinct
             ,
             and
             
               not
               Distinct
            
             ,
             namely
             what
             are
             the
             SAME
             and
             DIFFERENT
             ;
             for
             this
             also
             is
             common
             to
             all
             Bodies
             ,
             that
             they
             may
             be
             distinguished
             and
             differenced
             from
             one
             another
             .
             Now
             two
             Bodies
             are
             said
             to
             Differ
             from
             one
             another
             ,
             when
             something
             may
             be
             said
             of
             one
             of
             them
             ,
             which
             cannot
             be
             said
             of
             the
             other
             at
             the
             same
             time
             .
          
           
             2
             And
             first
             of
             all
             ,
             it
             is
             manifest
             that
             no
             Two
             Bodies
             are
             the
             Same
             ;
             for
             seeing
             they
             are
             Two
             ,
             they
             are
             in
             two
             places
             at
             the
             same
             time
             ;
             as
             that
             which
             is
             the
             Same
             ,
             is
             at
             the
             same
             time
             in
             one
             and
             the
             same
             place
             .
             All
             Bodies
             therefore
             differ
             from
             one
             another
             in
             Number
             ,
             namely
             ,
             as
             One
             and
             Another
             ;
             so
             that
             the
             Same
             and
             
               different
               in
               Number
            
             are
             Names
             opposed
             to
             one
             another
             by
             Contradiction
             .
          
           
             In
             Magnitude
             Bodies
             differ
             when
             One
             is
             greater
             then
             Another
             ,
             as
             
               a
               Cubit
               long
            
             ,
             and
             
               two
               Cubits
               long
               ,
               of
               two
               pound
               weight
               ,
            
             and
             of
             
               three
               pound
               weight
            
             .
             And
             to
             these
             ,
             Equals
             are
             opposed
             .
          
           
           
             Bodies
             which
             differ
             more
             then
             in
             Magnitude
             are
             called
             Unlike
             ,
             and
             those
             which
             differ
             onely
             in
             Magnitude
             ,
             Like
             .
             Also
             of
             Unlike
             Bodies
             some
             are
             said
             to
             differ
             in
             the
             Species
             ,
             other
             in
             the
             Genus
             ;
             in
             the
             Species
             when
             their
             difference
             is
             perceived
             by
             one
             and
             the
             same
             Sense
             ,
             as
             White
             and
             Black
             ;
             and
             in
             the
             Genus
             ,
             when
             their
             difference
             is
             not
             perceived
             but
             by
             divers
             Senses
             ,
             as
             White
             and
             Hot.
             
          
           
             3
             And
             the
             Likeness
             ,
             or
             
               Unlikeness
               ,
               Equality
            
             or
             Inequality
             of
             one
             Body
             to
             another
             ,
             is
             called
             their
             RELATION
             ;
             and
             the
             Bodies
             themselves
             Relatives
             or
             
               Correlatives
               ;
               Aristotle
            
             calls
             them
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             ;
             the
             first
             whereof
             is
             usually
             named
             the
             Antecedent
             ,
             and
             the
             second
             the
             Consequent
             ;
             and
             the
             Relation
             of
             the
             Antecedent
             to
             the
             Consequent
             according
             to
             Magnitude
             ,
             namely
             ,
             the
             Equality
             ,
             the
             Excess
             or
             Defect
             thereof
             ,
             is
             called
             the
             PROPORTION
             of
             the
             Antecedent
             to
             the
             Consequent
             ,
             so
             that
             Proportion
             is
             nothing
             but
             the
             Equality
             or
             Inequality
             of
             the
             Magnitude
             of
             the
             Antecedent
             compared
             to
             the
             Magnitude
             of
             the
             Consequent
             by
             their
             difference
             only
             ,
             or
             compared
             also
             with
             their
             difference
             .
             For
             Example
             ,
             the
             Proportion
             of
             Three
             to
             Two
             consists
             only
             in
             this
             ,
             that
             Three
             exceeds
             Two
             by
             Unity
             ;
             and
             the
             Proportion
             of
             Two
             to
             Five
             in
             this
             ,
             that
             Two
             compared
             with
             Five
             is
             deficient
             of
             it
             by
             Three
             ,
             either
             simply
             ,
             or
             compared
             with
             the
             numbers
             different
             ;
             and
             therefore
             in
             the
             Proportion
             of
             Unequals
             ,
             the
             Proportion
             of
             the
             Lesse
             to
             the
             Greater
             is
             called
             DEFECT
             ,
             and
             that
             of
             the
             Greater
             to
             the
             Lesse
             EXCESS
             .
          
           
             4
             Besides
             ,
             of
             Unequals
             ,
             some
             are
             more
             ,
             some
             lesse
             ,
             and
             some
             equally
             unequall
             ;
             so
             that
             there
             is
             
               Proportion
               of
               Proportions
            
             ,
             as
             well
             as
             of
             Magnitudes
             ,
             namely
             ,
             where
             two
             Unequals
             have
             relation
             to
             two
             other
             Unequals
             ;
             as
             when
             the
             Inequality
             which
             is
             between
             2
             and
             3
             is
             compared
             with
             the
             Inequality
             which
             is
             between
             4
             and
             5.
             
             In
             which
             Comparison
             there
             are
             alwayes
             four
             Magnitudes
             ,
             or
             (
             which
             is
             all
             one
             )
             if
             there
             be
             but
             three
             ,
             the
             midlemost
             is
             twice
             numbred
             ;
             and
             if
             the
             Proportion
             of
             the
             first
             to
             the
             second
             be
             equal
             to
             the
             Proportion
             of
             the
             third
             to
             the
             fourth
             ,
             then
             the
             four
             are
             said
             to
             be
             Proportionals
             ;
             otherwise
             they
             are
             not
             Proportionals
             .
          
           
             5
             The
             Proportion
             of
             the
             Antecedent
             to
             the
             Consequent
             ,
             consists
             in
             their
             Difference
             ,
             not
             onely
             simply
             taken
             ,
             but
             also
             as
             compared
             with
             one
             of
             the
             Relatives
             ;
             that
             is
             ,
             either
             in
             that
             part
             of
             the
             greater
             by
             which
             it
             exceeds
             the
             lesse
             ,
             or
             in
             the
             Remainder
             after
             
             the
             lesse
             is
             taken
             out
             of
             the
             greater
             ;
             as
             the
             Proportion
             of
             Two
             to
             Five
             ,
             consists
             in
             the
             Three
             by
             which
             Five
             exceeds
             Two
             ,
             not
             in
             Three
             ,
             simply
             onely
             ,
             but
             also
             as
             compared
             with
             Five
             or
             Two.
             For
             though
             there
             be
             the
             same
             difference
             between
             Two
             &
             Five
             ,
             which
             is
             between
             Nine
             and
             Twelve
             ,
             namely
             Three
             ,
             yet
             there
             is
             not
             the
             same
             Inequality
             ;
             and
             therefore
             the
             Proportion
             of
             Two
             to
             Five
             ,
             is
             not
             in
             all
             Relation
             the
             same
             with
             that
             of
             Nine
             to
             Twelve
             ,
             but
             onely
             in
             that
             which
             is
             called
             Arithmetical
             .
          
           
             6
             But
             we
             must
             not
             so
             think
             of
             Relation
             ,
             as
             if
             it
             were
             an
             Accident
             differing
             from
             all
             the
             other
             Accidents
             of
             the
             Relative
             ;
             but
             one
             of
             them
             ;
             namely
             that
             ,
             by
             which
             the
             Comparison
             is
             made
             .
             For
             example
             ,
             the
             likeness
             of
             one
             White
             to
             another
             White
             ,
             or
             its
             Unlikeness
             to
             Black
             ,
             is
             the
             same
             Accident
             with
             its
             Whiteness
             ;
             and
             Equality
             and
             Inequality
             ,
             the
             same
             Accident
             with
             the
             Magnitude
             of
             the
             thing
             compared
             ,
             though
             under
             another
             Name
             ;
             for
             that
             which
             is
             called
             White
             or
             Great
             ,
             when
             it
             is
             not
             compared
             with
             something
             else
             ,
             the
             same
             when
             it
             is
             compared
             is
             called
             Like
             or
             
               Unlike
               ,
               Equal
            
             or
             Unequal
             .
             And
             from
             this
             it
             follows
             ,
             that
             the
             Causes
             of
             the
             Accidents
             which
             are
             in
             Relatives
             ,
             are
             the
             Causes
             also
             of
             
               Likeness
               ,
               Unlikeness
               ,
               Equality
            
             and
             Inequality
             ;
             namely
             ,
             that
             he
             that
             makes
             two
             Unequal
             Bodies
             ,
             makes
             also
             their
             Inequality
             ;
             and
             he
             that
             makes
             a
             Rule
             and
             an
             Action
             ,
             makes
             also
             ,
             if
             the
             Action
             be
             congruous
             to
             the
             Rule
             ,
             their
             Congruity
             ;
             if
             Incongruous
             ,
             their
             Incongruity
             .
             And
             thus
             much
             concerning
             Comparison
             of
             one
             Body
             with
             another
             .
          
           
             7
             But
             the
             same
             Body
             may
             at
             different
             times
             be
             Compared
             with
             it self
             .
             And
             from
             hence
             springs
             a
             great
             controversie
             among
             Philosophers
             about
             the
             
               Beginning
               of
               Individuation
            
             ;
             namely
             ,
             in
             what
             sense
             it
             may
             be
             conceived
             that
             a
             Body
             is
             at
             one
             time
             the
             same
             ,
             at
             another
             time
             not
             the
             same
             it
             was
             formerly
             .
             For
             example
             ,
             whether
             a
             Man
             grown
             old
             be
             the
             same
             Man
             he
             was
             whilest
             he
             was
             young
             ,
             or
             another
             Man
             ;
             or
             whether
             a
             City
             be
             in
             different
             Ages
             the
             same
             ,
             or
             another
             City
             .
             Some
             place
             Individuity
             in
             the
             Unity
             of
             Matter
             ;
             others
             in
             the
             Unity
             of
             Form
             ;
             and
             one
             sayes
             it
             consists
             in
             the
             Unity
             of
             the
             
               Aggregate
               of
               all
               the
               Accidents
               together
               .
            
             For
             Matter
             ,
             it
             is
             pleaded
             ,
             that
             a
             lump
             of
             Wax
             ,
             whether
             it
             be
             Spherical
             or
             Cubical
             ,
             is
             the
             same
             Wax
             ,
             because
             the
             same
             Matter
             .
             For
             Form
             ,
             that
             when
             a
             Man
             is
             grown
             from
             an
             Infant
             to
             be
             an
             Old
             Man
             ,
             though
             his
             Matter
             be
             changed
             ,
             yet
             he
             is
             still
             the
             same
             Numerical
             Man
             ;
             for
             that
             
             Identity
             which
             cannot
             be
             attributed
             to
             the
             Matter
             ,
             ought
             probably
             to
             be
             ascribed
             to
             the
             Form.
             For
             the
             
               Aggregate
               of
               Accidents
            
             no
             Instance
             can
             be
             made
             ;
             but
             because
             when
             any
             new
             Accident
             is
             generated
             ,
             a
             new
             Name
             is
             commonly
             imposed
             on
             the
             Thing
             ,
             therefore
             he
             that
             assigned
             this
             cause
             of
             Individuity
             ,
             thought
             the
             thing
             it self
             also
             was
             become
             another
             thing
             .
             According
             to
             the
             first
             Opinion
             ,
             He
             that
             sins
             ,
             and
             he
             that
             is
             punished
             should
             not
             be
             the
             same
             Man
             ,
             by
             reason
             of
             the
             perpetual
             flux
             and
             change
             of
             Mans
             Body
             ;
             nor
             should
             the
             City
             which
             makes
             Lawes
             in
             one
             Age
             ,
             and
             abrogates
             them
             in
             another
             ,
             be
             the
             same
             City
             ;
             which
             were
             to
             confound
             all
             Civil
             Rights
             .
             According
             to
             the
             second
             Opinion
             ,
             two
             Bodies
             existing
             both
             at
             once
             ,
             would
             be
             one
             and
             the
             same
             Numerical
             Body
             ;
             for
             if
             (
             for
             example
             )
             that
             Ship
             of
             Theseus
             (
             concerning
             the
             Difference
             whereof
             ,
             made
             by
             continual
             reparation
             ,
             in
             taking
             out
             the
             old
             Planks
             ,
             and
             putting
             in
             new
             ,
             the
             Sophisters
             of
             Athens
             were
             wont
             to
             dispute
             )
             were
             ,
             after
             all
             the
             Planks
             were
             changed
             ,
             the
             same
             Numerical
             Ship
             it
             was
             at
             the
             beginning
             ;
             and
             if
             some
             Man
             had
             kept
             the
             Old
             Planks
             as
             they
             were
             taken
             out
             ,
             and
             by
             putting
             them
             afterwards
             together
             in
             the
             same
             order
             ,
             had
             again
             made
             a
             Ship
             of
             them
             ,
             this
             without
             doubt
             had
             also
             been
             the
             same
             Numerical
             Ship
             with
             that
             which
             was
             at
             the
             beginning
             ;
             and
             so
             there
             would
             have
             been
             two
             Ships
             Numerically
             the
             same
             ,
             which
             is
             absurd
             .
             But
             according
             to
             the
             third
             Opinion
             ,
             Nothing
             would
             be
             the
             same
             it
             was
             ;
             so
             that
             a
             Man
             standing
             ,
             would
             not
             be
             the
             same
             he
             was
             sitting
             ;
             nor
             the
             Water
             which
             is
             in
             the
             Vessel
             ,
             the
             same
             with
             that
             which
             is
             poured
             out
             of
             it
             .
             Wherefore
             the
             beginning
             of
             Individuation
             is
             not
             alwayes
             to
             be
             taken
             either
             from
             Matter
             alone
             ,
             or
             from
             Form
             alone
             .
          
           
             But
             we
             must
             consider
             by
             what
             name
             any
             thing
             is
             called
             ,
             when
             we
             enquire
             concerning
             the
             Identity
             of
             it
             ;
             for
             it
             is
             one
             thing
             to
             ask
             concerning
             Socrates
             whether
             he
             be
             the
             same
             Man
             ,
             and
             another
             to
             ask
             whether
             he
             be
             the
             same
             Body
             ;
             for
             his
             Body
             when
             he
             is
             Old
             ,
             cannot
             be
             the
             same
             it
             was
             when
             he
             was
             an
             Infant
             ,
             by
             reason
             of
             the
             difference
             of
             Magnitude
             ;
             for
             One
             Body
             has
             alwayes
             One
             and
             the
             same
             Magnitude
             ;
             yet
             nevertheless
             he
             may
             be
             the
             same
             Man.
             And
             therefore
             whensoever
             the
             Name
             by
             which
             it
             is
             asked
             whether
             a
             thing
             be
             the
             same
             it
             was
             ,
             is
             given
             it
             for
             the
             Matter
             onely
             ,
             then
             if
             the
             Matter
             be
             the
             same
             ,
             the
             thing
             also
             is
             Individually
             the
             same
             ;
             as
             the
             Wat●r
             which
             was
             in
             the
             Sea
             ,
             is
             the
             same
             which
             is
             afterwards
             
             in
             the
             Cloud
             ;
             and
             any
             Body
             is
             the
             same
             ,
             whether
             the
             parts
             of
             it
             be
             put
             together
             ,
             or
             dispersed
             ,
             or
             whether
             it
             be
             congealed
             or
             dissolved
             .
             Also
             if
             the
             Name
             be
             given
             for
             such
             Form
             as
             is
             the
             beginning
             of
             Motion
             ,
             then
             as
             long
             as
             that
             Motion
             remains
             it
             will
             be
             the
             same
             Individual
             thing
             ;
             as
             that
             Man
             will
             be
             alwayes
             the
             same
             ,
             whose
             Actions
             and
             Thoughts
             proceed
             all
             from
             the
             same
             beginning
             of
             Motion
             ,
             namely
             ,
             that
             which
             was
             in
             his
             generation
             ;
             and
             that
             will
             be
             the
             same
             River
             ,
             which
             flows
             from
             one
             and
             the
             same
             Fountain
             ,
             whether
             the
             same
             Water
             ,
             or
             other
             Water
             ,
             or
             something
             else
             then
             Water
             flow
             from
             thence
             ;
             and
             that
             the
             same
             City
             ,
             whose
             Acts
             proceed
             continually
             from
             the
             same
             Institution
             ,
             whether
             the
             Men
             be
             the
             same
             or
             no.
             Lastly
             ,
             if
             the
             Name
             be
             given
             for
             some
             Accident
             ,
             then
             the
             Identity
             of
             the
             thing
             will
             depend
             upon
             the
             Matter
             ;
             for
             by
             the
             taking
             away
             and
             supplying
             of
             Matter
             ,
             the
             Accidents
             that
             were
             are
             destroyed
             ,
             and
             other
             new
             ones
             are
             generated
             ,
             which
             cannot
             be
             the
             same
             Numerically
             ;
             so
             that
             a
             Ship
             ,
             which
             signifies
             Matter
             so
             figured
             ,
             will
             be
             the
             same
             ,
             as
             long
             as
             the
             Matter
             remains
             the
             same
             ;
             but
             if
             no
             part
             of
             the
             Matter
             be
             the
             same
             ,
             then
             it
             is
             Numerically
             another
             Ship
             ;
             and
             if
             part
             of
             the
             Matter
             remain
             ,
             and
             part
             be
             changed
             ,
             then
             the
             Ship
             will
             be
             partly
             the
             same
             ,
             and
             partly
             not
             the
             same
             .
          
        
         
           
           
             
               CHAP.
               XII
            
             .
             Of
             Quantity
             .
          
           
             
               
                 1
                 The
                 Definition
                 of
                 Quantity
                 .
              
               
                 2
                 The
                 Exposition
                 of
                 Quantity
                 what
                 it
                 is
                 .
              
               
                 3
                 How
                 Line
                 ,
                 Superficies
                 and
                 Solid
                 are
                 exposed
                 .
              
               
                 4
                 How
                 Time
                 is
                 exposed
                 .
              
               
                 5
                 How
                 Number
                 is
                 exposed
                 .
              
               
                 6
                 How
                 Velocity
                 is
                 exposed
                 .
              
               
                 7
                 How
                 Weight
                 is
                 exposed
                 .
              
               
                 8
                 How
                 the
                 Proportion
                 of
                 Magnitudes
                 is
                 exposed
                 .
              
               
                 9
                 How
                 the
                 Proportion
                 of
                 Times
                 and
                 Velocities
                 is
                 exposed
                 .
              
            
          
           
             1
             WHat
             ,
             and
             how
             manifold
             Dimension
             is
             ,
             has
             been
             said
             in
             the
             8th
             .
             Chapter
             ,
             namely
             ,
             that
             there
             are
             three
             Dimensions
             ,
             Line
             (
             or
             Length
             )
             Superficies
             and
             Solid
             ;
             Every
             one
             of
             which
             ,
             if
             it
             be
             determined
             ,
             that
             is
             ,
             if
             the
             limits
             of
             it
             be
             made
             known
             is
             commonly
             called
             Quantity
             ;
             For
             by
             Quantity
             all
             men
             understand
             that
             which
             is
             signified
             by
             that
             word
             ,
             by
             which
             answer
             is
             made
             to
             the
             question
             
               How
               much
               is
               it
            
             .
             Whensoever
             therefore
             it
             is
             asked
             (
             for
             example
             )
             
               How
               long
               is
               the
               Journey
            
             ,
             it
             is
             not
             answered
             indefinitely
             Length
             ;
             nor
             when
             it
             is
             asked
             ,
             
               How
               big
               is
               the
               Field
            
             is
             it
             answered
             indefinitely
             Superficies
             ;
             nor
             if
             a
             man
             ask
             
               How
               great
               is
               the
               bulk
            
             ,
             indefinitely
             Solid
             ;
             but
             it
             is
             answered
             determinately
             ,
             The
             Journey
             is
             a
             hundred
             Miles
             ;
             the
             Field
             is
             a
             hundred
             Acres
             ;
             the
             Bulk
             is
             a
             hundred
             Cubical
             Feet
             ;
             or
             at
             least
             in
             some
             such
             manner
             ,
             that
             the
             Magnitude
             of
             the
             thing
             enquired
             after
             may
             by
             certain
             Limits
             be
             comprehended
             in
             the
             Mind
             .
             QUANTITY
             therefore
             cannot
             otherwise
             be
             defined
             ,
             then
             to
             be
             
               a
               Dimension
               determined
            
             or
             
               a
               Dimension
               ,
               whose
               Limits
               are
               set
               out
               ,
               either
               by
               their
               Place
               ,
               or
               by
               some
               Comparison
               .
            
          
           
             2
             And
             Quantity
             is
             determined
             two
             wayes
             ;
             One
             ,
             by
             the
             Sense
             ,
             when
             some
             sensible
             Object
             is
             set
             before
             it
             ;
             as
             when
             a
             Line
             ,
             a
             Superficies
             or
             Solid
             ,
             of
             a
             Foot
             or
             Cubit
             ,
             marked
             out
             in
             some
             Matter
             ,
             is
             objected
             to
             the
             Eyes
             ;
             which
             way
             of
             Determining
             is
             called
             Exposition
             ,
             and
             the
             Quantity
             so
             known
             is
             called
             
               Exposed
               Quantity
            
             ;
             The
             Other
             ,
             by
             Memory
             ,
             that
             is
             ,
             by
             Comparison
             with
             some
             Exposed
             Quantity
             .
             In
             the
             first
             manner
             ,
             when
             it
             is
             asked
             of
             what
             Quantity
             a
             thing
             is
             ,
             it
             is
             answered
             ,
             of
             such
             Quantity
             as
             you
             see
             
             Exposed
             .
             In
             the
             second
             manner
             ,
             answer
             cannot
             be
             made
             but
             by
             Comparison
             with
             some
             ●xposed
             Quantity
             ;
             for
             if
             it
             be
             asked
             ,
             How
             long
             is
             the
             Way
             ,
             the
             answer
             is
             ,
             so
             many
             thousand
             Paces
             ;
             that
             is
             ,
             by
             Comparing
             the
             Way
             with
             a
             Pace
             ,
             or
             some
             other
             Measure
             determined
             and
             known
             by
             Exposition
             ;
             or
             the
             Quantity
             of
             it
             is
             to
             some
             other
             Quantity
             known
             by
             Exposition
             ,
             as
             the
             Diameter
             of
             a
             Square
             is
             to
             the
             Side
             of
             the
             same
             ,
             or
             by
             some
             other
             the
             like
             means
             .
             But
             it
             is
             to
             be
             understood
             that
             the
             Quantity
             Exposed
             must
             be
             some
             standing
             or
             permanent
             thing
             ,
             such
             as
             is
             marked
             out
             in
             consistent
             or
             durable
             matter
             ;
             or
             at
             least
             something
             which
             is
             revocable
             to
             sense
             ;
             for
             otherwise
             no
             Comparison
             can
             be
             made
             by
             it
             .
             Seeing
             therefore
             (
             by
             what
             has
             been
             said
             in
             the
             next
             preceding
             Chapter
             )
             Comparison
             of
             one
             Magnitude
             with
             another
             ,
             is
             the
             same
             thing
             with
             Proportion
             ;
             it
             is
             manifest
             ,
             that
             Quantity
             determined
             in
             the
             second
             manner
             ,
             is
             nothing
             else
             but
             the
             Proportion
             of
             a
             Dimension
             not
             Exposed
             to
             another
             which
             is
             Exposed
             ;
             that
             is
             ,
             the
             Comparison
             of
             the
             Equality
             or
             Inequality
             thereof
             with
             an
             Exposed
             Quantity
             .
          
           
             3
             
               Lines
               ,
               Superficies
            
             and
             Solids
             are
             Exposed
             ,
             First
             ,
             by
             Motion
             ,
             in
             such
             manner
             ,
             as
             (
             in
             the
             8th
             Chapter
             )
             I
             have
             said
             they
             are
             generated
             ;
             but
             so
             ,
             as
             that
             the
             Marks
             of
             such
             Motion
             be
             permanent
             ;
             as
             when
             they
             are
             designed
             upon
             some
             Matter
             ,
             as
             a
             Line
             upon
             Paper
             ;
             or
             graven
             in
             some
             durable
             Matter
             .
             Secondly
             ,
             by
             Apposition
             ;
             as
             when
             one
             Line
             or
             Length
             is
             applyed
             to
             another
             Line
             or
             Length
             ,
             one
             Breadth
             to
             another
             Breadth
             ,
             and
             one
             Thickness
             to
             another
             Thickness
             ;
             which
             is
             as
             much
             as
             to
             describe
             a
             Line
             by
             Points
             ,
             a
             Superficies
             by
             Lines
             ,
             and
             a
             Solid
             by
             Superficies
             ,
             saving
             that
             by
             Points
             in
             this
             place
             are
             to
             be
             understood
             very
             short
             Lines
             ,
             and
             by
             Superficies
             very
             Thin
             Solids
             .
             Thirdly
             ,
             Lines
             and
             Superficies
             may
             be
             Exposed
             by
             Section
             ;
             namely
             a
             Line
             may
             be
             made
             by
             Cutting
             an
             Exposed
             Superficies
             ,
             and
             a
             Superficies
             by
             the
             Cutting
             of
             an
             Exposed
             Solid
             .
          
           
             4
             Time
             is
             Exposed
             not
             onely
             by
             the
             Exposition
             of
             a
             Line
             ,
             but
             also
             of
             some
             Moveable
             thing
             ,
             which
             is
             moved
             Uniformly
             upon
             that
             Line
             ,
             or
             at
             least
             is
             supposed
             so
             to
             be
             Moved
             .
             For
             seeing
             Time
             is
             an
             Idea
             of
             Motion
             in
             which
             we
             consider
             Former
             and
             Later
             ,
             that
             is
             ,
             Succession
             ,
             it
             is
             not
             sufficient
             for
             the
             Exposition
             of
             Time
             that
             a
             Line
             be
             described
             ,
             but
             we
             must
             also
             have
             in
             our
             Minde
             an
             Imagination
             ▪
             
             of
             some
             Moveable
             thing
             passing
             over
             that
             Line
             ,
             and
             the
             Motion
             of
             it
             must
             be
             Uniform
             ,
             that
             Time
             may
             be
             divided
             and
             compounded
             as
             often
             as
             there
             shall
             be
             need
             .
             And
             therefore
             when
             Philosophers
             in
             their
             Demonstrations
             draw
             a
             Line
             ,
             and
             say
             ,
             
               Let
               that
               Line
               be
               Time
            
             ,
             it
             is
             to
             be
             understood
             as
             if
             they
             said
             ,
             
               Let
               the
               Conception
               of
               Uniform
               Motion
               upon
               that
               Line
               ,
               be
               Time.
            
             For
             though
             the
             Circles
             in
             Dials
             be
             Lines
             ,
             yet
             they
             are
             not
             of
             themselves
             sufficient
             to
             note
             Time
             by
             ,
             except
             also
             there
             be
             ,
             or
             be
             supposed
             to
             be
             a
             Motion
             of
             the
             Shadow
             or
             the
             Hand
             .
          
           
             5
             Number
             is
             Exposed
             either
             by
             the
             Exposition
             of
             Points
             ,
             or
             of
             the
             Names
             of
             Number
             
               One
               ,
               Two
               ,
               Three
               ,
               &c.
            
             and
             those
             Points
             must
             not
             be
             contiguous
             ,
             so
             as
             that
             they
             cannot
             be
             distinguished
             by
             Notes
             ,
             but
             they
             must
             be
             so
             placed
             that
             they
             may
             be
             discerned
             one
             from
             another
             ;
             for
             from
             this
             it
             is
             that
             Number
             is
             called
             
               Discrete
               Quantity
            
             ,
             whereas
             all
             Quantity
             which
             is
             designed
             by
             Motion
             ,
             is
             called
             
               Continual
               Quantity
            
             .
             But
             that
             Number
             may
             be
             Exposed
             by
             the
             Names
             of
             Number
             ,
             it
             is
             necessary
             that
             they
             be
             recited
             by
             heart
             and
             in
             order
             ,
             as
             One
             ,
             Two
             ,
             Three
             ,
             &c.
             for
             by
             saying
             One
             ,
             One
             ,
             One
             and
             so
             forward
             ,
             we
             know
             not
             what
             Number
             we
             are
             at
             beyond
             Two
             or
             Three
             ,
             which
             also
             appear
             to
             Us
             in
             this
             manner
             not
             as
             Number
             ,
             but
             as
             Figure
             .
          
           
           
             7
             Weight
             is
             Exposed
             by
             any
             Heavy
             Body
             ,
             of
             what
             Matter
             soever
             ,
             so
             it
             be
             alwayes
             alike
             Heavy
             .
          
           
           
        
         
           
           
             
               CHAP.
               XIII
            
             .
             Of
             Analogisme
             or
             
               the
               Same
               Proportion
            
             .
          
           
             
               
                 1
                 ,
                 2
                 ,
                 3
                 ,
                 4.
                 
                 The
                 Nature
                 and
                 Definition
                 of
                 Proportion
                 Arithmetical
                 and
                 Geometrical
                 .
              
               
                 5
                 The
                 Definition
                 and
                 some
                 properties
                 of
                 the
                 Same
                 Arithmetical
                 Proportion
                 .
              
               
                 6
                 ,
                 7.
                 
                 The
                 Definition
                 and
                 Transmutations
                 of
                 Analogisme
                 ,
                 or
                 The
                 Same
                 Geometrical
                 Proportion
                 .
              
               
                 8
                 ,
                 9.
                 
                 The
                 Definitions
                 of
                 Hyperlogisme
                 and
                 Hypologisme
                 ,
                 that
                 is
                 ,
                 of
                 Greater
                 and
                 Lesse
                 Proportion
                 ,
                 and
                 their
                 Transmutations
                 .
              
               
                 10
                 ,
                 11
                 ,
                 12.
                 
                 Comparison
                 of
                 Analogical
                 quantities
                 according
                 to
                 Magnitude
                 .
              
               
                 13
                 ,
                 14
                 ,
                 15.
                 
                 Composition
                 of
                 Proportions
                 .
              
               
                 16
                 ,
                 17
                 ,
                 18
                 ,
                 19
                 ,
                 20
                 ,
                 21
                 ,
                 22
                 ,
                 23
                 ,
                 24
                 ,
                 25.
                 
                 The
                 Definition
                 and
                 Properties
                 of
                 Continual
                 Proportion
                 .
              
               
                 26
                 ,
                 27
                 ,
                 28
                 ,
                 29.
                 
                 Comparison
                 of
                 Arithmetical
                 and
                 Geometrical
                 Proportions
                 .
              
            
             
               Note
               that
               in
               this
               Chapter
               ,
               the
               signe
               +
               ,
               signifies
               that
               the
               quantities
               betwixt
               which
               it
               is
               put
               ,
               are
               added
               together
               ;
               and
               this
               signe
               -
               ,
               the
               Remainder
               ,
               after
               the
               later
               quantity
               is
               taken
               out
               of
               the
               former
               .
               So
               that
               A+B
               is
               equal
               to
               both
               A
               and
               B
               together
               ;
               and
               where
               you
               see
               
                 A
                 −
                 B
              
               ,
               there
               A
               is
               the
               Whole
               ,
               B
               the
               part
               taken
               out
               of
               it
               ,
               and
               
                 A
                 −
                 B
              
               the
               Remainder
               .
               Also
               two
               letters
               set
               together
               without
               any
               sign
               ,
               signifies
               (
               unlesse
               they
               belong
               to
               a
               Figure
               )
               that
               one
               of
               the
               quantities
               is
               multiplyed
               by
               the
               other
               ;
               as
               
                 A
                 B
              
               signifies
               the
               Product
               of
               A
               multiplyed
               by
               B.
               
            
          
           
             1
             GReat
             &
             Little
             are
             not
             intelligible
             but
             by
             Comparison
             .
             Now
             that
             to
             which
             they
             are
             compared
             is
             something
             Exposed
             ,
             that
             is
             ,
             some
             Magnitude
             either
             perceived
             by
             Sense
             ,
             or
             so
             defined
             by
             Words
             ,
             that
             it
             may
             be
             comprehended
             by
             the
             Mind
             .
             Also
             that
             to
             which
             any
             Magnitude
             is
             compared
             ,
             is
             either
             Greater
             ,
             or
             Less
             ,
             or
             Equal
             to
             it
             .
             And
             therefore
             Proportion
             (
             which
             as
             I
             have
             shewn
             ,
             is
             the
             Estimation
             ,
             or
             Comprehension
             of
             Magnitudes
             by
             Comparison
             )
             is
             threefold
             ;
             namely
             Proportion
             of
             Equality
             ,
             that
             is
             ,
             of
             Equal
             to
             Equal
             ;
             or
             of
             Excesse
             ,
             which
             is
             of
             the
             Greater
             to
             the
             Lesse
             ;
             or
             of
             Defect
             ,
             which
             is
             the
             Proportion
             of
             the
             Lesse
             to
             the
             Greater
             .
          
           
           
             Again
             ,
             every
             one
             of
             these
             Proportions
             is
             twofold
             ;
             For
             if
             it
             be
             asked
             concerning
             any
             Magnitude
             given
             ,
             how
             Great
             it
             is
             ,
             the
             answer
             may
             be
             made
             by
             Comparing
             it
             two
             wayes
             ;
             First
             ,
             by
             saying
             it
             is
             greater
             or
             lesse
             then
             another
             Magnitude
             by
             so
             much
             ;
             as
             Seven
             is
             lesse
             then
             Ten
             by
             three
             Unities
             ;
             and
             this
             is
             called
             
               Arithmetical
               Proportion
            
             .
             Secondly
             ,
             by
             saying
             it
             is
             greater
             or
             lesse
             then
             another
             Magnitude
             by
             such
             a
             part
             or
             parts
             thereof
             ;
             as
             Seven
             is
             less
             then
             Ten
             by
             three
             tenth
             parts
             of
             the
             same
             Ten.
             And
             though
             this
             Proportion
             be
             not
             alwayes
             explicable
             by
             Number
             ,
             yet
             it
             is
             a
             determinate
             Proportion
             ,
             and
             of
             a
             different
             kind
             from
             the
             former
             ,
             and
             called
             
               Geometrical
               Proportion
            
             ,
             and
             most
             commonly
             
               Proportion
               simply
            
             .
          
           
             2
             Proportion
             ,
             whether
             it
             be
             Arithmetical
             or
             Geometrical
             ,
             cannot
             be
             Exposed
             but
             in
             two
             Magnitudes
             (
             of
             which
             the
             former
             is
             cōmonly
             called
             the
             Antecedent
             ,
             &
             the
             later
             the
             Consequent
             of
             the
             Proportion
             )
             as
             I
             have
             shewn
             in
             the
             8th
             .
             Article
             of
             the
             precedent
             Chapter
             .
             And
             therefore
             if
             two
             Proportions
             be
             to
             be
             compared
             ,
             there
             must
             be
             four
             Magnitudes
             Exposed
             ,
             namely
             two
             Antecedents
             and
             two
             Consequents
             ;
             for
             though
             it
             happen
             sometimes
             ,
             that
             the
             Consequent
             of
             the
             former
             Proportion
             be
             the
             same
             with
             the
             Antecedent
             of
             the
             later
             ,
             yet
             in
             that
             double
             Comparison
             it
             must
             of
             necessity
             be
             twice
             numbred
             ;
             so
             that
             there
             will
             be
             alwayes
             four
             Terms
             .
          
           
             3
             Of
             two
             Proportions
             ,
             whether
             they
             be
             Arithmetical
             or
             Geometrical
             ,
             when
             the
             Magnitudes
             compared
             in
             both
             (
             which
             Euclide
             in
             the
             fifth
             Definition
             of
             his
             sixth
             Book
             calls
             the
             
               Quantities
               of
               Proportions
            
             )
             are
             equal
             ,
             then
             one
             of
             the
             Proportions
             cannot
             be
             either
             greater
             or
             lesse
             then
             the
             other
             ;
             For
             one
             Equality
             is
             neither
             greater
             nor
             lesse
             then
             another
             Equality
             .
             But
             of
             two
             Proportions
             of
             Inequality
             ,
             whether
             they
             be
             Proportions
             of
             Excesse
             or
             of
             Defect
             ,
             one
             of
             them
             may
             be
             either
             greater
             or
             lesse
             then
             the
             other
             ,
             or
             they
             may
             both
             be
             equal
             ;
             for
             though
             there
             be
             propounded
             two
             Magnitudes
             that
             are
             unequal
             to
             one
             another
             ,
             yet
             there
             may
             be
             other
             two
             more
             unequal
             ,
             and
             other
             two
             equally
             unequall
             ,
             and
             other
             two
             less
             unequal
             then
             the
             two
             which
             were
             propounded
             .
             And
             from
             hence
             it
             may
             be
             understood
             that
             the
             Proportions
             of
             Excess
             and
             Defect
             are
             Quantity
             ,
             being
             capable
             of
             More
             &
             Less
             ;
             but
             the
             Proportion
             of
             Equality
             is
             not
             Quantity
             ,
             because
             not
             capable
             ,
             neither
             of
             More
             nor
             of
             Less
             .
             And
             therefore
             Proportions
             of
             Inequality
             may
             
             be
             added
             together
             or
             substracted
             from
             one
             another
             ,
             or
             be
             multiplyed
             or
             divided
             by
             one
             another
             ,
             or
             by
             Number
             ;
             but
             Proportions
             of
             Equality
             not
             so
             .
          
           
             4
             Two
             Equal
             Proportions
             are
             commonly
             called
             
               The
               Same
               Proportion
            
             ;
             and
             it
             is
             said
             ,
             that
             the
             Proportion
             of
             the
             first
             Antecedent
             to
             the
             first
             Consequent
             is
             the
             same
             with
             that
             of
             the
             second
             Antecedent
             to
             the
             second
             Consequent
             .
             And
             when
             four
             Magnitudes
             are
             thus
             to
             one
             another
             in
             Geometrical
             Proportion
             ,
             they
             are
             called
             Proportionals
             ,
             and
             by
             some
             more
             briefly
             Analogisme
             .
             And
             
               Greater
               Proportion
            
             ,
             is
             the
             Proportion
             of
             a
             Greater
             Antecedent
             to
             the
             same
             Consequent
             ,
             or
             of
             the
             same
             Antec●dent
             to
             a
             Less
             Consequent
             ;
             and
             when
             the
             Proportion
             of
             the
             first
             Antecedent
             to
             the
             first
             Consequent
             ,
             is
             greater
             then
             that
             of
             the
             second
             Antecedent
             to
             the
             second
             Consequent
             ,
             the
             four
             Magnitudes
             which
             are
             so
             to
             one
             another
             may
             be
             called
             Hyperlogisme
             .
          
           
             
               Less
               Proportion
            
             is
             the
             Proportion
             of
             a
             Less
             Antecedent
             to
             the
             same
             Consequent
             ,
             or
             of
             the
             same
             Antecedent
             to
             a
             Greater
             Consequent
             ;
             and
             when
             the
             Proportion
             of
             the
             first
             Antecedent
             to
             the
             first
             Consequent
             ,
             is
             less
             then
             that
             of
             the
             second
             to
             the
             second
             ,
             the
             four
             Magnitudes
             may
             be
             called
             Hypologisme
             .
          
           
             5
             One
             Arithmetical
             Proportion
             is
             the
             Same
             with
             another
             Arithmetical
             Proportion
             ,
             when
             one
             of
             the
             Antecedents
             exceeds
             its
             Consequent
             ,
             or
             is
             exceeded
             by
             it
             ,
             as
             much
             as
             the
             other
             Antecedent
             exceeds
             its
             Consequent
             ,
             or
             is
             exceeded
             by
             it
             .
             And
             therefore
             in
             four
             Magnitudes
             Arithmetically
             Proportional
             ,
             the
             sum
             of
             the
             Extremes
             is
             equal
             to
             the
             sum
             of
             the
             Means
             .
             For
             if
             A.
             B
             :
             :
             C.
             D
             be
             Arithmetically
             Proportional
             ,
             and
             the
             Difference
             on
             both
             sides
             be
             the
             same
             Excess
             or
             the
             same
             Defect
             E
             ;
             then
             B+C
             (
             if
             A
             be
             greater
             then
             B
             )
             will
             be
             equal
             to
             A
             −
             E+C
             ;
             and
             A+D
             will
             be
             equal
             to
             A+C
             −
             E
             ;
             But
             A
             −
             E+C
             and
             A+C
             −
             E
             are
             equal
             .
             Or
             if
             A
             be
             less
             then
             B
             ,
             then
             B+C
             will
             be
             equal
             to
             A+E+C
             ;
             and
             A+D
             will
             be
             equal
             to
             A+C+E
             ;
             But
             A+E+C
             ,
             and
             A+C+E
             are
             equal
             .
          
           
             Also
             if
             there
             be
             never
             so
             many
             Magnitudes
             Arithmetically
             Proportional
             ,
             the
             Sum
             of
             them
             all
             will
             be
             equal
             to
             the
             Product
             of
             half
             the
             number
             of
             the
             Terms
             multiplyed
             by
             the
             Sum
             of
             the
             Extremes
             .
             For
             if
             A.
             B
             :
             :
             C.
             D
             :
             :
             E.
             F
             be
             Arithmetically
             Proportional
             ,
             the
             Couples
             A+F
             ,
             B+E
             ,
             C+D
             will
             be
             equal
             to
             one
             another
             ;
             and
             
             their
             Sum
             will
             be
             equal
             to
             A+F
             multiplyed
             by
             the
             number
             of
             their
             Combinations
             ,
             that
             is
             ,
             by
             half
             the
             number
             of
             the
             Terms
             .
          
           
             If
             of
             four
             Unequal
             Magnitudes
             ,
             any
             two
             together
             taken
             be
             equal
             to
             the
             other
             two
             together
             taken
             ,
             then
             the
             greatest
             and
             the
             least
             of
             them
             will
             be
             in
             the
             same
             Combination
             .
             Let
             the
             Unequal
             Magnitudes
             be
             A
             ,
             B
             ,
             C
             ,
             D
             ;
             and
             let
             A+B
             be
             equal
             to
             C+D
             ;
             &
             let
             A
             be
             the
             greatest
             of
             them
             all
             ;
             I
             say
             B
             will
             be
             the
             least
             .
             For
             if
             it
             may
             be
             ,
             let
             any
             of
             the
             rest
             ,
             as
             D
             ,
             be
             the
             least
             .
             Seeing
             therefore
             A
             is
             greater
             then
             C
             ,
             and
             B
             then
             D
             ,
             A+B
             will
             be
             greater
             then
             C+D
             ;
             which
             is
             contrary
             to
             what
             was
             supposed
             .
          
           
             If
             there
             be
             any
             four
             Magnitudes
             ,
             the
             Sum
             of
             the
             greatest
             and
             least
             ,
             the
             Sum
             of
             the
             Means
             ,
             the
             difference
             of
             the
             two
             greatest
             ,
             and
             the
             difference
             of
             the
             two
             least
             will
             be
             Arithmetically
             Proportional
             .
             For
             let
             there
             be
             four
             Magnitudes
             ,
             whereof
             A
             is
             the
             greatest
             ,
             D
             the
             least
             ,
             and
             B
             and
             C
             the
             Means
             ;
             I
             say
             A+D
             .
             B+C
             :
             :
             A
             −
             B.
             C
             −
             D
             are
             Arithmetically
             Proportional
             .
             For
             the
             difference
             between
             the
             first
             Antecedent
             and
             its
             Consequent
             is
             this
             ,
             A+D
             −
             B
             −
             C
             ;
             and
             the
             difference
             between
             the
             second
             Antecedent
             and
             its
             Consequent
             this
             ,
             A
             −
             B
             −
             C+D
             ;
             but
             these
             two
             Differences
             are
             equal
             ,
             and
             therefore
             (
             by
             this
             5th
             .
             Article
             )
             A+D
             .
             B+C
             :
             :
             A
             −
             B.
             C
             −
             D
             are
             Arithmetically
             Proportional
             .
          
           
             If
             of
             four
             Magnitudes
             ,
             two
             be
             equal
             to
             the
             other
             two
             ,
             they
             will
             be
             in
             reciprocal
             Arithmetical
             Proportion
             .
             For
             let
             A+B
             be
             equal
             to
             C+D
             ;
             I
             say
             A.
             C
             :
             :
             D.
             B
             are
             Arithmetically
             Proportional
             .
             For
             if
             they
             be
             not
             ,
             let
             A.
             C
             :
             :
             D.
             E
             (
             supposing
             E
             to
             be
             greater
             or
             less
             then
             B
             )
             be
             Arithmetically
             Proportional
             ,
             and
             then
             A+E
             will
             be
             equal
             to
             C+D
             ;
             wherefore
             A+B
             and
             C+D
             are
             not
             equal
             ;
             which
             is
             contrary
             to
             what
             was
             supposed
             .
          
           
             6
             One
             Geometrical
             Proportion
             is
             the
             same
             with
             another
             Geometrical
             Proportion
             ,
             when
             the
             same
             Cause
             producing
             equal
             Effects
             in
             equal
             Times
             ,
             determines
             both
             the
             Proportions
             .
          
           
             If
             a
             Point
             Uniformly
             moved
             ,
             describe
             two
             Lines
             either
             with
             the
             same
             ,
             or
             different
             Velocity
             ,
             all
             the
             parts
             of
             them
             which
             are
             contemporary
             ,
             that
             is
             ,
             which
             are
             described
             in
             the
             same
             time
             ,
             will
             be
             Two
             to
             Two
             in
             Geometrical
             Proportion
             ,
             whether
             the
             Antecedents
             be
             taken
             in
             the
             same
             Line
             ,
             or
             not
             .
             For
             ,
             from
             the
             point
             A
             (
             in
             the
             10
             Figure
             at
             the
             end
             of
             the
             14
             Chapter
             )
             let
             the
             two
             Lines
             A
             D
             ,
             A
             G
             ,
             be
             described
             with
             Uniform
             Motion
             ;
             and
             let
             there
             be
             taken
             
             in
             them
             two
             parts
             AB
             ,
             AE
             ,
             and
             again
             two
             other
             parts
             AC
             ,
             AF
             ;
             in
             such
             manner
             ,
             that
             AB
             ,
             AE
             ,
             be
             contemporary
             ,
             and
             likewise
             AC
             ,
             AF
             contemporary
             .
             I
             say
             first
             (
             taking
             the
             Antecedents
             AB
             ,
             AC
             in
             the
             Line
             AD
             ,
             and
             the
             Consequents
             AE
             ,
             AF
             in
             the
             Line
             AG
             )
             that
             AB
             .
             AC
             :
             :
             AE
             ▪
             AF
             are
             Proportionals
             .
             For
             seeing
             (
             by
             the
             8th
             .
             Chapter
             and
             the
             15
             Article
             )
             Velocity
             is
             Motion
             considered
             as
             determined
             by
             a
             certain
             Length
             or
             Line
             ,
             in
             a
             certain
             Time
             transmitted
             by
             it
             ,
             the
             quantity
             of
             the
             Line
             AB
             will
             be
             determined
             by
             the
             Velocity
             and
             Time
             by
             which
             the
             same
             AB
             is
             described
             ;
             and
             for
             the
             same
             reason
             ,
             the
             quantity
             of
             the
             Line
             AC
             will
             be
             determined
             by
             the
             Velocity
             and
             Time
             ,
             by
             which
             the
             same
             AC
             is
             described
             ;
             and
             therefore
             the
             proportion
             of
             AB
             to
             AC
             ,
             whether
             it
             be
             Proportion
             of
             Equality
             ,
             or
             of
             Excess
             or
             Defect
             ,
             is
             determined
             by
             the
             Velocities
             and
             Times
             by
             which
             AB
             ,
             AC
             are
             described
             ;
             But
             seeing
             the
             Motion
             of
             the
             Point
             A
             upon
             AB
             and
             AC
             is
             Uniform
             ▪
             they
             are
             both
             desribed
             with
             equal
             Velocity
             ;
             and
             therefore
             whether
             one
             of
             them
             have
             to
             the
             other
             the
             Proportion
             of
             Majority
             or
             of
             Minority
             ,
             the
             sole
             cause
             of
             that
             Proportion
             is
             the
             difference
             of
             their
             Times
             ;
             and
             by
             the
             same
             reason
             it
             is
             evident
             ,
             that
             the
             proportion
             of
             AE
             to
             AF
             is
             determined
             by
             the
             difference
             of
             their
             Times
             onely
             .
             Seeing
             therefore
             AB
             ,
             AE
             ,
             as
             also
             AC
             ,
             AF
             are
             contemporary
             ,
             the
             difference
             of
             the
             Times
             in
             which
             AB
             and
             AC
             are
             described
             ,
             is
             the
             same
             with
             that
             in
             which
             AE
             and
             AF
             are
             described
             .
             Wherfore
             the
             proportion
             of
             AB
             to
             AC
             ,
             and
             the
             proportion
             of
             AE
             to
             AF
             are
             both
             determined
             by
             the
             same
             Cause
             .
             But
             the
             Cause
             which
             so
             determines
             the
             proportion
             of
             both
             ,
             works
             equally
             in
             equal
             Times
             ,
             for
             it
             is
             Uniform
             Motion
             ;
             and
             therefore
             (
             by
             the
             last
             precedent
             Definition
             )
             the
             proportion
             of
             AB
             to
             AC
             is
             the
             same
             with
             that
             of
             AE
             to
             AF
             ;
             and
             consequently
             AB
             .
             AC
             :
             :
             AF.
             AF
             are
             Proportionals
             ;
             which
             is
             the
             first
             .
          
           
             Secondly
             (
             taking
             the
             Antecedents
             in
             different
             Lines
             ,
             )
             I
             say
             ,
             AB
             .
             AE
             :
             :
             AC
             .
             AF
             are
             Proportionals
             ;
             For
             seeing
             AB
             ,
             AE
             are
             described
             in
             the
             same
             Time
             ,
             the
             difference
             of
             the
             Velocities
             in
             which
             they
             are
             described
             are
             the
             sole
             Cause
             of
             the
             proportion
             they
             have
             to
             one
             another
             .
             And
             the
             same
             may
             be
             said
             of
             the
             proportion
             of
             AC
             to
             AF.
             But
             seeing
             both
             the
             Lines
             AD
             and
             AG
             are
             passed
             over
             by
             Uniform
             Motion
             ,
             the
             difference
             of
             the
             Velocities
             in
             which
             AB
             ,
             AE
             are
             described
             ,
             will
             be
             the
             same
             with
             the
             
             difference
             of
             the
             Velocities
             ,
             in
             which
             AC
             ,
             AF
             are
             described
             .
             Wherefore
             the
             Cause
             which
             determines
             the
             proportion
             of
             AB
             to
             AE
             ,
             is
             the
             same
             with
             that
             which
             determines
             the
             proportion
             of
             AC
             to
             AF
             ;
             and
             therefore
             AB
             .
             AE
             :
             :
             AC
             .
             AF
             ,
             are
             Proportionals
             ;
             which
             remained
             to
             be
             proved
             .
          
           
             
               1
               Corollary
               .
               If
               four
               Magnitudes
               be
               in
               Geometrical
               Proportion
               ,
               they
               will
               also
               be
               Proportionals
               by
               Permutation
               ,
               (
               that
               is
               ,
               by
               transposing
               the
               Middle
               Terms
               .
               )
               For
               I
               have
               shewn
               ,
               that
               not
               onely
               AB
               .
               AC
               :
               :
               AE
               .
               AF
               ,
               but
               also
               that
               (
               by
               Permutation
               )
               AB
               .
               AE
               :
               :
               AC
               .
               AF
               are
               Proportionals
               .
            
             
               2
               Coroll
               .
               If
               there
               be
               four
               Proportionals
               ,
               they
               will
               also
               be
               Proportionals
               by
               Inversion
               or
               Conversion
               ,
               that
               is
               ,
               by
               turning
               the
               Antecedents
               into
               Consequents
               .
               For
               if
               in
               the
               last
               Analogisme
               ,
               I
               had
               for
               AB
               ,
               AC
               ,
               put
               by
               Inversion
               AC
               ,
               AB
               ,
               and
               in
               like
               manner
               converted
               AE
               ,
               AF
               into
               AF
               ,
               AE
               ,
               yet
               the
               same
               Demonstration
               had
               served
               .
               For
               as
               well
               AC
               ,
               AB
               ,
               as
               AB
               ,
               AC
               are
               of
               equal
               Velocity
               ;
               and
               AC
               ,
               AF
               ,
               as
               well
               as
               AF
               ,
               AC
               are
               Contemporary
               .
            
             
               3
               Coroll
               .
               If
               Proportionals
               be
               added
               to
               Proportionals
               ,
               or
               taken
               from
               them
               ,
               the
               Aggregates
               ,
               or
               Remainders
               will
               be
               Proportionals
               .
               For
               Contemporaries
               whether
               they
               be
               added
               to
               Contemporaries
               ,
               or
               taken
               from
               them
               ;
               make
               the
               Aggregates
               or
               Remainders
               Contemporary
               ,
               though
               the
               Addition
               or
               Substraction
               be
               of
               all
               the
               Terms
               ,
               or
               of
               the
               Antecedents
               alone
               ,
               or
               of
               the
               Consequents
               alone
               .
            
             
               4
               Coroll
               .
               If
               both
               the
               Antecedents
               of
               four
               Proportionals
               ,
               or
               both
               the
               Consequents
               ,
               or
               all
               the
               Terms
               ,
               be
               multiplyed
               or
               divided
               by
               the
               same
               Number
               or
               Quantity
               ,
               the
               Products
               or
               Quotients
               will
               be
               Proportionals
               .
               For
               the
               Multiplication
               and
               Division
               of
               Proportionals
               ,
               is
               the
               same
               with
               the
               Addition
               and
               Substraction
               of
               them
               .
            
             
               5
               Coroll
               .
               If
               there
               be
               four
               Proportionals
               ,
               they
               will
               also
               be
               Proportionals
               by
               Composition
               (
               that
               is
               ,
               by
               compounding
               an
               Antecedent
               of
               the
               Antecedent
               and
               Consequent
               put
               together
               ,
               and
               by
               taking
               for
               Consequent
               either
               the
               Consequent
               singly
               ,
               or
               the
               Antecedent
               singly
               )
               .
               For
               this
               Composition
               is
               nothing
               but
               Addition
               of
               Proportionals
               ,
               namely
               ,
               of
               Consequents
               to
               their
               own
               Antecedents
               ,
               which
               by
               Supposition
               are
               Proportionals
               .
            
             
               6
               Coroll
               .
               In
               like
               manner
               ,
               if
               the
               Antecedent
               singly
               ,
               or
               Consequent
               singly
               be
               put
               for
               Antecedent
               ,
               and
               the
               Consequent
               be
               made
               of
               both
               put
               together
               ,
               these
               also
               will
               be
               Proportionals
               .
               For
               it
               is
               the
               
                 Inversion
                 of
                 Porportion
                 by
                 Composition
              
               .
            
             
               
               7
               Coroll
               .
               If
               there
               be
               four
               Proportionals
               ,
               they
               will
               also
               be
               Proportionals
               by
               Division
               (
               that
               is
               ,
               by
               taking
               the
               Remainder
               after
               the
               Consequent
               is
               substracted
               from
               the
               Antecedent
               ,
               or
               the
               Difference
               between
               the
               Antecedent
               and
               Consequent
               for
               Antecedent
               ,
               and
               either
               the
               Whole
               or
               the
               Remainder
               for
               Consequent
               ;
               )
               As
               if
               A.
               B
               :
               :
               C.
               D
               be
               Proportionals
               ,
               they
               will
               by
               Division
               be
               A
               −
               B.
               B
               :
               :
               C
               −
               D.
               D
               ,
               and
               A
               −
               B.
               A
               :
               :
               C
               −
               D.
               C
               ;
               and
               when
               the
               Consequent
               is
               greater
               then
               the
               Antecedent
               ,
               B
               −
               A.
               A
               :
               :
               D
               −
               C.
               C
               ,
               and
               B
               −
               AB
               :
               :
               D
               −
               C.
               D.
               For
               in
               all
               these
               Divisions
               ,
               Proportionals
               are
               (
               by
               the
               very
               supposition
               of
               the
               Analogisme
               A.
               B
               :
               :
               C.
               D
               )
               taken
               from
               A
               and
               B
               ,
               and
               from
               C
               and
               D.
               
            
             
               8
               Coroll
               .
               If
               there
               be
               four
               Proportionals
               ,
               they
               will
               also
               be
               Proportionals
               by
               the
               
                 Conversion
                 of
                 Proportion
              
               ,
               (
               that
               is
               ,
               by
               In●erting
               the
               Divided
               Proportion
               ,
               or
               by
               taking
               the
               Whole
               for
               Antecedent
               ,
               &
               the
               Difference
               or
               Remainder
               for
               Consequent
               .
               )
               As
               ,
               if
               A.
               B
               :
               :
               C.
               D
               be
               Proportionals
               ,
               then
               A.
               A
               −
               B
               :
               :
               C.
               C
               −
               D
               ,
               as
               also
               B.
               A
               −
               B
               :
               :
               D.
               C
               −
               D
               will
               be
               Proportionals
               .
               For
               seeing
               these
               Inverted
               be
               Proportionals
               ,
               they
               are
               also
               themselves
               Proportionals
               .
            
             
               9
               Coroll
               .
               If
               there
               be
               two
               Analogismes
               which
               have
               their
               quantities
               equal
               ,
               the
               second
               to
               the
               second
               ,
               and
               the
               fourth
               to
               the
               fourth
               ,
               then
               either
               the
               Sum
               or
               Difference
               of
               the
               first
               quantities
               will
               be
               to
               the
               second
               ,
               as
               the
               Sum
               or
               Difference
               of
               the
               third
               quantities
               is
               to
               the
               fourth
               .
               Let
               A.
               B
               :
               :
               C.
               D
               and
               E.
               B
               :
               :
               F.
               D
               be
               Analogismes
               ;
               I
               say
               C+E.
               B
               :
               :
               F.
               D
               are
               Proportionals
               .
               For
               the
               said
               Analogismes
               will
               by
               Permutation
               be
               A.
               C
               :
               :
               B.
               D
               ,
               and
               E.
               F
               :
               :
               B.
               D
               ;
               and
               therefore
               A.
               C
               :
               :
               E.
               F
               will
               be
               Proportionals
               (
               for
               they
               have
               both
               the
               proportion
               of
               B
               to
               D
               common
               .
               )
               Wherefore
               if
               in
               the
               Permutation
               of
               the
               first
               Analogisme
               ,
               there
               be
               added
               E
               and
               F
               to
               A
               and
               C
               ,
               which
               E
               and
               F
               are
               proportional
               to
               A
               and
               C
               ,
               then
               (
               by
               the
               3d
               Coroll
               .
               )
               A+E
               .
               B
               :
               :
               C+F
               .
               D
               will
               be
               Proportionals
               ;
               which
               was
               to
               be
               proved
               .
            
          
           
             Also
             in
             the
             same
             manner
             it
             may
             be
             shewn
             ,
             that
             A
             −
             E.
             B
             :
             :
             C+F
             .
             D
             are
             Proportionals
             .
          
           
             7
             If
             there
             be
             two
             Analogismes
             ,
             where
             foure
             Antecedents
             make
             an
             Analogisme
             ,
             their
             Consequents
             also
             shall
             make
             an
             Analogisme
             ;
             as
             also
             the
             Sums
             of
             their
             Antecedents
             will
             be
             proportionall
             to
             the
             Sums
             of
             their
             Consequents
             .
             For
             if
             A.
             B
             :
             :
             C.
             D
             and
             E.
             F
             :
             :
             G.
             H
             be
             two
             Analogismes
             ;
             and
             A.
             E
             :
             :
             C.
             G
             be
             Proportionals
             ,
             then
             by
             Permutation
             A.
             C
             :
             :
             E.
             G
             ,
             and
             E.
             G
             :
             :
             F.
             H
             ,
             and
             A.
             C
             :
             :
             B.
             D
             will
             be
             Proportionals
             ;
             Wherefore
             B.
             D
             :
             :
             E.
             G
             ,
             that
             is
             ,
             B.
             D
             :
             :
             F.
             H
             ,
             and
             by
             
             Permutation
             B.
             F
             :
             :
             D.
             H
             are
             Proportionals
             ;
             which
             is
             the
             first
             .
             Secondly
             ,
             I
             say
             A+E
             .
             B+F
             :
             :
             C+G
             .
             D+H
             will
             be
             Proportionalls
             .
             For
             seeing
             A.
             E
             :
             :
             C.
             G
             are
             Proportionals
             ,
             A+E
             .
             E
             :
             :
             C+G
             .
             G
             will
             also
             by
             Composition
             be
             Proportionals
             ,
             and
             by
             Permutation
             A+E
             .
             C+G
             :
             :
             E.
             G
             will
             be
             Proportionals
             ;
             Wherefore
             also
             A+E
             .
             C+G
             :
             :
             F.
             H
             will
             be
             Proportionals
             .
             Again
             ,
             seeing
             (
             as
             is
             shewn
             above
             )
             P.
             F
             :
             :
             D.
             H
             are
             Proportionals
             ,
             B+F.
             F
             :
             :
             D+H
             .
             H
             will
             also
             by
             Composition
             be
             Proportionals
             ;
             and
             by
             Permutation
             B+F.
             D+H
             :
             :
             F.
             H
             will
             also
             be
             Proportionals
             ;
             Wherefore
             A+E
             .
             C+G
             :
             :
             B+F.
             D+H
             are
             Proportionals
             ;
             which
             remained
             to
             be
             proved
             .
          
           
             Coroll
             .
             By
             the
             same
             reason
             ,
             if
             there
             be
             never
             so
             many
             Analogismes
             ,
             and
             the
             Antecedents
             be
             proportional
             to
             the
             Antecedents
             ,
             it
             may
             be
             demonstrated
             also
             that
             the
             Consequents
             will
             be
             proportional
             to
             the
             Consequents
             ,
             as
             also
             the
             Sum
             of
             the
             Antecedents
             to
             the
             Sum
             of
             the
             Consequents
             .
          
           
             8
             In
             an
             Hyperlogisme
             ,
             that
             is
             ,
             where
             the
             Proportion
             of
             the
             first
             Antecedent
             to
             its
             Consequent
             ,
             is
             greater
             then
             the
             proportion
             of
             the
             second
             Antecedent
             to
             its
             Consequent
             ,
             the
             Permutation
             of
             the
             Proportionals
             ,
             and
             the
             Addition
             of
             Proportionals
             to
             Proportionals
             ,
             &
             Substraction
             of
             them
             from
             one
             another
             ;
             as
             also
             their
             Composition
             &
             Division
             ,
             &
             their
             Multiplication
             &
             Division
             by
             the
             same
             Number
             ,
             produce
             always
             an
             Hyperlogisme
             .
             For
             suppose
             A.
             B
             :
             :
             C.
             D
             &
             A.
             C
             :
             :
             E.
             F
             be
             Analogismes
             ,
             A+E
             .
             B
             :
             :
             C+F
             .
             D
             will
             be
             also
             an
             Analogisme
             ;
             But
             A+E
             .
             B
             :
             :
             C.
             D
             will
             be
             an
             Hyperlogisme
             ;
             Wherefore
             by
             Permutation
             ,
             A+E
             .
             C
             :
             :
             B.
             D
             is
             an
             Hyperlogisme
             ,
             because
             A.
             B
             :
             :
             C.
             D
             is
             an
             Analogisme
             .
             Secondly
             ,
             if
             to
             the
             Hyperlogisme
             A+E
             .
             B
             :
             :
             C.
             D
             the
             Proportionals
             G
             and
             H
             be
             added
             ,
             A+E+G
             .
             B
             :
             :
             ,
             C+H.
             D
             will
             be
             an
             Hyperlogisme
             ,
             by
             reason
             A+E+G
             .
             B
             :
             :
             C+F+H
             .
             D
             is
             an
             Analogisme
             .
             Also
             if
             G
             and
             H
             be
             taken
             away
             ,
             A+E
             ▪
             G.
             B
             :
             :
             C
             −
             H.
             D
             will
             he
             an
             Hyperlogisme
             ;
             for
             A+E
             −
             G.
             B
             :
             :
             C+F
             −
             H.
             D
             are
             an
             Analogisme
             .
             Thirdly
             ,
             by
             Composition
             A+E+B
             .
             B
             :
             :
             C+D
             .
             D
             will
             be
             an
             Hyperlogisme
             ,
             because
             A+E+B
             .
             B
             :
             :
             C+F+D
             .
             D
             is
             an
             Analogisme
             ,
             &
             so
             it
             will
             be
             in
             all
             the
             varieties
             of
             Composition
             .
             Fourthly
             ,
             by
             Division
             A+E
             −
             B.
             B
             :
             :
             C
             −
             D.
             D
             will
             be
             an
             Hyperlogisme
             ,
             by
             reason
             A+E
             −
             B.
             B
             :
             :
             C+F
             −
             D.
             D
             is
             an
             Analogisme
             .
             Also
             A+E
             −
             B.
             A+E
             :
             :
             C
             −
             D.
             C
             is
             an
             Hyperlogisme
             ;
             for
             A+E
             −
             B.
             A+E
             :
             :
             C+F
             −
             D.
             C
             is
             an
             Analogisme
             .
             Fifthly
             ,
             by
             Multiplication
             4
             A+E
             .
             B
             :
             :
             4
             C.
             D
             is
             an
             Hyperlogisme
             ,
             because
             4
             A.
             B
             :
             :
             4
             C.
             D
             is
             an
             Analogisme
             ;
             
             and
             by
             Division
             ¼A+¼E
             .
             B
             :
             :
             ¼C
             .
             D
             is
             an
             Hyperlogisme
             ,
             because
             ¼A
             .
             B
             :
             :
             ¼C
             .
             D
             is
             an
             Analogisme
             .
          
           
             9
             But
             if
             A+E
             .
             B
             :
             :
             C.
             D
             be
             an
             Hyperlogisme
             ,
             then
             by
             Inversion
             B.
             A+E
             :
             :
             D.
             C
             will
             be
             an
             Hypologisme
             ,
             because
             B.
             A
             :
             :
             D.
             C
             being
             an
             Analogisme
             ,
             the
             first
             Consequent
             will
             be
             too
             great
             .
             Also
             by
             Conversion
             of
             Proportion
             A+E
             .
             A+E
             −
             B
             :
             :
             C.
             C
             −
             D
             is
             an
             Hypologisme
             ,
             because
             the
             Inversion
             of
             it
             ,
             namely
             A+E
             −
             B.
             A+E
             :
             :
             C
             −
             D.
             C
             is
             an
             Hyperlogisme
             (
             as
             I
             have
             shewn
             but
             now
             )
             .
             So
             also
             B.
             A+E
             −
             B
             :
             :
             D.
             C
             −
             D
             is
             an
             Hypologisme
             ,
             because
             (
             as
             I
             have
             newly
             shewn
             )
             the
             Inversion
             of
             it
             ,
             namely
             A+E
             −
             B.
             B
             :
             :
             C
             −
             D.
             D
             is
             an
             Hyperlogisme
             .
             Note
             that
             this
             Hypologisme
             A+E
             .
             A+E
             −
             B
             :
             :
             C.
             C
             −
             D
             is
             commonly
             thus
             expressed
             ;
             If
             the
             proportion
             of
             the
             Whole
             (
             A+E
             )
             to
             that
             which
             is
             taken
             out
             of
             it
             (
             B
             )
             ,
             be
             greater
             then
             the
             proportion
             of
             the
             Whole
             (
             C
             )
             to
             that
             which
             is
             taken
             out
             of
             it
             (
             D
             ,
             )
             then
             the
             proportion
             of
             the
             whole
             (
             A+E
             )
             to
             the
             Remainder
             (
             A+E
             −
             B
             )
             wil
             be
             less
             then
             the
             proportion
             of
             the
             whole
             (
             C
             )
             to
             the
             Remainder
             (
             C
             −
             D.
             )
          
           
             10
             If
             there
             be
             four
             Proportionals
             ,
             the
             Difference
             of
             the
             two
             first
             ,
             to
             the
             Difference
             of
             the
             two
             last
             will
             be
             as
             the
             first
             Antecedent
             is
             to
             the
             second
             Antecedent
             ,
             or
             as
             the
             first
             Consequent
             to
             the
             second
             Consequent
             .
             For
             if
             A.
             B
             :
             :
             C.
             D
             be
             Proportionals
             ,
             then
             by
             Division
             A
             −
             B.
             B
             :
             :
             C
             −
             D.
             D
             will
             be
             Proportionals
             ;
             and
             by
             Permutation
             A
             −
             B.
             C
             −
             D
             :
             :
             B.
             D
             ;
             that
             is
             ,
             the
             Differences
             are
             proportional
             to
             the
             Consequents
             ,
             and
             therefore
             they
             are
             so
             also
             to
             the
             Antecedents
             .
          
           
             11
             Of
             four
             Proportionals
             ,
             if
             the
             first
             be
             greater
             then
             the
             second
             ,
             the
             third
             also
             shall
             be
             greater
             then
             the
             fourth
             .
             For
             seeing
             the
             first
             is
             greater
             then
             the
             second
             ,
             the
             proportion
             of
             the
             first
             to
             the
             second
             is
             the
             proportion
             of
             Excess
             ;
             But
             the
             proportion
             of
             the
             third
             to
             the
             fourth
             is
             the
             same
             with
             that
             of
             the
             first
             to
             the
             second
             ;
             and
             therefore
             also
             the
             proportion
             of
             the
             third
             to
             the
             fourth
             is
             the
             Proportion
             of
             Excess
             ;
             Wherefore
             the
             third
             is
             greater
             then
             the
             fourth
             .
             In
             the
             same
             manner
             it
             may
             be
             proved
             ,
             that
             whensoever
             the
             first
             is
             less
             then
             the
             second
             ,
             the
             third
             also
             is
             less
             then
             the
             fourth
             ;
             and
             when
             those
             are
             equal
             ,
             that
             these
             also
             are
             equal
             .
          
           
             12
             If
             there
             be
             four
             Proportionals
             whatsoever
             A.
             B
             :
             :
             C.
             D
             ,
             and
             the
             first
             and
             third
             be
             multiplyed
             by
             any
             one
             number
             ,
             as
             by
             2
             ;
             and
             again
             the
             second
             and
             fourth
             be
             multiplyed
             by
             any
             one
             number
             ,
             as
             
             by
             3
             ;
             and
             the
             product
             of
             the
             first
             2
             A
             ,
             be
             greater
             then
             the
             product
             of
             the
             second
             3
             B
             ;
             the
             product
             also
             of
             the
             third
             2
             C
             ,
             will
             be
             greater
             then
             the
             product
             of
             the
             fourth
             3
             D.
             But
             if
             the
             product
             of
             the
             first
             be
             less
             then
             the
             product
             of
             the
             second
             ,
             then
             the
             product
             of
             the
             third
             will
             be
             less
             then
             that
             of
             the
             fourth
             .
             And
             lastly
             ,
             if
             the
             products
             of
             the
             first
             and
             second
             be
             equal
             ,
             the
             products
             of
             the
             third
             and
             fourth
             shall
             also
             be
             equal
             .
             Now
             this
             Theoreme
             is
             all
             one
             with
             Euclides
             Definition
             of
             
               The
               Same
               Proportion
            
             ;
             and
             it
             may
             be
             demonstrated
             thus
             .
             Seeing
             A.
             B
             :
             :
             C.
             D
             are
             Proportionals
             ,
             by
             Permutation
             also
             (
             Art.
             6.
             
             Corol.
             1.
             )
             A.
             C
             :
             :
             B.
             D
             will
             be
             Proportionals
             ;
             Wherefore
             (
             by
             the
             4
             Corol.
             of
             the
             same
             6
             Article
             )
             2
             A.
             2
             C
             :
             :
             3
             B.
             3
             D
             will
             be
             Proportionals
             ;
             and
             again
             by
             Permutation
             2
             A.
             3
             B
             :
             :
             2
             C.
             3
             D
             will
             be
             Proportionals
             ;
             and
             therefore
             (
             by
             the
             last
             Article
             )
             If
             2
             A
             be
             greater
             then
             3
             B
             ,
             then
             2
             C
             will
             be
             greater
             then
             3
             D
             ;
             if
             less
             ,
             less
             ;
             and
             if
             equal
             ,
             equal
             ;
             which
             was
             to
             be
             demonstrated
             .
          
           
           
             In
             the
             second
             place
             ,
             let
             AD
             be
             the
             first
             ,
             AC
             the
             second
             ,
             &
             AB
             the
             third
             ,
             and
             let
             their
             proportion
             be
             the
             Porportion
             of
             Excess
             ,
             or
             of
             Greater
             to
             Less
             ;
             then
             ,
             as
             before
             ,
             the
             proportions
             of
             AD
             to
             AC
             ,
             and
             of
             AC
             to
             AB
             ,
             and
             of
             AD
             to
             AB
             will
             be
             determined
             by
             the
             difference
             of
             their
             Times
             ;
             which
             in
             the
             description
             of
             AD
             and
             AC
             ,
             and
             of
             AC
             and
             AB
             ▪
             together
             taken
             ,
             is
             the
             same
             with
             the
             difference
             of
             the
             Times
             in
             the
             description
             of
             AD
             and
             AB
             .
             Wherefore
             the
             proportion
             of
             AD
             to
             AB
             is
             equal
             to
             the
             two
             proportions
             of
             AD
             to
             AC
             and
             of
             AC
             to
             AB
             .
          
           
             In
             the
             last
             place
             .
             If
             one
             of
             the
             proportions
             ,
             namely
             of
             AD
             to
             AB
             be
             the
             Proportion
             of
             Excess
             ;
             and
             another
             of
             them
             ,
             as
             of
             AB
             to
             AC
             be
             the
             Proportion
             of
             Defect
             ,
             thus
             also
             the
             proportion
             of
             AD
             to
             AC
             will
             be
             equal
             to
             the
             two
             proportions
             together
             taken
             of
             AD
             to
             AB
             ,
             and
             of
             AB
             to
             AC
             .
             For
             the
             difference
             of
             the
             Times
             in
             which
             AD
             and
             AB
             are
             described
             is
             Excess
             of
             Time
             ;
             for
             there
             goes
             more
             time
             to
             the
             description
             of
             AD
             then
             of
             AB
             ;
             and
             the
             difference
             of
             the
             Times
             in
             which
             AB
             and
             AC
             are
             described
             is
             Defect
             of
             Time
             ,
             for
             less
             Time
             goes
             to
             the
             description
             of
             AB
             then
             of
             AC
             ;
             but
             this
             Excess
             ,
             and
             Defect
             being
             ad●ed
             together
             make
             DB
             −
             BC
             ,
             which
             is
             equal
             to
             DC
             ,
             by
             which
             the
             first
             AD
             exceeds
             the
             third
             AC
             ;
             and
             therefore
             the
             proportions
             of
             the
             first
             AD
             to
             the
             second
             AB
             ,
             and
             of
             the
             second
             AB
             to
             the
             third
             AC
             ,
             are
             determined
             by
             the
             same
             Cause
             which
             determines
             the
             Proportion
             of
             the
             first
             AD
             to
             the
             third
             AC
             .
             Wherefore
             ,
             If
             any
             three
             Magnitudes
             ,
             &c.
             
          
           
             
               1
               Coroll
               .
               If
               there
               be
               never
               so
               many
               Magnitudes
               having
               proportion
               to
               one
               another
               ,
               the
               proportion
               of
               the
               first
               to
               the
               last
               is
               compounded
               of
               the
               proportions
               of
               the
               first
               to
               the
               second
               ,
               of
               the
               second
               to
               the
               third
               ,
               &
               so
               on
               till
               you
               come
               to
               the
               last
               ;
               or
               ,
               the
               proportion
               of
               the
               first
               to
               the
               last
               ,
               is
               the
               same
               with
               the
               Sum
               of
               all
               the
               intermediate
               proportions
               .
               For
               any
               Number
               of
               Magnitudes
               having
               proportion
               to
               one
               another
               ,
               as
               A
               ,
               B
               ,
               C
               ,
               D
               ,
               E
               being
               propounded
               ,
               the
               proportion
               
               of
               A
               to
               E
               (
               as
               is
               newly
               shewn
               )
               is
               compounded
               of
               the
               Proportions
               of
               A
               to
               D
               and
               of
               D
               to
               E
               ;
               and
               again
               the
               proportion
               of
               A
               to
               D
               ,
               of
               the
               proportions
               of
               A
               to
               C
               ,
               and
               of
               C
               to
               D
               ;
               and
               lastly
               ,
               the
               proportion
               of
               A
               to
               C
               ,
               of
               the
               proportions
               of
               A
               to
               B
               ,
               and
               of
               B
               to
               C.
               
            
             
               2
               Coroll
               .
               From
               hence
               it
               may
               be
               understood
               how
               any
               two
               proportions
               may
               be
               compounded
               .
               For
               if
               the
               proportions
               of
               A
               to
               B
               ,
               and
               of
               C
               to
               D
               be
               propounded
               to
               be
               added
               together
               ,
               let
               B
               have
               to
               something
               else
               ,
               as
               to
               E
               ,
               the
               same
               proportion
               which
               C
               has
               to
               D
               ,
               and
               let
               them
               be
               set
               in
               this
               order
               A
               ,
               B
               ,
               E
               ;
               for
               so
               the
               proportion
               of
               A
               to
               E
               will
               evidently
               be
               the
               Sum
               of
               the
               two
               Proportions
               of
               A
               to
               B
               ,
               and
               of
               B
               to
               E
               ,
               that
               is
               ,
               of
               C
               to
               D.
               Or
               let
               it
               be
               as
               D
               to
               C
               ,
               so
               A
               to
               something
               else
               ,
               as
               to
               E
               ,
               and
               let
               them
               be
               ordered
               thus
               E
               ,
               A
               ,
               B
               ;
               for
               the
               proportion
               of
               E
               to
               B
               will
               be
               compounded
               of
               the
               proportions
               of
               E
               to
               A
               (
               that
               is
               ,
               of
               C
               to
               D
               )
               ,
               and
               of
               A
               to
               B.
               Also
               it
               may
               be
               understood
               how
               one
               Proportion
               may
               be
               taken
               out
               of
               another
               .
               For
               if
               the
               proportion
               of
               C
               to
               D
               be
               to
               be
               substracted
               out
               of
               the
               proportion
               of
               A
               to
               B
               ,
               let
               it
               be
               as
               C
               to
               D
               ,
               so
               A
               to
               something
               else
               ,
               as
               E
               ,
               and
               setting
               them
               in
               this
               order
               ,
               A
               ,
               E
               ,
               B
               ,
               and
               taking
               away
               the
               proportion
               of
               A
               to
               E
               ,
               that
               is
               ,
               of
               C
               to
               D
               ,
               there
               will
               remain
               the
               proportion
               of
               E
               to
               B.
               
            
             
               3
               Coroll
               .
               If
               there
               be
               two
               Orders
               of
               Magnitudes
               which
               have
               proportion
               to
               one
               another
               beginning
               and
               ending
               with
               the
               same
               Magnitudes
               ,
               and
               the
               several
               proportions
               of
               the
               first
               Order
               be
               the
               same
               and
               equal
               in
               number
               with
               the
               proportions
               of
               the
               second
               Order
               ;
               then
               ,
               whether
               the
               proportions
               in
               both
               Orders
               be
               successively
               answerable
               to
               one
               another
               ,
               which
               is
               called
               
                 Ordinate
                 Proportion
              
               ,
               or
               not
               successively
               answerable
               ,
               which
               is
               called
               
                 Perturbed
                 Proportion
              
               ,
               the
               first
               and
               the
               last
               in
               both
               will
               be
               Proportionals
               .
               For
               the
               Proportion
               of
               the
               first
               to
               the
               last
               is
               equal
               to
               all
               the
               intermediate
               proporons
               ;
               which
               being
               in
               both
               Orders
               the
               same
               ,
               and
               equal
               in
               number
               ,
               the
               Aggregates
               of
               those
               proportions
               will
               also
               be
               equal
               to
               one
               another
               ;
               but
               to
               their
               Aggregates
               the
               proportions
               of
               the
               first
               to
               the
               last
               are
               Equal
               ;
               and
               therefore
               the
               proportion
               of
               the
               first
               to
               the
               last
               in
               one
               Order
               ,
               is
               the
               same
               with
               the
               proportion
               of
               the
               first
               to
               the
               last
               in
               the
               other
               Order
               .
               Wherefore
               the
               first
               and
               the
               last
               in
               both
               are
               Proportionals
               .
            
          
           
             14
             If
             any
             two
             quantities
             be
             made
             of
             the
             mutual
             Multiplication
             of
             many
             quantities
             which
             have
             proportion
             to
             one
             another
             ,
             and
             
             the
             Efficient
             quantities
             on
             both
             sides
             be
             equal
             in
             number
             ,
             the
             proportion
             of
             the
             Products
             will
             be
             compounded
             of
             the
             several
             proportions
             which
             the
             Efficient
             quantities
             have
             to
             one
             another
             .
          
           
           
           
             From
             hence
             ariseth
             another
             way
             of
             Compounding
             many
             Proportions
             into
             One
             ,
             namely
             ,
             that
             which
             is
             supposed
             in
             the
             5
             Definition
             of
             the
             6
             Book
             of
             Euclide
             ;
             which
             is
             ,
             by
             multiplying
             all
             the
             Antecedents
             of
             the
             Proportions
             into
             one
             another
             ,
             and
             in
             like
             manner
             all
             the
             Consequents
             into
             one
             another
             .
             And
             from
             hence
             also
             it
             is
             evident
             ,
             in
             the
             first
             place
             ,
             That
             the
             Cause
             why
             Parallelograms
             ,
             which
             are
             made
             by
             the
             Duction
             of
             two
             straight
             Lines
             into
             one
             anther
             ,
             and
             all
             Solids
             which
             are
             equal
             to
             Figures
             so
             made
             ,
             have
             their
             proportions
             compounded
             of
             the
             proportions
             of
             the
             Efficients
             ;
             And
             in
             the
             second
             place
             ,
             why
             the
             Multiplication
             of
             two
             or
             more
             Fractions
             into
             one
             another
             ,
             is
             the
             same
             thing
             with
             the
             Composition
             of
             the
             proportions
             of
             their
             several
             Numerators
             to
             their
             several
             Denominators
             .
             For
             examp●●
             ,
             if
             these
             Fractions
             ½
             ,
             ⅔
             ,
             ¾
             be
             to
             be
             multiplyed
             into
             one
             another
             ,
             the
             Numerators
             1
             ,
             2
             ,
             3
             are
             first
             to
             be
             multiplyed
             into
             one
             another
             ,
             which
             make
             6
             ;
             and
             next
             the
             Denominators
             2
             ,
             3
             ,
             4
             ,
             which
             make
             24
             ;
             and
             these
             two
             Products
             make
             the
             Fraction
             6
             /
             24.
             
             In
             like
             manner
             ,
             if
             the
             proportions
             of
             1
             to
             2
             ,
             of
             2
             to
             3
             ,
             and
             of
             3
             to
             4
             be
             to
             be
             compounded
             ,
             by
             working
             as
             I
             have
             shewn
             above
             ,
             the
             same
             proportion
             ,
             of
             6
             to
             24
             will
             be
             produced
             .
          
           
             15
             If
             any
             Proportion
             be
             compounded
             with
             it self
             inverted
             ,
             the
             Compound
             will
             be
             the
             Proportion
             of
             Equality
             .
             For
             let
             any
             Proportion
             be
             given
             ,
             as
             of
             A
             to
             B
             ,
             and
             let
             the
             Inverse
             of
             it
             be
             that
             of
             C
             to
             D
             ;
             and
             as
             C
             to
             D
             ,
             so
             let
             B
             be
             to
             another
             quantity
             ;
             for
             thus
             they
             will
             be
             compounded
             (
             by
             the
             2
             Coroll
             .
             of
             the
             12
             Art.
             )
             Now
             seeing
             the
             proportion
             of
             C
             to
             D
             is
             the
             Inverse
             of
             the
             proportion
             of
             A
             to
             B
             ,
             it
             will
             be
             as
             C
             to
             D
             ,
             so
             B
             to
             A
             ;
             and
             therefore
             if
             they
             be
             placed
             in
             Order
             A
             ,
             B
             ,
             A
             ,
             the
             proportion
             compounded
             of
             the
             proportions
             of
             A
             to
             B
             ,
             and
             of
             C
             to
             D
             will
             be
             the
             proportion
             of
             A
             to
             A
             ,
             that
             is
             ,
             the
             proportion
             of
             Equality
             .
             And
             from
             hence
             the
             cause
             is
             evident
             ,
             why
             two
             equal
             products
             have
             their
             Efficients
             reciprocally
             proportional
             .
             For
             ,
             for
             the
             making
             of
             two
             products
             equal
             ,
             the
             proportions
             of
             their
             Efficients
             must
             be
             such
             ,
             as
             being
             compounded
             may
             make
             the
             proportion
             of
             Equality
             ,
             which
             cannot
             be
             ,
             except
             one
             be
             the
             Inverse
             of
             the
             other
             ;
             for
             if
             betwixt
             A
             and
             A
             any
             other
             quantity
             as
             C
             be
             interposed
             ,
             their
             order
             will
             be
             A
             ,
             C
             ,
             A
             ,
             and
             the
             later
             proportion
             of
             C
             to
             A
             will
             be
             the
             Inverse
             of
             the
             former
             proportion
             of
             A
             to
             C.
             
          
           
           
             16
             A
             Proportion
             is
             said
             to
             be
             multiplied
             by
             a
             Number
             when
             it
             is
             so
             often
             taken
             as
             there
             be
             Unities
             in
             that
             Number
             ;
             and
             if
             the
             Proportion
             be
             of
             the
             Greater
             to
             the
             Less
             ,
             then
             shall
             also
             the
             quantity
             of
             the
             Proportion
             be
             increased
             by
             the
             Multiplication
             ;
             but
             when
             the
             Proportion
             is
             of
             the
             Less
             to
             the
             Greater
             ,
             then
             as
             the
             Number
             increaseth
             ,
             the
             quantity
             of
             the
             Proportion
             diminisheth
             ;
             as
             in
             these
             three
             Numbers
             4
             ,
             2
             ,
             1
             ,
             the
             Proportion
             of
             4
             to
             1
             ,
             is
             not
             onely
             the
             Duplicate
             of
             4
             to
             2
             ,
             but
             also
             twice
             as
             great
             ;
             but
             inverting
             the
             order
             of
             those
             Numbers
             thus
             1
             ,
             2
             ,
             4
             ,
             the
             Proportion
             of
             1
             to
             2
             is
             greater
             then
             that
             of
             1
             to
             4
             ;
             and
             therefore
             though
             the
             proportion
             of
             1
             to
             4
             be
             the
             Duplicate
             of
             1
             to
             2
             ,
             yet
             it
             is
             not
             twice
             so
             great
             as
             that
             of
             1
             to
             2
             ,
             but
             contrarily
             the
             half
             of
             it
             .
             In
             like
             manner
             ,
             a
             Proportion
             is
             said
             to
             be
             Divided
             ,
             when
             between
             two
             quantities
             are
             interposed
             one
             or
             more
             Means
             in
             continual
             Proportion
             ,
             and
             then
             the
             Proportion
             of
             the
             first
             to
             the
             second
             is
             said
             to
             be
             Subduplicate
             of
             that
             of
             the
             first
             to
             the
             third
             ,
             and
             Subtriplicate
             of
             that
             of
             the
             first
             to
             the
             fourth
             ,
             &c.
             
          
           
             This
             mixture
             of
             Proportions
             ,
             where
             some
             are
             Proportions
             of
             Excess
             ,
             others
             of
             Defect
             (
             as
             in
             a
             Merchants
             accompt
             of
             Debitor
             and
             Creditor
             )
             is
             not
             so
             easily
             reckoned
             as
             some
             think
             ;
             but
             maketh
             the
             Composition
             of
             Proportions
             sometimes
             to
             be
             Addition
             ,
             sometimes
             Substraction
             ;
             which
             soundeth
             absurdly
             to
             such
             as
             have
             alwayes
             by
             Composition
             understood
             Addition
             ,
             and
             by
             Diminution
             Substraction
             .
             Therefore
             to
             make
             this
             account
             a
             little
             clearer
             ,
             we
             are
             to
             consider
             (
             that
             which
             is
             commonly
             assumed
             ,
             and
             truly
             )
             that
             if
             there
             be
             never
             so
             many
             Quantities
             ,
             the
             Proportion
             of
             the
             first
             to
             the
             last
             is
             compounded
             of
             the
             Proportions
             of
             the
             first
             to
             the
             second
             ,
             and
             of
             the
             second
             to
             the
             third
             ,
             and
             so
             on
             to
             the
             last
             ,
             without
             regarding
             their
             Equality
             ,
             Excess
             or
             Defect
             ;
             So
             that
             if
             two
             Proportions
             ,
             one
             of
             Inequality
             ,
             the
             other
             of
             Equality
             be
             added
             together
             ,
             the
             Proportion
             is
             not
             thereby
             made
             Greater
             nor
             Less
             ;
             as
             for
             example
             ,
             if
             the
             Proportions
             of
             A
             to
             B
             and
             of
             B
             to
             B
             be
             compounded
             ,
             the
             Proportion
             of
             the
             first
             to
             the
             second
             is
             as
             much
             as
             the
             Sum
             of
             both
             ,
             because
             Proportion
             of
             Equality
             (
             being
             not
             quantity
             )
             neither
             augmenteth
             quantity
             nor
             lesseneth
             it
             .
             But
             if
             there
             be
             three
             quantities
             A
             ,
             B
             ,
             C
             ,
             unequal
             ,
             and
             the
             first
             be
             greatest
             ,
             the
             last
             least
             ,
             then
             the
             Proportion
             of
             B
             to
             C
             is
             an
             addition
             to
             that
             of
             A
             to
             B
             ,
             and
             
             makes
             it
             greater
             ;
             and
             on
             the
             contrary
             ,
             if
             A
             be
             the
             least
             ,
             and
             C
             the
             greatest
             quantity
             ,
             then
             doth
             the
             addition
             of
             the
             Proportion
             of
             B
             to
             C
             make
             the
             cōpounded
             Proportion
             of
             A
             to
             C
             less
             then
             the
             Proportion
             of
             A
             to
             B
             ,
             that
             is
             ,
             the
             Whole
             less
             then
             the
             Part.
             The
             Compositiont
             herefore
             of
             Proportions
             is
             not
             in
             this
             case
             the
             Augmentation
             of
             them
             ,
             but
             the
             Diminution
             ;
             for
             the
             same
             quantity
             (
             Euclide
             the
             5
             ,
             8.
             )
             compared
             with
             two
             other
             quantities
             hath
             a
             greater
             Proportion
             to
             the
             lesser
             of
             them
             then
             to
             the
             greater
             .
             Likewise
             ,
             when
             the
             Proportions
             compounded
             are
             one
             of
             Excess
             ,
             the
             other
             of
             Defect
             ,
             if
             the
             first
             be
             of
             Excess
             ,
             as
             in
             these
             numbers
             8
             ,
             6
             ,
             9
             ,
             the
             Proportion
             compounded
             ,
             namely
             ,
             of
             8
             to
             9
             ,
             is
             less
             then
             the
             Proportion
             of
             one
             of
             the
             parts
             of
             it
             ,
             namely
             of
             8
             to
             6
             ;
             but
             if
             the
             Proportion
             of
             the
             first
             to
             the
             second
             be
             of
             Defect
             ,
             and
             that
             of
             the
             second
             to
             the
             third
             be
             of
             Excess
             ,
             as
             in
             these
             Numbers
             6
             ,
             8
             ,
             ●
             ,
             then
             shall
             the
             Proportion
             of
             the
             first
             to
             the
             third
             be
             greater
             then
             that
             of
             the
             first
             to
             the
             second
             ,
             as
             6
             hath
             a
             greater
             Proportion
             to
             4
             then
             to
             8
             ;
             the
             reason
             whereof
             is
             manifestly
             this
             ,
             that
             the
             less
             any
             quantity
             is
             deficient
             of
             another
             ,
             or
             the
             more
             one
             exceedeth
             another
             ,
             the
             proportion
             of
             it
             to
             that
             other
             is
             the
             greater
             .
          
           
             Suppose
             now
             three
             quantities
             in
             continual
             Proportion
             A
             B
             4
             ,
             A
             C
             6
             ,
             A
             D
             9.
             
             Because
             therefore
             A
             D
             is
             greater
             then
             A
             C
             ,
             but
             not
             greater
             then
             A
             D
             ,
             the
             proportion
             of
             A
             D
             to
             A
             C
             will
             be
             (
             by
             Euclide
             5
             ,
             8.
             )
             greater
             then
             that
             of
             A
             D
             to
             A
             D
             ;
             and
             likewise
             ,
             because
             the
             Proportions
             of
             A
             D
             to
             A
             C
             ,
             and
             of
             A
             C
             to
             A
             B
             are
             the
             same
             ,
             the
             proportions
             of
             A
             D
             to
             A
             C
             and
             of
             A
             C
             to
             A
             B
             (
             being
             both
             Proportions
             of
             Excess
             )
             make
             the
             whole
             Proportion
             of
             A
             D
             to
             A
             B
             (
             or
             of
             9
             to
             4
             )
             not
             onely
             the
             Duplicate
             of
             A
             D
             to
             A
             C
             (
             that
             is
             ,
             of
             9
             to
             6
             )
             but
             also
             the
             Double
             ,
             or
             twice
             so
             great
             .
             On
             the
             other
             side
             ,
             because
             the
             proportion
             of
             A
             D
             to
             A
             D
             (
             or
             9
             to
             9
             )
             being
             Proportion
             of
             Equality
             ,
             is
             no
             quantity
             ,
             &
             yet
             greater
             then
             that
             of
             A
             C
             to
             A
             D
             (
             or
             6
             to
             9
             )
             it
             will
             be
             as
             0-9
             to
             0-6
             ,
             so
             A
             C
             to
             A
             D
             ,
             and
             again
             ,
             as
             0-9
             to
             0-6
             ,
             so
             0-6
             to
             0-4
             ;
             but
             0-4
             ,
             0-6
             ,
             0-9
             are
             in
             continual
             proportion
             ;
             and
             because
             0-4
             is
             greater
             then
             0-6
             ,
             the
             protion
             of
             0-4
             to
             0-6
             will
             be
             Double
             to
             the
             proportion
             of
             0-4
             to
             0-9
             ,
             Double
             I
             say
             ,
             and
             yet
             not
             Duplicate
             ,
             but
             Subduplicate
             .
          
           
           
             By
             the
             same
             method
             ,
             if
             there
             be
             more
             quantities
             then
             three
             ,
             as
             A
             ,
             B
             ,
             C
             ,
             D
             in
             continual
             Proportion
             ,
             and
             A
             be
             the
             least
             ,
             it
             may
             be
             made
             appear
             ,
             that
             the
             Proportion
             of
             A
             to
             B
             is
             Triple
             Magnitude
             (
             though
             Subtriple
             in
             Multitude
             )
             to
             the
             Proportion
             of
             A
             to
             D.
             
          
           
             17
             If
             there
             be
             never
             so
             many
             quantities
             ,
             the
             number
             whereof
             is
             odd
             ,
             and
             their
             Order
             such
             ,
             that
             from
             the
             middlemost
             quantity
             both
             wayes
             they
             proceed
             in
             continual
             Proportion
             ,
             the
             proportion
             of
             the
             two
             which
             are
             next
             on
             either
             side
             to
             the
             middlemost
             ,
             is
             Subduplicate
             to
             the
             proportion
             of
             the
             two
             which
             are
             next
             to
             these
             on
             both
             sides
             ,
             and
             Subtriplicate
             of
             the
             proportion
             of
             the
             two
             which
             are
             yet
             one
             place
             more
             remote
             ,
             &c.
             
             For
             let
             the
             Magnitudes
             be
             C
             ,
             B
             ,
             A
             ,
             D
             ,
             E
             ,
             and
             let
             A
             ,
             B
             ,
             C
             ,
             as
             also
             A
             ,
             D
             ,
             E
             be
             in
             Continual
             Proportion
             ;
             I
             say
             the
             proportion
             of
             D
             to
             B
             is
             Subduplicate
             of
             the
             proportion
             of
             E
             to
             C.
             For
             the
             proportion
             of
             D
             to
             B
             is
             compounded
             of
             the
             proportions
             of
             D
             to
             A
             ,
             and
             of
             A
             to
             B
             once
             taken
             ;
             But
             the
             proportion
             of
             E
             to
             C
             is
             compounded
             of
             the
             same
             twice
             taken
             ;
             and
             therefore
             the
             proportion
             of
             D
             to
             B
             is
             Subduplicate
             of
             the
             proportion
             of
             E
             to
             C
             -
             And
             in
             the
             same
             manner
             ,
             if
             there
             were
             three
             Terms
             on
             either
             side
             ,
             it
             might
             be
             demonstrated
             that
             the
             proportion
             of
             D
             to
             B
             would
             be
             Subtriplicate
             of
             that
             of
             the
             Extremes
             ,
             &c.
             
          
           
             18
             If
             there
             be
             never
             so
             many
             continual
             Proportionals
             ,
             as
             the
             first
             ,
             second
             ,
             third
             ,
             &c.
             their
             Differences
             will
             be
             Proportional
             to
             them
             .
             For
             the
             second
             ,
             third
             ,
             &c.
             are
             severally
             Consequents
             of
             the
             preceding
             ,
             and
             Antecedents
             of
             the
             following
             Proportion
             .
             But
             (
             by
             the
             10
             Art.
             )
             the
             Difference
             of
             the
             first
             Antecedent
             and
             Consequent
             ,
             to
             Difference
             of
             the
             second
             Antecedent
             and
             Consequent
             ,
             is
             as
             the
             first
             Antecedent
             to
             the
             second
             Antecedent
             ,
             that
             is
             ,
             as
             the
             first
             
             Term
             to
             the
             second
             ,
             or
             as
             the
             second
             to
             the
             third
             ,
             &c.
             in
             continuall
             Proportionals
             .
          
           
             19
             If
             there
             be
             three
             Continual
             Proportionals
             ,
             The
             Sum
             of
             the
             Extremes
             ,
             together
             with
             the
             Mean
             twice
             taken
             ;
             The
             Sum
             of
             the
             Mean
             and
             either
             of
             the
             Extremes
             ;
             and
             the
             same
             Extreme
             ,
             are
             Continual
             Proportionals
             .
             For
             let
             A.
             B.
             C
             ∝
             be
             Continual
             Proportionals
             .
             Seeing
             therefore
             A.
             B
             :
             :
             B.
             C
             are
             Proportionals
             ;
             by
             Composition
             also
             A+B
             .
             B
             :
             :
             B+C
             .
             C
             will
             be
             Proportionals
             ;
             and
             by
             Permutation
             A+B
             .
             B+C
             :
             :
             B.
             C
             will
             also
             be
             Proportionals
             ;
             and
             again
             by
             Composition
             A+2
             B+C
             .
             B+C
             :
             :
             B+C
             .
             C
             ;
             which
             was
             to
             be
             proved
             .
          
           
             20
             In
             four
             Continual
             Proportionals
             ,
             the
             greatest
             and
             the
             least
             put
             together
             ,
             is
             a
             greater
             quantity
             then
             the
             other
             two
             put
             together
             .
             Let
             A.
             B
             :
             :
             C.
             D
             be
             Continual
             Proportionals
             ;
             whereof
             let
             the
             greatest
             be
             A
             ,
             and
             the
             least
             be
             D
             ;
             I
             say
             A+D
             is
             greater
             then
             B+C
             .
             For
             (
             by
             the
             10
             Art.
             )
             A
             −
             B.
             C
             −
             D
             :
             :
             A.
             C
             are
             Proportionals
             ;
             and
             therefore
             A
             −
             B
             is
             (
             by
             the
             11
             Art.
             )
             greater
             then
             C
             −
             D.
             Add
             B
             on
             both
             sides
             ,
             and
             A
             will
             be
             greater
             then
             C+B
             −
             D.
             And
             again
             ,
             add
             D
             on
             both
             sides
             ,
             and
             A+D
             will
             be
             greater
             then
             B+C
             ;
             which
             was
             to
             be
             proved
             .
          
           
             21
             If
             there
             be
             four
             Proportionals
             ,
             the
             Extremes
             multiplyed
             into
             one
             another
             ,
             &
             the
             Means
             multiplied
             into
             one
             another
             wil
             make
             equal
             Products
             .
             Let
             A.
             B
             :
             :
             C.
             D
             be
             Proportionals
             ;
             I
             say
             A
             D
             is
             equal
             to
             B
             C.
             For
             the
             Proportion
             of
             A
             D
             to
             B
             C
             is
             compounded
             (
             by
             the
             13
             Art.
             )
             of
             the
             Proportions
             of
             A
             to
             B
             ,
             and
             D
             to
             C
             ,
             that
             is
             ,
             its
             Inverse
             B
             to
             A
             ;
             and
             therefore
             (
             by
             the
             14
             Art
             )
             this
             Compounded
             Proportion
             is
             the
             Proportion
             of
             Equality
             ;
             and
             therefore
             also
             the
             Proportion
             of
             A
             D
             to
             B
             C
             is
             the
             Proportion
             of
             Equality
             .
             Wherefore
             they
             are
             equal
             .
          
           
             22
             If
             there
             be
             four
             quantities
             ,
             and
             the
             Proportion
             of
             the
             first
             to
             the
             second
             be
             Duplicate
             of
             the
             Proportion
             of
             the
             third
             to
             the
             fourth
             ,
             the
             Product
             of
             the
             Extremes
             to
             the
             Product
             of
             the
             Means
             will
             be
             as
             the
             third
             to
             the
             fourth
             .
             Let
             the
             four
             Quantities
             be
             A
             ,
             B
             C
             and
             D
             ;
             and
             let
             the
             Proportion
             of
             A
             to
             B
             be
             Duplicate
             of
             the
             Proportion
             of
             C
             to
             D
             ,
             I
             say
             A
             D
             ,
             that
             is
             ,
             the
             Product
             of
             A
             into
             D
             ,
             is
             to
             BC
             ,
             that
             is
             ,
             to
             the
             Product
             of
             the
             Means
             ,
             as
             C
             to
             D.
             For
             seeing
             the
             Proportion
             of
             A
             to
             B
             is
             Duplicate
             of
             the
             proportion
             of
             C
             to
             D
             ,
             if
             it
             be
             as
             C
             to
             D
             ,
             so
             D
             to
             another
             E
             ,
             then
             A.
             B
             :
             :
             C.
             E
             will
             be
             Proportionals
             ;
             
             for
             the
             proportion
             of
             A
             to
             B
             is
             by
             supposition
             duplicate
             of
             the
             Proportion
             of
             C
             to
             D
             ;
             and
             C
             to
             E
             duplicate
             also
             of
             that
             of
             C
             to
             D
             by
             the
             Definition
             (
             15
             Art.
             )
             Wherefore
             (
             by
             the
             last
             Article
             )
             A
             E
             or
             A
             into
             E
             ,
             is
             equal
             to
             B
             C
             or
             B
             into
             C
             ;
             But
             (
             by
             the
             4
             Coroll
             .
             of
             the
             6
             Art.
             )
             A
             D
             is
             to
             A
             E
             as
             D
             to
             E
             ,
             that
             is
             ,
             as
             C
             to
             D
             ;
             and
             therefore
             A
             D
             is
             to
             B
             C
             (
             which
             as
             I
             have
             shewn
             ,
             is
             equal
             to
             A
             E
             )
             as
             C
             to
             D
             ;
             which
             was
             to
             be
             proved
             .
          
           
             Moreover
             ,
             If
             the
             proportion
             of
             the
             first
             A
             ,
             to
             the
             second
             B
             be
             triplicate
             of
             the
             proportion
             of
             the
             third
             C
             to
             the
             fourth
             D
             ,
             the
             Product
             of
             the
             Extremes
             to
             the
             product
             of
             the
             Means
             will
             be
             duplicate
             of
             the
             Proportion
             of
             the
             third
             to
             the
             fourth
             .
             For
             if
             it
             be
             as
             C
             to
             D
             so
             D
             to
             E
             ,
             and
             again
             ,
             as
             D
             to
             E
             so
             E
             to
             another
             F
             ,
             then
             the
             proportion
             of
             C
             to
             F
             will
             be
             triplicate
             of
             the
             proportion
             of
             C
             to
             D
             ;
             and
             consequently
             A.
             B
             :
             :
             C.
             F
             will
             be
             proportionals
             ,
             and
             A
             F
             equal
             to
             B
             C.
             But
             as
             A
             D
             to
             A
             F
             ,
             so
             is
             D
             to
             F
             ;
             and
             therefore
             also
             as
             A
             D
             to
             B
             C
             ,
             so
             D
             to
             F
             ,
             that
             is
             ,
             so
             C
             to
             E
             ;
             But
             the
             proportion
             of
             C
             to
             E
             is
             duplicate
             of
             the
             proportion
             of
             C
             to
             D
             ;
             Wherefore
             also
             the
             proportion
             of
             A
             D
             to
             B
             C
             is
             duplicate
             of
             that
             of
             C
             to
             D
             ,
             as
             was
             propounded
             .
          
           
             23
             If
             there
             be
             four
             Proportionals
             ,
             and
             a
             Mean
             be
             interposed
             betwixt
             the
             first
             and
             second
             ,
             and
             another
             betwixt
             the
             third
             and
             fourth
             ,
             the
             first
             of
             these
             Means
             will
             be
             to
             the
             second
             ,
             as
             the
             first
             of
             the
             Proportionals
             is
             to
             the
             third
             ,
             or
             as
             the
             second
             of
             them
             is
             to
             the
             fourth
             .
             For
             let
             A.
             B
             :
             :
             C.
             D
             be
             Proportionals
             ,
             and
             let
             E
             be
             a
             Mean
             betwixt
             A
             and
             B
             ,
             &
             F
             a
             Mean
             betwixt
             C
             and
             D
             ;
             I
             say
             A.
             C
             :
             :
             E.
             F
             are
             Proportionals
             .
             For
             the
             proportion
             of
             A
             to
             E
             is
             Subduplicate
             of
             the
             proportion
             of
             A
             to
             B
             ,
             or
             of
             C
             to
             D.
             Also
             the
             proportion
             of
             C
             to
             F
             is
             Subduplicate
             of
             that
             of
             C
             to
             D
             ;
             and
             therefore
             A.
             E
             :
             :
             C.
             F
             are
             Proportionals
             ;
             and
             by
             Permutation
             A.
             C
             :
             :
             E.
             F
             are
             also
             Proportionals
             ;
             which
             was
             to
             be
             proved
             .
          
           
           
             25
             If
             there
             be
             three
             continual
             Proportionals
             ,
             and
             again
             ,
             three
             other
             continual
             Proportionals
             which
             have
             the
             same
             Middle
             Term
             ,
             their
             Extremes
             will
             be
             in
             reciprocal
             Proportion
             .
             For
             let
             A.
             B.
             C
             :
             :
             and
             D.
             B.
             E
             :
             :
             be
             continual
             Proportionals
             ,
             I
             say
             A.
             D
             :
             :
             E.
             C
             shall
             be
             Proportionals
             .
             For
             the
             Proportion
             of
             A
             to
             D
             is
             compounded
             of
             the
             Proportions
             of
             A
             to
             B
             ,
             and
             of
             B
             to
             D
             ;
             and
             the
             Proportion
             of
             E
             to
             C
             is
             compounded
             of
             those
             of
             E
             to
             B
             ,
             that
             is
             ,
             of
             B
             to
             A
             ,
             and
             of
             B
             to
             C
             ,
             that
             is
             ,
             of
             A
             to
             B.
             Wherefore
             (
             by
             Equality
             )
             A.
             D
             :
             :
             E.
             C.
             are
             Proportionals
             .
          
           
           
           
             From
             the
             consideration
             hereof
             it
             is
             manifest
             ,
             that
             B
             ,
             that
             is
             ,
             A
             together
             with
             something
             else
             which
             is
             less
             then
             a
             fourth
             part
             of
             the
             difference
             of
             the
             Extremes
             A
             and
             E
             ,
             is
             less
             then
             F
             ,
             that
             is
             ,
             then
             the
             same
             A
             with
             something
             else
             which
             is
             equal
             to
             the
             said
             fourth
             part
             .
             Also
             ,
             that
             C
             ,
             that
             is
             A
             with
             something
             else
             which
             is
             less
             then
             two
             fourth
             parts
             of
             the
             said
             difference
             ,
             is
             less
             then
             G
             ,
             that
             is
             ,
             then
             A
             together
             with
             the
             said
             two
             fourths
             .
             And
             lastly
             ,
             that
             D
             which
             exceeds
             A
             by
             less
             then
             three
             fourths
             of
             the
             said
             difference
             ,
             is
             less
             then
             H
             ,
             which
             exceeds
             the
             same
             A
             by
             three
             entire
             fourths
             of
             the
             said
             difference
             .
             And
             in
             the
             same
             manner
             it
             would
             be
             if
             there
             were
             four
             Means
             ,
             saving
             that
             in
             stead
             of
             fourths
             of
             the
             difference
             of
             the
             Extremes
             ,
             we
             are
             to
             take
             fifth
             parts
             ;
             and
             so
             on
             .
          
           
             27
             Lemma
             .
             If
             a
             quantity
             being
             given
             ,
             first
             one
             quantity
             be
             both
             added
             to
             it
             and
             substracted
             from
             it
             ,
             and
             then
             another
             greater
             or
             lesse
             ,
             the
             proportion
             of
             the
             Remainder
             to
             the
             Aggregate
             ,
             is
             greater
             where
             the
             less
             quantity
             is
             added
             and
             substracted
             ,
             then
             where
             the
             greater
             quantity
             is
             added
             and
             substracted
             .
             Let
             B
             be
             added
             to
             ,
             and
             substracted
             from
             the
             quantity
             A
             ;
             so
             that
             A
             −
             B
             be
             the
             Remainder
             ,
             and
             A+B
             the
             Aggregate
             ;
             And
             again
             let
             C
             ,
             a
             greater
             quantitity
             then
             B
             be
             added
             to
             ,
             and
             substracted
             from
             the
             same
             A
             ,
             so
             that
             A
             −
             C
             be
             the
             Remainder
             and
             A+C
             the
             Aggregate
             ;
             I
             say
             A
             −
             B.
             A+B
             :
             :
             A
             −
             C.
             A+C
             will
             be
             an
             Hyperlogisme
             .
             For
             A
             −
             B.
             A
             :
             :
             A
             −
             C.
             A
             is
             an
             Hyperlogisme
             of
             a
             less
             Antecedent
             to
             the
             same
             Consequent
             ;
             and
             therefore
             A
             −
             B.
             A+B
             :
             :
             A
             −
             C.
             A+C
             is
             a
             much
             greater
             Hyperlogisme
             ,
             being
             made
             of
             a
             less
             Antecedent
             to
             a
             greater
             Consequent
             .
          
           
           
             
               Then
               by
               Composition
               we
               have
               this
               AB+AG
               .
               AB
               :
               :
               BG+GE
               (
               that
               is
               BE
               )
               .
               BG
               .
            
             
               And
               by
               taking
               the
               halves
               of
               the
               Antecedents
               this
               third
               ½AB+½AG
               .
               AB
               :
               :
               ½BG+½GE
               (
               that
               is
               BH
               )
               .
               BG
               .
            
             
               And
               by
               Conversion
               a
               fourth
               AB
               .
               ½AB+½AG
               :
               :
               BG
               .
               BH
            
             
               And
               by
               Division
               this
               fifth
               ½AB
               −
               ½AG
               .
               ½AB+½AG
               :
               :
               HG
               .
               BH
               .
            
             
               And
               by
               doubling
               the
               first
               Antecedent
               and
               the
               first
               Consequent
               AB
               −
               AG.
               AB+AG
               :
               :
               HG
               .
               BH
               .
            
             
               Also
               by
               the
               same
               method
               may
               be
               found
               out
               this
               Analogisme
               AB
               −
               AI.
               AB+AI
               :
               :
               KI
               .
               BK
               .
            
          
           
             Now
             seeing
             the
             proportion
             of
             AB
             to
             AE
             is
             greater
             then
             that
             of
             AB
             to
             AF
             ,
             the
             proportion
             of
             AB
             to
             AG
             ,
             which
             is
             half
             the
             greater
             proportion
             ,
             is
             greater
             then
             the
             proportion
             of
             AB
             to
             AI
             the
             half
             of
             the
             less
             Proportion
             ;
             and
             therefore
             AI
             is
             greater
             then
             AG.
             Wherefore
             the
             proportion
             of
             AB
             −
             AG
             to
             AB+AG
             (
             by
             the
             precedent
             Lemma
             )
             will
             be
             greater
             then
             the
             proportion
             of
             AB
             −
             AI
             to
             AB+AI
             ;
             &
             therefore
             also
             the
             proportion
             of
             HG
             to
             BH
             will
             be
             greater
             then
             that
             of
             KI
             to
             BK
             ,
             and
             much
             greater
             then
             the
             proportion
             of
             KI
             to
             BH
             ,
             which
             is
             greater
             then
             BK
             ,
             (
             for
             BH
             is
             the
             half
             of
             BE
             ,
             as
             BK
             is
             the
             half
             of
             BF
             ,
             which
             (
             by
             supposition
             )
             is
             less
             less
             then
             BE
             )
             .
             Wherefore
             HG
             is
             greater
             then
             KI
             ;
             which
             was
             to
             be
             proved
             .
          
           
             Coroll
             .
             It
             is
             manifest
             from
             hence
             ,
             that
             if
             any
             quantity
             be
             supposed
             to
             be
             divided
             into
             equal
             parts
             infinite
             in
             number
             ,
             the
             difference
             between
             the
             Arithmetical
             and
             Geometrical
             Means
             will
             be
             infinitely
             little
             ,
             that
             is
             ,
             none
             at
             all
             .
             And
             upon
             this
             foundation
             chiefly
             ,
             the
             Art
             of
             making
             those
             Numbers
             which
             are
             called
             Logarithmes
             seems
             to
             have
             been
             built
             .
          
           
             29
             If
             any
             number
             of
             quantities
             be
             propounded
             ,
             whether
             they
             be
             unequall
             ,
             or
             equall
             to
             one
             another
             ;
             and
             there
             be
             another
             quantity
             which
             multiplied
             by
             the
             number
             of
             the
             propounded
             quantities
             is
             equall
             to
             them
             all
             ;
             that
             other
             Quantity
             is
             a
             mean
             in
             Arithmeticall
             Proportion
             to
             all
             those
             propounded
             quantities
             .
          
        
         
           
           
             
               CHAP.
               XIV
            
             .
             Of
             Straight
             and
             
               Crooked
               ,
               Angle
            
             and
             Figure
             .
          
           
             
               
                 1
                 The
                 Definition
                 and
                 Properties
                 of
                 a
                 Straight
                 Line
              
               
                 2
                 The
                 Definition
                 and
                 Properties
                 of
                 a
                 Plain
                 Superficies
                 .
              
               
                 3
                 Several
                 sorts
                 of
                 Crooked
                 Lines
                 .
              
               
                 4
                 The
                 Definition
                 and
                 Properties
                 of
                 a
                 Circular
                 Line
                 .
              
               
                 5
                 The
                 Properties
                 of
                 a
                 Straight
                 Line
                 taken
                 in
                 a
                 Plain
                 .
              
               
                 6
                 The
                 Definition
                 of
                 Tangent
                 Lines
                 .
              
               
                 7
                 The
                 Definition
                 of
                 an
                 Angle
                 ,
                 and
                 the
                 kindes
                 thereof
                 .
              
               
                 8
                 In
                 Concentrick
                 Circles
                 ,
                 Arches
                 of
                 the
                 same
                 Angle
                 are
                 to
                 one
                 another
                 as
                 the
                 whole
                 Circumferences
                 are
                 .
              
               
                 9
                 The
                 Quantity
                 of
                 an
                 Angle
                 in
                 what
                 it
                 consists
                 .
              
               
                 10
                 The
                 Distinction
                 of
                 Angles
                 simply
                 so
                 called
                 .
              
               
                 11
                 Of
                 Straight
                 Lines
                 from
                 the
                 Center
                 of
                 a
                 Circle
                 to
                 a
                 Tangent
                 of
                 the
                 same
                 .
              
               
                 12
                 The
                 general
                 Definition
                 of
                 Parallels
                 ;
                 and
                 the
                 Properties
                 of
                 Straight
                 Parallels
                 .
              
               
                 13
                 The
                 Circumferences
                 of
                 Circles
                 are
                 to
                 one
                 another
                 as
                 their
                 Diameters
                 are
                 .
              
               
                 14
                 In
                 Triangles
                 ,
                 Straight
                 Lines
                 parallel
                 to
                 the
                 Bases
                 ,
                 are
                 to
                 one
                 another
                 ,
                 as
                 the
                 parts
                 of
                 the
                 Sides
                 which
                 they
                 cut
                 off
                 from
                 the
                 Vertex
                 .
              
               
                 15
                 By
                 what
                 Fraction
                 of
                 a
                 Straight
                 Line
                 the
                 Circumference
                 of
                 a
                 Circle
                 is
                 made
                 .
              
               
                 16
                 That
                 an
                 Angle
                 of
                 Contingence
                 is
                 Quantity
                 ,
                 but
                 of
                 a
                 Different
                 kinde
                 from
                 that
                 of
                 an
                 Angle
                 simply
                 so
                 called
                 ;
                 and
                 that
                 it
                 can
                 neither
                 add
                 nor
                 take
                 away
                 any
                 thing
                 from
                 the
                 same
                 .
              
               
                 17
                 That
                 the
                 Inclination
                 of
                 Plains
                 is
                 Angle
                 simply
                 so
                 called
                 .
              
               
                 18
                 A
                 Solid
                 Angle
                 what
                 it
                 is
                 .
              
               
                 19
                 What
                 is
                 the
                 Nature
                 of
                 Asymptotes
                 .
              
               
                 20
                 Situation
                 ,
                 by
                 what
                 it
                 is
                 determined
                 .
              
               
                 21
                 What
                 is
                 like
                 Situation
                 ;
                 What
                 is
                 Figure
                 ;
                 and
                 what
                 are
                 like
                 Figures
                 .
              
            
          
           
             1
             BEtween
             two
             points
             given
             ,
             the
             shortest
             Line
             is
             that
             ,
             whose
             extreme
             points
             cannot
             be
             drawn
             further
             asunder
             ,
             withour
             altering
             the
             quantity
             ,
             that
             is
             ,
             without
             altering
             the
             proportion
             of
             that
             line
             to
             any
             other
             line
             given
             .
             For
             the
             Magnitude
             of
             a
             Line
             is
             computed
             by
             the
             greatest
             distance
             which
             may
             be
             between
             its
             extreme
             points
             ;
             So
             that
             any
             one
             Line
             ,
             whether
             it
             be
             extended
             ,
             or
             bowed
             ,
             has
             alwayes
             one
             and
             the
             same
             Length
             ,
             because
             it
             can
             have
             but
             one
             greatest
             distance
             between
             its
             extreme
             points
             .
          
           
           
             And
             seeing
             the
             action
             by
             which
             a
             Straight
             Line
             is
             made
             Crooked
             ,
             or
             contrarily
             a
             Crooked
             Line
             is
             made
             Straight
             ,
             is
             nothing
             but
             the
             bringing
             of
             its
             extreme
             points
             neerer
             to
             one
             another
             ,
             or
             the
             setting
             of
             them
             further
             asunder
             ,
             a
             CROOKED
             Line
             may
             rightly
             be
             defined
             to
             be
             
               That
               ,
               whose
               extreme
               points
               may
               be
               understood
               to
               be
               drawn
               further
               asunder
            
             ;
             and
             a
             STRAIGHT
             Line
             to
             be
             
               That
               ,
               whose
               extreme
               points
               cannot
               be
               drawn
               further
               asunder
               ;
               and
               comparatively
               ,
               A
               more
               Crooked
               ,
               to
               be
               That
               line
               whose
               extreme
               points
               are
               neerer
               to
               one
               another
               then
               those
               of
               the
               other
               ,
               (
               supposing
               both
               the
               Lines
               to
               be
               of
               equal
               Length
               .
               )
            
             Now
             howsoever
             a
             Line
             be
             bowed
             ,
             it
             makes
             alwayes
             a
             Sinus
             or
             Cavity
             ,
             sometimes
             on
             one
             side
             ,
             sometimes
             on
             another
             ;
             So
             that
             the
             same
             Crooked
             Line
             may
             either
             have
             its
             whole
             Cavity
             on
             one
             Side
             onely
             ,
             or
             it
             may
             have
             it
             part
             on
             one
             side
             and
             part
             on
             other
             sides
             .
             Which
             being
             well
             understood
             ,
             it
             will
             be
             easie
             to
             understand
             the
             following
             Comparisons
             of
             Straight
             and
             Crooked
             Lines
             .
          
           
             First
             ,
             If
             a
             Straight
             &
             a
             Crooked
             Line
             have
             their
             Extreme
             points
             common
             ,
             the
             Crooked
             Line
             is
             longer
             then
             the
             Straight
             Line
             .
             For
             if
             the
             extreme
             points
             of
             the
             Crooked
             Line
             be
             drawn
             out
             to
             their
             greatest
             distance
             ,
             it
             wil
             be
             made
             a
             straight
             line
             ,
             of
             which
             that
             which
             was
             a
             Straight
             Line
             from
             the
             beginning
             will
             be
             but
             a
             part
             ;
             and
             therefore
             the
             Straight
             Line
             was
             shorter
             then
             the
             Crooked
             Line
             which
             had
             the
             same
             extreme
             points
             .
             And
             for
             the
             same
             reason
             ,
             if
             two
             Crooked
             Lines
             have
             their
             extreme
             points
             common
             ,
             and
             both
             of
             them
             have
             all
             their
             cavity
             on
             one
             and
             the
             same
             side
             ,
             the
             outermost
             of
             the
             two
             will
             be
             the
             longest
             Line
             .
          
           
             Secondly
             ,
             A
             Straight
             Line
             and
             a
             perpetually
             Crooked
             Line
             cānot
             be
             coincident
             ,
             no
             not
             in
             the
             least
             part
             .
             For
             if
             they
             should
             ,
             then
             not
             onely
             some
             Straight
             Line
             would
             have
             its
             extreme
             points
             common
             with
             some
             Crooked
             Line
             ,
             but
             also
             they
             would
             by
             reason
             of
             their
             coincidence
             ,
             be
             equal
             to
             one
             another
             ;
             which
             ,
             as
             I
             have
             newly
             shewn
             ,
             cannot
             be
             .
          
           
             Thirdly
             ,
             Between
             two
             points
             given
             there
             can
             be
             understood
             but
             one
             straight
             Line
             ;
             because
             there
             cannot
             be
             more
             then
             one
             least
             Interval
             or
             Length
             between
             the
             same
             points
             .
             For
             if
             there
             may
             be
             two
             ,
             they
             will
             either
             be
             coincident
             ,
             and
             so
             both
             of
             them
             will
             be
             one
             Straight
             Line
             ;
             or
             if
             they
             be
             not
             coincident
             ,
             then
             the
             application
             of
             one
             to
             the
             other
             by
             extension
             ,
             will
             make
             the
             extended
             Line
             
             have
             its
             extreme
             points
             at
             greater
             distance
             then
             the
             other
             ;
             and
             consequently
             it
             was
             Crooked
             from
             the
             beginning
             .
          
           
             Fourthly
             ,
             From
             this
             last
             it
             follows
             ,
             that
             two
             Straight
             Lines
             cannot
             include
             a
             Superficies
             .
             For
             if
             they
             have
             both
             their
             extreme
             points
             common
             ,
             they
             are
             coincident
             ;
             and
             if
             they
             have
             but
             one
             ,
             or
             neither
             of
             them
             common
             ,
             then
             at
             one
             ,
             or
             both
             ends
             ,
             the
             extreme
             points
             will
             be
             disjoyned
             ,
             and
             include
             no
             Superficies
             ,
             but
             leave
             all
             open
             and
             undetermined
             .
          
           
             Fifthly
             ,
             Every
             part
             of
             a
             Straight
             Line
             is
             a
             Straight
             Line
             .
             For
             seeing
             every
             part
             of
             a
             Straight
             Line
             is
             the
             least
             that
             can
             be
             drawn
             between
             its
             own
             extreme
             points
             ,
             if
             all
             the
             parts
             should
             not
             constitute
             a
             Straight
             Line
             ,
             they
             would
             all
             together
             be
             longer
             then
             the
             whole
             Line
             .
          
           
             2
             APLAIN
             ,
             or
             a
             
               Plain
               Superficies
               ,
               is
               that
               which
               is
               described
               by
               a
               Straight
               Line
               so
               moved
               that
               all
               the
               several
               points
               thereof
               describe
               several
               Straight
               Lines
               .
            
             A
             straight
             line
             therefore
             is
             necessarily
             all
             of
             it
             in
             the
             same
             Plain
             which
             it
             describes
             .
             Also
             the
             Straight
             Lines
             which
             are
             made
             by
             the
             points
             that
             describe
             a
             Plain
             ,
             are
             all
             of
             them
             in
             the
             Same
             Plain
             .
             Moreover
             ,
             if
             any
             Line
             whatsoever
             be
             moved
             in
             a
             Plain
             ,
             the
             Lines
             which
             are
             described
             by
             it
             are
             all
             of
             them
             in
             the
             same
             Plain
             .
          
           
             All
             other
             Superficies
             which
             are
             not
             Plain
             ,
             are
             Crooked
             ,
             that
             is
             ,
             are
             either
             Concave
             or
             Convex
             .
             And
             the
             same
             Comparisons
             which
             were
             made
             of
             Straight
             and
             Crooked
             Lines
             :
             may
             also
             be
             made
             of
             Plain
             and
             Crooked
             Superficies
             .
             For
          
           
             First
             ,
             If
             a
             Plain
             and
             a
             Crooked
             Superficies
             be
             terminated
             with
             the
             same
             Lines
             ,
             the
             Crooked
             Superficies
             is
             greater
             then
             the
             Plain
             Superficies
             .
             For
             if
             the
             Lines
             of
             which
             the
             Crooked
             Superficies
             consists
             be
             extended
             ,
             they
             will
             be
             found
             to
             be
             longer
             then
             those
             of
             which
             the
             Plain
             Superficies
             consists
             ,
             which
             cannot
             be
             extended
             because
             they
             are
             Straight
             .
          
           
             Secondly
             ,
             Two
             Superficies
             ,
             wherof
             the
             one
             is
             Plain
             ,
             and
             the
             other
             continually
             Crooked
             ,
             cannot
             be
             coincident
             ,
             no
             not
             in
             the
             least
             part
             .
             For
             if
             they
             were
             coincident
             they
             would
             be
             equal
             ;
             nay
             ,
             the
             same
             Superficies
             would
             be
             both
             Plain
             and
             Crooked
             ,
             which
             is
             impossible
             .
          
           
             Thirdly
             ,
             Within
             the
             same
             terminating
             Lines
             ,
             there
             can
             be
             no
             more
             then
             one
             Plain
             Superficies
             ;
             because
             there
             can
             be
             but
             one
             least
             Superficies
             within
             the
             same
             .
          
           
           
             Fourthly
             ,
             No
             number
             of
             Plain
             Superficies
             can
             include
             a
             Solid
             ,
             unless
             more
             then
             two
             of
             them
             end
             in
             a
             Common
             Vertex
             .
             For
             if
             two
             Plains
             have
             both
             the
             same
             terminating
             Lines
             ,
             they
             are
             coincident
             ,
             that
             is
             ,
             they
             are
             but
             one
             Superficies
             ;
             and
             if
             their
             terminating
             Lines
             be
             not
             the
             same
             ,
             they
             leave
             one
             or
             more
             sides
             open
             .
          
           
             Fifthly
             ,
             Every
             part
             of
             a
             Plain
             Superficies
             is
             a
             Plain
             Superficies
             .
             For
             seeing
             the
             whole
             Plain
             Superficies
             is
             the
             least
             of
             all
             those
             that
             have
             the
             same
             terminating
             Lines
             ;
             and
             also
             every
             part
             of
             the
             same
             Superficies
             is
             the
             least
             of
             all
             those
             that
             are
             terminated
             with
             the
             same
             Lines
             ;
             if
             every
             part
             should
             not
             constitute
             a
             Plain
             Superficies
             ,
             all
             the
             parts
             put
             together
             would
             not
             be
             equal
             to
             the
             whole
             .
          
           
             3
             Of
             Straightness
             ,
             whether
             it
             be
             in
             Lines
             ,
             or
             in
             Superficies
             there
             is
             but
             one
             kinde
             ;
             but
             of
             Crookedness
             there
             are
             many
             kindes
             ;
             for
             of
             Crooked
             Magnitudes
             ,
             some
             are
             Congruous
             ,
             that
             is
             ,
             are
             coincident
             when
             they
             are
             applyed
             to
             one
             another
             ;
             others
             are
             Incongruous
             .
             Again
             ,
             some
             are
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             or
             Uniform
             ,
             that
             is
             ,
             have
             their
             parts
             howsoever
             taken
             ,
             congruous
             to
             one
             another
             ;
             others
             are
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             or
             of
             
               several
               Forms
            
             .
             Moreover
             ,
             of
             such
             as
             are
             Crooked
             ,
             some
             are
             
               Continually
               Crooked
            
             ,
             others
             
               have
               parts
               which
               are
               not
               Crooked
               .
            
          
           
             4
             If
             a
             Straight
             Line
             be
             moved
             in
             a
             Plain
             ,
             in
             such
             manner
             ,
             that
             while
             one
             end
             of
             it
             stands
             still
             ,
             the
             whole
             Line
             be
             carried
             round
             about
             ,
             til
             it
             come
             again
             into
             the
             same
             place
             from
             whence
             it
             was
             first
             moved
             ,
             it
             will
             describe
             a
             plain
             Superficies
             ,
             which
             will
             be
             terminated
             every
             way
             by
             that
             Crooked
             Line
             which
             is
             made
             by
             that
             end
             of
             the
             Straight
             Line
             which
             was
             carried
             round
             .
             Now
             this
             Superficies
             is
             called
             a
             CIRCLE
             ;
             and
             of
             this
             Circle
             ,
             the
             Unmoved
             Point
             ,
             is
             the
             the
             Center
             ;
             the
             Crooked
             Line
             which
             terminates
             it
             ,
             the
             Perimeter
             ;
             and
             every
             part
             of
             that
             Crooked
             Line
             ,
             a
             Circumference
             or
             Arch
             ;
             the
             straight
             Line
             which
             generated
             the
             Circle
             ,
             is
             the
             Semidiameter
             or
             Radius
             ;
             and
             any
             straight
             Line
             which
             passeth
             through
             the
             Center
             ,
             and
             is
             terminated
             on
             both
             sides
             in
             the
             Circumference
             is
             called
             the
             Diameter
             .
             Moreover
             ,
             every
             point
             of
             the
             Radius
             which
             describes
             the
             Circle
             ,
             describes
             in
             the
             same
             time
             it
             s
             own
             Perimeter
             ,
             terminating
             its
             own
             Circle
             ,
             which
             is
             said
             to
             be
             Concentrick
             to
             all
             the
             other
             Circles
             ,
             because
             this
             and
             all
             those
             have
             one
             common
             Center
             .
          
           
             Wherefore
             in
             every
             Circle
             ,
             all
             Straight
             Lines
             from
             the
             Center
             
             to
             the
             Circumference
             are
             equal
             .
             For
             they
             are
             all
             coincident
             with
             the
             Radius
             which
             generates
             the
             Circle
             .
          
           
             Also
             the
             Diameter
             divides
             both
             the
             Perimeter
             and
             the
             Circle
             it self
             into
             two
             equal
             parts
             .
             For
             if
             those
             two
             parts
             be
             applyed
             to
             one
             another
             ,
             and
             the
             Semiperimeters
             be
             coincident
             ,
             then
             seeing
             they
             have
             one
             common
             Diameter
             ,
             they
             will
             be
             equal
             ;
             and
             the
             Semicircles
             will
             be
             equal
             also
             ;
             for
             these
             also
             will
             be
             coincident
             .
             But
             if
             the
             Semiperimeters
             be
             not
             coincident
             ,
             then
             some
             one
             straight
             Line
             which
             passes
             through
             the
             Center
             (
             which
             Center
             is
             in
             the
             Diameter
             )
             will
             be
             cut
             by
             them
             in
             two
             points
             .
             Wherefore
             ,
             seeing
             all
             the
             straight
             Lines
             from
             the
             Center
             to
             the
             Circumference
             are
             equal
             ,
             a
             part
             of
             the
             same
             straight
             Line
             will
             be
             equal
             to
             the
             whole
             ;
             which
             is
             impossible
             .
          
           
             For
             the
             same
             reason
             the
             Perimeter
             of
             a
             Circle
             will
             be
             Uniform
             ,
             that
             is
             ,
             any
             one
             part
             of
             it
             will
             be
             coincident
             with
             any
             other
             equal
             part
             of
             the
             same
             .
          
           
             5
             From
             hence
             may
             be
             collected
             this
             property
             of
             a
             Straight
             Line
             ,
             namely
             ,
             that
             it
             is
             all
             conteined
             in
             that
             Plain
             which
             conteins
             both
             its
             extreme
             points
             .
             For
             seeing
             both
             its
             extreme
             points
             are
             in
             the
             Plain
             ,
             that
             Straight
             Line
             which
             describes
             the
             Plain
             will
             pass
             through
             them
             both
             ;
             and
             if
             one
             of
             them
             be
             made
             a
             Center
             ,
             and
             at
             the
             distance
             between
             both
             ,
             a
             Circumference
             be
             described
             ,
             whose
             Radius
             is
             the
             Straight
             Line
             which
             describes
             the
             Plain
             ,
             that
             Circumference
             will
             pass
             through
             the
             other
             point
             .
             Wherefore
             between
             the
             two
             propounded
             points
             ,
             there
             is
             one
             straight
             line
             (
             by
             the
             Definition
             of
             a
             Circle
             )
             conteined
             wholly
             in
             the
             propounded
             Plain
             ;
             and
             therefore
             if
             another
             straight
             Line
             might
             be
             drawn
             between
             the
             same
             points
             ,
             and
             yet
             not
             be
             conteined
             in
             the
             same
             Plain
             ,
             it
             would
             follow
             ,
             that
             between
             two
             points
             two
             straight
             lines
             may
             be
             drawn
             ;
             which
             has
             been
             demonstrated
             to
             be
             impossible
             .
          
           
             It
             may
             also
             be
             collected
             ,
             That
             if
             two
             Plains
             cut
             one
             another
             ,
             their
             common
             section
             will
             be
             a
             straight
             Line
             .
             For
             the
             two
             extreme
             points
             of
             the
             intersection
             are
             in
             both
             the
             intersecting
             Plains
             ;
             and
             between
             those
             points
             a
             straight
             Line
             may
             be
             drawn
             ;
             but
             a
             straight
             Line
             between
             any
             two
             points
             is
             in
             the
             same
             Plain
             in
             which
             the
             Points
             are
             ;
             and
             seeing
             these
             are
             in
             both
             the
             Plains
             ,
             the
             straight
             line
             which
             connects
             them
             will
             also
             be
             in
             both
             the
             same
             Plains
             ,
             and
             therefore
             it
             is
             the
             cōmon
             section
             of
             both
             .
             And
             every
             other
             Line
             that
             
             can
             be
             drawn
             between
             those
             points
             will
             be
             either
             coincident
             with
             that
             Line
             ,
             that
             is
             ,
             it
             will
             be
             the
             Same
             Line
             ;
             or
             it
             will
             not
             be
             coincident
             ;
             and
             then
             it
             wil
             be
             in
             neither
             ,
             or
             but
             in
             one
             of
             those
             Plains
             .
          
           
             As
             a
             straight
             Line
             may
             be
             understood
             to
             be
             moved
             round
             about
             whilest
             one
             end
             thereof
             remains
             fixed
             ,
             as
             the
             Center
             ;
             so
             in
             like
             manner
             it
             is
             easie
             to
             understand
             ,
             that
             a
             Plain
             may
             be
             circumduced
             about
             a
             straight
             line
             ,
             whilest
             the
             straight
             line
             remaines
             still
             in
             one
             and
             the
             same
             place
             ,
             as
             the
             Axis
             of
             that
             motion
             .
             Now
             from
             hence
             it
             is
             manifest
             ,
             that
             any
             three
             Points
             are
             in
             some
             one
             Plain
             .
             For
             as
             any
             two
             Points
             ,
             if
             they
             be
             connected
             by
             a
             straight
             Line
             ,
             are
             understood
             to
             be
             in
             the
             same
             Plaine
             in
             which
             the
             straight
             Line
             is
             ;
             so
             ,
             if
             that
             Plaine
             be
             circumduced
             about
             the
             same
             straight
             Line
             ,
             it
             will
             in
             its
             revolution
             take
             in
             any
             third
             Point
             ,
             howsoever
             it
             be
             situate
             ;
             and
             then
             the
             three
             Points
             will
             be
             all
             in
             that
             Plaine
             ;
             and
             consequently
             the
             three
             straight
             Lines
             which
             connect
             those
             Points
             ,
             will
             also
             be
             in
             the
             same
             Plain
             .
          
           
             6
             Two
             Lines
             are
             said
             to
             Touch
             one
             another
             ,
             which
             being
             both
             drawne
             to
             one
             and
             the
             same
             point
             ,
             will
             not
             cut
             one
             another
             ,
             though
             they
             be
             produced
             ,
             produced
             I
             say
             in
             the
             same
             manner
             in
             which
             they
             were
             generated
             .
             And
             therefore
             if
             two
             straight
             Lines
             touch
             one
             another
             in
             any
             one
             point
             ,
             they
             wil
             be
             contiguous
             through
             their
             whole
             length
             .
             Also
             two
             Lines
             continually
             crooked
             wil
             do
             the
             same
             ,
             if
             they
             be
             congruous
             ,
             and
             be
             applyed
             to
             one
             another
             according
             to
             their
             congruity
             ;
             otherwise
             ,
             if
             they
             be
             incongruously
             applyed
             ,
             they
             will
             ,
             as
             all
             other
             crooked
             Lines
             ,
             touch
             one
             another
             (
             where
             they
             touch
             )
             but
             in
             one
             point
             onely
             .
             Which
             is
             manifest
             from
             this
             ,
             that
             there
             can
             be
             no
             congruity
             between
             a
             straight
             line
             and
             a
             line
             that
             is
             continually
             crooked
             ;
             for
             otherwise
             the
             same
             line
             might
             be
             both
             straight
             and
             crooked
             .
             Besides
             when
             a
             straight
             line
             touches
             a
             crooked
             line
             ,
             if
             the
             straight
             line
             be
             never
             so
             little
             moved
             about
             upon
             the
             point
             of
             contact
             ,
             it
             will
             cut
             the
             crooked
             line
             ;
             for
             seeing
             it
             touches
             it
             but
             in
             one
             point
             ,
             if
             it
             incline
             any
             way
             ,
             it
             will
             do
             more
             then
             touch
             it
             ;
             that
             is
             ,
             it
             will
             either
             be
             congruous
             to
             it
             ,
             or
             it
             will
             cut
             it
             ;
             but
             it
             cannot
             be
             congruous
             to
             it
             ;
             and
             therefore
             it
             will
             cut
             it
             .
          
           
             7
             An
             Angle
             ,
             according
             to
             the
             most
             general
             acception
             of
             the
             word
             ,
             may
             be
             thus
             defined
             ;
             
               When
               two
               Lines
               ,
               or
               many
               Superficies
               concurre
               in
               one
               sole
               point
               ,
               and
               diverge
               every
               where
               else
               ,
               the
               quantity
               of
            
             
             
               that
               divergence
               is
               an
            
             ANGLE
             .
             And
             an
             Angle
             is
             of
             two
             ●orts
             ;
             for
             first
             it
             may
             be
             made
             by
             the
             concurrence
             of
             Lines
             ,
             and
             then
             it
             is
             a
             
               Superficiall
               Angle
            
             ;
             or
             by
             the
             concurrence
             of
             Superficies
             ,
             and
             then
             it
             is
             called
             a
             
               Solid
               Angle
            
             .
          
           
             Again
             ,
             from
             the
             two
             wayes
             by
             which
             two
             lines
             may
             diverge
             from
             one
             another
             ,
             Superficial
             Angles
             are
             divided
             into
             two
             kindes
             .
             For
             two
             straight
             lines
             which
             are
             applyed
             to
             one
             another
             ,
             and
             are
             contiguous
             in
             their
             whole
             length
             ,
             may
             be
             separated
             or
             pulied
             open
             in
             such
             manner
             ,
             that
             their
             concurrence
             in
             one
             point
             will
             still
             remain
             ;
             And
             this
             Separation
             or
             Opening
             may
             be
             either
             by
             Circular
             Motion
             ,
             the
             Center
             whereof
             is
             their
             point
             of
             concurrence
             ,
             and
             the
             Lines
             will
             still
             ret●in
             their
             straightness
             ,
             the
             quantity
             of
             which
             Separation
             ,
             or
             Divergence
             is
             an
             Angle
             simply
             so
             called
             ;
             Or
             they
             may
             be
             separated
             by
             continual
             Flexion
             or
             Curvation
             in
             every
             imaginable
             point
             ;
             and
             the
             quantity
             of
             this
             Separation
             is
             that
             which
             is
             called
             an
             
               Angle
               of
               Contingence
            
             .
          
           
             Besides
             of
             Superficial
             Angles
             simply
             so
             called
             ,
             those
             which
             are
             in
             a
             plain
             Superficies
             are
             Plain
             ;
             and
             those
             which
             are
             not
             plain
             ,
             are
             denominated
             from
             the
             Superficies
             in
             which
             they
             are
             ▪
          
           
             Lastly
             ,
             those
             are
             
               Straight-lined
               Angles
            
             which
             are
             made
             by
             straight
             lines
             ;
             as
             those
             which
             are
             made
             by
             crooked
             lines
             are
             Crooked-lined
             ;
             and
             those
             which
             are
             made
             both
             of
             straight
             and
             crooked
             lines
             ,
             are
             
               Mixed
               Angles
            
             .
          
           
             8
             Two
             Arches
             inter
             cepted
             between
             two
             Radii
             of
             Concentrick
             Circles
             ,
             have
             the
             same
             proportion
             to
             one
             another
             ,
             which
             their
             whole
             Perimeters
             have
             to
             one
             another
             .
             For
             let
             the
             point
             A
             (
             in
             the
             first
             Figure
             )
             be
             the
             Center
             of
             the
             two
             Circles
             B
             C
             D
             and
             E
             F
             G
             ;
             in
             which
             the
             Radii
             A
             E
             B
             and
             A
             F
             C
             intercept
             the
             Arches
             B
             C
             and
             E
             F
             ;
             I
             say
             the
             proportion
             of
             the
             Arch
             B
             C
             to
             the
             Arch
             E
             F
             ,
             is
             the
             same
             with
             that
             of
             the
             Perimet
             B
             C
             D
             to
             the
             Perimerer
             er
             E
             F
             G.
             For
             if
             the
             Radius
             A
             F
             C
             be
             understood
             to
             be
             moved
             about
             the
             Center
             A
             with
             Circular
             &
             Uniform
             Motion
             ,
             that
             is
             ,
             with
             equal
             Swiftness
             every
             where
             ,
             the
             point
             C
             will
             in
             a
             certain
             time
             describe
             the
             Perimeter
             B
             C
             D
             ,
             and
             in
             a
             part
             of
             that
             time
             the
             Arch
             B
             C
             ;
             and
             because
             the
             Velocities
             are
             equal
             by
             which
             both
             the
             Arch
             and
             the
             whole
             Perimeter
             are
             described
             ,
             the
             proportion
             of
             the
             magnitude
             of
             the
             Perimeter
             B
             C
             D
             to
             the
             magnitude
             of
             the
             Arch
             B
             C
             ,
             is
             determined
             by
             nothing
             
             but
             the
             difference
             of
             the
             times
             in
             which
             the
             Perimeter
             and
             the
             Arch
             are
             described
             .
             But
             both
             the
             Perimeters
             are
             described
             in
             one
             and
             the
             same
             time
             ,
             and
             both
             the
             Arches
             in
             one
             and
             the
             same
             time
             ;
             and
             therefore
             the
             proportions
             of
             the
             Perimeter
             B
             C
             D
             to
             the
             Arch
             B
             C
             ,
             and
             of
             the
             Perimeter
             E
             F
             G
             to
             the
             Arch
             E
             F
             ,
             are
             both
             determined
             by
             the
             same
             cause
             .
             Wherefore
             B
             C
             D
             ,
             B
             C
             :
             :
             E
             F
             G.
             E
             F
             are
             Proportionals
             (
             by
             the
             6
             Art.
             of
             the
             last
             Chapter
             )
             ,
             and
             by
             Permutation
             B
             C
             D.
             E
             F
             G
             :
             :
             B
             C.
             E
             F
             will
             also
             be
             Proportionals
             ,
             which
             was
             to
             be
             demonstrated
             .
          
           
             9
             Nothing
             is
             contributed
             towards
             the
             Quantity
             of
             an
             Angle
             ,
             neither
             by
             the
             Length
             ,
             nor
             by
             the
             Equality
             ,
             nor
             by
             the
             Inequality
             of
             the
             lines
             which
             comprehend
             it
             .
             For
             the
             lines
             A
             B
             and
             A
             C
             comprehend
             the
             same
             Angle
             which
             is
             comprehended
             by
             the
             lines
             A
             E
             and
             A
             F
             ,
             or
             A
             B
             and
             A
             F.
             Nor
             is
             an
             Angle
             either
             increased
             or
             diminished
             by
             the
             absolute
             quantity
             of
             the
             Arch
             which
             subtends
             the
             same
             ;
             for
             both
             the
             greater
             Arch
             B
             C
             ,
             and
             the
             lesser
             Arch
             E
             F
             are
             subtended
             to
             the
             same
             Angle
             .
             But
             the
             quantity
             of
             an
             Angle
             is
             estimated
             by
             the
             quantity
             of
             the
             subtending
             Arch
             compared
             with
             the
             quantity
             of
             the
             whole
             Perimeter
             .
             And
             therefore
             the
             Quantity
             of
             an
             Angle
             simply
             so
             called
             may
             be
             thus
             defined
             .
          
           
             
               The
               Quantity
               of
               an
               Angle
               ,
               is
               an
               Arch
               or
               Circumference
               of
               a
               Circle
               ,
               determined
               by
               its
               proportion
               to
               the
               whole
               Perimeter
               .
            
             So
             that
             when
             an
             Arch
             is
             intercepted
             between
             two
             straight
             lines
             drawn
             from
             the
             Center
             ,
             look
             how
             great
             a
             portion
             that
             Arch
             is
             of
             the
             whole
             Perimeter
             ,
             so
             great
             is
             the
             Angle
             .
             From
             whence
             it
             may
             be
             understood
             ,
             that
             when
             the
             lines
             which
             contein
             an
             Angle
             are
             straight
             lines
             ,
             the
             quantity
             of
             that
             Angle
             may
             be
             taken
             at
             any
             distance
             from
             the
             Center
             .
             But
             if
             one
             ,
             or
             both
             of
             the
             conteining
             lines
             be
             crooked
             ,
             then
             the
             quantity
             of
             the
             Angle
             is
             to
             be
             taken
             in
             the
             least
             distance
             from
             the
             Center
             ,
             or
             from
             their
             concurrence
             ;
             for
             the
             least
             distance
             is
             to
             be
             considered
             as
             a
             straight
             line
             ,
             seeing
             no
             crooked
             line
             can
             be
             imagined
             so
             little
             ,
             but
             that
             there
             may
             he
             a
             less
             straight
             line
             .
             And
             although
             the
             least
             straight
             line
             cannot
             be
             given
             ,
             because
             the
             least
             given
             line
             may
             still
             be
             divided
             ,
             yet
             we
             may
             come
             to
             a
             part
             so
             small
             ,
             as
             is
             not
             at
             all
             considerable
             ;
             which
             we
             call
             a
             Point
             .
             And
             this
             point
             may
             be
             understood
             to
             be
             in
             a
             straight
             line
             which
             touches
             a
             crooked
             line
             ;
             for
             an
             Angle
             is
             generated
             by
             separating
             by
             circular
             motion
             one
             straight
             line
             from
             another
             which
             touches
             it
             ,
             (
             as
             has
             been
             said
             above
             in
             the
             
             7th
             Article
             )
             .
             Wherefore
             an
             Angle
             which
             two
             crooked
             lines
             make
             is
             the
             same
             with
             that
             which
             is
             made
             by
             two
             straight
             lines
             which
             touch
             them
             .
          
           
             10
             From
             hence
             it
             follows
             ,
             that
             
               Vertical
               Angles
            
             ,
             such
             as
             are
             ABC
             ,
             D
             B
             F
             (
             in
             the
             second
             Figure
             )
             are
             equal
             to
             one
             another
             .
             For
             if
             from
             the
             two
             Semiperimeters
             D
             A
             C
             ,
             F
             D
             A
             which
             are
             equal
             to
             one
             another
             ,
             the
             common
             Arch
             D
             A
             be
             taken
             away
             ,
             the
             remayning
             Arches
             A
             C
             ,
             D
             F
             will
             be
             equal
             to
             one
             another
             .
          
           
             Another
             distinction
             of
             Angles
             is
             ,
             into
             Right
             and
             
               Oblique
               .
               A
               Right
               Angle
               is
               that
               ,
               whose
               quantity
               is
               the
               fourth
               part
               of
               the
               Perimeter
               .
            
             And
             the
             lines
             which
             make
             a
             Right
             Angle
             are
             said
             to
             be
             Perpendicular
             to
             one
             another
             .
             Also
             of
             Oblique
             Angles
             ,
             that
             which
             is
             greater
             then
             a
             Right
             ,
             is
             called
             an
             
               Obtuse
               Angle
            
             ;
             and
             that
             which
             is
             less
             ,
             an
             
               Acute
               Angle
            
             .
             From
             whence
             it
             follows
             ,
             that
             all
             the
             Angles
             that
             can
             possibly
             be
             made
             at
             one
             and
             the
             same
             Point
             ,
             together
             taken
             ,
             are
             equal
             to
             four
             Right
             Angles
             ;
             because
             the
             quantities
             of
             them
             all
             put
             together
             make
             the
             whole
             Perimeter
             .
             Also
             ,
             that
             all
             the
             Angles
             which
             are
             made
             on
             one
             side
             of
             a
             straight
             line
             from
             any
             one
             point
             taken
             in
             the
             same
             ,
             are
             equal
             to
             two
             Right
             Angles
             ;
             for
             if
             that
             point
             be
             made
             the
             Center
             ,
             that
             straight
             line
             will
             be
             the
             Diameter
             of
             a
             Circle
             ,
             by
             whose
             Circumference
             the
             quantity
             of
             an
             Angle
             is
             determined
             ;
             and
             that
             Diameter
             will
             divide
             the
             Perimeter
             into
             two
             equal
             parts
             .
          
           
             11
             If
             a
             Tangent
             be
             made
             the
             Diameter
             of
             a
             Circle
             whose
             Center
             is
             the
             point
             of
             Contact
             ,
             a
             straight
             line
             drawn
             from
             the
             Center
             of
             the
             former
             Circle
             to
             the
             Center
             of
             the
             later
             Circle
             ,
             will
             make
             two
             Angles
             with
             the
             Tangent
             (
             that
             is
             ,
             with
             the
             Diameter
             of
             the
             later
             Circle
             )
             equal
             to
             two
             Right
             Angles
             (
             by
             the
             last
             Article
             )
             .
             And
             because
             (
             by
             the
             6th
             Article
             )
             the
             Tangent
             has
             on
             both
             sides
             equal
             inclination
             to
             the
             Circle
             ,
             each
             of
             them
             will
             be
             a
             Right
             Angle
             ;
             as
             also
             the
             Semidiameter
             will
             be
             perpendicular
             to
             the
             same
             Tangent
             .
             Moreover
             ,
             the
             Semidiameter
             ,
             in
             as
             much
             as
             it
             is
             the
             Semidiameter
             ,
             is
             the
             least
             straight
             line
             which
             can
             be
             drawn
             frō
             the
             Center
             to
             the
             Tangent
             ;
             and
             every
             other
             straight
             line
             that
             reaches
             the
             Tangent
             will
             pass
             out
             of
             the
             Circle
             ,
             and
             will
             therefore
             be
             greater
             then
             the
             Semidiameter
             .
             In
             like
             manner
             ,
             of
             all
             the
             straight
             lines
             which
             may
             be
             drawn
             from
             the
             Center
             to
             the
             Tangent
             ,
             that
             is
             the
             greatest
             which
             makes
             the
             greatest
             Angle
             with
             the
             Perpenpicular
             ;
             which
             
             will
             be
             manifest
             ,
             if
             about
             the
             same
             Center
             another
             Circle
             be
             described
             ,
             whose
             Semidiameter
             is
             a
             straight
             line
             taken
             neerer
             to
             the
             Perpendicular
             ,
             and
             there
             be
             drawn
             a
             Perpendicular
             (
             that
             is
             ,
             a
             Tangent
             )
             to
             the
             same
             .
          
           
             From
             whence
             it
             is
             also
             manifest
             ,
             that
             if
             two
             straight
             lines
             which
             make
             equal
             Angles
             on
             either
             side
             of
             the
             Perpendicular
             ,
             be
             produced
             to
             the
             Tangent
             ,
             they
             will
             be
             equal
             .
          
           
             12
             There
             is
             in
             Euclide
             a
             Definition
             of
             Straight-lined
             Parallels
             ;
             but
             I
             do
             not
             find
             that
             Parallels
             in
             general
             ,
             are
             any
             where
             defined
             ;
             and
             therefore
             for
             an
             Universal
             definition
             of
             them
             ,
             I
             say
             ,
             that
             
               Any
               two
               lines
               whatsoever
               (
               Straight
               or
               Crooked
               )
               as
               also
               any
               two
               Superficies
               ,
               are
            
             PARALLEL
             ,
             
               when
               two
               equal
               straight
               lines
               wheresoever
               they
               fall
               upon
               them
               ,
               make
               always
               equal
               Angles
               with
               each
               of
               them
               .
            
             From
             which
             Definition
             it
             follows
             .
          
           
             First
             ,
             that
             any
             two
             straight
             lines
             not
             enclined
             opposite
             wayes
             ,
             falling
             upon
             two
             other
             straight
             lines
             which
             are
             Parallel
             ,
             and
             intercepting
             equal
             parts
             in
             both
             of
             them
             ,
             are
             themselves
             also
             equal
             and
             Parallel
             .
             As
             if
             AB
             and
             CD
             (
             in
             the
             third
             Figure
             )
             enclined
             both
             the
             same
             way
             (
             fall
             upon
             the
             Parallels
             AC
             and
             BD
             ,
             and
             AC
             and
             BD
             be
             equal
             ,
             AB
             and
             CD
             will
             also
             be
             equal
             and
             Parallel
             .
             For
             the
             Perpendiculars
             BE
             and
             DF
             being
             drawn
             ,
             the
             Right
             Angles
             EBD
             and
             FDH
             will
             be
             equal
             .
             Wherefore
             seeing
             EF
             and
             BD
             are
             parallel
             ,
             the
             angles
             EBA
             and
             FDC
             will
             be
             equal
             .
             Now
             if
             DC
             be
             not
             equal
             to
             BA
             ,
             let
             any
             other
             straight
             line
             equal
             to
             BA
             be
             drawn
             from
             the
             point
             D
             ;
             which
             seeing
             it
             cannot
             fall
             upon
             the
             point
             C
             ,
             let
             it
             fall
             upon
             G.
             Whereore
             AG
             will
             be
             either
             greater
             or
             less
             then
             BD
             ;
             and
             therefore
             the
             angles
             EBA
             and
             FDG
             are
             not
             equal
             ,
             as
             was
             supposed
             .
             Wherefore
             AB
             and
             CD
             are
             equal
             ;
             which
             is
             the
             first
             .
          
           
             Again
             ,
             because
             they
             make
             equal
             Angles
             with
             the
             Perpendiculars
             BE
             and
             DF
             ;
             therefore
             the
             Angle
             CDH
             will
             be
             equal
             to
             the
             Angle
             ABD
             ,
             and
             (
             by
             the
             Definition
             of
             Parallels
             )
             AB
             and
             CD
             will
             be
             parallel
             ;
             which
             is
             the
             second
             .
          
           
             
               That
               Plain
               which
               is
               included
               both
               wayes
               within
               parallel
               lines
               is
               called
               a
            
             PARALLELOGRAM
             .
          
           
             
               1
               Corollary
               .
               From
               this
               last
               it
               follows
               ,
               That
               the
               Angles
               ABD
               and
               CDH
               are
               equal
               ;
               that
               is
               ,
               that
               a
               straight
               line
               (
               as
               BH
               )
               falling
               upon
               two
               Parallels
               (
               as
               AB
               and
               CD
               )
               makes
               the
               internal
               Angle
               ABD
               equal
               to
               the
               external
               and
               opposite
               Angle
               CDH
               .
            
             
               
               2
               Coroll
               .
               And
               from
               hence
               again
               it
               follows
               ,
               that
               a
               straight
               line
               falling
               upon
               two
               Parallels
               ,
               makes
               the
               alternate
               Angles
               equal
               ;
               that
               is
               ,
               the
               Angle
               AGF
               (
               in
               the
               fourth
               figure
               )
               equal
               to
               the
               Angle
               GFD
               .
               For
               seeing
               GFD
               is
               equal
               to
               the
               external
               opposite
               Angle
               EGB
               ,
               it
               will
               be
               also
               equal
               to
               its
               vertical
               Angle
               AGF
               ,
               which
               is
               alternate
               to
               GFD
               .
            
             
               3
               Coroll
               .
               That
               the
               internal
               Angles
               on
               the
               same
               side
               of
               the
               line
               FG
               are
               equal
               to
               two
               Right
               Angles
               .
               For
               the
               Angles
               at
               F
               ,
               namely
               GFC
               and
               GFD
               are
               equal
               to
               two
               Right
               Angles
               .
               But
               GFD
               is
               equal
               to
               its
               alternate
               Angle
               AGF
               .
               Wherefore
               both
               the
               Angles
               GFC
               and
               AGF
               which
               are
               internal
               on
               the
               same
               side
               of
               the
               line
               FG
               ,
               are
               equal
               to
               two
               Right
               Angles
               .
            
             
               4
               Coroll
               .
               That
               the
               three
               Angles
               of
               a
               straight-lined
               plain
               Triangle
               are
               equal
               to
               two
               Right
               Angles
               ;
               and
               any
               side
               being
               produced
               ,
               the
               external
               Angle
               will
               be
               equal
               to
               the
               two
               opposite
               internal
               Angles
               .
               For
               if
               there
               be
               drawn
               by
               the
               Vertex
               of
               the
               plain
               Triangle
               ABC
               (
               figure
               5.
               )
               a
               Parallel
               to
               any
               of
               the
               sides
               ,
               as
               to
               AB
               ,
               the
               Angles
               A
               and
               B
               will
               be
               equal
               to
               their
               alternate
               Angles
               E
               &
               F
               ,
               &
               the
               Angle
               C
               is
               common
               .
               But
               (
               by
               the
               10th
               Article
               )
               the
               three
               Angles
               E
               ,
               C
               and
               F
               are
               equal
               to
               two
               Right
               Angles
               ;
               and
               therefore
               the
               three
               Angles
               of
               the
               Triangle
               are
               equal
               to
               the
               same
               ;
               which
               is
               the
               first
               .
               Again
               ,
               the
               two
               Angles
               B
               and
               D
               are
               equal
               to
               two
               Right
               Angles
               (
               by
               the
               10th
               Article
               )
               .
               Wherefore
               taking
               away
               B
               ,
               there
               will
               remain
               the
               Angles
               A
               and
               C
               equal
               to
               the
               Angle
               D
               ;
               which
               is
               the
               second
               .
            
             
               5
               Coroll
               .
               If
               the
               Angles
               A
               and
               B
               be
               equal
               ,
               the
               sides
               AC
               and
               CB
               will
               also
               be
               equal
               ,
               because
               AB
               and
               EF
               are
               parallel
               ;
               And
               on
               the
               contrary
               ,
               if
               the
               sides
               AC
               and
               CB
               be
               equal
               ,
               the
               Angles
               A
               and
               B
               will
               also
               be
               equal
               .
               For
               if
               they
               be
               not
               equal
               ,
               let
               the
               Angles
               B
               and
               G
               be
               equal
               .
               Wherefore
               seeing
               GB
               and
               EF
               are
               parallels
               ,
               and
               the
               Angles
               G
               and
               B
               equal
               ,
               the
               sides
               GC
               and
               CB
               will
               also
               be
               equal
               ;
               and
               because
               CB
               and
               AC
               are
               equal
               by
               supposition
               ,
               CG
               and
               CA
               will
               also
               be
               equal
               ;
               which
               cannot
               be
               (
               by
               the
               11th
               Article
               .
               )
            
             
               6
               Coroll
               .
               From
               hence
               it
               is
               manifest
               ,
               that
               if
               two
               Radii
               of
               a
               Circle
               be
               connected
               by
               a
               straight
               Line
               ,
               the
               Angles
               they
               make
               with
               that
               connecting
               Line
               will
               be
               equal
               to
               one
               another
               ;
               and
               if
               there
               be
               added
               that
               segment
               of
               the
               Circle
               which
               is
               subtended
               by
               the
               same
               line
               which
               connects
               the
               Radii
               ,
               then
               the
               Angles
               which
               those
               Radii
               make
               with
               the
               circumference
               wil
               also
               be
               equal
               to
               one
               another
               .
               For
               
               a
               straight
               line
               which
               subtends
               any
               Arch
               ,
               makes
               equal
               Angles
               with
               the
               same
               ,
               because
               if
               the
               Arch
               and
               the
               Subtense
               be
               divided
               in
               the
               middle
               ,
               the
               two
               halves
               of
               the
               segment
               wil
               be
               congruous
               to
               one
               another
               ,
               by
               reason
               of
               the
               Uniformity
               both
               of
               the
               Circumference
               of
               the
               Circle
               and
               of
               the
               straight
               Line
               .
            
          
           
             13
             Perimeters
             of
             Circles
             are
             to
             one
             another
             as
             their
             Semidiameters
             are
             .
             For
             let
             there
             be
             any
             two
             Circles
             ,
             as
             (
             in
             the
             first
             figure
             )
             BCD
             the
             greater
             ,
             and
             EFG
             the
             lesser
             ,
             having
             their
             common
             Center
             at
             A
             ;
             and
             let
             their
             Semidiameters
             be
             AC
             and
             AE
             .
             I
             say
             AC
             has
             the
             same
             proportion
             to
             AE
             which
             the
             Perimeter
             BCD
             has
             to
             the
             Perimeter
             EFG
             .
             For
             the
             magnitude
             of
             the
             Semidiameters
             AC
             and
             AE
             is
             determined
             by
             the
             distances
             of
             the
             points
             C
             and
             E
             from
             the
             Center
             A
             ;
             and
             the
             same
             distances
             are
             acquired
             by
             the
             uniform
             motion
             of
             a
             point
             from
             A
             to
             C
             in
             such
             manner
             that
             in
             equal
             times
             the
             distances
             acquired
             be
             equal
             .
             But
             the
             Perimeters
             BCD
             and
             EFG
             are
             also
             determined
             by
             the
             same
             distances
             of
             the
             points
             C
             and
             E
             from
             the
             Center
             A
             ;
             and
             therefore
             the
             Perimeters
             BCD
             and
             EFG
             ,
             as
             well
             as
             the
             Semidiameters
             AC
             and
             AE
             have
             their
             magnitudes
             determined
             by
             the
             same
             cause
             ,
             which
             cause
             makes
             in
             equal
             times
             equal
             spaces
             .
             Wherefore
             (
             by
             the
             13
             Chapter
             and
             6th
             Article
             )
             the
             Perimeters
             of
             Circles
             and
             their
             Semidiameters
             are
             Proportionals
             ;
             which
             was
             to
             be
             proved
             .
          
           
             14
             If
             two
             straight
             Lines
             w
             ch
             cōstitute
             an
             Angle
             ,
             be
             cut
             by
             straight-lined
             Parallels
             ,
             the
             intercepted
             Parallels
             will
             be
             to
             one
             another
             ,
             as
             the
             parts
             w
             ch
             they
             cut
             off
             frō
             the
             Vertex
             .
             Let
             the
             straight
             lines
             AB
             and
             AC
             (
             in
             the
             6
             figure
             )
             make
             an
             Angle
             at
             A
             ,
             &
             be
             cut
             by
             the
             two
             straight-lined
             Parallels
             BC
             and
             DE
             ,
             so
             that
             the
             parts
             cut
             off
             from
             the
             Vertex
             in
             either
             of
             those
             Lines
             (
             as
             in
             AB
             )
             may
             be
             AB
             and
             AD.
             I
             say
             the
             Parallels
             BC
             and
             DE
             are
             to
             one
             another
             as
             the
             parts
             AB
             and
             AD.
             For
             let
             AB
             be
             divided
             into
             any
             number
             of
             equal
             parts
             ,
             as
             into
             AF
             ,
             FD
             ,
             DB
             ;
             and
             by
             the
             points
             F
             and
             D
             ,
             let
             FG
             and
             DE
             be
             drawn
             Parallel
             to
             the
             base
             BC
             ,
             and
             cut
             AC
             in
             G
             and
             E
             ;
             and
             again
             by
             the
             points
             G
             and
             E
             let
             other
             straight
             lines
             be
             drawn
             Parallel
             to
             AB
             ,
             and
             cut
             BC
             in
             H
             and
             I.
             If
             now
             the
             point
             A
             be
             understood
             to
             be
             moved
             uniformly
             over
             AB
             ,
             and
             in
             the
             same
             time
             B
             be
             moved
             to
             C
             ,
             and
             all
             the
             points
             F
             ,
             D
             and
             B
             be
             moved
             uniformly
             and
             with
             equal
             Swiftness
             over
             FG
             ,
             DE
             ,
             and
             BC
             ;
             then
             shall
             B
             pass
             over
             BH
             (
             equal
             to
             FG
             )
             in
             the
             same
             time
             that
             A
             passes
             over
             AF
             ;
             and
             AF
             and
             FG
             will
             be
             to
             one
             another
             as
             their
             Velocities
             
             are
             ;
             and
             when
             A
             is
             in
             F
             ,
             D
             will
             be
             in
             K
             ;
             when
             A
             is
             in
             D
             ,
             D
             will
             be
             in
             E
             ;
             and
             in
             what
             manner
             the
             point
             A
             passes
             by
             the
             points
             F
             ,
             D
             and
             B
             ,
             in
             the
             same
             manner
             the
             point
             B
             will
             pass
             by
             the
             points
             H
             ,
             I
             and
             C
             ;
             &
             the
             straight
             lines
             FG
             ,
             DK
             ,
             KE
             ,
             BH
             ,
             HI
             &
             IC
             are
             equal
             by
             reason
             of
             their
             Parallelisme
             ;
             and
             therefore
             ,
             as
             the
             velocitie
             in
             AB
             is
             to
             the
             velocity
             〈◊〉
             BC
             ,
             so
             is
             AD
             to
             DE
             ;
             but
             as
             the
             velocity
             in
             AB
             is
             to
             the
             velocity
             in
             BC
             ,
             so
             is
             AB
             to
             BC
             ;
             that
             is
             to
             say
             ,
             all
             the
             Parallels
             will
             be
             severally
             to
             all
             the
             parts
             cut
             off
             from
             the
             Vertex
             ,
             as
             AF
             is
             to
             FG.
             Wherefore
             ,
             AF.
             GF
             :
             :
             AD.
             DE
             :
             :
             AB
             .
             BC
             are
             Proportionals
             .
          
           
             The
             Subtenses
             of
             equal
             Angles
             in
             different
             Circles
             (
             as
             the
             straight
             lines
             BC
             and
             FE
             (
             in
             the
             1
             figure
             )
             are
             to
             one
             another
             as
             the
             Arches
             which
             they
             subtend
             .
             For
             (
             by
             the
             8th
             Article
             )
             the
             Arches
             of
             equal
             Angles
             are
             to
             one
             another
             as
             their
             Perimeters
             are
             ;
             and
             (
             by
             the
             13th
             Art.
             )
             the
             Perimeters
             as
             their
             Semidiameters
             ;
             But
             the
             the
             Subtenses
             BC
             and
             FE
             are
             parallel
             to
             one
             another
             by
             reason
             of
             the
             equality
             of
             the
             Angles
             which
             they
             make
             with
             the
             Semidiameters
             ;
             and
             therefore
             the
             same
             Subtenses
             (
             by
             the
             last
             precedent
             Article
             )
             will
             be
             proportional
             to
             the
             Semidiameters
             ,
             that
             is
             ,
             to
             the
             Perimeters
             ,
             that
             is
             ,
             to
             the
             Arches
             which
             they
             subtend
             .
          
           
             15
             If
             in
             a
             Circle
             any
             number
             of
             equal
             Subtenses
             be
             placed
             immediatly
             after
             one
             another
             ,
             and
             straight
             lines
             be
             drawn
             from
             the
             extreme
             point
             of
             the
             first
             Subtense
             to
             the
             extreme
             points
             of
             all
             the
             rest
             ,
             The
             first
             Subtense
             being
             produced
             will
             make
             with
             the
             second
             Subtense
             an
             external
             Angle
             double
             to
             that
             which
             is
             made
             by
             the
             same
             first
             Subtense
             and
             a
             Tangent
             to
             the
             Circle
             touching
             it
             in
             the
             extreme
             point
             thereof
             ;
             and
             if
             a
             straight
             line
             which
             subtends
             two
             of
             those
             Arches
             be
             produced
             ,
             it
             will
             make
             an
             external
             Angle
             with
             the
             third
             Subtense
             ,
             triple
             to
             the
             Angle
             which
             is
             made
             by
             the
             Tangent
             with
             the
             first
             Subtense
             ;
             and
             so
             continually
             .
             For
             with
             the
             Radius
             AB
             (
             in
             the
             7th
             figure
             )
             let
             a
             circle
             be
             described
             ,
             &
             in
             it
             let
             any
             number
             of
             equal
             Subtenses
             BC
             ,
             CD
             &
             DE
             be
             placed
             ;
             also
             let
             BD
             &
             BE
             be
             drawn
             ;
             &
             by
             producing
             BC
             ,
             BD
             &
             BE
             to
             any
             distance
             in
             G
             ,
             H
             and
             I
             ,
             let
             them
             make
             Angles
             with
             the
             Subtenses
             which
             succeed
             one
             another
             ,
             namely
             the
             external
             Angles
             GCD
             ,
             and
             HDE
             .
             Lastly
             ,
             let
             the
             Tangent
             KB
             be
             drawn
             ,
             making
             with
             the
             first
             Subtense
             the
             Angle
             KBC
             .
             I
             say
             the
             Angle
             GCD
             is
             double
             to
             the
             Angle
             KBC
             ,
             and
             the
             Angle
             HDE
             triple
             to
             the
             same
             Angle
             KBC
             .
             For
             if
             AC
             be
             drawn
             cutting
             BD
             in
             M
             ,
             and
             from
             the
             point
             C
             
             there
             be
             drawn
             LC
             perpendicular
             to
             the
             same
             AC
             ,
             then
             CL
             and
             MD
             will
             be
             parallel
             by
             reason
             of
             the
             right
             Angles
             at
             C
             and
             M
             ;
             and
             therefore
             the
             alterne
             Angles
             LCD
             and
             BDC
             wil
             be
             equal
             ;
             as
             also
             the
             Angles
             BDC
             and
             CBD
             will
             be
             equal
             because
             of
             the
             equality
             of
             the
             straight
             lines
             BC
             and
             CD
             .
             Wherefore
             the
             Angle
             GCD
             is
             double
             to
             either
             of
             the
             A●gles
             CBD
             or
             CDB
             ;
             and
             therefore
             also
             the
             Angle
             GCD
             is
             double
             to
             the
             Angle
             LCD
             ,
             that
             is
             ,
             to
             the
             Angle
             KBC
             .
             Again
             ,
             CD
             is
             parallel
             to
             BE
             by
             reason
             of
             the
             equality
             of
             the
             Angles
             CBE
             and
             DEB
             ,
             and
             of
             the
             straight
             lines
             CB
             and
             DE
             ;
             and
             therefore
             the
             Angles
             GCD
             and
             GBE
             are
             equal
             ;
             and
             consequently
             GBE
             ,
             as
             also
             DEB
             is
             double
             to
             the
             Angle
             KBC
             .
             But
             the
             external
             Angle
             HDE
             is
             equal
             to
             the
             two
             internal
             DEB
             and
             DBE
             ;
             and
             therefore
             the
             Angle
             HDE
             is
             triple
             to
             the
             Angle
             KBC
             ,
             &c.
             which
             was
             to
             be
             proved
             .
          
           
             
               1
               Corollary
               .
               From
               hence
               it
               is
               manifest
               ,
               that
               the
               Angles
               KBC
               and
               CBD
               ,
               as
               also
               ,
               that
               all
               the
               Angles
               that
               are
               comprehended
               by
               two
               straight
               lines
               meeting
               in
               the
               circumference
               of
               a
               Circle
               and
               insisting
               upon
               equal
               Arches
               ,
               are
               equal
               to
               one
               another
               .
            
             
               2
               Coroll
               .
               If
               the
               Tangent
               BK
               be
               moved
               in
               the
               Circumference
               with
               Uniform
               motion
               about
               the
               Center
               B
               ,
               it
               will
               in
               equal
               times
               cut
               off
               equal
               Arches
               ;
               and
               will
               pass
               over
               the
               whole
               Perimeter
               in
               the
               same
               time
               in
               which
               it self
               describes
               a
               semiperimeter
               about
               the
               Center
               B.
               
            
             
               3
               Coroll
               .
               From
               hence
               also
               we
               may
               understand
               what
               it
               is
               that
               determines
               the
               bending
               or
               Curvation
               of
               a
               straight
               line
               into
               the
               circumference
               of
               a
               Circle
               ;
               namely
               ,
               that
               it
               is
               Fraction
               continually
               encreasing
               in
               the
               same
               manner
               as
               Nūbers
               from
               One
               upwards
               encrease
               by
               the
               continual
               addition
               of
               Unity
               .
               For
               the
               indefinite
               straight
               Line
               KB
               being
               broken
               in
               B
               according
               to
               any
               Angle
               ,
               as
               that
               of
               KBC
               ,
               &
               again
               in
               C
               according
               to
               a
               double
               Angle
               ,
               and
               in
               D
               according
               to
               an
               Angle
               which
               is
               triple
               ,
               and
               in
               E
               according
               to
               an
               Angle
               which
               is
               quadruple
               ●o
               the
               first
               Angle
               ,
               and
               so
               continually
               ,
               there
               will
               be
               described
               a
               figure
               which
               will
               indeed
               be
               rectilineal
               if
               the
               broken
               parts
               be
               considered
               as
               ha●ing
               magnitude
               ;
               but
               if
               they
               be
               understood
               to
               be
               the
               least
               t●a●
               can
               〈◊〉
               ,
               ●●at
               is
               ,
               as
               so
               many
               Points
               ;
               then
               the
               figure
               described
               will
               ●ot
               be
               rectilineal
               ,
               but
               a
               Circle
               ,
               whose
               Circumference
               w●…
               broken
               line
               .
            
             
               4
               〈…〉
               been
               said
               in
               this
               present
               Article
               it
               may
               〈…〉
               
                 Angle
                 in
                 the
                 center
                 is
                 double
                 to
                 an
                 Angle
              
               
               
                 in
                 the
                 Circumference
                 of
                 the
                 same
                 Circle
                 ,
                 if
                 the
                 intercepted
                 Arches
                 be
                 equal
                 .
              
               For
               seeing
               that
               straight
               Line
               by
               whose
               motion
               an
               Angle
               is
               determined
               ,
               passes
               over
               equal
               Arches
               in
               equal
               times
               ,
               as
               well
               from
               the
               Center
               as
               from
               the
               Circumference
               ;
               and
               while
               that
               which
               is
               from
               the
               Circumference
               is
               passing
               over
               half
               its
               own
               Perimeter
               ,
               it
               passes
               in
               the
               same
               time
               over
               the
               whole
               Perimeter
               of
               that
               which
               is
               from
               the
               Center
               ,
               the
               Arches
               w
               ch
               it
               cuts
               off
               in
               the
               Perimeter
               whose
               Center
               is
               A
               ,
               wil
               be
               double
               to
               those
               which
               it
               makes
               in
               its
               own
               Semiperimeter
               whose
               Center
               is
               B.
               But
               in
               equal
               Circles
               ,
               as
               Arches
               are
               to
               one
               another
               ,
               so
               also
               are
               Angles
               .
            
          
           
             It
             may
             also
             be
             demonstrated
             that
             the
             external
             Angle
             made
             by
             a
             Subtense
             produced
             and
             the
             next
             equal
             Subtense
             ,
             is
             equal
             to
             an
             Angle
             from
             the
             Center
             insisting
             upon
             the
             same
             Arch
             ;
             As
             in
             the
             last
             Diagram
             ,
             the
             Angle
             GCD
             is
             equal
             to
             the
             Angle
             CAD
             ;
             For
             the
             external
             Angle
             GCD
             is
             double
             to
             the
             Angle
             CBD
             ;
             and
             the
             Angle
             CAD
             insisting
             upon
             the
             same
             Arch
             CD
             ,
             is
             also
             double
             to
             the
             same
             Angle
             CBD
             or
             KBC
             .
          
           
             16
             An
             Angle
             of
             Contingence
             ,
             if
             it
             be
             compared
             with
             an
             Angle
             simply
             so
             called
             how
             little
             soever
             ,
             has
             such
             proportion
             to
             it
             as
             a
             Point
             has
             to
             a
             Line
             ;
             that
             is
             ,
             no
             proportion
             at
             all
             ,
             nor
             any
             quantity
             .
             For
             first
             ,
             an
             Angle
             of
             cōtingence
             is
             made
             by
             cōtinual
             flexion
             ,
             so
             that
             in
             the
             generation
             of
             it
             there
             is
             no
             circular
             motion
             at
             all
             ,
             in
             which
             consists
             the
             nature
             of
             an
             Angle
             simply
             so
             called
             ;
             and
             therefore
             it
             cannot
             be
             compared
             with
             it
             according
             to
             Quantity
             .
             Secondly
             ,
             seeing
             the
             external
             Angle
             made
             by
             a
             Subtense
             produced
             and
             the
             next
             Subtense
             ,
             is
             equal
             to
             an
             Angle
             from
             the
             Center
             insisting
             upon
             the
             same
             Arch
             (
             as
             in
             the
             last
             figure
             the
             Angle
             GCD
             is
             equal
             to
             the
             Angle
             CAD
             )
             the
             Angle
             of
             Contingence
             wil
             be
             equal
             to
             that
             Angle
             from
             the
             Center
             which
             is
             made
             by
             AB
             and
             the
             same
             AB
             ;
             for
             no
             part
             of
             a
             Tangent
             can
             subtend
             any
             Arch
             ;
             but
             as
             the
             point
             of
             Contact
             is
             to
             be
             taken
             for
             the
             Subtense
             ,
             so
             the
             Angle
             of
             Contingence
             is
             to
             be
             accounted
             for
             the
             external
             Angle
             ,
             and
             equal
             to
             that
             Angle
             whose
             Arch
             is
             the
             same
             point
             B.
             
          
           
             Now
             seeing
             an
             Angle
             in
             general
             is
             defined
             to
             be
             the
             Opening
             o●
             Divergence
             of
             two
             lines
             which
             concurre
             in
             one
             sole
             point
             ;
             &
             seeing
             one
             Opening
             is
             greater
             then
             another
             ,
             it
             cānot
             be
             denied
             but
             that
             by
             the
             very
             generation
             of
             it
             an
             Angle
             of
             Contingence
             is
             Quantity
             ;
             for
             wheresoever
             there
             is
             Greater
             and
             Less
             ,
             there
             is
             also
             Quantity
             ;
             
             but
             this
             quantity
             consists
             in
             greater
             and
             less
             Flexion
             ;
             for
             how
             much
             the
             greater
             a
             Circle
             is
             ,
             so
             much
             the
             neerer
             comes
             the
             Circumference
             of
             it
             to
             the
             nature
             of
             a
             straight
             Line
             ;
             for
             the
             Circumference
             of
             a
             Circle
             being
             made
             by
             the
             curvation
             of
             a
             straight
             line
             ,
             the
             less
             that
             straight
             line
             is
             ,
             the
             greater
             is
             the
             curvation
             ;
             &
             therfore
             when
             one
             straight
             line
             is
             a
             Tangent
             to
             many
             Circles
             ,
             the
             Angle
             of
             Contingence
             which
             it
             makes
             with
             a
             less
             Circle
             is
             greater
             then
             that
             which
             it
             makes
             with
             a
             greater
             Circle
             .
          
           
             Nothing
             therefore
             is
             added
             to
             ,
             or
             taken
             from
             an
             Angle
             simply
             so
             called
             ,
             by
             the
             addition
             to
             it
             or
             taking
             from
             it
             of
             never
             so
             many
             Angles
             of
             Contingence
             .
             And
             as
             an
             Angle
             of
             one
             sort
             can
             never
             be
             equal
             to
             an
             Angle
             of
             the
             other
             sort
             ,
             so
             they
             cannot
             be
             either
             greater
             or
             less
             then
             one
             another
             .
          
           
             From
             whence
             it
             follows
             ,
             that
             an
             Angle
             of
             a
             Segment
             ,
             that
             is
             ,
             the
             Angle
             which
             any
             straight
             line
             makes
             with
             any
             Arch
             ,
             is
             equal
             to
             the
             Angle
             which
             is
             made
             by
             the
             same
             straight
             line
             ,
             &
             another
             which
             touches
             the
             Circle
             in
             the
             point
             of
             their
             Concurrence
             ;
             as
             in
             the
             last
             figure
             ,
             the
             Angle
             which
             is
             made
             between
             GB
             and
             BK
             is
             equal
             to
             that
             which
             is
             made
             between
             GB
             and
             the
             Arch
             BC.
             
          
           
             17
             An
             Angle
             which
             is
             made
             by
             two
             Plains
             ,
             is
             commonly
             called
             the
             Inclination
             of
             those
             Plains
             ;
             And
             because
             Plains
             have
             equal
             Inclination
             in
             all
             their
             parts
             ,
             instead
             of
             their
             Inclination
             an
             Angle
             is
             taken
             which
             is
             made
             by
             two
             straight
             lines
             ,
             one
             of
             which
             is
             in
             one
             ,
             the
             other
             in
             the
             other
             of
             those
             Plains
             ,
             but
             both
             perpendicular
             to
             the
             common
             Section
             .
          
           
             18
             A
             Solid
             Angle
             may
             be
             conceived
             two
             wayes
             .
             First
             ,
             for
             the
             aggregate
             of
             all
             the
             Angles
             which
             are
             made
             by
             the
             motion
             of
             a
             straight
             line
             ,
             while
             one
             extreme
             point
             thereof
             remayning
             fixed
             ,
             it
             is
             carried
             about
             any
             plain
             figure
             in
             which
             the
             fixed
             point
             of
             the
             straight
             line
             is
             not
             conteined
             .
             And
             in
             this
             sense
             it
             seems
             to
             be
             understood
             by
             Euclide
             .
             Now
             it
             is
             manifest
             that
             the
             quantity
             of
             a
             Solid
             Angle
             so
             conceived
             is
             no
             other
             then
             the
             aggregate
             of
             all
             the
             Angles
             in
             a
             Superficies
             so
             described
             ,
             that
             is
             ,
             in
             the
             Superficies
             of
             a
             Pyramidal
             Solid
             .
             Secondly
             ,
             when
             a
             Py●amis
             or
             Cone
             has
             its
             Vertex
             in
             the
             Center
             of
             a
             Sphere
             ,
             a
             Solid
             Angle
             may
             be
             understood
             to
             be
             the
             proportion
             of
             a
             Spherical
             Superficies
             subtending
             that
             Vertex
             ,
             to
             the
             whole
             Superficies
             of
             the
             Sphere
             .
             In
             which
             sense
             solid
             Angles
             are
             to
             one
             another
             as
             
             the
             Spherical
             Bases
             of
             Solids
             which
             have
             their
             Vertex
             in
             the
             Center
             of
             the
             same
             Sphere
             .
          
           
             19
             All
             the
             waye●
             by
             which
             two
             lines
             respect
             one
             another
             ,
             or
             all
             the
             variety
             of
             their
             position
             may
             be
             comprehended
             under
             four
             kindes
             ;
             For
             any
             two
             lines
             whatsoever
             are
             either
             Parallels
             or
             being
             produced
             (
             if
             need
             be
             )
             or
             moved
             one
             of
             them
             to
             the
             other
             parallelly
             to
             it self
             ,
             they
             make
             an
             Angle
             ;
             or
             else
             (
             by
             the
             like
             production
             and
             motion
             )
             they
             Touch
             one
             another
             ;
             or
             lastly
             ,
             they
             are
             Asymptotes
             .
             The
             nature
             of
             Parallels
             ,
             Angles
             and
             Tangents
             has
             been
             already
             declared
             .
             It
             remains
             that
             I
             speak
             briefly
             of
             the
             nature
             of
             Asymptotes
             .
          
           
             Asymptosy
             depends
             upon
             this
             ,
             that
             Quantity
             is
             infinitly
             divisible
             .
             And
             from
             hence
             it
             follows
             ,
             that
             any
             line
             being
             given
             ,
             and
             a
             Body
             supposed
             to
             be
             moved
             from
             one
             extreme
             thereof
             towards
             the
             other
             ,
             it
             is
             possible
             (
             by
             taking
             degrees
             of
             Velocity
             alwayes
             lesse
             and
             lesse
             in
             such
             proportion
             as
             the
             parts
             of
             the
             Line
             are
             made
             lesse
             by
             continual
             division
             )
             that
             the
             same
             Body
             may
             be
             alwayes
             moved
             forwards
             in
             that
             Line
             ,
             and
             yet
             never
             reach
             the
             end
             of
             it
             .
             For
             it
             is
             manifest
             that
             if
             any
             straight
             Line
             ,
             as
             AF
             (
             in
             the
             8th
             figure
             )
             be
             cut
             any
             where
             in
             B
             ,
             and
             again
             BF
             be
             cut
             in
             C
             ,
             and
             CF
             in
             D
             ,
             and
             DF
             in
             E
             ,
             and
             so
             eternally
             ,
             and
             there
             be
             drawn
             from
             the
             point
             F
             the
             straight
             Line
             FF
             at
             any
             Angle
             AFF
             ;
             and
             lastly
             ,
             if
             the
             straight
             Lines
             AF
             ,
             BF
             ,
             CF
             ,
             DF
             ,
             EF
             ,
             &c.
             having
             the
             same
             proportion
             to
             one
             another
             with
             the
             Segments
             of
             the
             Line
             AF
             ,
             be
             set
             in
             order
             and
             parallel
             to
             the
             same
             AF
             ,
             the
             crooked
             Line
             ABCDE
             and
             the
             straight
             Line
             FF
             will
             be
             Asymptotes
             ,
             that
             is
             ,
             they
             will
             alwayes
             come
             neerer
             and
             neerer
             together
             ,
             but
             never
             touch
             one
             another
             .
             Now
             because
             any
             Line
             may
             be
             cut
             eternally
             according
             to
             the
             proportions
             which
             the
             Segments
             have
             to
             one
             another
             ,
             therefore
             the
             divers
             kindes
             of
             Asymptotes
             are
             infinite
             in
             number
             ,
             and
             not
             necessary
             to
             be
             further
             spoken
             of
             in
             this
             place
             .
             In
             the
             nature
             of
             Asymptotes
             in
             general
             there
             is
             no
             more
             ,
             then
             that
             they
             come
             still
             neerer
             and
             neerer
             but
             never
             touch
             .
             But
             in
             special
             in
             the
             Asymptosie
             of
             Hyperbolique
             Lines
             ,
             it
             is
             understood
             they
             should
             approach
             to
             a
             distance
             lesse
             then
             any
             given
             quantity
             .
          
           
             20
             SITUATION
             
               is
               the
               relation
               of
               one
               place
               to
               another
            
             ;
             &
             where
             there
             are
             many
             places
             ,
             their
             Situation
             is
             determined
             by
             four
             things
             ;
             By
             
             
               their
               Distances
               from
               one
               another
            
             ;
             By
             
               several
               Distances
               from
               a
               place
               assigned
            
             ;
             By
             
               the
               order
               of
               straight
               lines
               drawn
               from
               a
               place
               assigned
               to
               the
               places
               of
               them
               all
            
             ;
             and
             by
             
               the
               Angles
               which
               are
               made
               by
               the
               lines
               so
               drawn
               .
            
             For
             if
             their
             Distances
             ,
             Order
             ,
             and
             Angles
             be
             given
             ,
             that
             is
             ,
             be
             certainly
             known
             ,
             their
             several
             places
             will
             also
             be
             so
             certainly
             known
             ,
             as
             that
             they
             can
             be
             no
             other
             .
          
           
             21
             Points
             ,
             how
             many
             soever
             they
             be
             ,
             have
             
               Like
               Situation
            
             with
             an
             equal
             number
             of
             other
             Points
             ,
             when
             all
             the
             straight
             lines
             that
             are
             drawn
             from
             some
             one
             point
             to
             all
             these
             ,
             have
             severally
             the
             same
             proportion
             to
             those
             that
             are
             drawn
             in
             the
             same
             order
             and
             at
             equal
             Angles
             from
             some
             one
             point
             to
             all
             those
             .
             For
             let
             there
             be
             any
             number
             of
             Points
             as
             A
             ,
             B
             and
             C
             ,
             (
             in
             the
             9
             figure
             )
             ,
             to
             which
             from
             some
             one
             point
             D
             let
             the
             straight
             Lines
             DA
             ,
             DB
             and
             DC
             be
             drawn
             ;
             and
             let
             there
             be
             an
             equal
             number
             of
             other
             Points
             as
             E
             ,
             F
             and
             G
             ,
             and
             from
             some
             point
             H
             let
             the
             straight
             Lines
             HE
             ,
             HF
             and
             HG
             be
             drawn
             ,
             so
             that
             the
             Angles
             ADB
             and
             BDC
             be
             severally
             and
             in
             the
             same
             order
             equal
             to
             the
             Angles
             EHF
             and
             FHG
             ,
             and
             the
             straight
             Lines
             DA
             ,
             DB
             and
             DC
             proportional
             to
             the
             straight
             Lines
             HE
             ,
             HF
             and
             HG
             ;
             I
             say
             the
             three
             points
             A
             ,
             B
             and
             C
             ,
             have
             Like
             Situation
             with
             the
             three
             points
             E
             ,
             F
             &
             G
             ,
             or
             are
             placed
             Alike
             .
             For
             if
             HE
             be
             understood
             to
             be
             layed
             upon
             DA
             ,
             so
             that
             the
             point
             H
             be
             in
             D
             ,
             the
             point
             F
             will
             be
             in
             the
             straight
             Line
             DB
             by
             reason
             of
             the
             equality
             of
             the
             Angles
             ADB
             and
             EHF
             ;
             and
             the
             point
             G
             will
             be
             in
             the
             straight
             Line
             DC
             by
             reason
             of
             the
             equality
             of
             the
             Angles
             BDC
             and
             FHG
             ;
             and
             the
             strright
             Lines
             AB
             and
             EF
             ,
             as
             also
             BC
             and
             FG
             will
             be
             parallel
             ,
             because
             AD.
             ED
             :
             :
             BH
             .
             FH
             :
             :
             CD
             .
             GH
             are
             Proportionals
             by
             construction
             ;
             and
             therefore
             the
             distances
             between
             the
             points
             A
             and
             B
             ,
             and
             the
             points
             B
             and
             C
             ,
             will
             be
             proportional
             to
             the
             distances
             between
             the
             points
             E
             and
             F
             ,
             and
             the
             points
             F
             and
             G.
             Wherefore
             in
             the
             situation
             of
             the
             points
             A
             ,
             B
             and
             C
             ,
             and
             the
             situation
             of
             the
             points
             E
             ,
             F
             and
             G
             the
             Angles
             in
             the
             same
             order
             are
             equal
             ,
             so
             that
             their
             situations
             differ
             in
             nothing
             but
             the
             inequality
             of
             their
             distances
             from
             one
             another
             ,
             and
             of
             their
             distances
             from
             the
             points
             D
             and
             H.
             Now
             in
             both
             the
             orders
             of
             Points
             those
             inequalities
             are
             equal
             ;
             for
             AB
             .
             BC
             :
             :
             EF.
             FG
             ,
             which
             are
             their
             distances
             from
             one
             another
             ,
             as
             also
             DA.
             DB.
             DC
             :
             :
             HE.
             HF.
             HG
             which
             are
             their
             distances
             from
             the
             assumed
             points
             D
             and
             H
             ,
             are
             Proportionals
             .
             Their
             difference
             therefore
             
             consists
             solely
             in
             the
             magnitude
             of
             their
             distances
             .
             But
             by
             the
             definition
             of
             Like
             (
             Chap.
             11.
             
             Art
             ,
             2
             )
             those
             things
             which
             differ
             onely
             in
             Magnitude
             are
             Like
             .
             Wherefore
             the
             points
             A
             ,
             B
             and
             C
             have
             to
             one
             another
             Like
             Situation
             with
             the
             points
             E
             ,
             F
             and
             G
             ,
             or
             are
             placed
             Alike
             ;
             which
             was
             to
             be
             proved
             .
          
           
             FIGURE
             ,
             
               is
               quantity
               determined
               by
               the
               Situation
               ,
               or
               placing
               of
               all
               its
               extreme
               Points
               .
            
             Now
             I
             call
             those
             points
             Extreme
             which
             are
             contiguous
             to
             the
             place
             which
             is
             without
             the
             figure
             .
             In
             Lines
             therefore
             and
             Superficies
             all
             Points
             may
             be
             called
             Extreme
             ,
             but
             in
             Solids
             onely
             those
             which
             are
             in
             the
             Superficies
             that
             includes
             them
             .
          
           
             
               Like
               Figures
            
             ,
             are
             those
             ,
             whose
             extreme
             points
             in
             one
             of
             them
             ,
             are
             all
             placed
             like
             all
             the
             extreme
             points
             in
             the
             other
             ;
             for
             such
             Figures
             differ
             in
             nothing
             but
             Magnitude
             .
          
           
             And
             like
             Figures
             are
             
               alike
               placed
            
             ,
             when
             in
             both
             of
             them
             the
             homologal
             straight
             lines
             ,
             that
             is
             ,
             the
             straight
             lines
             which
             connect
             the
             points
             which
             answer
             one
             another
             are
             parallel
             ,
             and
             have
             their
             proportional
             sides
             enclined
             the
             same
             way
             .
          
           
             And
             seeing
             every
             Straight
             Line
             is
             like
             every
             other
             Straight
             Line
             ,
             and
             every
             Plain
             like
             every
             other
             Plain
             when
             nothing
             but
             Plainness
             is
             considered
             ;
             if
             the
             Lines
             which
             include
             Plains
             ,
             or
             the
             Superficies
             which
             include
             Solids
             have
             their
             proportions
             known
             ,
             it
             will
             not
             be
             hard
             to
             know
             whether
             any
             Figure
             be
             like
             or
             unlike
             to
             another
             propounded
             Figure
             .
          
           
             And
             thus
             much
             concerning
             the
             
               First
               Grounds
               of
               Philosophy
            
             .
             The
             next
             place
             belongs
             to
             Geometry
             ;
             in
             which
             the
             Quantities
             of
             Figures
             are
             sought
             out
             from
             the
             Proportions
             of
             Lines
             and
             Angles
             .
             Wherefore
             it
             is
             necessary
             for
             him
             that
             would
             study
             Geometry
             to
             know
             first
             what
             is
             the
             nature
             of
             Quantity
             ,
             Proportion
             ,
             Angle
             and
             Figure
             .
             Having
             therefore
             explained
             these
             in
             the
             three
             last
             Chapters
             ,
             I
             thought
             ●it
             to
             add
             them
             to
             this
             Part
             ;
             and
             so
             passe
             to
             the
             next
             .
          
        
      
       
         
         
         
           OF
           THE
           PROPORTIONS
           OF
           MOTIONS
           AND
           MAGNITVDES
           .
        
         
           
             
               CHAP.
               XV.
            
             Of
             the
             Nature
             ,
             Properties
             ,
             and
             diverse
             Considerations
             of
             Motion
             and
             Endeavour
             .
          
           
             
               
                 1
                 Repetition
                 of
                 some
                 Principles
                 of
                 the
                 doctrine
                 of
                 Motion
                 formerly
                 set
                 down
                 .
              
               
                 2
                 Other
                 Principles
                 added
                 to
                 them
                 .
              
               
                 3
                 Certain
                 Theoremes
                 concerning
                 the
                 nature
                 of
                 Motion
                 .
              
               
                 4
                 Diverse
                 Considerations
                 of
                 Motion
                 ▪
              
               
                 5
                 The
                 way
                 by
                 which
                 the
                 first
                 Endeavour
                 of
                 Bodies
                 Moved
                 ●endoth
                 .
              
               
                 6
                 In
                 Motion
                 which
                 is
                 made
                 by
                 Concourse
                 ,
                 one
                 of
                 the
                 Movents
                 ceasing
                 ,
                 the
                 Endeavour
                 is
                 made
                 by
                 the
                 way
                 by
                 which
                 the
                 rest
                 tend
                 .
              
               
                 7
                 The
                 Endeavour
                 of
                 any
                 Moved
                 Body
                 ,
                 which
                 having
                 its
                 Motion
                 in
                 the
                 Circumference
                 of
                 a
                 Circle
                 ,
                 parts
                 from
                 the
                 same
                 ,
                 proceeds
                 afterwards
                 in
                 a
                 straight
                 line
                 which
                 toucheth
                 the
                 Circle
                 .
              
               
                 8
                 How
                 much
                 greater
                 ,
                 the
                 Velocity
                 or
                 Magnitude
                 is
                 of
                 a
                 Movent
                 ,
                 so
                 much
                 greater
                 is
                 the
                 Efficacy
                 thereof
                 upon
                 any
                 other
                 Body
                 in
                 its
                 way
                 .
              
            
          
           
             1
             THe
             next
             things
             in
             order
             to
             be
             treated
             of
             are
             MOTION
             and
             MAGNITUDE
             ,
             which
             are
             the
             most
             common
             Accidents
             of
             all
             Bodies
             .
             This
             place
             therefore
             most
             properly
             belongs
             to
             the
             Elements
             of
             Geometry
             .
             But
             because
             this
             part
             of
             Philosophy
             having
             been
             improved
             by
             the
             best
             Wits
             of
             all
             Ages
             has
             afforded
             greater
             plenty
             of
             matter
             then
             can
             well
             be
             thrust
             together
             within
             the
             narrow
             limits
             of
             this
             discourse
             ;
             I
             thought
             fit
             to
             admonish
             the
             Reader
             ,
             that
             before
             he
             
             proceed
             further
             ,
             he
             take
             into
             his
             hands
             the
             Works
             of
             
               Euclide
               ,
               Archimedes
               ,
               Apollonius
            
             and
             other
             as
             well
             Ancient
             as
             Modern
             Writers
             .
             For
             to
             what
             end
             is
             it
             to
             do
             over
             again
             that
             which
             is
             already
             done
             ?
             The
             little
             therefore
             that
             I
             shall
             say
             concerning
             Geometry
             in
             some
             of
             the
             following
             Chapters
             ,
             shall
             be
             such
             onely
             as
             is
             new
             ,
             and
             conducing
             to
             Natural
             Philosophy
             .
          
           
             I
             have
             already
             delivered
             some
             of
             the
             Principles
             of
             this
             doctrine
             in
             the
             8
             &
             9
             Chapters
             ,
             which
             I
             shall
             briefly
             put
             together
             here
             ,
             that
             the
             Reader
             in
             going
             on
             may
             have
             their
             light
             neerer
             at
             hand
             .
          
           
             First
             therefore
             in
             the
             8th
             Chap.
             and
             10
             
               Article
               ,
               Motion
            
             is
             defined
             to
             be
             
               the
               continual
               privation
               of
               one
               place
               ,
               and
               acquisition
               of
               another
               .
            
          
           
             Secondly
             ,
             it
             is
             there
             shewn
             ,
             that
             
               Whatsoever
               is
               Moved
               is
               Moved
               in
               Time.
               
            
          
           
             Thirdly
             ,
             in
             the
             same
             Chap.
             11.
             
             Article
             ,
             I
             have
             defined
             
               Rest
               to
               be
               when
               a
               Body
               remains
               for
               some
               time
               in
               one
               place
               .
            
          
           
             Fourthly
             ,
             it
             is
             there
             shewn
             ,
             that
             
               Whatsoever
               is
               Moved
               is
               not
               in
               any
               determined
               place
            
             ;
             as
             also
             that
             the
             same
             
               has
               been
               Moved
               ,
               is
               still
               Moved
               ,
               and
               will
               yet
               be
               Moved
            
             ;
             So
             that
             in
             every
             part
             of
             that
             Space
             in
             which
             Motion
             is
             made
             ,
             we
             may
             consider
             three
             Times
             ,
             namely
             the
             Past
             ,
             the
             Present
             ,
             and
             the
             
               Future
               Time.
            
             
          
           
             Fiftly
             ,
             in
             the
             15
             Article
             of
             the
             same
             Chapter
             ,
             I
             have
             defined
             Velocity
             or
             
               Swiftness
               to
               be
               Motion
               considered
               as
               Power
               ,
               namely
               ,
               that
               Power
               by
               which
               a
               Body
               Moved
               may
               in
               a
               certain
               Time
               transmit
               a
               certain
               Length
            
             ;
             which
             also
             may
             more
             briefly
             be
             enunciated
             thus
             ,
             
               Velocity
               is
               the
               quantity
               of
               Motion
               determined
               by
               Time
               and
               Line
               .
            
          
           
             Sixthly
             ,
             in
             the
             same
             Chap.
             16.
             
             Article
             ,
             I
             have
             shewn
             that
             
               Motion
               is
               the
               Measure
               of
               Time.
               
            
          
           
             Seventhly
             ,
             in
             the
             same
             Chap.
             17th
             Art.
             I
             have
             defined
             Motions
             to
             be
             Equally
             Swift
             ,
             when
             in
             Equal
             Times
             Equal
             Lengths
             are
             transmitted
             by
             them
             .
          
           
             Eighthly
             ,
             in
             the
             18
             Article
             of
             the
             same
             Chapter
             ,
             Motions
             are
             defined
             to
             be
             
               Equal
               ,
               when
               the
               Swiftness
               of
               one
               Moved
               Body
               computed
               in
               every
               part
               of
               its
               magnitude
               ,
               is
               equal
               to
               the
               Swiftness
               of
               another
               computed
               also
               in
               every
               part
               of
               its
               magnitude
               .
            
             From
             whence
             it
             is
             to
             be
             noted
             ,
             that
             
               Motions
               Equal
               to
               one
               another
               ,
               and
               Motions
               Equally
               Swift
               ,
            
             do
             not
             signifie
             the
             same
             thing
             ;
             for
             when
             two
             horses
             draw
             abrest
             ,
             the
             Motion
             of
             both
             is
             greater
             then
             the
             Motion
             of
             either
             of
             them
             singly
             ;
             but
             the
             Swiftness
             of
             both
             together
             is
             but
             Equal
             to
             that
             of
             either
             .
          
           
             Ninthly
             ,
             in
             the
             19
             Article
             of
             the
             same
             Chapter
             ,
             I
             have
             shewn
             ,
             
             that
             
               Whatsoever
               is
               at
               Rest
               will
               alwayes
               be
               at
               Rest
               ,
               unless
               there
               be
               some
               other
               Body
               besides
               it
               ,
               which
               by
               getting
               into
               its
               place
               ,
               suffers
               it
               no
               longer
               to
               remain
               at
               Rest.
            
             And
             that
             
               Whatsoever
               is
               Moved
               ,
               will
               alwayes
               be
               Moved
               ,
               unless
               there
               be
               some
               other
               Body
               besides
               it
               ,
               which
               hinders
               its
               Motion
               .
            
          
           
             Tenthly
             ,
             In
             the
             9
             Chapter
             and
             7
             Article
             ,
             I
             have
             demonstrated
             ,
             that
             
               When
               any
               Body
               is
               moved
               which
               was
               formerly
               at
               Rest
               ,
               the
               immediate
               efficient
               cause
               of
               that
               Motion
               is
               in
               some
               other
               Moved
               and
               Contiguou●
               Body
               .
            
          
           
             Eleventhly
             ,
             I
             have
             shewn
             in
             the
             same
             place
             ,
             that
             
               Whatsoever
               is
               Moved
               ,
               will
               always
               be
               Moved
               in
               the
               same
               way
               ,
               and
               with
               the
               same
               Swiftness
               ,
               if
               it
               be
               not
               hindered
               by
               some
               other
               Moved
               and
               Contiguou●
               Body
               .
            
          
           
             2
             To
             which
             Principles
             I
             shall
             here
             add
             these
             that
             follow
             .
             First
             ,
             I
             define
             ENDEAVOUR
             
               to
               be
               Motion
               made
               in
               less
               Space
               and
               Time
               then
               can
               be
               given
            
             ;
             that
             is
             ,
             
               less
               then
               can
               be
               determined
               or
               assigned
               by
               Exposition
               or
               Number
            
             ;
             that
             is
             ,
             
               Motion
               made
               through
               the
               length
               of
               a
               Point
               ,
               and
               in
               an
               Instant
               or
               Point
               of
               Time.
            
             For
             the
             explayning
             of
             which
             Definition
             it
             must
             be
             remembred
             ,
             that
             by
             a
             Point
             is
             not
             to
             be
             understood
             that
             which
             has
             no
             quantity
             ,
             or
             which
             cannot
             by
             any
             means
             be
             divided
             (
             for
             there
             is
             no
             such
             thing
             in
             Nature
             )
             ;
             but
             that
             whose
             quantity
             is
             not
             at
             all
             considered
             ,
             that
             is
             ,
             whereof
             neither
             quantity
             nor
             any
             part
             is
             computed
             in
             demonstration
             ;
             so
             that
             a
             Point
             is
             not
             to
             be
             taken
             for
             an
             Indivisible
             ,
             but
             for
             an
             Undivided
             thing
             ;
             as
             also
             an
             Instant
             is
             to
             be
             taken
             for
             an
             Undivided
             ,
             and
             not
             for
             an
             Indivisible
             Time.
             
          
           
             In
             like
             manner
             Endeavour
             is
             to
             be
             conceived
             as
             Motion
             ;
             but
             so
             ,
             as
             that
             neither
             the
             quantity
             of
             the
             Time
             in
             which
             ,
             nor
             of
             the
             Line
             in
             which
             it
             is
             made
             may
             in
             demonstration
             be
             at
             all
             brought
             into
             comparison
             with
             the
             quantity
             of
             that
             Time
             ,
             or
             of
             that
             Line
             of
             which
             it
             is
             a
             part
             .
             And
             yet
             ,
             as
             a
             Point
             may
             be
             compared
             with
             a
             Point
             ,
             so
             one
             Endeavour
             may
             be
             compared
             with
             another
             Endeavour
             ,
             and
             one
             may
             be
             found
             to
             be
             greater
             or
             lesse
             then
             another
             .
             For
             if
             the
             Vertical
             points
             of
             two
             Angles
             be
             compared
             ,
             they
             will
             be
             equal
             or
             unequal
             in
             the
             same
             proportion
             which
             the
             Angles
             themselves
             have
             to
             one
             another
             .
             Or
             if
             a
             straight
             Line
             cut
             many
             Circumferences
             of
             Concentrick
             Circles
             ,
             the
             inequality
             of
             the
             points
             of
             intersection
             will
             be
             in
             the
             same
             proportion
             which
             the
             Perimeters
             have
             to
             one
             another
             .
             And
             in
             the
             same
             manner
             ,
             if
             two
             Motions
             begin
             and
             end
             both
             together
             ,
             their
             Endeavours
             will
             be
             Equal
             or
             Unequal
             according
             to
             the
             proportion
             of
             their
             Velocities
             ;
             as
             we
             
             see
             a
             bullet
             of
             Lead
             descend
             with
             greater
             Endeavour
             then
             a
             ball
             of
             Wooll
             ,
          
           
             Secondly
             ,
             I
             define
             IMPETUS
             
               or
               Quickness
               of
               Motion
               ,
               to
               be
               the
               Swiftness
               or
               Velocity
               of
               the
               Body
               moved
               ,
               but
               considered
               in
               the
               several
               points
               of
               that
               time
               in
               which
               it
               is
               moved
               ;
               In
               which
               sense
            
             Impetus
             
               is
               nothing
               else
               but
               the
               quantity
               or
               velocity
               of
               Endeavour
               .
               But
               considered
               with
               the
               whole
               time
               ,
               it
               is
               the
               whole
               velocity
               of
               the
               Body
               moved
               ,
               taken
               together
               throughout
               all
               the
               time
               ,
               and
               equal
               to
               the
               Product
               of
               a
               Line
               representing
               the
               time
               multiplyed
               into
               a
               Line
               representing
               the
               arithmetically
               mean
            
             Impetus
             
               or
               Quickness
            
             .
             Which
             Arithmetical
             Mean
             what
             it
             is
             ,
             is
             defined
             in
             the
             29th
             Article
             of
             the
             13th
             Chapter
             .
          
           
             And
             because
             in
             equal
             times
             the
             wayes
             that
             are
             passed
             are
             as
             the
             Velocities
             ,
             and
             the
             Impetus
             is
             the
             Velocity
             they
             go
             withal
             reckoned
             in
             all
             the
             several
             points
             of
             the
             times
             ,
             it
             followeth
             that
             during
             any
             time
             whatsoever
             ,
             howsoever
             the
             Impetus
             be
             encreased
             or
             decreased
             ,
             the
             length
             of
             the
             way
             passed
             over
             shall
             be
             encreased
             or
             decreased
             in
             the
             same
             proportion
             ;
             and
             the
             same
             Line
             shall
             represent
             both
             the
             way
             of
             the
             Body
             moved
             ,
             and
             the
             several
             Impetus
             or
             degrees
             of
             Swiftness
             wherewith
             the
             way
             is
             passed
             over
             .
          
           
             And
             if
             the
             Body
             moved
             be
             not
             a
             point
             ,
             but
             a
             straight
             line
             moved
             so
             as
             that
             every
             point
             thereof
             make
             a
             several
             straight
             line
             ,
             the
             Plain
             described
             by
             its
             motion
             ,
             whether
             Uniform
             ,
             Accelerated
             or
             Retarded
             ,
             shall
             be
             greater
             or
             less
             (
             the
             time
             being
             the
             same
             )
             in
             the
             same
             proportion
             with
             that
             of
             the
             Impetus
             reckoned
             in
             one
             motion
             to
             the
             Impetus
             reckoned
             in
             the
             other
             .
             For
             the
             reason
             is
             the
             same
             in
             Parallelograms
             and
             their
             Sides
             ,
          
           
             For
             the
             same
             cause
             also
             if
             the
             Body
             moved
             be
             a
             Plain
             ,
             the
             Solid
             described
             shall
             be
             still
             greater
             or
             less
             in
             the
             proportions
             of
             the
             several
             Impetus
             or
             Quicknesses
             reckoned
             through
             one
             Line
             ,
             to
             the
             several
             Impetus
             reckoned
             through
             another
             .
          
           
             This
             understood
             ,
             let
             ABCD
             (
             in
             the
             first
             figure
             of
             the
             17th
             Chapter
             )
             be
             a
             Parallelogram
             ;
             in
             which
             suppose
             the
             side
             AB
             to
             be
             moved
             parallelly
             to
             the
             opposite
             side
             CD
             ,
             decreasing
             al
             the
             way
             till
             it
             vanish
             in
             the
             point
             C
             ,
             and
             so
             describing
             the
             figure
             ABEFC
             ;
             the
             point
             B
             as
             AB
             decreaseth
             ,
             will
             therefore
             describe
             the
             Line
             BEFC
             ;
             and
             suppose
             the
             time
             of
             this
             motion
             designed
             by
             the
             line
             CD
             ;
             and
             in
             the
             same
             time
             CD
             suppose
             the
             side
             AC
             to
             be
             moved
             parallelly
             and
             uniformly
             to
             BD.
             From
             the
             point
             O
             taken
             at
             adventure
             
             in
             the
             Line
             CD
             ,
             draw
             OR
             parallel
             to
             BD
             ,
             cutting
             the
             Line
             BEFC
             in
             E
             ,
             and
             the
             side
             AB
             in
             R.
             And
             again
             from
             the
             point
             Q
             taken
             also
             at
             adventure
             in
             the
             Line
             CD
             ,
             draw
             QS
             parallel
             to
             BD
             ,
             cutting
             the
             Line
             BEFC
             in
             F
             ,
             and
             the
             side
             AB
             in
             S
             ;
             and
             draw
             EG
             and
             FH
             parallel
             to
             CD
             ,
             cutting
             AC
             in
             G
             and
             H.
             Lastly
             ,
             suppose
             the
             same
             construction
             done
             in
             all
             the
             points
             possible
             of
             the
             Line
             BEFC
             .
             I
             sa●
             ,
             that
             as
             the
             proportions
             of
             the
             Swiftnesses
             wherewith
             QF
             ,
             OE
             ,
             DB
             ,
             and
             all
             the
             rest
             supopsed
             to
             be
             drawn
             parallel
             to
             DB
             ,
             and
             terminated
             in
             the
             Line
             BEFC
             ,
             are
             to
             the
             proportions
             of
             their
             several
             Times
             designed
             by
             the
             several
             parallels
             HF
             ,
             GE
             ,
             AB
             and
             all
             the
             rest
             supposed
             to
             be
             drawn
             parallel
             to
             the
             Line
             of
             time
             CD
             ,
             and
             terminated
             in
             the
             Line
             BEFC
             (
             the
             aggregate
             to
             the
             aggregate
             )
             so
             is
             the
             Area
             or
             Plain
             DBEFC
             to
             the
             Area
             or
             Plain
             ACFEB
             .
             For
             as
             AB
             decreasing
             continually
             by
             the
             line
             BEFC
             vanisheth
             in
             the
             time
             CD
             into
             the
             point
             C
             ,
             so
             in
             the
             same
             time
             the
             line
             DC
             continually
             decreasing
             vanisheth
             by
             the
             same
             line
             CFEB
             into
             the
             point
             B
             ;
             and
             the
             point
             D
             describeth
             in
             that
             decreasing
             motion
             the
             line
             DB
             equall
             to
             the
             line
             AC
             described
             by
             the
             point
             A
             in
             the
             decreasing
             motion
             of
             A
             &
             B
             ;
             &
             their
             swiftnesses
             are
             therefore
             equal
             .
             Again
             ,
             because
             in
             the
             time
             GE
             the
             point
             O
             describeth
             the
             line
             OE
             ,
             and
             in
             the
             same
             time
             the
             point
             R
             describeth
             the
             line
             RE
             ,
             the
             line
             OE
             shall
             be
             to
             the
             line
             RE
             ,
             as
             the
             swiftness
             wherewith
             OE
             is
             described
             to
             the
             swiftness
             wherwith
             RE
             is
             described
             .
             In
             like
             māner
             ,
             because
             in
             the
             same
             time
             HF
             the
             point
             Q
             describeth
             the
             Line
             QF
             ,
             and
             the
             point
             S
             the
             Line
             SF
             ,
             it
             shall
             be
             as
             the
             swiftness
             by
             which
             QF
             is
             described
             to
             the
             swiftness
             by
             which
             SF
             is
             described
             ,
             so
             the
             Line
             it self
             QF
             to
             the
             Line
             it self
             SF
             ;
             and
             so
             in
             all
             the
             Lines
             that
             can
             possibly
             be
             drawn
             parallel
             to
             BD
             in
             the
             points
             where
             they
             cut
             the
             Line
             BEFC
             .
             But
             all
             the
             parallels
             to
             BD
             ,
             as
             RE
             ,
             SF
             ,
             AC
             and
             the
             rest
             that
             can
             possibly
             be
             drawn
             from
             the
             Line
             AB
             to
             the
             Line
             BEFC
             make
             the
             Area
             of
             the
             Plain
             ABEFC
             ;
             and
             all
             the
             parallels
             to
             the
             same
             BD
             ,
             as
             QF
             ,
             OE
             ,
             DB
             &
             the
             rest
             drawn
             to
             the
             points
             where
             they
             cut
             the
             same
             Line
             BEFC
             make
             the
             Area
             of
             the
             Plain
             BEFCD
             .
             As
             therefore
             the
             aggregate
             of
             the
             Swiftnesses
             wherwith
             the
             Plain
             BEFCD
             is
             described
             is
             to
             the
             aggregate
             of
             the
             Swiftnesses
             wherewith
             the
             Plain
             ACFEB
             is
             described
             ,
             so
             is
             the
             Plain
             it self
             BEFCD
             to
             the
             
             Plain
             it self
             ACFEB
             .
             But
             the
             aggregate
             of
             the
             Times
             represented
             by
             the
             parallels
             AB
             ,
             GE
             ,
             HF
             and
             the
             rest
             ,
             maketh
             also
             the
             Area
             ACFEB
             .
             And
             therefore
             as
             the
             aggregate
             of
             all
             the
             Lines
             QF
             ,
             OE
             ,
             DB
             and
             all
             the
             rest
             of
             the
             Lines
             parallel
             to
             BD
             and
             terminated
             in
             the
             Line
             BEFC
             is
             to
             the
             aggregate
             of
             all
             the
             Lines
             HF
             ,
             GE
             ,
             AB
             and
             all
             the
             rest
             of
             the
             Lines
             parallel
             to
             CD
             &
             terminated
             in
             the
             same
             Line
             BEFC
             ;
             that
             is
             ,
             as
             the
             aggregate
             of
             the
             Lines
             of
             Swiftness
             to
             the
             aggregate
             of
             the
             lines
             of
             Time
             ,
             or
             as
             the
             whole
             Swiftness
             in
             the
             parallels
             to
             DB
             to
             the
             whole
             Time
             in
             the
             parallels
             to
             CD
             ,
             so
             is
             the
             Plain
             BEFCD
             to
             the
             Plain
             ACFEB
             .
             And
             the
             proportions
             of
             QF
             to
             FH
             ,
             and
             of
             OE
             to
             EG
             ,
             and
             of
             DB
             to
             BA
             ,
             and
             so
             of
             all
             the
             rest
             taken
             together
             ,
             are
             the
             proportion
             of
             the
             Plain
             DBEFC
             to
             the
             Plain
             ABEFC
             .
             But
             the
             Lines
             QF
             ,
             OE
             ,
             DB
             and
             the
             rest
             are
             the
             Lines
             that
             designe
             the
             Swifness
             ;
             and
             the
             Lines
             HF
             ,
             GE
             ,
             AB
             &
             the
             rest
             are
             the
             Lines
             that
             designe
             the
             Times
             of
             the
             motions
             ;
             and
             therefore
             the
             proportion
             of
             the
             Plain
             DBEFC
             to
             the
             Plain
             ABEFC
             is
             the
             proportions
             of
             all
             the
             Velocities
             taken
             together
             ,
             to
             all
             the
             Times
             taken
             together
             .
             Wherefore
             as
             the
             proportions
             of
             the
             Swiftnesses
             ,
             &c.
             which
             was
             to
             be
             demonstrated
             .
          
           
             The
             same
             holds
             also
             in
             the
             diminution
             of
             the
             Circles
             whereof
             the
             lines
             of
             Time
             are
             the
             Semidiameters
             ,
             as
             may
             easily
             be
             conceived
             by
             imagining
             the
             whole
             Plain
             ABCD
             turned
             round
             upon
             the
             Axis
             BD
             ;
             for
             the
             Line
             BEFC
             will
             be
             every
             where
             in
             the
             Superficies
             so
             made
             ,
             and
             the
             Lines
             HG
             ,
             GE
             ,
             AB
             which
             here
             are
             Parallelograms
             will
             be
             there
             Cylinders
             ,
             the
             Diameters
             of
             whose
             bases
             are
             the
             lines
             HF
             ,
             GE
             ,
             AB
             ,
             &c.
             and
             the
             Altitude
             a
             point
             ,
             that
             is
             to
             say
             ,
             a
             quantity
             less
             then
             any
             quantity
             that
             can
             possibly
             be
             named
             ;
             and
             the
             Lines
             QF
             ,
             OE
             ,
             DB
             ,
             &c.
             small
             solids
             whose
             lengths
             and
             breadths
             are
             less
             then
             any
             quantity
             that
             can
             be
             named
             .
          
           
             But
             this
             is
             to
             be
             noted
             ,
             that
             unless
             the
             proportion
             of
             the
             summe
             of
             the
             Swiftnesses
             to
             the
             proportion
             of
             the
             summe
             of
             the
             Times
             be
             determined
             ,
             the
             proportion
             of
             the
             Figure
             DBEFC
             to
             the
             Figure
             ABEFC
             cannot
             be
             determined
             .
          
           
             Thirdly
             ,
             I
             define
             RESISTANCE
             
               to
               be
               the
               endeavour
               of
               one
               moved
               Body
               ,
               either
               wholly
               or
               in
               part
               contrary
               to
               the
               endeavour
               of
               another
               moved
               Body
               ,
               which
               toucheth
               the
               same
               .
            
             I
             say
             wholly
             contrary
             ,
             when
             the
             endeavour
             of
             two
             Bodies
             proceeds
             in
             the
             same
             straight
             Line
             from
             the
             opposite
             extremes
             ,
             and
             contrary
             in
             part
             ,
             when
             two
             Bodies
             
             have
             their
             endeavour
             in
             two
             Lines
             ,
             which
             proceeding
             from
             the
             extreme
             points
             of
             a
             straight
             Line
             ,
             meet
             without
             the
             same
             .
          
           
             Fourthly
             ,
             that
             I
             may
             define
             what
             it
             is
             to
             PRESSE
             ,
             I
             say
             that
             
               Of
               two
               moved
               Bodies
               one
               Presses
               the
               other
               ,
               when
               with
               its
               Endeavour
               it
               makes
               either
               all
               or
               part
               of
               the
               other
               Body
               to
               go
               out
               of
               its
               place
               .
            
          
           
             Fifthly
             ,
             
               A
               Body
               which
               is
               pressed
               and
               not
               wholly
               removed
               is
               said
               to
            
             RESTORE
             
               it self
               ,
               when
               (
               the
               pressing
               Body
               being
               taken
               away
               )
               the
               parts
               which
               were
               moved
               ,
               do
               by
               reason
               of
               the
               internal
               constitution
               of
               the
               pressed
               Body
               ,
               return
               every
               one
               into
               its
               own
               place
               .
            
             And
             this
             we
             may
             observe
             in
             Springs
             ,
             in
             blown
             Bladders
             ,
             and
             in
             many
             other
             Bodies
             ,
             whose
             parts
             yeild
             more
             or
             less
             to
             the
             Endeavour
             which
             the
             pressing
             Body
             makes
             at
             the
             first
             arrival
             ;
             but
             afterwards
             (
             when
             the
             pressing
             Body
             is
             removed
             )
             they
             do
             by
             some
             force
             within
             them
             Restore
             themselves
             ,
             and
             give
             their
             whole
             Body
             the
             same
             figure
             it
             had
             before
             .
          
           
             Sixthly
             ,
             I
             define
             FORCE
             
               to
               be
               the
            
             Impetus
             
               or
               Quickness
               of
               Motion
               multiplyed
               either
               into
               it self
               ,
               or
               into
               the
               Magnitude
               of
               the
               Movent
               ,
               by
               means
               wherof
               the
               said
               Movent
               works
               more
               or
               less
               upon
               the
               Body
               that
               resists
               it
               .
            
          
           
             3
             Having
             premised
             thus
             much
             ,
             I
             shal
             now
             demonstrate
             ,
             First
             ,
             That
             if
             a
             point
             moved
             come
             to
             touch
             another
             point
             which
             is
             at
             rest
             ,
             how
             little
             soever
             the
             Impetus
             or
             quickness
             of
             its
             motion
             be
             ,
             it
             shall
             move
             that
             other
             point
             .
             For
             if
             by
             that
             Impetus
             it
             do
             not
             at
             all
             move
             it
             out
             of
             its
             place
             ,
             neither
             shall
             it
             move
             it
             ,
             with
             double
             the
             same
             Impetus
             ;
             for
             nothing
             doubled
             is
             still
             nothing
             ;
             and
             for
             the
             same
             reason
             it
             shall
             never
             move
             it
             with
             that
             Impetus
             how
             many
             times
             soever
             it
             be
             multiplyed
             ,
             because
             nothing
             how
             soever
             it
             be
             multiplyed
             will
             for
             ever
             be
             nothing
             .
             Wherefore
             when
             a
             point
             is
             at
             rest
             ,
             if
             it
             do
             not
             yeild
             to
             the
             least
             Impetus
             ,
             it
             will
             yeild
             to
             none
             ,
             and
             consequently
             it
             will
             be
             impossible
             that
             that
             which
             is
             at
             rest
             should
             ever
             be
             moved
             .
          
           
             Secondly
             ,
             that
             when
             a
             point
             moved
             ,
             how
             little
             soever
             the
             Impetus
             thereof
             be
             ,
             falls
             upon
             a
             point
             of
             any
             Body
             at
             rest
             ,
             how
             hard
             soever
             that
             Body
             be
             ,
             it
             will
             at
             the
             first
             touch
             make
             it
             yeild
             a
             little
             .
             For
             if
             it
             do
             not
             yeild
             to
             the
             Impetus
             which
             is
             in
             that
             point
             ,
             neither
             will
             it
             yeild
             to
             the
             Impetus
             of
             never
             so
             many
             points
             ,
             which
             have
             all
             their
             Impetus
             severally
             equal
             to
             the
             Impetus
             of
             that
             point
             .
             For
             seeing
             all
             those
             points
             together
             work
             equally
             ,
             if
             any
             one
             of
             them
             have
             no
             effect
             ,
             the
             aggregate
             of
             them
             all
             together
             shall
             have
             no
             effect
             as
             many
             times
             told
             as
             there
             are
             points
             in
             the
             whole
             Body
             ,
             that
             is
             ,
             still
             no
             effect
             at
             all
             ;
             and
             by
             consequent
             
             there
             would
             be
             some
             Bodies
             so
             hard
             that
             it
             would
             be
             impossible
             to
             break
             them
             ;
             that
             is
             ,
             a
             finite
             hardnesse
             ,
             or
             a
             finite
             force
             would
             not
             yeild
             to
             that
             which
             is
             infinite
             ;
             which
             is
             absurd
             .
          
           
             Corollary
             .
             It
             is
             therefore
             manifest
             ,
             that
             Rest
             does
             nothing
             at
             all
             ,
             nor
             is
             of
             any
             efficacy
             ;
             and
             that
             nothing
             but
             Motion
             gives
             Motion
             to
             such
             things
             as
             be
             at
             Rest
             ,
             and
             takes
             it
             from
             things
             moved
             ,
          
           
             Thirdly
             ,
             that
             Cessation
             in
             the
             Movent
             does
             not
             cause
             Cessation
             in
             that
             which
             was
             moved
             by
             it
             .
             For
             (
             by
             the
             11th
             Number
             of
             the
             1
             Article
             of
             this
             Chapter
             )
             whatsoever
             is
             moved
             ,
             persevers
             in
             the
             same
             way
             ,
             &
             with
             the
             same
             Swiftness
             ,
             as
             long
             as
             it
             is
             not
             hindered
             by
             some
             thing
             that
             is
             moved
             against
             it
             .
             Now
             it
             is
             manifest
             ,
             that
             Cessation
             is
             not
             contrary
             Motion
             ;
             and
             therefore
             it
             follows
             ,
             that
             the
             standing
             still
             of
             the
             Movent
             ,
             does
             not
             make
             it
             necessary
             that
             the
             thing
             moved
             should
             also
             stand
             still
             .
          
           
             Corollary
             .
             They
             are
             therefore
             deceived
             ,
             that
             reckon
             the
             taking
             away
             of
             the
             impediment
             or
             resistance
             ,
             for
             one
             of
             the
             causes
             of
             Motion
             .
          
           
             4
             Motion
             is
             brought
             into
             account
             for
             divers
             respects
             ;
             First
             ,
             as
             in
             a
             Body
             Undivided
             ,
             (
             that
             is
             ,
             considered
             as
             a
             point
             )
             ;
             or
             ,
             as
             in
             a
             Divided
             Body
             .
             In
             an
             Undivided
             Body
             ,
             when
             we
             suppose
             the
             way
             by
             which
             the
             Motion
             is
             made
             ,
             to
             be
             a
             Line
             ;
             and
             in
             a
             Divided
             Body
             ,
             when
             we
             compute
             the
             Motion
             of
             the
             several
             parts
             of
             that
             Body
             ,
             as
             of
             Parts
             .
          
           
             Secondly
             ,
             From
             the
             diversity
             of
             the
             regulation
             of
             Motion
             ,
             it
             is
             in
             a
             Body
             considered
             as
             Undivided
             ,
             sometimes
             Uniform
             ,
             and
             sometimes
             
               Multiform
               .
               Uniform
            
             is
             that
             by
             which
             equal
             Lines
             are
             alwayes
             transmitted
             in
             equal
             times
             ;
             &
             Multiform
             ,
             when
             in
             one
             time
             more
             ,
             in
             another
             time
             less
             space
             is
             transmitted
             .
             Again
             ,
             of
             Multiform
             Motions
             ,
             there
             are
             some
             in
             which
             the
             degrees
             of
             Acceleration
             and
             Retardation
             proceed
             in
             the
             same
             proportions
             which
             the
             Spaces
             transmitted
             have
             ,
             whether
             duplicate
             ,
             or
             triplicate
             ,
             or
             by
             whatsoever
             number
             multiplyed
             ;
             and
             others
             in
             which
             it
             is
             otherwise
             .
          
           
             Thirdly
             ,
             from
             the
             number
             of
             the
             Movents
             ;
             that
             is
             ,
             one
             Motion
             is
             made
             by
             one
             Movent
             onely
             ,
             and
             another
             by
             the
             concourse
             of
             many
             Movents
             .
          
           
             Fourthly
             ,
             from
             the
             position
             of
             that
             Line
             in
             which
             a
             Body
             is
             moved
             ,
             in
             respect
             of
             some
             other
             Line
             ;
             and
             from
             hence
             one
             Motion
             is
             called
             Perpendicular
             ,
             another
             Oblique
             ,
             another
             Parallel
             .
          
           
           
             Fifthly
             ,
             from
             the
             position
             of
             the
             Movent
             in
             respect
             of
             the
             Moved
             Body
             ;
             from
             whence
             one
             Motion
             is
             Pulsion
             or
             Driving
             ;
             another
             Traction
             or
             Drawing
             .
             PULSION
             ,
             when
             the
             Movent
             makes
             the
             Moved
             Body
             goe
             before
             it
             ;
             and
             TRACTION
             ,
             when
             it
             makes
             it
             follow
             .
             Again
             ,
             there
             are
             two
             sorts
             of
             Pulsion
             ;
             one
             ,
             when
             the
             motions
             of
             the
             Movent
             and
             Moved
             Body
             begin
             both
             together
             ,
             which
             may
             be
             called
             TRUSION
             or
             Thrusting
             and
             VECTION
             ;
             the
             other
             ,
             when
             the
             Movent
             is
             first
             moved
             ,
             and
             afterwards
             the
             Moved
             Body
             ,
             which
             Motion
             is
             called
             PERCUSSION
             or
             Stroke
             .
          
           
             Sixthly
             ,
             Motion
             is
             considered
             sometimes
             from
             the
             Effect
             onely
             which
             the
             Movent
             works
             in
             the
             Moved
             Body
             ,
             which
             is
             usually
             called
             Moment
             .
             Now
             MOMENT
             
               is
               the
               Excess
               of
               Motion
               which
               the
               Movent
               has
               ,
               above
               the
               Motion
               or
               Endeavour
               of
               the
               Resisting
               Body
               .
            
          
           
             Seventhly
             ,
             it
             may
             be
             considered
             from
             the
             diversity
             of
             the
             Medium
             ;
             as
             one
             Motion
             may
             be
             made
             in
             Vacuity
             or
             
               empty
               Place
            
             ;
             another
             in
             a
             fluid
             ;
             another
             in
             a
             
               consistent
               Medium
            
             ,
             that
             is
             ,
             a
             Medium
             whose
             parts
             are
             by
             some
             power
             so
             consistent
             and
             cohering
             ,
             that
             no
             part
             of
             the
             same
             will
             yeild
             to
             the
             Movent
             ,
             unless
             the
             whole
             yeild
             also
             .
          
           
             Eighthly
             ,
             when
             a
             Moved
             Body
             is
             considered
             as
             having
             parts
             ,
             there
             arises
             another
             distinction
             of
             Motion
             into
             Simple
             and
             
               Compounded
               .
               Simple
            
             ,
             when
             all
             the
             several
             parts
             describe
             several
             equal
             lines
             ;
             Compounded
             ,
             when
             the
             lines
             described
             are
             Unequal
             .
          
           
             5
             All
             Endeavour
             tends
             towards
             that
             part
             ,
             that
             is
             to
             say
             ,
             in
             that
             way
             which
             is
             determined
             by
             the
             Motion
             of
             the
             Movent
             ,
             if
             the
             Movent
             be
             but
             one
             ;
             or
             ,
             if
             there
             be
             many
             Movents
             ,
             in
             that
             way
             which
             their
             concourse
             determines
             .
             For
             example
             ,
             if
             a
             Moved
             Body
             have
             direct
             Motion
             ,
             its
             first
             Endeavour
             will
             be
             in
             a
             Straight
             line
             ;
             if
             it
             have
             Circular
             Motion
             ,
             its
             first
             Endeavour
             will
             be
             in
             the
             Circumference
             of
             a
             Circle
             ;
             &
             whatsoever
             the
             line
             be
             in
             which
             a
             Body
             has
             its
             Motion
             from
             the
             concourse
             of
             two
             Movents
             ,
             as
             soon
             as
             in
             any
             point
             thereof
             the
             force
             of
             one
             of
             the
             Movents
             ceases
             ,
             there
             immediately
             the
             former
             Endeavour
             of
             that
             Body
             will
             be
             changed
             into
             an
             Endeavour
             in
             the
             line
             of
             the
             other
             Movent
             .
          
           
             6
             Wherefore
             ,
             when
             any
             Body
             is
             carried
             on
             by
             the
             concourse
             of
             two
             Winds
             ,
             one
             of
             those
             Winds
             ceasing
             ,
             the
             Endeavour
             and
             Motion
             of
             that
             Body
             will
             be
             in
             that
             line
             ,
             in
             which
             it
             would
             have
             been
             carried
             by
             that
             Wind
             alone
             which
             blows
             still
             .
             And
             in
             the
             describing
             
             of
             a
             Circle
             ,
             where
             that
             which
             is
             moved
             has
             its
             Motion
             determined
             by
             a
             Movent
             in
             a
             Tangent
             ,
             and
             by
             the
             Radius
             which
             keeps
             it
             in
             a
             certain
             distance
             from
             the
             Center
             ,
             if
             the
             retention
             of
             the
             Radius
             cease
             ,
             that
             Endeavour
             which
             was
             in
             the
             Circumference
             of
             the
             Circle
             ,
             will
             now
             be
             in
             the
             Tangent
             ,
             that
             is
             ,
             in
             a
             Straight
             line
             .
             For
             seeing
             Endeavour
             is
             computed
             in
             a
             lesse
             part
             of
             the
             Circumference
             then
             can
             be
             given
             ,
             that
             is
             ,
             in
             a
             point
             ,
             the
             way
             by
             which
             a
             Body
             is
             moved
             in
             the
             Circumference
             is
             compounded
             of
             innumerable
             Straight
             lines
             ;
             of
             which
             every
             one
             is
             less
             then
             can
             be
             given
             ,
             which
             are
             therefore
             called
             Points
             .
             Wherefore
             when
             any
             Body
             which
             is
             moved
             in
             the
             Circumference
             of
             a
             Circle
             ,
             is
             freed
             from
             the
             retention
             of
             the
             Radius
             ,
             it
             will
             proceed
             in
             one
             of
             those
             Straight
             lines
             ,
             that
             is
             ,
             in
             a
             Tangent
             .
          
           
             7
             All
             Endeavour
             ,
             whether
             strong
             or
             weak
             ,
             is
             propagated
             to
             infinite
             distance
             ;
             for
             it
             is
             Motion
             .
             If
             therefore
             the
             first
             Endeavour
             of
             a
             Body
             be
             made
             in
             Space
             which
             is
             empty
             ,
             it
             will
             alwayes
             proceed
             with
             the
             same
             Velocity
             ;
             for
             it
             cannot
             be
             supposed
             that
             it
             can
             receive
             any
             resistance
             at
             all
             from
             empty
             Space
             ;
             and
             therefore
             (
             by
             the
             7
             Article
             of
             the
             9
             Chapter
             )
             it
             will
             alwayes
             proceed
             in
             the
             same
             way
             and
             with
             the
             same
             Swiftness
             .
             And
             if
             its
             Endeavour
             be
             in
             Space
             which
             is
             filled
             ,
             yet
             seeing
             Endeavour
             is
             Motion
             ,
             that
             which
             stands
             next
             in
             its
             way
             shall
             be
             removed
             ,
             and
             endeavour
             further
             ,
             and
             again
             remove
             that
             which
             stands
             next
             ,
             &
             so
             infinitely
             .
             Wherefore
             the
             propagation
             of
             Endeavour
             from
             one
             part
             of
             full
             Space
             to
             another
             ,
             proceeds
             infinitely
             .
             Besides
             ,
             it
             reaches
             in
             any
             instant
             to
             any
             distance
             ,
             how
             great
             soever
             ;
             For
             in
             the
             same
             instant
             in
             which
             the
             first
             part
             of
             the
             full
             Medium
             removes
             that
             which
             is
             next
             it
             ,
             the
             second
             also
             removes
             that
             part
             which
             is
             next
             to
             it
             ;
             and
             therefore
             all
             Endeavour
             ,
             whether
             it
             be
             in
             empty
             or
             in
             full
             Space
             ,
             proceeds
             not
             onely
             to
             any
             distance
             how
             great
             soever
             ,
             but
             also
             in
             any
             time
             how
             little
             soever
             ,
             that
             is
             ,
             in
             an
             instant
             .
             Nor
             makes
             it
             any
             matter
             ,
             that
             Endeavour
             by
             proceeding
             growes
             weaker
             and
             weaker
             ,
             till
             at
             last
             it
             can
             no
             longer
             be
             perceived
             by
             Sense
             ;
             for
             Motion
             may
             be
             insensible
             ;
             and
             I
             do
             not
             here
             examine
             things
             by
             Sense
             and
             Experience
             ,
             but
             by
             Reason
             .
          
           
             8
             When
             two
             Movents
             are
             of
             equal
             Magnitude
             ,
             the
             swifter
             of
             them
             works
             with
             greater
             force
             then
             the
             slower
             upon
             a
             Body
             that
             
             resists
             their
             Motion
             .
             Also
             if
             two
             Movents
             have
             equal
             Velocity
             ,
             the
             greater
             of
             them
             works
             with
             more
             force
             then
             the
             less
             .
             For
             where
             the
             Magnitude
             is
             equal
             ,
             the
             Movent
             of
             greater
             Velocity
             makes
             the
             greater
             impression
             upon
             that
             Body
             upon
             which
             it
             falls
             ;
             and
             where
             the
             Velocity
             is
             equal
             ,
             the
             Movent
             of
             greater
             Magnitude
             falling
             upon
             the
             same
             point
             ,
             or
             an
             equal
             part
             of
             another
             Body
             ,
             loses
             less
             of
             its
             Velocity
             ,
             because
             the
             resisting
             Body
             works
             onely
             upon
             that
             part
             of
             the
             Movent
             which
             it
             touches
             ,
             and
             therefore
             abates
             the
             Impetus
             of
             that
             part
             onely
             ,
             whereas
             in
             the
             mean
             time
             the
             parts
             which
             are
             not
             touched
             proceed
             ,
             and
             retein
             their
             whole
             force
             till
             they
             also
             come
             to
             be
             touched
             ,
             and
             their
             force
             has
             some
             effect
             .
             Wherfore
             (
             for
             example
             )
             in
             Batteries
             ,
             a
             longer
             then
             a
             shorter
             piece
             of
             Timber
             of
             the
             same
             thickness
             and
             velocity
             ,
             and
             a
             thicker
             then
             a
             slenderer
             piece
             of
             the
             same
             length
             and
             velocity
             ,
             works
             a
             greater
             effect
             upon
             the
             Wall.
             
          
        
         
           
           
             
               CHAP.
               XVI
            
             .
             Of
             
               Motion
               Accelerated
            
             and
             Vniform
             ,
             and
             of
             Motion
             by
             Concourse
             .
          
           
             
               
                 1
                 The
                 Velocity
                 of
                 any
                 Body
                 ,
                 in
                 what
                 Time
                 soever
                 it
                 be
                 computed
                 ,
                 is
                 that
                 which
                 is
                 made
                 of
                 the
                 multiplication
                 of
                 the
                 Impetus
                 ,
                 or
                 Quickness
                 of
                 its
                 Motion
                 into
                 the
                 Time.
                 2
                 ,
                 &c.
                 
                 In
                 all
                 Motion
                 ,
                 the
                 Lengths
                 which
                 are
                 passed
                 through
                 ,
                 are
                 to
                 one
                 another
                 ,
                 as
                 the
                 Products
                 made
                 by
                 the
                 Impetus
                 multipyed
                 into
                 the
                 Time.
                 
              
               
                 6
                 If
                 two
                 Bodies
                 be
                 moved
                 with
                 Uniform
                 Motion
                 through
                 two
                 Lengths
                 ,
                 the
                 proportion
                 of
                 those
                 Lengths
                 to
                 one
                 another
                 will
                 be
                 compounded
                 of
                 the
                 proportions
                 of
                 Time
                 to
                 Time
                 ,
                 and
                 Impetus
                 to
                 Impetus
                 ,
                 directly
                 taken
                 .
              
               
                 7
                 If
                 two
                 Bodies
                 pass
                 through
                 two
                 Lengths
                 with
                 Uniform
                 Motion
                 ,
                 the
                 proportion
                 of
                 their
                 Times
                 to
                 one
                 another
                 will
                 be
                 compounded
                 of
                 the
                 proportions
                 of
                 Length
                 to
                 Length
                 and
                 Impetus
                 to
                 Impetus
                 ,
                 reciprocally
                 taken
                 ;
                 also
                 the
                 proportion
                 of
                 their
                 Impetus
                 to
                 one
                 another
                 will
                 be
                 compounded
                 of
                 the
                 proportions
                 of
                 Length
                 to
                 Length
                 and
                 Time
                 to
                 Time
                 ,
                 reciprocally
                 taken
                 .
              
               
                 8
                 If
                 a
                 Body
                 be
                 carried
                 on
                 with
                 Uniform
                 Motion
                 by
                 two
                 Movents
                 together
                 ,
                 which
                 meet
                 in
                 an
                 Angle
                 ,
                 the
                 line
                 by
                 which
                 it
                 passes
                 will
                 be
                 a
                 straight
                 line
                 subtending
                 the
                 complement
                 of
                 that
                 Angle
                 to
                 two
                 right
                 Angles
                 .
                 9
                 ,
                 &c.
                 If
                 a
                 Body
                 be
                 carried
                 by
                 two
                 Movents
                 together
                 ,
                 one
                 of
                 them
                 being
                 moved
                 with
                 Uniform
                 ,
                 the
                 other
                 with
                 Accelerated
                 Motion
                 ,
                 and
                 the
                 proportion
                 of
                 their
                 Lengths
                 to
                 their
                 Times
                 being
                 explicable
                 in
                 numbers
                 ,
                 How
                 to
                 find
                 out
                 what
                 line
                 that
                 Body
                 describes
                 .
              
            
          
           
             1
             THe
             Velocity
             of
             any
             Body
             ,
             in
             whatsoever
             Time
             it
             be
             moved
             ,
             has
             its
             quantity
             determined
             by
             the
             sum
             of
             all
             the
             several
             Quicknesses
             or
             Impetus
             which
             it
             hath
             in
             the
             several
             points
             of
             the
             Time
             of
             the
             Bodies
             Motion
             .
             For
             seeing
             Velocity
             (
             by
             the
             Definition
             of
             it
             Chap.
             8.
             
             Art.
             15.
             )
             is
             that
             Power
             by
             which
             a
             Body
             can
             in
             a
             certain
             time
             pass
             through
             a
             certain
             length
             ;
             and
             Quickness
             of
             Motion
             ,
             or
             Impetus
             (
             by
             the
             15
             
               Chap.
               Artic.
            
             2.
             
             Numb
             .
             2.
             )
             is
             Velocity
             taken
             in
             one
             point
             of
             time
             onely
             ,
             all
             the
             Impetus
             together
             taken
             in
             all
             the
             points
             of
             time
             ,
             will
             be
             the
             same
             thing
             with
             the
             Mean
             Impetus
             multiplyed
             into
             the
             whole
             Time
             ,
             or
             which
             is
             all
             one
             ,
             will
             be
             the
             Velocity
             of
             the
             whole
             Motion
             .
          
           
           
             Corollary
             .
             If
             the
             Impetus
             be
             the
             same
             in
             every
             point
             ,
             any
             straight
             line
             representing
             it
             may
             be
             taken
             for
             the
             measure
             of
             Time
             ;
             and
             the
             Quicknesses
             or
             Impetus
             applyed
             ordinately
             to
             any
             straight
             line
             making
             an
             Angle
             with
             it
             ,
             and
             representing
             the
             way
             of
             the
             Bodies
             motion
             ,
             will
             designe
             a
             parallelogram
             which
             shall
             represent
             the
             Velocity
             of
             the
             whole
             Motion
             .
             But
             if
             the
             Impetus
             or
             Quickness
             of
             Motion
             begin
             from
             Rest
             ,
             and
             increase
             Uniformly
             ,
             that
             is
             ,
             in
             the
             same
             proportion
             continually
             with
             the
             times
             which
             are
             passed
             ,
             the
             whole
             .
             Velocity
             of
             the
             Motion
             shall
             be
             represented
             by
             a
             Triangle
             ,
             one
             side
             whereof
             is
             the
             whole
             time
             ,
             and
             the
             other
             the
             greatest
             Impetus
             acquired
             in
             that
             time
             ;
             or
             else
             by
             a
             parallelogram
             ,
             one
             of
             whose
             sides
             is
             the
             whole
             time
             of
             Motion
             ,
             and
             the
             other
             ,
             half
             the
             greatest
             Impetus
             ;
             or
             lastly
             by
             a
             parallelogram
             having
             for
             one
             side
             a
             mean
             proportional
             between
             the
             whole
             time
             &
             the
             half
             of
             that
             time
             ,
             &
             for
             the
             other
             side
             the
             half
             of
             the
             greatest
             Impetus
             .
             For
             both
             these
             parallelograms
             are
             equal
             to
             one
             another
             ,
             &
             severally
             equal
             to
             the
             triangle
             which
             is
             made
             of
             the
             whole
             line
             of
             time
             ,
             and
             the
             greatest
             acquired
             Impetus
             ;
             as
             is
             demonstrated
             in
             the
             Elements
             of
             Geometry
             .
          
           
             2
             In
             all
             Uniform
             Motions
             the
             Lengths
             which
             are
             transmitted
             are
             to
             one
             another
             ,
             as
             the
             product
             of
             the
             mean
             Impetus
             multiplyed
             into
             its
             time
             ,
             to
             the
             product
             of
             the
             mean
             Impetus
             multiplyed
             also
             into
             its
             time
             .
          
           
             For
             let
             AB
             (
             in
             the
             first
             Figure
             )
             be
             the
             Time
             ,
             and
             AC
             the
             Impetus
             by
             which
             any
             Body
             passes
             with
             Uniform
             Motion
             through
             the
             Length
             DE
             ;
             &
             in
             any
             part
             of
             the
             time
             AB
             ,
             as
             in
             the
             time
             AF
             ,
             let
             another
             Body
             be
             moved
             with
             Uniform
             Motion
             ,
             first
             ,
             with
             the
             same
             Impetus
             AC
             .
             This
             Body
             therefore
             in
             the
             time
             AB
             with
             the
             Impetus
             AC
             will
             pass
             through
             the
             length
             AF.
             Seeing
             therefore
             ,
             when
             Bodies
             are
             moved
             in
             the
             same
             Time
             ,
             &
             with
             the
             same
             Velocity
             &
             Impetus
             in
             every
             part
             of
             their
             motion
             ,
             the
             proportion
             of
             one
             Length
             transmitted
             to
             another
             Length
             trāsmitted
             ,
             is
             the
             same
             w
             th
             that
             of
             Time
             to
             Time
             ,
             it
             followeth
             ,
             that
             the
             Length
             transmitted
             in
             the
             time
             AB
             with
             the
             Impetus
             AC
             will
             be
             to
             the
             Length
             transmitted
             in
             the
             time
             AF
             with
             the
             same
             Impetus
             AC
             ,
             as
             AB
             it self
             is
             to
             AF
             ,
             that
             is
             ,
             as
             the
             parallelogram
             AI
             is
             to
             the
             parallelogram
             AH
             ,
             that
             is
             ,
             as
             the
             product
             of
             the
             time
             AB
             into
             the
             mean
             Impetus
             AC
             is
             to
             the
             product
             of
             the
             time
             AF
             into
             the
             same
             Impetus
             AC
             .
             Again
             ,
             let
             it
             be
             supposed
             that
             a
             Body
             be
             moved
             in
             the
             time
             AF
             ,
             
             not
             with
             the
             same
             but
             with
             some
             other
             Uniform
             Impetus
             ,
             as
             A
             L.
             Seeing
             therfore
             one
             of
             the
             Bodies
             has
             in
             all
             the
             parts
             of
             its
             motion
             the
             Impetus
             A
             C
             ,
             and
             the
             other
             in
             like
             manner
             the
             Impetus
             A
             L
             ,
             the
             Length
             trāsmitted
             by
             the
             Body
             moved
             with
             the
             Impetus
             A
             C
             will
             be
             to
             the
             Length
             transmitted
             by
             the
             Body
             moved
             with
             the
             Impetus
             A
             L
             ,
             as
             A
             C
             it self
             is
             to
             A
             L
             ,
             that
             is
             ,
             as
             the
             parallelogram
             A
             H
             is
             to
             the
             parallelogram
             F
             L.
             Wherefore
             ,
             by
             ordinate
             proportion
             it
             will
             be
             ,
             as
             the
             parallelogram
             A
             I
             to
             the
             parallelogram
             F
             L
             ,
             that
             is
             ,
             as
             the
             product
             of
             the
             mean
             Impetus
             into
             the
             Time
             is
             to
             the
             product
             of
             the
             mean
             Impetus
             into
             the
             Time
             ,
             so
             the
             Length
             transmitted
             in
             the
             time
             A
             B
             with
             the
             Impetus
             A
             C
             ,
             to
             the
             length
             transmitted
             in
             the
             time
             A
             F
             with
             the
             Impetus
             ,
             A
             L
             ;
             which
             was
             to
             be
             demonstrated
             .
          
           
             Corollary
             .
             Seeing
             therefore
             in
             Uniform
             Motion
             (
             as
             has
             been
             shewn
             )
             the
             Lengths
             transmitted
             are
             to
             one
             another
             as
             the
             parallelograms
             which
             are
             made
             by
             the
             multiplication
             of
             the
             mean
             Impetus
             into
             the
             Times
             ,
             that
             is
             ,
             (
             by
             reason
             of
             the
             equality
             of
             the
             Impetus
             all
             the
             way
             )
             as
             the
             Times
             themselves
             ,
             it
             will
             also
             be
             by
             permutation
             ,
             as
             to
             Time
             to
             Length
             ,
             so
             Time
             to
             Length
             ;
             and
             in
             general
             ,
             to
             this
             place
             are
             applicable
             all
             the
             properties
             and
             transmutations
             of
             Analogismes
             which
             I
             have
             set
             down
             and
             demonstrated
             in
             the
             13
             Chapter
             .
          
           
             3
             In
             Motion
             begun
             from
             Rest
             ,
             and
             Uniformly
             Accelerated
             (
             that
             is
             ,
             where
             the
             Impetus
             encreaseth
             continually
             according
             to
             the
             proportion
             of
             the
             Times
             )
             it
             will
             also
             be
             ,
             as
             one
             product
             made
             by
             the
             Mean
             Impetus
             multiplyed
             into
             the
             Time
             ,
             to
             another
             product
             made
             likewise
             by
             the
             Mean
             Impetus
             multiplyed
             into
             the
             Time
             ,
             so
             the
             Length
             transmitted
             in
             the
             one
             Time
             ,
             to
             the
             Length
             transmitted
             in
             the
             other
             Time.
             
          
           
             For
             let
             A
             B
             (
             in
             the
             same
             1
             figure
             )
             represent
             a
             Time
             ;
             in
             the
             beginning
             of
             which
             Time
             A
             ,
             let
             the
             Impetus
             be
             as
             the
             point
             A
             ;
             but
             as
             the
             Time
             goes
             on
             ,
             so
             let
             the
             Impetus
             encrease
             Uniformly
             till
             in
             the
             last
             point
             of
             that
             Time
             A
             B
             ,
             namely
             in
             B
             ,
             the
             Impetus
             acquired
             be
             B
             I.
             Again
             ,
             let
             A
             F
             represent
             another
             Time
             ,
             in
             whose
             beginning
             A
             ,
             let
             the
             Impetus
             be
             as
             the
             point
             it self
             A
             ;
             but
             as
             the
             Time
             proceeds
             ,
             so
             let
             the
             Impetus
             encrease
             Uniformly
             till
             in
             the
             last
             point
             F
             of
             the
             Time
             A
             F
             the
             Impetus
             acquired
             be
             F
             K
             ;
             and
             let
             D
             E
             be
             the
             Length
             passed
             through
             in
             the
             Time
             A
             B
             with
             Impetus
             Uniformly
             encreased
             .
             I
             say
             the
             Length
             D
             E
             ,
             is
             to
             the
             Length
             transmitted
             in
             the
             Time
             A
             F
             ,
             as
             the
             Time
             A
             B
             multiplyed
             into
             the
             Mean
             of
             the
             Impetus
             encreasing
             
             through
             the
             time
             A
             B
             ,
             is
             to
             the
             Time
             A
             F
             multiplyed
             into
             the
             Mean
             of
             the
             Impetus
             encreasing
             through
             the
             time
             A
             F.
             
          
           
             For
             seeing
             the
             Triangle
             A
             B
             I
             is
             the
             whole
             Velocity
             of
             the
             Body
             moved
             in
             the
             Time
             A
             B
             till
             the
             Impetus
             acquired
             be
             B
             I
             ;
             and
             the
             Triangle
             A
             F
             K
             the
             whole
             Velocity
             of
             the
             Body
             moved
             in
             the
             Time
             A
             F
             with
             Impetus
             encreasing
             till
             there
             be
             acquired
             the
             Impetus
             F
             K
             ;
             the
             Length
             D
             E
             to
             the
             Length
             acquired
             in
             the
             Time
             A
             F
             with
             Impetus
             encreasing
             from
             Rest
             in
             A
             till
             there
             be
             acquired
             the
             Impetus
             F
             K
             ,
             will
             be
             as
             the
             Triangle
             A
             B
             I
             to
             the
             Triangle
             A
             F
             K
             ,
             that
             is
             ,
             if
             the
             Triangles
             A
             B
             I
             and
             A
             F
             K
             be
             like
             ,
             in
             duplicate
             proportion
             of
             the
             Time
             A
             B
             to
             the
             Time
             A
             F
             ;
             but
             if
             unlike
             ,
             in
             the
             proportion
             compounded
             of
             the
             proportions
             of
             A
             B
             to
             B
             I
             ,
             &
             of
             A
             K
             to
             A
             F.
             Wherefore
             ,
             as
             A
             B
             I
             is
             to
             A
             F
             K
             ,
             so
             let
             D
             E
             be
             to
             D
             P
             ;
             for
             so
             ,
             the
             Length
             transmitted
             in
             the
             Time
             A
             B
             with
             Impetus
             encreasing
             to
             B
             I
             ,
             will
             be
             to
             the
             Length
             transmitted
             in
             the
             Time
             A
             F
             with
             Impetus
             encreasing
             to
             F
             K
             ,
             as
             the
             triangle
             A
             B
             I
             is
             to
             the
             triangle
             A
             F
             K
             ;
             But
             the
             triangle
             A
             B
             I
             is
             made
             by
             the
             multiplication
             of
             the
             Time
             A
             B
             into
             the
             Mean
             of
             the
             Impetus
             encreasing
             to
             B
             I
             ,
             and
             the
             triangle
             A
             F
             K
             is
             made
             by
             the
             multiplication
             of
             the
             Time
             A
             F
             into
             the
             Mean
             of
             the
             Impetus
             encreasing
             to
             F
             K
             ;
             and
             therefore
             the
             Length
             D
             E
             which
             is
             transmitted
             in
             the
             Time
             A
             B
             with
             Impetus
             encreasing
             to
             B
             I
             ,
             to
             the
             Length
             D
             P
             which
             is
             transmitted
             in
             the
             Time
             A
             F
             with
             Impetus
             encreasing
             to
             F
             K
             ,
             is
             as
             the
             product
             which
             is
             made
             of
             the
             Time
             A
             B
             multiplyed
             into
             its
             mean
             Impetus
             ,
             to
             the
             product
             of
             the
             Time
             A
             F
             multiplyed
             also
             into
             its
             mean
             Impetus
             ;
             which
             was
             to
             be
             proved
             .
          
           
             
               Corol.
               1
               In
               Motion
               Uniformly
               accelerated
               ,
               the
               proportion
               of
               the
               Lengths
               transmitted
               ,
               to
               that
               of
               their
               Times
               ,
               is
               compounded
               of
               the
               proportions
               of
               their
               Times
               to
               their
               Times
               and
               Impetus
               to
               Impetus
               .
            
             
               Corol.
               2
               In
               Motion
               Uniformly
               accelerated
               ,
               the
               Lengths
               transmitted
               in
               equal
               times
               taken
               in
               continual
               succession
               from
               the
               beginning
               of
               Motion
               ,
               are
               as
               the
               differences
               of
               square
               numbers
               beginning
               from
               Unity
               ,
               namely
               ,
               as
               3
               ,
               5
               ,
               7
               ,
               &c.
               
               For
               if
               in
               the
               first
               time
               the
               Length
               transmitted
               be
               as
               1
               ,
               in
               the
               first
               and
               second
               times
               the
               Length
               transmitted
               will
               be
               as
               4
               ,
               which
               is
               the
               Square
               of
               2
               ,
               and
               in
               the
               three
               first
               times
               ,
               it
               will
               be
               as
               9
               ,
               which
               is
               the
               Square
               of
               3
               ,
               and
               in
               the
               four
               first
               times
               as
               16
               ,
               and
               so
               on
               .
               Now
               the
               differences
               of
               these
               Squares
               are
               3
               ,
               5
               ,
               7
               ,
               &c.
               
            
             
               Corol.
               3
               In
               Motion
               Uniformly
               accelerated
               from
               Rest
               ,
               the
               Length
               
               transmitted
               ,
               is
               to
               another
               Length
               transmitted
               vniformly
               in
               the
               same
               Time
               ,
               but
               with
               such
               Impetus
               as
               was
               acquired
               by
               the
               accelerated
               Motion
               in
               the
               last
               point
               of
               that
               Time
               ,
               as
               a
               triangle
               to
               a
               parallelogram
               which
               have
               their
               altitude
               and
               base
               common
               .
               For
               seeing
               the
               Length
               D
               E
               (
               in
               the
               same
               1
               figure
               )
               is
               passed
               through
               with
               Velocity
               as
               the
               triangle
               A
               B
               I
               ,
               it
               is
               necessary
               that
               for
               the
               passing
               through
               of
               a
               Length
               which
               is
               double
               to
               D
               E
               ,
               the
               Velocity
               be
               as
               the
               parallelogram
               A
               I
               ;
               for
               the
               parallelogram
               A
               I
               is
               double
               to
               the
               triangle
               A
               B
               I.
               
            
          
           
             4
             In
             Motion
             which
             beginning
             from
             Rest
             ,
             is
             so
             accelerated
             ,
             that
             the
             Impetus
             thereof
             encrease
             continually
             in
             proportion
             duplicate
             to
             the
             proportion
             of
             the
             times
             in
             which
             it
             is
             made
             ,
             a
             Length
             transmitted
             in
             one
             time
             will
             be
             to
             a
             Length
             transmitted
             in
             another
             time
             ,
             as
             the
             product
             made
             by
             the
             Mean
             Impetus
             multiplyed
             into
             the
             time
             of
             one
             of
             those
             Motions
             ,
             to
             the
             product
             of
             the
             Mean
             Impetus
             multiplyed
             into
             the
             time
             of
             the
             other
             Motion
             .
          
           
             For
             let
             A
             B
             (
             in
             the
             2d
             .
             figure
             )
             represent
             a
             Time
             ,
             in
             whose
             first
             instant
             A
             let
             the
             Impetus
             be
             as
             the
             point
             A
             ;
             but
             as
             the
             time
             proceeds
             ,
             so
             let
             the
             Impetus
             encrease
             continually
             in
             duplicate
             proportion
             to
             that
             of
             the
             times
             ,
             till
             in
             the
             last
             point
             of
             time
             B
             the
             Impetus
             acquired
             be
             B
             I
             ;
             then
             taking
             the
             point
             F
             any
             where
             in
             the
             time
             A
             B
             ,
             let
             the
             Impetus
             F
             K
             acquired
             in
             the
             time
             A
             F
             be
             ordinately
             applyed
             to
             that
             point
             F.
             Seeing
             therefore
             the
             proportion
             of
             F
             K
             to
             B
             I
             is
             supposed
             to
             be
             duplicate
             to
             that
             of
             A
             F
             to
             A
             B
             ,
             the
             proportion
             of
             A
             F
             to
             A
             B
             will
             be
             subduplicate
             to
             that
             of
             F
             K
             to
             B
             I
             ;
             and
             that
             of
             A
             B
             to
             A
             F
             will
             be
             (
             by
             Chap.
             13.
             
             Article
             16
             )
             duplicate
             to
             that
             of
             B
             I
             to
             F
             K
             ,
             and
             consequently
             the
             point
             K
             will
             be
             in
             a
             parabolical
             line
             whose
             diameter
             is
             A
             B
             and
             base
             B
             I
             ;
             and
             for
             the
             same
             reason
             ,
             to
             what
             point
             soever
             of
             the
             time
             A
             B
             the
             Impetus
             acquired
             in
             that
             time
             be
             ordinately
             applyed
             ,
             the
             straight
             line
             designing
             that
             Impetus
             will
             be
             in
             the
             same
             parabolical
             line
             A
             K
             I.
             Wherefore
             the
             mean
             Impetus
             multiplyed
             into
             the
             whole
             time
             A
             B
             will
             be
             the
             Parabola
             A
             K
             I
             B
             ,
             equal
             to
             the
             parallelogram
             A
             M
             ,
             which
             parallelogram
             has
             for
             one
             side
             the
             line
             of
             time
             A
             B
             and
             for
             the
             other
             the
             line
             of
             the
             Impetus
             A
             L
             ,
             which
             is
             two
             thirds
             of
             the
             Impetus
             B
             I
             ;
             for
             every
             Parabola
             is
             equal
             to
             two
             thirds
             of
             that
             parallelogram
             with
             which
             it
             has
             its
             altitude
             and
             base
             common
             .
             Wherefore
             the
             whole
             Velocity
             in
             A
             B
             will
             be
             the
             parallelogram
             A
             M
             ,
             as
             being
             made
             by
             the
             multiplication
             of
             the
             Impetus
             A
             L
             into
             the
             time
             A
             B.
             And
             in
             like
             manner
             ,
             if
             F
             N
             
             be
             taken
             ,
             which
             is
             two
             thirds
             of
             the
             Impetus
             F
             K
             ,
             and
             the
             parallelogram
             F
             O
             be
             completed
             ,
             F
             O
             will
             be
             the
             whole
             Velocity
             in
             the
             time
             A
             F
             ,
             as
             being
             made
             by
             the
             Uniform
             Impetus
             A
             O
             or
             F
             N
             multiplyed
             into
             the
             time
             A
             F.
             Let
             now
             the
             length
             transmitted
             in
             the
             time
             A
             B
             and
             with
             the
             Velocity
             A
             M
             be
             the
             straight
             line
             D
             E
             ;
             and
             lastly
             ,
             let
             the
             Length
             transmitted
             in
             the
             time
             A
             F
             with
             the
             Velocity
             A
             N
             ,
             be
             D
             P
             ;
             I
             say
             that
             as
             A
             M
             is
             to
             A
             N
             ,
             or
             as
             the
             Parabola
             A
             K
             I
             B
             to
             the
             Parabola
             A
             F
             K
             ,
             so
             is
             D
             E
             to
             D
             P.
             For
             as
             A
             M
             is
             to
             F
             L
             (
             that
             is
             ,
             as
             A
             B
             is
             to
             A
             F
             )
             so
             let
             D
             E
             be
             to
             D
             G.
             Now
             the
             proportion
             of
             A
             M
             to
             A
             N
             is
             compounded
             of
             the
             proportions
             of
             A
             M
             to
             F
             L
             ,
             and
             of
             F
             L
             to
             A
             N.
             But
             as
             A
             M
             to
             F
             L
             ,
             so
             (
             by
             construction
             )
             is
             D
             E
             to
             D
             G
             ;
             and
             as
             F
             L
             is
             to
             A
             N
             (
             seeing
             the
             time
             in
             both
             is
             the
             same
             ,
             namely
             ,
             A
             F
             )
             ,
             so
             is
             the
             Length
             D
             G
             to
             the
             Length
             D
             P
             ;
             for
             Lengths
             transmitted
             in
             the
             same
             time
             are
             to
             one
             another
             as
             their
             Velocities
             are
             .
             Wherefore
             by
             ordinate
             proportion
             ,
             as
             A
             M
             is
             to
             A
             N
             ,
             that
             is
             ,
             as
             the
             mean
             Impetus
             A
             L
             multiplyed
             into
             its
             time
             A
             B
             ,
             is
             to
             the
             mean
             Impetus
             A
             O
             multiplyed
             into
             A
             F
             ,
             so
             is
             D
             E
             to
             D
             P
             ;
             which
             was
             to
             be
             proved
             .
          
           
             
               Corol.
               1
               Lengths
               transmitted
               with
               Motion
               so
               accelerated
               that
               the
               Impetus
               encrease
               continually
               in
               duplicate
               proportion
               to
               that
               of
               their
               times
               ,
               if
               the
               base
               represent
               the
               Impetus
               ,
               are
               in
               triplicate
               proportion
               of
               their
               Impetus
               acquired
               in
               the
               last
               point
               of
               their
               times
               .
               For
               as
               the
               Length
               D
               E
               is
               to
               the
               Length
               D
               P
               ,
               so
               is
               the
               parallelogram
               A
               M
               to
               the
               parallelogram
               A
               N
               ,
               and
               so
               the
               Parabola
               A
               B
               I
               K
               to
               the
               Parabola
               A
               F
               K
               ;
               But
               the
               proportion
               of
               the
               Parabola
               A
               B
               I
               K
               to
               the
               Parabola
               A
               F
               K
               is
               triplicate
               to
               the
               proportion
               which
               the
               base
               B
               I
               has
               to
               the
               base
               F
               K.
               Wherefore
               also
               the
               proportion
               of
               D
               E
               to
               D
               P
               ,
               is
               triplicate
               to
               that
               of
               B
               I
               to
               F
               K.
               
            
             
               Corol.
               2
               Lengths
               transmitted
               in
               equal
               Times
               succeeding
               one
               another
               from
               the
               beginning
               ,
               by
               Motion
               so
               accelerated
               ,
               that
               the
               proportiō
               of
               the
               Impetus
               be
               duplicate
               to
               the
               proportiō
               of
               the
               times
               ,
               are
               to
               one
               another
               as
               the
               differences
               of
               Cubique
               Numbers
               beginning
               at
               Unity
               ,
               that
               is
               ,
               as
               7
               ,
               19
               ,
               37
               ,
               &c.
               
               For
               if
               in
               the
               first
               time
               the
               Length
               transmitted
               be
               as
               1
               ,
               the
               Length
               at
               the
               end
               of
               the
               second
               time
               will
               be
               as
               8
               ,
               at
               the
               end
               of
               the
               third
               time
               as
               27
               ,
               and
               at
               the
               end
               of
               the
               fourth
               time
               as
               64
               ,
               &c.
               which
               are
               Cubique
               Numbers
               ,
               whose
               differences
               are
               7
               ,
               19
               ,
               37
               ,
               &c.
               
            
             
               Corol.
               3
               In
               Motion
               so
               accelerated
               ,
               as
               that
               the
               Length
               transmitted
               
               be
               alwayes
               to
               the
               Length
               transmitted
               in
               duplicate
               proportion
               to
               their
               Times
               ,
               the
               Length
               Uniformly
               transmitted
               in
               the
               whole
               time
               and
               with
               Impetus
               all
               the
               way
               equal
               to
               that
               which
               is
               last
               acquired
               ,
               is
               as
               a
               Parabola
               to
               a
               parallelogram
               of
               the
               same
               altitude
               &
               base
               ,
               that
               is
               ,
               as
               2
               to
               3.
               
               For
               the
               Parabola
               A
               B
               I
               K
               is
               the
               Impetus
               encreasing
               in
               the
               time
               A
               B
               ;
               and
               the
               parallelogram
               A
               I
               is
               the
               greatest
               Uniform
               Impetus
               multiplyed
               into
               the
               same
               time
               A
               B.
               Wherefore
               the
               Lengths
               transmitted
               will
               be
               as
               a
               Parabola
               to
               a
               parallelogram
               &c.
               that
               is
               ,
               as
               2
               to
               3.
               
            
          
           
             5
             If
             I
             should
             proceed
             to
             the
             explication
             of
             such
             Motions
             as
             are
             made
             by
             Impetus
             encreasing
             in
             proportion
             triplicate
             ,
             quadruplicate
             ,
             quintuplicate
             ,
             &c.
             to
             that
             of
             their
             times
             ,
             it
             would
             be
             a
             labour
             infinite
             and
             unnecessary
             .
             For
             by
             the
             same
             method
             by
             which
             I
             have
             computed
             such
             Lengths
             as
             are
             transmitted
             with
             Impetus
             encreasing
             in
             single
             and
             duplicate
             proportion
             ,
             any
             man
             may
             compute
             such
             as
             are
             transmitted
             with
             Impetus
             encreasing
             in
             triplicate
             ,
             quadruplicate
             or
             what
             other
             proportion
             he
             pleases
             .
          
           
             In
             making
             which
             computation
             he
             shall
             finde
             ,
             that
             where
             the
             Impetus
             encrease
             in
             proportion
             triplicate
             to
             that
             of
             the
             times
             ,
             there
             the
             whole
             Velocity
             will
             be
             designed
             by
             the
             first
             Parabolaster
             (
             of
             which
             see
             the
             next
             Chapter
             )
             ;
             and
             the
             Lengths
             transmitted
             will
             be
             in
             proportion
             quadruplicate
             to
             that
             of
             the
             times
             .
             And
             in
             like
             manner
             ,
             where
             the
             Impetus
             encrease
             in
             quadruplicate
             proportion
             to
             that
             of
             the
             times
             ,
             that
             there
             the
             whole
             Velocity
             will
             be
             designed
             by
             the
             second
             Parabolaster
             ,
             and
             the
             Lengths
             transmitted
             will
             be
             in
             quintuplicate
             proportion
             to
             that
             of
             the
             times
             ;
             and
             so
             on
             continually
             .
          
           
             6
             If
             two
             Bodies
             with
             Uniform
             Motion
             transmit
             two
             Lengths
             ,
             each
             with
             its
             own
             Impetus
             and
             Time
             ,
             the
             proportion
             of
             the
             Lengths
             transmitted
             will
             be
             compounded
             of
             the
             proportions
             of
             Time
             to
             Time
             ,
             and
             Impetus
             to
             Impetus
             ,
             directly
             taken
             .
          
           
             Let
             two
             Bodies
             be
             moved
             Uniformly
             (
             as
             in
             the
             3d
             figure
             )
             One
             in
             the
             time
             A
             B
             with
             the
             Impetus
             A
             C
             ,
             the
             other
             in
             the
             time
             A
             D
             with
             the
             Impetus
             A
             E.
             I
             say
             the
             Lengths
             transmitted
             have
             their
             proportion
             to
             one
             another
             compounded
             of
             the
             proportions
             of
             A
             B
             to
             A
             D
             ,
             and
             of
             A
             C
             to
             A
             E.
             For
             let
             any
             Length
             whatsoever
             ,
             as
             Z
             ,
             be
             transmitted
             by
             one
             of
             the
             Bodies
             in
             the
             time
             A
             B
             with
             the
             Impetus
             A
             C
             ;
             and
             any
             other
             Length
             ,
             as
             X
             ,
             be
             transmitted
             by
             the
             other
             Body
             in
             the
             time
             A
             D
             with
             the
             Impetus
             A
             E
             ;
             and
             let
             the
             parallelograms
             
             A
             F
             and
             A
             G
             be
             completed
             .
             Seeing
             now
             Z
             is
             to
             X
             (
             by
             the
             2d
             Article
             )
             as
             the
             Impetus
             A
             C
             multiplyed
             into
             the
             time
             A
             B
             is
             to
             the
             Impetus
             A
             E
             multiplyed
             into
             the
             time
             A
             D
             ,
             that
             is
             ,
             as
             A
             F
             to
             A
             G
             ;
             the
             proportion
             of
             Z
             to
             X
             will
             be
             compounded
             of
             the
             same
             proportions
             ,
             of
             which
             the
             proportion
             of
             A
             F
             to
             A
             G
             is
             compounded
             ;
             But
             the
             proportion
             of
             A
             F
             to
             A
             G
             is
             compounded
             of
             the
             proportions
             of
             the
             side
             A
             B
             to
             the
             side
             A
             D
             ,
             and
             of
             the
             side
             A
             C
             to
             the
             side
             A
             E
             (
             as
             is
             evident
             by
             the
             Elements
             of
             Euclide
             )
             ,
             that
             is
             ,
             of
             the
             proportions
             of
             the
             time
             A
             B
             to
             the
             time
             A
             D
             ,
             and
             of
             the
             Impetus
             A
             C
             to
             the
             Impetus
             A
             E.
             Wherefore
             also
             the
             proportion
             of
             Z
             to
             X
             is
             compounded
             of
             the
             same
             proportions
             of
             the
             time
             A
             B
             to
             the
             time
             A
             D
             ,
             and
             of
             the
             Impetus
             A
             C
             to
             the
             Impetus
             A
             E
             ;
             which
             was
             to
             be
             demonstrated
             .
          
           
             
               Corol.
               1
               When
               two
               Bodies
               are
               moved
               with
               Uniform
               Motion
               ,
               if
               the
               Times
               and
               Impetus
               be
               in
               reciprocal
               proportion
               ,
               the
               Lengths
               transmitted
               shall
               be
               equal
               .
               For
               if
               it
               were
               as
               A
               B
               to
               A
               D
               (
               in
               the
               same
               3d
               figure
               )
               so
               reciprocally
               A
               E
               to
               A
               C
               ,
               the
               proportion
               of
               A
               F
               to
               A
               G
               would
               be
               compounded
               of
               the
               proportions
               of
               A
               B
               to
               A
               D
               and
               of
               A
               C
               to
               A
               E
               ,
               that
               is
               ,
               of
               the
               proportions
               of
               A
               B
               to
               A
               D
               and
               of
               A
               D
               to
               A
               B.
               Wherefore
               ,
               A
               F
               would
               be
               to
               A
               G
               as
               A
               B
               to
               A
               B
               ,
               that
               is
               equal
               ;
               and
               so
               the
               two
               products
               made
               by
               the
               multiplication
               of
               Impetus
               into
               Time
               would
               be
               equal
               ;
               and
               by
               consequent
               ,
               Z
               would
               be
               equal
               to
               X.
               
            
             
               Corol.
               2
               If
               two
               Bodies
               be
               moved
               in
               the
               same
               Time
               ,
               but
               with
               different
               Impetus
               ,
               the
               Lengths
               transmitted
               will
               be
               as
               Impetus
               to
               Impetus
               .
               For
               if
               the
               Time
               of
               both
               of
               them
               be
               A
               D
               ,
               and
               their
               different
               Impetus
               be
               A
               E
               and
               A
               C
               ,
               the
               proportion
               of
               A
               G
               to
               D
               C
               will
               be
               compounded
               of
               the
               proportions
               of
               A
               E
               to
               A
               C
               and
               of
               A
               D
               to
               A
               D
               ,
               that
               is
               ,
               of
               the
               proportions
               of
               A
               E
               to
               A
               C
               and
               of
               A
               C
               to
               A
               C
               ;
               and
               so
               the
               proportion
               of
               A
               G
               to
               D
               C
               ,
               that
               is
               ,
               the
               proportion
               of
               Length
               to
               Length
               will
               be
               as
               A
               E
               to
               A
               C
               ,
               that
               is
               ,
               as
               that
               of
               Impetus
               to
               Impetus
               .
               In
               like
               manner
               ,
               if
               two
               Bodies
               be
               moved
               Uniformly
               ,
               and
               both
               of
               them
               with
               the
               same
               Impetus
               ,
               but
               in
               different
               times
               ,
               the
               proportion
               of
               the
               Lengths
               transmitted
               by
               them
               will
               be
               as
               that
               of
               their
               times
               .
               For
               if
               they
               have
               both
               the
               same
               Impetus
               A
               C
               ,
               and
               their
               different
               times
               be
               A
               B
               &
               A
               D
               ,
               the
               proportion
               ●f
               A
               F
               to
               D
               C
               will
               be
               compounded
               of
               the
               proportions
               of
               A
               B
               to
               A
               D
               and
               of
               A
               C
               to
               
               A
               C
               ;
               that
               is
               ,
               of
               the
               proportions
               of
               A
               B
               to
               A
               D
               and
               of
               A
               D
               ,
               to
               A
               D
               ;
               and
               therefore
               the
               proportion
               of
               A
               F
               to
               D
               C
               ,
               that
               is
               ,
               of
               Length
               to
               Length
               ,
               will
               be
               the
               same
               with
               that
               of
               A
               B
               to
               A
               D
               ,
               which
               is
               the
               proportion
               of
               Time
               to
               Time.
               
            
          
           
             7
             If
             two
             Bodies
             pass
             through
             two
             Lengths
             with
             Uniform
             Motion
             ,
             the
             proportion
             of
             the
             Times
             in
             which
             they
             are
             moved
             will
             be
             compounded
             of
             the
             proportions
             of
             Length
             to
             Length
             and
             Impetus
             to
             Impetus
             reciprocally
             taken
             .
          
           
             For
             let
             any
             two
             Lengths
             be
             given
             ,
             as
             (
             in
             the
             same
             3d
             figure
             )
             Z
             and
             X
             ,
             and
             let
             one
             of
             them
             be
             transmitted
             with
             the
             Impetus
             A
             C
             ,
             the
             other
             with
             the
             Impetus
             A
             E.
             I
             say
             the
             proportion
             of
             the
             Times
             in
             which
             they
             are
             transmitted
             ,
             will
             be
             compounded
             of
             the
             proportions
             of
             Z
             to
             X
             ,
             and
             of
             A
             E
             (
             which
             is
             the
             Impetus
             with
             which
             X
             is
             transmitted
             )
             to
             A
             C
             (
             the
             Impetus
             with
             which
             Z
             is
             transmitted
             .
             )
             For
             seeing
             A
             F
             is
             the
             product
             of
             the
             Impetus
             A
             C
             multiplyed
             into
             the
             Time
             A
             B
             ,
             the
             time
             of
             Motion
             through
             Z
             will
             be
             a
             line
             w
             ch
             is
             made
             by
             the
             applicatiō
             of
             the
             parallelogram
             A
             F
             to
             the
             straight
             line
             A
             C
             ,
             which
             line
             is
             A
             B
             ;
             and
             therefore
             A
             B
             is
             the
             time
             of
             motion
             through
             Z.
             In
             like
             manner
             ,
             seeing
             A
             G
             is
             the
             product
             of
             the
             Impetus
             A
             E
             multiplied
             into
             the
             Time
             A
             D
             ,
             the
             time
             of
             motion
             through
             X
             wil
             be
             a
             line
             which
             is
             made
             by
             the
             application
             of
             A
             G
             to
             the
             straight
             line
             A
             D
             ;
             but
             A
             D
             is
             the
             time
             of
             motiō
             through
             X.
             Now
             the
             proportion
             of
             A
             B
             to
             A
             D
             is
             cōpounded
             of
             the
             proportions
             of
             the
             parallelogram
             A
             F
             to
             the
             parallelogram
             A
             G
             ,
             and
             of
             the
             Impetus
             A
             E
             to
             the
             Impetus
             A
             C
             ;
             which
             may
             be
             demonstrated
             thus
             .
             Put
             the
             parallelograms
             in
             order
             A
             F
             ,
             A
             G
             ,
             D
             C
             ;
             and
             it
             will
             be
             manifest
             that
             the
             proportion
             of
             A
             F
             to
             D
             C
             is
             compounded
             of
             the
             proportions
             of
             A
             F
             to
             A
             G
             and
             of
             A
             G
             to
             D
             C
             ;
             but
             A
             F
             is
             to
             D
             C
             as
             A
             B
             to
             A
             D
             ;
             wherefore
             also
             the
             proportion
             of
             A
             B
             to
             A
             D
             is
             compounded
             of
             the
             propotrions
             of
             A
             F
             to
             A
             G
             &
             of
             A
             G
             to
             D
             C.
             And
             because
             the
             Length
             Z
             is
             to
             the
             Length
             X
             as
             A
             F
             is
             to
             A
             G
             ,
             &
             the
             Impetus
             A
             E
             to
             the
             Impetus
             A
             C
             as
             A
             G
             to
             D
             C
             ,
             therefore
             the
             proportion
             of
             A
             B
             to
             A
             D
             will
             be
             compounded
             of
             the
             proportions
             of
             the
             Length
             Z
             to
             the
             Length
             X
             ,
             and
             of
             the
             Impetus
             A
             E
             to
             the
             Impetus
             A
             C
             ;
             which
             was
             to
             be
             demonstrated
             .
          
           
             In
             the
             same
             manner
             it
             may
             be
             proved
             ,
             that
             in
             two
             Uniform
             Motions
             the
             proportion
             of
             the
             Impetus
             is
             compounded
             of
             the
             proportions
             ,
             
             of
             Length
             to
             Length
             ,
             and
             of
             Time
             to
             Time
             reciprocally
             taken
             .
          
           
             For
             if
             we
             suppose
             A
             C
             (
             in
             the
             same
             3d
             figure
             )
             to
             be
             the
             Time
             ,
             and
             A
             B
             the
             Impetus
             with
             which
             the
             Length
             Z
             is
             passed
             through
             ;
             and
             A
             E
             to
             be
             the
             Time
             ,
             and
             A
             D
             the
             Impetus
             with
             which
             the
             Length
             X
             is
             passed
             through
             ,
             the
             demonstration
             will
             proceed
             as
             in
             the
             last
             Article
             .
          
           
             8
             If
             a
             Body
             be
             carried
             by
             two
             Movents
             together
             which
             move
             with
             straight
             and
             Uniform
             Motion
             ,
             and
             concurre
             in
             any
             given
             angle
             ,
             the
             line
             by
             which
             that
             Body
             passes
             will
             be
             a
             straight
             line
             .
          
           
             Let
             the
             Movent
             A
             B
             (
             in
             the
             4th
             figure
             )
             have
             straight
             and
             Uniform
             Motion
             ,
             and
             be
             moved
             till
             it
             come
             into
             the
             place
             C
             D
             ;
             and
             let
             another
             Movent
             A
             C
             ,
             having
             likewise
             straight
             and
             Uniform
             Motion
             ,
             and
             making
             with
             the
             Movent
             A
             B
             any
             given
             angle
             C
             A
             B
             ,
             be
             understood
             to
             be
             moved
             in
             the
             same
             time
             to
             D
             B
             ;
             and
             let
             the
             Body
             be
             placed
             in
             the
             point
             of
             their
             concourse
             A.
             I
             say
             the
             line
             which
             that
             Body
             describes
             with
             its
             Motion
             is
             a
             straight
             line
             .
             For
             let
             the
             parallelogram
             A
             B
             D
             C
             be
             completed
             ,
             and
             its
             diagonal
             A
             D
             be
             drawn
             ;
             and
             in
             the
             straight
             line
             A
             B
             let
             any
             point
             E
             be
             taken
             ;
             and
             from
             it
             let
             E
             F
             be
             drawn
             parallel
             to
             the
             straight
             lines
             A
             C
             and
             B
             D
             ,
             cutting
             A
             D
             in
             G
             ;
             and
             through
             the
             point
             G
             let
             H
             I
             be
             drawn
             parallel
             to
             the
             straight
             lines
             A
             B
             and
             C
             D
             ;
             and
             lastly
             ,
             let
             the
             measure
             of
             the
             time
             be
             A
             C.
             Seeing
             therefore
             both
             the
             Motions
             are
             made
             in
             the
             same
             time
             ,
             when
             A
             B
             is
             in
             C
             D
             ,
             the
             Body
             also
             will
             be
             in
             C
             D
             ;
             and
             in
             like
             manner
             ,
             when
             A
             C
             is
             in
             B
             D
             ,
             the
             Body
             will
             be
             in
             B
             D.
             But
             A
             B
             is
             in
             C
             D
             at
             the
             same
             time
             when
             A
             C
             is
             in
             B
             D
             ;
             and
             therefore
             the
             Body
             will
             be
             in
             C
             D
             and
             B
             D
             at
             the
             same
             time
             ;
             Wherefore
             it
             will
             be
             in
             the
             common
             point
             D.
             Again
             ,
             seeing
             the
             Motion
             from
             A
             C
             to
             B
             D
             is
             Uniform
             ,
             that
             is
             ,
             the
             Spaces
             transmitted
             by
             it
             are
             in
             proportion
             to
             one
             another
             as
             the
             Times
             in
             which
             they
             are
             transmitted
             ,
             when
             A
             C
             is
             in
             E
             F
             ,
             the
             proportion
             of
             A
             B
             to
             A
             E
             will
             be
             same
             with
             that
             of
             E
             F
             to
             E
             G
             ,
             that
             is
             ,
             of
             the
             Time
             A
             C
             to
             the
             Time
             A
             H.
             Wherefore
             A
             B
             will
             be
             in
             H
             I
             in
             the
             same
             time
             in
             which
             A
             C
             is
             in
             E
             F
             ,
             so
             that
             the
             Body
             will
             at
             the
             same
             time
             be
             in
             E
             F
             and
             in
             H
             I
             ,
             and
             therefore
             in
             their
             common
             point
             G.
             And
             in
             the
             same
             manner
             it
             will
             ,
             be
             wheresoever
             the
             point
             E
             be
             taken
             between
             A
             and
             B.
             Wherefore
             the
             Body
             will
             alwayes
             be
             in
             the
             Diagonal
             A
             D
             ;
             which
             was
             to
             be
             demonstrated
             .
          
           
           
             Corollary
             .
             From
             hence
             it
             is
             manifest
             ,
             that
             the
             Body
             will
             be
             carried
             through
             the
             same
             straight
             line
             A
             D
             ,
             though
             the
             Motion
             be
             not
             Uniform
             ,
             provided
             it
             have
             like
             acceleration
             ;
             for
             the
             proportion
             of
             A
             B
             to
             A
             E
             will
             alwayes
             be
             the
             same
             with
             that
             of
             A
             C
             to
             A
             H.
             
          
           
             9
             If
             a
             Body
             be
             carried
             by
             two
             Movents
             together
             ,
             which
             meet
             in
             any
             given
             angle
             ,
             and
             are
             moved
             ,
             the
             one
             Uniformly
             ,
             the
             other
             with
             Motion
             Uniformly
             accelerated
             from
             Rest
             (
             that
             is
             ,
             that
             the
             proportion
             of
             their
             Impetus
             be
             as
             that
             of
             their
             Times
             )
             that
             is
             ,
             that
             the
             proportion
             of
             their
             Lēgths
             be
             duplicate
             to
             that
             of
             the
             lines
             of
             their
             Times
             ,
             till
             the
             line
             of
             greatest
             Impetus
             acquired
             by
             acceleration
             be
             equal
             to
             that
             of
             the
             line
             of
             Time
             of
             the
             Uniform
             Motion
             ;
             the
             line
             in
             which
             the
             Body
             is
             carried
             will
             be
             the
             crooked
             line
             of
             a
             Semiparabola
             ,
             whose
             base
             is
             the
             Impetus
             last
             acquired
             ,
             and
             Vertex
             the
             point
             of
             Rest.
             
          
           
             Let
             the
             straight
             line
             A
             B
             (
             in
             the
             5th
             Figure
             )
             be
             understood
             to
             be
             moved
             with
             Uniform
             Motion
             to
             C
             D
             ;
             and
             let
             another
             Movent
             in
             the
             straight
             line
             A
             C
             be
             supposed
             to
             be
             moved
             in
             the
             same
             time
             to
             B
             D
             ,
             but
             with
             motion
             Uniformly
             accelerted
             ,
             that
             is
             ,
             with
             such
             motion
             ,
             that
             the
             proportion
             of
             the
             spaces
             which
             are
             transmitted
             be
             alwayes
             duplicate
             to
             that
             of
             the
             Times
             ,
             till
             the
             Impetus
             acquired
             be
             B
             D
             equal
             to
             the
             straight
             line
             A
             C
             ;
             and
             let
             the
             Semiparabola
             A
             G
             D
             B
             be
             described
             .
             I
             say
             that
             by
             the
             concourse
             of
             those
             two
             Movents
             ,
             the
             Body
             will
             be
             carried
             through
             the
             Semipabolical
             crooked
             line
             A
             G
             D.
             For
             let
             the
             parallelelogram
             A
             B
             D
             C
             be
             completed
             ;
             &
             from
             the
             point
             E
             taken
             any
             where
             in
             the
             straight
             line
             A
             B
             let
             E
             F
             be
             drawn
             parallel
             to
             A
             C
             ,
             and
             cutting
             the
             crooked
             line
             in
             G
             ,
             and
             lastly
             ,
             through
             the
             point
             G
             let
             A
             I
             be
             drawn
             parallel
             to
             the
             straight
             lines
             A
             B
             and
             C
             D.
             Seeing
             therefore
             the
             proportion
             of
             A
             B
             to
             A
             E
             is
             by
             supposition
             duplicate
             to
             the
             proportion
             of
             E
             F
             to
             E
             G
             ,
             that
             is
             ,
             of
             the
             time
             A
             C
             to
             the
             time
             A
             H
             ,
             at
             the
             same
             time
             when
             A
             C
             is
             in
             E
             F
             ,
             A
             B
             will
             be
             in
             H
             I
             ;
             and
             therefore
             the
             moved
             Body
             will
             be
             in
             the
             common
             point
             G.
             And
             so
             it
             will
             alwayes
             be
             in
             what
             part
             soever
             of
             A
             B
             the
             point
             E
             be
             taken
             .
             Wherefore
             the
             moved
             Body
             will
             always
             be
             found
             in
             the
             parabolical
             line
             A
             G
             D
             ;
             which
             was
             to
             be
             demonstr●ted
             .
          
           
             10
             If
             a
             Body
             be
             carried
             by
             two
             Movents
             together
             ,
             which
             meet
             in
             any
             given
             angle
             ,
             and
             are
             moved
             ,
             the
             one
             Uniformly
             ,
             the
             other
             with
             Impetus
             encreasing
             from
             Rest
             till
             it
             be
             equal
             to
             that
             of
             the
             Uniform
             
             Motion
             ,
             and
             with
             such
             acceleration
             ,
             that
             the
             proportion
             of
             the
             Lengths
             transmitted
             be
             every
             where
             triplicate
             to
             that
             of
             the
             Times
             in
             which
             they
             are
             transmitted
             ,
             The
             line
             in
             which
             that
             Body
             is
             moved
             ,
             will
             be
             the
             crooked
             line
             of
             the
             first
             Semiparabolaster
             of
             two
             Means
             ,
             whose
             ba●e
             is
             the
             Impetus
             last
             acquired
             .
          
           
             Let
             the
             straight
             line
             A
             B
             (
             in
             the
             6th
             .
             Figure
             )
             be
             moved
             Uniformly
             to
             C
             D
             ;
             and
             let
             another
             Movent
             A
             C
             be
             moved
             at
             the
             same
             time
             to
             B
             D
             with
             motion
             so
             accelerated
             ,
             that
             the
             proportion
             of
             the
             Lengths
             transmitted
             by
             every
             where
             triplicate
             to
             the
             proportion
             of
             their
             Times
             ;
             and
             let
             the
             Impetus
             acquired
             in
             the
             end
             of
             that
             motion
             be
             B
             D
             ,
             equal
             to
             the
             straight
             line
             A
             C
             ;
             &
             lastly
             ,
             let
             A
             D
             be
             the
             crooked
             line
             of
             the
             first
             Semiparabolaster
             of
             two
             Means
             .
             I
             say
             that
             by
             the
             concourse
             of
             the
             two
             Movents
             together
             ,
             the
             Body
             will
             be
             alwayes
             in
             that
             crooked
             line
             A
             D.
             For
             let
             the
             parallelogram
             A
             B
             D
             C
             be
             completed
             ;
             and
             from
             the
             point
             E
             taken
             any
             where
             in
             the
             straight
             line
             A
             B
             let
             E
             F
             be
             drawn
             parallel
             to
             A
             C
             and
             cutting
             the
             crooked
             line
             in
             G
             ;
             and
             through
             the
             point
             G
             let
             H
             I
             be
             drawn
             parallel
             to
             the
             straight
             lines
             A
             B
             and
             C
             D.
             Seeing
             therefore
             the
             proportion
             of
             A
             B
             to
             A
             E
             is
             (
             by
             supposition
             )
             triplicate
             to
             the
             proportion
             of
             E
             F
             to
             E
             G
             ,
             that
             is
             ,
             of
             the
             time
             A
             C
             to
             the
             time
             A
             H
             ,
             at
             the
             same
             time
             when
             A
             C
             is
             in
             E
             F
             ,
             A
             B
             will
             be
             in
             H
             I
             ;
             and
             therefore
             the
             moved
             Body
             will
             be
             in
             the
             common
             point
             G.
             And
             so
             it
             will
             alwayes
             be
             in
             what
             part
             soever
             of
             A
             B
             the
             point
             E
             be
             taken
             ;
             and
             by
             consequent
             the
             Body
             will
             always
             be
             in
             the
             crooked
             line
             A
             G
             D
             ;
             which
             was
             to
             be
             demonstrated
             .
          
           
             11
             By
             the
             same
             method
             it
             may
             be
             shewn
             what
             line
             it
             is
             that
             it
             made
             by
             the
             motion
             of
             a
             Body
             carried
             by
             the
             concourse
             of
             any
             two
             Movents
             ,
             which
             are
             moved
             ,
             one
             of
             them
             Uniformly
             ,
             the
             other
             with
             acceleration
             ,
             but
             in
             such
             proportions
             of
             Spaces
             and
             Times
             as
             are
             explicable
             by
             Numbers
             ,
             as
             
               duplicate
               ,
               triplicate
            
             &c.
             or
             such
             as
             may
             be
             designed
             by
             any
             broken
             number
             whatsoever
             .
             For
             which
             this
             is
             the
             Rule
             .
          
           
             Let
             the
             two
             numbers
             of
             the
             Length
             &
             Time
             be
             added
             together
             ;
             &
             let
             their
             Sum
             be
             the
             Denominator
             of
             a
             Fraction
             ,
             whose
             Numerator
             must
             be
             the
             number
             of
             the
             Length
             .
             Seek
             this
             Fraction
             in
             the
             Table
             of
             the
             third
             Article
             of
             the
             17th
             Chapter
             ,
             and
             the
             line
             sought
             will
             be
             that
             which
             denominates
             the
             three-sided
             Figure
             noted
             on
             the
             left
             hand
             ,
             and
             the
             kind
             of
             it
             will
             be
             that
             which
             is
             numbred
             
             above
             over
             the
             Fraction
             .
             For
             example
             ,
             Let
             there
             be
             a
             concourse
             of
             two
             Movements
             ,
             whereof
             one
             is
             moved
             Uniformly
             ,
             the
             other
             with
             motion
             so
             accelerated
             that
             the
             Spaces
             are
             to
             the
             Times
             as
             5
             to
             3.
             
             Let
             a
             Fraction
             be
             made
             whose
             Denominator
             is
             the
             Sum
             of
             5
             and
             3
             ,
             and
             the
             Numerator
             5
             ,
             namely
             the
             Fraction
             ⅝
             .
             Seek
             in
             the
             Table
             ,
             and
             you
             will
             find
             ⅝
             to
             be
             the
             third
             in
             that
             row
             which
             belongs
             to
             the
             three-sided
             Figure
             of
             four
             Means
             .
             Wherfore
             the
             line
             of
             Motion
             made
             by
             the
             concourse
             of
             two
             such
             Movents
             as
             are
             last
             of
             all
             described
             ,
             will
             be
             the
             crooked
             line
             of
             the
             third
             Parabolaster
             of
             four
             Means
             .
          
           
             12
             If
             Motion
             be
             made
             by
             the
             concourse
             of
             two
             Movents
             whereof
             one
             is
             moved
             Uniformly
             ,
             the
             other
             beginning
             from
             Rest
             in
             the
             Angle
             of
             concourse
             with
             any
             acceleration
             whatsoever
             ;
             the
             Movent
             which
             is
             Moved
             Uniformly
             shall
             put
             forward
             the
             moved
             Body
             in
             the
             several
             parallel
             Spaces
             ,
             lesse
             ,
             then
             if
             both
             the
             Movents
             had
             Uniform
             motion
             ;
             and
             still
             lesse
             and
             lesse
             ,
             as
             the
             Motion
             of
             the
             other
             Movent
             is
             more
             and
             more
             accelerated
             .
          
           
             Let
             the
             Body
             be
             placed
             in
             A
             (
             in
             the
             7th
             figure
             )
             and
             be
             moved
             by
             two
             Movents
             ,
             by
             one
             with
             Uniform
             Motion
             from
             the
             straight
             line
             A
             B
             to
             the
             straight
             line
             C
             D
             parallel
             to
             it
             ;
             and
             by
             the
             other
             with
             any
             acceleration
             from
             the
             straight
             line
             A
             C
             to
             the
             straight
             line
             B
             D
             parallel
             to
             it
             ;
             and
             in
             the
             parallelogram
             A
             B
             D
             C
             let
             a
             Space
             be
             taken
             between
             any
             two
             parallels
             E
             F
             and
             G
             H.
             I
             say
             ,
             that
             whilest
             the
             Movent
             A
             C
             passes
             through
             the
             latitude
             which
             is
             between
             E
             F
             and
             G
             H
             ,
             the
             Body
             is
             lesse
             moved
             forwards
             from
             A
             B
             towards
             C
             D
             ,
             then
             it
             would
             have
             been
             ,
             if
             the
             Motion
             from
             A
             C
             to
             B
             D
             had
             been
             Uniform
             .
          
           
             For
             suppose
             that
             whilest
             the
             Body
             is
             made
             to
             descend
             to
             the
             parallel
             E
             F
             by
             the
             power
             of
             the
             Movent
             from
             A
             C
             towards
             B
             D
             ,
             the
             same
             Body
             in
             the
             same
             time
             is
             moved
             forwards
             to
             any
             point
             F
             in
             the
             line
             E
             F
             by
             the
             power
             of
             the
             Movent
             from
             A
             B
             towards
             C
             D
             ;
             and
             let
             the
             straight
             line
             A
             F
             be
             drawn
             and
             produced
             indeterminately
             ,
             cutting
             G
             H
             in
             H.
             Seeing
             therefore
             it
             is
             as
             A
             E
             to
             A
             G
             ,
             so
             E
             F
             to
             G
             H
             ;
             if
             A
             C
             should
             descend
             towards
             B
             D
             with
             uniform
             Motion
             ,
             the
             Body
             in
             the
             time
             G
             H
             (
             for
             I
             make
             A
             C
             and
             its
             parallels
             the
             measure
             of
             time
             )
             would
             be
             found
             in
             the
             point
             H.
             But
             because
             A
             C
             is
             supposed
             to
             be
             moued
             towards
             B
             D
             which
             motion
             continually
             accelerated
             ,
             that
             is
             ,
             in
             greater
             proportion
             of
             Space
             to
             
             Space
             then
             of
             Time
             to
             Time
             ,
             in
             the
             time
             G
             H
             the
             Body
             will
             be
             in
             some
             parallel
             beyond
             it
             ,
             as
             between
             G
             H
             and
             B
             D.
             Suppose
             now
             that
             in
             the
             end
             of
             the
             time
             G
             H
             it
             be
             in
             the
             parallel
             I
             K
             ,
             &
             in
             I
             K
             let
             I
             L
             be
             taken
             equal
             to
             G
             H.
             When
             therefore
             the
             Body
             is
             in
             the
             parallel
             I
             K
             ,
             it
             will
             be
             in
             the
             point
             L.
             Wherefore
             when
             it
             was
             in
             the
             parallel
             G
             H
             ,
             it
             was
             in
             some
             point
             between
             G
             and
             H
             ,
             as
             in
             the
             point
             M
             ;
             but
             if
             both
             the
             Motions
             had
             been
             Uniform
             it
             had
             been
             in
             the
             point
             H
             ;
             and
             therefore
             whilest
             the
             Movent
             A
             C
             passes
             over
             the
             latitude
             which
             is
             between
             E
             F
             and
             G
             A
             ,
             the
             Body
             is
             less
             moved
             forwards
             from
             A
             B
             towards
             C
             D
             ,
             then
             it
             would
             have
             been
             if
             both
             the
             Motions
             had
             been
             Uniform
             ;
             which
             was
             to
             be
             demonstrated
             .
          
           
             13
             Any
             Length
             being
             given
             whch
             is
             passed
             through
             in
             a
             given
             time
             with
             uniform
             motion
             ,
             To
             find
             out
             what
             Length
             shall
             be
             passed
             through
             in
             the
             same
             time
             with
             Motion
             uniformly
             accelerated
             ,
             that
             is
             ,
             with
             such
             Motion
             ,
             that
             the
             proportion
             of
             the
             Lengths
             passed
             through
             be
             continually
             duplicate
             to
             that
             of
             their
             Times
             ,
             and
             that
             the
             line
             of
             the
             Impetus
             last
             acquired
             ,
             be
             equal
             to
             the
             line
             of
             the
             whole
             time
             of
             the
             Motion
             .
          
           
             Let
             A
             B
             (
             in
             the
             8th
             .
             figure
             )
             be
             a
             Length
             transmitted
             with
             Uniform
             Motion
             in
             the
             time
             A
             C
             ;
             and
             let
             it
             be
             required
             to
             find
             another
             Length
             which
             shall
             be
             transmitted
             in
             the
             same
             time
             with
             Motion
             Uniformly
             accelerated
             ,
             so
             ,
             that
             the
             line
             of
             the
             Impetus
             last
             acquired
             be
             equal
             to
             the
             straight
             line
             A
             C.
             
          
           
             Let
             the
             parallelogram
             A
             B
             D
             C
             be
             completed
             ;
             and
             let
             B
             D
             be
             divided
             in
             the
             middle
             at
             E
             ;
             and
             between
             B
             E
             and
             B
             D
             let
             B
             F
             be
             a
             mean
             proportional
             ;
             and
             let
             A
             F
             be
             drawn
             and
             produced
             till
             it
             meet
             with
             C
             D
             produced
             in
             G
             ;
             and
             lastly
             ,
             let
             the
             parallelogram
             A
             C
             G
             H
             be
             completed
             .
             I
             say
             A
             H
             is
             the
             Length
             required
             .
          
           
             For
             as
             duplicate
             proportion
             is
             to
             single
             proportion
             ,
             so
             let
             A
             H
             be
             to
             A
             I
             ,
             that
             is
             ,
             let
             A
             I
             be
             the
             half
             of
             A
             H
             ;
             and
             let
             I
             K
             be
             drawn
             parallel
             to
             the
             straight
             line
             A
             C
             ,
             and
             cutting
             the
             Diagonal
             A
             D
             in
             K
             ,
             and
             the
             straight
             line
             A
             G
             in
             L.
             Seeing
             therefore
             A
             I
             is
             the
             half
             of
             A
             H
             ,
             I
             L
             will
             also
             be
             the
             half
             of
             B
             D
             ,
             that
             is
             ,
             equal
             to
             B
             E
             ,
             and
             I
             K
             equal
             to
             B
             F
             ;
             for
             B
             D
             ,
             (
             that
             is
             ,
             G
             H
             )
             ,
             B
             F
             ,
             and
             B
             E
             (
             that
             is
             ,
             I
             L
             )
             being
             continual
             proportionals
             ,
             A
             H
             ,
             A
             B
             ,
             and
             A
             I
             will
             also
             be
             continual
             proportionals
             .
             But
             as
             A
             B
             is
             to
             A
             I
             ,
             that
             is
             ,
             as
             A
             H
             is
             to
             A
             B
             ,
             so
             is
             B
             D
             to
             I
             K
             ,
             and
             so
             also
             is
             G
             H
             ,
             that
             is
             ,
             B
             D
             to
             B
             F
             ;
             and
             therefore
             B
             F
             and
             I
             K
             are
             equal
             .
             Now
             the
             proportion
             of
             A
             H
             to
             A
             I
             is
             duplicate
             to
             the
             proportion
             of
             A
             B
             to
             A
             I
             ,
             that
             is
             ,
             to
             that
             
             of
             BD
             to
             IK
             ,
             or
             of
             GH
             to
             IK
             .
             Wherefore
             the
             point
             K
             will
             be
             in
             a
             Parabola
             ,
             whose
             diameter
             is
             AH
             &
             ba●e
             GH
             ,
             which
             GH
             is
             equal
             to
             AC
             .
             The
             Body
             therefore
             proceeding
             from
             Rest
             in
             A
             with
             motion
             Uniformly
             accelerated
             in
             the
             time
             AC
             ,
             when
             it
             has
             passed
             through
             the
             Length
             AH
             ,
             will
             acquire
             the
             Impetus
             GH
             equal
             to
             the
             time
             AC
             ;
             that
             is
             ,
             such
             Impetus
             ,
             as
             that
             with
             it
             the
             Body
             will
             pass
             through
             the
             Length
             AC
             in
             the
             time
             AC
             .
             Wherefore
             any
             Length
             being
             given
             ,
             &c.
             which
             was
             propounded
             to
             be
             done
             .
          
           
             14
             Any
             Length
             being
             given
             which
             in
             a
             given
             Time
             is
             transmitted
             with
             Uniform
             Motion
             ,
             To
             find
             out
             what
             Length
             shall
             be
             transmitted
             in
             the
             same
             Time
             with
             Motion
             so
             accelerated
             ,
             that
             the
             Lengths
             transmitted
             be
             continually
             in
             triplicate
             proportion
             to
             that
             of
             their
             Times
             ,
             and
             the
             line
             of
             the
             Impetus
             last
             of
             all
             acquired
             be
             equal
             to
             the
             Line
             of
             Time
             given
             .
          
           
             Let
             the
             given
             Length
             AB
             (
             in
             the
             9th
             figure
             )
             be
             transmitted
             with
             Uniform
             motion
             in
             the
             Time
             AC
             ;
             and
             let
             it
             be
             required
             to
             find
             what
             Length
             shall
             be
             transmitted
             in
             the
             same
             time
             with
             motion
             so
             accelerated
             ,
             that
             the
             Lengths
             transmitted
             be
             continually
             in
             triplicate
             proportion
             to
             that
             of
             their
             Times
             ,
             and
             the
             Impetus
             last
             acquired
             be
             equal
             to
             the
             Time
             given
             .
          
           
             Let
             the
             parallelogram
             ABDC
             be
             completed
             ;
             and
             let
             BD
             be
             so
             divided
             in
             E
             ,
             that
             BE
             be
             a
             third
             part
             of
             the
             whole
             BD
             ;
             and
             let
             BF
             be
             a
             mean
             proportional
             between
             BD
             and
             BE
             ;
             and
             let
             AF
             be
             drawn
             and
             produced
             till
             it
             meet
             the
             straight
             line
             CD
             in
             G
             ;
             and
             lastly
             ,
             let
             the
             parallelogram
             ACGH
             be
             completed
             .
             I
             say
             AH
             is
             the
             Length
             required
             .
          
           
             For
             as
             triplicate
             proportion
             is
             to
             single
             proportion
             ,
             so
             let
             AH
             be
             to
             another
             line
             AI
             ,
             that
             is
             ,
             make
             AI
             a
             third
             part
             of
             the
             whole
             AH
             ;
             and
             let
             IK
             be
             drawn
             parallel
             to
             the
             straight
             line
             AC
             ,
             cutting
             the
             Diagonal
             AD
             in
             K
             ,
             and
             the
             straight
             line
             AG
             in
             L
             ;
             then
             ,
             as
             AB
             is
             to
             AI
             ,
             so
             let
             AI
             be
             to
             another
             AN
             ;
             and
             from
             the
             point
             N
             let
             NQ
             be
             drawn
             parallel
             to
             AC
             ,
             cutting
             AG
             ,
             AD
             ,
             and
             FK
             produced
             ,
             in
             P
             ,
             M
             and
             O
             ;
             and
             last
             of
             all
             let
             FO
             and
             LM
             be
             drawn
             ,
             which
             will
             be
             equal
             and
             parallel
             to
             the
             straight
             lines
             BN
             and
             IN.
             By
             this
             construction
             ,
             the
             Lengths
             transmitted
             AH
             ,
             AB
             ,
             AI
             and
             AN
             will
             be
             continual
             proportionals
             ;
             and
             in
             like
             manner
             ,
             the
             Times
             GH
             ,
             BF
             ,
             IL
             and
             NP
             ,
             that
             is
             ,
             NQ
             ,
             NO
             ,
             NM
             and
             NP
             will
             be
             continual
             proportionals
             ,
             and
             in
             the
             same
             proportion
             with
             
             AH
             ,
             AB
             ,
             AI
             ,
             and
             AN.
             Wherefore
             the
             proportion
             of
             AH
             to
             AN
             is
             the
             same
             with
             that
             of
             BD
             ,
             that
             is
             ,
             of
             NQ
             to
             NP
             ;
             and
             the
             proportion
             of
             NQ
             to
             NP
             triplicate
             to
             that
             of
             NQ
             to
             NO
             ,
             that
             is
             ,
             triplicate
             to
             that
             of
             BD
             to
             IK
             ;
             Wherefore
             also
             the
             Length
             AH
             is
             to
             the
             Length
             AN
             in
             triplicate
             proportion
             to
             that
             of
             the
             Time
             BD
             to
             the
             Time
             IK
             ;
             and
             therefore
             the
             crooked
             line
             of
             the
             first
             three
             sided
             figure
             of
             two
             means
             ,
             whose
             Diameter
             is
             AH
             ,
             and
             base
             GH
             equal
             to
             AC
             ,
             shall
             pass
             through
             the
             point
             O
             ;
             and
             consequently
             AH
             shall
             be
             transmitted
             in
             the
             time
             AC
             ,
             and
             shall
             have
             its
             last
             acquired
             Impetus
             GH
             equal
             to
             AC
             ,
             and
             the
             proportions
             of
             the
             Lengths
             acquired
             in
             any
             of
             the
             times
             triplicate
             to
             the
             proportions
             of
             the
             times
             themselves
             .
             Wherefore
             AH
             is
             the
             Length
             required
             to
             be
             found
             out
             ,
          
           
             By
             the
             same
             method
             ,
             if
             a
             Length
             be
             given
             which
             is
             transmitted
             with
             Uniform
             Motion
             in
             any
             given
             Time
             ,
             another
             Length
             may
             be
             found
             out
             ,
             which
             shall
             be
             transmitted
             in
             the
             same
             Time
             with
             motion
             so
             accelerated
             ,
             that
             the
             Lengths
             transmitted
             shall
             be
             to
             the
             Times
             in
             which
             they
             are
             transmitted
             ,
             in
             proportion
             quadruplicate
             ,
             quintuplicate
             ,
             and
             so
             on
             infinitely
             .
             For
             if
             BD
             be
             divided
             in
             E
             ,
             so
             ,
             that
             BD
             be
             to
             BE
             as
             4
             to
             1
             ;
             and
             there
             be
             taken
             between
             BD
             and
             BE
             a
             mean
             proportional
             FB
             ;
             and
             as
             AH
             is
             to
             AB
             ,
             so
             AB
             be
             made
             to
             a
             third
             ,
             and
             again
             so
             that
             third
             to
             a
             fourth
             ,
             and
             that
             fourth
             to
             a
             fifth
             AN
             ,
             so
             that
             the
             proportion
             of
             AH
             to
             AN
             be
             quadruplicate
             to
             that
             of
             AH
             to
             AB
             ,
             and
             the
             parallelogram
             NBFO
             be
             completed
             ;
             the
             crooked
             line
             of
             the
             first
             three-sided
             Figure
             of
             three
             Means
             will
             pass
             through
             the
             point
             O
             ▪
             and
             consequently
             the
             Body
             moved
             will
             acquire
             the
             Impetus
             GH
             equal
             to
             AC
             in
             the
             time
             AC
             .
             And
             so
             of
             the
             rest
             .
          
           
             15
             Also
             ,
             if
             the
             proportion
             of
             the
             Lengths
             transmitted
             ,
             be
             to
             that
             of
             their
             Times
             ,
             as
             any
             number
             to
             any
             number
             ,
             the
             same
             method
             serves
             for
             the
             finding
             out
             of
             the
             Length
             transmitted
             with
             such
             Impetus
             ,
             and
             in
             such
             Time.
             
          
           
             For
             let
             AC
             (
             in
             the
             10
             figure
             )
             be
             the
             time
             ,
             in
             which
             a
             Body
             is
             transmitted
             with
             Uniform
             Motion
             from
             A
             to
             B
             ;
             and
             the
             parallelogram
             ABDC
             being
             completed
             ,
             let
             it
             be
             required
             to
             find
             out
             a
             Length
             in
             which
             that
             Body
             may
             be
             moved
             in
             the
             same
             time
             AC
             ,
             frō
             A
             w
             th
             motion
             so
             accelerated
             ,
             that
             the
             proportion
             of
             the
             Lengths
             transmitted
             ,
             to
             that
             of
             the
             Times
             be
             continually
             as
             3
             to
             2.
             
          
           
           
             Let
             BD
             be
             so
             divided
             in
             E
             ,
             that
             BD
             be
             to
             BE
             as
             3
             to
             2
             ;
             and
             between
             BD
             and
             BE
             let
             BF
             be
             a
             mean
             proportionall
             ;
             and
             let
             AF
             be
             drawn
             and
             produced
             till
             it
             meet
             with
             CD
             produced
             in
             G
             ;
             and
             making
             AM
             a
             mean
             proportional
             between
             AH
             and
             AB
             ,
             let
             it
             be
             as
             AM
             to
             AB
             ,
             so
             AB
             to
             AI
             ;
             and
             so
             the
             proportion
             of
             AH
             to
             AI
             will
             be
             to
             that
             of
             AH
             to
             AB
             ,
             as
             3
             to
             2.
             (
             for
             of
             the
             proportions
             of
             which
             that
             of
             AH
             to
             AM
             is
             one
             ,
             that
             of
             AH
             to
             AB
             is
             two
             ,
             and
             that
             of
             AH
             to
             AI
             is
             three
             ;
             )
             &
             consequently
             as
             3
             to
             2
             to
             that
             of
             GH
             to
             BF
             ,
             &
             (
             FK
             being
             drawn
             parallel
             to
             BI
             ,
             and
             cutting
             AD
             in
             K
             )
             so
             likewise
             to
             that
             of
             GH
             or
             BD
             to
             IK
             ;
             Wherefore
             the
             proportion
             of
             the
             Length
             AH
             to
             AI
             is
             to
             the
             proportion
             of
             the
             Time
             BD
             to
             IK
             ,
             as
             3
             to
             2
             ;
             and
             therefore
             ,
             if
             in
             the
             time
             AC
             ,
             the
             Body
             be
             moved
             with
             accelerated
             motion
             ,
             as
             was
             propounded
             ,
             till
             it
             acquire
             the
             Impetus
             HG
             equal
             to
             AC
             ,
             the
             Length
             transmitted
             in
             the
             same
             Time
             will
             be
             AH
             .
          
           
             16
             But
             if
             the
             proportion
             of
             the
             Lengths
             to
             that
             of
             the
             Times
             had
             been
             as
             4
             to
             3
             ,
             there
             should
             then
             have
             been
             taken
             two
             mean
             proportionals
             between
             AH
             and
             AB
             ,
             and
             their
             proportion
             should
             have
             been
             continued
             one
             term
             further
             ,
             so
             that
             AH
             to
             AB
             might
             have
             three
             of
             the
             same
             proportions
             ,
             of
             which
             AH
             to
             AI
             has
             four
             ;
             and
             all
             things
             else
             should
             have
             been
             done
             as
             is
             already
             shewn
             .
             Now
             the
             way
             how
             to
             interpose
             any
             number
             of
             Means
             between
             two
             Lines
             given
             ,
             is
             not
             yet
             found
             out
             .
             Nevertheless
             ,
             this
             may
             stand
             for
             a
             general
             Rule
             ;
             
               If
               there
               be
               a
               Time
               given
               ,
               and
               a
               Length
               be
               transmitted
               in
               that
               Time
               with
               Uniform
               Motion
               ;
               as
               for
               example
               ,
               if
               the
               Time
               be
            
             AC
             ,
             
               and
               the
               Length
            
             AB
             ;
             
               the
               straight
               line
            
             AG
             ,
             
               which
               determines
               the
               Length
            
             CG
             or
             AH
             
               transmitted
               in
               the
               same
               Time
            
             AC
             
               with
               any
               accelerated
               motion
               ,
               shall
               so
               cut
            
             BD
             in
             F
             ,
             that
             BF
             
               shall
               be
               a
               mean
               proportional
               between
            
             BD
             and
             BE
             ,
             BE
             
               being
               so
               taken
               in
            
             BD
             ,
             
               that
               the
               proportion
               of
               Length
               to
               Length
               be
               every
               where
               to
               the
               proportion
               of
               Time
               to
               Time
               ,
               as
               the
               whole
            
             BD
             
               is
               to
               its
               part
            
             BE.
             
          
           
             17
             If
             in
             a
             given
             Time
             ,
             two
             Lengths
             be
             transmitted
             ,
             One
             with
             uniform
             motion
             ,
             the
             other
             with
             motion
             accelerated
             in
             any
             proportion
             of
             the
             Lengths
             to
             the
             Times
             ;
             and
             again
             in
             part
             of
             the
             same
             Time
             ,
             parts
             of
             the
             same
             Lengths
             be
             transmitted
             with
             the
             same
             motions
             ,
             the
             whole
             Length
             will
             exceed
             the
             other
             Length
             in
             the
             same
             proportion
             in
             which
             one
             part
             exceeds
             the
             other
             part
             .
          
           
             For
             example
             ,
             let
             AB
             (
             in
             the
             8th
             .
             figure
             )
             be
             a
             Length
             transmitted
             
             in
             the
             time
             AC
             with
             uniform
             Motion
             ;
             and
             let
             AH
             be
             another
             Length
             transmitted
             in
             the
             same
             time
             with
             Motion
             uniformly
             accelerated
             ,
             so
             that
             the
             Impetus
             last
             acquired
             be
             GH
             equal
             to
             AC
             ;
             and
             in
             AH
             let
             any
             part
             AI
             be
             taken
             ,
             and
             transmitted
             in
             part
             of
             the
             time
             AC
             with
             uniform
             Motion
             ;
             and
             let
             another
             part
             AB
             be
             taken
             ,
             and
             transmitted
             in
             the
             same
             part
             of
             the
             time
             AC
             with
             Motion
             uniformly
             accelerated
             .
             I
             say
             ,
             that
             as
             AH
             is
             to
             AB
             ,
             so
             will
             AB
             be
             to
             AI.
             
          
           
             Let
             BD
             be
             drawn
             parallel
             and
             equal
             to
             HG
             ,
             and
             divided
             in
             the
             midst
             at
             E
             ,
             and
             between
             BD
             and
             BE
             ,
             let
             a
             mean
             proportional
             be
             taken
             as
             BF
             ;
             &
             the
             straight
             line
             AG
             (
             by
             the
             demonstration
             of
             the
             13th
             Art.
             )
             shal
             pass
             through
             F.
             And
             dividing
             AH
             in
             the
             midst
             at
             I
             ,
             AB
             shall
             be
             a
             mean
             proportional
             between
             AH
             and
             AI.
             Again
             (
             because
             AI
             and
             AB
             are
             described
             by
             the
             same
             Motions
             )
             if
             IK
             be
             drawn
             parallel
             and
             equal
             to
             BF
             or
             AM
             ,
             and
             divided
             in
             the
             midst
             at
             N
             ,
             and
             between
             IK
             and
             IN
             be
             taken
             the
             mean
             proportional
             IL
             ,
             the
             straight
             line
             AF
             will
             (
             by
             the
             demonstration
             of
             the
             same
             13th
             Ar●
             .
             )
             pass
             through
             L.
             And
             dividing
             AB
             in
             the
             midst
             at
             O
             ,
             the
             line
             AI
             will
             be
             a
             mean
             proportional
             between
             AB
             and
             AO
             .
             Where
             ▪
             AB
             is
             divided
             in
             I
             and
             O
             in
             like
             manner
             as
             AH
             is
             divided
             in
             B
             and
             I
             ;
             and
             as
             AH
             to
             AB
             so
             is
             AB
             to
             AI.
             Which
             was
             to
             be
             proved
             .
          
           
             Coroll
             .
             Also
             as
             AH
             to
             AB
             ,
             so
             is
             HB
             to
             BI
             ;
             and
             so
             also
             BI
             to
             IO.
             
          
           
             And
             as
             this
             (
             where
             one
             of
             the
             Motions
             is
             uniformly
             accelerated
             )
             is
             proved
             out
             of
             the
             demonstration
             of
             the
             13th
             Article
             ;
             so
             (
             when
             the
             accelerations
             are
             in
             double
             proportion
             to
             the
             times
             )
             the
             same
             may
             be
             proved
             by
             the
             demonstration
             of
             the
             14th
             Art.
             and
             by
             the
             same
             method
             in
             all
             other
             accelerations
             ,
             whose
             proportions
             to
             the
             times
             are
             explicable
             in
             numbers
             .
          
           
             18
             If
             two
             sides
             which
             contain
             an
             Angle
             in
             any
             Parallelogram
             ,
             be
             moved
             in
             the
             same
             time
             to
             the
             sides
             opposite
             to
             them
             ,
             one
             of
             them
             with
             Uniform
             Motion
             ,
             the
             other
             with
             Motion
             Uniformly
             accelerated
             ;
             that
             side
             which
             is
             moved
             Uniformly
             will
             effect
             as
             much
             with
             its
             concourse
             through
             the
             whole
             Length
             transmitted
             ,
             as
             it
             would
             do
             if
             the
             other
             Motion
             were
             also
             Uniform
             ,
             and
             the
             Length
             transmitted
             by
             it
             in
             the
             same
             time
             were
             a
             mean
             proportional
             between
             the
             whole
             and
             the
             half
             .
          
           
             Let
             the
             side
             AB
             of
             the
             Parallelogram
             ABDC
             (
             in
             the
             11th
             Figure
             )
             be
             understood
             to
             be
             moved
             with
             Uniform
             Motion
             till
             it
             be
             coincident
             with
             CD
             ;
             and
             let
             the
             time
             of
             that
             Motion
             be
             AC
             
             or
             BD.
             Also
             in
             the
             same
             time
             let
             the
             side
             AC
             be
             understood
             to
             be
             moved
             with
             Motion
             uniformly
             accelerated
             ,
             till
             it
             be
             coincident
             with
             BD
             ;
             then
             dividing
             AB
             in
             the
             middle
             in
             E
             ,
             let
             AF
             be
             made
             a
             mean
             proportional
             between
             AB
             and
             AE
             ;
             and
             drawing
             FG
             parallel
             to
             AC
             ,
             let
             the
             side
             AC
             be
             understood
             to
             be
             moved
             in
             the
             same
             time
             AC
             with
             uniform
             Motion
             till
             it
             be
             coincident
             with
             FG.
             I
             say
             the
             whole
             AB
             confers
             as
             much
             to
             the
             velocity
             of
             the
             Body
             placed
             in
             A
             when
             the
             Motion
             of
             AC
             is
             uniformly
             accelerated
             till
             it
             come
             to
             BD
             ,
             as
             the
             part
             AF
             confers
             to
             the
             same
             when
             the
             side
             AC
             is
             moved
             Uniformly
             and
             in
             the
             same
             time
             to
             FG.
             
          
           
             For
             seeing
             AF
             is
             a
             mean
             proportional
             between
             the
             whole
             AB
             &
             it
             is
             half
             AE
             ,
             BD
             wil
             (
             by
             the
             13th
             Art.
             )
             be
             the
             last
             Impetus
             acquired
             by
             AC
             with
             motion
             uniformly
             accelerated
             till
             it
             come
             to
             the
             same
             BD
             ;
             and
             consequently
             the
             straight
             line
             FB
             will
             be
             the
             excess
             by
             w
             ch
             the
             Length
             transmitted
             by
             AC
             with
             motion
             uniformly
             accelerated
             ,
             will
             exceed
             the
             Length
             transmitted
             by
             the
             same
             AC
             in
             the
             same
             time
             with
             Uniform
             Motion
             ,
             and
             with
             Impetus
             every
             where
             equal
             to
             BD.
             Wherefore
             if
             the
             whole
             AB
             be
             moved
             Uniformly
             to
             CD
             in
             the
             same
             time
             in
             which
             AC
             is
             moved
             Uniformly
             to
             FG
             ,
             the
             part
             FB
             (
             seeing
             it
             concurs
             not
             at
             all
             with
             the
             Motion
             of
             the
             side
             AC
             which
             is
             supposed
             to
             be
             moved
             onely
             to
             FG
             )
             will
             cōfer
             nothing
             to
             its
             motion
             .
             Again
             ,
             supposing
             the
             side
             AC
             to
             be
             moved
             to
             BD
             with
             Motion
             Uniformly
             accelerated
             ,
             the
             side
             AB
             with
             its
             uniform
             Motion
             to
             CD
             will
             less
             put
             forwards
             the
             Body
             when
             it
             ▪
             is
             accelerated
             in
             all
             the
             parallels
             ,
             then
             when
             it
             is
             not
             at
             all
             accelerated
             ;
             &
             by
             how
             much
             the
             greater
             the
             acceleration
             is
             ,
             by
             so
             much
             the
             less
             it
             will
             put
             it
             forwards
             (
             as
             is
             shewn
             in
             the
             12th
             Artic.
             )
             When
             therefore
             AC
             is
             in
             FG
             with
             accelerated
             Motion
             ,
             the
             Body
             will
             not
             be
             in
             the
             side
             CD
             at
             the
             point
             G
             ,
             but
             at
             the
             point
             D
             ;
             so
             that
             GD
             wil
             be
             the
             excess
             by
             which
             the
             Length
             transmitted
             with
             accelerated
             Motion
             to
             BD
             ,
             exceeds
             the
             Length
             transmitted
             with
             Uniform
             Motion
             to
             FG
             ;
             so
             that
             the
             Body
             by
             its
             acceleration
             avoids
             the
             action
             of
             the
             part
             AF
             ,
             &
             comes
             to
             the
             side
             CD
             in
             the
             time
             AC
             ,
             and
             makes
             the
             Length
             CD
             ,
             which
             is
             equal
             to
             the
             Length
             AB
             .
             Wherefore
             Uniform
             Motion
             from
             AB
             to
             CD
             in
             the
             time
             AC
             works
             no
             more
             in
             the
             whole
             Length
             AB
             upon
             the
             Body
             uniformly
             accelerated
             from
             AC
             to
             BD
             ,
             then
             if
             AC
             were
             moved
             in
             the
             same
             time
             with
             uniform
             Motion
             to
             FG
             ;
             the
             difference
             consisting
             onely
             in
             this
             ,
             that
             when
             AB
             works
             upon
             the
             Body
             uniformly
             moved
             from
             AC
             to
             
             FG
             ,
             that
             by
             which
             the
             accelerated
             Motion
             exceeds
             the
             Uniform
             Motion
             ,
             is
             altogether
             in
             FB
             ,
             or
             GD
             ;
             but
             when
             the
             same
             AB
             works
             upon
             the
             Body
             acclerated
             ,
             that
             by
             which
             the
             accelerated
             Motion
             exceeds
             the
             Uniform
             Motion
             ,
             is
             dispersed
             through
             the
             whole
             Length
             AB
             or
             CD
             ,
             yet
             so
             that
             if
             it
             were
             collected
             and
             put
             together
             ,
             it
             would
             be
             equal
             to
             the
             same
             FB
             or
             GD
             .
             Wherefore
             ,
             If
             two
             sides
             which
             contain
             an
             angle
             which
             was
             to
             be
             demonstrated
             .
          
           
             19
             If
             two
             transmitted
             Lengths
             have
             to
             their
             Times
             any
             other
             proportion
             explicable
             by
             number
             ,
             &
             the
             side
             AB
             be
             so
             divided
             in
             E
             ,
             that
             AB
             be
             to
             AE
             in
             the
             same
             proportion
             which
             the
             Lengths
             transmitted
             have
             to
             the
             Times
             in
             which
             they
             are
             transmitted
             ,
             and
             between
             AB
             and
             AE
             there
             be
             taken
             a
             mean
             proportional
             AF
             ,
             it
             may
             be
             shewn
             by
             the
             same
             method
             ,
             that
             the
             side
             which
             is
             moved
             with
             Uniform
             Motion
             ,
             works
             as
             much
             with
             its
             concourse
             through
             the
             whole
             Length
             AB
             ,
             as
             it
             would
             do
             if
             the
             other
             Motion
             were
             also
             Uniform
             ,
             and
             the
             Length
             transmitted
             in
             the
             same
             Time
             AC
             were
             that
             mean
             proportional
             AF.
             
          
           
             And
             thus
             much
             concerning
             Motion
             by
             concourse
             .
          
        
         
           
           
           
             
               CHAP.
               XVII
            
             .
             Of
             
               Figures
               Deficient
            
             .
          
           
             
               
                 1
                 Definitions
                 of
                 a
                 Deficient
                 Figure
                 ;
                 of
                 a
                 Complete
                 Figure
                 ;
                 of
                 the
                 Complement
                 of
                 a
                 Deficient
                 Figure
                 ;
                 and
                 of
                 Pro●o●tions
                 which
                 are
                 Proportional
                 and
                 Commensurable
                 to
                 one
                 another
                 .
              
               
                 2
                 The
                 proportion
                 of
                 a
                 Deficient
                 Figure
                 to
                 its
                 Complement
                 .
              
               
                 3
                 The
                 proportions
                 of
                 Deficient
                 Figures
                 to
                 the
                 Parallelograms
                 in
                 which
                 they
                 are
                 described
                 ,
                 set
                 forth
                 in
                 a
                 Table
                 .
              
               
                 4
                 The
                 Description
                 and
                 Production
                 of
                 the
                 same
                 Figures
                 .
              
               
                 5
                 The
                 drawing
                 of
                 Tangents
                 to
                 them
                 .
              
               
                 6
                 In
                 what
                 proportion
                 the
                 same
                 Figures
                 exceed
                 a
                 straight
                 lined
                 Triangle
                 of
                 the
                 same
                 Altitude
                 and
                 Base
                 .
              
               
                 7
                 A
                 Table
                 of
                 Solid
                 Deficient
                 Figures
                 described
                 in
                 a
                 Cylinder
                 .
              
               
                 8
                 In
                 what
                 proportion
                 the
                 same
                 Figures
                 exceed
                 a
                 Cone
                 of
                 the
                 same
                 Altitude
                 and
                 Base
                 .
              
               
                 9
                 How
                 a
                 plain
                 Deficient
                 Figure
                 may
                 be
                 described
                 in
                 a
                 Parallelogram
                 ,
                 so
                 ,
                 that
                 it
                 be
                 to
                 a
                 Triangle
                 of
                 the
                 same
                 Base
                 and
                 Altitude
                 ,
                 as
                 another
                 Deficient
                 Figure
                 (
                 plain
                 or
                 solid
                 )
                 twice
                 taken
                 ,
                 is
                 to
                 the
                 same
                 Deficient
                 Figure
                 together
                 with
                 the
                 Complete
                 Figure
                 in
                 which
                 it
                 is
                 described
                 .
              
               
                 10
                 The
                 transferring
                 of
                 certain
                 properties
                 of
                 Deficient
                 Figures
                 described
                 in
                 a
                 Parallelogram
                 to
                 the
                 proportions
                 of
                 the
                 Spaces
                 transmitted
                 with
                 several
                 degrees
                 of
                 Velocity
                 .
              
               
                 11
                 Of
                 Deficient
                 Figures
                 described
                 in
                 a
                 Circle
                 .
              
               
                 12
                 The
                 proposition
                 demonstrated
                 in
                 the
                 2d
                 .
                 Article
                 ,
                 confirmed
                 from
                 the
                 Elements
                 of
                 Philosophy
                 .
              
               
                 13
                 An
                 unusual
                 way
                 of
                 reasoning
                 concerning
                 the
                 Equality
                 between
                 the
                 superficies
                 of
                 a
                 portion
                 of
                 a
                 Sphere
                 ,
                 and
                 a
                 Circle
                 .
              
               
                 14
                 How
                 from
                 the
                 description
                 of
                 Deficient
                 Figures
                 in
                 a
                 Parallelogram
                 ,
                 any
                 number
                 of
                 mean
                 Proportionals
                 may
                 be
                 found
                 out
                 between
                 two
                 given
                 straight
                 lines
                 .
              
            
          
           
             1
             I
             Call
             those
             
               Deficient
               Figures
            
             ,
             which
             may
             be
             understood
             to
             be
             generated
             by
             the
             Uniform
             Motion
             of
             some
             quantity
             ,
             which
             decreases
             continually
             ,
             till
             at
             last
             it
             have
             no
             magnitude
             at
             all
             .
          
           
             And
             I
             call
             that
             a
             
               Complete
               Figure
            
             ,
             answering
             to
             a
             Deficient
             Figure
             ,
             which
             is
             generated
             with
             the
             same
             motion
             ,
             and
             in
             the
             same
             time
             ,
             by
             a
             Quantity
             which
             retaines
             alwayes
             its
             whole
             magnitude
             .
          
           
             The
             Complement
             of
             a
             Deficient
             Figure
             is
             that
             ,
             which
             being
             added
             to
             the
             Deficient
             Figure
             ,
             makes
             it
             Complete
             .
          
           
             Four
             Proportions
             are
             said
             to
             be
             Proportionall
             ,
             when
             the
             first
             of
             
             them
             is
             to
             the
             second
             ,
             as
             the
             third
             is
             to
             the
             fourth
             .
             For
             example
             ,
             if
             the
             first
             proportion
             be
             duplicate
             to
             the
             second
             ;
             and
             again
             the
             third
             be
             duplicate
             to
             the
             fourth
             ,
             those
             Proportions
             are
             said
             to
             be
             Proportionall
             .
          
           
             And
             Commensurable
             Proportions
             are
             those
             ,
             which
             are
             to
             one
             another
             as
             number
             to
             number
             .
             As
             when
             to
             a
             proportion
             given
             ,
             one
             proportion
             is
             duplicate
             ,
             another
             triplicate
             ,
             the
             duplicate
             proportion
             will
             be
             to
             the
             triplicate
             proportion
             as
             2
             to
             3
             ;
             but
             to
             the
             given
             proportion
             it
             will
             be
             as
             2
             to
             ●
             ;
             and
             therefore
             I
             call
             those
             three
             proportions
             Commensurable
             .
          
           
             2
             A
             Deficient
             Figure
             ,
             which
             is
             made
             by
             a
             Quantity
             continually
             decreasing
             to
             nothing
             by
             proportions
             every
             where
             proportionall
             and
             commensurable
             ,
             is
             to
             its
             Complement
             ,
             as
             the
             proportion
             of
             the
             whole
             altitude
             ,
             to
             an
             altitude
             diminished
             in
             any
             time
             ,
             is
             to
             the
             proportion
             of
             the
             whole
             Quantity
             which
             describes
             the
             Figure
             ,
             to
             the
             same
             Quantity
             diminished
             in
             the
             same
             time
             .
          
           
             Let
             the
             quantity
             AB
             (
             in
             the
             1
             figure
             )
             by
             its
             motion
             through
             the
             altitude
             AC
             ,
             describe
             the
             Complete
             Figure
             AD
             ;
             and
             againe
             ,
             let
             the
             same
             quantity
             ,
             by
             decreasing
             continually
             to
             nothing
             in
             C
             ,
             describe
             the
             Deficient
             Figure
             ABEFC
             ,
             whose
             Complement
             will
             be
             the
             Figure
             BDCFE
             .
             Now
             let
             AB
             be
             supposed
             to
             be
             moved
             till
             it
             lie
             in
             GK
             ,
             so
             that
             the
             altitude
             diminished
             be
             GC
             ,
             and
             AB
             diminished
             be
             GE
             ;
             and
             let
             the
             proportion
             of
             the
             whole
             altitude
             AC
             to
             the
             diminished
             altitude
             GC
             ,
             be
             (
             for
             example
             )
             triplicate
             to
             the
             proportion
             of
             the
             whole
             quantity
             AB
             or
             GK
             ,
             to
             the
             diminished
             quantity
             GE.
             And
             in
             like
             manner
             ,
             let
             HI
             be
             taken
             equal
             to
             GE
             ,
             &
             let
             it
             be
             diminished
             to
             HF
             ;
             and
             let
             the
             proportion
             of
             GC
             to
             HC
             be
             triplicate
             to
             that
             of
             HI
             to
             HF
             ;
             &
             let
             the
             same
             be
             done
             in
             as
             many
             parts
             of
             the
             straight
             line
             AC
             as
             is
             possible
             ;
             and
             a
             line
             be
             drawn
             through
             the
             points
             B
             ,
             E
             ,
             F
             and
             C.
             I
             say
             the
             Deficient
             Figure
             ABEFC
             ,
             is
             to
             its
             Complement
             BDCEF
             as
             3
             to
             ●
             ,
             or
             as
             the
             proportion
             of
             AC
             to
             GC
             is
             to
             the
             proportion
             of
             AB
             ,
             that
             is
             ,
             of
             GK
             to
             GE.
             
          
           
             For
             (
             by
             the
             second
             Article
             of
             the
             15.
             
             Chap.
             )
             the
             proportion
             of
             the
             complement
             BEFCD
             to
             the
             deficient
             figure
             ABEFC
             ,
             is
             all
             the
             proportions
             of
             DB
             to
             OE
             ,
             and
             of
             DB
             to
             QF
             ,
             and
             of
             all
             the
             lines
             parallel
             to
             DB
             terminated
             in
             the
             line
             BEFC
             ,
             to
             all
             the
             parallels
             to
             AB
             terminated
             in
             the
             same
             points
             of
             the
             line
             BEFC
             .
             
             And
             seeing
             the
             proportions
             of
             DB
             to
             OE
             ,
             and
             of
             DB
             to
             QF
             &c
             are
             every
             where
             triplicate
             of
             the
             proportions
             of
             AB
             to
             GE
             ,
             and
             of
             AB
             to
             HF
             &c.
             the
             proportions
             of
             HF
             to
             AB
             ,
             and
             of
             GE
             to
             AB
             &c.
             (
             by
             the
             16
             Article
             of
             the
             13
             Chap.
             )
             are
             triplicate
             of
             the
             proportions
             of
             QF
             to
             DB
             ,
             and
             of
             OE
             to
             DB
             &c.
             and
             therefore
             the
             deficient
             figure
             ABEFC
             which
             is
             the
             aggregate
             of
             all
             the
             lines
             HF
             ,
             GE
             ,
             AB
             ,
             &c.
             is
             triple
             to
             the
             complement
             BEFCD
             made
             of
             all
             the
             lines
             QF
             ,
             OE
             ,
             DB
             ,
             &c.
             which
             was
             to
             be
             proved
             .
          
           
             It
             follows
             from
             hence
             ,
             That
             the
             same
             complement
             BEFCD
             is
             ¼
             of
             the
             whole
             Parallelogram
             .
             And
             by
             the
             same
             method
             may
             be
             calculated
             in
             all
             other
             Deficient
             Figures
             generated
             as
             above
             declared
             ,
             the
             proportion
             of
             the
             Parallelogram
             to
             either
             of
             its
             parts
             ;
             as
             that
             when
             the
             parallels
             encrease
             fom
             a
             point
             in
             the
             same
             proportion
             ,
             the
             Parallelogram
             will
             be
             divided
             into
             two
             equal
             Triangles
             ;
             when
             one
             encrease
             is
             double
             to
             the
             other
             ,
             it
             will
             be
             divided
             into
             a
             Semiparabola
             and
             its
             Complement
             ,
             or
             into
             2
             and
             1.
             
          
           
             The
             same
             construction
             standing
             ,
             the
             same
             conclusion
             may
             otherwise
             be
             demonstrated
             ,
             thus
             .
          
           
             Let
             the
             straight
             line
             CB
             be
             drawn
             cutting
             GK
             in
             L
             ,
             &
             through
             L
             let
             MN
             be
             drawn
             parallel
             to
             the
             straight
             line
             AC
             ;
             wherefore
             the
             Parallelograms
             GM
             and
             LD
             will
             be
             equal
             .
             Then
             let
             LK
             be
             divided
             into
             three
             equal
             parts
             ,
             so
             that
             it
             may
             be
             to
             one
             of
             those
             parts
             in
             the
             same
             proportion
             which
             the
             proportion
             of
             AC
             to
             GC
             or
             of
             GK
             to
             GL
             hath
             to
             the
             proportion
             of
             GK
             to
             GE.
             Therefore
             LK
             will
             be
             to
             one
             of
             those
             three
             parts
             as
             the
             Arithmetical
             proportion
             between
             GK
             and
             GL
             is
             to
             the
             Arithmetical
             proportion
             between
             GK
             and
             the
             same
             GK
             want
             the
             third
             part
             of
             LK
             ;
             and
             KE
             will
             be
             somwhat
             greater
             then
             a
             third
             of
             LK
             .
             Seeing
             now
             the
             altitude
             AG
             or
             ML
             is
             by
             reason
             of
             the
             continual
             decrease
             ,
             to
             be
             supposed
             less
             then
             any
             quantity
             that
             can
             be
             given
             ;
             LK
             (
             which
             is
             intercepted
             between
             the
             Diagonal
             BC
             and
             the
             side
             BD
             )
             will
             be
             also
             less
             then
             any
             quantity
             that
             can
             be
             given
             ;
             and
             consequently
             ,
             if
             G
             be
             put
             so
             neer
             to
             A
             in
             g
             ,
             as
             that
             the
             difference
             between
             Cg
             and
             CA
             be
             less
             then
             any
             quantity
             that
             can
             be
             assigned
             ,
             the
             difference
             also
             between
             Cl
             (
             removing
             L
             to
             l
             )
             and
             CB
             ,
             will
             be
             less
             then
             any
             quantity
             that
             can
             be
             assigned
             ;
             and
             the
             line
             gl
             being
             drawn
             &
             produced
             to
             the
             line
             BD
             in
             k
             cutting
             the
             crooked
             line
             
             in
             e
             ,
             the
             proportion
             of
             Gk
             to
             Gl
             will
             still
             be
             triplicate
             to
             the
             proportion
             of
             Gk
             to
             Ge
             ,
             and
             the
             difference
             between
             k
             and
             e
             the
             third
             part
             of
             kl
             will
             be
             less
             then
             any
             quantity
             that
             can
             be
             given
             ;
             and
             therefore
             the
             Parallelogram
             eD
             will
             differ
             from
             a
             third
             part
             of
             the
             Parallelogram
             Ae
             by
             a
             less
             difference
             then
             any
             quantity
             that
             can
             be
             assigned
             .
             Again
             ,
             let
             HI
             be
             drawn
             parallel
             and
             equal
             to
             ge
             ,
             cutting
             CB
             in
             P
             ,
             the
             crooked
             line
             in
             F
             ,
             and
             BD
             in
             I
             ,
             and
             the
             proportion
             of
             Cg
             ,
             to
             CH
             will
             be
             triplicate
             to
             the
             proportion
             of
             HF
             to
             HP
             ,
             and
             IF
             will
             be
             greater
             then
             the
             third
             part
             of
             PI.
             But
             again
             ,
             setting
             H
             in
             h
             so
             neer
             to
             g
             ,
             as
             that
             the
             difference
             between
             Ch
             and
             Cg
             may
             be
             but
             as
             a
             point
             ,
             the
             point
             P
             will
             also
             in
             p
             be
             so
             neer
             to
             l
             ,
             as
             that
             the
             difference
             between
             Cp
             and
             Cl
             will
             be
             but
             as
             a
             point
             ;
             and
             drawing
             hp
             till
             it
             meet
             with
             gk
             in
             i
             ,
             cutting
             the
             crooked
             line
             in
             f
             ,
             and
             having
             drawn
             eo
             parallel
             to
             BD
             ,
             cutting
             DC
             in
             o
             ,
             the
             Parallelogram
             fo
             will
             differ
             less
             from
             the
             third
             part
             of
             the
             Parallelogram
             gf
             ,
             then
             by
             any
             quantity
             that
             can
             be
             given
             .
             And
             so
             it
             will
             be
             in
             all
             other
             Spaces
             generated
             in
             the
             same
             manner
             .
             Wherefore
             the
             differences
             of
             the
             Arithmetical
             and
             Geometrical
             Means
             ,
             which
             are
             but
             as
             so
             many
             points
             B
             ,
             
               e
               ,
               f
            
             ,
             &c.
             (
             seeing
             the
             whole
             Figure
             is
             made
             up
             of
             so
             many
             indivisible
             Spaces
             )
             will
             constitute
             a
             certain
             line
             ,
             such
             as
             is
             the
             line
             BEFC
             ,
             which
             will
             divide
             the
             complete
             Figure
             AD
             into
             two
             parts
             ;
             whereof
             one
             ,
             namely
             ABEFC
             ,
             which
             I
             call
             a
             Deficient
             Figure
             ,
             is
             triple
             to
             the
             other
             ,
             namely
             BDCEF
             ,
             which
             I
             call
             the
             Complement
             thereof
             .
             And
             whereas
             the
             proportion
             of
             the
             altitudes
             to
             one
             another
             ,
             is
             in
             this
             case
             everywhere
             triplicate
             to
             that
             of
             the
             decreasing
             quantities
             to
             one
             another
             ;
             in
             the
             same
             manner
             if
             the
             proportion
             of
             the
             altitudes
             had
             been
             every
             where
             quadruplicate
             to
             that
             of
             the
             decreasing
             quantities
             it
             might
             have
             been
             demonstrated
             ,
             that
             the
             Deficient
             Figure
             had
             been
             quadruple
             to
             its
             Complement
             ;
             and
             so
             in
             any
             other
             proportion
             ,
             Wherefore
             ,
             a
             Deficient
             Figure
             ,
             which
             is
             made
             ,
             &c.
             
             Which
             was
             to
             be
             demonstrated
             .
          
           
             The
             same
             rule
             ●oldeth
             also
             in
             the
             diminution
             of
             the
             Bases
             of
             Cylinders
             ,
             as
             is
             demonstrated
             Chap.
             15.
             
             Art.
             2.
             
          
           
             ●
             By
             this
             Proposition
             ,
             the
             magnitudes
             of
             all
             Deficient
             Figures
             (
             when
             the
             proportions
             by
             which
             their
             bases
             decrease
             continually
             ,
             are
             proportionall
             to
             those
             by
             which
             their
             altitudes
             decrease
             )
             may
             be
             compared
             with
             the
             magnitudes
             of
             their
             Complements
             ;
             
             and
             consequently
             ,
             with
             the
             magnitudes
             of
             their
             Complete
             Figures
             .
             And
             they
             will
             be
             found
             to
             be
             as
             I
             have
             set
             them
             down
             in
             the
             following
             Tables
             ;
             in
             which
             I
             compare
             a
             Parallelogram
             with
             three-sided
             Figures
             ;
             and
             first
             with
             a
             straight
             lined
             triangle
             ,
             made
             by
             the
             base
             of
             the
             Parallelogram
             continually
             decreasing
             in
             such
             manner
             ;
             that
             the
             altitudes
             be
             alwayes
             in
             proportion
             to
             one
             another
             as
             the
             bases
             are
             ,
             and
             so
             the
             triangle
             will
             be
             equal
             to
             its
             Complement
             ;
             or
             the
             proportions
             of
             the
             altitudes
             and
             bases
             wil
             be
             as
             1
             to
             1
             ,
             and
             then
             the
             triangle
             will
             be
             half
             the
             Parallelogram
             .
             Secondly
             ,
             with
             that
             three-sided
             Figure
             which
             is
             made
             by
             the
             continual
             decreasing
             of
             the
             bases
             in
             subduplicate
             proportion
             to
             that
             of
             the
             altitudes
             ;
             and
             so
             the
             Deficient
             Figure
             will
             be
             double
             to
             its
             Complement
             ,
             and
             to
             the
             Parallelogram
             as
             2
             to
             3.
             
             Then
             ,
             with
             that
             ,
             where
             the
             proportion
             of
             the
             altitudes
             is
             triplicate
             to
             that
             of
             the
             bases
             ;
             and
             then
             the
             Deficient
             Figure
             will
             be
             triple
             to
             its
             Complement
             ,
             and
             to
             the
             Parallelogram
             as
             3
             to
             4.
             
             Also
             the
             proportion
             of
             the
             altitudes
             to
             that
             of
             the
             bases
             may
             be
             as
             3
             to
             2
             ;
             and
             then
             the
             Deficient
             Figure
             will
             be
             to
             its
             Complement
             as
             3
             to
             2
             ,
             &
             to
             the
             Parallelogram
             as
             3
             to
             5
             ;
             and
             so
             forwards
             according
             as
             more
             mean
             proportionals
             are
             taken
             ,
             or
             as
             the
             proportions
             are
             more
             multiplyed
             ,
             as
             may
             be
             seen
             in
             the
             following
             Table
             .
             For
             example
             ,
             if
             the
             bases
             decrease
             so
             ,
             that
             the
             proportion
             of
             the
             altitudes
             to
             that
             of
             the
             bases
             be
             alwayes
             as
             5
             to
             2
             ,
             and
             it
             be
             demanded
             what
             proportion
             the
             Figure
             made
             has
             to
             the
             Parallelogram
             ,
             which
             is
             supposed
             to
             be
             Unity
             ;
             then
             ,
             seeing
             that
             where
             the
             proportion
             is
             taken
             five
             times
             ,
             there
             must
             be
             four
             Means
             ;
             look
             in
             the
             Table
             amongst
             the
             three-sided
             figures
             of
             four
             Means
             ,
             and
             seeing
             the
             proportion
             was
             as
             5
             to
             2
             ,
             look
             in
             the
             uppermost
             row
             for
             the
             number
             2
             ,
             and
             descending
             in
             the
             2d
             Columne
             till
             you
             meet
             with
             that
             three-sided
             Figure
             ,
             you
             will
             finde
             5
             /
             7
             ;
             which
             shews
             that
             the
             Deficient
             Figure
             is
             to
             the
             Parallelogram
             as
             5
             /
             ●
             to
             1
             ,
             or
             as
             5
             to
             7.
             
          
           
           
             
               
                 
                    
                
                 
                   1
                
                 
                   2
                
                 
                   3
                
                 
                   4
                
                 
                   5
                
                 
                   6
                
                 
                   7
                
              
               
                 
                   Parallelogram
                   :
                   ..........
                
                 
                   1
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   Straight-sided
                   Triangle
                   .....
                
                 
                   ½
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   Three-sided
                   figure
                   of
                   1
                   Mean
                
                 
                   ⅔
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   Three-sided
                   figure
                   of
                   2
                   Means
                
                 
                   ¾
                
                 
                   ⅗
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   Three-sided
                   figure
                   of
                   3
                   Means
                
                 
                   ⅘
                
                 
                   4
                   /
                   6
                
                 
                   4
                   /
                   7
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   Three-sided
                   figure
                   of
                   4
                   Means
                
                 
                   ⅚
                
                 
                   5
                   /
                   7
                
                 
                   ⅝
                
                 
                   5
                   9
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   Three-sided
                   figure
                   of
                   5
                   Means
                
                 
                   6
                   /
                   7
                
                 
                   6
                   /
                   8
                
                 
                   6
                   /
                   9
                
                 
                   6
                   /
                   10
                
                 
                   6
                   /
                   11
                
                 
                    
                
                 
                    
                
              
               
                 
                   Three-sided
                   figure
                   of
                   6
                   Means
                
                 
                   ⅞
                
                 
                   7
                   /
                   9
                
                 
                   7
                   /
                   10
                
                 
                   7
                   /
                   11
                
                 
                   7
                   /
                   12
                
                 
                   7
                   /
                   13
                
                 
                    
                
              
               
                 
                   Three-sided
                   figure
                   of
                   7
                   Means
                
                 
                   8
                   /
                   9
                
                 
                   8
                   /
                   10
                
                 
                   8
                   /
                   11
                
                 
                   8
                   /
                   12
                
                 
                   8
                   /
                   13
                
                 
                   8
                   /
                   14
                
                 
                   8
                   /
                   15
                
              
            
          
           
             4
             Now
             for
             the
             better
             understanding
             of
             the
             nature
             of
             these
             three-sided
             figures
             ,
             I
             will
             shew
             how
             they
             may
             be
             described
             by
             points
             ;
             and
             first
             ,
             those
             which
             are
             in
             the
             first
             column
             of
             the
             Table
             .
             Any
             Parallelogram
             being
             described
             ,
             as
             ABCD
             (
             in
             the
             2d
             .
             figure
             ,
             )
             let
             the
             Diagonal
             BD
             be
             drawn
             ;
             and
             the
             straight-lined
             triangle
             BCD
             will
             be
             half
             the
             Parallelogram
             ;
             Then
             let
             any
             number
             of
             lines
             ,
             as
             EF
             ,
             be
             drawn
             parallel
             to
             the
             Side
             BC
             ,
             and
             cutting
             the
             Diagonal
             BD
             in
             G
             ;
             &
             let
             it
             be
             every
             where
             ,
             as
             EF
             to
             EG
             ,
             so
             EG
             to
             another
             EH
             ;
             and
             through
             all
             the
             points
             H
             let
             the
             line
             BHHD
             be
             drawn
             ;
             and
             the
             Figure
             BHHDC
             will
             be
             that
             which
             I
             call
             a
             Three-sided
             Figure
             of
             one
             Mean
             ,
             because
             in
             three
             proportionals
             ,
             as
             EF
             ,
             EG
             and
             EH
             ,
             there
             is
             but
             one
             Mean
             ,
             namely
             ,
             EG
             ;
             and
             this
             three-sided
             figure
             will
             be
             ⅔
             of
             the
             Parallelogram
             ,
             and
             is
             called
             a
             Parabola
             .
             Again
             ,
             let
             it
             be
             as
             EG
             to
             EH
             ,
             so
             EH
             to
             another
             EI
             ,
             and
             let
             the
             line
             BIID
             be
             drawn
             ,
             making
             the
             three-sided
             figure
             BIIDC
             ;
             &
             this
             will
             be
             ¾
             of
             the
             Parallelogram
             ,
             and
             is
             by
             many
             called
             a
             
               Cubique
               Parabola
            
             .
             In
             like
             manner
             ,
             if
             the
             proportions
             be
             further
             continued
             in
             EF
             ,
             there
             will
             be
             made
             the
             rest
             of
             the
             three-sided
             
             figures
             of
             the
             first
             Column
             ;
             which
             I
             thus
             demonstrate
             .
             Let
             there
             be
             drawn
             straight
             lines
             ,
             as
             HK
             and
             GL
             parallel
             to
             the
             base
             DC
             .
             Seeing
             therefore
             the
             proportion
             of
             EF
             to
             EH
             is
             duplicate
             of
             that
             of
             EF
             to
             EG
             ,
             or
             of
             BC
             to
             BL
             ,
             that
             is
             ,
             of
             CD
             to
             LG
             ,
             or
             of
             KM
             (
             producing
             KH
             to
             AD
             in
             M
             )
             to
             KH
             ,
             the
             proportion
             of
             BC
             to
             BK
             will
             be
             duplicate
             to
             that
             of
             KM
             to
             KH
             ;
             but
             as
             BC
             is
             to
             BK
             ,
             so
             is
             DC
             ,
             or
             KM
             to
             KN
             ;
             and
             therefore
             the
             proportion
             of
             KM
             to
             KN
             is
             duplicate
             to
             that
             of
             KM
             to
             KH
             ;
             and
             so
             it
             will
             be
             wheresoever
             the
             parallel
             KM
             be
             placed
             .
             Wherefore
             the
             Figure
             BHHDC
             is
             double
             to
             its
             Complement
             BHHDA
             ,
             and
             consequently
             ⅔
             of
             the
             whole
             Parallelogram
             .
             In
             the
             same
             manner
             if
             through
             I
             ,
             be
             drawn
             OPIQ
             parallel
             and
             equal
             to
             CD
             ,
             it
             may
             be
             demonstrated
             that
             the
             proportion
             of
             OQ
             to
             OP
             ,
             that
             is
             ,
             of
             BC
             to
             BO
             ,
             is
             triplicate
             to
             that
             of
             OQ
             to
             OI
             ,
             and
             therefore
             that
             the
             Figure
             BIIDC
             is
             triple
             to
             its
             Complement
             BIIDA
             ,
             and
             consequently
             ¾
             of
             the
             whole
             Parallelogram
             ,
             &c.
             
          
           
             Secondly
             ,
             such
             three-sided
             figures
             as
             are
             in
             any
             of
             the
             transverse
             rowes
             ,
             may
             be
             thus
             described
             .
             Let
             ABCD
             (
             in
             the
             3d.
             Figure
             )
             be
             a
             Parallelogram
             ,
             whose
             Diagonal
             is
             BD.
             I
             would
             describe
             in
             it
             such
             figures
             ,
             as
             in
             the
             preceding
             Table
             I
             call
             Three-sided
             Figures
             of
             three
             Means
             .
             Parallel
             to
             DC
             ,
             I
             draw
             EF
             as
             often
             as
             is
             necessary
             ,
             cutting
             BD
             in
             G
             ;
             and
             between
             EF
             and
             EG
             I
             take
             three
             proportionals
             EH
             ,
             EI
             and
             EK
             .
             If
             now
             there
             be
             drawn
             lines
             through
             all
             the
             points
             H
             ,
             I
             &
             K
             ;
             that
             through
             all
             the
             points
             H
             will
             make
             the
             figure
             BHDC
             ,
             which
             is
             the
             first
             of
             those
             three-sided
             figures
             ;
             and
             that
             through
             all
             the
             points
             I
             ,
             will
             make
             the
             figure
             BIDC
             ,
             which
             is
             the
             second
             ;
             and
             that
             which
             is
             drawn
             through
             all
             the
             points
             K
             ,
             will
             make
             the
             figure
             BKDC
             the
             third
             of
             those
             three-sided
             figures
             .
             The
             first
             of
             these
             (
             seeing
             the
             proportion
             of
             EF
             to
             EC
             is
             quadruplicate
             of
             that
             of
             EF
             to
             EH
             )
             will
             be
             to
             its
             Complement
             as
             4
             to
             1
             ,
             and
             to
             the
             Parallelogram
             as
             4
             to
             5.
             
             The
             second
             (
             seeing
             the
             proportion
             of
             EF
             to
             EG
             is
             to
             that
             of
             EF
             to
             EI
             as
             4
             to
             2
             )
             will
             be
             double
             to
             its
             Complement
             ,
             and
             4
             /
             6
             or
             ⅔
             of
             the
             Parallelogram
             .
             The
             third
             (
             seeing
             the
             proportion
             of
             EF
             to
             EG
             is
             to
             that
             of
             EF
             to
             EK
             as
             4
             to
             3
             )
             will
             be
             to
             its
             Complement
             as
             4
             to
             3
             ,
             and
             to
             the
             Parallelogram
             as
             4
             to
             7.
             
          
           
             Any
             of
             these
             ,
             figures
             being
             described
             ,
             may
             be
             produced
             at
             pleasure
             ,
             thus
             ;
             Let
             ABCD
             (
             in
             the
             4th
             figure
             )
             be
             a
             Parallelogram
             ,
             and
             in
             it
             let
             the
             figure
             BKDC
             be
             described
             ,
             namely
             ,
             the
             third
             three-sided
             figure
             of
             three
             Means
             .
             Let
             BD
             be
             produced
             indefinitely
             to
             E
             ,
             and
             let
             EF
             be
             made
             parallel
             to
             the
             base
             DC
             ,
             cutting
             
             AD
             produced
             in
             G
             ,
             and
             BC
             produced
             in
             F
             ;
             and
             in
             GE
             let
             the
             point
             H
             be
             so
             taken
             ,
             that
             the
             proportion
             of
             FE
             to
             FG
             may
             be
             quadruplicate
             to
             that
             of
             FE
             to
             FH
             (
             which
             may
             be
             done
             by
             making
             FH
             the
             greatest
             of
             three
             proportionals
             between
             FE
             and
             FG
             )
             ;
             the
             crooked
             line
             BKD
             produced
             ,
             will
             pass
             through
             the
             point
             H.
             For
             if
             the
             straight
             line
             BH
             be
             drawn
             ,
             cutting
             CD
             in
             I
             ,
             and
             HL
             be
             drawn
             parallel
             to
             GD
             ,
             and
             meeting
             CD
             produced
             in
             L
             ;
             it
             will
             be
             as
             FE
             to
             FG
             ,
             so
             CL
             to
             CI
             ;
             that
             is
             ,
             in
             quadruplicate
             proportion
             to
             that
             of
             FE
             to
             FH
             ,
             or
             of
             CD
             to
             CI.
             Wherefore
             if
             the
             line
             BKD
             be
             produced
             according
             to
             its
             generation
             ,
             it
             will
             fall
             upon
             the
             point
             H.
             
          
           
             5
             A
             straight
             line
             may
             be
             drawn
             so
             ,
             as
             to
             touch
             the
             crooked
             line
             of
             the
             said
             figure
             in
             any
             point
             ,
             in
             this
             manner
             .
             Let
             it
             be
             required
             to
             draw
             a
             Tangent
             to
             the
             line
             BKDH
             (
             in
             the
             4th
             figure
             )
             in
             the
             point
             D.
             Let
             the
             points
             B
             and
             D
             be
             connected
             ,
             and
             drawing
             DA
             equal
             and
             parallel
             to
             BC
             ,
             let
             B
             and
             A
             be
             connected
             ;
             and
             because
             this
             figure
             is
             by
             construction
             the
             third
             of
             three
             Means
             ,
             let
             there
             be
             taken
             in
             AB
             three
             points
             ,
             so
             ,
             that
             by
             them
             the
             same
             AB
             be
             divided
             into
             four
             equal
             parts
             ;
             of
             which
             take
             three
             ,
             namely
             ,
             AM
             ,
             so
             that
             AB
             may
             be
             to
             AM
             ,
             as
             the
             figure
             BKDC
             is
             to
             its
             Complement
             .
             I
             say
             the
             straight
             line
             MD
             ,
             will
             touch
             the
             figure
             in
             the
             point
             given
             D.
             For
             let
             there
             be
             drawn
             any
             where
             between
             AB
             and
             DC
             a
             parallel
             ,
             as
             RQ
             ,
             cutting
             the
             straight
             line
             BD
             ,
             the
             crooked
             line
             BD
             ,
             the
             straight
             line
             MD
             ,
             and
             the
             straight
             line
             AD
             in
             the
             points
             P
             ,
             K
             ,
             O
             and
             Q.
             RK
             will
             therefore
             (
             by
             construction
             )
             be
             the
             least
             of
             three
             Means
             in
             Geometrical
             proportion
             between
             RQ
             and
             RP
             .
             Wherefore
             (
             by
             the
             Coroll
             .
             of
             the
             28th
             Article
             of
             the
             13th
             Chapter
             )
             RK
             will
             be
             less
             then
             RO
             ;
             and
             therefore
             MD
             will
             fall
             without
             the
             figure
             .
             Now
             if
             MD
             be
             produced
             to
             N
             ,
             FN
             will
             be
             the
             least
             of
             three
             Means
             in
             Arithmetical
             proportion
             between
             FE
             and
             FG
             ;
             and
             FH
             will
             be
             the
             greatest
             of
             three
             Means
             in
             Geometrical
             proportion
             between
             the
             same
             FE
             and
             FG.
             Wherefore
             (
             by
             the
             same
             Coroll
             .
             of
             the
             28
             Artic.
             of
             the
             13th
             Chap.
             )
             FH
             will
             be
             less
             then
             FN
             ;
             and
             therefore
             DN
             will
             fall
             without
             the
             figure
             ,
             and
             the
             straight
             line
             MN
             will
             touch
             the
             same
             figure
             onely
             in
             the
             point
             D.
             
          
           
             6
             The
             proportion
             of
             a
             Deficient
             Figure
             to
             its
             Complement
             being
             known
             ,
             it
             may
             also
             be
             known
             what
             proportion
             a
             straight-lined
             
             triangle
             has
             to
             the
             excess
             of
             the
             Deficient
             Figure
             above
             the
             same
             triangle
             ;
             and
             these
             proportions
             I
             have
             set
             down
             in
             the
             following
             Table
             ;
             where
             if
             you
             seek
             (
             for
             example
             )
             how
             much
             the
             fourth
             three-sided
             figure
             of
             five
             Means
             exceeds
             a
             triangle
             of
             the
             same
             altitude
             and
             base
             ,
             you
             will
             find
             in
             the
             concourse
             of
             the
             fourth
             column
             with
             the
             three-sided
             figures
             of
             five
             Means
             ,
             2
             /
             10
             ;
             by
             which
             is
             signified
             ,
             that
             that
             three-sided
             figure
             exceeds
             the
             triangle
             by
             two
             tenths
             ,
             or
             by
             one
             fifth
             part
             of
             the
             same
             triangle
             .
          
           
             
               
                 
                    
                
                 
                   1
                
                 
                   2
                
                 
                   3
                
                 
                   4
                
                 
                   5
                
                 
                   6
                
                 
                   7
                
              
               
                 
                   The
                   Triangle
                   ............
                
                 
                   1
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   The
                   Excess
                   of
                   a
                   Three-sided
                   figure
                   of
                   one
                   Mean
                
                 
                   ⅓
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   The
                   Excess
                   of
                   a
                   Three-sided
                   figure
                   of
                   2
                   Means
                
                 
                   2
                   /
                   4
                
                 
                   ⅕
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   The
                   Excess
                   of
                   a
                   Three-sided
                   figure
                   of
                   3
                   Means
                
                 
                   ⅗
                
                 
                   2
                   /
                   6
                
                 
                   1
                   /
                   7
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   The
                   Excess
                   of
                   a
                   Three-sided
                   figure
                   of
                   4
                   Means
                
                 
                   4
                   /
                   6
                
                 
                   3
                   /
                   7
                
                 
                   2
                   /
                   8
                
                 
                   1
                   /
                   9
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   The
                   Excess
                   of
                   a
                   Three-sided
                   figure
                   of
                   5
                   Means
                
                 
                   5
                   /
                   7
                
                 
                   4
                   /
                   8
                
                 
                   3
                   /
                   9
                
                 
                   2
                   /
                   10
                
                 
                   1
                   /
                   11
                
                 
                    
                
                 
                    
                
              
               
                 
                   The
                   Excess
                   of
                   a
                   Three-sided
                   figure
                   of
                   6
                   Means
                
                 
                   6
                   /
                   8
                
                 
                   5
                   /
                   9
                
                 
                   4
                   /
                   10
                
                 
                   3
                   /
                   11
                
                 
                   2
                   /
                   12
                
                 
                   1
                   /
                   13
                
                 
                    
                
              
               
                 
                   The
                   Excess
                   of
                   a
                   Three-sided
                   figure
                   of
                   7
                   Means
                
                 
                   7
                   /
                   9
                
                 
                   6
                   /
                   10
                
                 
                   5
                   /
                   11
                
                 
                   4
                   /
                   12
                
                 
                   3
                   /
                   13
                
                 
                   2
                   /
                   14
                
                 
                   1
                   /
                   15
                
              
            
          
           
             7
             In
             the
             next
             Table
             are
             set
             down
             the
             proportion
             of
             a
             Cone
             ,
             and
             the
             Solids
             of
             the
             said
             three-sided
             figures
             ,
             namely
             ,
             the
             proportions
             between
             them
             and
             a
             Cylinder
             .
             As
             for
             example
             ,
             in
             the
             concourse
             of
             the
             second
             Column
             with
             the
             three-sided
             figures
             of
             four
             Means
             ,
             you
             have
             ●
             /
             9
             ;
             which
             gives
             you
             to
             understand
             ,
             that
             the
             Solid
             of
             the
             second
             three-sided
             figure
             of
             four
             Means
             is
             to
             the
             Cylinder
             as
             ●
             /
             9
             to
             1
             ,
             or
             as
             5
             to
             9
             ,
          
           
           
             
               
                 
                    
                
                 
                   1
                
                 
                   2
                
                 
                   3
                
                 
                   4
                
                 
                   5
                
                 
                   6
                
                 
                   7
                
              
               
                 
                   The
                   Cylinder
                
                 
                   1
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   A
                   Cone
                
                 
                   ⅓
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   The
                   Solids
                   of
                   a
                   Three-sided
                   figure
                   of
                   one
                   Mean
                
                 
                   2
                   /
                   4
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   The
                   Solids
                   of
                   a
                   Three-sided
                   figure
                   of
                   2
                   Means
                
                 
                   ⅗
                
                 
                   3
                   /
                   7
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   The
                   Solids
                   of
                   a
                   Three-sided
                   figure
                   of
                   3
                   Means
                
                 
                   4
                   /
                   6
                
                 
                   4
                   /
                   8
                
                 
                   4
                   /
                   10
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   The
                   Solids
                   of
                   a
                   Three-sided
                   figure
                   of
                   4
                   Means
                
                 
                   5
                   /
                   7
                
                 
                   5
                   /
                   9
                
                 
                   5
                   /
                   11
                
                 
                   5
                   /
                   13
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   The
                   Solids
                   of
                   a
                   Three-sided
                   figure
                   of
                   5
                   Means
                
                 
                   6
                   /
                   8
                
                 
                   6
                   /
                   10
                
                 
                   6
                   /
                   12
                
                 
                   6
                   /
                   14
                
                 
                   6
                   /
                   16
                
                 
                    
                
                 
                    
                
              
               
                 
                   The
                   Solids
                   of
                   a
                   Three-sided
                   figure
                   of
                   6
                   Means
                
                 
                   7
                   /
                   9
                
                 
                   7
                   /
                   11
                
                 
                   7
                   /
                   13
                
                 
                   7
                   /
                   15
                
                 
                   7
                   /
                   17
                
                 
                   7
                   /
                   19
                
                 
                    
                
              
               
                 
                   The
                   Solids
                   of
                   a
                   Three-sided
                   figure
                   of
                   7
                   Means
                
                 
                   8
                   /
                   10
                
                 
                   8
                   /
                   12
                
                 
                   8
                   /
                   14
                
                 
                   8
                   /
                   16
                
                 
                   8
                   /
                   18
                
                 
                   8
                   /
                   20
                
                 
                   8
                   /
                   22
                
              
            
          
           
             8
             Lastly
             ,
             the
             Excess
             of
             the
             Solids
             of
             the
             said
             three-sided
             figures
             ,
             above
             a
             Cone
             of
             the
             same
             altitude
             and
             base
             ,
             are
             set
             down
             in
             the
             Table
             which
             follows
          
           
             
               
                 
                    
                
                 
                   1
                
                 
                   2
                
                 
                   3
                
                 
                   4
                
                 
                   5
                
                 
                   6
                
                 
                   7
                
              
               
                 
                   The
                   Cone
                
                 
                   1
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   The
                   Excesses
                   of
                   the
                   Solids
                   of
                   these
                   three-sided
                   figures
                   above
                   a
                   C
                   n●
                   Of
                   the
                   Solid
                   of
                   a
                   three-sided
                   figure
                   of
                   one
                   Mean
                
                 
                   6
                   /
                   12
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   The
                   Excesses
                   of
                   the
                   Solids
                   of
                   these
                   three-sided
                   figures
                   above
                   a
                   C
                   n●
                   Of
                   the
                   Solid
                   of
                   a
                   three-sided
                   figure
                   of
                   2
                   Means
                
                 
                   12
                   /
                   15
                
                 
                   6
                   /
                   21
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   The
                   Excesses
                   of
                   the
                   Solids
                   of
                   these
                   three-sided
                   figures
                   above
                   a
                   C
                   n●
                   Of
                   the
                   Solid
                   of
                   a
                   three-sided
                   figure
                   of
                   3
                   Means
                
                 
                   18
                   /
                   18
                
                 
                   12
                   /
                   24
                
                 
                   6
                   /
                   30
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   The
                   Excesses
                   of
                   the
                   Solids
                   of
                   these
                   three-sided
                   figures
                   above
                   a
                   C
                   n●
                   Of
                   the
                   Solid
                   of
                   a
                   three-sided
                   figure
                   of
                   4
                   Means
                
                 
                   24
                   /
                   21
                
                 
                   18
                   /
                   27
                
                 
                   12
                   /
                   33
                
                 
                   6
                   /
                   39
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   The
                   Excesses
                   of
                   the
                   Solids
                   of
                   these
                   three-sided
                   figures
                   above
                   a
                   C
                   n●
                   Of
                   the
                   Solid
                   of
                   a
                   three-sided
                   figure
                   of
                   5
                   Means
                
                 
                   30
                   /
                   24
                
                 
                   24
                   /
                   30
                
                 
                   18
                   /
                   36
                
                 
                   12
                   /
                   42
                
                 
                   6
                   /
                   48
                
                 
                    
                
                 
                    
                
              
               
                 
                   The
                   Excesses
                   of
                   the
                   Solids
                   of
                   these
                   three-sided
                   figures
                   above
                   a
                   C
                   n●
                   Of
                   the
                   Solid
                   of
                   a
                   three-sided
                   figure
                   of
                   6
                   Means
                
                 
                   36
                   /
                   27
                
                 
                   30
                   /
                   33
                
                 
                   24
                   /
                   39
                
                 
                   18
                   /
                   45
                
                 
                   12
                   /
                   51
                
                 
                   6
                   /
                   57
                
                 
                    
                
              
               
                 
                   The
                   Excesses
                   of
                   the
                   Solids
                   of
                   these
                   three-sided
                   figures
                   above
                   a
                   C
                   n●
                   Of
                   the
                   Solid
                   of
                   a
                   three-sided
                   figure
                   of
                   ●
                   Means
                
                 
                   42
                   /
                   30
                
                 
                   36
                   /
                   36
                
                 
                   20
                   /
                   42
                
                 
                   24
                   /
                   48
                
                 
                   18
                   /
                   54
                
                 
                   12
                   /
                   60
                
                 
                   6
                   /
                   66
                
              
            
          
           
           
             9
             If
             any
             of
             these
             Deficient
             Figures
             ,
             of
             which
             I
             have
             now
             spoken
             ,
             as
             A
             B
             C
             D
             (
             in
             the
             5th
             figure
             )
             be
             inscribed
             within
             the
             Complete
             figure
             B
             E
             ,
             having
             A
             D
             C
             E
             for
             its
             Complement
             ;
             and
             there
             be
             made
             upon
             C
             B
             produced
             ,
             the
             triangle
             A
             B
             I
             ;
             and
             the
             Parallelogram
             A
             B
             I
             K
             be
             completed
             ;
             and
             there
             be
             drawn
             parallel
             to
             the
             straight
             line
             C
             I
             ,
             any
             number
             of
             lines
             as
             M
             F
             ,
             cutting
             every
             one
             of
             them
             the
             crooked
             line
             of
             the
             Deficient
             Figure
             in
             D
             ,
             and
             the
             straight
             lines
             A
             C
             ,
             A
             B
             and
             A
             I
             in
             H
             ,
             G
             and
             L
             ;
             and
             as
             G
             F
             is
             to
             G
             D
             ,
             so
             G
             L
             be
             made
             to
             another
             G
             N
             ;
             and
             through
             all
             the
             points
             N
             there
             be
             drawn
             the
             line
             A
             N
             I
             ,
             there
             will
             be
             a
             Deficient
             Figure
             A
             N
             I
             B
             ,
             whose
             Complement
             will
             be
             A
             N
             I
             K.
             I
             say
             the
             figure
             A
             N
             I
             B
             is
             to
             the
             triangle
             A
             B
             I
             ,
             as
             the
             Deficient
             Figure
             A
             B
             C
             D
             twice
             taken
             ,
             is
             to
             the
             same
             Deficient
             Figure
             together
             with
             the
             Complete
             figure
             B
             E.
             
          
           
             For
             as
             the
             proportion
             of
             A
             B
             to
             A
             G
             ,
             that
             is
             ,
             of
             G
             M
             to
             G
             L
             ,
             is
             to
             the
             proportion
             of
             G
             M
             to
             G
             N
             ;
             so
             is
             the
             magnitude
             of
             the
             figure
             A
             N
             I
             B
             ,
             to
             that
             of
             its
             Complement
             A
             N
             I
             K
             (
             by
             the
             2d
             .
             Art.
             of
             this
             Chapter
             .
             )
          
           
             But
             (
             by
             the
             same
             Article
             )
             ,
             As
             the
             proportion
             of
             A
             B
             to
             A
             G
             ,
             that
             is
             ,
             of
             G
             M
             to
             G
             L
             ,
             is
             to
             the
             proportion
             of
             G
             F
             to
             G
             D
             ,
             that
             is
             ,
             (
             by
             construction
             )
             of
             G
             L
             to
             G
             N
             ;
             so
             is
             the
             figure
             A
             B
             C
             D
             to
             its
             Complement
             A
             D
             C
             E.
             
          
           
             And
             by
             Composition
             ,
             As
             the
             proportion
             of
             G
             M
             to
             G
             L
             ,
             together
             with
             that
             of
             G
             L
             to
             G
             N
             ,
             is
             to
             the
             proportion
             of
             G
             M
             to
             G
             L
             ;
             so
             is
             the
             complete
             figure
             B
             E
             ,
             to
             the
             Deficient
             Figure
             A
             B
             C
             D.
             
          
           
             And
             by
             Conversion
             ,
             As
             the
             proportion
             of
             G
             M
             to
             G
             L
             ,
             is
             to
             both
             the
             proportions
             of
             G
             M
             to
             G
             L
             and
             of
             G
             L
             to
             G
             N
             ,
             that
             is
             ,
             to
             the
             proportion
             of
             G
             M
             to
             G
             N
             (
             which
             is
             the
             proportion
             compounded
             of
             both
             )
             ;
             so
             is
             the
             Deficient
             Figure
             A
             B
             C
             D
             ,
             to
             the
             complete
             Figure
             B
             E.
             
          
           
             But
             it
             was
             ,
             As
             the
             proportion
             of
             G
             M
             to
             G
             L
             ,
             to
             that
             of
             G
             M
             to
             G
             N
             ;
             so
             the
             figure
             A
             N
             I
             B
             to
             its
             Complement
             A
             N
             I
             K.
             And
             therefore
             ,
             A
             B
             C
             D.
             B
             E
             :
             :
             A
             N
             I
             B.
             A
             N
             I
             K
             are
             proportionals
             .
             And
             by
             Composition
             ,
             A
             B
             C
             D+B
             E.
             A
             B
             C
             D
             :
             :
             B
             K.
             A
             N
             I
             B
             are
             proportionals
             .
          
           
           
             And
             by
             doubling
             the
             Consequents
             A
             B
             C
             D+B
             E.
             2
             A
             B
             C
             D
             :
             :
             B
             K.
             2
             A
             N
             I
             B
             are
             proportionals
             .
          
           
             And
             by
             taking
             the
             halfes
             of
             the
             third
             &
             the
             fourth
             A
             B
             C
             D+B
             E.
             2
             A
             B
             C
             D
             :
             :
             A
             B
             I.
             A
             N
             I
             B
             are
             also
             proportionals
             ;
             which
             was
             to
             be
             proved
             .
          
           
             10
             From
             what
             has
             been
             said
             of
             Deficient
             Figures
             described
             in
             a
             Parallelogram
             ,
             may
             be
             found
             out
             what
             proportions
             Spaces
             transmitted
             with
             accelerated
             Motion
             in
             determined
             times
             ,
             have
             to
             the
             times
             themselves
             ,
             according
             as
             the
             moved
             Body
             is
             accelerated
             in
             the
             several
             times
             with
             one
             or
             more
             degrees
             of
             Velocity
             .
          
           
             For
             ,
             let
             the
             Parallelogram
             A
             B
             C
             D
             (
             in
             the
             6th
             figure
             )
             and
             in
             it
             the
             three-sided
             figure
             D
             E
             B
             C
             be
             described
             ;
             and
             let
             F
             G
             be
             drawn
             any
             where
             parallel
             to
             the
             base
             ,
             cutting
             the
             Diagonal
             B
             D
             in
             H
             ,
             and
             the
             crooked
             line
             B
             E
             D
             in
             E
             ;
             &
             let
             the
             proportion
             of
             B
             C
             to
             B
             F
             be
             (
             for
             example
             )
             triplicate
             to
             that
             of
             F
             G
             to
             F
             E
             ;
             whereupon
             the
             figure
             D
             E
             B
             C
             will
             be
             triple
             to
             its
             Complement
             B
             E
             D
             A
             ;
             and
             in
             like
             manner
             ,
             I
             F
             being
             drawn
             parallel
             to
             B
             C
             ,
             the
             three-sided
             figure
             E
             K
             B
             F
             will
             be
             triple
             to
             its
             Complement
             B
             K
             E
             I.
             Wherefore
             ,
             the
             parts
             of
             the
             Deficient
             Figure
             cut
             off
             from
             the
             Vertex
             by
             straight
             lines
             parallel
             to
             the
             base
             ,
             namely
             D
             E
             B
             C
             and
             E
             K
             B
             F
             ,
             will
             be
             to
             one
             another
             as
             the
             Parallelograms
             A
             C
             and
             I
             F
             ;
             that
             is
             ,
             in
             proportion
             compounded
             of
             the
             proportions
             of
             the
             altitudes
             and
             bases
             .
             Seeing
             therefore
             the
             proportion
             of
             the
             altitude
             B
             C
             to
             the
             altitude
             B
             F
             is
             triplicate
             to
             the
             proportion
             of
             the
             base
             D
             C
             to
             the
             base
             F
             E
             ,
             the
             figure
             D
             E
             B
             C
             to
             the
             figure
             E
             K
             B
             F
             will
             be
             quadruplicate
             to
             the
             proportion
             of
             the
             same
             D
             C
             to
             F
             E.
             And
             by
             the
             same
             method
             ,
             may
             be
             found
             out
             ,
             what
             proportion
             any
             of
             the
             said
             three-sided
             figures
             ,
             has
             to
             any
             part
             of
             the
             same
             cut
             off
             from
             the
             Vertex
             by
             a
             straight
             line
             parallel
             to
             the
             base
             .
          
           
             Now
             as
             the
             said
             figures
             are
             understood
             to
             be
             described
             by
             the
             continual
             decreasing
             of
             the
             base
             ,
             as
             of
             C
             D
             (
             for
             example
             )
             till
             it
             end
             in
             a
             point
             ,
             as
             in
             B
             ;
             so
             also
             they
             may
             be
             understood
             to
             be
             described
             by
             the
             continual
             encreasing
             of
             a
             point
             ,
             as
             of
             B
             ,
             till
             it
             acquire
             any
             magnitude
             ,
             as
             that
             of
             C
             D.
             
          
           
             Suppose
             now
             the
             figure
             B
             E
             D
             C
             to
             be
             described
             by
             the
             encreasing
             of
             the
             point
             B
             to
             the
             magnitude
             C
             D.
             Seeing
             therefore
             the
             propor
             ion
             of
             B
             C
             to
             B
             F
             is
             triplicate
             to
             that
             of
             C
             D
             to
             F
             E
             ,
             the
             
             proportion
             of
             F
             E
             to
             C
             D
             will
             by
             Conversion
             (
             as
             I
             shall
             presently
             demonstrate
             )
             be
             triplicate
             to
             that
             B
             F
             to
             B
             C.
             Wherefore
             ,
             if
             the
             straight
             line
             B
             C
             be
             taken
             for
             the
             measure
             of
             the
             time
             in
             which
             the
             point
             B
             is
             moved
             ,
             the
             Figure
             E
             K
             B
             F
             will
             represent
             the
             Sum
             of
             all
             the
             encreasing
             Velocities
             in
             the
             time
             B
             F
             ;
             and
             the
             figure
             D
             E
             B
             C
             will
             in
             like
             manner
             represent
             the
             Summe
             of
             all
             the
             encreasing
             Velocities
             in
             the
             time
             B
             C.
             Seeing
             therefore
             the
             proportion
             of
             the
             figure
             E
             K
             B
             F
             to
             the
             figure
             D
             E
             B
             C
             ,
             is
             compounded
             of
             the
             proportions
             of
             altitude
             to
             altitude
             ,
             and
             base
             to
             base
             ;
             and
             seeing
             the
             proportion
             of
             F
             E
             to
             C
             D
             is
             triplicate
             to
             that
             of
             B
             F
             to
             B
             C
             ;
             the
             proportion
             of
             the
             figure
             E
             K
             B
             F
             to
             the
             figure
             D
             E
             B
             C
             ,
             will
             be
             quadruplicate
             to
             that
             of
             B
             F
             to
             B
             C
             ;
             that
             is
             ,
             the
             proportion
             of
             the
             Sum
             of
             the
             Velocities
             in
             the
             time
             B
             F
             ,
             to
             the
             Sum
             of
             the
             Velocities
             in
             the
             time
             B
             C
             wil
             be
             quadruplicate
             to
             the
             proportion
             of
             B
             F
             to
             B
             C.
             Wherfore
             if
             a
             Body
             be
             moved
             from
             B
             with
             Velocity
             so
             encreasing
             ,
             that
             the
             Velocity
             acquired
             in
             the
             time
             B
             F
             ,
             be
             to
             the
             Velocity
             acquired
             in
             the
             time
             B
             C
             in
             triplicate
             proportion
             to
             that
             of
             the
             times
             themselves
             B
             F
             to
             B
             C
             ,
             and
             the
             Body
             be
             carried
             to
             F
             in
             the
             time
             B
             F
             ;
             the
             same
             Body
             in
             the
             time
             B
             C
             will
             be
             carried
             through
             a
             line
             equal
             to
             the
             fifth
             proportional
             from
             B
             F
             in
             the
             continual
             proportion
             of
             B
             F
             to
             B
             C.
             And
             by
             the
             same
             manner
             of
             working
             ,
             we
             may
             determine
             ,
             what
             Spaces
             are
             transmitted
             by
             Velocities
             encreasing
             according
             to
             any
             other
             proportions
             .
          
           
             It
             remains
             ,
             that
             I
             demonstrate
             the
             proportion
             of
             F
             E
             to
             C
             D
             ,
             to
             be
             triplicate
             to
             that
             of
             B
             F
             to
             B
             C.
             Seeing
             therefore
             the
             proportion
             of
             C
             D
             ,
             that
             is
             of
             F
             G
             to
             F
             E
             is
             subtriplicate
             to
             that
             of
             B
             C
             to
             B
             F
             ;
             the
             proportion
             of
             F
             G
             to
             F
             E
             will
             also
             be
             subtriplicate
             to
             that
             of
             F
             G
             to
             F
             H.
             Wherefore
             the
             proportion
             of
             F
             G
             to
             F
             H
             is
             triplicate
             to
             that
             of
             F
             G
             ,
             that
             is
             ,
             of
             C
             D
             to
             F
             E.
             But
             in
             four
             continual
             proportionals
             ,
             of
             which
             the
             least
             is
             the
             first
             ,
             the
             proportion
             of
             the
             first
             to
             the
             fourth
             (
             by
             the
             16
             Art.
             of
             the
             13
             Chap.
             )
             is
             subtriplicate
             to
             the
             proportion
             of
             the
             third
             to
             the
             same
             fourh
             .
             Wherefore
             the
             proportion
             of
             F
             H
             to
             G
             F
             is
             subtriplicate
             to
             that
             of
             F
             E
             to
             C
             D
             ;
             and
             therefore
             the
             proportion
             of
             F
             E
             to
             C
             D
             is
             triplicate
             to
             that
             of
             F
             H
             to
             F
             G
             ,
             that
             is
             ,
             of
             B
             F
             to
             B
             C
             ,
             which
             was
             to
             be
             proved
             .
          
           
             It
             may
             from
             hence
             be
             collected
             ,
             that
             when
             the
             Velocity
             of
             a
             
             Body
             ,
             is
             encreased
             in
             the
             same
             proportion
             with
             that
             of
             the
             times
             ,
             the
             degrees
             of
             Velocity
             above
             one
             another
             proceed
             as
             numbers
             do
             in
             immediate
             succession
             from
             Unity
             ,
             namely
             ,
             as
             1
             ,
             2
             ,
             3
             ,
             4
             ,
             &c.
             
             And
             when
             the
             Velocity
             is
             encreased
             in
             proportion
             duplicate
             to
             that
             of
             the
             times
             ,
             the
             degrees
             proceed
             as
             numbers
             from
             Unity
             skipping
             One
             ,
             as
             1
             ,
             3
             ,
             5
             ,
             7
             ,
             &c.
             
             Lastly
             ,
             when
             the
             proportions
             of
             the
             Velocities
             are
             triplicate
             to
             those
             of
             the
             times
             ,
             the
             progression
             of
             the
             degrees
             is
             as
             that
             of
             numbers
             from
             Unity
             skipping
             Two
             in
             every
             place
             ,
             as
             1
             ,
             4
             ,
             7
             ,
             10
             ,
             &c.
             and
             so
             of
             other
             proportions
             .
             For
             Geometrical
             proportionals
             ,
             when
             they
             are
             taken
             in
             every
             point
             ,
             are
             the
             same
             with
             Arithmetical
             proportionals
             .
          
           
             11
             Moreover
             ,
             it
             is
             to
             be
             noted
             ,
             that
             as
             in
             quantities
             which
             are
             made
             by
             any
             magnitudes
             decreasing
             ,
             the
             proportions
             of
             the
             figures
             to
             one
             another
             ,
             are
             as
             the
             proportions
             of
             the
             altitudes
             to
             those
             of
             the
             bases
             ;
             so
             also
             it
             is
             in
             those
             which
             are
             made
             with
             motion
             decreasing
             ,
             which
             motion
             is
             nothing
             else
             but
             that
             power
             by
             which
             the
             described
             figures
             are
             greater
             or
             less
             .
             And
             therefore
             in
             the
             description
             of
             Archimedes
             his
             Spiral
             ,
             which
             is
             done
             by
             the
             continual
             diminution
             of
             the
             Semidiameter
             of
             a
             Circle
             in
             the
             same
             proportion
             in
             which
             the
             Circumference
             is
             diminished
             ,
             the
             Space
             which
             is
             contained
             within
             the
             Semidiameter
             and
             the
             Spiral
             Line
             ,
             is
             a
             third
             part
             of
             the
             whole
             Circle
             .
             For
             the
             Semidiameters
             of
             Circles
             ,
             in
             as
             much
             as
             Circles
             are
             understood
             to
             be
             made
             up
             of
             the
             aggregate
             of
             them
             ,
             are
             so
             many
             Sectors
             ;
             and
             therefore
             in
             the
             description
             of
             a
             Spiral
             ,
             the
             Sector
             which
             describes
             it
             ,
             is
             diminished
             in
             duplicate
             proportions
             to
             the
             diminutions
             of
             the
             Circumference
             of
             the
             Circle
             in
             which
             it
             is
             inscribed
             ;
             so
             that
             the
             Complement
             of
             the
             Spiral
             (
             that
             is
             ,
             that
             space
             in
             the
             Circle
             which
             is
             without
             the
             Spiral
             Line
             ,
             )
             is
             double
             to
             the
             space
             within
             the
             Spiral
             Line
             .
             In
             the
             same
             manner
             ,
             if
             there
             be
             taken
             a
             mean
             proportional
             every
             where
             between
             the
             Semidiameter
             of
             the
             Circle
             which
             contains
             the
             Spiral
             ,
             and
             that
             part
             of
             the
             Semidiameter
             which
             is
             within
             the
             same
             ,
             there
             will
             be
             made
             another
             figure
             ,
             which
             will
             be
             half
             the
             Circle
             .
             And
             to
             conclude
             ,
             this
             Rule
             serves
             for
             all
             such
             Spaces
             as
             may
             be
             described
             by
             a
             Line
             or
             Superficies
             decreasing
             either
             in
             magnitude
             or
             power
             ;
             so
             that
             if
             the
             proportions
             in
             which
             they
             decrease
             ,
             be
             commensurable
             to
             the
             proportions
             of
             the
             times
             in
             which
             they
             decrease
             ,
             
             the
             magnitudes
             of
             the
             figures
             they
             describe
             will
             be
             known
             .
          
           
             12
             The
             truth
             of
             that
             proposition
             which
             I
             demonstrated
             in
             the
             second
             Article
             (
             which
             is
             the
             foundation
             of
             all
             that
             has
             been
             said
             concerning
             Deficient
             Figures
             )
             may
             be
             derived
             from
             the
             Elements
             of
             Philosophy
             ,
             as
             having
             i●●
             original
             in
             this
             ;
             
               That
               all
               equality
               and
               inequality
               between
               two
               effects
               ,
               (
               that
               is
               ,
               all
               Proportion
               )
               proceeds
               from
               ,
               and
               is
               determined
               by
               the
               equal
               and
               unequal
               causes
               of
               those
               effects
               ,
               or
               from
               the
               proportion
               which
               the
               causes
               concurring
               to
               one
               effect
               ,
               have
               to
               the
               causes
               which
               concurre
               to
               the
               producing
               of
               the
               other
               effect
            
             ;
             and
             that
             therefore
             the
             proportions
             of
             Quantities
             are
             the
             same
             with
             the
             proportions
             of
             their
             causes
             .
             Seeing
             therefore
             two
             Deficient
             Figures
             (
             of
             which
             one
             is
             the
             Complement
             of
             the
             other
             )
             are
             made
             ,
             one
             by
             motion
             decreasing
             in
             a
             certain
             time
             and
             proportion
             ,
             the
             other
             by
             the
             loss
             of
             Motion
             in
             the
             same
             time
             ,
             the
             causes
             which
             make
             and
             determine
             the
             quantities
             of
             both
             the
             figures
             ,
             so
             ,
             that
             they
             can
             be
             no
             other
             then
             they
             are
             ,
             differ
             onely
             in
             this
             ,
             that
             the
             proportions
             by
             which
             the
             quantity
             which
             generates
             the
             figure
             proceeds
             in
             describing
             of
             the
             same
             ,
             (
             that
             is
             ,
             the
             proportions
             of
             the
             remainders
             of
             all
             the
             times
             and
             altitudes
             )
             may
             be
             other
             proportions
             then
             those
             by
             which
             the
             same
             generating
             quantity
             decreases
             in
             making
             the
             Complement
             of
             that
             Figure
             ,
             (
             that
             is
             ,
             the
             proportions
             of
             the
             quantity
             which
             generates
             the
             Figure
             continually
             diminished
             .
             )
             Wherefore
             ,
             as
             the
             proportions
             of
             the
             quantity
             in
             which
             Motion
             is
             lost
             ,
             is
             to
             that
             of
             the
             decreasing
             quantities
             by
             which
             the
             Deficient
             Figure
             is
             generated
             ,
             so
             will
             the
             Defect
             or
             Complement
             be
             to
             the
             Figure
             it self
             which
             is
             generated
             .
          
           
             13
             There
             are
             also
             other
             quantities
             which
             are
             determinable
             from
             the
             knowledge
             of
             their
             causes
             ,
             namely
             ,
             from
             the
             comparison
             of
             the
             Motions
             by
             which
             they
             are
             made
             ,
             and
             that
             more
             easily
             then
             from
             the
             common
             Elements
             of
             Geometry
             .
             For
             example
             ,
             That
             the
             Superficies
             of
             any
             portion
             of
             a
             Sphere
             ,
             is
             equal
             to
             that
             Circle
             ,
             whose
             Radius
             is
             a
             straight
             Line
             drawn
             from
             the
             Pole
             of
             the
             portion
             to
             the
             Circumference
             of
             its
             base
             ,
             I
             may
             demonstrate
             in
             this
             manner
             .
             Let
             B
             A
             C
             (
             in
             the
             7
             Figure
             )
             be
             a
             portion
             of
             a
             Sphere
             ,
             whose
             Axis
             is
             A
             E
             ,
             &
             whose
             base
             is
             B
             C
             ;
             &
             let
             A
             B
             be
             the
             straight
             line
             drawn
             from
             the
             Pole
             A
             to
             the
             base
             in
             B
             ;
             and
             let
             A
             D
             ,
             equal
             to
             A
             B
             ,
             touch
             the
             great
             Circle
             B
             A
             C
             in
             the
             Pole
             A.
             It
             is
             to
             be
             proved
             
             that
             the
             Circle
             made
             by
             the
             Radius
             A
             D
             ,
             is
             equal
             to
             the
             Superficies
             of
             the
             portion
             B
             A
             C.
             Let
             the
             plain
             A
             E
             B
             D
             be
             understood
             to
             make
             a
             revolution
             about
             the
             Axis
             A
             E
             ;
             &
             it
             is
             manifest
             that
             by
             the
             straight
             line
             A
             D
             a
             Circle
             will
             be
             described
             ;
             and
             by
             the
             arch
             A
             B
             the
             Superficies
             of
             a
             portion
             of
             a
             Sphere
             ,
             and
             lastly
             ,
             by
             the
             Subtense
             A
             B
             the
             Superficies
             of
             a
             right
             Cone
             .
             Now
             seeing
             both
             the
             straight
             line
             A
             B
             ,
             and
             the
             arch
             A
             B
             make
             one
             and
             the
             same
             revolution
             ,
             and
             both
             of
             them
             have
             the
             same
             extreme
             points
             A
             and
             B
             ,
             the
             cause
             why
             the
             the
             Spherical
             Superficies
             which
             is
             made
             by
             the
             arch
             ,
             is
             greater
             then
             the
             Conical
             Superficies
             which
             is
             made
             by
             the
             Subtense
             ,
             is
             ,
             that
             A
             B
             the
             arch
             ,
             is
             greater
             then
             A
             B
             the
             Subtense
             ;
             and
             the
             cause
             why
             it
             is
             greater
             consists
             in
             this
             ,
             that
             although
             they
             be
             both
             drawn
             from
             A
             to
             B
             ,
             yet
             the
             Subtense
             is
             drawn
             straight
             ,
             but
             the
             arch
             angularly
             ,
             namely
             according
             to
             that
             angle
             which
             the
             arch
             makes
             with
             the
             Subtense
             ,
             which
             angle
             is
             equal
             to
             the
             angle
             D
             A
             B
             (
             for
             an
             angle
             of
             contingence
             adds
             nothing
             to
             an
             angle
             of
             a
             Segment
             ,
             as
             has
             been
             shewn
             in
             the
             14
             Chapter
             at
             the
             16th
             Article
             .
             )
             Wherefore
             the
             magnitude
             of
             the
             angle
             D
             A
             B
             is
             the
             cause
             why
             the
             Superficies
             of
             the
             portion
             described
             by
             the
             arch
             A
             B
             ,
             is
             greater
             then
             the
             Superficies
             of
             the
             right
             Cone
             described
             by
             the
             Subtense
             A
             B.
             
          
           
             Again
             ,
             the
             cause
             why
             the
             Circle
             described
             by
             the
             Tangent
             A
             D
             is
             greater
             then
             the
             Superficies
             of
             the
             right
             Cone
             described
             by
             the
             Subtense
             A
             B
             (
             notwitstanding
             that
             the
             Tangent
             and
             the
             Subtense
             are
             equal
             ,
             and
             both
             moved
             round
             in
             the
             same
             time
             )
             is
             this
             ,
             that
             A
             D
             stands
             at
             right
             angles
             to
             the
             Axis
             ,
             but
             A
             B
             obliquely
             ;
             which
             obliquity
             consists
             in
             the
             same
             angle
             D
             A
             B.
             Seeing
             therefore
             the
             quantity
             of
             the
             angle
             D
             A
             B
             is
             that
             which
             makes
             the
             excess
             both
             of
             the
             Superficies
             of
             the
             Portion
             ,
             and
             of
             the
             Circle
             made
             by
             the
             Radius
             A
             D
             ,
             above
             the
             superficies
             of
             the
             Right
             Cone
             described
             by
             the
             subtense
             A
             B
             ;
             it
             follows
             ,
             that
             both
             the
             Superficies
             of
             the
             Portion
             ,
             and
             that
             of
             the
             Circle
             ,
             do
             equally
             exceed
             the
             Superficies
             of
             the
             Cone
             .
             Wherefore
             ,
             the
             Circle
             made
             by
             A
             D
             ,
             or
             A
             B
             ,
             and
             the
             Spherical
             Superficies
             made
             by
             the
             arch
             A
             B
             ,
             are
             equal
             to
             one
             another
             ;
             which
             was
             to
             be
             proved
             .
          
           
             ●4
             If
             these
             Deficient
             Figures
             which
             I
             have
             described
             in
             a
             〈◊〉
             ,
             were
             capable
             of
             exact
             description
             ,
             then
             any
             number
             
             of
             mean
             proportionals
             might
             be
             found
             out
             between
             two
             straight
             lines
             given
             .
             For
             example
             ,
             in
             the
             Parallelogram
             A
             B
             C
             D
             ,
             (
             in
             the
             8th
             .
             Figure
             )
             let
             the
             three-sided
             figure
             of
             two
             Means
             be
             described
             ,
             (
             which
             many
             call
             a
             
               Cubical
               Parabola
            
             )
             ;
             and
             let
             R
             and
             S
             be
             two
             given
             straight
             lines
             ;
             between
             which
             ,
             if
             it
             be
             required
             to
             find
             two
             mean
             proportionals
             ,
             it
             may
             be
             done
             thus
             .
             Let
             it
             be
             as
             R
             to
             S
             ,
             so
             B
             C
             to
             B
             F
             ;
             and
             let
             F
             E
             be
             drawn
             parallel
             to
             B
             A
             ,
             and
             cut
             the
             crooked
             line
             in
             E
             ;
             then
             through
             E
             let
             G
             H
             be
             drawn
             parallel
             and
             equal
             to
             the
             straight
             line
             A
             D
             ,
             and
             cut
             the
             Diagonal
             B
             D
             in
             I
             ;
             for
             thus
             we
             have
             G
             I
             the
             greatest
             of
             two
             Means
             between
             G
             H
             and
             G
             E
             ,
             as
             appears
             by
             the
             description
             of
             the
             figure
             in
             the
             4th
             Article
             .
             Wherefore
             if
             it
             be
             as
             G
             H
             to
             G
             I
             ,
             so
             R
             to
             another
             line
             T
             ,
             that
             T
             will
             be
             the
             greatest
             of
             two
             Means
             between
             R
             and
             S.
             And
             therefore
             if
             it
             be
             again
             as
             R
             to
             T
             ,
             so
             T
             to
             another
             line
             X
             ,
             that
             will
             be
             done
             which
             was
             required
             .
          
           
             In
             the
             same
             manner
             ,
             four
             mean
             proportionals
             may
             be
             found
             out
             ,
             by
             the
             description
             of
             a
             three-sided
             figure
             of
             four
             Means
             ;
             and
             so
             ,
             any
             other
             number
             of
             Means
             ,
             &c.
             
          
        
         
           
           
           
             
               CHAP.
               XVIII
            
             .
             Of
             the
             Equation
             of
             Straight
             Lines
             with
             the
             Crooked
             Lines
             of
             Parabolas
             and
             other
             Figures
             made
             in
             imitation
             of
             Parabolas
             .
          
           
             
               
                 1
                 To
                 find
                 a
                 straight
                 Line
                 equal
                 to
                 the
                 crooked
                 Line
                 of
                 a
                 Semiparabola
                 .
              
               
                 2
                 To
                 find
                 a
                 straight
                 Line
                 equal
                 to
                 the
                 Crooked
                 Line
                 of
                 the
                 first
                 Semiparabolaster
                 ,
                 or
                 to
                 the
                 Crooked
                 Line
                 of
                 any
                 other
                 of
                 the
                 Deficient
                 Figures
                 of
                 the
                 Table
                 of
                 the
                 3d.
                 Article
                 of
                 the
                 pr●●edent
                 Chapter
                 .
              
            
          
           
             1
             AParabola
             being
             given
             ,
             to
             find
             a
             Straight
             Line
             equal
             to
             the
             Crooked
             Line
             of
             the
             Semiparabola
             .
          
           
             Let
             the
             Parabolical
             Line
             given
             be
             ABC
             (
             in
             the
             first
             Figure
             )
             ,
             and
             the
             Diameter
             found
             be
             AD
             ,
             and
             the
             base
             drawn
             DC
             ,
             and
             the
             Parallelogram
             ADCE
             being
             completed
             ,
             draw
             the
             straight
             Line
             AC
             .
             Then
             dividing
             AD
             into
             two
             equal
             parts
             in
             F
             ,
             draw
             FH
             equal
             and
             parallel
             to
             DC
             ,
             cutting
             AC
             in
             K
             ,
             and
             the
             parabolical
             line
             in
             O
             ;
             and
             between
             FH
             and
             FO
             take
             a
             mean
             proportional
             FP
             ,
             and
             draw
             AO
             ,
             AP
             and
             PC
             .
             I
             say
             that
             the
             two
             Lines
             AP
             and
             PC
             taken
             together
             as
             one
             Line
             ,
             is
             equal
             to
             the
             parabolical
             line
             ABOC
             .
          
           
             For
             the
             line
             ABOC
             being
             a
             parabolical
             line
             ,
             is
             generated
             by
             the
             concourse
             of
             two
             Motions
             ,
             one
             Uniform
             from
             A
             to
             E
             ,
             the
             other
             in
             the
             same
             time
             uniformly
             accelerated
             from
             rest
             in
             A
             to
             D.
             And
             because
             the
             motion
             from
             A
             to
             E
             is
             uniform
             ,
             AE
             may
             represent
             the
             times
             of
             both
             those
             motions
             from
             the
             beginning
             to
             the
             end
             .
             Let
             therefore
             AE
             be
             the
             time
             ;
             and
             consequently
             the
             lines
             
             ordinately
             applyed
             in
             the
             Semiparabola
             ,
             will
             designe
             the
             parts
             of
             time
             wherein
             the
             Body
             that
             describeth
             the
             line
             ABOC
             is
             in
             every
             point
             of
             the
             same
             ;
             so
             that
             as
             at
             the
             end
             of
             the
             time
             AE
             or
             DC
             it
             is
             in
             C
             ,
             so
             at
             the
             end
             of
             the
             time
             FO
             it
             will
             be
             in
             O.
             And
             because
             the
             Velocity
             in
             AD
             is
             encreased
             uniformly
             ,
             that
             is
             ,
             in
             the
             same
             proportion
             with
             the
             time
             ,
             the
             same
             lines
             ordinately
             applyed
             in
             the
             Semiparabola
             will
             designe
             also
             the
             continual
             augmentations
             of
             the
             Impetus
             ,
             till
             it
             be
             at
             the
             greatest
             ,
             designed
             by
             the
             base
             DC
             .
             Therefore
             supposing
             Uniform
             motion
             in
             the
             line
             AF
             ,
             in
             the
             time
             FK
             the
             Body
             in
             A
             by
             the
             concourse
             of
             the
             two
             uniform
             motions
             in
             AF
             and
             FK
             will
             be
             moved
             uniformly
             in
             the
             line
             AK
             ;
             and
             KO
             wil
             be
             the
             encrease
             of
             the
             Impetus
             or
             Swiftness
             gained
             in
             the
             time
             FK
             ;
             and
             the
             line
             AO
             will
             be
             uniformly
             described
             by
             the
             concourse
             of
             the
             two
             uniform
             motions
             in
             AF
             and
             FO
             in
             the
             time
             FO
             .
             From
             O
             draw
             OL
             parallel
             to
             EC
             ,
             cutting
             AC
             in
             L
             ;
             &
             draw
             LN
             parallel
             to
             DC
             ,
             cutting
             EC
             in
             N
             ,
             and
             the
             parabolical
             line
             in
             M
             ;
             and
             produce
             it
             on
             the
             other
             side
             to
             AD
             in
             I
             ;
             and
             IN
             ,
             IM
             and
             IL
             will
             be
             (
             by
             the
             construction
             of
             a
             Parabola
             )
             in
             continual
             proportion
             ,
             &
             equal
             to
             the
             three
             lines
             FH
             ,
             FP
             and
             FO
             ;
             and
             a
             straight
             line
             parallel
             to
             EC
             passing
             through
             M
             will
             fall
             on
             P
             ;
             and
             therefore
             OP
             will
             be
             the
             encrease
             of
             Impetus
             gained
             in
             the
             time
             FO
             or
             IL.
             Lastly
             ,
             produce
             PM
             to
             CD
             in
             Q
             ;
             and
             QC
             ,
             or
             MN
             ,
             or
             PH
             will
             be
             the
             encrease
             of
             Impetus
             proportional
             to
             the
             time
             FP
             ,
             or
             IM
             ,
             or
             DQ
             .
             Suppose
             now
             uniform
             motion
             from
             H
             to
             C
             in
             the
             time
             PH.
             Seeing
             therefore
             in
             the
             time
             FP
             with
             uniform
             motion
             and
             the
             Impetus
             encreased
             in
             proportion
             to
             the
             times
             ,
             is
             described
             the
             straight
             line
             AP
             ;
             and
             in
             the
             rest
             of
             the
             time
             and
             Impetus
             ,
             namely
             PH
             ,
             is
             described
             the
             line
             CP
             uniformly
             ;
             it
             followeth
             that
             the
             whole
             line
             APC
             is
             described
             with
             the
             whole
             Impetus
             ,
             and
             in
             the
             same
             time
             wherewith
             is
             described
             the
             parabolicall
             line
             ABC
             ;
             and
             therefore
             the
             line
             APC
             ,
             made
             of
             the
             two
             straight
             lines
             AP
             and
             PC
             ,
             is
             equal
             to
             the
             parabolical
             line
             ABC
             ;
             which
             was
             to
             be
             proved
             .
          
           
             2
             To
             find
             a
             Straight
             line
             equal
             to
             the
             Crooked
             line
             of
             the
             first
             Semiparabolaster
             .
          
           
             Let
             ABC
             be
             the
             Crooked
             line
             of
             the
             first
             Semiparabolaster
             ;
             AD
             the
             Diameter
             ;
             DC
             the
             Base
             ;
             and
             let
             the
             Parallelogram
             
             completed
             be
             ADCE
             ,
             whose
             Diagonal
             is
             AC
             .
             Divide
             the
             Diameter
             into
             two
             equal
             parts
             in
             F
             ,
             and
             draw
             FH
             equal
             and
             parallel
             to
             DC
             ,
             ●utting
             AC
             in
             K
             ,
             the
             Crooked
             line
             in
             O
             ,
             and
             EC
             in
             H.
             Then
             draw
             OL
             parallel
             to
             EC
             ,
             cutting
             AC
             in
             L
             ;
             and
             draw
             LN
             parallel
             to
             the
             base
             DC
             ,
             cutting
             the
             Crooked
             line
             in
             M
             ,
             and
             the
             straight
             line
             EC
             in
             N
             ;
             and
             produce
             it
             on
             the
             other
             side
             to
             AD
             in
             I.
             Lastly
             ,
             through
             the
             point
             M
             draw
             PMQ
             parallel
             and
             equal
             to
             HC
             ,
             cutting
             FH
             in
             P
             ;
             and
             joyn
             CP
             ,
             AP
             and
             AO
             .
             I
             say
             the
             two
             Straight
             lines
             AP
             and
             PC
             are
             equal
             to
             the
             Crooked
             line
             ABOC
             .
          
           
             For
             the
             line
             ABOC
             being
             the
             Crooked
             line
             of
             the
             first
             Semiparabolaster
             ,
             is
             generated
             by
             the
             concourse
             of
             two
             Motions
             ,
             one
             uniform
             from
             A
             to
             E
             ,
             the
             other
             in
             the
             same
             time
             accelerated
             from
             rest
             in
             A
             to
             D
             ,
             so
             as
             that
             the
             Impetus
             encreaseth
             in
             proportion
             perpetually
             triplicate
             to
             that
             of
             the
             encrease
             of
             the
             time
             ,
             or
             (
             which
             is
             all
             one
             )
             the
             lengths
             transmitted
             are
             in
             proportion
             triplicate
             to
             that
             of
             the
             times
             of
             their
             transmission
             ;
             for
             as
             the
             Impetus
             or
             Quicknesses
             encrease
             ,
             so
             the
             Lengths
             transmitted
             encrease
             also
             .
             And
             because
             the
             motion
             from
             A
             to
             E
             is
             uniform
             ,
             the
             line
             AE
             may
             serve
             to
             represent
             the
             time
             ,
             and
             consequently
             the
             lines
             ordinately
             drawn
             in
             the
             Semiparabolaster
             ,
             will
             designe
             the
             parts
             of
             time
             wherein
             the
             Body
             beginning
             from
             rest
             in
             A
             ,
             describeth
             by
             its
             motion
             the
             Crooked
             line
             ABOC
             .
             And
             because
             DC
             which
             represents
             the
             greatest
             acquired
             Impetus
             is
             equal
             to
             AE
             ,
             the
             same
             ordinate
             lines
             will
             represent
             the
             several
             augmentations
             of
             the
             Impetus
             encreasing
             from
             rest
             in
             A.
             Therefore
             supposing
             uniform
             Motion
             from
             A
             to
             F
             in
             the
             time
             FK
             ,
             there
             will
             be
             described
             by
             the
             concourse
             of
             the
             two
             uniform
             Motions
             AF
             and
             FK
             the
             line
             AK
             uniformly
             ,
             and
             KO
             will
             be
             the
             encrease
             of
             Impetus
             in
             the
             time
             FK
             ;
             And
             by
             the
             concourse
             of
             the
             two
             uniform
             Motions
             in
             AF
             and
             FO
             ,
             will
             be
             described
             the
             line
             AO
             uniformly
             .
             Through
             the
             point
             L
             draw
             the
             straight
             line
             LMN
             parallel
             to
             DC
             ,
             cutting
             the
             straight
             line
             AD
             in
             I
             ,
             the
             crooked
             line
             ABC
             in
             M
             ,
             and
             the
             straight
             line
             EC
             in
             N
             ;
             and
             through
             the
             point
             M
             the
             straight
             line
             PMQ
             parallel
             and
             equal
             to
             HC
             ,
             cutting
             DC
             in
             Q
             ,
             and
             FH
             in
             P.
             By
             the
             concourse
             therefore
             of
             the
             two
             uniform
             Motions
             in
             AF
             and
             FP
             in
             the
             time
             FP
             will
             be
             uniformly
             described
             the
             straight
             line
             AP
             ;
             and
             LM
             or
             OP
             will
             be
             the
             encrease
             of
             Impetus
             to
             be
             added
             for
             the
             time
             FO
             .
             And
             because
             the
             proportion
             if
             IN
             
             to
             I
             L
             is
             triplicate
             to
             the
             proportion
             of
             I
             N
             to
             I
             M
             ,
             the
             proportion
             of
             F
             H
             to
             F
             O
             will
             also
             be
             triplicate
             to
             the
             proportion
             of
             F
             H
             to
             F
             P
             ;
             and
             the
             proportional
             Impetus
             gained
             in
             the
             time
             F
             P
             is
             P
             H.
             So
             that
             F
             H
             being
             equal
             to
             P
             C
             which
             designed
             the
             whole
             Impetus
             acquired
             by
             the
             acceleration
             ,
             there
             is
             no
             more
             encrease
             of
             Impetus
             to
             be
             computed
             .
             Now
             in
             the
             time
             P
             H
             suppose
             an
             uniform
             motion
             from
             H
             to
             C
             ;
             and
             by
             the
             two
             uniform
             motions
             in
             C
             H
             and
             H
             P
             will
             be
             described
             uniformly
             the
             Straight
             line
             P
             C.
             Seeing
             therefore
             the
             two
             Straight
             lines
             A
             P
             and
             P
             C
             are
             described
             in
             the
             time
             A
             E
             with
             the
             same
             encrease
             of
             Impetus
             wherewith
             the
             Crooked
             line
             A
             B
             C
             is
             described
             in
             the
             same
             time
             A
             E
             ,
             that
             is
             ,
             seeing
             the
             Line
             A
             P
             C
             and
             the
             Line
             A
             B
             C
             are
             transmitted
             by
             the
             same
             Body
             in
             the
             same
             Time
             ,
             &
             with
             equal
             Velocities
             ,
             the
             Lines
             themselves
             are
             equal
             ;
             which
             was
             to
             be
             demonstrated
             .
          
           
             By
             the
             same
             method
             ,
             if
             any
             of
             the
             Semiparabolasters
             in
             the
             Table
             of
             the
             3d
             Article
             of
             the
             precedent
             Chapter
             be
             exhibited
             ,
             may
             be
             found
             a
             Straight
             line
             equal
             to
             the
             Crooked
             line
             thereof
             ,
             namely
             ,
             by
             dividing
             the
             Diameter
             into
             two
             equal
             parts
             ,
             and
             proceeding
             as
             before
             .
             Yet
             no
             man
             hitherto
             hath
             compared
             any
             Crooked
             with
             any
             Straight
             Line
             ,
             though
             many
             Geometricians
             of
             every
             Age
             have
             endeavoured
             it
             .
             But
             the
             cause
             why
             they
             have
             not
             done
             it
             may
             be
             this
             ,
             that
             there
             being
             in
             Euclide
             no
             Definition
             of
             Equality
             ,
             nor
             any
             mark
             by
             which
             to
             judge
             of
             it
             besides
             Congruity
             (
             which
             is
             the
             8th
             .
             Axiome
             of
             the
             first
             Book
             of
             his
             Elements
             )
             a
             thing
             of
             no
             use
             at
             all
             in
             the
             comparing
             of
             Straight
             and
             Crooked
             ;
             and
             others
             after
             Euclide
             (
             except
             Archimedes
             and
             Apollonius
             ,
             and
             in
             our
             time
             Bo●a●entura
             )
             thinking
             the
             industry
             of
             the
             Ancients
             had
             reached
             to
             all
             that
             was
             to
             be
             done
             in
             Geometry
             ,
             thought
             also
             ,
             that
             all
             that
             could
             be
             propounded
             ,
             was
             either
             to
             be
             deduced
             from
             what
             they
             had
             written
             ,
             or
             else
             that
             it
             was
             not
             at
             all
             to
             be
             done
             .
             It
             was
             therefore
             disputed
             by
             some
             of
             those
             Ancients
             themselves
             ,
             whether
             there
             might
             be
             any
             Equality
             at
             all
             between
             Crooked
             and
             Straight
             Lines
             ;
             Which
             question
             Archimedes
             (
             who
             assumed
             that
             some
             Straight
             line●
             was
             equal
             to
             the
             Circumference
             of
             a
             Circle
             )
             seems
             to
             have
             despised
             ,
             as
             he
             had
             reason
             .
             And
             there
             is
             a
             late
             Writer
             that
             granteth
             that
             between
             a
             Straight
             and
             a
             Crooked
             Line
             there
             is
             Equality
             ;
             but
             now
             ,
             now
             sayes
             he
             ,
             since
             the
             fall
             of
             Adam
             ,
             without
             the
             special
             assistance
             of
             Divine
             Grace
             ,
             it
             is
             not
             to
             be
             found
             ▪
          
        
         
           
           
           
           
             
               CHAP.
               XIX
            
             .
             Of
             Angles
             of
             Incidence
             and
             Reflection
             ,
             equal
             by
             supposition
             .
          
           
             
               
                 1
                 If
                 two
                 straight
                 lines
                 falling
                 upon
                 another
                 straight
                 line
                 be
                 parallel
                 ,
                 the
                 lines
                 reflected
                 from
                 them
                 shall
                 also
                 be
                 parallel
                 .
              
               
                 2
                 If
                 two
                 straight
                 lines
                 drawn
                 from
                 one
                 point
                 ,
                 fall
                 upon
                 another
                 straight
                 line
                 ,
                 the
                 lines
                 reflected
                 from
                 them
                 ,
                 if
                 they
                 be
                 drawn
                 out
                 the
                 other
                 way
                 ,
                 will
                 me●t
                 in
                 an
                 angle
                 equal
                 to
                 the
                 angle
                 made
                 by
                 the
                 lines
                 of
                 Incidence
                 .
              
               
                 3
                 If
                 two
                 straight
                 parallel
                 lines
                 drawn
                 (
                 not
                 oppositely
                 but
                 )
                 from
                 the
                 same
                 parts
                 ,
                 fall
                 upon
                 the
                 Circumference
                 of
                 a
                 Circle
                 ,
                 the
                 lines
                 reflected
                 from
                 them
                 ,
                 if
                 produced
                 they
                 meet
                 within
                 the
                 Circle
                 ,
                 will
                 make
                 an
                 angle
                 double
                 ,
                 to
                 that
                 which
                 is
                 made
                 by
                 two
                 straight
                 lines
                 drawn
                 from
                 the
                 Center
                 to
                 the
                 points
                 of
                 Incidence
                 .
              
               
                 4
                 If
                 two
                 straight
                 lines
                 drawn
                 from
                 the
                 same
                 point
                 without
                 a
                 Circle
                 fall
                 upon
                 the
                 Circumference
                 ,
                 and
                 the
                 lines
                 reflected
                 from
                 them
                 ,
                 being
                 produced
                 meet
                 within
                 the
                 Circle
                 ,
                 they
                 will
                 make
                 an
                 angle
                 equal
                 to
                 twice
                 that
                 angle
                 which
                 is
                 made
                 by
                 two
                 straight
                 lines
                 drawn
                 from
                 the
                 Center
                 to
                 the
                 points
                 of
                 Incidence
                 ,
                 together
                 with
                 the
                 angle
                 which
                 the
                 incident
                 lines
                 themselves
                 make
                 .
              
               
                 5
                 If
                 two
                 straight
                 lines
                 drawn
                 from
                 one
                 point
                 fall
                 upon
                 the
                 concave
                 Circumference
                 of
                 a
                 Circle
                 ,
                 and
                 the
                 angle
                 they
                 make
                 be
                 less
                 then
                 twice
                 the
                 angle
                 at
                 the
                 Center
                 ,
                 the
                 lines
                 reflected
                 from
                 them
                 ,
                 and
                 meeting
                 within
                 the
                 Circle
                 ,
                 will
                 make
                 an
                 angle
                 which
                 being
                 added
                 to
                 the
                 angle
                 of
                 the
                 incident
                 lines
                 ,
                 will
                 be
                 equal
                 to
                 twice
                 the
                 angle
                 at
                 the
                 Center
                 .
              
               
                 6
                 If
                 through
                 any
                 one
                 point
                 two
                 unequal
                 Chords
                 be
                 drawn
                 cutting
                 one
                 another
                 ,
                 and
                 the
                 Center
                 of
                 the
                 Circle
                 be
                 not
                 placed
                 between
                 them
                 ,
                 and
                 the
                 lines
                 reflected
                 from
                 them
                 concurre
                 wheresoever
                 ,
                 there
                 cannot
                 through
                 the
                 point
                 through
                 which
                 the
                 two
                 former
                 lines
                 were
                 drawn
                 ,
                 be
                 drawn
                 any
                 other
                 straight
                 line
                 ,
                 whose
                 reflected
                 line
                 shall
                 pass
                 through
                 the
                 common
                 point
                 of
                 the
                 two
                 former
                 lines
                 reflected
                 .
              
               
                 7
                 In
                 equal
                 Chords
                 the
                 same
                 is
                 not
                 true
                 .
              
               
                 8
                 Two
                 points
                 being
                 given
                 in
                 the
                 Circumference
                 of
                 a
                 Circle
                 ,
                 to
                 draw
                 two
                 straight
                 lines
                 to
                 them
                 ,
                 so
                 as
                 that
                 their
                 reflected
                 lines
                 may
                 contain
                 any
                 angle
                 given
                 .
              
               
                 9
                 If
                 a
                 straight
                 line
                 falling
                 upon
                 the
                 Circumference
                 of
                 a
                 Circle
                 be
                 produced
                 till
                 it
                 reach
                 the
                 Semidiameter
                 ,
                 and
                 that
                 part
                 of
                 it
                 which
                 is
                 intercepted
                 between
                 the
                 Circumference
                 and
                 the
                 Semidiameter
                 ,
                 be
                 equal
                 to
                 that
                 part
                 of
                 the
                 Semidiameter
                 which
                 is
                 between
                 the
                 point
                 of
                 concourse
                 &
                 the
                 center
                 ,
                 the
                 reflected
                 line
                 will
                 be
                 parallel
                 to
                 the
                 Semidiameter
                 .
              
               
                 10
                 If
                 from
                 a
                 point
                 within
                 a
                 
                 Circle
                 ,
                 two
                 straight
                 lines
                 be
                 drawn
                 to
                 the
                 Circumference
                 ,
                 and
                 their
                 reflected
                 lines
                 meet
                 in
                 the
                 Circumference
                 of
                 the
                 same
                 Circle
                 ,
                 the
                 angle
                 made
                 by
                 the
                 reflected
                 lines
                 ,
                 will
                 be
                 a
                 third
                 part
                 of
                 the
                 angle
                 made
                 by
                 the
                 incident
                 lines
                 .
              
            
          
           
             WHether
             a
             Body
             ,
             falling
             upon
             the
             superficies
             of
             another
             Body
             and
             being
             reflected
             from
             it
             ,
             do
             make
             equal
             angles
             at
             that
             superficies
             ,
             it
             belongs
             not
             to
             this
             place
             to
             dispute
             ,
             being
             a
             knowledge
             which
             depends
             upon
             the
             natural
             causes
             of
             Reflection
             ;
             of
             which
             hitherto
             nothing
             has
             been
             said
             ,
             but
             shall
             be
             spoken
             of
             hereafter
             .
          
           
             In
             this
             place
             therefore
             let
             it
             be
             supposed
             ,
             that
             the
             angle
             of
             Incidence
             is
             equal
             to
             the
             angle
             of
             Reflection
             ,
             that
             our
             present
             search
             may
             be
             applyed
             not
             to
             the
             finding
             out
             of
             the
             causes
             ,
             but
             some
             consequences
             of
             the
             same
             .
          
           
             I
             call
             an
             
               Angle
               of
               Incidence
            
             ,
             that
             which
             is
             made
             between
             a
             straight
             line
             and
             another
             line
             (
             straight
             or
             crooked
             )
             upon
             which
             it
             falls
             ,
             and
             which
             I
             call
             the
             
               Line
               Reflecting
            
             ;
             and
             an
             
               Angle
               of
               Reflection
            
             equal
             to
             it
             ,
             that
             which
             is
             made
             at
             the
             same
             point
             between
             the
             straight
             line
             which
             is
             reflected
             ,
             and
             the
             line
             reflecting
             .
          
           
             1
             If
             two
             straight
             lines
             which
             fall
             upon
             another
             straight
             line
             be
             be
             parallel
             ,
             their
             reflected
             lines
             shall
             be
             also
             parallel
             .
          
           
             Let
             the
             two
             straight
             lines
             AB
             and
             CD
             (
             in
             the
             1
             figure
             )
             which
             fall
             upon
             the
             straight
             line
             EF
             ,
             at
             the
             points
             B
             and
             D
             ,
             be
             parallel
             ;
             and
             let
             the
             lines
             reflected
             from
             them
             be
             BG
             and
             DH
             .
             I
             say
             BG
             and
             DH
             are
             also
             parallel
             .
          
           
             For
             the
             angles
             ABE
             and
             CDE
             are
             equal
             by
             reason
             of
             the
             parallellelisme
             of
             AB
             and
             CD
             ;
             and
             the
             angles
             GBF
             and
             HDF
             are
             equal
             to
             them
             by
             supposition
             ;
             for
             the
             lines
             BG
             and
             DH
             are
             reflected
             from
             the
             lines
             AB
             and
             CD
             .
             Wherefore
             BG
             and
             DH
             are
             parallel
             .
          
           
             2
             If
             two
             straight
             lines
             drawn
             from
             the
             same
             point
             ,
             fall
             upon
             another
             straight
             line
             ,
             the
             lines
             reflected
             from
             them
             ,
             if
             they
             be
             drawn
             out
             the
             other
             way
             ,
             will
             meet
             in
             an
             angle
             equal
             to
             the
             angle
             of
             the
             Incident
             lines
             .
          
           
             From
             the
             point
             AC
             (
             in
             the
             2d
             .
             figure
             )
             let
             the
             two
             straight
             lines
             AB
             and
             AD
             be
             drawn
             ;
             and
             let
             them
             fall
             upon
             the
             straight
             line
             EK
             at
             the
             points
             B
             and
             D
             ;
             and
             let
             the
             lines
             BI
             and
             DG
             be
             reflected
             from
             them
             .
             I
             say
             ,
             IB
             and
             GD
             do
             converge
             ,
             and
             that
             if
             they
             
             be
             produced
             on
             the
             other
             side
             of
             the
             line
             EK
             they
             shall
             meet
             ,
             as
             in
             F
             ;
             and
             that
             the
             angle
             BFD
             shal
             be
             equal
             to
             the
             angle
             BAD
             .
          
           
             For
             the
             angle
             of
             Reflection
             IBK
             is
             equal
             to
             the
             angle
             of
             Incidence
             ABE
             ;
             and
             to
             the
             angle
             IBK
             ,
             its
             vertical
             angle
             EBF
             is
             equal
             ;
             and
             therefore
             the
             angle
             ABE
             is
             equal
             to
             the
             angle
             EBF
             .
             Again
             the
             angle
             ADE
             is
             equal
             to
             the
             angle
             of
             Reflection
             GDK
             ,
             that
             is
             ,
             to
             its
             vertical
             angle
             EDF
             ;
             and
             therefore
             the
             two
             angles
             ABD
             and
             ADB
             of
             the
             triangle
             ABD
             ,
             are
             one
             by
             one
             equal
             to
             the
             two
             angles
             FBD
             and
             FDB
             of
             the
             triangle
             FBD
             ;
             Wherfore
             also
             the
             third
             angle
             BAD
             is
             equal
             to
             the
             third
             angle
             BFD
             ,
             which
             was
             to
             be
             proved
             .
          
           
             
               Corollary
               1.
               
               If
               the
               straight
               line
               AF
               be
               drawn
               ,
               it
               will
               be
               perpendicular
               to
               the
               straight
               line
               EK
               .
               For
               both
               the
               angles
               at
               E
               will
               be
               equal
               ,
               by
               reason
               of
               the
               equality
               of
               the
               two
               angles
               ABE
               and
               FBE
               ,
               and
               of
               the
               two
               sides
               AB
               and
               FB
               .
            
             
               Corollary
               2.
               
               If
               upon
               any
               point
               between
               B
               and
               D
               there
               fall
               a
               straight
               line
               ,
               as
               AC
               ,
               whose
               reflected
               line
               is
               CH
               ,
               this
               also
               produced
               beyond
               C
               ,
               will
               fall
               upon
               F
               ;
               which
               is
               evident
               by
               the
               demonstration
               above
               .
            
          
           
             3
             If
             from
             two
             points
             taken
             without
             a
             Circle
             ,
             two
             straight
             parallel
             lines
             drawn
             (
             not
             oppositely
             but
             )
             from
             the
             same
             parts
             ,
             fall
             upon
             the
             Circumference
             ;
             the
             lines
             reflected
             from
             them
             ,
             if
             produced
             they
             meet
             within
             the
             Circle
             ,
             will
             make
             an
             angle
             double
             to
             that
             which
             is
             made
             by
             two
             straight
             lines
             drawn
             from
             the
             Center
             to
             the
             points
             of
             Incidence
             .
          
           
             Let
             the
             two
             straight
             parallels
             AB
             and
             DC
             (
             in
             the
             3d
             figure
             )
             fall
             upon
             the
             Circumference
             BC
             at
             the
             points
             B
             and
             C
             ;
             and
             let
             the
             Center
             of
             the
             Circle
             be
             E
             ;
             and
             let
             AB
             reflected
             be
             BF
             ,
             and
             DC
             reflected
             be
             CG
             ;
             and
             let
             the
             lines
             FB
             and
             GC
             produced
             meet
             within
             the
             Circle
             in
             H
             ;
             and
             let
             EB
             and
             EC
             be
             connected
             .
             I
             say
             the
             angle
             FHG
             is
             double
             to
             the
             angle
             BEC
             .
          
           
             For
             seeing
             AB
             and
             DC
             are
             parallels
             ,
             and
             EB
             cuts
             AB
             in
             B
             ,
             the
             same
             EB
             produced
             will
             cut
             DC
             somewhere
             ;
             let
             it
             cut
             it
             in
             D
             ,
             &
             let
             DC
             be
             produced
             howsoever
             to
             I
             ,
             and
             let
             the
             intersection
             of
             DC
             &
             BF
             be
             at
             K.
             The
             angle
             therefore
             ICH
             (
             being
             external
             to
             the
             triangle
             CKH
             ,
             )
             will
             be
             equal
             to
             the
             two
             opposite
             angles
             CKH
             and
             CHK
             .
             Again
             ,
             ICE
             (
             being
             external
             to
             the
             triangle
             CDE
             )
             
             is
             equal
             to
             the
             two
             angles
             at
             D
             and
             E.
             Wherefore
             the
             angle
             ICH
             ,
             being
             double
             to
             the
             angle
             ICE
             ,
             is
             equal
             to
             the
             angles
             at
             D
             and
             E
             twice
             taken
             ;
             and
             therefore
             the
             two
             angles
             CKH
             and
             CHK
             are
             equal
             to
             the
             two
             angles
             at
             D
             and
             E
             twice
             taken
             .
             But
             the
             angle
             CKH
             is
             equal
             to
             the
             angles
             D
             and
             ABD
             ,
             that
             is
             ,
             D
             twice
             taken
             ,
             (
             for
             AB
             and
             DC
             being
             parallels
             ,
             the
             altern
             angles
             D
             ,
             and
             ABD
             are
             equal
             )
             .
             Wherefore
             CHK
             ,
             that
             is
             the
             angle
             ,
             FHG
             is
             also
             equal
             to
             the
             angle
             at
             E
             twice
             taken
             ;
             which
             was
             to
             be
             proved
             .
          
           
             Corollary
             .
             If
             from
             two
             points
             taken
             within
             a
             circle
             ,
             two
             straight
             parallels
             fall
             upon
             the
             circumference
             ,
             the
             lines
             reflected
             from
             them
             shall
             meet
             in
             an
             angle
             ,
             double
             to
             that
             which
             is
             made
             by
             two
             straight
             lines
             drawn
             from
             the
             center
             to
             the
             points
             of
             Incidence
             .
             For
             the
             parallels
             LB
             and
             IC
             falling
             upon
             the
             points
             B
             and
             C
             ,
             are
             reflected
             in
             the
             lines
             BH
             and
             CH
             ,
             and
             make
             the
             angle
             at
             H
             double
             to
             the
             angle
             at
             E
             ,
             as
             was
             but
             now
             demonstrated
             -
          
           
             4
             If
             two
             straight
             lines
             drawn
             from
             the
             same
             point
             without
             a
             circle
             ,
             fall
             upon
             the
             circumference
             ,
             and
             the
             lines
             reflected
             from
             them
             being
             produced
             meet
             within
             the
             circle
             ,
             they
             will
             make
             an
             angle
             equal
             to
             twice
             that
             angle
             which
             is
             made
             by
             two
             straight
             lines
             drawn
             from
             the
             center
             to
             the
             points
             of
             Incidence
             together
             with
             the
             angle
             which
             the
             incident
             lines
             themselves
             make
             .
          
           
             Let
             the
             two
             straight
             lines
             AB
             and
             AC
             (
             in
             the
             4th
             figure
             )
             be
             drawn
             from
             the
             point
             A
             to
             the
             circumference
             of
             the
             circle
             ,
             whose
             center
             is
             D
             ;
             and
             let
             the
             lines
             reflected
             from
             them
             be
             BE
             and
             CG
             ,
             and
             being
             produced
             make
             within
             the
             circle
             the
             angle
             H
             ;
             also
             let
             the
             two
             straight
             lines
             DB
             and
             DC
             be
             drawn
             from
             the
             center
             D
             to
             the
             points
             of
             Incidence
             B
             and
             C.
             I
             say
             the
             angle
             H
             is
             equal
             to
             twice
             the
             angle
             at
             D
             together
             with
             the
             angle
             at
             A.
             
          
           
             For
             let
             AC
             be
             produced
             howsoever
             to
             I.
             Therefore
             the
             angle
             CH
             (
             which
             is
             external
             to
             the
             triangle
             CKH
             )
             will
             be
             equal
             to
             the
             two
             angles
             GKH
             and
             CHK
             .
             Again
             ,
             the
             angle
             ICD
             (
             which
             is
             external
             to
             the
             triangle
             CLD
             )
             wil
             be
             equal
             to
             the
             two
             angles
             CLD
             and
             CDL
             .
             But
             the
             angle
             ICH
             is
             double
             to
             the
             angle
             ICD
             ,
             and
             is
             therefore
             equal
             to
             the
             angles
             CLD
             and
             CDL
             twice
             taken
             .
             Wherefore
             the
             angles
             CKH
             and
             CHK
             are
             equal
             to
             the
             angles
             CLD
             and
             CDL
             twice
             taken
             .
             But
             the
             angle
             CLD
             (
             being
             external
             
             to
             the
             triangle
             ALB
             )
             is
             equal
             to
             the
             two
             angles
             LAB
             &
             LBA
             ;
             &
             consequently
             CLD
             twice
             taken
             is
             equal
             to
             LAB
             &
             LAB
             twice
             taken
             .
             Wherefore
             CKH
             &
             CHK
             are
             equal
             to
             the
             angle
             CDL
             together
             with
             LAB
             and
             LBA
             twice
             taken
             .
             Also
             the
             angle
             CKH
             is
             equal
             to
             the
             angle
             LAB
             once
             ,
             and
             ABK
             ,
             that
             is
             ,
             LBA
             twice
             taken
             .
             Wherefore
             the
             Angle
             CHK
             is
             equal
             to
             the
             remaining
             angle
             CDL
             (
             that
             is
             ,
             to
             the
             angle
             at
             D
             )
             twice
             taken
             ,
             and
             the
             angle
             LAB
             (
             that
             is
             ,
             the
             angle
             at
             A
             )
             once
             taken
             ;
             which
             was
             to
             be
             proved
             .
          
           
             Corollary
             .
             If
             two
             straight
             converging
             lines
             ,
             as
             IC
             and
             MB
             fall
             upon
             the
             concave
             circumference
             of
             a
             circle
             ,
             their
             reflected
             lines
             ,
             as
             CH
             and
             BH
             ,
             will
             meet
             in
             the
             angle
             H
             ,
             equal
             to
             twice
             the
             angle
             D
             ,
             together
             with
             the
             angle
             at
             A
             made
             by
             the
             ●ncident
             lines
             produced
             .
             Or
             ,
             if
             the
             Incident
             lines
             be
             HB
             and
             IC
             ,
             whose
             reflected
             lines
             CH
             and
             BM
             meet
             in
             the
             point
             N
             ,
             the
             angle
             CNB
             will
             be
             equal
             to
             twice
             the
             angle
             D
             ,
             together
             with
             the
             angle
             CKH
             made
             by
             the
             lines
             of
             Incidence
             .
             For
             the
             angle
             CNB
             is
             equal
             to
             the
             angle
             H
             (
             that
             is
             ,
             to
             twice
             the
             angle
             D
             )
             together
             with
             the
             two
             angles
             A
             and
             NBH
             (
             that
             is
             KBA
             )
             .
             But
             the
             angles
             KBA
             and
             A
             are
             equal
             to
             the
             angle
             CKH
             .
             Wherefore
             the
             angle
             CNB
             is
             equal
             to
             twice
             the
             angle
             D
             ,
             together
             with
             the
             angle
             CKH
             made
             by
             the
             lines
             of
             Incidence
             IC
             and
             HB
             produced
             to
             K.
             
          
           
             5
             If
             two
             straight
             lines
             drawn
             from
             one
             point
             ,
             fall
             upon
             the
             concave
             circumference
             of
             a
             circle
             ,
             and
             the
             angle
             they
             make
             be
             lesse
             then
             twice
             the
             angle
             at
             the
             center
             ;
             the
             lines
             reflected
             from
             them
             ,
             and
             meeting
             within
             the
             circle
             ,
             will
             make
             an
             angle
             ,
             which
             being
             added
             to
             the
             angle
             of
             the
             incident
             lines
             ,
             will
             be
             equal
             to
             twice
             the
             angle
             at
             the
             center
             .
          
           
             Let
             the
             two
             Lines
             AB
             and
             AC
             (
             in
             the
             5th
             figure
             )
             drawn
             from
             the
             point
             A
             ,
             fall
             upon
             the
             concave
             circumference
             of
             the
             circle
             whose
             center
             is
             D
             ;
             &
             let
             their
             reflected
             Lines
             BE
             and
             CE
             meet
             in
             the
             point
             E
             ;
             also
             let
             the
             angle
             A
             be
             less
             then
             twice
             the
             angle
             D.
             I
             say
             the
             angles
             A
             and
             E
             together
             taken
             are
             equal
             to
             twice
             the
             angle
             D.
             
          
           
             For
             let
             the
             straight
             Lines
             AB
             and
             EC
             cut
             the
             straight
             Lines
             DC
             and
             DB
             in
             the
             points
             G
             and
             H
             ;
             and
             the
             angle
             BHC
             will
             be
             equal
             to
             the
             two
             angles
             EBH
             and
             E
             ;
             also
             the
             same
             angle
             BHC
             will
             be
             equal
             to
             the
             two
             angles
             D
             and
             DCH
             ;
             and
             in
             like
             manner
             the
             angle
             BGC
             will
             be
             equal
             to
             the
             two
             angles
             ACD
             &
             A
             ,
             &
             the
             
             same
             angle
             BGC
             will
             be
             also
             equal
             to
             the
             two
             angles
             DBG
             and
             D.
             Wherefore
             the
             four
             angles
             EBH
             ,
             E
             ,
             ACD
             and
             A
             are
             equal
             to
             the
             four
             angles
             D
             ,
             DCH
             ,
             DBG
             and
             D.
             If
             therefore
             equals
             be
             taken
             away
             on
             both
             sides
             ,
             namely
             ,
             on
             one
             side
             ACD
             and
             EBH
             ,
             and
             on
             the
             other
             side
             DCH
             and
             DBG
             (
             for
             the
             angle
             EBH
             is
             equal
             to
             the
             angle
             DBG
             ,
             and
             the
             angle
             ACD
             equal
             to
             the
             angle
             DCH
             )
             the
             remainders
             on
             both
             sides
             will
             be
             equal
             ,
             namely
             ,
             on
             one
             side
             the
             angles
             A
             and
             E
             ,
             and
             on
             the
             other
             the
             angle
             D
             twice
             taken
             .
             Wherefore
             the
             angles
             A
             and
             E
             are
             equal
             to
             twice
             the
             angle
             D.
             
          
           
             Corollary
             .
             If
             the
             angle
             A
             be
             greater
             then
             twice
             the
             angle
             D
             ,
             their
             reflected
             ●●ines
             will
             diverge
             .
             For
             ,
             by
             the
             Corollary
             of
             the
             third
             Proposition
             ,
             if
             the
             angle
             A
             be
             equal
             to
             twice
             the
             angle
             D
             ,
             the
             reflected
             Lines
             BE
             and
             CE
             will
             be
             parallel
             ;
             and
             if
             it
             be
             lesse
             ,
             they
             will
             concurre
             ,
             as
             has
             now
             been
             demonstrated
             ;
             and
             therefore
             if
             it
             be
             greater
             ,
             the
             reflected
             Lines
             BE
             and
             CE
             will
             diverge
             ,
             and
             consequently
             ,
             if
             they
             be
             produced
             the
             other
             way
             ,
             they
             will
             concurre
             ,
             and
             make
             an
             angle
             equal
             to
             the
             excesse
             of
             the
             angle
             A
             above
             twice
             the
             angle
             D
             ;
             as
             is
             evident
             by
             the
             fourth
             Article
             .
          
           
             6
             If
             through
             any
             one
             point
             ,
             two
             unequal
             chords
             be
             drawn
             ,
             cutting
             one
             another
             ,
             either
             within
             the
             circle
             ,
             or
             (
             if
             they
             be
             produced
             )
             without
             it
             ,
             and
             the
             center
             of
             the
             circle
             be
             not
             placed
             between
             them
             ,
             and
             the
             Lines
             reflected
             from
             them
             concurre
             wheresoever
             ;
             there
             cannot
             through
             the
             point
             through
             which
             the
             former
             Lines
             were
             drawn
             ,
             be
             drawn
             another
             straight
             Line
             ,
             whose
             reflected
             Line
             shall
             passe
             through
             the
             point
             where
             the
             two
             former
             reflected
             Lines
             concurre
             .
          
           
             Let
             any
             two
             unequal
             chords
             ,
             as
             BK
             and
             CH
             (
             in
             the
             6th
             Figure
             )
             be
             drawn
             through
             the
             point
             A
             in
             the
             circle
             BC
             ;
             and
             let
             their
             reflected
             Lines
             BD
             and
             CE
             meet
             in
             F
             ;
             and
             let
             the
             center
             not
             be
             between
             AB
             and
             AC
             ;
             and
             from
             the
             point
             A
             let
             any
             other
             straight
             Line
             as
             AG
             be
             drawn
             to
             the
             circumference
             between
             B
             and
             C.
             I
             say
             GN
             ,
             which
             passes
             through
             the
             point
             F
             ,
             where
             the
             reflected
             Lines
             BD
             and
             CE
             meet
             ,
             will
             not
             be
             the
             reflected
             Line
             of
             AG.
             
          
           
             For
             let
             the
             arch
             BL
             be
             taken
             equal
             to
             the
             arch
             BG
             ,
             and
             the
             straight
             Line
             BM
             equal
             to
             the
             straight
             Line
             BA
             ;
             and
             LM
             being
             drawn
             ,
             let
             it
             be
             produced
             to
             the
             circūmference
             in
             O.
             Seeing
             therefore
             
             BA
             and
             BM
             are
             equal
             ,
             and
             the
             arch
             BL
             equal
             to
             the
             arch
             BG
             ,
             and
             the
             angle
             MBL
             equal
             to
             the
             angle
             ABG
             ,
             AG
             and
             ML
             will
             also
             be
             equal
             ,
             and
             (
             producing
             GA
             to
             the
             circumference
             in
             I
             )
             the
             whole
             lines
             LO
             and
             GI
             will
             in
             like
             manner
             be
             equal
             .
             But
             LO
             is
             greater
             then
             GFN
             (
             as
             shall
             presently
             be
             demonstrated
             )
             and
             therefore
             also
             GI
             is
             greater
             then
             GN
             .
             Wherefore
             the
             angles
             NGC
             and
             IGB
             are
             not
             equal
             .
             Wherefore
             the
             Line
             GFN
             is
             not
             reflected
             from
             the
             Line
             of
             Incidence
             AG
             ,
             and
             consequently
             no
             other
             straight
             Line
             (
             besides
             AB
             and
             AC
             )
             which
             is
             drawn
             through
             the
             point
             A
             ,
             and
             fa●ls
             upon
             the
             circumference
             BC
             ,
             can
             be
             reflected
             to
             the
             point
             F
             ,
             which
             was
             to
             be
             demonstrated
             .
          
           
             It
             remains
             that
             I
             prove
             LO
             to
             be
             greater
             then
             GN
             ;
             which
             I
             shall
             do
             in
             this
             manner
             .
             LO
             and
             GN
             cut
             one
             another
             in
             P
             ;
             and
             PL
             is
             greater
             then
             PG.
             Seeing
             now
             LP
             .
             PG
             :
             :
             PN
             .
             PO
             are
             proportionals
             ,
             therefore
             the
             two
             Extremes
             LP
             and
             PO
             together
             taken
             ,
             (
             that
             is
             LO
             )
             ,
             are
             greater
             then
             PG
             and
             PN
             together
             taken
             ,
             (
             that
             is
             ,
             GN
             ,
             )
             which
             remained
             to
             be
             proved
             .
          
           
             7
             But
             if
             two
             equal
             chords
             be
             drawn
             through
             one
             point
             within
             a
             circle
             ,
             and
             the
             Lines
             reflected
             from
             them
             meet
             in
             another
             point
             ,
             then
             another
             straight
             Line
             may
             be
             drawn
             between
             them
             through
             the
             former
             point
             ,
             whose
             reflected
             Line
             shall
             pass
             through
             the
             later
             point
             .
          
           
             Let
             the
             two
             equal
             chords
             BC
             and
             ED
             (
             in
             the
             7th
             figure
             )
             cut
             one
             another
             in
             the
             point
             A
             within
             the
             circle
             BCD
             ;
             and
             let
             their
             reflected
             Lines
             CH
             and
             DI
             meet
             in
             the
             point
             F.
             Then
             dividing
             the
             arch
             CD
             equally
             in
             G
             ,
             let
             the
             two
             chords
             GK
             and
             GL
             be
             drawn
             through
             the
             points
             A
             and
             F.
             I
             say
             GL
             will
             be
             the
             Line
             reflected
             from
             the
             chord
             KG
             .
             For
             the
             four
             chords
             BC
             ,
             CH
             ,
             ED
             and
             DI
             ,
             are
             by
             supposition
             all
             equal
             to
             one
             another
             ;
             and
             therefore
             the
             arch
             BCH
             is
             equal
             to
             the
             arch
             EDI
             ;
             as
             also
             the
             angle
             BCH
             to
             the
             angle
             EDI
             ;
             &
             the
             angle
             AMC
             to
             its
             vertical
             angle
             FMD
             ;
             and
             the
             straight
             Line
             DM
             to
             the
             straight
             Line
             CM
             ;
             and
             in
             like
             manner
             ,
             the
             straight
             Line
             AC
             to
             the
             straight
             Line
             FD
             ;
             and
             the
             chords
             CG
             and
             GD
             being
             drawn
             ,
             will
             also
             be
             equal
             ;
             as
             also
             the
             angles
             FDG
             and
             ACG
             ,
             in
             the
             equal
             Segments
             GDI
             and
             GCB
             .
             Wherefore
             the
             straight
             Lines
             FG
             and
             AG
             are
             equal
             ;
             and
             therefore
             the
             angle
             FGD
             is
             equal
             to
             the
             angle
             AGC
             ,
             that
             is
             ,
             the
             angle
             
             of
             Incidence
             equal
             to
             the
             angle
             of
             Reflection
             .
             Wherefore
             the
             line
             GL
             is
             reflected
             from
             the
             incident
             Line
             KG
             ;
             which
             was
             to
             be
             proved
             .
          
           
             Corollary
             .
             By
             the
             very
             sight
             of
             the
             figure
             ,
             it
             is
             manifest
             ,
             that
             if
             G
             be
             not
             the
             middle
             point
             between
             C
             and
             D
             ,
             the
             reflected
             Line
             GL
             will
             not
             pass
             through
             the
             point
             F.
             
          
           
             8
             Two
             points
             in
             the
             circumference
             of
             a
             circle
             being
             given
             ,
             to
             draw
             two
             straight
             Lines
             to
             them
             ,
             so
             as
             that
             their
             reflected
             Lines
             may
             be
             parallel
             ,
             or
             contain
             any
             angle
             given
             .
          
           
             In
             the
             circumference
             of
             the
             circle
             whose
             center
             is
             A
             (
             in
             the
             8th
             .
             figure
             )
             let
             the
             two
             points
             B
             and
             C
             be
             given
             ;
             and
             let
             it
             be
             required
             to
             draw
             to
             them
             from
             two
             points
             taken
             without
             the
             circle
             ,
             two
             incident
             Lines
             ,
             so
             ,
             that
             their
             reflected
             Lines
             may
             (
             first
             )
             be
             parallel
             .
          
           
             Let
             AB
             and
             AC
             be
             drawn
             ;
             as
             also
             any
             incident
             Line
             DC
             ,
             with
             its
             reflected
             Line
             CF
             ;
             and
             let
             the
             angle
             ECD
             be
             made
             double
             to
             the
             angle
             A
             ;
             and
             let
             HB
             be
             drawn
             parallel
             to
             EC
             ,
             and
             produced
             till
             it
             meet
             with
             DC
             produced
             in
             I.
             Lastly
             ,
             producing
             AB
             indefinitely
             to
             K
             ,
             let
             GB
             be
             drawn
             ,
             so
             ,
             that
             the
             angle
             GBK
             may
             be
             equal
             to
             the
             angle
             HBK
             ,
             and
             then
             GB
             will
             be
             the
             reflected
             Line
             of
             the
             incident
             Line
             HB
             .
             I
             say
             DC
             and
             HB
             are
             two
             incident
             Lines
             ,
             whose
             reflected
             Lines
             CF
             and
             BG
             are
             parallel
             .
          
           
             For
             seeing
             the
             angle
             ECD
             is
             double
             to
             the
             angle
             BAC
             ,
             the
             angle
             HIC
             is
             also
             (
             by
             reason
             of
             the
             parallels
             EC
             and
             HI
             )
             double
             to
             the
             same
             BAC
             ;
             Therefore
             also
             FC
             and
             GB
             (
             namely
             the
             lines
             reflected
             from
             the
             incident
             lines
             DC
             and
             HB
             are
             parallel
             .
             Wherefore
             the
             first
             thing
             required
             ,
             is
             done
             .
          
           
             Secondly
             ,
             let
             it
             be
             required
             to
             draw
             to
             the
             points
             B
             &
             C
             two
             straight
             lines
             of
             Incidence
             ,
             so
             ,
             that
             the
             lines
             reflected
             from
             them
             may
             contain
             the
             given
             angle
             Z.
             
          
           
             To
             the
             angle
             ECD
             made
             at
             the
             point
             C
             ,
             let
             there
             be
             added
             on
             one
             side
             the
             angle
             DCL
             equal
             to
             half
             Z
             ,
             and
             on
             the
             other
             side
             the
             angle
             ECM
             equal
             to
             the
             angle
             DCL
             ;
             and
             let
             the
             straight
             Line
             BN
             be
             drawn
             parallel
             to
             the
             straight
             line
             CM
             ;
             and
             let
             the
             angle
             KBO
             be
             made
             equal
             to
             the
             angle
             NBK
             ;
             which
             being
             done
             ,
             BO
             will
             be
             the
             Line
             of
             Reflection
             from
             the
             Line
             of
             Incidence
             NB.
             Lastly
             ,
             from
             the
             incident
             Line
             LC
             ,
             let
             the
             reflected
             Line
             CO
             be
             drawn
             ,
             cutting
             BO
             at
             O
             ,
             and
             making
             the
             angle
             COB
             .
             I
             say
             the
             angle
             COB
             is
             equal
             to
             the
             angle
             Z.
             
          
           
           
             Let
             NB
             be
             produced
             till
             it
             meet
             with
             the
             straight
             line
             LC
             produced
             in
             P.
             Seeing
             therefore
             the
             angle
             LCM
             is
             by
             construction
             equal
             to
             twice
             the
             angle
             BAC
             together
             with
             the
             angle
             Z
             ;
             the
             angle
             NPL
             (
             which
             is
             equal
             to
             LCM
             by
             reason
             of
             the
             parallels
             NP
             and
             MC
             )
             will
             also
             be
             equal
             to
             twice
             the
             same
             angle
             BAC
             together
             with
             the
             angle
             Z.
             And
             seeing
             the
             two
             straight
             lines
             OC
             and
             OB
             fall
             from
             the
             point
             O
             upon
             the
             points
             C
             and
             B
             ;
             and
             their
             reflected
             lines
             LC
             and
             NB
             meet
             in
             the
             point
             P
             ;
             the
             angle
             NPL
             will
             be
             equal
             to
             twice
             the
             angle
             BAC
             together
             with
             the
             angle
             COP
             .
             But
             I
             have
             already
             proved
             the
             angle
             NPL
             to
             be
             equal
             to
             twice
             the
             angle
             BAC
             together
             with
             the
             angle
             Z.
             Therefore
             the
             angle
             COP
             is
             equal
             to
             the
             angle
             Z
             ;
             Wherefore
             ,
             Two
             points
             in
             the
             circumference
             of
             a
             Circle
             being
             given
             ,
             I
             have
             drawn
             ,
             &c.
             which
             was
             to
             be
             done
             .
          
           
             But
             if
             it
             be
             required
             to
             draw
             the
             incident
             Lines
             from
             a
             point
             within
             the
             circle
             ,
             so
             ,
             that
             the
             Lines
             reflected
             from
             them
             may
             contain
             an
             angle
             equal
             to
             the
             angle
             Z
             ,
             the
             same
             method
             is
             to
             be
             used
             ,
             saving
             that
             in
             this
             case
             the
             angle
             Z
             is
             not
             to
             be
             added
             to
             twice
             the
             angle
             BAC
             ,
             but
             to
             be
             taken
             from
             it
             .
          
           
             9
             If
             a
             straight
             line
             falling
             upon
             the
             circumference
             of
             a
             circle
             ,
             be
             produced
             till
             it
             reach
             the
             Semidiameter
             ,
             and
             that
             part
             of
             it
             which
             is
             intercepted
             between
             the
             circumference
             and
             the
             Semidiameter
             ,
             be
             equal
             to
             that
             part
             of
             the
             Semidiameter
             which
             is
             between
             the
             point
             of
             concourse
             and
             the
             center
             ,
             the
             reflected
             Line
             will
             be
             parallel
             to
             the
             Semidiameter
             .
          
           
             Let
             any
             Line
             AB
             (
             in
             the
             9th
             figure
             )
             be
             the
             Semidiameter
             of
             the
             circle
             whose
             center
             is
             A
             ;
             and
             upon
             the
             circumference
             BD
             let
             the
             straight
             Line
             CD
             fall
             ,
             and
             be
             produced
             till
             it
             cut
             AB
             in
             E
             ,
             so
             ,
             that
             ED
             and
             EA
             may
             be
             equal
             ;
             &
             from
             the
             incident
             Line
             CD
             let
             the
             Line
             DF
             be
             reflected
             .
             I
             say
             AB
             and
             DF
             will
             be
             parallel
             .
          
           
             Let
             AG
             be
             drawn
             through
             the
             point
             D.
             Seeing
             therefore
             ED
             and
             EA
             are
             equal
             ,
             the
             angles
             EDA
             and
             EAD
             will
             also
             be
             equal
             .
             But
             the
             angles
             FDG
             and
             EDA
             are
             equal
             (
             for
             each
             of
             them
             is
             half
             the
             angle
             EDH
             or
             FDC
             .
             )
             Wherefore
             the
             angles
             FDG
             and
             EAD
             are
             equal
             ;
             and
             consequently
             DF
             and
             AB
             are
             parallel
             ;
             which
             was
             to
             be
             proved
             .
          
           
             Corollahy
             .
             If
             EA
             be
             greater
             then
             ED
             ,
             then
             DF
             and
             AB
             being
             produced
             will
             concurre
             ;
             but
             if
             EA
             be
             less
             then
             ED
             ,
             then
             BA
             and
             DH
             being
             produced
             will
             concurre
             .
          
           
           
             10
             If
             from
             a
             point
             within
             a
             circle
             ,
             two
             straight
             Lines
             be
             drawn
             to
             the
             Circumference
             ,
             and
             their
             reflected
             Lines
             meet
             in
             the
             Circumference
             of
             the
             same
             circle
             ,
             the
             angle
             made
             by
             the
             Lines
             of
             Reflection
             ,
             will
             be
             a
             third
             part
             of
             the
             angle
             made
             by
             the
             Lines
             of
             Incidence
             .
          
           
             From
             the
             point
             B
             (
             in
             the
             10th
             figure
             )
             taken
             within
             the
             circle
             whose
             center
             is
             A
             ,
             let
             the
             two
             straight
             lines
             BC
             and
             BD
             be
             drawn
             to
             the
             circumference
             ;
             and
             let
             their
             reflected
             Lines
             CE
             and
             DE
             meet
             in
             the
             circumference
             of
             the
             same
             circle
             at
             the
             point
             E.
             I
             say
             the
             angle
             CED
             will
             be
             a
             third
             part
             of
             the
             angle
             CBD
             .
          
           
             Let
             AC
             and
             AD
             be
             drawn
             .
             Seeing
             therefore
             the
             angles
             CED
             and
             CBD
             together
             taken
             ,
             are
             equal
             to
             twice
             the
             angle
             CAD
             (
             as
             has
             been
             demonstrated
             in
             the
             5th
             article
             )
             ;
             and
             the
             angle
             CAD
             twice
             taken
             is
             quadruple
             to
             the
             angle
             CED
             ;
             the
             angles
             CED
             and
             CBD
             together
             taken
             ,
             will
             also
             be
             equal
             to
             the
             angle
             CED
             four
             times
             taken
             ;
             and
             therefore
             if
             the
             angle
             CED
             be
             taken
             away
             on
             both
             sides
             ,
             there
             will
             remain
             the
             angle
             CBD
             on
             one
             side
             ,
             equal
             to
             the
             angle
             CED
             thrice
             taken
             on
             the
             other
             side
             ;
             which
             was
             to
             be
             demonstrated
             .
          
           
             Coroll
             .
             Therefore
             a
             point
             being
             given
             within
             a
             Circle
             ,
             there
             may
             be
             drawn
             two
             Lines
             from
             it
             to
             the
             Circumference
             ,
             so
             as
             their
             reflected
             Lines
             may
             meet
             in
             the
             Circumference
             .
             For
             it
             is
             but
             trisecting
             the
             Angle
             CBD
             ;
             which
             how
             it
             may
             be
             done
             ,
             shall
             be
             shewn
             in
             the
             following
             Chapter
             .
          
        
         
           
           
           
           
             
               CHAP.
               XX.
            
             Of
             the
             Dimension
             of
             a
             Circle
             ,
             and
             the
             Division
             of
             Angles
             or
             Arches
             .
          
           
             
               
                 1
                 The
                 Dimension
                 of
                 a
                 Circle
                 neer
                 determined
                 in
                 Numbers
                 by
                 Archimedes
                 and
                 others
                 .
              
               
                 2
                 The
                 first
                 attempt
                 for
                 the
                 finding
                 out
                 of
                 the
                 Dimension
                 of
                 a
                 Circle
                 by
                 Lines
                 .
              
               
                 3
                 The
                 second
                 attempt
                 for
                 the
                 finding
                 out
                 of
                 the
                 Dimension
                 of
                 a
                 Circle
                 from
                 the
                 consideration
                 of
                 the
                 nature
                 of
                 Crookedness
                 .
              
               
                 4
                 The
                 third
                 attempt
                 ;
                 and
                 some
                 things
                 propounded
                 to
                 be
                 further
                 searched
                 into
                 .
              
               
                 5
                 The
                 Equation
                 of
                 the
                 Spiral
                 of
                 Archimedes
                 with
                 a
                 straight
                 Line
                 .
              
               
                 6
                 Of
                 the
                 Analysis
                 of
                 Geometricians
                 by
                 the
                 Powers
                 of
                 Lines
                 .
              
            
          
           
             1
             IN
             the
             comparing
             of
             an
             Arch
             of
             a
             Circle
             with
             a
             Straight
             Line
             ,
             many
             and
             great
             Geometricians
             ,
             even
             from
             the
             most
             ancient
             times
             ,
             have
             exercised
             their
             wits
             ;
             and
             more
             had
             done
             the
             same
             ,
             if
             they
             had
             not
             seen
             their
             pains
             ,
             though
             undertaken
             for
             the
             common
             good
             ,
             if
             not
             brought
             to
             perfection
             ,
             vilified
             by
             those
             that
             envy
             the
             prayses
             of
             other
             men
             .
             Amongst
             those
             Ancient
             Writers
             whose
             Works
             are
             come
             to
             our
             hands
             ,
             Archimedes
             was
             the
             first
             that
             brought
             the
             Length
             of
             the
             Perimeter
             of
             a
             Circle
             within
             the
             limits
             of
             Numbers
             very
             litle
             differing
             from
             the
             truth
             ;
             demonstrating
             the
             same
             to
             be
             less
             then
             three
             Diameters
             and
             a
             seventh
             part
             ,
             but
             greater
             then
             three
             Diameters
             and
             ten
             seventy
             one
             parts
             of
             the
             Diameter
             .
             So
             that
             supposing
             the
             Radius
             to
             consist
             of
             10000000
             equal
             parts
             ,
             the
             Arch
             of
             a
             Quadrant
             will
             be
             between
             15714285
             and
             15
             04225
             of
             the
             same
             parts
             .
             In
             our
             times
             
               Ludovicus
               Van
               Cullen
            
             &
             
               Willebrordus
               Snellius
            
             with
             joint
             endeavour
             have
             come
             yet
             neerer
             to
             the
             truth
             ;
             and
             pronounced
             from
             true
             Principles
             ,
             that
             the
             Arch
             of
             a
             Quadrant
             (
             putting
             ,
             as
             before
             10000000
             for
             Radius
             )
             differs
             not
             one
             whole
             Unity
             from
             the
             number
             15707963
             ;
             which
             ,
             if
             they
             had
             
             exhibited
             their
             arithmetical
             operations
             (
             and
             no
             man
             had
             discovered
             any
             errour
             in
             that
             long
             work
             of
             theirs
             )
             had
             been
             demonstrated
             by
             them
             .
             This
             is
             the
             furthest
             progress
             that
             has
             been
             made
             by
             the
             way
             of
             Numbers
             ;
             and
             they
             that
             have
             proceeded
             thus
             far
             deserve
             the
             praise
             of
             Industry
             .
             Nevertheless
             ,
             if
             we
             consider
             the
             benefit
             (
             which
             is
             the
             scope
             at
             which
             all
             Speculation
             should
             aime
             )
             the
             improvement
             they
             have
             made
             has
             been
             little
             ,
             or
             none
             .
             For
             any
             ordinary
             man
             may
             much
             sooner
             ,
             &
             more
             accurately
             find
             a
             Straight
             Line
             equal
             to
             the
             Perimeter
             of
             a
             Circle
             ,
             and
             consequently
             square
             the
             Circle
             ,
             by
             winding
             a
             small
             thred
             about
             a
             given
             Cylinder
             ,
             then
             any
             Geometrician
             shall
             do
             the
             same
             by
             dividing
             the
             Radius
             into
             10000000
             equal
             parts
             .
             But
             though
             the
             length
             of
             the
             Circumference
             were
             exactly
             set
             out
             ,
             either
             by
             Numbers
             ,
             or
             mechanically
             ,
             or
             onely
             by
             chance
             ,
             yet
             this
             would
             contribute
             no
             help
             at
             all
             towards
             the
             Section
             of
             Angles
             ,
             unless
             happily
             these
             two
             Problemes
             ,
             
               To
               divide
               a
               given
               Angle
               according
               to
               any
               proportion
               assigned
               ,
            
             and
             
               To
               finde
               a
               Straight
               Line
               equal
               to
               the
               Arch
               of
               a
               Circle
               ,
            
             were
             reciprocal
             ,
             and
             followed
             one
             another
             .
             Seeing
             therefore
             the
             benefit
             proceeding
             from
             the
             knowledge
             of
             the
             Length
             of
             the
             Arch
             of
             a
             Quadrant
             ,
             consists
             in
             this
             ,
             that
             we
             may
             there
             by
             divide
             an
             Angle
             according
             to
             any
             proportion
             ,
             either
             accurately
             ,
             or
             at
             least
             accurately
             enough
             for
             common
             use
             ;
             and
             seeing
             this
             cannot
             be
             done
             by
             Arithmetick
             ,
             I
             thought
             fit
             to
             attempt
             the
             same
             by
             Geometry
             ;
             and
             in
             this
             Chapter
             to
             make
             trial
             whether
             it
             might
             not
             be
             performed
             by
             the
             drawing
             of
             Straight
             and
             Circular
             Lines
             .
          
           
             2
             Let
             the
             Square
             A
             B
             C
             D
             (
             in
             the
             first
             figure
             )
             be
             described
             ;
             and
             with
             the
             Radii
             A
             B
             ,
             B
             C
             and
             D
             C
             the
             three
             Arches
             B
             D
             ,
             C
             A
             and
             A
             C
             ;
             of
             which
             let
             the
             two
             B
             D
             and
             C
             A
             cut
             one
             another
             in
             E
             ,
             and
             the
             two
             B
             D
             and
             A
             C
             in
             F.
             The
             Diagonals
             therefore
             B
             D
             and
             A
             C
             being
             drawn
             will
             cut
             one
             another
             in
             the
             center
             of
             the
             Square
             G
             ,
             and
             the
             two
             Arches
             B
             D
             and
             C
             A
             into
             two
             equal
             parts
             in
             H
             and
             Y
             ;
             and
             the
             Arch
             B
             H
             D
             will
             be
             trisected
             in
             F
             and
             E.
             Through
             the
             Center
             G
             let
             the
             two
             Straight
             Lines
             K
             G
             L
             and
             M
             G
             N
             be
             drawn
             parallel
             and
             equal
             to
             the
             sides
             of
             the
             Square
             A
             B
             and
             A
             D
             ,
             cutting
             the
             four
             sides
             of
             the
             same
             Square
             in
             the
             points
             K
             ,
             L
             ,
             M
             and
             N
             ;
             which
             being
             done
             ,
             K
             L
             will
             pass
             through
             F
             ,
             and
             M
             N
             through
             E.
             Then
             let
             O
             P
             be
             drawn
             parallel
             and
             equal
             to
             the
             side
             B
             C
             ,
             cutting
             the
             Arch
             B
             F
             D
             in
             F
             ,
             and
             the
             sides
             A
             B
             and
             D
             C
             
             in
             O
             and
             P.
             Therefore
             O
             F
             will
             be
             the
             Sine
             of
             the
             arch
             B
             F
             ,
             which
             is
             an
             arch
             of
             30
             degrees
             ;
             and
             the
             same
             O
             F
             will
             be
             equal
             to
             half
             the
             Radius
             .
             Lastly
             ,
             dividing
             the
             arch
             B
             F
             in
             the
             middle
             in
             Q
             ,
             let
             R
             Q
             the
             Sine
             of
             the
             arch
             B
             Q
             be
             drawn
             and
             produced
             to
             S
             ▪
             so
             that
             Q
             S
             be
             equal
             to
             R
             Q
             ,
             and
             consequently
             R
             S
             be
             equal
             to
             the
             chord
             of
             the
             arch
             B
             F
             ;
             and
             let
             F
             S
             be
             drawn
             and
             produced
             to
             T
             in
             the
             side
             B
             C.
             I
             say
             ,
             the
             Straight
             Line
             B
             T
             is
             equal
             to
             the
             Arch
             B
             F
             ;
             and
             consequently
             that
             B
             V
             the
             triple
             of
             B
             T
             is
             equal
             to
             the
             Arch
             of
             the
             Quadrant
             B
             F
             E
             D.
             
          
           
             Let
             T
             F
             be
             produced
             till
             it
             meet
             the
             side
             B
             A
             produced
             in
             X
             ;
             and
             dividing
             O
             F
             in
             the
             middle
             in
             Z
             ,
             let
             .
             Q
             Z
             be
             drawn
             and
             produced
             till
             it
             meet
             with
             the
             side
             B
             A
             produced
             .
             Seeing
             therefore
             the
             Straight
             Lines
             R
             S
             and
             O
             F
             are
             parallel
             ,
             and
             divided
             in
             the
             midst
             in
             Q
             and
             Z
             ,
             Q
             Z
             produced
             will
             fall
             upon
             X
             ,
             and
             X
             Z
             Q
             produced
             to
             the
             side
             B
             C
             will
             cut
             B
             T
             in
             the
             midst
             in
             α.
             
          
           
             Upon
             the
             Straight
             line
             F
             Z
             the
             fourth
             part
             of
             the
             Radius
             A
             B
             let
             the
             equilateral
             triangle
             a
             Z
             F
             be
             constituted
             ;
             &
             upon
             the
             center
             a
             ,
             with
             the
             Radius
             a
             Z
             let
             the
             arch
             Z
             F
             be
             drawn
             ;
             which
             arch
             Z
             F
             will
             therefore
             be
             equal
             to
             the
             arch
             Q
             F
             the
             half
             of
             the
             arch
             B
             F.
             Again
             ,
             let
             the
             straight
             line
             Z
             O
             be
             cut
             in
             the
             midst
             in
             b
             ,
             and
             the
             straight
             line
             b
             O
             in
             the
             midst
             in
             c
             ;
             and
             let
             the
             bisection
             be
             continued
             in
             this
             manner
             till
             the
             last
             part
             O
             c
             be
             the
             least
             that
             can
             possibly
             be
             taken
             ;
             and
             upon
             it
             ,
             and
             all
             the
             rest
             of
             the
             parts
             equal
             to
             it
             into
             which
             the
             straight
             line
             O
             F
             may
             be
             cut
             ,
             let
             so
             many
             equilateral
             triangles
             be
             understood
             to
             be
             constituted
             ;
             of
             which
             let
             the
             last
             be
             d
             O
             c.
             If
             therefore
             upon
             the
             center
             d
             ,
             with
             the
             Radius
             d
             O
             be
             drawn
             the
             arch
             O
             c
             ,
             and
             upon
             the
             rest
             of
             the
             equal
             parts
             of
             the
             straight
             line
             O
             F
             be
             drawn
             in
             like
             manner
             so
             many
             equal
             arches
             ,
             all
             those
             arches
             together
             taken
             will
             be
             equal
             to
             the
             whole
             arch
             B
             F
             ;
             &
             the
             half
             of
             them
             ,
             namely
             ,
             those
             that
             are
             comprehended
             between
             O
             &
             Z
             ,
             or
             between
             Z
             &
             F
             will
             be
             equal
             to
             the
             arch
             B
             Q
             or
             Q
             F
             and
             in
             summe
             ,
             what
             part
             soever
             the
             straight
             line
             O
             c
             be
             of
             the
             straight
             line
             O
             F
             ,
             the
             same
             part
             will
             the
             arch
             O
             c
             be
             of
             the
             arch
             O
             F
             ,
             though
             both
             the
             arch
             and
             the
             chord
             be
             infinitely
             bisected
             .
             Now
             seeing
             the
             arch
             O
             c
             is
             more
             crooked
             then
             that
             part
             of
             the
             arch
             B
             F
             which
             is
             equal
             to
             it
             ;
             and
             seeing
             also
             that
             the
             more
             the
             straight
             line
             X
             c
             is
             produced
             the
             more
             it
             diverges
             from
             the
             straight
             line
             X
             O
             ,
             if
             
             the
             points
             O
             and
             c
             be
             understood
             to
             be
             moved
             forwards
             with
             straight
             motion
             in
             X
             O
             and
             X
             c
             ,
             the
             arch
             O
             ●
             will
             thereby
             be
             extended
             by
             little
             and
             little
             ,
             till
             at
             the
             last
             it
             come
             some-where
             ●o
             have
             the
             same
             crookedness
             with
             that
             part
             of
             the
             arch
             B
             F
             which
             is
             equal
             to
             it
             .
             In
             like
             manner
             ,
             if
             the
             straight
             line
             X
             b
             be
             drawn
             ,
             and
             the
             point
             b
             be
             understood
             to
             be
             moved
             forwards
             at
             the
             same
             time
             ,
             the
             arch
             
               c
               b
            
             will
             also
             by
             little
             and
             little
             be
             extended
             ,
             till
             its
             crookedness
             come
             to
             be
             equal
             to
             the
             crookedness
             of
             that
             part
             of
             the
             arch
             B
             F
             which
             is
             equal
             to
             it
             .
             And
             the
             same
             will
             happen
             in
             all
             those
             smal
             equal
             arches
             which
             are
             described
             upon
             so
             many
             equal
             parts
             of
             the
             straight
             line
             O
             F.
             It
             is
             also
             manifest
             ,
             that
             by
             straight
             motion
             in
             X
             O
             and
             X
             Z
             all
             those
             small
             arches
             will
             lie
             in
             the
             arch
             B
             F
             in
             the
             points
             B
             ,
             Q
             and
             F.
             And
             though
             the
             same
             small
             equall
             arches
             should
             not
             be
             coincident
             with
             the
             equall
             parts
             of
             the
             arch
             B
             F
             in
             all
             the
             other
             points
             thereof
             ,
             yet
             certainly
             they
             will
             constitute
             two
             crooked
             lines
             ,
             not
             onely
             equall
             to
             the
             two
             arches
             B
             Q
             and
             Q
             F
             and
             equally
             crooked
             ,
             but
             also
             having
             their
             cavity
             towards
             the
             same
             parts
             ;
             which
             how
             it
             should
             be
             ,
             unlesse
             all
             those
             small
             arches
             should
             be
             coincident
             with
             the
             arch
             B
             F
             in
             all
             its
             points
             ,
             is
             not
             imaginable
             .
             They
             are
             therefore
             coincident
             ,
             and
             all
             the
             straight
             lines
             drawne
             from
             X
             &
             passing
             through
             the
             points
             of
             division
             of
             the
             straight
             line
             O
             F
             ,
             will
             also
             divide
             the
             arch
             B
             F
             into
             the
             same
             proportions
             into
             which
             O
             F
             is
             divided
             .
          
           
             Now
             seeing
             X
             b
             cuts
             off
             from
             the
             point
             B
             the
             fourth
             part
             of
             the
             arch
             B
             F
             ,
             let
             that
             fourth
             part
             be
             B
             e
             ;
             and
             let
             the
             Sine
             thereof
             
               f
               e
            
             be
             produced
             to
             F
             T
             in
             g
             ,
             for
             so
             
               f
               e
            
             will
             be
             the
             fourth
             part
             of
             the
             straight
             line
             
               f
               g
            
             ,
             because
             as
             O
             b
             is
             to
             O
             F
             ,
             so
             is
             
               f
               e
            
             to
             
               f
               g.
            
             But
             B
             T
             is
             greater
             then
             
               f
               g
            
             ;
             and
             therefore
             the
             same
             B
             T
             is
             greater
             then
             four
             Sines
             of
             the
             fourth
             part
             of
             the
             arch
             B
             F.
             And
             in
             like
             m●nner
             ,
             if
             the
             arch
             B
             F
             be
             subdivided
             into
             any
             number
             of
             equal
             par●●
             whatsoever
             ,
             it
             may
             be
             proved
             that
             the
             straight
             line
             B
             T
             is
             greater
             then
             the
             Sine
             of
             one
             of
             those
             small
             arches
             so
             many
             times
             〈◊〉
             as
             ●here
             be
             parts
             made
             of
             the
             whole
             arch
             B
             F.
             Wherefore
             the
             ●traight
             line
             B
             T
             is
             not
             lesse
             then
             the
             Arch
             B
             F.
             But
             neither
             can
             it
             be
             greater
             ,
             because
             if
             any
             straight
             line
             whatsoever
             ,
             lesse
             then
             B
             T
             ,
             be
             draw●
             below
             B
             T
             parallel
             to
             it
             and
             terminated
             in
             the
             straight
             line●
             X
             B
             and
             X
             T
             ,
             it
             would
             cut
             the
             arch
             B
             F
             ;
             and
             
             so
             the
             Sine
             of
             some
             one
             of
             the
             parts
             of
             the
             arch
             B
             F
             taken
             so
             often
             as
             that
             small
             arch
             is
             found
             in
             the
             whole
             arch
             B
             F
             ,
             would
             be
             greater
             then
             so
             many
             of
             the
             same
             arches
             ;
             which
             is
             absurd
             .
             Wherefore
             the
             Straight
             line
             B
             T
             is
             equal
             to
             the
             Arch
             B
             F
             ;
             &
             the
             Straight
             line
             B
             V
             equal
             to
             the
             Arch
             of
             the
             Quadrant
             B
             F
             D
             ;
             and
             B
             V
             four
             times
             taken
             ,
             equal
             to
             the
             Perimeter
             of
             the
             Circle
             described
             with
             the
             Radius
             A
             B.
             Also
             the
             Arch
             B
             F
             and
             the
             Straight
             line
             B
             T
             are
             every
             where
             divided
             into
             the
             same
             proportions
             ;
             and
             consequently
             any
             given
             Angle
             ,
             whether
             greater
             or
             less
             then
             B
             A
             F
             may
             be
             divided
             into
             any
             proportion
             given
             .
          
           
             But
             the
             straight
             line
             B
             V
             (
             though
             its
             magnitude
             fall
             within
             the
             terms
             assigned
             by
             Archimedes
             )
             is
             found
             ,
             if
             computed
             by
             the
             Canon
             of
             Sines
             ,
             to
             be
             somwhat
             greater
             then
             that
             w
             ch
             is
             exhibited
             by
             the
             Ludolphine
             numbers
             .
             Nevertheless
             ,
             if
             in
             the
             place
             of
             B
             T
             ,
             another
             straight
             line
             ,
             though
             never
             so
             little
             less
             ,
             be
             substituted
             ,
             the
             division
             of
             Angles
             is
             immediatly
             lost
             ,
             as
             may
             by
             any
             man
             be
             demonstrated
             by
             this
             very
             Scheme
             .
          
           
             Howsoever
             ,
             if
             any
             man
             think
             this
             my
             Straight
             line
             B
             V
             to
             be
             too
             great
             ,
             yet
             ,
             seeing
             the
             Arch
             and
             all
             the
             Parallels
             are
             every
             where
             so
             exactly
             divided
             ,
             and
             B
             V
             comes
             so
             neer
             to
             the
             truth
             ,
             I
             desire
             he
             would
             seach
             out
             the
             reason
             ,
             Why
             (
             granting
             B
             V
             to
             be
             precisely
             true
             )
             the
             Arches
             cut
             off
             should
             not
             be
             equal
             .
          
           
             But
             some
             man
             may
             yet
             ask
             the
             reason
             why
             the
             straight
             lines
             drawn
             from
             X
             through
             the
             equal
             parts
             of
             the
             arch
             B
             F
             should
             cut
             off
             in
             the
             Tangent
             B
             V
             so
             many
             straight
             lines
             equal
             to
             them
             ,
             seeing
             the
             connected
             straight
             line
             X
             V
             passes
             not
             through
             the
             point
             D
             ,
             but
             cuts
             the
             straight
             line
             A
             D
             produced
             in
             l
             ;
             and
             consequently
             require
             some
             determination
             of
             this
             Probleme
             .
             Concerning
             which
             ,
             I
             will
             say
             what
             I
             think
             to
             be
             the
             reason
             ,
             namely
             ,
             that
             whilest
             the
             magnitude
             of
             the
             Arch
             doth
             not
             exceed
             the
             magnitude
             of
             the
             Radius
             ,
             that
             is
             ,
             the
             magnitude
             of
             the
             Tangent
             B
             C
             ,
             both
             the
             Arch
             and
             the
             Tangent
             are
             cut
             alike
             by
             the
             straight
             lines
             drawn
             from
             X
             ;
             otherwise
             not
             .
             For
             A
             V
             being
             connected
             ,
             cutting
             the
             arch
             B
             H
             D
             in
             I
             ,
             if
             X
             C
             being
             drawn
             should
             cut
             the
             same
             arch
             in
             the
             same
             point
             I
             ,
             it
             would
             be
             as
             true
             that
             the
             Arch
             B
             I
             is
             equal
             to
             the
             Radius
             B
             C
             ,
             as
             it
             is
             true
             that
             the
             Arch
             B
             F
             is
             equal
             to
             the
             straight
             line
             B
             T
             ,
             and
             drawing
             X
             K
             it
             would
             cut
             the
             arch
             B
             I
             in
             the
             midst
             in
             i
             ;
             Also
             
             drawing
             A
             i
             and
             producing
             it
             to
             the
             Tangent
             B
             C
             in
             k
             ,
             the
             straight
             line
             B
             k
             will
             be
             the
             Tangent
             of
             the
             arch
             B
             i
             ,
             (
             which
             arch
             is
             equal
             to
             half
             the
             Radius
             )
             and
             the
             same
             straight
             line
             B
             k
             will
             be
             equal
             to
             the
             straight
             line
             k
             I.
             I
             say
             all
             this
             is
             true
             ,
             if
             the
             preceding
             demonstration
             be
             true
             ;
             and
             consequently
             the
             proportional
             section
             of
             the
             Arch
             and
             its
             Tangent
             proceeds
             hitherto
             .
             But
             it
             is
             manifest
             by
             the
             Golden
             Rule
             ,
             that
             taking
             B
             h
             double
             to
             B
             T
             ,
             the
             line
             X
             h
             shall
             not
             cut
             off
             the
             arch
             B
             E
             which
             is
             double
             to
             the
             arch
             B
             F
             ,
             but
             a
             much
             greater
             .
             For
             the
             magnitude
             of
             the
             straight
             lines
             X
             M
             ,
             X
             B
             and
             M
             E
             being
             known
             (
             in
             numbers
             )
             the
             magnitude
             of
             the
             straight
             line
             cut
             off
             in
             the
             Tangent
             by
             the
             straight
             line
             X
             E
             produced
             to
             the
             Tangent
             may
             also
             be
             known
             ;
             and
             it
             will
             be
             found
             to
             be
             less
             then
             B
             h
             ;
             Wherfore
             the
             straight
             line
             Xh
             being
             drawn
             will
             cut
             off
             a
             part
             of
             the
             arch
             of
             the
             Quadrant
             greater
             then
             the
             arch
             B
             E.
             But
             I
             shall
             speak
             more
             fully
             in
             the
             next
             Article
             concerning
             the
             magnitude
             of
             the
             arch
             B
             I.
             
          
           
             And
             let
             this
             be
             the
             first
             attempt
             for
             the
             finding
             out
             of
             the
             dimension
             of
             a
             Circle
             by
             the
             Section
             of
             the
             arch
             B
             F.
             
          
           
             3
             I
             shall
             now
             attempt
             the
             same
             by
             arguments
             drawn
             from
             the
             nature
             of
             the
             Crookedness
             of
             the
             Circle
             it self
             ;
             but
             I
             shall
             first
             set
             down
             some
             Premisses
             necessary
             for
             this
             speculation
             ;
             and
          
           
             First
             ,
             If
             a
             Straight
             line
             be
             bowed
             into
             an
             Arch
             of
             a
             Circle
             equal
             to
             it
             ,
             as
             when
             a
             stretched
             thred
             which
             toucheth
             a
             Right
             Cylinder
             ,
             is
             so
             bowed
             in
             every
             point
             ,
             that
             it
             be
             every
             where
             coincident
             with
             the
             Perimeter
             of
             the
             base
             of
             the
             Cylinder
             ,
             the
             Flexion
             of
             that
             line
             will
             be
             equal
             ,
             in
             all
             its
             points
             ;
             and
             consequently
             the
             Crookedness
             of
             the
             Arch
             of
             a
             Circle
             is
             every
             where
             Uniform
             ;
             which
             needs
             no
             other
             demonstration
             then
             this
             ,
             That
             the
             Perimeter
             of
             a
             Circle
             is
             an
             Uniform
             line
             .
          
           
             Secondly
             ,
             and
             consequently
             ,
             If
             two
             unequal
             Arches
             of
             the
             same
             Circle
             be
             made
             by
             the
             bowing
             of
             two
             straight
             lines
             equal
             to
             them
             ,
             the
             Flexion
             of
             the
             longer
             line
             (
             whilest
             it
             is
             bowed
             into
             the
             greater
             Arch
             )
             is
             greater
             then
             the
             Flexion
             of
             the
             shorter
             line
             (
             whilest
             it
             is
             bowed
             into
             the
             lesser
             Arch
             )
             according
             to
             the
             proportion
             of
             the
             Arches
             themselves
             ;
             and
             consequently
             ,
             the
             Crookedness
             of
             the
             greater
             Arch
             is
             to
             the
             Crookedness
             of
             the
             lesser
             Arch
             ;
             as
             the
             greater
             Arch
             is
             to
             the
             lesser
             Arch.
             
          
           
             Thirdly
             ,
             If
             two
             unequal
             Circles
             and
             a
             straight
             line
             touch
             one
             
             another
             in
             the
             same
             point
             ,
             the
             Crookedness
             of
             any
             Arch
             taken
             in
             the
             lesser
             Circle
             ,
             will
             be
             greater
             then
             the
             Crookedness
             of
             an
             Arch
             equal
             to
             it
             taken
             in
             the
             greater
             Circle
             ,
             in
             reciprocal
             proportion
             to
             that
             of
             the
             Radii
             with
             which
             the
             Circles
             are
             described
             ;
             or
             ,
             which
             is
             all
             one
             ,
             any
             straight
             line
             being
             drawn
             from
             the
             point
             of
             Contact
             till
             it
             cut
             both
             the
             circumferences
             ,
             as
             the
             part
             of
             that
             straight
             line
             cut
             off
             by
             the
             circumference
             of
             the
             greater
             Circle
             to
             that
             part
             which
             is
             cut
             off
             by
             the
             circumference
             of
             the
             lesser
             Circle
             .
          
           
             For
             let
             A
             B
             and
             A
             C
             (
             in
             the
             second
             figure
             )
             be
             two
             Circles
             ,
             touching
             one
             another
             and
             the
             straight
             line
             A
             D
             in
             the
             point
             A
             ;
             and
             let
             their
             Centers
             be
             E
             and
             F
             ;
             and
             let
             it
             be
             supposed
             ,
             that
             as
             A
             E
             is
             to
             A
             F
             ,
             so
             is
             the
             Arch
             A
             B
             to
             the
             Arch
             A
             H.
             I
             say
             the
             Crookedness
             of
             the
             Arch
             A
             C
             is
             to
             the
             Crookedness
             of
             the
             Arch
             A
             H
             ,
             as
             A
             E
             is
             to
             A
             F.
             For
             let
             the
             straight
             line
             A
             D
             be
             supposed
             to
             be
             equal
             to
             the
             Arch
             A
             B
             ,
             and
             the
             straight
             line
             A
             G
             to
             the
             Arch
             A
             C
             ;
             and
             let
             A
             D
             (
             for
             example
             )
             be
             double
             to
             A
             G.
             Therefore
             by
             reason
             of
             the
             likeness
             of
             the
             Arches
             A
             B
             and
             A
             C
             ,
             the
             straight
             line
             A
             B
             will
             be
             double
             to
             the
             straight
             line
             A
             C
             ,
             and
             the
             Radius
             A
             E
             double
             to
             the
             Radius
             A
             F
             ,
             and
             the
             Arch
             A
             B
             double
             to
             the
             Arch
             A
             H.
             And
             because
             the
             straight
             line
             A
             D
             is
             so
             bowed
             to
             be
             coincident
             with
             the
             Arch
             A
             B
             equal
             to
             it
             ,
             as
             the
             straight
             line
             A
             G
             is
             bowed
             to
             be
             coincident
             with
             the
             Arch
             A
             C
             equal
             also
             to
             it
             ,
             the
             Flexion
             of
             the
             straight
             line
             A
             G
             into
             the
             Crooked
             line
             A
             C
             will
             be
             equal
             to
             the
             Flexion
             of
             the
             straight
             Line
             A
             D
             into
             the
             Crooked
             line
             A
             B.
             But
             the
             Flexion
             of
             the
             straight
             line
             A
             D
             into
             the
             Crooked
             line
             A
             B
             is
             double
             to
             the
             the
             Flexion
             of
             the
             straight
             line
             A
             G
             into
             the
             Crooked
             line
             A
             H
             ;
             and
             therefore
             the
             Flexion
             of
             the
             straight
             line
             A
             G
             into
             the
             Crooked
             line
             A
             C
             is
             double
             to
             the
             Flexion
             of
             the
             same
             straight
             line
             A
             G
             into
             the
             Crooked
             line
             A
             H.
             Wherefore
             ,
             as
             the
             Arch
             A
             B
             is
             to
             the
             Arch
             A
             C
             or
             A
             H
             ;
             or
             as
             the
             Radius
             A
             E
             is
             to
             the
             Radius
             A
             F
             ;
             or
             as
             the
             Chord
             A
             B
             is
             to
             the
             Chord
             A
             C
             ;
             so
             reciprocally
             is
             the
             Flexion
             or
             Uniform
             Crookedness
             of
             the
             Arch
             A
             C
             ,
             to
             the
             Flexion
             or
             Uniform
             Crookedness
             of
             the
             Arch
             A
             H
             ,
             namely
             ,
             here
             double
             .
             And
             this
             may
             by
             the
             same
             method
             be
             demonstrated
             in
             Circles
             whose
             Perimeters
             are
             to
             one
             another
             triple
             ,
             quadruple
             ,
             or
             in
             whatsoever
             given
             proportion
             .
             The
             Crookedness
             therefore
             of
             two
             equal
             Arches
             
             taken
             in
             several
             Circles
             are
             in
             proportion
             reciprocall
             to
             that
             of
             their
             Radii
             ,
             or
             like
             Arches
             ,
             or
             like
             Chords
             ;
             which
             was
             to
             be
             demonstrated
             .
          
           
             Let
             the
             Square
             A
             B
             C
             D
             be
             again
             described
             (
             in
             the
             third
             Figure
             ,
             )
             and
             in
             it
             the
             Quadrants
             A
             B
             D
             ,
             B
             C
             A
             and
             D
             A
             C
             ;
             and
             dividing
             each
             side
             of
             the
             Square
             A
             B
             C
             D
             in
             the
             midst
             in
             E
             ,
             F
             ,
             G
             and
             H
             ,
             let
             E
             G
             and
             F
             H
             be
             connected
             ,
             which
             will
             cut
             one
             another
             in
             the
             center
             of
             the
             Square
             at
             I
             ,
             and
             divide
             the
             arch
             of
             the
             Quadrant
             A
             B
             D
             into
             three
             equal
             parts
             in
             K
             and
             L.
             Also
             the
             Diagonals
             A
             C
             and
             B
             D
             being
             drawn
             will
             cut
             one
             another
             in
             I
             ,
             and
             divide
             the
             arches
             B
             K
             D
             and
             C
             L
             A
             into
             two
             equal
             parts
             in
             M
             and
             N.
             Then
             with
             the
             Radius
             B
             F
             let
             the
             arch
             F
             E
             be
             drawn
             ,
             cutting
             the
             Diagonal
             B
             D
             in
             O
             ;
             and
             dividing
             the
             arch
             B
             M
             in
             the
             midst
             in
             P
             ,
             let
             the
             straight
             line
             E
             a
             equal
             to
             the
             chord
             B
             P
             be
             set
             off
             from
             the
             point
             E
             in
             the
             arch
             E
             F
             ,
             and
             let
             the
             arch
             
               a
               b
            
             be
             taken
             equal
             to
             the
             arch
             
               O
               a
            
             ,
             and
             let
             B
             a
             and
             B
             b
             be
             drawn
             and
             produced
             to
             the
             arch
             A
             N
             in
             c
             and
             d
             ;
             and
             lastly
             ,
             let
             the
             straight
             line
             A
             d
             be
             drawn
             .
             I
             say
             the
             Straight
             line
             A
             d
             is
             equal
             to
             the
             Arch
             A
             N
             or
             B
             M.
             
          
           
             I
             have
             proved
             in
             the
             preceding
             article
             ,
             that
             the
             arch
             E
             O
             is
             twice
             as
             crooked
             as
             the
             arch
             B
             P
             ,
             that
             is
             to
             say
             ,
             that
             the
             arch
             E
             O
             is
             so
             much
             more
             crooked
             then
             the
             arch
             B
             P
             ,
             as
             the
             arch
             B
             P
             is
             more
             crooked
             then
             the
             straight
             line
             E
             a.
             The
             crookedness
             therefore
             of
             the
             chord
             E
             a
             ,
             of
             the
             arch
             B
             P
             ,
             and
             of
             the
             arch
             E
             O
             are
             as
             0
             ,
             1
             ,
             2.
             
             Also
             the
             difference
             between
             the
             arches
             E
             O
             and
             E
             O
             ,
             the
             difference
             between
             the
             arches
             E
             O
             and
             E
             a
             ,
             and
             the
             difference
             between
             the
             arches
             E
             O
             and
             E
             b
             are
             as
             0
             ,
             1
             ,
             2.
             
             So
             also
             the
             difference
             between
             the
             arches
             A
             N
             and
             A
             N
             ,
             the
             difference
             between
             the
             arches
             A
             N
             and
             A
             c
             ,
             and
             the
             difference
             between
             the
             arches
             A
             N
             and
             A
             d
             are
             as
             0
             ,
             1
             ,
             2
             ;
             and
             the
             straight
             line
             A
             c
             is
             double
             to
             the
             chord
             B
             P
             or
             E
             a
             ,
             and
             the
             straight
             line
             A
             d
             double
             to
             the
             chord
             E
             b.
             
          
           
             Again
             ,
             let
             the
             straight
             line
             B
             F
             be
             divided
             in
             the
             midst
             in
             Q
             ,
             and
             the
             arch
             B
             P
             in
             the
             midst
             in
             R
             ;
             and
             describing
             the
             Quadrant
             B
             Q
             S
             (
             whose
             arch
             Q
             S
             is
             a
             fourth
             part
             of
             the
             arch
             of
             the
             Quadrant
             B
             M
             D
             ,
             as
             the
             arch
             B
             R
             is
             a
             fourth
             part
             of
             the
             arch
             B
             M
             which
             is
             the
             arch
             of
             the
             Semiquadrant
             A
             B
             M
             )
             let
             the
             chord
             S
             e
             equal
             to
             the
             chord
             B
             R
             be
             set
             off
             from
             the
             point
             S
             in
             the
             arch
             S
             Q
             ;
             and
             let
             B
             e
             be
             drawn
             and
             produced
             to
             the
             arch
             A
             N
             in
             f
             ;
             which
             being
             done
             ,
             the
             straight
             line
             A
             f
             will
             be
             quadruple
             to
             the
             
             chord
             B
             R
             or
             S
             e.
             And
             seeing
             the
             crookedness
             of
             the
             arch
             S
             e
             or
             of
             the
             arch
             A
             c
             is
             double
             to
             the
             crookedness
             of
             the
             arch
             B
             R
             ,
             the
             excess
             of
             the
             crookedness
             of
             the
             arch
             A
             f
             above
             the
             crookedness
             of
             the
             arch
             A
             c
             will
             be
             subduple
             to
             the
             excess
             of
             the
             crookedness
             of
             the
             arch
             A
             c
             above
             the
             crookedness
             of
             the
             arch
             A
             N
             ;
             and
             therefore
             the
             arch
             N
             c
             will
             be
             double
             to
             the
             arch
             
               c
               f.
            
             Wherefore
             the
             arch
             
               c
               d
            
             is
             divided
             in
             the
             midst
             in
             f
             ,
             and
             the
             arch
             N
             f
             is
             ¾
             of
             the
             arch
             N
             d.
             And
             in
             like
             manner
             if
             the
             arch
             B
             R
             be
             bisected
             in
             V
             ,
             and
             the
             straight
             Line
             B
             Q
             in
             X
             ,
             and
             the
             quadrant
             B
             X
             Y
             be
             described
             ,
             and
             the
             straight
             Line
             Y
             g
             equal
             to
             the
             chord
             B
             V
             be
             set
             off
             from
             the
             point
             Y
             in
             the
             arch
             Y
             X
             ,
             it
             may
             be
             demonstrated
             that
             the
             straight
             Line
             B
             g
             being
             drawn
             and
             produced
             to
             the
             arch
             A
             N
             will
             cut
             the
             arch
             
               f
               d
            
             into
             two
             equal
             parts
             ,
             and
             that
             a
             straight
             Line
             drawn
             from
             A
             to
             the
             point
             of
             that
             Section
             ,
             will
             be
             equal
             to
             eight
             chords
             of
             the
             arch
             B
             V
             ,
             and
             so
             on
             perpetually
             ;
             and
             consequently
             ,
             that
             the
             straight
             Line
             A
             d
             is
             equal
             to
             so
             many
             equal
             chords
             of
             equal
             parts
             of
             the
             arch
             B
             M
             ,
             as
             may
             be
             made
             by
             infinite
             bisections
             .
             Wherefore
             the
             Straight
             Line
             A
             d
             is
             equal
             to
             the
             Arch
             B
             M
             or
             A
             N
             ,
             that
             is
             ,
             to
             half
             the
             Arch
             of
             the
             Quadrant
             A
             B
             D
             or
             B
             C
             A.
             
          
           
             Corollary
             .
             An
             Arch
             being
             given
             not
             greater
             then
             the
             arch
             of
             a
             Quadrant
             (
             for
             being
             made
             greater
             it
             comes
             again
             towards
             the
             Radius
             B
             A
             produced
             ,
             from
             which
             it
             receded
             before
             )
             if
             a
             straight
             Line
             double
             to
             the
             chord
             of
             half
             the
             given
             arch
             be
             adapted
             from
             the
             beginning
             of
             the
             arch
             ,
             and
             by
             how
             much
             the
             arch
             that
             is
             subtended
             by
             it
             is
             greater
             then
             the
             given
             arch
             ,
             by
             so
             much
             a
             greater
             arch
             be
             subtended
             by
             another
             straight
             Line
             ,
             this
             Straight
             Line
             shall
             be
             equal
             to
             the
             first
             given
             Arch.
             
          
           
             Supposing
             the
             Straight
             Line
             B
             V
             (
             in
             the
             first
             Figure
             )
             be
             equal
             to
             the
             arch
             of
             the
             Quadrant
             B
             H
             D
             ,
             and
             A
             V
             be
             connected
             cutting
             the
             arch
             B
             H
             D
             in
             I
             ,
             it
             may
             be
             asked
             what
             proportion
             the
             arch
             B
             I
             has
             to
             the
             arch
             I
             D.
             Let
             therefore
             the
             arch
             A
             Y
             be
             divided
             in
             the
             midst
             in
             o
             ,
             and
             in
             the
             straight
             line
             A
             D
             let
             A
             p
             be
             taken
             equal
             ,
             and
             A
             q
             double
             to
             the
             drawn
             chord
             A
             o.
             Then
             upon
             the
             center
             A
             ,
             with
             the
             Radius
             A
             q
             let
             an
             arch
             of
             a
             circle
             be
             drawne
             cutting
             the
             arch
             A
             Y
             in
             r
             ,
             and
             let
             the
             arch
             Y
             r
             be
             doubled
             at
             t
             ;
             which
             being
             done
             ,
             the
             drawne
             straight
             line
             A
             t
             (
             by
             what
             has
             been
             last
             demonstrated
             )
             will
             be
             equall
             to
             the
             arch
             A
             Y.
             
             Again
             ,
             upon
             the
             Center
             A
             with
             the
             Radius
             A
             t
             let
             the
             arch
             tu
             be
             drawne
             cutting
             AD
             in
             u
             ;
             and
             the
             straight
             line
             A
             u
             will
             be
             equall
             to
             the
             arch
             AY
             .
             From
             the
             point
             u
             let
             the
             straight
             line
             us
             be
             drawn
             equal
             and
             parallell
             to
             the
             straight
             line
             AB
             ,
             cutting
             MN
             in
             x
             ,
             and
             bisected
             by
             MN
             in
             the
             same
             point
             x.
             Therefore
             the
             straight
             line
             A
             x
             being
             drawn
             and
             produced
             till
             it
             meet
             with
             BC
             produced
             in
             V
             ,
             it
             will
             cut
             off
             BV
             double
             to
             B
             s
             ,
             that
             is
             ,
             equal
             to
             the
             arch
             BHD
             .
             Now
             let
             the
             point
             where
             the
             straight
             line
             AV
             cuts
             the
             arch
             BHD
             ,
             be
             I
             ;
             and
             let
             the
             arch
             DI
             be
             divided
             in
             the
             midst
             in
             y
             ;
             and
             in
             the
             straight
             line
             DC
             ,
             let
             D
             z
             be
             taken
             equal
             ,
             and
             D
             δ
             double
             to
             the
             drawn
             chord
             D
             y
             ;
             and
             upon
             the
             center
             D
             with
             the
             Radius
             D
             δ
             let
             an
             arch
             of
             a
             circle
             be
             drawn
             cutting
             the
             arch
             BHD
             in
             the
             point
             n
             ;
             and
             let
             the
             arch
             nm
             be
             taken
             equal
             to
             the
             arch
             I
             n
             ;
             which
             being
             done
             ,
             the
             straight
             line
             D
             m
             will
             (
             by
             the
             last
             foregoing
             Corollary
             )
             be
             equal
             to
             the
             arch
             DI.
             If
             now
             the
             straight
             lines
             D
             m
             and
             CV
             be
             equal
             ,
             the
             arch
             BI
             will
             be
             equal
             to
             the
             Radius
             AB
             or
             BC
             ;
             and
             consequently
             XC
             being
             drawn
             will
             pass
             through
             the
             point
             I.
             Moreover
             ,
             if
             the
             semicircle
             BHD
             β
             being
             completed
             ,
             the
             straight
             lines
             β
             I
             and
             BI
             be
             drawn
             making
             a
             right
             angle
             (
             in
             the
             Semicircle
             )
             at
             I
             ,
             and
             the
             arch
             BI
             be
             divided
             in
             the
             midst
             at
             i
             ,
             it
             will
             follow
             that
             A
             i
             being
             connected
             will
             be
             parallel
             to
             the
             straight
             line
             β
             I
             ,
             and
             being
             produced
             to
             BC
             in
             k
             ,
             will
             cut
             off
             the
             straight
             line
             B
             k
             equal
             to
             the
             straight
             line
             k
             I
             ,
             and
             equal
             also
             to
             the
             straight
             line
             Aγ
             cut
             off
             in
             AD
             by
             the
             straight
             line
             β
             I.
             All
             which
             is
             manifest
             ,
             supposing
             the
             arch
             BI
             and
             the
             Radius
             BC
             to
             be
             equal
             .
          
           
             But
             that
             the
             arch
             BI
             and
             the
             Radius
             BC
             are
             precisely
             equal
             ,
             cannot
             (
             how
             true
             soever
             it
             he
             )
             be
             demonstrated
             ,
             unless
             that
             be
             first
             proved
             w
             ch
             is
             contained
             in
             the
             first
             article
             ,
             namely
             ,
             that
             the
             straight
             lines
             drawn
             from
             X
             through
             the
             equal
             parts
             of
             OF
             (
             produced
             to
             a
             certain
             length
             )
             cut
             off
             so
             many
             parts
             also
             in
             the
             Tangent
             BC
             severally
             equal
             to
             the
             several
             arches
             cut
             off
             ;
             which
             they
             do
             most
             exactly
             as
             far
             as
             BC
             in
             the
             Tangent
             ,
             and
             BI
             in
             the
             arch
             BE
             ;
             in
             so
             much
             that
             no
             inequality
             between
             the
             arch
             BI
             and
             the
             Radius
             BC
             can
             be
             discovered
             either
             by
             the
             hand
             or
             by
             ratiocination
             .
             It
             is
             therefore
             to
             be
             further
             enquired
             ,
             whether
             the
             straight
             line
             AV
             cut
             the
             arch
             of
             the
             Quadrant
             in
             I
             in
             the
             same
             proportion
             as
             the
             point
             C
             divides
             
             the
             straight
             line
             BV
             which
             is
             equal
             to
             the
             arch
             of
             the
             Quadrant
             .
             But
             however
             this
             be
             ,
             it
             has
             been
             demonstrated
             that
             the
             straight
             line
             BV
             is
             equal
             to
             the
             arch
             BHD
             .
          
           
             4
             I
             shall
             now
             attempt
             the
             same
             dimension
             of
             a
             Circle
             another
             way
             ,
             assuming
             the
             two
             following
             Lemma's
             .
          
           
             Lemma
             1.
             
             If
             to
             the
             Arch
             of
             a
             Quadrant
             ,
             and
             the
             Radius
             ,
             there
             be
             taken
             in
             continual
             proportion
             a
             third
             Line
             Z
             ;
             then
             the
             Arch
             of
             the
             Semiquadrant
             ,
             Half
             the
             chord
             of
             the
             Quadrant
             ,
             and
             Z
             will
             also
             be
             in
             continual
             proportion
             .
          
           
             For
             seeing
             the
             Radius
             is
             a
             mean
             proportional
             between
             the
             Chord
             of
             a
             Quadrant
             and
             its
             Semichord
             ,
             and
             the
             same
             Radius
             a
             mean
             proportional
             between
             the
             Arch
             of
             the
             Quadrant
             and
             Z
             ,
             the
             Square
             of
             the
             Radius
             will
             be
             equal
             as
             well
             to
             the
             Rectangle
             made
             of
             the
             Chord
             and
             Semichord
             of
             the
             Quadrant
             ,
             as
             to
             the
             Rectangle
             made
             of
             the
             Arch
             of
             the
             Quadrant
             and
             Z
             ;
             and
             these
             two
             Rectangles
             will
             be
             equal
             to
             one
             another
             .
             Wherfore
             ,
             as
             the
             Arch
             of
             a
             Quadrant
             is
             to
             its
             Chord
             ,
             so
             reciprocally
             is
             half
             the
             Chord
             of
             the
             Quadrant
             to
             Z.
             But
             as
             the
             Arch
             of
             the
             Quadrant
             is
             to
             its
             Chord
             ,
             so
             is
             half
             the
             Arch
             of
             the
             Quadrant
             to
             half
             the
             Chord
             of
             the
             Quadrant
             .
             Wherefore
             ,
             as
             half
             the
             arch
             of
             the
             Quadrant
             is
             to
             half
             the
             Chord
             of
             the
             Quadrant
             (
             or
             to
             the
             Sine
             of
             45
             degrees
             )
             so
             is
             half
             the
             Chord
             of
             the
             Quadrant
             to
             Z
             ;
             which
             was
             to
             be
             proved
             .
          
           
             Lemma
             2.
             
             The
             Radius
             ,
             the
             Arch
             of
             the
             Semiquadrant
             ,
             the
             Sine
             of
             45
             degrees
             ,
             and
             the
             Semiradius
             are
             proportional
             .
          
           
             For
             seeing
             the
             Sine
             of
             45
             degrees
             is
             a
             mean
             proportional
             between
             the
             Radius
             and
             the
             Semiradius
             ;
             and
             the
             same
             Sine
             of
             45
             degrees
             is
             also
             a
             mean
             proportional
             (
             by
             the
             precedent
             Lemma
             )
             between
             the
             Arch
             of
             45
             degrees
             and
             Z
             ;
             the
             Square
             of
             the
             Sine
             of
             45
             degrees
             will
             be
             equal
             as
             well
             to
             the
             Rectangle
             made
             of
             the
             Radius
             and
             Semiradius
             ,
             as
             to
             the
             Rectangle
             made
             of
             the
             Arch
             of
             45
             degrees
             and
             Z.
             Wherefore
             ,
             as
             the
             Radius
             is
             to
             the
             Arch
             of
             45
             degrees
             ,
             so
             reciprocally
             is
             Z
             to
             the
             Semiradius
             ;
             which
             was
             to
             be
             demonstrated
             .
          
           
             Let
             now
             ABCD
             (
             in
             the
             fourth
             Figure
             )
             be
             a
             Square
             ;
             and
             with
             the
             Radii
             AB
             ,
             BC
             and
             DA
             let
             the
             three
             Quadrants
             ABD
             ,
             BCA
             and
             DAC
             be
             described
             ;
             and
             let
             the
             straight
             lines
             EF
             and
             GH
             drawn
             parallel
             to
             the
             Sides
             BC
             &
             AB
             ,
             divide
             the
             Square
             ABCD
             into
             foure
             equal
             Squares
             .
             They
             will
             therefore
             cut
             the
             arch
             of
             the
             
             Quadrant
             ABD
             into
             three
             equal
             parts
             in
             I
             and
             K
             ,
             and
             the
             arch
             of
             the
             Quadrant
             BCA
             into
             three
             equal
             parts
             in
             K
             and
             L.
             Also
             let
             the
             Diagonals
             AC
             and
             BD
             be
             drawn
             ,
             cutting
             the
             arches
             BID
             and
             ALC
             in
             M
             and
             N.
             Then
             upon
             the
             center
             H
             with
             the
             Radius
             HF
             equal
             to
             half
             the
             Chord
             of
             the
             arch
             BMD
             ,
             or
             to
             the
             Sine
             of
             45
             degrees
             ,
             let
             the
             arch
             FO
             be
             drawn
             cutting
             the
             arc●
             CK
             in
             O
             ;
             and
             let
             AO
             be
             drawn
             and
             produced
             till
             it
             meet
             with
             BC
             produced
             in
             P
             ;
             also
             let
             it
             cut
             the
             arch
             BMD
             in
             Q
             ,
             and
             the
             straight
             line
             DC
             in
             R.
             If
             now
             the
             straight
             line
             H
             Q
             be
             equal
             to
             the
             straight
             line
             DR
             ,
             and
             being
             produced
             to
             DC
             in
             S
             cut
             off
             DS
             equal
             to
             half
             the
             straight
             line
             BP
             ;
             I
             say
             then
             the
             Straight
             Line
             BP
             will
             be
             equal
             to
             the
             Arch
             BMD
             .
          
           
             For
             seeing
             PBA
             and
             ADR
             are
             like
             triangles
             ,
             it
             will
             be
             as
             PB
             to
             the
             Radius
             BA
             or
             AD
             ,
             so
             AD
             to
             DR
             ;
             and
             therefore
             as
             well
             PB
             ,
             AD
             and
             DR
             ,
             as
             PB
             ,
             AD
             (
             or
             A
             Q
             )
             and
             Q
             H
             are
             in
             continuall
             proportion
             ;
             and
             producing
             HO
             to
             DC
             in
             T
             ,
             DT
             will
             be
             equal
             to
             the
             Sine
             of
             45
             degrees
             ,
             as
             shall
             by
             and
             by
             be
             demonstrated
             .
             Now
             DS
             ,
             DT
             and
             DR
             are
             in
             continual
             porportion
             by
             the
             first
             Lemma
             ;
             and
             by
             the
             second
             Lemma
             DC
             .
             DS
             :
             :
             DR
             .
             DF
             are
             proportionals
             .
             And
             thus
             it
             will
             be
             ,
             whether
             BP
             be
             equal
             or
             not
             equal
             to
             the
             arch
             of
             the
             Quadrant
             BMD
             .
             But
             if
             they
             be
             equal
             ,
             it
             will
             then
             be
             ,
             as
             that
             part
             of
             the
             arch
             BMD
             which
             is
             equal
             to
             the
             Radius
             ,
             is
             to
             the
             remainder
             of
             the
             same
             arch
             BMD
             ;
             so
             A
             Q
             to
             H
             Q
             ,
             or
             so
             BC
             to
             CP
             .
             And
             then
             will
             BP
             and
             the
             arch
             BMD
             be
             equal
             .
             But
             it
             is
             not
             demonstrated
             that
             the
             Straight
             Lines
             H
             Q
             and
             DR
             are
             equal
             ;
             though
             if
             from
             the
             point
             B
             there
             be
             drawn
             (
             by
             the
             construction
             of
             the
             first
             figure
             )
             a
             Straight
             Line
             equal
             to
             the
             arch
             BMD
             ,
             then
             DR
             to
             H
             Q
             ,
             and
             also
             the
             half
             of
             the
             Straight
             Line
             BP
             to
             DS
             ,
             will
             always
             be
             so
             equal
             ,
             that
             no
             inequality
             can
             be
             discovered
             between
             them
             .
             I
             will
             therefore
             leave
             this
             to
             be
             further
             searched
             into
             .
             For
             though
             it
             be
             almost
             out
             of
             doubt
             ,
             that
             the
             Straight
             Line
             BP
             and
             the
             arch
             BMD
             are
             equal
             ,
             yet
             that
             may
             not
             be
             received
             without
             demonstration
             ;
             and
             means
             of
             Demonstration
             the
             Circular
             Line
             admitteth
             none
             that
             is
             not
             grounded
             upon
             the
             nature
             of
             Flexion
             ,
             or
             of
             Angles
             .
             But
             by
             that
             way
             I
             have
             already
             exhibited
             a
             Straight
             Line
             equal
             to
             the
             Arch
             of
             a
             Quadrant
             in
             the
             First
             and
             Second
             aggression
             .
          
           
           
             It
             remains
             that
             I
             prove
             DT
             to
             be
             equal
             to
             the
             Sine
             of
             45
             degrees
             .
          
           
             In
             BA
             produced
             let
             AV
             he
             taken
             equal
             to
             the
             Sine
             of
             45
             degrees
             ;
             and
             drawing
             and
             producing
             VH
             ,
             it
             will
             cut
             the
             arch
             of
             the
             Quadrant
             CNA
             in
             the
             midst
             in
             N
             ,
             and
             the
             same
             arch
             again
             in
             O
             ,
             and
             the
             Straight
             line
             DC
             in
             T
             ,
             so
             ,
             that
             DT
             will
             be
             equal
             to
             the
             Sine
             of
             45
             degrees
             ,
             or
             to
             the
             straight
             line
             AV
             ;
             also
             the
             Straight
             line
             VH
             will
             be
             equal
             to
             the
             straight
             line
             HI
             or
             the
             Sine
             of
             60
             degrees
             .
          
           
             For
             the
             square
             of
             AV
             is
             equal
             to
             two
             squares
             of
             the
             Semiradius
             ;
             and
             consequently
             the
             square
             of
             VH
             is
             equal
             to
             three
             Squares
             of
             the
             Semiradius
             .
             But
             HI
             is
             a
             mean
             proportional
             between
             the
             Semiradius
             and
             three
             Semiradii
             ;
             and
             therefore
             the
             square
             of
             HI
             is
             equal
             to
             three
             Squares
             of
             the
             Semiradius
             .
             Wherefore
             HI
             is
             eqval
             to
             HV
             .
             But
             because
             AD
             is
             cut
             in
             the
             midst
             in
             H
             ,
             therefore
             VH
             and
             HT
             are
             equal
             ;
             and
             therefore
             also
             DT
             is
             equal
             to
             the
             Sine
             of
             45
             degrees
             .
             In
             the
             Radius
             BA
             let
             BX
             be
             taken
             equal
             to
             the
             Sine
             of
             45
             degrees
             ;
             for
             so
             VX
             will
             be
             equal
             to
             the
             Radius
             ;
             and
             it
             will
             be
             as
             VA
             to
             AH
             the
             Semiradius
             ,
             so
             VX
             the
             Radius
             to
             XN
             the
             Sine
             of
             45
             degrees
             .
             Wherefore
             VH
             produced
             passes
             through
             N.
             Lastly
             ,
             upon
             the
             center
             V
             with
             the
             Radius
             VA
             let
             the
             arch
             of
             a
             circle
             be
             drawn
             cutting
             VH
             in
             Y
             ;
             which
             being
             done
             ,
             VY
             will
             be
             equal
             to
             HO
             (
             for
             HO
             is
             by
             construction
             equal
             to
             the
             Sine
             of
             45
             degrees
             )
             and
             YH
             will
             be
             equal
             to
             OT
             ;
             &
             therefore
             VT
             passes
             through
             O.
             All
             which
             was
             to
             be
             demonstrated
             .
          
           
             I
             will
             here
             add
             certain
             Problemes
             ,
             of
             which
             if
             any
             Analyst
             can
             make
             the
             construction
             ,
             he
             will
             thereby
             be
             able
             to
             judge
             clearly
             of
             what
             I
             have
             now
             said
             concerning
             the
             dimension
             of
             a
             Circle
             .
             Now
             these
             Problems
             are
             nothing
             else
             (
             at
             least
             to
             sense
             )
             but
             certain
             symptomes
             accompanying
             the
             construction
             of
             the
             first
             and
             third
             figure
             of
             this
             Chapter
             .
          
           
             Describing
             therefore
             again
             the
             Square
             ABCD
             (
             in
             the
             fifth
             figure
             )
             and
             the
             three
             Quadrants
             ABD
             ,
             BCA
             and
             DAC
             ,
             let
             the
             Diagonals
             AC
             &
             BD
             be
             drawn
             ,
             cutting
             the
             arches
             BHD
             &
             CIA
             in
             the
             middle
             in
             H
             and
             I
             ;
             &
             the
             straight
             lines
             EF
             and
             GL
             ,
             dividing
             the
             square
             ABCD
             into
             four
             equal
             squares
             ,
             and
             trisecting
             the
             arches
             BHD
             and
             CIA
             ,
             namely
             ,
             BHD
             in
             K
             and
             M
             ,
             and
             CIA
             
             in
             M
             and
             O.
             Then
             dividing
             the
             arch
             BK
             in
             the
             midst
             in
             P
             ,
             let
             QP
             the
             Sine
             of
             the
             arch
             BP
             be
             drawn
             and
             produced
             to
             R
             ,
             so
             that
             QR
             be
             double
             to
             QP
             ;
             and
             connecting
             KR
             ,
             let
             it
             be
             produced
             one
             way
             to
             BC
             in
             S
             ,
             and
             the
             other
             way
             to
             BA
             produced
             in
             T.
             Also
             let
             BV
             be
             made
             triple
             to
             BS
             ,
             and
             consequently
             (
             by
             the
             second
             article
             of
             this
             Chapter
             )
             equall
             to
             the
             arch
             BD.
             This
             construction
             is
             the
             same
             with
             that
             of
             the
             first
             figure
             ,
             which
             I
             thought
             fit
             to
             renew
             discharged
             of
             all
             lines
             but
             such
             as
             are
             necessary
             for
             my
             present
             purpose
             .
          
           
             In
             the
             first
             place
             therefore
             ,
             if
             AV
             be
             drawn
             ,
             cutting
             the
             arch
             BHD
             in
             X
             ,
             and
             the
             side
             DC
             in
             Z
             ,
             I
             desire
             some
             Analyst
             would
             (
             if
             he
             can
             )
             give
             a
             reason
             ,
             Why
             the
             straight
             lines
             TE
             and
             TC
             should
             cut
             the
             arch
             BD
             the
             one
             in
             Y
             ,
             the
             other
             in
             X
             ,
             so
             as
             to
             make
             the
             arch
             BY
             equal
             to
             the
             arch
             YX
             ;
             or
             if
             they
             be
             not
             equal
             ,
             that
             he
             would
             determine
             their
             difference
             .
          
           
             Secondly
             ,
             if
             in
             the
             side
             DA
             ,
             the
             straight
             line
             Da
             be
             taken
             equal
             to
             DZ
             ,
             and
             Va
             be
             drawn
             ;
             Why
             Va
             and
             VB
             should
             be
             equal
             ;
             or
             if
             they
             be
             not
             equal
             ,
             What
             is
             the
             difference
             .
          
           
             Thirdly
             ,
             drawing
             Zb
             parallel
             and
             equal
             to
             the
             side
             CB
             ,
             cutting
             the
             arch
             BHD
             in
             c
             ,
             and
             drawing
             the
             straight
             line
             Ac
             ,
             and
             producing
             it
             to
             BV
             in
             d
             ;
             Why
             Ad
             should
             be
             equal
             and
             parallel
             to
             the
             straight
             line
             aV
             ,
             and
             consequently
             equal
             also
             to
             the
             arch
             BD.
             
          
           
             Fourthly
             ,
             drawing
             eK
             the
             Sine
             of
             the
             arch
             BK
             ,
             &
             taking
             (
             in
             eA
             produced
             )
             ef
             equal
             to
             the
             Diagonal
             AC
             ,
             and
             connecting
             fC
             ;
             Why
             fC
             should
             pass
             through
             a
             (
             which
             point
             being
             given
             ,
             the
             length
             of
             the
             arch
             BHD
             is
             also
             given
             )
             and
             c
             ;
             and
             why
             fe
             and
             fc
             should
             be
             equal
             ;
             or
             if
             not
             ,
             why
             unequal
             .
          
           
             Fifthly
             ,
             drawing
             fZ
             ,
             I
             desire
             he
             would
             shew
             ,
             Why
             it
             is
             equal
             to
             BV
             ,
             or
             to
             the
             arch
             BD
             ;
             or
             if
             they
             be
             not
             equal
             ,
             What
             is
             their
             difference
             .
          
           
             Sixtly
             ,
             granting
             fZ
             to
             be
             equal
             to
             the
             arch
             BD
             ,
             I
             desire
             he
             would
             determine
             whether
             it
             fall
             all
             without
             the
             arch
             BCA
             ,
             or
             cut
             the
             same
             ;
             or
             touch
             it
             ,
             and
             in
             what
             point
             .
          
           
             Seventhly
             ,
             the
             Semicircle
             BDg
             being
             completed
             ;
             Why
             gI
             being
             drawn
             and
             produced
             ,
             should
             pass
             through
             X
             (
             by
             which
             point
             X
             the
             length
             of
             the
             arch
             BD
             is
             determined
             )
             .
             And
             the
             same
             gI
             being
             yet
             further
             produced
             to
             DC
             in
             h
             ;
             Why
             Ad
             (
             which
             is
             equal
             
             to
             the
             arch
             BD
             )
             should
             pass
             through
             that
             point
             h.
             
          
           
             Eighthly
             ,
             upon
             the
             Center
             of
             the
             square
             ABCD
             ,
             which
             let
             be
             k
             ,
             the
             arch
             of
             the
             quadrant
             EiL
             being
             drawn
             ,
             cutting
             eK
             produced
             in
             i
             ;
             Why
             the
             drawn
             straight
             line
             iX
             should
             be
             parallel
             to
             the
             side
             CD
             .
          
           
             Ninthly
             ,
             in
             the
             sides
             BA
             and
             BC
             taking
             Bl
             and
             Bm
             severally
             equal
             to
             half
             BV
             ,
             or
             to
             the
             arch
             BH
             ,
             and
             drawing
             mn
             parallel
             and
             equal
             to
             the
             side
             BA
             ,
             cutting
             the
             arch
             BD
             in
             o
             ;
             Why
             the
             straight
             line
             wich
             connects
             Vl
             should
             pass
             through
             the
             point
             o
             ,
          
           
             Tenthly
             ,
             I
             would
             know
             of
             him
             ,
             Why
             the
             straight
             line
             which
             connects
             aH
             should
             be
             equal
             to
             Bm
             ;
             or
             if
             not
             ,
             how
             much
             it
             differs
             from
             it
             .
          
           
             The
             Analyst
             that
             can
             solve
             these
             Problemes
             without
             knowing
             first
             the
             length
             of
             the
             arch
             BD
             ,
             or
             using
             any
             other
             known
             Method
             then
             that
             which
             proceeds
             by
             perpetual
             bisection
             of
             an
             angle
             ,
             or
             is
             drawn
             from
             the
             consideration
             of
             the
             nature
             of
             Flexion
             ,
             shall
             do
             more
             then
             ordinary
             Geometry
             is
             able
             to
             perform
             .
             But
             if
             the
             Dimension
             of
             a
             Circle
             cannot
             be
             found
             by
             any
             other
             Method
             ;
             then
             I
             have
             either
             found
             it
             ,
             or
             it
             is
             not
             at
             all
             to
             be
             found
             .
          
           
             From
             the
             known
             Length
             of
             the
             Arch
             of
             a
             Quadrant
             ,
             and
             from
             the
             proportional
             Division
             of
             the
             Arch
             and
             of
             the
             Tangent
             BC
             ,
             may
             be
             deduced
             the
             Section
             of
             an
             Angle
             into
             any
             given
             proportion
             ;
             as
             also
             the
             Squaring
             of
             the
             Circle
             ,
             the
             Squaring
             of
             a
             given
             Sector
             ,
             and
             many
             the
             like
             propositions
             ,
             which
             it
             is
             not
             necessary
             here
             to
             demonstrate
             .
             I
             will
             therefore
             onely
             exhibit
             a
             Straight
             line
             equal
             to
             the
             Spiral
             of
             Archimedes
             ,
             and
             so
             dismiss
             this
             speculation
             .
          
           
             5
             The
             length
             of
             the
             Perimeter
             of
             a
             Circle
             being
             found
             ,
             that
             Straight
             line
             is
             also
             found
             ,
             which
             touches
             a
             Spiral
             at
             the
             end
             of
             its
             first
             conversion
             .
             For
             upon
             the
             center
             A
             (
             in
             the
             sixth
             figure
             )
             let
             the
             circle
             BCDE
             be
             described
             ;
             and
             in
             it
             let
             Archimedes
             his
             Spiral
             AFGHB
             be
             drawn
             ,
             beginning
             at
             A
             and
             ending
             at
             B.
             Through
             the
             center
             A
             let
             the
             straight
             line
             CE
             be
             drawn
             ,
             cutting
             the
             Diameter
             BD
             at
             right
             angles
             ;
             and
             let
             it
             be
             produced
             to
             I
             ,
             so
             ,
             that
             AI
             be
             equal
             to
             the
             Perimeter
             BCDEB
             .
             Therefore
             IB
             being
             drawn
             will
             touch
             the
             Spiral
             AFGHB
             in
             B
             ;
             which
             is
             demonstrated
             by
             Archimedes
             in
             his
             book
             
               de
               Spiralibus
            
             .
          
           
             And
             for
             a
             Straight
             Line
             equal
             to
             the
             given
             Spiral
             AFGHB
             ,
             it
             may
             be
             found
             thus
             .
          
           
           
             Let
             the
             straight
             line
             AI
             (
             which
             is
             equal
             to
             the
             Perimeter
             BCDE
             )
             be
             bisected
             in
             K
             ;
             and
             taking
             KL
             equal
             to
             the
             Radius
             AB
             ,
             let
             the
             rectangle
             IL
             be
             completed
             .
             Let
             ML
             be
             understood
             to
             be
             the
             axis
             ,
             and
             KL
             the
             base
             of
             a
             Parabola
             ,
             and
             let
             MK
             be
             the
             crooked
             line
             thereof
             .
             Now
             if
             the
             point
             M
             be
             conceived
             to
             be
             so
             moved
             by
             the
             concourse
             of
             two
             movents
             ,
             the
             one
             frō
             IM
             to
             KL
             with
             velocity
             encreasing
             continually
             in
             the
             same
             proportion
             with
             the
             Times
             ,
             the
             other
             from
             ML
             to
             IK
             uniformly
             ,
             that
             both
             those
             motions
             begin
             together
             in
             M
             and
             end
             in
             K
             ;
             Galilaeus
             has
             demonstrated
             that
             by
             such
             motion
             of
             the
             point
             M
             ,
             the
             crooked
             line
             of
             a
             Parabola
             will
             be
             described
             .
             Again
             ,
             if
             the
             point
             A
             be
             conceived
             to
             be
             moved
             uniformly
             in
             the
             straight
             line
             AB
             ,
             and
             in
             the
             same
             time
             to
             be
             carried
             round
             upon
             the
             center
             A
             by
             the
             circular
             motion
             of
             all
             the
             points
             between
             A
             and
             B
             ;
             Archimedes
             has
             demonstrated
             that
             by
             such
             motion
             will
             be
             described
             a
             Spiral
             line
             .
             And
             seeing
             the
             circles
             of
             all
             these
             motions
             are
             concentrick
             in
             A
             ;
             and
             the
             interiour
             circle
             is
             alwayes
             lesse
             then
             the
             exteriour
             in
             the
             proportion
             of
             the
             times
             in
             which
             AB
             is
             passed
             over
             with
             uniform
             motion
             ;
             the
             velocity
             also
             of
             the
             circular
             motion
             of
             the
             point
             A
             ,
             will
             continually
             encrease
             proportionally
             to
             the
             times
             .
             And
             thus
             far
             the
             generations
             of
             the
             Parabolical
             line
             MK
             ,
             and
             of
             the
             Spiral
             line
             AFGHB
             ,
             are
             like
             .
             But
             the
             Uniform
             motion
             in
             AB
             concurring
             with
             circular
             motion
             in
             the
             Perimeters
             of
             all
             the
             concentrick
             circles
             ,
             describes
             that
             circle
             ,
             whose
             center
             is
             A
             ,
             and
             Perimeter
             BCDE
             ;
             and
             therefore
             that
             circle
             is
             (
             by
             the
             Coroll
             .
             of
             the
             first
             article
             of
             the
             16
             Chapter
             )
             the
             aggregate
             of
             all
             the
             Velocities
             together
             taken
             of
             the
             point
             A
             whilst
             it
             describes
             the
             Spiral
             AFGHB
             .
             Also
             the
             rectangle
             IKLM
             is
             the
             aggregate
             of
             all
             the
             Velocities
             together
             taken
             of
             the
             point
             M
             ,
             whilest
             it
             describes
             the
             crooked
             line
             MK
             .
             And
             therefore
             the
             whole
             velocity
             ,
             by
             which
             the
             Parabolicall
             line
             MK
             is
             described
             ▪
             is
             to
             the
             whole
             velocity
             with
             which
             the
             Spiral
             line
             AFGHB
             is
             described
             in
             the
             same
             time
             ,
             as
             the
             rectangle
             IKLM
             ,
             is
             to
             the
             Circle
             BCDE
             ,
             that
             is
             to
             the
             triangle
             AIB
             .
             But
             because
             AI
             is
             bisected
             in
             K
             &
             the
             straight
             lines
             IM
             &
             AB
             are
             equal
             ,
             therefore
             the
             rectangle
             IKLM
             and
             the
             triangle
             AIB
             are
             also
             equal
             .
             Wherefore
             the
             Spiral
             line
             AFGHB
             ,
             and
             the
             Parabolical
             line
             MK
             ,
             being
             described
             with
             equal
             velocity
             and
             in
             equal
             times
             ,
             are
             equal
             to
             one
             another
             .
             Now
             in
             the
             first
             article
             of
             the
             18
             Chapter
             a
             straight
             line
             is
             found
             
             out
             equal
             to
             any
             Parabolical
             line
             .
             Wherefore
             also
             a
             Straight
             line
             is
             found
             out
             ,
             equal
             to
             a
             given
             Spiral
             line
             of
             the
             first
             revolution
             described
             by
             Archimedes
             ;
             which
             was
             to
             be
             done
             .
          
           
             6
             In
             the
             sixth
             Chapter
             ,
             which
             is
             of
             Method
             ,
             that
             which
             I
             should
             there
             have
             spoken
             of
             the
             Analyticks
             of
             Geometricians
             ,
             I
             thought
             fit
             to
             deferre
             ,
             because
             I
             could
             not
             there
             have
             been
             understood
             ,
             as
             not
             having
             then
             so
             much
             as
             named
             
               Lines
               ,
               Superficies
               ,
               Solids
               ,
               Equal
               and
               Unequal
               &c.
               
            
             Wherefore
             I
             will
             in
             this
             place
             set
             down
             my
             thoughts
             concerning
             it
             .
          
           
             Analysis
             ,
             is
             continual
             Reasoning
             from
             the
             Definitions
             of
             the
             terms
             of
             a
             proposition
             we
             suppose
             true
             ,
             and
             again
             from
             the
             Definitions
             of
             the
             terms
             of
             those
             Definitions
             ,
             and
             so
             on
             ,
             till
             we
             come
             to
             some
             things
             known
             ,
             the
             Composition
             whereof
             is
             the
             demonstration
             of
             the
             truth
             or
             falsity
             of
             the
             first
             supposition
             ;
             and
             this
             Composition
             or
             Demonstration
             is
             that
             we
             call
             
               Synthesis
               .
               Analytica
            
             therefore
             is
             that
             art
             ,
             by
             which
             our
             reason
             proceeds
             from
             something
             supposed
             ,
             to
             Principles
             ,
             that
             is
             ,
             to
             prime
             Propositions
             ,
             or
             to
             such
             as
             are
             known
             by
             these
             ,
             till
             we
             have
             so
             many
             known
             Propositions
             as
             are
             sufficient
             for
             the
             demonstration
             of
             the
             truth
             or
             falsity
             of
             the
             thing
             supposed
             .
             Synthetica
             is
             the
             art
             it self
             of
             Demonstration
             .
             Synthesis
             therefore
             and
             Analysis
             differ
             in
             nothing
             ,
             but
             in
             proceeding
             forwards
             or
             backwards
             ;
             and
             Logistica
             comprehends
             both
             .
             So
             that
             in
             the
             Analysis
             or
             Synthesis
             of
             any
             question
             ,
             that
             is
             to
             say
             ,
             of
             any
             Probleme
             ,
             the
             Terms
             of
             all
             the
             Propositions
             ought
             to
             be
             convertible
             ;
             or
             if
             they
             be
             enunciated
             Hypothetically
             ,
             the
             truth
             of
             the
             Consequent
             ought
             not
             onely
             to
             follow
             out
             of
             the
             truth
             of
             its
             Antecedent
             ,
             but
             contrarily
             also
             the
             truth
             of
             the
             Antecedent
             must
             necessarily
             be
             inferred
             from
             the
             truth
             of
             the
             Consequent
             .
             For
             otherwise
             ,
             when
             by
             Resolution
             we
             are
             arrived
             at
             Principles
             ,
             we
             cannot
             by
             Composition
             return
             directly
             back
             to
             the
             thing
             sought
             for
             .
             For
             those
             Terms
             which
             are
             the
             first
             in
             Analysis
             ,
             will
             be
             the
             last
             in
             Synthesis
             ;
             as
             for
             example
             ,
             when
             in
             Resol●ing
             ,
             we
             say
             ,
             these
             two
             Rectangles
             are
             equal
             and
             therefore
             their
             sides
             are
             reciprocally
             proportional
             ,
             we
             must
             necessarily
             in
             Compounding
             say
             ,
             the
             sides
             of
             these
             Rectangles
             are
             reciprocally
             proportional
             and
             therefore
             the
             Rectangles
             themselves
             are
             equal
             ;
             Which
             we
             could
             not
             say
             ,
             ●…ss
             
               Rectangles
               have
               their
               sides
               reciprocally
               proportional
               ,
               and
               Rectangles
               are
               equal
               ,
            
             were
             Terms
             convertible
             .
          
           
           
             Now
             in
             every
             Analysis
             ,
             that
             which
             is
             sought
             ,
             is
             the
             Proportion
             of
             two
             quantities
             ;
             by
             which
             proportion
             (
             a
             figure
             being
             described
             )
             the
             quantity
             sought
             for
             may
             be
             exposed
             to
             Sense
             .
             And
             this
             Exposition
             is
             the
             end
             and
             Solution
             of
             the
             question
             ,
             or
             the
             construction
             of
             the
             Probleme
             .
          
           
             And
             seeing
             Analysis
             is
             reasoning
             from
             something
             supposed
             ,
             till
             we
             come
             to
             Principles
             ,
             that
             is
             ,
             to
             Definitions
             ,
             or
             to
             Theoremes
             formerly
             known
             ;
             and
             seeing
             the
             same
             reasoning
             tends
             in
             the
             last
             place
             to
             some
             Equation
             ;
             we
             can
             therefore
             make
             no
             end
             of
             Resolving
             ,
             till
             we
             come
             at
             last
             to
             the
             causes
             themselves
             of
             Equality
             and
             Inequality
             ,
             or
             to
             Theoremes
             formerly
             demonstrated
             from
             those
             causes
             ;
             and
             so
             have
             a
             sufficient
             number
             of
             those
             Theoremes
             for
             the
             demonstration
             of
             the
             thing
             sought
             for
             .
          
           
             And
             seeing
             also
             ,
             that
             the
             end
             of
             the
             Analyticks
             ,
             is
             either
             the
             construction
             of
             such
             a
             Probleme
             as
             is
             possible
             ,
             or
             the
             detection
             of
             the
             impossibility
             thereof
             ;
             whensoever
             the
             Probleme
             may
             be
             solved
             ,
             the
             Analyst
             must
             not
             stay
             ,
             till
             he
             come
             to
             those
             things
             which
             contain
             the
             efficient
             cause
             of
             that
             whereof
             he
             is
             to
             make
             construction
             .
             But
             he
             must
             of
             necessity
             stay
             when
             he
             comes
             to
             prime
             Propositions
             ;
             and
             these
             are
             Definitions
             .
             These
             Definitions
             therefore
             must
             contain
             the
             efficient
             cause
             of
             his
             Construction
             ;
             I
             say
             of
             his
             Construction
             ,
             not
             of
             the
             Conclusion
             which
             he
             demonstrates
             ;
             for
             the
             cause
             of
             the
             Conclusion
             is
             contained
             in
             the
             premised
             propositions
             ;
             that
             is
             to
             say
             ,
             the
             truth
             of
             the
             proposition
             he
             proves
             ,
             is
             drawn
             from
             the
             propositions
             which
             prove
             the
             same
             .
             But
             the
             cause
             of
             his
             construction
             is
             in
             the
             things
             themselves
             ,
             and
             consists
             in
             motion
             ,
             or
             in
             the
             concourse
             of
             motions
             .
             Wherefore
             those
             propositions
             in
             which
             Analysis
             ends
             ,
             are
             Definitions
             ,
             but
             such
             ,
             as
             signifie
             in
             what
             manner
             the
             construction
             ,
             or
             generation
             of
             the
             thing
             proceeds
             .
             For
             otherwise
             ,
             when
             he
             goes
             back
             by
             Synthesis
             to
             the
             proofe
             of
             his
             Probleme
             ,
             he
             will
             come
             to
             no
             Demonstration
             at
             all
             ;
             there
             being
             no
             true
             Demonstration
             but
             such
             as
             is
             scientificall
             ;
             and
             no
             Demonstration
             is
             scientifical
             but
             that
             which
             proceeds
             from
             the
             knowledge
             of
             the
             causes
             from
             which
             the
             construction
             of
             the
             Probleme
             is
             drawne
             .
             To
             collect
             therefore
             what
             has
             been
             said
             into
             few
             words
             ;
             ANALYSIS
             
               is
               Ratiocination
               from
               the
               supposed
               construction
               or
               generation
               of
               a
               thing
               to
               the
               efficient
               cause
               ,
               or
               coefficient
               causes
               of
               that
               which
               is
               constructed
               or
               generated
               .
               And
            
             SYNTHESIS
             
               is
               Ratiocination
               from
               the
               first
               causes
            
             
             
               of
               the
               Construction
               ,
               continued
               through
               all
               the
               middle
               causes
               till
               we
               come
               to
               the
               thing
               it selfe
               which
               is
               constructed
               or
               generated
               .
            
          
           
             But
             because
             there
             are
             many
             means
             by
             which
             the
             same
             thing
             may
             be
             generated
             ,
             or
             the
             same
             Probleme
             be
             constructed
             ,
             therefore
             neither
             do
             all
             Geometricians
             ,
             nor
             doth
             the
             same
             Geometrician
             alwayes
             use
             one
             and
             the
             same
             Method
             .
             For
             if
             to
             a
             certain
             quantity
             given
             ,
             it
             be
             required
             to
             construct
             another
             quantity
             equal
             ,
             there
             may
             be
             some
             that
             will
             enquire
             whether
             this
             may
             not
             be
             done
             by
             means
             of
             some
             motion
             .
             For
             there
             are
             quantities
             ,
             whose
             equality
             and
             inequality
             may
             be
             argued
             from
             Motion
             and
             Time
             ,
             as
             well
             as
             from
             Congruence
             ;
             and
             there
             is
             motion
             ,
             by
             which
             two
             quantities
             ,
             whether
             Lines
             or
             Superficies
             ,
             though
             one
             of
             them
             be
             crooked
             ,
             the
             other
             straight
             ,
             may
             be
             made
             congruous
             or
             coincident
             .
             And
             this
             method
             Archimedes
             made
             use
             of
             in
             his
             Book
             
               de
               Spiralibus
            
             .
             Also
             the
             equality
             or
             inequality
             of
             two
             quantities
             may
             be
             found
             out
             and
             demonstrated
             from
             the
             consideration
             of
             Waight
             ,
             as
             the
             same
             Archimedes
             did
             in
             his
             Quadrature
             of
             the
             Parabola
             .
             Besides
             ,
             equality
             and
             equality
             are
             found
             out
             often
             by
             the
             division
             of
             the
             two
             quantityes
             into
             parts
             which
             are
             considered
             as
             undivisible
             ;
             as
             
               Cavallerius
               Bonaventura
            
             has
             done
             in
             our
             time
             ,
             and
             Archimedes
             often
             .
             Lastly
             ,
             the
             same
             is
             performed
             by
             the
             consideration
             of
             the
             Powers
             of
             lines
             ,
             or
             the
             roots
             of
             those
             Powers
             ,
             and
             by
             the
             multiplication
             ,
             division
             ,
             addition
             and
             substraction
             ,
             as
             also
             by
             the
             extraction
             of
             the
             roots
             of
             those
             Powers
             ,
             or
             by
             finding
             where
             straight
             lines
             of
             the
             same
             proportion
             terminate
             .
             For
             example
             ,
             when
             any
             number
             of
             straight
             lines
             ,
             how
             many
             soever
             ,
             are
             drawne
             from
             a
             straight
             line
             ,
             and
             passe
             all
             through
             the
             same
             point
             ,
             looke
             what
             proportion
             they
             have
             ,
             and
             if
             their
             parts
             continued
             from
             the
             point
             retaine
             every
             where
             the
             same
             proportion
             ,
             they
             shall
             all
             terminate
             in
             a
             straight
             line
             .
             And
             the
             same
             happens
             if
             the
             point
             be
             taken
             between
             two
             Circles
             .
             So
             that
             the
             places
             of
             all
             their
             points
             of
             termination
             make
             either
             straight
             lines
             ,
             or
             circumferences
             of
             Circles
             ,
             and
             are
             called
             
               Plain
               Places
            
             .
             So
             also
             when
             straight
             parallel
             lines
             are
             applyed
             to
             one
             straight
             line
             ,
             if
             the
             parts
             of
             the
             straight
             line
             to
             which
             they
             are
             applyed
             be
             to
             one
             another
             in
             proportion
             duplicate
             to
             that
             of
             the
             contiguous
             applyed
             lines
             ,
             they
             will
             all
             terminate
             in
             a
             Conical
             Section
             ;
             which
             Section
             being
             the
             place
             of
             their
             
             termination
             ,
             is
             called
             a
             
               Solid
               Place
            
             ,
             because
             it
             serves
             for
             the
             finding
             out
             of
             the
             quantity
             of
             any
             Equation
             which
             consists
             of
             three
             dimensions
             .
             There
             are
             therfore
             three
             ways
             of
             finding
             out
             the
             cause
             of
             Equality
             or
             Inequality
             between
             two
             given
             quantities
             ;
             namely
             ,
             First
             by
             the
             Computation
             of
             Motions
             (
             for
             by
             equal
             Motion
             ,
             &
             equal
             Time
             equal
             Spaces
             are
             described
             ,
             )
             and
             Ponderation
             is
             motion
             .
             Secondly
             By
             Indivisibles
             ;
             because
             all
             the
             parts
             together
             taken
             are
             equal
             to
             the
             whole
             .
             And
             thirdly
             by
             the
             Powers
             ;
             for
             when
             they
             are
             equall
             ,
             their
             roots
             also
             are
             equall
             ;
             and
             contrarily
             ,
             the
             Powers
             are
             equall
             ,
             when
             their
             roots
             are
             equal
             .
             But
             if
             the
             question
             be
             much
             complicated
             ,
             there
             cannot
             by
             any
             of
             these
             wayes
             be
             constituted
             a
             certaine
             Rule
             ,
             from
             the
             supposition
             of
             which
             of
             the
             unknown
             quantities
             the
             Analysis
             may
             best
             begin
             ;
             nor
             out
             of
             the
             variety
             of
             Equations
             that
             at
             first
             appeare
             ,
             which
             we
             were
             best
             to
             choose
             ;
             but
             the
             successe
             will
             depend
             upon
             dexterity
             ,
             upon
             formerly
             acquired
             Science
             ,
             and
             many
             times
             upon
             fortune
             .
          
           
             For
             no
             man
             can
             ever
             be
             a
             good
             Analyst
             without
             being
             first
             a
             good
             Geometrician
             ;
             nor
             do
             the
             rules
             of
             Analysis
             make
             a
             Geometrician
             ,
             as
             Synthesis
             doth
             ;
             which
             begins
             at
             the
             very
             Elements
             ,
             and
             proceeds
             by
             a
             Logical
             Use
             of
             the
             same
             .
             For
             the
             true
             teaching
             of
             Geometry
             is
             by
             Synthesis
             ,
             according
             to
             Euclides
             method
             ;
             and
             he
             that
             hath
             Euclide
             for
             his
             Master
             ,
             may
             be
             a
             Geometrician
             without
             Vieta
             (
             though
             Vieta
             was
             a
             most
             admirable
             Geometrician
             )
             ;
             but
             he
             that
             has
             Vieta
             for
             his
             master
             ,
             not
             so
             ,
             without
             Euclide
             .
          
           
             And
             as
             for
             that
             part
             of
             Analysis
             which
             works
             by
             the
             Powers
             ,
             though
             it
             be
             esteemed
             by
             some
             Geometricians
             (
             not
             the
             chiefest
             )
             to
             be
             the
             best
             way
             of
             solving
             all
             Problemes
             ,
             yet
             it
             is
             a
             thing
             of
             no
             great
             extent
             ;
             it
             being
             all
             contained
             in
             the
             doctrine
             of
             rectangles
             ,
             and
             rectangled
             Solids
             .
             So
             that
             although
             they
             come
             to
             an
             Equation
             which
             determines
             the
             quantity
             sought
             ,
             yet
             they
             cannot
             sometimes
             by
             art
             exhibit
             that
             quantity
             in
             a
             Plain
             ,
             but
             in
             some
             Conique
             Section
             ;
             that
             is
             ,
             as
             Geometricians
             say
             ,
             not
             Geometrically
             ,
             but
             mechanically
             .
             Now
             such
             Problemes
             as
             these
             ,
             they
             call
             Solid
             ;
             and
             when
             they
             cannot
             exhibit
             the
             quantity
             sought
             for
             with
             the
             helpe
             of
             a
             conique
             Section
             ,
             they
             call
             it
             a
             Lineary
             Probleme
             .
             And
             therefore
             in
             the
             quantities
             of
             angles
             ,
             and
             of
             the
             arches
             of
             Circles
             ,
             there
             is
             no
             use
             at
             all
             of
             the
             Analyticks
             which
             proceed
             by
             the
             Powers
             ;
             so
             that
             
             the
             Antients
             pronounced
             it
             impossible
             ,
             to
             exhibit
             in
             a
             plaine
             the
             Division
             of
             Angles
             ,
             except
             bisection
             ,
             and
             the
             bisection
             of
             the
             bisected
             parts
             ,
             otherwise
             then
             mechanically
             .
             For
             Pappus
             ,
             (
             before
             the
             31
             proposition
             of
             his
             fourth
             Book
             )
             distinguishing
             and
             defining
             the
             several
             kinds
             of
             Problemes
             ,
             says
             that
             
               some
               are
            
             Plain
             ,
             others
             Solid
             ,
             
               and
               others
            
             Lineary
             .
             
               Those
               therefore
               which
               may
               be
               solved
               by
               straight
               lines
               and
               the
               circumferences
               of
               Circles
            
             (
             that
             is
             ,
             which
             may
             be
             described
             with
             the
             Rule
             and
             Compass
             ,
             without
             any
             other
             Instrument
             )
             
               are
               fitly
               called
            
             Plain
             ;
             
               for
               the
               lines
               by
               which
               such
               Problemes
               are
               found
               out
               ,
               have
               their
               generation
               in
               a
               Plain
               .
               But
               those
               which
               are
               solved
               by
               the
               using
               of
               some
               one
               or
               more
               Conique
               Sections
               in
               their
               construction
               ,
               are
               called
            
             Solid
             ,
             
               because
               their
               construction
               cannot
               be
               made
               without
               using
               the
               superficies
               of
               solid
               figures
               ,
               namely
               of
               Cones
               .
               There
               remains
               the
               third
               kinde
               ,
               which
               is
               called
            
             Lineary
             ,
             
               because
               other
               lines
               besides
               those
               already
               mentioned
               are
               made
               use
               of
               in
               their
               construction
               ,
               &c.
               
            
             And
             a
             little
             after
             he
             sayes
             ,
             
               Of
               this
               kinde
               are
               the
            
             Spiral
             
               lines
               ,
               the
            
             Quadratrices
             ,
             the
             Conchoeides
             ,
             
               and
               the
            
             Cissoeides
             .
             
               And
               Geometricians
               think
               it
               no
               small
               fault
               ,
               when
               for
               the
               finding
               out
               of
               a
            
             Plain
             
               Probleme
               any
               man
               makes
               use
               of
            
             Coniques
             ,
             
               or
               new
            
             Lines
             .
             Now
             he
             ranks
             the
             Trisection
             of
             an
             angle
             among
             Solid
             Problemes
             ,
             and
             the
             Quinquesection
             among
             Lineary
             .
             But
             what
             !
             are
             the
             ancient
             Geometricians
             to
             be
             blamed
             ,
             who
             made
             use
             of
             the
             Quadratrix
             for
             the
             finding
             out
             of
             a
             straight
             line
             equal
             to
             the
             arch
             of
             a
             Circle
             ?
             and
             Pappus
             himself
             ,
             was
             he
             faulty
             when
             he
             found
             out
             the
             trisection
             of
             an
             Angle
             by
             the
             help
             of
             an
             Hyperbole
             ?
             Or
             am
             I
             in
             the
             wrong
             ,
             who
             think
             I
             have
             found
             out
             the
             construction
             of
             both
             these
             Problemes
             by
             the
             Rule
             and
             Compass
             onely
             ?
             Neither
             they
             ,
             nor
             I.
             For
             the
             Ancients
             made
             use
             of
             this
             Analysis
             which
             proceeds
             by
             the
             Powers
             ;
             and
             with
             them
             it
             was
             a
             fault
             to
             do
             that
             by
             a
             more
             remote
             Power
             ,
             which
             might
             be
             done
             by
             a
             neerer
             ;
             as
             being
             an
             argument
             that
             they
             did
             not
             sufficiently
             understand
             the
             nature
             of
             the
             thing
             .
             The
             virtue
             of
             this
             kind
             of
             Analysis
             consists
             in
             the
             changing
             and
             turning
             and
             tossing
             of
             Rectangles
             and
             Analogismes
             ;
             and
             the
             skill
             of
             Analysts
             is
             meer
             Logick
             ,
             by
             which
             they
             are
             able
             methodically
             to
             find
             out
             whatsoever
             lies
             hid
             either
             in
             the
             Subject
             or
             Predicate
             of
             the
             Conclusiō
             sought
             for
             .
             But
             this
             doth
             not
             properly
             belong
             to
             Algebra
             ,
             or
             the
             
               Analyticks
               Specious
               ,
               Symbolical
            
             or
             Cossick
             ;
             which
             are
             ,
             as
             I
             may
             say
             ,
             the
             Brachygraphy
             of
             the
             Analyticks
             ,
             and
             an
             art
             ,
             neither
             of
             teaching
             nor
             
             learning
             Geometry
             ,
             but
             of
             registring
             with
             brevity
             and
             celerity
             ,
             the
             inventions
             of
             Geometricians
             .
             For
             though
             it
             be
             easie
             to
             discourse
             by
             Symbols
             of
             very
             remote
             propositions
             ;
             yet
             whether
             such
             discourse
             deserve
             to
             be
             thought
             very
             profitable
             ,
             when
             it
             is
             made
             without
             any
             Ideas
             of
             the
             things
             themselves
             ,
             I
             know
             not
             .
          
        
         
           
           
           
           
             
               CHAP.
               XXI
            
             .
             Of
             
               Circular
               Motion
            
             .
          
           
             
               
                 1
                 In
                 Simple
                 Motion
                 ,
                 every
                 Straight
                 Line
                 taken
                 in
                 the
                 Body
                 moved
                 ,
                 is
                 so
                 carried
                 ,
                 that
                 it
                 is
                 always
                 parallel
                 to
                 the
                 places
                 in
                 which
                 it
                 formerly
                 was
                 .
              
               
                 2
                 If
                 Circular
                 Motion
                 be
                 made
                 about
                 a
                 resting
                 Center
                 ,
                 and
                 in
                 that
                 Circle
                 there
                 be
                 an
                 Epicyle
                 ,
                 whose
                 revolution
                 is
                 made
                 the
                 contrary
                 way
                 ,
                 in
                 such
                 manner
                 ,
                 that
                 in
                 equal
                 times
                 it
                 make
                 equal
                 angles
                 ,
                 every
                 Straight
                 Line
                 taken
                 in
                 that
                 Epicycle
                 will
                 be
                 so
                 carried
                 ,
                 that
                 it
                 will
                 alwayes
                 be
                 parallel
                 to
                 the
                 places
                 in
                 which
                 it
                 formerly
                 was
                 .
              
               
                 3
                 The
                 properties
                 of
                 Simple
                 Motion
                 .
              
               
                 4
                 If
                 a
                 fluid
                 Body
                 be
                 moved
                 with
                 simple
                 Circular
                 Motion
                 ,
                 all
                 the
                 points
                 taken
                 in
                 it
                 will
                 describe
                 their
                 Circles
                 in
                 times
                 proportional
                 to
                 the
                 distances
                 from
                 the
                 Center
                 .
              
               
                 5
                 Simple
                 Motion
                 dissipates
                 Heterogeneous
                 and
                 congregates
                 Homogeneous
                 Bodies
                 .
              
               
                 6
                 If
                 a
                 Circle
                 made
                 by
                 a
                 Movent
                 moved
                 with
                 Simple
                 Motion
                 ,
                 be
                 commensurable
                 to
                 another
                 Circle
                 made
                 by
                 a
                 point
                 which
                 is
                 carried
                 about
                 by
                 the
                 same
                 Movent
                 ,
                 all
                 the
                 points
                 of
                 both
                 the
                 Circles
                 will
                 at
                 some
                 time
                 return
                 to
                 the
                 same
                 situation
                 .
              
               
                 7
                 If
                 a
                 Sphere
                 have
                 Simple
                 Motion
                 ,
                 its
                 Motion
                 will
                 more
                 dissipate
                 Heterogeneous
                 Bodies
                 by
                 how
                 much
                 it
                 is
                 more
                 remote
                 from
                 the
                 Poles
                 .
              
               
                 8
                 If
                 the
                 Simple
                 Circular
                 Motion
                 of
                 a
                 fluid
                 Body
                 ,
                 be
                 hindered
                 by
                 a
                 Body
                 which
                 is
                 not
                 fluid
                 ,
                 the
                 fluid
                 Body
                 will
                 spread
                 it self
                 upon
                 the
                 Superficies
                 of
                 that
                 Body
                 .
              
               
                 9
                 Circular
                 Motion
                 about
                 a
                 fixed
                 Center
                 ,
                 casteth
                 off
                 by
                 the
                 Tangent
                 such
                 things
                 as
                 lie
                 upon
                 the
                 Circumference
                 and
                 stick
                 not
                 to
                 it
                 .
              
               
                 10
                 Such
                 things
                 as
                 are
                 moved
                 with
                 Simple
                 Circular
                 Motition
                 ,
                 beget
                 Simple
                 Circular
                 Motion
                 .
              
               
                 11
                 If
                 that
                 which
                 is
                 so
                 moved
                 have
                 one
                 side
                 hard
                 ,
                 and
                 the
                 other
                 side
                 fluid
                 ,
                 its
                 Motion
                 will
                 not
                 be
                 perfectly
                 Circular
                 .
              
            
          
           
             1
             I
             Have
             already
             defined
             
               Simple
               Motion
            
             to
             be
             that
             ,
             in
             which
             the
             several
             points
             taken
             in
             a
             moved
             Body
             ,
             do
             in
             several
             equal
             times
             describe
             several
             equal
             arches
             .
             And
             therefore
             in
             Simple
             Circular
             Motion
             it
             is
             necessary
             that
             every
             Straight
             Line
             taken
             in
             the
             Moved
             Body
             be
             alwayes
             carried
             parallel
             to
             itself
             ;
             which
             I
             thus
             demonstrate
             .
          
           
             First
             ,
             let
             A
             B
             (
             in
             the
             first
             figure
             )
             be
             any
             Straight
             Line
             taken
             in
             
             any
             Solid
             Body
             ;
             and
             let
             AD
             be
             any
             arch
             drawn
             upon
             any
             Center
             C
             and
             Radius
             CA.
             Let
             the
             point
             B
             be
             understood
             to
             describe
             towards
             the
             same
             parts
             the
             arch
             BE
             ,
             like
             and
             equall
             to
             the
             arch
             AD.
             Now
             in
             the
             same
             time
             in
             which
             the
             point
             A
             transmits
             the
             arch
             AD
             ,
             the
             point
             B
             (
             which
             by
             reason
             of
             its
             simple
             motion
             is
             supposed
             to
             be
             carried
             with
             velocity
             equall
             to
             that
             of
             A
             )
             will
             transmit
             the
             arch
             BE
             ;
             and
             at
             the
             end
             of
             the
             same
             time
             the
             whole
             AB
             will
             be
             in
             DE
             ;
             and
             therefore
             AB
             and
             DE
             are
             equall
             .
             And
             seeing
             the
             arches
             AD
             and
             BE
             are
             like
             and
             equall
             ,
             their
             subtending
             straight
             lines
             AD
             and
             BE
             will
             also
             be
             equall
             ;
             and
             therefore
             the
             four
             sided
             figure
             ABDE
             will
             be
             a
             parallelogram
             .
             Wherefore
             AB
             is
             carried
             parallel
             to
             it selfe
             .
             And
             the
             same
             may
             be
             proved
             by
             the
             same
             method
             ,
             if
             any
             other
             straight
             line
             be
             taken
             in
             the
             same
             moved
             Body
             in
             which
             the
             straight
             line
             AB
             was
             taken
             .
             So
             that
             all
             straight
             lines
             taken
             in
             a
             Body
             moved
             with
             Simple
             Ci●cular
             Motion
             will
             be
             carried
             parallel
             to
             themselves
             .
          
           
             
               Coroll
               .
               1
               It
               is
               manifest
               that
               the
               same
               will
               also
               happen
               in
               any
               Body
               which
               hath
               Simple
               Motion
               ,
               though
               not
               Circular
               .
               For
               all
               the
               points
               of
               any
               straight
               line
               whatsoever
               ,
               will
               describe
               lines
               though
               not
               Circular
               ,
               yet
               equall
               ;
               so
               that
               though
               the
               crooked
               lines
               AD
               and
               BE
               were
               not
               arches
               of
               Circles
               ,
               but
               of
               
                 Parabolas
                 ,
                 Ellipses
              
               ,
               or
               of
               any
               other
               figures
               ;
               yet
               both
               they
               ,
               and
               their
               Subtenses
               ,
               and
               the
               straight
               lines
               which
               joyne
               them
               ,
               would
               be
               equal
               and
               parallel
               .
            
             
               Coroll
               .
               2
               It
               is
               also
               manifest
               ,
               that
               the
               Radii
               of
               the
               equall
               circles
               AD
               and
               BE
               ,
               or
               the
               Axis
               of
               a
               Sphere
               ,
               will
               be
               so
               carried
               ,
               as
               to
               be
               allwayes
               parallel
               to
               the
               places
               in
               which
               they
               formerly
               were
               .
               For
               the
               straight
               line
               BF
               drawn
               to
               the
               center
               of
               the
               arch
               BE
               being
               equall
               to
               the
               Radius
               AC
               ,
               will
               also
               be
               equall
               to
               the
               straight
               line
               FE
               or
               CD
               ;
               and
               the
               angle
               BFE
               will
               be
               equall
               to
               the
               angle
               ACD
               .
               Now
               the
               intersection
               of
               the
               straight
               lines
               CA
               and
               BE
               ,
               being
               at
               G
               ,
               the
               angle
               CGE
               (
               seeing
               BE
               and
               AD
               are
               parallel
               )
               will
               be
               equal
               to
               the
               angle
               DAC
               .
               But
               the
               angle
               EBF
               is
               equal
               to
               the
               same
               angle
               DAC
               ;
               and
               therefore
               the
               angles
               CGE
               and
               EBF
               are
               also
               equal
               .
               Wherefore
               AC
               and
               BF
               are
               parallel
               ;
               which
               was
               to
               be
               demonstrated
               .
            
          
           
             2
             Let
             there
             be
             a
             Circle
             given
             (
             in
             the
             second
             figure
             )
             ,
             whose
             
             center
             is
             A
             ,
             and
             Radius
             AB
             ;
             and
             upon
             the
             center
             B
             and
             any
             Radius
             BC
             let
             the
             Epicycle
             CDE
             be
             described
             .
             Let
             the
             center
             B
             be
             understood
             to
             be
             carried
             about
             the
             center
             A
             ,
             and
             the
             whole
             Epicycle
             with
             it
             till
             it
             be
             coincident
             with
             the
             Circle
             FGH
             ,
             whose
             center
             is
             I
             ;
             and
             let
             BAI
             be
             any
             angle
             given
             .
             But
             in
             the
             time
             that
             the
             center
             B
             is
             moved
             to
             I
             ,
             let
             the
             Epicycle
             CDE
             have
             a
             contrary
             revolution
             upon
             its
             own
             center
             ,
             namely
             from
             E
             by
             D
             to
             C
             according
             to
             the
             same
             proportions
             ;
             that
             is
             ,
             in
             such
             manner
             ,
             that
             in
             both
             the
             Circles
             ,
             equal
             angles
             be
             made
             in
             equal
             times
             .
             I
             say
             EC
             the
             Axis
             of
             the
             Epicycle
             will
             be
             alwayes
             carried
             parallel
             to
             it self
             .
             Let
             the
             angle
             FIG
             be
             made
             equal
             to
             the
             angle
             BAI
             ;
             IF
             and
             AB
             will
             therefore
             be
             parallel
             ;
             and
             how
             much
             the
             Axis
             AG
             has
             departed
             from
             its
             former
             place
             AC
             (
             the
             measure
             of
             which
             progression
             is
             the
             angle
             CAG
             ,
             or
             CBD
             which
             I
             suppose
             equal
             to
             it
             )
             ,
             so
             much
             in
             the
             same
             time
             has
             the
             Axis
             IG
             (
             the
             same
             with
             BC
             )
             departed
             from
             its
             own
             former
             situation
             .
             Wherefore
             ,
             in
             what
             time
             BC
             comes
             to
             IG
             by
             the
             motion
             from
             B
             to
             I
             upon
             the
             center
             A
             ,
             in
             the
             same
             time
             G
             will
             come
             to
             F
             by
             the
             contrary
             motion
             of
             the
             Epicycle
             ;
             that
             is
             ,
             it
             will
             be
             turned
             backwards
             to
             F
             ,
             &
             IG
             will
             lie
             in
             IF
             .
             But
             the
             angles
             FIG
             and
             GAC
             are
             equal
             ;
             and
             therefore
             AC
             ,
             that
             is
             ,
             BC
             )
             ,
             and
             IG
             ,
             (
             that
             is
             the
             Axis
             ,
             though
             in
             different
             places
             )
             will
             be
             parallel
             .
             Wherefore
             ,
             the
             Axis
             of
             the
             Epicycle
             EDC
             will
             be
             carried
             alwayes
             parallel
             to
             it self
             ;
             which
             was
             to
             be
             proved
             .
          
           
             Coroll
             .
             From
             hence
             it
             is
             manifest
             ,
             that
             those
             two
             annual
             Motions
             which
             Copernicus
             ascribes
             to
             the
             Earth
             ,
             are
             reducible
             to
             this
             one
             Circular
             Simple
             Motion
             ,
             by
             which
             all
             the
             points
             of
             the
             moved
             Body
             are
             carried
             always
             with
             equal
             velocity
             ,
             that
             is
             ,
             in
             equal
             times
             they
             make
             equal
             revolutions
             uniformly
             .
          
           
             This
             ,
             as
             it
             is
             the
             most
             simple
             ,
             so
             it
             is
             the
             most
             frequent
             of
             all
             Circular
             Motions
             ;
             being
             the
             same
             which
             is
             used
             by
             all
             men
             when
             they
             turn
             any
             thing
             round
             with
             their
             arms
             ,
             as
             they
             do
             in
             grinding
             or
             sifting
             .
             For
             all
             the
             points
             of
             the
             thing
             moved
             ,
             describe
             lines
             which
             are
             like
             and
             equal
             to
             one
             another
             .
             So
             that
             if
             a
             man
             had
             a
             Ruler
             ,
             in
             which
             many
             Pens
             points
             of
             equal
             length
             were
             fastned
             ,
             he
             might
             with
             this
             one
             Motion
             write
             many
             lines
             at
             once
             .
          
           
             3
             Having
             shewed
             what
             Simple
             Motion
             is
             ,
             I
             will
             here
             also
             set
             down
             some
             properties
             of
             the
             same
             .
          
           
           
             First
             ,
             when
             a
             Body
             is
             moved
             with
             Simple
             Motion
             in
             a
             fluid
             Medium
             which
             hath
             no
             vacuity
             ,
             it
             changes
             the
             situation
             of
             all
             the
             parts
             of
             the
             fluid
             ambient
             which
             resist
             its
             motion
             ;
             I
             say
             there
             are
             no
             parts
             so
             small
             of
             the
             fluid
             ambient
             ,
             how
             farre
             soever
             it
             be
             continued
             ,
             but
             do
             change
             their
             situation
             ,
             in
             such
             manner
             ,
             as
             that
             they
             leave
             their
             places
             continually
             to
             other
             small
             parts
             that
             come
             into
             the
             same
             .
          
           
             For
             (
             in
             the
             same
             second
             figure
             )
             let
             any
             Body
             ,
             as
             KLMN
             ,
             be
             understood
             to
             be
             moved
             with
             Simple
             Circular
             Motion
             ;
             and
             let
             the
             Circle
             which
             every
             point
             thereof
             describes
             have
             any
             determined
             quantity
             ,
             suppose
             that
             of
             the
             same
             KLMN
             .
             Wherefore
             the
             Center
             A
             ,
             and
             every
             other
             point
             ,
             and
             consequently
             the
             moved
             Body
             it self
             ,
             will
             be
             carried
             sometimes
             towards
             the
             side
             where
             is
             K
             ,
             and
             sometimes
             towards
             the
             other
             side
             where
             is
             M.
             When
             therefore
             it
             is
             carried
             to
             K
             ,
             the
             parts
             of
             the
             fluid
             Medium
             on
             that
             side
             will
             go
             back
             ;
             and
             (
             supposing
             all
             space
             to
             be
             full
             )
             others
             on
             the
             other
             side
             will
             succeed
             .
             And
             so
             it
             will
             be
             when
             the
             Body
             is
             carried
             to
             the
             side
             M
             ,
             and
             to
             N
             ,
             and
             every
             way
             .
             Now
             when
             the
             neerest
             parts
             of
             the
             fluid
             Medium
             go
             back
             ,
             it
             is
             necessary
             that
             the
             parts
             next
             to
             those
             neerest
             parts
             go
             back
             also
             ;
             and
             (
             supposing
             still
             all
             space
             to
             be
             full
             )
             other
             parts
             will
             come
             into
             their
             places
             with
             succession
             perpetual
             and
             infinite
             .
             Wherefore
             all
             ,
             even
             the
             least
             parts
             of
             the
             fluid
             Medium
             change
             their
             places
             ,
             &c.
             which
             was
             to
             be
             proved
             .
          
           
             It
             is
             evident
             from
             hence
             ,
             that
             Simple
             Motion
             ,
             whether
             Circular
             ,
             or
             not
             Circular
             ,
             of
             Bodies
             which
             make
             perpetual
             returns
             to
             their
             former
             places
             ,
             hath
             greater
             or
             less
             force
             to
             dissipate
             the
             parts
             of
             resisting
             Bodies
             ,
             as
             it
             is
             more
             or
             less
             swift
             ,
             and
             as
             the
             lines
             described
             have
             greater
             or
             less
             magnitude
             .
             Now
             the
             greatest
             Velocity
             that
             can
             be
             ,
             may
             be
             understood
             to
             be
             in
             the
             least
             circuit
             ,
             and
             the
             least
             in
             the
             greatest
             ;
             and
             may
             be
             so
             supposed
             when
             there
             is
             need
             .
          
           
             4
             Secondly
             ,
             supposing
             the
             same
             Simple
             Motion
             in
             the
             Aire
             ,
             Water
             ,
             or
             other
             fluid
             Medium
             ;
             the
             parts
             of
             the
             Medium
             which
             adhere
             to
             the
             Moved
             Body
             will
             be
             carried
             about
             with
             the
             same
             Motion
             and
             Velocity
             ,
             so
             that
             in
             what
             time
             soever
             any
             point
             of
             the
             Movent
             finishes
             its
             Circle
             ,
             in
             the
             same
             time
             every
             part
             of
             the
             Medium
             which
             adheres
             to
             the
             Movent
             ,
             shall
             also
             describe
             
             such
             a
             part
             of
             its
             Circle
             ,
             as
             is
             equal
             to
             the
             whole
             Circle
             of
             the
             Movent
             ;
             I
             say
             it
             shall
             describe
             a
             part
             ,
             and
             not
             the
             whole
             Circle
             ,
             because
             all
             its
             parts
             receive
             their
             motion
             from
             an
             interiour
             concentrique
             Movent
             ,
             and
             of
             Concentrique
             Circles
             the
             exteriour
             are
             alwayes
             greater
             then
             the
             interiour
             ;
             nor
             can
             the
             motion
             imprinted
             by
             any
             Movent
             be
             of
             greater
             Velocity
             then
             that
             of
             the
             Movent
             it self
             .
             From
             whence
             it
             follows
             ,
             that
             the
             more
             remote
             parts
             of
             the
             fluid
             ambient
             ,
             shall
             finish
             their
             Circles
             in
             times
             which
             have
             to
             one
             another
             the
             same
             proportion
             with
             their
             distances
             from
             the
             Movent
             .
             For
             every
             point
             of
             the
             fluid
             ambient
             ,
             as
             long
             as
             it
             toucheth
             the
             Body
             which
             carries
             it
             about
             ,
             is
             carried
             about
             with
             it
             ,
             and
             would
             make
             the
             same
             Circle
             ,
             but
             that
             it
             is
             left
             behind
             so
             much
             as
             the
             exteriour
             Circle
             exceeds
             the
             interiour
             .
             So
             that
             if
             we
             suppose
             some
             thing
             which
             is
             not
             fluid
             to
             float
             in
             that
             part
             of
             the
             fluid
             ambient
             which
             is
             neerest
             to
             the
             Movent
             ,
             it
             will
             together
             with
             the
             Movent
             be
             carried
             about
             .
             Now
             that
             part
             of
             the
             fluid
             ambient
             which
             is
             not
             the
             neerest
             but
             almost
             the
             neerest
             ,
             receiving
             its
             degree
             of
             velocity
             from
             the
             neerest
             ,
             (
             which
             degree
             cannot
             be
             greater
             then
             it
             was
             in
             the
             giver
             )
             doth
             therefore
             in
             the
             same
             time
             make
             a
             Circular
             Line
             ,
             not
             a
             whole
             Circle
             ,
             yet
             equal
             to
             the
             whole
             Circle
             of
             the
             neerest
             .
             Therefore
             in
             the
             same
             time
             that
             the
             Movent
             describes
             its
             Circle
             ,
             that
             which
             doth
             not
             touch
             it
             shall
             not
             describe
             its
             Circle
             ;
             yet
             it
             shall
             describe
             such
             a
             part
             of
             it
             ,
             as
             is
             equal
             to
             the
             whole
             Circle
             of
             the
             Movent
             .
             And
             after
             the
             same
             manner
             ,
             the
             more
             remote
             parts
             of
             the
             ambient
             will
             describe
             in
             the
             same
             time
             such
             parts
             of
             their
             Circles
             as
             shall
             be
             severally
             equal
             to
             the
             whole
             Circle
             of
             the
             Movent
             ;
             and
             by
             consequent
             they
             shall
             finish
             their
             whole
             Circles
             in
             times
             proportional
             to
             their
             distances
             from
             the
             Movent
             ;
             which
             was
             to
             be
             proved
             .
          
           
             5
             Thirdly
             ,
             The
             same
             Simple
             Motion
             of
             a
             Body
             placed
             in
             a
             fluid
             Medium
             ,
             congregates
             ,
             or
             gathers
             into
             one
             place
             such
             things
             as
             naturally
             float
             in
             that
             Medium
             ,
             if
             they
             be
             Homogeneous
             ;
             and
             if
             they
             be
             Heterogeneous
             ,
             it
             separates
             and
             dissipates
             them
             .
             But
             if
             such
             things
             as
             be
             Heterogeneous
             do
             not
             float
             ,
             but
             settle
             ,
             then
             the
             same
             Motion
             stirs
             and
             mingles
             them
             disorderly
             together
             .
             For
             seeing
             Bodies
             which
             are
             unlike
             to
             one
             another
             ,
             that
             is
             ,
             Heterogeneous
             Bodies
             ,
             are
             not
             unlike
             in
             that
             they
             are
             Bodies
             (
             for
             Bodies
             ,
             as
             Bodies
             ,
             
             have
             no
             difference
             )
             but
             onely
             from
             some
             special
             Cause
             ,
             that
             is
             ,
             from
             some
             internal
             Motion
             ,
             or
             Motions
             of
             their
             smallest
             parts
             (
             for
             I
             have
             shewn
             in
             the
             9th
             Chapter
             and
             9th
             Article
             ,
             that
             all
             Mutation
             is
             such
             Motion
             )
             ,
             it
             remains
             that
             Heterogeneous
             Bodies
             have
             their
             unlikeness
             or
             difference
             from
             one
             another
             from
             their
             internal
             or
             specifical
             Motions
             .
             Now
             Bodies
             w
             ch
             have
             such
             difference
             ,
             receive
             unlike
             &
             different
             Motions
             from
             the
             same
             external
             common
             Movent
             ;
             and
             therefore
             they
             will
             not
             be
             moved
             together
             ,
             that
             is
             to
             say
             ,
             they
             will
             be
             dissipated
             .
             And
             being
             dissipated
             they
             will
             necessarily
             at
             some
             time
             or
             other
             meet
             with
             Bodies
             like
             themselves
             ,
             and
             be
             moved
             alike
             and
             together
             with
             them
             ;
             and
             afterwards
             meeting
             with
             more
             Bodies
             like
             themselves
             ,
             they
             will
             unite
             and
             become
             greater
             Bodies
             .
             Wherefore
             Homogeneous
             Bodies
             are
             congregated
             ,
             and
             Heterogenous
             dissipated
             by
             Simple
             Motion
             in
             a
             Medium
             where
             they
             naturally
             float
             .
             Again
             ,
             such
             as
             being
             in
             a
             fluid
             Medium
             ,
             do
             not
             float
             ,
             but
             sink
             ,
             if
             the
             Motion
             of
             the
             fluid
             Medium
             be
             strong
             enough
             ,
             will
             be
             stirred
             up
             and
             carried
             away
             by
             that
             Motion
             ,
             and
             consequently
             they
             will
             be
             hindred
             from
             returning
             to
             that
             place
             to
             which
             they
             sink
             naturally
             ,
             and
             in
             which
             onely
             they
             would
             unite
             ,
             and
             out
             of
             which
             they
             are
             promiscuously
             carried
             ;
             that
             is
             ,
             they
             are
             disorderly
             mingled
             .
          
           
             Now
             this
             Motion
             by
             which
             Homogeneous
             Bodies
             are
             congregated
             ,
             and
             Heterogeneous
             are
             scattered
             ,
             is
             that
             which
             is
             commonly
             called
             Fermentation
             ,
             from
             the
             Latine
             Fervere
             ;
             as
             the
             Greeks
             have
             their
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             (
             which
             signifies
             the
             same
             )
             from
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             Ferveo
             .
             For
             Seething
             makes
             all
             the
             parts
             of
             the
             Water
             change
             their
             places
             ;
             and
             the
             parts
             of
             any
             thing
             that
             is
             thrown
             into
             it
             ,
             will
             go
             several
             wayes
             according
             to
             their
             several
             natures
             .
             And
             yet
             all
             Fervor
             or
             Seething
             is
             not
             caused
             by
             Fire
             ;
             for
             New
             Wine
             and
             many
             other
             things
             have
             also
             their
             Fermentation
             and
             Fervor
             ,
             to
             which
             Fire
             contributes
             little
             ,
             and
             some
             times
             nothing
             .
             But
             when
             in
             Fermentation
             we
             find
             Heat
             ,
             it
             is
             made
             by
             the
             Fermentation
             .
          
           
             6
             Fourthly
             ,
             in
             what
             time
             soever
             the
             Movent
             whose
             Center
             is
             A
             (
             in
             the
             2d
             figure
             )
             moved
             in
             KLN
             shall
             by
             any
             number
             of
             revolutions
             (
             that
             is
             ,
             when
             the
             Perimeters
             BI
             and
             KLN
             be
             commensurable
             )
             have
             described
             a
             Line
             equal
             to
             the
             Circle
             which
             passes
             through
             the
             points
             B
             and
             I
             ;
             in
             the
             same
             time
             all
             the
             points
             of
             the
             
             floating
             Body
             whose
             Center
             is
             B
             ,
             shall
             return
             to
             have
             the
             same
             situation
             in
             respect
             of
             the
             Movent
             ,
             from
             which
             they
             departed
             .
             For
             seeing
             it
             is
             as
             the
             distance
             BA
             ,
             that
             is
             ,
             as
             the
             Radius
             of
             the
             Circle
             which
             passes
             through
             BI
             ,
             is
             to
             the
             Perimeter
             it self
             BI
             ,
             so
             the
             Radius
             of
             the
             Circle
             KLN
             is
             to
             the
             Perimeter
             KLN
             ;
             and
             seeing
             the
             velocities
             of
             the
             points
             B
             and
             K
             are
             equal
             ,
             the
             time
             also
             of
             the
             revolution
             in
             IB
             to
             the
             time
             of
             one
             revolution
             in
             KLN
             ,
             will
             be
             as
             the
             Perimeter
             BI
             to
             the
             Perimeter
             KLN
             ;
             and
             therefore
             so
             many
             revolutions
             in
             KLN
             as
             together
             taken
             are
             equal
             to
             the
             Perimeter
             BI
             ,
             will
             be
             finished
             in
             the
             same
             time
             in
             which
             the
             whole
             Perimeter
             BI
             is
             finished
             ;
             &
             therefore
             also
             the
             points
             L
             ,
             N
             ,
             F
             &
             H
             ,
             or
             any
             of
             the
             rest
             ,
             will
             in
             the
             same
             time
             return
             to
             the
             same
             situation
             from
             which
             they
             departed
             ;
             and
             this
             may
             be
             demonstrated
             whatsoever
             be
             the
             points
             considered
             .
             Wherefore
             all
             the
             points
             shall
             in
             that
             time
             return
             to
             the
             same
             situation
             ;
             which
             was
             to
             be
             proved
             .
          
           
             From
             hence
             it
             follows
             ,
             that
             if
             the
             Perimeters
             BI
             and
             LKN
             be
             not
             commensurable
             ,
             then
             all
             the
             points
             wil
             never
             return
             to
             have
             the
             same
             situation
             or
             configuration
             in
             respect
             of
             one
             another
             .
          
           
             7
             In
             Simple
             Motion
             ,
             if
             the
             Body
             moved
             be
             of
             a
             Spherical
             figure
             ,
             it
             hath
             less
             force
             towards
             its
             Poles
             then
             towards
             its
             middle
             ,
             to
             dissipate
             Heterogeneous
             ,
             or
             to
             congregate
             Homogeneous
             Bodies
             .
          
           
             Let
             there
             be
             a
             Sphere
             (
             as
             in
             the
             third
             figure
             )
             whose
             Center
             is
             A
             and
             Diameter
             BC
             ;
             &
             let
             it
             be
             conceived
             to
             be
             moved
             with
             Simple
             Circular
             Motion
             ;
             of
             which
             Motion
             let
             the
             Axis
             be
             the
             Straight
             Line
             DE
             ,
             cutting
             the
             Diameter
             BC
             at
             right
             angles
             in
             A.
             Let
             now
             the
             Circle
             which
             is
             described
             by
             any
             point
             B
             of
             the
             Sphere
             ,
             have
             BF
             for
             its
             Diameter
             ;
             and
             taking
             FG
             equal
             to
             BC
             ,
             and
             dividing
             it
             in
             the
             middle
             in
             H
             ,
             the
             Center
             of
             the
             Sphere
             A
             ,
             will
             when
             half
             a
             revolution
             is
             finished
             ,
             lie
             in
             H.
             And
             seeing
             HF
             and
             AB
             are
             equal
             ,
             a
             Circle
             described
             upon
             the
             Center
             H
             with
             the
             Radius
             HF
             or
             HG
             ,
             will
             be
             equal
             to
             the
             Circle
             whose
             Center
             is
             A
             and
             Radius
             AB
             .
             And
             if
             the
             same
             Motion
             be
             continued
             ,
             the
             point
             B
             will
             at
             the
             end
             of
             another
             half
             revolution
             return
             to
             the
             place
             from
             whence
             it
             began
             to
             be
             moved
             ;
             and
             therefore
             at
             the
             end
             of
             half
             a
             revolution
             ,
             the
             point
             B
             will
             be
             carried
             to
             F
             ,
             and
             the
             whole
             Hemisphere
             DBE
             into
             that
             Hemisphere
             in
             which
             are
             the
             points
             L
             ,
             K
             and
             F.
             Wherfore
             that
             part
             of
             the
             fluid
             Medium
             which
             is
             cōtiguous
             to
             the
             
             point
             F
             ,
             will
             in
             the
             same
             time
             go
             back
             the
             length
             of
             the
             Straight
             Line
             BF
             ;
             and
             in
             the
             return
             of
             the
             point
             F
             to
             B
             ,
             that
             is
             ,
             of
             G
             to
             C
             ,
             the
             fluid
             Medium
             wil
             go
             back
             as
             much
             in
             a
             Straight
             Line
             from
             the
             point
             C.
             And
             this
             is
             the
             effect
             of
             Simple
             Motion
             in
             the
             middle
             of
             the
             Sphere
             ,
             where
             the
             distance
             from
             the
             Poles
             is
             greatest
             .
             Let
             now
             the
             point
             I
             be
             taken
             in
             the
             same
             Sphere
             neerer
             to
             the
             Pole
             E
             ,
             and
             through
             it
             let
             the
             Straight
             Line
             IK
             be
             drawn
             parallel
             to
             the
             Straight
             Line
             BF
             ,
             cutting
             the
             arch
             FL
             in
             K
             ,
             &
             the
             Axis
             HL
             in
             M
             ;
             then
             connecting
             HK
             ,
             upon
             HF
             let
             the
             perpendicular
             KN
             be
             drawn
             .
             In
             the
             same
             time
             therefore
             that
             B
             comes
             to
             F
             ,
             the
             point
             I
             will
             come
             to
             K
             ,
             BF
             and
             IK
             being
             equal
             ,
             and
             described
             with
             the
             same
             velocity
             .
             Now
             the
             Motion
             in
             IK
             to
             the
             fluid
             Medium
             upon
             which
             it
             works
             ,
             namely
             to
             that
             part
             of
             the
             Medium
             which
             is
             contiguous
             to
             the
             point
             K
             ,
             is
             oblique
             ,
             whereas
             if
             it
             proceeded
             in
             the
             Straight
             Line
             HK
             ,
             it
             would
             be
             perpendicular
             ;
             and
             therefore
             the
             Motion
             which
             proceeds
             in
             IK
             has
             less
             power
             ,
             then
             that
             which
             proceeds
             in
             HK
             with
             the
             same
             velocity
             .
             But
             the
             Motions
             in
             HK
             and
             HF
             do
             equally
             thrust
             back
             the
             Medium
             ;
             and
             therefore
             the
             part
             of
             the
             Sphere
             at
             K
             ,
             moves
             the
             Medium
             less
             ,
             then
             the
             part
             at
             F
             ;
             namely
             so
             much
             less
             ,
             as
             KN
             is
             less
             then
             HF.
             Wherefore
             also
             the
             same
             Motion
             hath
             less
             power
             to
             disperse
             Heterogeneous
             ,
             and
             to
             congregate
             Homogeneous
             Bodies
             ,
             when
             it
             is
             neerer
             ,
             then
             when
             it
             is
             more
             remote
             from
             the
             Poles
             ;
             which
             was
             to
             be
             proved
             .
          
           
             Corollary
             .
             It
             is
             also
             necessary
             ,
             that
             in
             Plains
             which
             are
             perpendicular
             to
             the
             Axis
             ,
             and
             more
             remote
             then
             the
             Pole
             it self
             from
             the
             middle
             of
             the
             Sphere
             ,
             this
             Simple
             Motion
             have
             no
             effect
             .
             For
             the
             Axis
             DE
             with
             Simple
             Motion
             describes
             the
             Superficies
             of
             a
             Cylinder
             ;
             and
             towards
             the
             Bases
             of
             the
             Cylinder
             there
             is
             in
             this
             Motion
             no
             endeavour
             at
             all
             .
          
           
             8
             If
             in
             a
             fluid
             Medium
             ,
             moved
             about
             (
             as
             hath
             been
             said
             )
             with
             Simple
             Motion
             ,
             there
             be
             conceived
             to
             float
             some
             other
             Spherical
             Body
             which
             is
             not
             fluid
             ,
             the
             parts
             of
             the
             Medium
             which
             are
             stopped
             by
             that
             Body
             ,
             will
             endeavour
             to
             spread
             themselves
             every
             way
             upon
             the
             Superficies
             of
             it
             .
             And
             this
             is
             manifest
             enough
             by
             experience
             ,
             namely
             by
             the
             spreading
             of
             water
             poured
             out
             upon
             a
             pavement
             .
             But
             the
             reason
             of
             it
             may
             be
             this
             .
             Seeing
             the
             Sphere
             A
             (
             in
             the
             3d
             figure
             )
             is
             moved
             towards
             B
             ,
             the
             Medium
             also
             in
             which
             it
             is
             moved
             ,
             will
             have
             the
             same
             Motion
             .
             But
             because
             in
             this
             Motion
             
             it
             falls
             upon
             a
             Body
             not
             liquid
             ,
             as
             G
             ,
             so
             that
             it
             cannot
             go
             on
             ;
             and
             seeing
             the
             small
             parts
             of
             the
             Medium
             can
             not
             go
             forwards
             ,
             nor
             can
             they
             go
             directly
             backwards
             ;
             against
             the
             force
             of
             the
             Movent
             ;
             it
             remayns
             therefore
             that
             they
             diffuse
             themselves
             upon
             the
             Superficies
             of
             that
             Body
             ,
             as
             towards
             O
             and
             P
             ,
             Which
             was
             to
             be
             proved
             .
          
           
             9
             Compounded
             Circular
             Motion
             (
             in
             which
             all
             the
             parts
             of
             the
             moved
             Body
             do
             at
             once
             describe
             Circumferences
             ,
             some
             greater
             ,
             others
             less
             ,
             according
             to
             the
             proportion
             of
             their
             several
             distances
             from
             the
             common
             Center
             )
             carries
             about
             with
             it
             such
             Bodies
             ,
             as
             being
             not
             fluid
             ,
             adhere
             to
             the
             Body
             so
             moved
             ;
             and
             such
             as
             do
             not
             adhere
             ,
             it
             casteth
             forwards
             in
             a
             Straight
             Line
             which
             is
             a
             Tangent
             to
             the
             point
             from
             which
             they
             are
             cast
             off
             .
          
           
             For
             let
             there
             be
             a
             Circle
             whose
             Radius
             is
             AB
             (
             in
             the
             fourth
             figure
             )
             ;
             and
             let
             a
             Body
             be
             placed
             in
             the
             Circumference
             in
             B
             ,
             which
             if
             it
             be
             fixed
             there
             ,
             will
             necessarily
             be
             carried
             about
             with
             it
             ,
             as
             is
             manifest
             of
             it self
             .
             But
             whilest
             the
             motion
             proceeds
             ,
             let
             us
             suppose
             that
             Body
             to
             be
             unfixed
             in
             B.
             I
             say
             the
             Body
             wil
             cōtinue
             its
             motion
             in
             the
             Tangent
             BC.
             For
             let
             both
             the
             Radius
             AB
             ,
             and
             the
             Sphere
             B
             ,
             be
             conceived
             to
             consist
             of
             hard
             matter
             ;
             and
             let
             us
             suppose
             the
             Radius
             AB
             to
             be
             stricken
             in
             the
             point
             B
             by
             some
             other
             Body
             which
             falls
             upon
             it
             in
             the
             Tangent
             DB.
             Now
             therefore
             there
             will
             be
             a
             motion
             made
             by
             the
             concourse
             of
             two
             things
             ,
             the
             one
             ,
             Endeavour
             towards
             C
             in
             the
             Straight
             Line
             DB
             produced
             ,
             (
             in
             which
             the
             Body
             B
             would
             proceed
             ,
             if
             it
             were
             not
             retained
             by
             the
             Radius
             AB
             )
             ;
             the
             other
             ,
             the
             Retention
             it self
             .
             But
             the
             Retention
             alone
             causeth
             no
             endeavour
             towards
             the
             Center
             ;
             and
             therefore
             the
             Retention
             being
             taken
             away
             ,
             (
             which
             is
             done
             by
             the
             unfixing
             of
             B
             )
             there
             will
             remain
             but
             one
             Endeavour
             in
             B
             ,
             namely
             ,
             that
             in
             the
             Tangent
             BC.
             Wherefore
             the
             Motion
             of
             the
             Body
             B
             unfixed
             ,
             will
             proceed
             in
             the
             Tangent
             BC
             ;
             which
             was
             to
             be
             proved
             .
          
           
             By
             this
             demonstration
             it
             is
             manifest
             ,
             that
             Circular
             Motion
             about
             an
             unmoved
             Axis
             ,
             shakes
             off
             ,
             and
             puts
             further
             from
             the
             Center
             of
             its
             motion
             such
             things
             as
             touch
             ,
             but
             do
             not
             stick
             fast
             to
             its
             Superficies
             ;
             and
             the
             more
             ,
             by
             how
             much
             the
             distance
             is
             greater
             from
             the
             Poles
             of
             the
             Circular
             Motion
             ;
             and
             so
             much
             the
             more
             also
             ,
             by
             how
             much
             the
             things
             that
             are
             shaken
             off
             ,
             are
             less
             driven
             towards
             the
             Center
             by
             the
             fluid
             ambient
             ,
             for
             other
             Causes
             .
          
           
           
             10
             If
             in
             a
             fluid
             Medium
             a
             Spherical
             Body
             be
             moved
             with
             simple
             Circular
             Motion
             ;
             and
             in
             the
             same
             Medium
             there
             float
             another
             Sphere
             whose
             matter
             is
             not
             fluid
             ;
             this
             Sphere
             also
             shall
             be
             moved
             with
             simple
             Circular
             Motion
             .
          
           
             Let
             BCD
             (
             in
             the
             5th
             figure
             )
             be
             a
             Circle
             ,
             whose
             Center
             is
             A
             ,
             and
             in
             whose
             Circumference
             there
             is
             a
             Sphere
             so
             moved
             that
             it
             describes
             with
             Simple
             Motion
             the
             Perimeter
             BCD
             .
             Let
             also
             EFG
             be
             another
             Sphere
             of
             Consistent
             matter
             ,
             whose
             Semidiameter
             is
             EH
             ,
             and
             Center
             H
             ;
             and
             with
             the
             Radius
             AH
             let
             the
             Circle
             HI
             be
             described
             .
             I
             say
             the
             Sphere
             EFG
             will
             (
             by
             the
             Motion
             of
             the
             Body
             in
             BCD
             )
             be
             moved
             in
             the
             Circumference
             HI
             with
             Simple
             Motion
             .
          
           
             For
             seeing
             the
             Motion
             in
             BCD
             (
             by
             the
             4th
             Article
             of
             this
             Chapter
             )
             makes
             all
             the
             points
             of
             the
             fluid
             Medium
             describe
             in
             the
             same
             time
             Circular
             Lines
             equal
             to
             one
             another
             ,
             the
             points
             E
             ,
             H
             and
             G
             of
             the
             Straight
             Line
             EHG
             will
             in
             the
             same
             time
             describe
             with
             equal
             Radii
             equal
             Circles
             .
             Let
             EB
             be
             drawn
             equal
             and
             parallel
             to
             the
             Straight
             Line
             AH
             ;
             and
             let
             AB
             be
             connected
             ,
             which
             will
             therefore
             be
             equal
             and
             parallel
             to
             EH
             ;
             and
             therefore
             also
             ,
             if
             upon
             the
             Center
             B
             and
             Radius
             BE
             the
             arch
             EK
             be
             drawn
             equal
             to
             the
             arch
             HI
             ,
             and
             the
             straight
             Lines
             AI
             ,
             BK
             and
             IK
             be
             drawn
             ,
             BK
             and
             AI
             will
             be
             equal
             ;
             and
             they
             will
             also
             be
             parallel
             ,
             because
             the
             two
             arches
             EK
             and
             HI
             ,
             that
             is
             ,
             the
             two
             angles
             KBE
             and
             IAH
             are
             equal
             ;
             and
             consequently
             the
             Straight
             Lines
             AB
             and
             KI
             which
             connect
             them
             will
             also
             be
             equal
             and
             parallel
             .
             Wherefore
             KI
             and
             EH
             are
             parallel
             .
             Seeing
             therefore
             E
             and
             H
             are
             carried
             in
             the
             same
             time
             to
             K
             and
             I
             ,
             the
             whole
             Straight
             Line
             IK
             will
             be
             parallel
             to
             EH
             ,
             from
             whence
             it
             departed
             .
             And
             therefore
             ,
             (
             seeing
             the
             Sphere
             EFG
             is
             supposed
             to
             be
             of
             consistent
             matter
             ,
             so
             as
             all
             its
             points
             keep
             alwayes
             the
             same
             situation
             )
             it
             is
             necessary
             that
             every
             other
             Straight
             Line
             taken
             in
             the
             same
             Sphere
             ,
             be
             carried
             alwayes
             parallel
             to
             the
             places
             in
             which
             it
             formerly
             was
             .
             Wherefore
             the
             Sphere
             EFG
             is
             moved
             with
             simple
             Circular
             Motion
             ;
             which
             was
             to
             be
             demonstrated
             .
          
           
             11
             If
             in
             a
             fluid
             Medium
             ,
             whose
             parts
             are
             stirred
             by
             a
             Body
             moved
             with
             Simple
             Motion
             ,
             there
             float
             annother
             Body
             ,
             which
             hath
             its
             Superficies
             either
             wholly
             hard
             ,
             or
             wholly
             fluid
             ;
             the
             parts
             of
             
             this
             Body
             shall
             approach
             the
             Center
             equally
             on
             all
             sides
             ,
             that
             is
             to
             say
             ,
             the
             motion
             of
             the
             Body
             shall
             be
             Circular
             ,
             and
             Concentrique
             with
             the
             motion
             of
             the
             Movent
             .
             But
             if
             it
             have
             one
             side
             hard
             ,
             and
             the
             other
             side
             fluid
             ,
             then
             both
             those
             Motions
             shall
             not
             have
             the
             same
             center
             ,
             nor
             shall
             the
             floating
             Body
             be
             moved
             in
             the
             Circumference
             of
             a
             perfect
             Circle
             .
          
           
             Let
             a
             Body
             be
             moved
             in
             the
             Circumference
             of
             the
             Circle
             KL
             MN
             (
             in
             the
             2d
             figure
             )
             whose
             center
             is
             A.
             And
             let
             there
             be
             another
             Body
             at
             I
             ,
             whose
             Superficies
             is
             either
             all
             hard
             ,
             or
             all
             fluid
             .
             Also
             let
             the
             Medium
             in
             which
             both
             the
             Bodies
             are
             placed
             ,
             be
             fluid
             .
             I
             say
             the
             Body
             at
             I
             will
             be
             moved
             in
             the
             Circle
             IB
             about
             the
             Center
             A.
             For
             this
             has
             been
             demonstrated
             in
             the
             last
             Article
             .
          
           
             Wherefore
             let
             the
             Superficies
             of
             the
             Body
             at
             I
             ,
             be
             fluid
             on
             one
             side
             ,
             and
             hard
             on
             the
             other
             .
             And
             first
             ,
             let
             the
             fluid
             side
             be
             towards
             the
             Center
             .
             Seeing
             therefore
             the
             Motion
             of
             the
             Medium
             is
             such
             ,
             as
             that
             its
             parts
             do
             continually
             change
             their
             places
             ,
             (
             as
             hath
             been
             shewn
             in
             the
             5th
             Article
             )
             ;
             if
             this
             change
             of
             place
             be
             considered
             in
             those
             parts
             of
             the
             Medium
             which
             are
             contiguous
             to
             the
             fluid
             Superficies
             ,
             it
             must
             needs
             be
             ,
             that
             the
             small
             parts
             of
             that
             Superficies
             enter
             into
             the
             places
             of
             the
             small
             parts
             of
             the
             Medium
             which
             are
             contiguous
             to
             them
             ;
             And
             the
             like
             change
             of
             place
             will
             be
             made
             with
             the
             next
             contiguous
             parts
             towards
             A.
             And
             if
             the
             fluid
             parts
             of
             the
             Body
             at
             I
             ,
             have
             any
             degree
             at
             all
             of
             tenacity
             (
             for
             there
             are
             degrees
             of
             tenacity
             ,
             as
             in
             the
             Aire
             and
             Water
             )
             the
             whole
             fluid
             side
             will
             be
             lifted
             up
             a
             little
             ;
             but
             so
             much
             the
             less
             ,
             as
             its
             parts
             have
             less
             tenacity
             ;
             whereas
             the
             hard
             part
             of
             the
             Superficies
             which
             is
             contiguous
             to
             the
             fluid
             part
             ,
             has
             no
             cause
             at
             all
             of
             elevation
             ,
             that
             is
             to
             say
             ,
             no
             endeavour
             towards
             A.
             
          
           
             Secondly
             ,
             let
             the
             hard
             Superficies
             of
             the
             Body
             at
             I
             ,
             be
             towards
             A.
             By
             reason
             therefore
             of
             the
             said
             change
             of
             place
             of
             the
             parts
             which
             are
             contiguous
             to
             it
             ,
             the
             hard
             Superficies
             must
             of
             necessity
             (
             seeing
             by
             Supposition
             there
             is
             no
             empty
             Space
             )
             either
             come
             neerer
             to
             A
             ,
             or
             else
             its
             smallest
             parts
             must
             supply
             the
             contiguous
             places
             of
             the
             Medium
             ,
             which
             otherwise
             would
             be
             empty
             .
             But
             this
             cannot
             be
             by
             reason
             of
             the
             supposed
             hardness
             ;
             and
             therefore
             the
             other
             must
             needs
             be
             ,
             namely
             ,
             that
             the
             Body
             come
             neerer
             to
             A.
             Wherefore
             the
             Body
             at
             I
             ,
             has
             greater
             endeavour
             towards
             the
             center
             
             A
             ,
             when
             its
             hard
             side
             is
             next
             it
             ,
             then
             when
             it
             is
             averted
             from
             it
             .
             But
             the
             Body
             in
             I
             ,
             while
             it
             is
             moving
             in
             the
             circumference
             of
             the
             Circle
             IB
             ,
             has
             sometimes
             one
             side
             ,
             sometimes
             another
             turned
             towards
             the
             center
             ;
             and
             therefore
             it
             is
             sometimes
             neerer
             ,
             sometimes
             further
             off
             from
             the
             center
             A.
             Wherefore
             the
             Body
             at
             I
             ,
             is
             not
             carried
             in
             the
             circumference
             of
             a
             perfect
             Circle
             ;
             which
             was
             to
             be
             demonstrated
             .
          
        
         
           
           
           
           
             
               CHAP.
               XXII
            
             .
             Of
             other
             Variety
             of
             Motion
             .
          
           
             
               
                 1
                 Endeavour
                 and
                 Pressure
                 how
                 they
                 differ
                 .
              
               
                 2
                 Two
                 kinds
                 of
                 Mediums
                 in
                 which
                 Bodies
                 are
                 moved
                 .
              
               
                 3
                 Propagation
                 of
                 Motion
                 what
                 it
                 is
              
               
                 4
                 What
                 motion
                 Bodies
                 have
                 when
                 they
                 press
                 one
                 another
                 .
              
               
                 5
                 Fluid
                 Bodies
                 ,
                 when
                 they
                 are
                 pressed
                 together
                 ,
                 penetrate
                 one
                 another
                 .
              
               
                 6
                 When
                 one
                 Body
                 presseth
                 another
                 ,
                 and
                 doth
                 not
                 penetrate
                 it
                 ,
                 the
                 action
                 of
                 the
                 pressing
                 Body
                 is
                 perpendicular
                 to
                 the
                 Superficies
                 of
                 the
                 Body
                 pressed
                 .
              
               
                 7
                 When
                 a
                 hard
                 Body
                 ,
                 pressing
                 another
                 Body
                 ,
                 penetrates
                 the
                 same
                 ,
                 it
                 doth
                 not
                 penetrate
                 it
                 perpendicularly
                 ,
                 unless
                 it
                 fall
                 perpendicularly
                 upon
                 it
                 .
              
               
                 8
                 Motion
                 sometimes
                 opposite
                 to
                 that
                 of
                 the
                 Movent
                 .
              
               
                 9
                 In
                 a
                 full
                 Medium
                 ,
                 Motion
                 is
                 propagated
                 to
                 any
                 distance
                 .
              
               
                 10
                 Dilatation
                 and
                 Contraction
                 what
                 they
                 are
                 .
              
               
                 11
                 Dilatation
                 and
                 Contraction
                 suppose
                 Mutation
                 of
                 the
                 smallest
                 parts
                 in
                 respect
                 of
                 their
                 situation
                 .
              
               
                 12
                 All
                 Traction
                 is
                 Pulsion
                 .
              
               
                 13
                 Such
                 things
                 as
                 being
                 pressed
                 ,
                 or
                 bent
                 ,
                 restore
                 themselves
                 ,
                 have
                 motion
                 in
                 their
                 internal
                 parts
                 .
              
               
                 14
                 Though
                 that
                 which
                 carrieth
                 another
                 be
                 stopped
                 ,
                 the
                 Body
                 carried
                 will
                 proceed
                 .
              
               
                 15
                 ,
                 16
                 The
                 effects
                 of
                 Percussion
                 not
                 to
                 be
                 compared
                 with
                 those
                 of
                 Waight
                 .
              
               
                 17
                 ,
                 18
                 Motion
                 cannot
                 begin
                 first
                 in
                 the
                 internal
                 parts
                 of
                 a
                 Body
                 .
              
               
                 19
                 Action
                 and
                 Reaction
                 proceed
                 in
                 the
                 same
                 Line
                 .
              
               
                 20
                 Habit
                 what
                 it
                 is
                 .
              
            
          
           
             1
             I
             Have
             already
             (
             in
             the
             15th
             Chap.
             at
             the
             2d
             Article
             )
             defined
             Endeavour
             to
             be
             Motion
             through
             some
             Length
             ,
             though
             not
             considered
             as
             Length
             ,
             but
             as
             a
             Point
             .
             Whether
             therefore
             there
             be
             resistance
             or
             no
             resistance
             ,
             the
             Endeavour
             will
             still
             be
             the
             same
             .
             For
             simply
             to
             Endeavour
             ,
             is
             to
             Go.
             But
             when
             two
             Bodies
             having
             opposite
             Endeavours
             press
             one
             another
             ,
             then
             the
             Endeavour
             of
             either
             of
             them
             is
             that
             which
             we
             call
             Pressure
             ,
             and
             is
             mutual
             when
             their
             pressures
             are
             opposite
             .
          
           
             2.
             
             Bodies
             moved
             ,
             and
             also
             the
             Mediums
             in
             which
             they
             are
             moved
             ,
             are
             of
             two
             kinds
             .
             For
             either
             they
             have
             their
             parts
             coherent
             in
             such
             manner
             ,
             as
             no
             part
             of
             the
             Moved
             Body
             will
             easily
             yeild
             to
             the
             Mouent
             ,
             except
             the
             whole
             Body
             yeild
             also
             ,
             and
             such
             are
             the
             things
             
             we
             call
             Hard
             ;
             Or
             else
             their
             parts
             ,
             while
             the
             whole
             remains
             unmoved
             ,
             will
             easily
             yeild
             to
             the
             Movent
             ;
             and
             these
             we
             call
             Fluid
             or
             Soft
             Bodies
             .
             For
             the
             words
             
               Fluid
               ,
               Soft
               ,
               Tough
            
             and
             Hard
             (
             in
             the
             same
             manner
             as
             Great
             and
             Little
             )
             are
             used
             onely
             comparatively
             ;
             and
             are
             not
             different
             kinds
             ,
             but
             different
             degrees
             of
             Quality
             .
          
           
             3
             To
             Do
             ,
             and
             to
             Suffer
             is
             to
             Move
             and
             to
             be
             moved
             ;
             and
             nothing
             is
             moved
             ,
             but
             by
             that
             which
             toucheth
             it
             ,
             and
             is
             also
             moved
             ,
             (
             as
             has
             been
             formerly
             shewn
             )
             .
             And
             how
             great
             sover
             the
             distance
             be
             ,
             we
             say
             the
             first
             Movent
             moveth
             the
             last
             moved
             Body
             ;
             but
             mediately
             ;
             namely
             so
             ,
             as
             that
             the
             first
             moveth
             the
             second
             ,
             the
             second
             the
             third
             ,
             and
             so
             on
             ,
             till
             the
             last
             of
             all
             be
             touched
             .
             When
             therefore
             one
             Body
             having
             opposite
             Endeavour
             to
             another
             Body
             ,
             moveth
             the
             same
             ,
             and
             that
             moveth
             a
             third
             ,
             and
             so
             on
             ,
             I
             call
             that
             action
             
               Propagation
               of
               Motion
            
             .
          
           
             4
             When
             two
             fluid
             Bodies
             which
             are
             in
             a
             free
             and
             open
             Space
             ,
             press
             one
             another
             ,
             their
             parts
             will
             endeavour
             ,
             or
             be
             moved
             towards
             the
             sides
             ,
             not
             onely
             those
             parts
             which
             are
             there
             where
             the
             mutual
             contact
             is
             ,
             but
             all
             the
             other
             parts
             .
             For
             in
             the
             first
             contact
             ,
             the
             parts
             which
             are
             pressed
             by
             both
             the
             endeavouring
             Bodies
             ,
             have
             no
             place
             either
             forwards
             or
             backwards
             in
             which
             they
             can
             be
             moved
             ;
             and
             therefore
             they
             are
             pressed
             out
             towards
             the
             sides
             .
             And
             this
             expressure
             ,
             when
             the
             forces
             are
             equal
             ,
             is
             in
             a
             line
             perpendicular
             to
             the
             Bodies
             pressing
             .
             But
             whensoever
             the
             formost
             parts
             of
             both
             the
             Bodies
             are
             pressed
             ,
             the
             hindermost
             also
             must
             be
             pressed
             at
             the
             same
             time
             ;
             for
             the
             motion
             of
             the
             hindermost
             parts
             cannot
             in
             an
             instant
             be
             stopped
             by
             the
             resistance
             of
             the
             formost
             parts
             ,
             but
             proceeds
             for
             some
             time
             ;
             and
             therefore
             seeing
             they
             must
             have
             some
             place
             in
             which
             they
             may
             be
             moved
             ,
             and
             that
             there
             is
             no
             place
             at
             all
             for
             them
             forwards
             ,
             it
             is
             necessary
             that
             they
             be
             moved
             into
             the
             places
             which
             are
             towards
             the
             sides
             every
             way
             .
             And
             this
             effect
             followes
             of
             necessity
             ,
             not
             onely
             in
             Fluid
             ,
             but
             in
             Consistent
             and
             Hard
             Bodies
             ,
             though
             it
             be
             not
             alwayes
             manifest
             to
             sense
             .
             For
             though
             from
             the
             compression
             of
             two
             stones
             we
             cannot
             with
             our
             eyes
             discerne
             any
             swelling
             outwards
             towards
             the
             sides
             ,
             (
             as
             we
             perceive
             in
             two
             Bodies
             of
             wax
             ;
             )
             yet
             we
             know
             well
             enough
             by
             reason
             ,
             that
             some
             tumor
             must
             needs
             be
             there
             ,
             though
             it
             be
             but
             little
             .
          
           
           
             5
             But
             when
             the
             Space
             is
             enclosed
             ,
             and
             both
             the
             Bodies
             be
             fluid
             ,
             they
             will
             (
             if
             they
             be
             pressed
             together
             )
             penetrate
             one
             anoteer
             ,
             though
             differently
             according
             to
             their
             different
             endeavours
             .
             For
             suppose
             a
             hollow
             Cylinder
             of
             hard
             matter
             ,
             well
             stopped
             at
             both
             ends
             ,
             but
             filled
             first
             ,
             below
             with
             some
             heavy
             fluid
             Body
             ,
             as
             Quicksilver
             ;
             and
             above
             with
             Water
             or
             Aire
             .
             If
             now
             the
             bottome
             of
             the
             Cylinder
             be
             turned
             upwards
             ,
             the
             heaviest
             fluid
             Body
             which
             is
             now
             at
             the
             top
             ,
             having
             the
             greatest
             endeavour
             downwards
             ,
             and
             being
             by
             the
             hard
             sides
             of
             the
             vessel
             hindered
             from
             extending
             it selfe
             sidewayes
             ,
             must
             of
             necessity
             either
             be
             received
             by
             the
             lighter
             Body
             ,
             that
             it
             may
             sink
             through
             it
             ,
             or
             else
             it
             must
             open
             a
             passage
             through
             it selfe
             ,
             by
             which
             the
             lighter
             Body
             may
             ascend
             .
             For
             of
             the
             two
             Bodies
             ,
             that
             whose
             parts
             are
             most
             easily
             separated
             ,
             will
             the
             first
             be
             divided
             ;
             which
             being
             done
             ,
             it
             is
             not
             necessary
             that
             the
             parts
             of
             the
             other
             ,
             suffer
             any
             separation
             at
             all
             .
             And
             therefore
             when
             two
             Liquours
             which
             are
             enclosed
             in
             the
             same
             vessel
             ,
             change
             their
             places
             ,
             there
             is
             no
             need
             that
             their
             smallest
             parts
             should
             be
             mingled
             with
             one
             another
             ;
             for
             a
             way
             being
             opened
             through
             one
             of
             them
             ,
             the
             parts
             of
             the
             other
             need
             not
             be
             separated
             .
          
           
             Now
             if
             a
             fluid
             Body
             which
             is
             not
             enclosed
             press
             a
             hard
             Body
             ,
             its
             endeavour
             will
             indeed
             be
             towards
             the
             internal
             parts
             of
             that
             hard
             Body
             ;
             but
             (
             being
             excluded
             by
             the
             resistance
             of
             it
             )
             the
             parts
             of
             the
             fluid
             Body
             will
             be
             moved
             every
             way
             according
             to
             the
             Superficies
             of
             the
             hard
             Body
             ,
             and
             that
             equally
             ,
             if
             the
             pressure
             be
             perpendicular
             ;
             for
             when
             all
             the
             parts
             of
             the
             Cause
             are
             equal
             ,
             the
             Effects
             will
             be
             equal
             also
             .
             But
             if
             the
             pressure
             be
             not
             perpendicular
             ,
             then
             the
             angles
             of
             Incidence
             being
             unequal
             ,
             the
             expansion
             also
             will
             be
             unequal
             ,
             namely
             ,
             greater
             on
             that
             side
             where
             the
             angle
             is
             greater
             ,
             because
             that
             motion
             is
             most
             direct
             which
             proceeds
             by
             the
             directest
             Line
             .
          
           
             6
             If
             a
             Body
             ,
             pressing
             another
             Body
             do
             not
             penetrate
             it
             ,
             it
             will
             nevertheless
             give
             to
             the
             part
             it
             presseth
             ,
             an
             endeavour
             to
             yeild
             and
             recede
             in
             a
             straight
             line
             perpendicular
             to
             its
             Superficies
             in
             that
             point
             in
             which
             it
             is
             pressed
             .
          
           
             Let
             ABCD
             (
             in
             the
             first
             figure
             )
             be
             a
             hard
             Body
             ;
             and
             let
             another
             Body
             ,
             falling
             upon
             it
             in
             the
             straight
             line
             EA
             ,
             with
             any
             inclination
             ,
             or
             without
             inclination
             ,
             press
             it
             in
             the
             point
             A.
             I
             say
             the
             Body
             so
             
             pressing
             ,
             &
             not
             penetrating
             it
             ,
             will
             give
             to
             the
             part
             A
             an
             endeavour
             to
             yeild
             or
             recede
             in
             a
             straight
             Line
             perpendicular
             to
             the
             line
             AD.
             
          
           
             For
             let
             AB
             be
             perpendicular
             to
             AD
             ;
             and
             let
             BA
             be
             produced
             to
             F.
             If
             therefore
             AF
             be
             coincident
             with
             AE
             ,
             it
             is
             of
             it self
             manifest
             that
             the
             motion
             in
             EA
             will
             make
             A
             to
             endeavour
             in
             the
             line
             AB
             .
             Let
             now
             EA
             be
             oblique
             to
             AD
             ;
             and
             from
             the
             point
             E
             let
             the
             straight
             line
             EC
             be
             drawn
             ,
             cutting
             AD
             at
             right
             angles
             in
             D
             ;
             and
             let
             the
             rectangles
             ABCD
             and
             ADEF
             be
             completed
             .
             I
             have
             shewn
             (
             in
             the
             8th
             Article
             of
             the
             16th
             Chapter
             )
             that
             the
             Body
             will
             be
             carried
             from
             E
             to
             A
             by
             the
             concourse
             of
             two
             Uniform
             Motions
             ,
             the
             one
             in
             EF
             and
             its
             parallels
             ,
             the
             other
             in
             ED
             and
             its
             parallels
             .
             But
             the
             motion
             in
             EF
             and
             its
             parallels
             (
             whereof
             DA
             is
             one
             )
             contributes
             nothing
             to
             the
             Body
             in
             A
             ,
             to
             make
             it
             endeavour
             or
             press
             towards
             B
             ;
             and
             therefore
             the
             whole
             endeavour
             which
             the
             Body
             hath
             in
             the
             inclined
             line
             EA
             ,
             to
             pass
             ,
             or
             press
             the
             Straight
             line
             AD
             ,
             it
             hath
             it
             all
             from
             the
             perpendicular
             motion
             or
             endeavour
             in
             FA.
             Wherefore
             the
             Body
             E
             after
             it
             is
             in
             A
             ,
             will
             have
             onely
             that
             perpendicular
             endeavour
             which
             proceeds
             from
             the
             motion
             in
             FA
             ,
             that
             is
             ,
             in
             AB
             ;
             which
             was
             to
             be
             proved
             .
          
           
             7
             If
             a
             hard
             Body
             falling
             upon
             ,
             or
             pressing
             another
             Body
             ,
             penetrate
             the
             same
             ,
             its
             endeavour
             after
             its
             first
             penetration
             will
             be
             neither
             in
             the
             inclined
             line
             produced
             ,
             nor
             in
             the
             perpendicular
             ,
             but
             sometimes
             betwixt
             both
             ,
             sometimes
             without
             them
             .
          
           
             Let
             EAG
             (
             in
             the
             same
             ●
             figure
             )
             be
             the
             inclined
             line
             produced
             ;
             and
             First
             ,
             let
             the
             passage
             through
             the
             Medium
             in
             which
             EA
             is
             ,
             be
             easier
             then
             the
             passage
             through
             the
             Medium
             in
             which
             AG
             is
             .
             As
             soon
             therefore
             as
             the
             Body
             is
             within
             the
             Medium
             in
             which
             is
             AG
             ,
             it
             will
             finde
             greater
             resistance
             to
             its
             motion
             in
             DA
             and
             its
             parallels
             ,
             then
             it
             did
             whilest
             it
             was
             above
             AD
             ;
             and
             therefore
             below
             AD
             it
             will
             proceed
             with
             slower
             motion
             in
             the
             parallels
             of
             DA
             ,
             then
             above
             it
             .
             Wherefore
             the
             motion
             which
             is
             compounded
             of
             the
             two
             motions
             in
             EF
             and
             ED
             will
             be
             slower
             below
             AD
             ,
             then
             above
             it
             ;
             and
             therefore
             also
             ,
             the
             Body
             will
             not
             proceed
             from
             A
             in
             EA
             produced
             ,
             but
             below
             it
             .
             Seeing
             therefore
             the
             endeavour
             in
             AB
             is
             generated
             by
             the
             endeavour
             in
             FA
             ;
             if
             to
             the
             endeavour
             in
             FA
             there
             be
             added
             the
             endeavour
             in
             DA
             ,
             (
             which
             is
             not
             all
             taken
             away
             by
             the
             immersion
             of
             the
             point
             A
             into
             the
             lower
             Medium
             )
             the
             Body
             will
             not
             proceed
             from
             A
             in
             the
             perpendicular
             AB
             ,
             but
             beyond
             
             it
             ,
             namely
             ,
             in
             some
             straight
             line
             between
             AB
             and
             AG
             ,
             as
             in
             the
             line
             AH
             .
          
           
             Secondly
             ,
             let
             the
             passage
             through
             the
             Medium
             EA
             ,
             be
             less
             easie
             then
             that
             through
             AG.
             The
             motion
             therefore
             which
             is
             made
             by
             the
             concourse
             of
             the
             motions
             in
             EF
             and
             FB
             ,
             is
             slower
             above
             AD
             then
             below
             it
             ;
             and
             consequently
             ,
             the
             endeavour
             will
             not
             proceed
             from
             A
             in
             EA
             produced
             ,
             but
             beyond
             it
             ,
             as
             in
             AI.
             Wherefore
             ,
             If
             a
             hard
             Body
             falling
             ,
             which
             was
             to
             be
             proved
             .
          
           
             This
             Divergency
             of
             the
             Straight
             line
             AH
             from
             the
             straight
             line
             AG
             ,
             is
             that
             which
             the
             Writers
             of
             Opticks
             commonly
             call
             Refraction
             ;
             which
             ,
             when
             the
             passage
             is
             ea●ier
             in
             the
             first
             then
             in
             the
             second
             Medium
             ,
             is
             made
             by
             diverging
             from
             the
             line
             of
             Inclination
             towards
             the
             perpendicular
             ;
             and
             contrarily
             ,
             when
             the
             passage
             is
             not
             so
             easie
             in
             the
             first
             Medium
             ,
             by
             departing
             farther
             from
             the
             perpendicular
             .
          
           
             8
             By
             the
             6th
             Theoreme
             it
             is
             manifest
             ,
             that
             the
             force
             of
             the
             Movent
             may
             be
             so
             placed
             ,
             as
             that
             the
             Body
             moved
             by
             it
             ,
             may
             proceed
             in
             a
             way
             almost
             directly
             contrary
             to
             that
             of
             the
             Movent
             ;
             as
             we
             see
             in
             the
             motion
             of
             Ships
             .
          
           
             For
             let
             AB
             (
             in
             the
             2d
             figure
             )
             represent
             a
             Ship
             ,
             whose
             length
             from
             the
             prow
             to
             the
             poop
             is
             AB
             ;
             and
             let
             the
             winde
             lie
             upon
             it
             in
             the
             straight
             parallel
             lines
             CB
             ,
             DE
             and
             FG
             ;
             and
             let
             DE
             and
             FG
             be
             cut
             in
             E
             and
             G
             by
             a
             straight
             Line
             drawn
             from
             B
             perpendicular
             to
             AB
             ;
             also
             let
             BE
             and
             EG
             be
             equal
             ,
             and
             the
             angle
             ABC
             any
             angle
             how
             small
             soever
             .
             Then
             between
             BC
             and
             BA
             let
             the
             straight
             line
             BI
             be
             drawn
             ;
             and
             let
             the
             Sail
             be
             conceived
             to
             be
             spred
             in
             the
             same
             line
             BI
             ,
             and
             the
             winde
             to
             fall
             upon
             it
             in
             the
             points
             L
             ,
             M
             and
             B
             ;
             from
             which
             points
             ,
             perpendicular
             to
             BI
             ,
             let
             BK
             ,
             MQ
             and
             LP
             be
             drawn
             .
             Lastly
             ,
             let
             EN
             and
             GO
             be
             drawn
             perpendicular
             to
             BG
             ,
             and
             cutting
             BK
             in
             H
             and
             K
             ;
             and
             let
             HN
             and
             KO
             be
             made
             equal
             to
             one
             another
             ,
             and
             severally
             equal
             to
             BA
             .
             I
             say
             the
             Ship
             BA
             by
             the
             winde
             falling
             upon
             it
             in
             CB
             ,
             DE
             ,
             FG
             ,
             and
             other
             lines
             parallel
             to
             them
             ,
             will
             be
             carried
             forwards
             almost
             opposite
             to
             the
             winde
             ,
             that
             is
             to
             say
             ,
             in
             a
             way
             almost
             contrary
             to
             the
             way
             of
             the
             Movent
             .
          
           
             For
             the
             Winde
             that
             blowes
             in
             the
             Line
             CB
             ,
             will
             (
             as
             hath
             been
             shewn
             in
             the
             6th
             Article
             )
             give
             to
             the
             point
             B
             an
             endeavour
             to
             proceed
             in
             a
             straight
             line
             perpendicular
             to
             the
             straight
             line
             BI
             ,
             that
             
             is
             ,
             in
             the
             straight
             line
             BK
             ;
             and
             to
             the
             points
             M
             and
             L
             an
             endeavour
             to
             proceed
             in
             the
             straight
             lines
             MQ
             and
             LP
             ,
             which
             are
             parallel
             to
             BK
             .
             Let
             now
             the
             measure
             of
             the
             time
             be
             BG
             ,
             which
             is
             divided
             in
             the
             middle
             in
             E
             ;
             &
             let
             the
             point
             B
             be
             carried
             to
             H
             in
             the
             time
             BE.
             In
             the
             same
             time
             therefore
             by
             the
             wind
             blowing
             in
             DM
             &
             FL
             (
             and
             as
             many
             other
             lines
             as
             may
             be
             drawn
             parallel
             to
             them
             )
             the
             whole
             Ship
             will
             be
             applyed
             to
             the
             straight
             line
             HN.
             Also
             at
             the
             end
             of
             the
             second
             time
             EG
             ,
             it
             will
             be
             applyed
             to
             the
             straight
             line
             KO
             .
             Wherefore
             the
             Ship
             will
             always
             go
             forwards
             ;
             and
             the
             angle
             it
             makes
             with
             the
             winde
             will
             be
             equal
             to
             the
             angle
             ABC
             ,
             how
             small
             soever
             that
             angle
             be
             ;
             and
             the
             way
             it
             makes
             will
             in
             every
             time
             be
             equal
             to
             the
             straight
             line
             EH
             .
             I
             say
             thus
             it
             would
             be
             ,
             if
             the
             Ship
             might
             be
             moved
             with
             as
             great
             celerity
             sidewayes
             from
             BA
             towards
             KO
             ,
             as
             it
             may
             be
             moved
             forwards
             in
             the
             line
             BA
             .
             But
             this
             is
             impossible
             ,
             by
             reason
             of
             the
             resistance
             made
             by
             the
             great
             quantity
             of
             water
             which
             presseth
             the
             side
             ,
             much
             exceeding
             the
             resistance
             made
             by
             the
             much
             smaller
             quantity
             which
             presseth
             the
             prow
             of
             the
             Ship
             ;
             so
             that
             the
             way
             the
             Ship
             makes
             sidewayes
             is
             scarce
             sensible
             ;
             and
             therefore
             the
             point
             B
             will
             proceed
             almost
             in
             the
             very
             line
             BA
             ,
             making
             with
             the
             winde
             the
             angle
             ABC
             ,
             how
             acute
             soever
             ,
             that
             is
             to
             say
             ,
             it
             will
             proceed
             almost
             in
             the
             straight
             line
             BC
             ,
             that
             is
             ,
             in
             a
             way
             almost
             contrary
             to
             the
             way
             of
             the
             Movent
             ;
             which
             was
             to
             be
             demonstrated
             .
          
           
             But
             the
             Sayl
             in
             BI
             must
             be
             so
             stretched
             ,
             as
             that
             there
             be
             left
             in
             it
             no
             bo●ome
             at
             all
             ;
             for
             otherwise
             the
             straight
             lines
             LP
             ,
             MQ
             &
             BK
             will
             not
             be
             perpendicular
             to
             the
             plain
             of
             the
             Sayl
             ,
             but
             falling
             below
             P
             ,
             Q
             and
             K
             will
             drive
             the
             Ship
             backwards
             .
             But
             by
             making
             use
             of
             a
             small
             Board
             for
             a
             Sayl
             ,
             a
             little
             Waggon
             with
             wheels
             for
             the
             Ship
             ,
             and
             of
             a
             smooth
             Pavement
             for
             the
             Sea
             ,
             I
             have
             by
             experience
             found
             this
             to
             be
             so
             true
             ,
             that
             I
             could
             scarce
             oppose
             the
             board
             to
             the
             winde
             in
             any
             obliquity
             though
             never
             so
             small
             ,
             but
             the
             Waggon
             was
             carried
             forwards
             by
             it
             .
          
           
             By
             the
             same
             6th
             .
             Theoreme
             ,
             it
             may
             be
             found
             ,
             how
             much
             a
             stroke
             which
             falls
             obliquely
             ,
             is
             weaker
             then
             a
             stroke
             falling
             perpendicularly
             ,
             they
             being
             like
             and
             equal
             in
             all
             other
             respects
             .
          
           
             Let
             a
             stroke
             fall
             upon
             the
             Wall
             AB
             obliquely
             ,
             as
             (
             for
             example
             )
             in
             the
             straight
             line
             CA
             (
             in
             the
             3d
             figure
             )
             .
             Let
             CE
             be
             drawn
             parallel
             
             to
             AB
             ,
             &
             DA
             perpendicular
             to
             the
             same
             AB
             &
             equal
             to
             CA
             ;
             &
             let
             both
             the
             velocity
             &
             time
             of
             the
             motion
             in
             CA
             be
             equal
             to
             the
             velocity
             &
             time
             of
             the
             motion
             in
             DA.
             I
             say
             the
             stroke
             in
             CA
             will
             be
             weaker
             then
             that
             in
             DA
             in
             the
             proportion
             of
             EA
             to
             DA.
             For
             producing
             DA
             howsoever
             to
             F
             ,
             the
             endeavour
             of
             both
             the
             strokes
             will
             (
             by
             the
             6th
             Art.
             )
             proceed
             from
             A
             in
             the
             perpendicular
             AF.
             But
             the
             stroke
             in
             CA
             is
             made
             by
             the
             concourse
             of
             two
             motions
             in
             CE
             and
             EA
             ;
             of
             which
             that
             in
             CE
             contributes
             nothing
             to
             the
             stroke
             in
             A
             ,
             because
             CE
             and
             BA
             are
             parallels
             ;
             and
             therefore
             the
             stroke
             in
             CA
             is
             made
             by
             the
             motion
             which
             is
             in
             EA
             onely
             .
             But
             the
             velocity
             or
             force
             of
             the
             perpendicular
             stroke
             in
             EA
             ,
             to
             the
             velocity
             or
             force
             of
             the
             stroke
             in
             DA
             ,
             is
             as
             EA
             to
             DA.
             Wherefore
             the
             oblique
             stroke
             in
             CA
             is
             weaker
             then
             the
             perpendicular
             stroke
             in
             DA
             ,
             in
             the
             proportion
             of
             EA
             to
             DA
             or
             CA
             ;
             Which
             was
             to
             be
             proved
             .
          
           
             9
             In
             a
             full
             Medium
             ,
             all
             Endeavour
             proceeds
             as
             far
             as
             the
             Medium
             it self
             reacheth
             ;
             that
             is
             to
             say
             ,
             if
             the
             Medium
             be
             infinite
             ,
             the
             Endeavour
             will
             proceed
             infinitely
             .
          
           
             For
             whatsoever
             Endeavoureth
             ,
             is
             Moved
             ,
             and
             therefore
             whatsoever
             standeth
             in
             its
             way
             ,
             it
             maketh
             it
             yeild
             ,
             at
             least
             a
             little
             ,
             namely
             so
             far
             as
             the
             Movent
             it self
             is
             moved
             forwards
             .
             But
             that
             which
             yeildeth
             is
             also
             moved
             ,
             and
             consequently
             maketh
             that
             to
             yeild
             which
             is
             in
             its
             way
             ,
             and
             so
             on
             successively
             as
             long
             as
             the
             Medium
             is
             full
             ;
             that
             is
             to
             say
             ,
             infinitely
             ,
             if
             the
             full
             Medium
             be
             infinite
             ,
             which
             was
             to
             be
             proved
             .
          
           
             Now
             although
             Endeavour
             thus
             perpetually
             propagated
             ,
             do
             not
             alwayes
             appear
             to
             the
             Senses
             as
             Motion
             ;
             yet
             it
             appears
             as
             Action
             ,
             or
             as
             the
             efficient
             cause
             of
             some
             Mutation
             .
             For
             if
             there
             be
             placed
             before
             our
             Eyes
             some
             very
             little
             object
             ,
             as
             (
             for
             example
             )
             a
             small
             grain
             of
             sand
             ,
             which
             at
             a
             certain
             distance
             is
             visible
             ;
             it
             is
             manifest
             that
             it
             may
             be
             removed
             to
             such
             a
             distance
             as
             not
             to
             be
             any
             longer
             seen
             ,
             though
             by
             its
             action
             it
             still
             work
             upon
             the
             organs
             of
             sight
             ,
             as
             is
             manifest
             from
             that
             (
             which
             was
             last
             proved
             )
             that
             all
             Endeavour
             proceeds
             infinitely
             .
             Let
             it
             be
             conceived
             therefore
             to
             be
             removed
             from
             our
             Eyes
             to
             any
             distance
             how
             great
             soever
             ,
             and
             a
             sufficient
             number
             of
             other
             grains
             of
             sand
             of
             the
             same
             bigness
             added
             to
             it
             ;
             it
             is
             evident
             that
             the
             aggregate
             of
             all
             those
             sands
             will
             be
             visible
             ;
             and
             though
             none
             of
             them
             can
             be
             seen
             when
             it
             is
             single
             and
             severed
             
             from
             the
             rest
             ,
             yet
             the
             whole
             heap
             or
             hill
             which
             they
             make
             wil
             manifestly
             appear
             to
             the
             sight
             ;
             which
             would
             be
             impossible
             if
             some
             action
             did
             not
             proceed
             from
             each
             several
             part
             of
             the
             whole
             heap
             .
          
           
             10
             Between
             the
             degrees
             of
             Hard
             and
             Soft
             ,
             are
             those
             things
             which
             we
             call
             
               Tough
               ,
               Tough
            
             being
             that
             ,
             which
             may
             be
             bended
             without
             being
             altered
             from
             what
             it
             was
             ;
             and
             the
             Bending
             of
             a
             Line
             ,
             is
             either
             the
             adduction
             or
             diduction
             of
             the
             extreme
             parts
             ,
             that
             is
             ,
             a
             morion
             from
             Straightness
             to
             Crookedness
             ,
             or
             contrarily
             ,
             whilest
             the
             line
             remains
             still
             the
             same
             it
             was
             ;
             for
             by
             drawing
             out
             the
             extreme
             points
             of
             a
             line
             to
             their
             greatest
             distance
             ,
             the
             line
             is
             made
             straight
             ,
             which
             otherwise
             is
             Crooked
             .
             So
             also
             the
             Bending
             of
             a
             Superficies
             ,
             is
             the
             diduction
             or
             adduction
             of
             its
             extreme
             lines
             ,
             that
             is
             ,
             their
             Dilatation
             and
             Contraction
             .
          
           
             11
             Dilatation
             and
             Contraction
             ,
             as
             also
             all
             Flexion
             supposes
             necessarily
             that
             the
             internal
             parts
             of
             the
             Body
             bowed
             do
             either
             come
             neerer
             to
             the
             external
             parts
             ,
             or
             go
             further
             from
             them
             .
             For
             though
             Flexion
             be
             considered
             onely
             in
             the
             length
             of
             a
             Body
             ,
             yet
             when
             that
             Body
             is
             bowed
             ,
             the
             line
             which
             is
             made
             on
             one
             side
             will
             be
             convex
             ,
             and
             the
             line
             on
             the
             other
             side
             will
             be
             concave
             ;
             of
             which
             the
             concave
             being
             the
             interiour
             line
             ,
             will
             (
             unless
             something
             be
             taken
             from
             it
             and
             added
             to
             the
             convex
             line
             )
             be
             the
             more
             crooked
             ,
             that
             is
             ,
             the
             greater
             of
             the
             two
             .
             But
             they
             are
             equal
             ;
             and
             therefore
             in
             Flexion
             there
             is
             an
             accession
             made
             from
             the
             interiour
             to
             the
             exteriour
             parts
             ;
             and
             on
             the
             contrary
             ,
             in
             Tension
             ,
             from
             the
             exteriour
             to
             the
             interiour
             parts
             .
             And
             as
             for
             those
             things
             which
             do
             not
             easily
             suffer
             such
             transposition
             of
             their
             parts
             ,
             they
             are
             called
             Brittle
             ;
             and
             the
             great
             force
             they
             require
             to
             make
             them
             yield
             ,
             makes
             them
             also
             with
             sudden
             motion
             to
             leap
             asunder
             ,
             and
             break
             in
             pieces
             .
          
           
             12
             Also
             Motion
             is
             distinguished
             into
             Pulsion
             and
             Traction
             .
             And
             Pulsion
             ,
             as
             I
             have
             already
             defined
             it
             ,
             is
             when
             that
             which
             is
             moved
             ,
             goes
             before
             that
             which
             moveth
             it
             .
             But
             contrarily
             ,
             in
             Traction
             the
             Movent
             goes
             before
             that
             which
             is
             moved
             .
             Nevertheless
             ,
             considering
             it
             with
             greater
             attention
             ,
             it
             seemeth
             to
             be
             the
             same
             with
             Pulsion
             .
             For
             of
             two
             parts
             of
             a
             hard
             Body
             ,
             when
             that
             which
             is
             foremost
             drives
             before
             it
             the
             Medium
             in
             which
             the
             motion
             is
             made
             ,
             at
             the
             same
             time
             that
             which
             is
             thrust
             forwards
             ,
             thrusteth
             the
             next
             ,
             and
             this
             again
             the
             next
             ,
             and
             so
             on
             successively
             .
             In
             which
             action
             ,
             if
             we
             
             suppose
             that
             there
             is
             no
             place
             void
             ,
             it
             must
             needs
             be
             ,
             that
             by
             continual
             Pulsion
             ,
             namely
             ,
             when
             that
             action
             has
             gone
             round
             ,
             the
             Movent
             will
             be
             behind
             that
             part
             which
             at
             the
             first
             seemed
             not
             to
             be
             thrust
             forwards
             ,
             but
             to
             be
             drawn
             ;
             so
             that
             now
             the
             Body
             which
             was
             drawn
             ,
             goes
             before
             the
             Body
             which
             gives
             it
             motion
             ;
             and
             its
             motion
             is
             no
             longer
             Traction
             ,
             but
             Pulsion
             .
          
           
             13
             Such
             things
             as
             are
             removed
             from
             their
             places
             by
             forcible
             Compression
             or
             Extension
             ,
             and
             as
             soon
             as
             the
             force
             is
             taken
             away
             ,
             doe
             presently
             return
             and
             restore
             themselves
             to
             their
             former
             situation
             ,
             have
             the
             beginning
             of
             their
             restitution
             within
             themselves
             ,
             namely
             ,
             a
             certain
             motion
             in
             their
             internal
             parts
             ,
             which
             was
             there
             ,
             when
             ,
             before
             the
             taking
             away
             of
             the
             force
             ,
             they
             were
             compressed
             ,
             or
             extended
             .
             For
             that
             Restitution
             is
             motion
             ,
             and
             that
             which
             is
             at
             rest
             cannot
             be
             moved
             ,
             but
             by
             a
             moved
             and
             a
             Contiguous
             Movent
             .
             Nor
             doth
             the
             cause
             of
             their
             Restitution
             proceed
             from
             the
             taking
             away
             of
             the
             force
             by
             which
             they
             were
             compressed
             or
             extended
             ;
             for
             the
             removing
             of
             impediments
             hath
             not
             the
             efficacy
             of
             a
             cause
             (
             as
             has
             been
             shewn
             at
             the
             end
             of
             the
             3d
             Article
             of
             the
             15th
             Chapter
             )
             .
             The
             Cause
             therefore
             of
             their
             Restitution
             ,
             is
             some
             motion
             either
             of
             the
             parts
             of
             the
             Ambient
             ;
             or
             of
             the
             parts
             of
             the
             Body
             compressed
             or
             extended
             .
             But
             the
             parts
             of
             the
             Ambient
             have
             no
             endeavour
             which
             contributes
             to
             their
             Compression
             or
             Extension
             ,
             nor
             to
             the
             setting
             of
             them
             at
             liberty
             ,
             or
             Restitution
             .
             It
             remayns
             therefore
             that
             from
             the
             time
             of
             their
             Compression
             or
             Extension
             there
             be
             left
             some
             endeavour
             (
             or
             motion
             )
             by
             which
             ,
             the
             impediment
             being
             removed
             ,
             every
             part
             resumes
             its
             former
             place
             ;
             that
             is
             to
             say
             ,
             the
             whole
             Restores
             it self
             .
          
           
             14
             In
             the
             Carriage
             of
             Bodies
             if
             that
             Body
             which
             carries
             another
             ,
             hit
             upon
             any
             obstacle
             ,
             or
             be
             by
             any
             means
             suddenly
             stopped
             ,
             and
             that
             which
             is
             carried
             be
             not
             stopped
             ,
             it
             will
             go
             on
             ,
             till
             its
             motion
             be
             by
             some
             external
             impediment
             taken
             away
             .
          
           
             For
             I
             have
             demonstrated
             in
             the
             8th
             Chapter
             at
             the
             19th
             Article
             ,
             that
             Motion
             ,
             unless
             it
             be
             hindred
             by
             some
             external
             resistance
             ,
             will
             be
             continued
             eternally
             with
             the
             same
             celerity
             ;
             and
             in
             the
             7th
             Article
             of
             the
             9th
             Chap.
             that
             the
             action
             of
             an
             external
             Agent
             is
             of
             no
             effect
             without
             contact
             .
             When
             therefore
             that
             which
             carrieth
             another
             thing
             ,
             is
             stopped
             ,
             that
             stop
             doth
             not
             presently
             take
             away
             the
             motion
             of
             that
             which
             is
             carried
             .
             It
             will
             therefore
             proceed
             ,
             
             till
             its
             motion
             be
             by
             little
             and
             little
             extinguished
             by
             some
             external
             resistance
             ;
             Which
             was
             to
             be
             proved
             ;
             Though
             experience
             alone
             had
             been
             sufficient
             to
             prove
             this
             .
          
           
             In
             like
             manner
             ,
             if
             that
             Body
             which
             carrieth
             another
             be
             put
             from
             rest
             into
             sudden
             motion
             ;
             that
             which
             is
             carried
             will
             not
             be
             moved
             forwards
             together
             with
             it
             ,
             but
             will
             be
             left
             behind
             .
             For
             the
             contiguous
             part
             of
             the
             Body
             carried
             ,
             hath
             almost
             the
             same
             motion
             with
             the
             Body
             which
             carries
             it
             ;
             and
             the
             remote
             parts
             will
             receive
             different
             Velocities
             according
             to
             their
             different
             distances
             from
             the
             Body
             that
             carries
             them
             ;
             namely
             ,
             the
             more
             remote
             the
             parts
             are
             ,
             the
             less
             will
             be
             their
             degrees
             of
             Velocity
             .
             It
             is
             necessary
             therefore
             that
             the
             Body
             which
             is
             carried
             ,
             be
             left
             accordingly
             more
             or
             less
             behind
             .
             And
             this
             also
             is
             manifest
             by
             experience
             ,
             when
             at
             the
             starting
             forward
             of
             the
             Horse
             ,
             the
             Rider
             falleth
             backwards
             .
          
           
             15
             In
             Percussion
             therefore
             ,
             when
             one
             hard
             Body
             is
             in
             some
             small
             ●art
             of
             it
             stricken
             by
             another
             with
             great
             force
             ,
             it
             is
             not
             necessary
             that
             the
             whole
             Body
             should
             yeild
             to
             the
             stroke
             with
             the
             same
             celerity
             with
             which
             the
             stricken
             part
             yeilds
             .
             For
             the
             rest
             of
             the
             parts
             receive
             their
             motion
             from
             the
             motion
             of
             the
             part
             stricken
             and
             yeilding
             ,
             which
             motion
             is
             less
             propagated
             every
             way
             towards
             the
             sides
             then
             it
             is
             directly
             forwards
             .
             And
             hence
             it
             is
             ,
             that
             sometimes
             very
             hard
             Bodies
             ,
             which
             being
             erected
             can
             hardly
             be
             made
             to
             stand
             ,
             are
             more
             easily
             broken
             ,
             then
             thrown
             down
             by
             a
             violent
             stroke
             ;
             when
             nevertheless
             ,
             if
             all
             their
             parts
             together
             were
             by
             any
             weak
             motion
             thrust
             forwards
             they
             would
             easily
             be
             cast
             down
             .
          
           
             16
             Though
             the
             difference
             between
             Trusion
             and
             Percussion
             consist
             onely
             in
             this
             ,
             that
             in
             Trusion
             the
             motion
             both
             of
             the
             Movent
             and
             Moved
             Body
             begin
             both
             together
             in
             their
             very
             contact
             ;
             and
             in
             Percussion
             the
             striking
             Body
             is
             first
             moved
             ,
             and
             afterwards
             the
             Body
             stricken
             ;
             Yet
             their
             Effects
             are
             so
             different
             ,
             that
             it
             seems
             scarce
             possible
             to
             compare
             their
             forces
             with
             one
             another
             .
             I
             say
             ,
             any
             effect
             of
             Percussion
             being
             propounded
             ,
             as
             for
             example
             the
             stroke
             of
             a
             Beetle
             of
             any
             weight
             assigned
             ,
             by
             which
             a
             Pile
             of
             any
             given
             length
             ,
             is
             to
             be
             driven
             into
             earth
             of
             any
             tenacity
             given
             ,
             it
             seems
             to
             me
             very
             hard
             if
             not
             impossible
             to
             define
             ,
             with
             what
             weight
             ,
             or
             with
             what
             stroke
             ,
             and
             in
             what
             time
             ,
             the
             same
             pile
             may
             be
             driven
             〈◊〉
             a
             depth
             assigned
             into
             the
             same
             earth
             .
             The
             cause
             of
             which
             difficulty
             
             is
             this
             ,
             that
             the
             velocity
             of
             the
             Percutient
             is
             to
             be
             compared
             with
             the
             magnitude
             of
             the
             Ponderant
             .
             Now
             Velocity
             ,
             seeing
             it
             is
             computed
             by
             the
             length
             of
             space
             transmitted
             ,
             is
             to
             be
             accounted
             but
             as
             one
             Dimension
             ;
             but
             Waight
             ,
             is
             as
             a
             solid
             thing
             ,
             being
             measured
             by
             the
             dimension
             of
             the
             whole
             Body
             .
             And
             there
             is
             no
             comparison
             to
             be
             made
             of
             a
             Solid
             Body
             with
             a
             Length
             ,
             that
             is
             ,
             with
             a
             Line
             .
          
           
             17
             If
             the
             internal
             parts
             of
             a
             Body
             be
             at
             rest
             ,
             or
             retain
             the
             same
             situation
             with
             one
             another
             for
             any
             time
             how
             little
             soever
             ,
             there
             cannot
             in
             those
             parts
             be
             generated
             any
             new
             motion
             ,
             or
             endeavour
             ,
             whereof
             the
             efficient
             cause
             is
             not
             without
             the
             Body
             of
             which
             they
             are
             parts
             .
             For
             if
             any
             small
             part
             which
             is
             comprehended
             within
             the
             Superficies
             of
             the
             whole
             Body
             ,
             be
             supposed
             to
             be
             now
             at
             rest
             ,
             and
             by
             and
             by
             to
             be
             moved
             ,
             that
             part
             must
             of
             necessity
             receive
             its
             motion
             from
             some
             moved
             and
             contiguous
             Body
             .
             But
             (
             by
             supposition
             )
             there
             is
             no
             such
             moved
             and
             contiguous
             part
             within
             the
             Body
             .
             Wherefore
             ,
             if
             there
             be
             any
             Endeavour
             or
             Motion
             ,
             or
             change
             of
             situation
             ,
             in
             the
             internal
             parts
             of
             that
             Body
             ,
             it
             must
             needs
             arise
             from
             some
             efficient
             cause
             that
             is
             without
             the
             Body
             which
             contains
             them
             ;
             Which
             was
             to
             be
             proved
             .
          
           
             18
             In
             hard
             Bodies
             therefore
             which
             are
             compressed
             or
             extended
             ,
             if
             that
             which
             compresseth
             or
             extendeth
             them
             being
             taken
             away
             ,
             they
             restore
             themselves
             to
             their
             former
             place
             or
             situation
             ,
             it
             must
             needs
             be
             ,
             that
             that
             Endeavour
             (
             or
             Motion
             )
             of
             their
             internal
             parts
             ,
             by
             which
             they
             were
             able
             to
             recover
             their
             former
             places
             or
             situations
             ,
             was
             not
             extinguished
             when
             the
             force
             by
             which
             they
             were
             compressed
             or
             extended
             was
             taken
             away
             .
             Therefore
             when
             the
             Lath
             of
             a
             Cross-bow
             bent
             ,
             doth
             ,
             as
             soon
             as
             it
             is
             at
             liberty
             ,
             restore
             it self
             ,
             though
             to
             him
             that
             judges
             by
             Sense
             ,
             both
             it
             and
             all
             its
             parts
             seem
             to
             be
             at
             rest
             ;
             yet
             he
             that
             judging
             by
             Reason
             ,
             doth
             not
             account
             the
             taking
             away
             of
             impediment
             for
             an
             efficient
             cause
             ,
             nor
             conceives
             that
             without
             an
             efficient
             cause
             any
             thing
             can
             pass
             from
             Rest
             to
             Motion
             ,
             will
             conclude
             ,
             that
             the
             parts
             were
             already
             in
             motion
             before
             they
             began
             to
             restore
             themselves
             .
          
           
             19
             Action
             and
             Reaction
             proceed
             in
             the
             same
             Line
             ,
             but
             from
             opposite
             Terms
             .
             For
             seeing
             Reaction
             is
             nothing
             but
             Endeavour
             in
             the
             Patient
             to
             restore
             it self
             to
             that
             situation
             from
             which
             it
             was
             forced
             
             by
             the
             Agent
             ;
             the
             endeavour
             or
             motion
             both
             of
             the
             Agent
             and
             Patient
             (
             or
             Reagent
             )
             will
             be
             propagated
             between
             the
             same
             terms
             ,
             (
             yet
             so
             ,
             as
             that
             in
             Action
             the
             Term
             
               from
               which
            
             ,
             is
             in
             Reaction
             the
             Term
             
               to
               which
            
             )
             .
             And
             seeing
             all
             Action
             proceeds
             in
             this
             manner
             ,
             not
             onely
             between
             the
             opposite
             Terms
             of
             the
             whole
             line
             in
             which
             it
             is
             propagated
             ,
             but
             also
             in
             all
             the
             parts
             of
             that
             line
             ,
             the
             Terms
             
               from
               which
            
             and
             
               to
               which
            
             ,
             both
             of
             the
             Action
             and
             Reaction
             ,
             will
             be
             in
             the
             same
             line
             .
             Wherefore
             Action
             and
             Reaction
             proceed
             in
             the
             same
             line
             ,
             &c.
             
          
           
             20
             To
             what
             has
             been
             said
             of
             Motion
             ,
             I
             will
             add
             what
             I
             have
             to
             say
             concerning
             Habit.
             Habit
             therefore
             is
             a
             generation
             of
             Motion
             ,
             not
             of
             Motion
             simply
             ,
             but
             an
             easie
             conducting
             of
             the
             moved
             Body
             in
             a
             certain
             and
             designed
             way
             .
             And
             seeing
             it
             is
             attained
             by
             the
             weakning
             of
             such
             endeavours
             as
             divert
             its
             motion
             ,
             therefore
             such
             endeavours
             are
             to
             be
             weakned
             by
             little
             and
             little
             .
             But
             this
             cannot
             be
             done
             but
             by
             the
             long
             continuance
             of
             action
             ,
             or
             by
             actions
             often
             repeated
             ;
             and
             therefore
             Custome
             begets
             that
             Facicility
             ,
             which
             is
             commonly
             and
             rightly
             called
             Habit
             ;
             and
             it
             may
             be
             defined
             thus
             ;
             HABIT
             
               is
               Motion
               made
               more
               easie
               and
               ready
               by
               Custome
               ;
               that
               is
               to
               say
               ,
               by
               perpetual
               endeavour
               ,
               or
               by
               iterated
               endevours
               in
               a
               way
               differing
               from
               that
               in
               which
               the
               Motion
               proceeded
               from
               the
               beginning
               ,
               and
               opposing
               such
               endeavours
               as
               resist
               .
            
             And
             to
             make
             this
             more
             perspicuous
             by
             example
             ,
             We
             may
             observe
             ,
             that
             when
             one
             that
             has
             no
             skill
             in
             Musique
             ,
             first
             ,
             puts
             his
             hand
             to
             an
             Instrument
             ,
             he
             cannot
             after
             the
             first
             stroke
             carry
             to
             his
             hand
             to
             the
             place
             where
             he
             would
             make
             the
             second
             stroke
             ,
             without
             taking
             it
             back
             by
             a
             new
             endeavour
             ,
             and
             as
             it
             were
             beginning
             again
             ,
             pass
             from
             the
             first
             to
             the
             second
             .
             Nor
             will
             he
             be
             able
             to
             go
             on
             to
             the
             third
             place
             without
             another
             new
             endeavour
             ,
             but
             he
             will
             be
             forced
             to
             draw
             back
             his
             hand
             again
             ,
             and
             so
             successively
             ,
             by
             renewing
             his
             endeavour
             at
             every
             stroke
             ,
             till
             at
             the
             last
             by
             doing
             this
             often
             ,
             and
             by
             compounding
             many
             interrupted
             motions
             or
             endeavours
             into
             one
             equal
             endeavour
             ,
             he
             be
             able
             to
             make
             his
             hand
             go
             readily
             on
             from
             stroke
             to
             stroke
             in
             that
             order
             and
             way
             which
             was
             at
             the
             first
             designed
             .
             Nor
             are
             Habits
             to
             be
             observed
             in
             living
             creatures
             only
             ,
             but
             also
             in
             Bodies
             inanimate
             .
             For
             we
             find
             ,
             that
             when
             the
             Lath
             of
             a
             Crossbow
             is
             strongly
             bent
             ,
             and
             would
             if
             the
             impediment
             
             were
             removed
             return
             again
             with
             great
             force
             ,
             if
             it
             remain
             a
             long
             time
             bent
             ,
             it
             will
             get
             such
             a
             Habit
             ,
             that
             when
             it
             is
             loosed
             and
             left
             to
             its
             own
             freedome
             ,
             it
             will
             not
             onely
             not
             restore
             it self
             ,
             but
             will
             require
             as
             much
             force
             for
             the
             bringing
             of
             it
             back
             to
             its
             first
             posture
             ,
             as
             it
             did
             for
             the
             bending
             of
             it
             at
             the
             first
             .
          
        
         
           
           
           
             
               CHAP.
               XXIII
            
             .
             Of
             the
             Center
             of
             Equiponderation
             of
             Bodies
             pressing
             do●●ards
             in
             straight
             Parallel
             Lines
             .
          
           
             
               
                 1
                 Definitions
                 and
                 Suppositions
                 .
              
               
                 2
                 Two
                 Plains
                 of
                 Equiponderation
                 are
                 n●●
                 parallel
                 .
              
               
                 3
                 The
                 Center
                 of
                 Equiponderation
                 is
                 in
                 every
                 Plain
                 of
                 Equiponderation
                 .
              
               
                 4
                 The
                 Moments
                 of
                 equal
                 Ponderants
                 are
                 to
                 one
                 another
                 as
                 their
                 distances
                 from
                 the
                 center
                 of
                 the
                 Scale
                 .
              
               
                 5,6
                 .
                 The
                 Moments
                 of
                 unequal
                 Ponderants
                 have
                 their
                 proportion
                 to
                 one
                 another
                 compounded
                 of
                 the
                 proportions
                 of
                 their
                 Waights
                 and
                 distances
                 from
                 the
                 center
                 of
                 the
                 Scale
                 reciprocally
                 taken
                 .
              
               
                 7.
                 
                 If
                 two
                 Ponderants
                 have
                 their
                 Moments
                 and
                 Distances
                 from
                 the
                 Center
                 of
                 the
                 Scale
                 in
                 reciprocal
                 proportion
                 ;
                 they
                 are
                 equally
                 poised
                 ;
                 and
                 contrarily
                 .
              
               
                 8
                 If
                 the
                 parts
                 of
                 any
                 Ponderant
                 press
                 the
                 Beam
                 of
                 the
                 Scale
                 every
                 where
                 equally
                 ,
                 all
                 the
                 parts
                 cut
                 out
                 off
                 reckoned
                 from
                 the
                 Center
                 of
                 the
                 Scale
                 ▪
                 will
                 have
                 their
                 Moments
                 in
                 the
                 same
                 proportion
                 with
                 that
                 of
                 the
                 parts
                 of
                 a
                 Triangle
                 cut
                 off
                 from
                 the
                 Vertex
                 by
                 straight
                 Lines
                 parallel
                 to
                 the
                 base
                 .
              
               
                 9
                 The
                 Diameter
                 of
                 Equiponderation
                 of
                 Figures
                 which
                 are
                 deficient
                 ▪
                 according
                 to
                 commensurable
                 proportions
                 of
                 their
                 altitudes
                 and
                 bases
                 ,
                 divides
                 the
                 Axis
                 ,
                 so
                 ,
                 that
                 the
                 part
                 taken
                 next
                 the
                 vertex
                 is
                 to
                 the
                 other
                 part
                 as
                 the
                 complete
                 figure
                 to
                 the
                 deficient
                 figure
                 .
              
               
                 10
                 The
                 diameter
                 of
                 Equiponderation
                 of
                 the
                 Complement
                 of
                 the
                 half
                 of
                 any
                 of
                 the
                 said
                 deficient
                 figures
                 ,
                 divides
                 that
                 line
                 which
                 is
                 drawn
                 ▪
                 through
                 the
                 vertex
                 parallel
                 to
                 the
                 base
                 ,
                 so
                 ,
                 that
                 the
                 part
                 next
                 the
                 vertex
                 is
                 to
                 the
                 other
                 part
                 as
                 the
                 complete
                 figure
                 to
                 the
                 Complement
                 .
              
               
                 11
                 The
                 Center
                 of
                 Equiponderation
                 of
                 the
                 half
                 of
                 any
                 of
                 the
                 desicient
                 figures
                 in
                 the
                 first
                 row
                 of
                 the
                 Table
                 of
                 the
                 3d.
                 Article
                 of
                 the
                 17th
                 Chapter
                 ,
                 may
                 be
                 found
                 out
                 by
                 the
                 numbers
                 of
                 the
                 second
                 row
                 .
              
               
                 12
                 The
                 center
                 of
                 Equiponderation
                 of
                 the
                 half
                 of
                 any
                 of
                 the
                 figures
                 in
                 the
                 second
                 row
                 of
                 the
                 same
                 Table
                 ,
                 may
                 be
                 found
                 out
                 by
                 the
                 numbers
                 of
                 the
                 fourth
                 row
                 .
              
               
                 13
                 The
                 Center
                 of
                 Equiponderation
                 of
                 the
                 half
                 of
                 any
                 of
                 the
                 figures
                 in
                 the
                 same
                 Table
                 ,
                 being
                 known
                 ,
                 the
                 Center
                 of
                 the
                 Excess
                 of
                 the
                 same
                 figure
                 above
                 a
                 Triangle
                 of
                 the
                 same
                 altitude
                 and
                 base
                 is
                 also
                 known
                 .
              
               
                 14
                 The
                 Center
                 of
                 Equiponderation
                 of
                 a
                 solid
                 Sector
                 ,
                 is
                 in
                 
                 the
                 Axis
                 ,
                 so
                 divided
                 ,
                 that
                 the
                 part
                 next
                 the
                 Vertex
                 be
                 to
                 the
                 whole
                 Axis
                 want
                 half
                 the
                 Axis
                 of
                 the
                 portion
                 of
                 the
                 Sphere
                 ,
                 as
                 3
                 to
                 4.
                 
              
            
          
           
             1
          
           
             
               Definitions
               .
            
             
               1
               A
               Scale
               ,
               is
               a
               straight
               line
               ,
               whose
               middle
               point
               is
               immoveable
               ,
               all
               the
               rest
               of
               its
               points
               being
               at
               liberty
               ;
               and
               that
               part
               of
               the
               Scale
               which
               reaches
               from
               the
               center
               to
               either
               of
               the
               waights
               ,
               is
               called
               the
               Beam.
               
            
             
               2
               Equiponderation
               is
               ,
               when
               the
               endeavour
               of
               one
               Body
               which
               presses
               one
               of
               the
               Beams
               ,
               resists
               the
               endeavour
               of
               another
               Body
               pressing
               the
               other
               Beam
               ,
               so
               ,
               that
               neither
               of
               them
               is
               moved
               ;
               and
               the
               Bodies
               when
               neither
               of
               them
               is
               moved
               ,
               are
               said
               to
               be
               
                 Equally
                 poised
              
               .
            
             
               3
               Waight
               ,
               is
               the
               aggregate
               of
               all
               the
               Endeavours
               ,
               by
               which
               all
               the
               points
               of
               that
               Body
               which
               presses
               the
               Beam
               ,
               tend
               downwards
               in
               lines
               parallel
               to
               one
               another
               ;
               and
               the
               Body
               which
               presses
               ,
               is
               called
               the
               Ponderant
               .
            
             
               4
               Moment
               ,
               is
               the
               Power
               which
               the
               Ponderant
               has
               to
               move
               the
               Beam
               ,
               by
               reason
               of
               a
               determined
               situation
               .
            
             
               5
               
                 The
                 plain
                 of
                 Equiponderation
              
               ,
               is
               that
               ,
               by
               which
               the
               Ponderant
               is
               so
               divided
               ,
               that
               the
               Moments
               on
               both
               sides
               remain
               equal
               .
            
             
               6
               
                 The
                 Diameter
                 of
                 Equiponderation
              
               ,
               is
               the
               common
               Section
               of
               the
               two
               Plains
               of
               Equiponderation
               ;
               and
               is
               in
               the
               straight
               line
               by
               which
               the
               waight
               is
               hanged
               .
            
             
               7
               
                 The
                 Center
                 of
                 Equiponderation
              
               ,
               is
               the
               common
               point
               of
               the
               two
               Diameters
               of
               Equiponderation
               .
            
          
           
             
               Suppositions
            
             
               1
               When
               two
               Bodies
               are
               equally
               pois'd
               ,
               if
               waight
               be
               added
               to
               one
               of
               them
               ,
               and
               not
               to
               the
               other
               ,
               their
               Equiponderation
               ceases
               .
            
             
               2
               When
               two
               Ponderants
               of
               equal
               magnitude
               ,
               and
               of
               the
               same
               Species
               or
               matter
               ,
               press
               the
               Beam
               on
               both
               sides
               at
               equal
               distances
               from
               the
               center
               of
               the
               Scale
               ,
               their
               Moments
               are
               equal
               .
               Also
               when
               two
               Bodies
               endeavour
               at
               equal
               distances
               from
               the
               center
               of
               the
               Scale
               ,
               if
               they
               be
               of
               equal
               magnitude
               and
               of
               the
               same
               Species
               ,
               their
               Moments
               are
               equal
               .
            
          
           
           
             2
             No
             two
             Plains
             of
             Equiponderation
             are
             parallel
             .
          
           
             Let
             A
             B
             C
             D
             (
             in
             the
             first
             figure
             )
             be
             any
             Ponderant
             whatsoever
             ;
             and
             in
             it
             let
             E
             F
             be
             a
             Plain
             of
             Equiponderation
             ;
             parallel
             to
             which
             ,
             let
             any
             other
             Plain
             be
             drawn
             ,
             as
             G
             H.
             I
             say
             G
             H
             is
             not
             a
             Plain
             of
             Equiponderation
             .
             For
             seeing
             the
             parts
             A
             E
             F
             D
             and
             E
             B
             C
             F
             of
             the
             Ponderant
             A
             B
             C
             D
             ,
             are
             equally
             pois'd
             ;
             and
             the
             weight
             E
             G
             H
             F
             is
             added
             to
             the
             part
             A
             E
             F
             D
             ,
             and
             nothing
             is
             added
             to
             the
             part
             E
             B
             C
             F
             ,
             but
             the
             weight
             E
             G
             H
             F
             is
             taken
             from
             it
             ;
             therefore
             (
             by
             the
             first
             Supposition
             )
             the
             parts
             A
             G
             H
             D
             and
             G
             B
             C
             H
             will
             not
             be
             equally
             pois'd
             ;
             and
             consequently
             G
             H
             is
             not
             a
             Plain
             of
             Equiponderation
             .
             Wherefore
             ,
             No
             two
             Plains
             of
             Equiponderation
             ,
             &c.
             Which
             was
             to
             be
             proved
             .
          
           
             3
             The
             Center
             of
             Equiponderation
             is
             in
             every
             Plain
             of
             Equiponderation
             .
          
           
             For
             if
             another
             Plain
             of
             Equiponderation
             be
             taken
             ,
             it
             will
             not
             (
             by
             the
             last
             Article
             )
             be
             parallel
             to
             the
             former
             Plain
             ;
             and
             therefore
             both
             those
             Plains
             will
             cut
             one
             another
             .
             Now
             that
             Section
             (
             by
             the
             6th
             Definition
             )
             is
             the
             Diameter
             of
             Equiponderation
             .
             Again
             ,
             if
             another
             Diameter
             of
             Equiponderation
             be
             taken
             ,
             it
             will
             cut
             that
             former
             Diameter
             ;
             and
             in
             that
             Section
             (
             by
             the
             7th
             Definition
             )
             is
             the
             Center
             of
             Equiponderation
             .
             Wherefore
             the
             Center
             of
             Equiponderation
             is
             in
             that
             Diameter
             which
             lies
             in
             the
             said
             Plain
             of
             Equiponderation
             .
          
           
             4
             The
             Moment
             of
             any
             Ponderant
             applyed
             to
             one
             point
             of
             the
             Beam
             ,
             to
             the
             Moment
             of
             the
             same
             ,
             or
             an
             equal
             Ponderant
             applyed
             to
             any
             other
             point
             of
             the
             Beam
             ,
             is
             as
             the
             distance
             of
             the
             former
             point
             from
             the
             Center
             of
             the
             Scale
             ,
             to
             the
             distance
             of
             the
             later
             point
             from
             the
             same
             Center
             .
             Or
             thus
             ,
             Those
             Moments
             are
             to
             one
             another
             ,
             as
             the
             Arches
             of
             Circles
             which
             are
             made
             upon
             the
             Center
             of
             the
             Scale
             through
             those
             points
             ,
             in
             the
             same
             time
             .
             Or
             lastly
             thus
             ;
             They
             are
             ,
             as
             the
             parallel
             bases
             of
             two
             Triangles
             ,
             which
             have
             a
             common
             angle
             at
             the
             Center
             of
             the
             Scale
             .
          
           
             Let
             A
             (
             in
             the
             2d
             figure
             )
             be
             the
             Center
             of
             the
             Scale
             ;
             and
             let
             the
             equal
             Poderants
             D
             and
             E
             press
             the
             Beam
             A
             B
             in
             the
             points
             B
             and
             C
             ;
             also
             let
             the
             straight
             lines
             B
             D
             and
             C
             E
             be
             Diameters
             of
             Equiponderation
             ;
             and
             the
             points
             D
             and
             E
             in
             the
             Ponderants
             D
             and
             E
             be
             their
             Centers
             of
             Equiponderation
             .
             Let
             A
             G
             F
             be
             drawn
             howsoever
             ,
             
             cutting
             D
             B
             produced
             in
             F
             ,
             and
             E
             C
             in
             G
             ;
             and
             lastly
             ,
             upon
             the
             common
             Center
             A
             ,
             let
             the
             two
             arches
             B
             H
             and
             C
             I
             be
             described
             ,
             cutting
             A
             G
             F
             in
             H
             and
             I.
             I
             say
             the
             Moment
             of
             the
             Ponderant
             D
             to
             the
             Moment
             of
             the
             Ponderant
             E
             ,
             is
             as
             A
             B
             to
             A
             C
             ,
             or
             as
             B
             H
             to
             C
             I
             ,
             or
             as
             B
             F
             to
             C
             G.
             For
             the
             effect
             of
             the
             Ponderant
             D
             in
             the
             point
             B
             ,
             is
             circular
             motion
             in
             the
             arch
             B
             H
             ;
             and
             the
             effect
             of
             the
             Ponderant
             E
             in
             the
             point
             C
             ,
             circular
             motion
             in
             the
             arch
             C
             I
             ;
             and
             by
             reason
             of
             the
             equality
             of
             the
             Ponderants
             D
             and
             E
             ,
             these
             motions
             are
             to
             one
             another
             as
             the
             Quicknesses
             or
             Velocities
             with
             which
             the
             points
             B
             and
             C
             describe
             the
             arches
             B
             H
             and
             C
             I
             ,
             that
             is
             ,
             as
             the
             arches
             themselves
             B
             H
             and
             C
             I
             ,
             or
             as
             the
             straight
             parallels
             B
             F
             and
             C
             G
             ,
             or
             as
             the
             parts
             of
             the
             Beam
             A
             B
             and
             A
             C
             ;
             for
             A
             B.
             A
             C
             :
             :
             B
             F.
             C
             G
             :
             :
             B
             H.
             C
             I.
             are
             proportionals
             ;
             and
             therefore
             the
             effects
             ,
             that
             is
             ,
             (
             by
             the
             4th
             Definition
             )
             the
             Moments
             of
             the
             equal
             Ponderants
             applyed
             to
             several
             points
             of
             the
             Beam
             ,
             are
             to
             one
             another
             ,
             as
             A
             B
             and
             A
             C
             ;
             or
             as
             the
             distances
             of
             those
             points
             from
             the
             center
             of
             the
             Scale
             ;
             or
             as
             the
             parallel
             bases
             of
             the
             Triangles
             which
             have
             a
             common
             angle
             at
             A
             ;
             or
             as
             the
             concentrick
             arches
             B
             H
             and
             C
             I
             ;
             which
             was
             to
             be
             demonstrated
             .
          
           
             5
             Unequal
             Ponderants
             ,
             when
             they
             are
             applyed
             to
             several
             points
             of
             the
             Beam
             ,
             and
             hang
             at
             liberty
             (
             that
             is
             ,
             so
             as
             the
             line
             by
             which
             they
             hang
             be
             the
             Diameter
             of
             Equiponderation
             ,
             whatsoever
             be
             the
             figure
             of
             the
             Ponderant
             )
             ,
             have
             their
             Moments
             to
             one
             another
             in
             proportion
             compounded
             of
             the
             proportions
             of
             their
             distances
             from
             the
             center
             of
             the
             Scale
             ,
             and
             of
             their
             Waights
             .
          
           
             Let
             A
             (
             in
             the
             3d
             figure
             )
             be
             the
             center
             of
             the
             Scale
             ,
             and
             A
             B
             the
             Beam
             ;
             to
             which
             let
             the
             two
             Ponderants
             C
             &
             D
             be
             applied
             at
             the
             points
             B
             and
             E.
             I
             say
             the
             proportion
             of
             the
             Moment
             of
             the
             Ponderant
             C
             ,
             to
             the
             Moment
             of
             the
             Ponderant
             D
             ,
             is
             compounded
             of
             the
             proportions
             of
             A
             B
             to
             A
             E
             and
             of
             the
             Waight
             C
             to
             the
             Waight
             D
             ;
             or
             (
             if
             C
             and
             D
             be
             of
             the
             same
             species
             )
             of
             the
             magnitude
             C
             to
             the
             magnitude
             D.
             
          
           
             Let
             either
             of
             them
             ,
             as
             C
             ,
             be
             supposed
             to
             be
             bigger
             then
             the
             other
             D.
             If
             therefore
             by
             the
             addition
             of
             F
             ,
             F
             and
             D
             together
             be
             as
             one
             Body
             equal
             to
             C
             ,
             the
             Moment
             of
             C
             to
             the
             Moment
             of
             F
             +
             D
             will
             be
             (
             by
             the
             last
             article
             )
             as
             B
             G
             is
             to
             E
             H.
             Now
             as
             F
             +
             D
             is
             to
             D
             ,
             so
             let
             E
             H
             be
             to
             another
             E
             I
             ;
             and
             the
             moment
             of
             F
             +
             D
             ,
             that
             is
             
             of
             C
             ,
             to
             the
             moment
             of
             D
             ,
             will
             be
             as
             B
             G
             to
             E
             I.
             But
             the
             proportion
             of
             B
             G
             to
             E
             I
             is
             compounded
             of
             the
             proportions
             (
             of
             B
             G
             to
             E
             H
             that
             is
             )
             of
             A
             B
             to
             A
             E
             ,
             and
             (
             of
             E
             H
             to
             E
             I
             ,
             that
             is
             )
             of
             the
             waight
             C
             to
             the
             waight
             D.
             Wherefore
             unequal
             Ponderants
             ,
             when
             they
             are
             applied
             ,
             &c.
             which
             was
             to
             be
             proved
             .
          
           
             6
             The
             same
             figure
             remaining
             ,
             if
             I
             K
             be
             drawn
             parallel
             to
             the
             Beame
             A
             B
             ,
             and
             cutting
             A
             G
             in
             K
             ;
             and
             K
             L
             be
             drawn
             parallel
             to
             to
             B
             G
             ,
             cutting
             A
             B
             in
             L
             ,
             the
             distances
             A
             B
             and
             A
             L
             from
             the
             center
             ,
             will
             be
             proportional
             to
             the
             moments
             of
             C
             and
             D.
             For
             the
             moment
             of
             C
             is
             B
             G
             ,
             and
             the
             moment
             of
             D
             is
             E
             I
             ,
             to
             which
             K
             L
             is
             equal
             .
             But
             as
             the
             distance
             A
             B
             from
             the
             center
             ,
             is
             to
             the
             distance
             A
             L
             from
             the
             center
             ,
             so
             is
             B
             G
             the
             moment
             of
             the
             Ponderant
             C
             ,
             to
             L
             K
             ,
             or
             E
             I
             the
             moment
             of
             the
             Ponderant
             D.
             
          
           
             7
             If
             two
             Ponderants
             have
             their
             waights
             and
             distances
             from
             the
             center
             in
             reciprocal
             proportion
             ,
             and
             the
             center
             of
             the
             Scale
             be
             between
             the
             points
             to
             which
             the
             Ponderants
             are
             applied
             ,
             they
             will
             be
             equally
             poised
             .
             And
             contrarily
             ,
             if
             they
             be
             equally
             poised
             ,
             their
             waights
             and
             distances
             from
             the
             center
             of
             the
             Scale
             will
             be
             in
             reciprocall
             proportion
             .
          
           
             Let
             the
             center
             of
             the
             Scale
             (
             in
             the
             same
             3d
             figure
             )
             be
             A
             ,
             the
             Beam
             A
             B
             ;
             and
             let
             any
             Ponderant
             C
             ,
             having
             B
             G
             for
             its
             moment
             ,
             be
             applied
             to
             the
             point
             B
             ;
             also
             let
             any
             other
             Ponderant
             D
             ,
             whose
             moment
             is
             E
             I
             ,
             be
             applied
             to
             the
             point
             E.
             Through
             the
             point
             I
             ,
             let
             I
             K
             be
             drawn
             parallel
             to
             the
             Beam
             A
             B
             ,
             cutting
             A
             G
             in
             K
             ;
             also
             let
             K
             L
             be
             drawn
             parallel
             to
             B
             G.
             K
             L
             will
             then
             be
             the
             Moment
             of
             the
             Ponderant
             D
             ;
             and
             (
             by
             the
             last
             Article
             )
             it
             will
             be
             as
             B
             G
             the
             Moment
             of
             the
             Ponderant
             C
             in
             the
             point
             B
             ,
             to
             L
             K
             the
             Moment
             of
             the
             Ponderant
             D
             in
             the
             point
             E
             ▪
             so
             A
             B
             to
             A
             L.
             On
             the
             other
             side
             of
             the
             center
             of
             the
             Scale
             ,
             let
             A
             N
             be
             taken
             equal
             to
             A
             L
             ;
             and
             to
             the
             point
             N
             let
             there
             be
             applyed
             the
             Ponderant
             O
             ,
             having
             to
             the
             Ponderant
             C
             the
             proportion
             of
             A
             B
             to
             A
             N.
             I
             say
             the
             Ponderants
             in
             B
             and
             N
             will
             be
             equally
             poised
             .
             For
             the
             proportion
             of
             the
             Moment
             of
             the
             Ponderant
             O
             in
             the
             point
             N
             ,
             to
             the
             Moment
             of
             the
             Ponderant
             C
             in
             the
             point
             B
             ,
             is
             (
             by
             the
             5th
             Article
             )
             cōpounded
             of
             the
             proportions
             of
             the
             waight
             O
             to
             the
             waight
             B
             ,
             &
             of
             the
             distance
             (
             from
             the
             center
             of
             the
             Scale
             )
             A
             N
             or
             A
             L
             to
             the
             distāce
             (
             frō
             the
             center
             of
             the
             Scale
             )
             A
             B.
             But
             seeing
             we
             have
             supposed
             ,
             that
             the
             distance
             A
             B
             to
             the
             distance
             A
             N
             ,
             is
             in
             reciprocal
             proportion
             
             of
             the
             Waight
             O
             to
             the
             waight
             C
             ,
             the
             proportion
             of
             the
             Moment
             of
             the
             Ponderant
             O
             in
             the
             point
             N
             ,
             to
             the
             Moment
             of
             the
             Ponderant
             C
             in
             the
             point
             B
             ,
             will
             be
             compounded
             of
             the
             proportions
             of
             A
             B
             to
             A
             N
             ,
             and
             of
             A
             N
             to
             A
             B.
             Wherefore
             ,
             setting
             in
             order
             A
             B
             ,
             A
             N
             ,
             A
             B
             ,
             the
             Moment
             of
             O
             to
             the
             Moment
             of
             C
             will
             be
             as
             the
             first
             to
             the
             last
             ,
             that
             is
             ,
             as
             A
             B
             to
             A
             B.
             Their
             Moments
             therefore
             are
             equal
             ;
             and
             consequently
             the
             Plain
             which
             passes
             through
             A
             ,
             will
             (
             by
             the
             fifth
             Definition
             )
             be
             a
             Plain
             of
             Equiponderation
             .
             Wherefore
             they
             will
             be
             equally
             poised
             ;
             as
             was
             to
             be
             proved
             .
          
           
             Now
             the
             converse
             of
             this
             is
             manifest
             .
             For
             if
             there
             be
             Equiponderation
             ,
             and
             the
             proportion
             of
             the
             Waights
             and
             Distances
             be
             not
             reciprocal
             ,
             then
             both
             the
             Waights
             will
             alwayes
             have
             the
             same
             Moments
             ,
             although
             one
             of
             them
             have
             more
             waight
             added
             to
             it
             ,
             or
             its
             distance
             changed
             .
          
           
             Corollary
             .
             When
             Ponderants
             are
             of
             the
             same
             Species
             ,
             and
             their
             Moments
             be
             equal
             ;
             their
             Magnitudes
             and
             Distances
             from
             the
             center
             of
             the
             Scale
             will
             be
             reciprocally
             proportional
             .
             For
             in
             Homogeneous
             Bodies
             ,
             it
             is
             as
             Waight
             to
             Waight
             ,
             so
             Magnitude
             to
             Magaltude
             .
          
           
             8
             If
             to
             the
             whole
             length
             of
             the
             Beam
             there
             be
             applyed
             a
             Parallelogram
             ,
             or
             a
             Parallelopipedum
             ,
             or
             a
             Prisma
             ,
             or
             a
             Cylinder
             ,
             or
             the
             Superficies
             of
             a
             Cylinder
             ,
             ot
             of
             a
             Sphere
             ,
             or
             of
             any
             portion
             of
             a
             Sphere
             or
             Prisma
             ;
             the
             parts
             of
             any
             of
             them
             cut
             off
             with
             plains
             parallel
             to
             the
             base
             ,
             will
             have
             their
             Moments
             in
             the
             same
             proportion
             with
             the
             parts
             of
             a
             Triangle
             which
             has
             its
             Vertex
             in
             the
             center
             of
             the
             Scale
             ,
             and
             for
             one
             of
             its
             sides
             the
             Beam
             it self
             ,
             which
             parts
             are
             cut
             off
             by
             Plains
             parallel
             to
             the
             base
             .
          
           
             First
             ,
             let
             the
             rectangled
             Parallelogram
             A
             B
             C
             D
             (
             in
             the
             4th
             figure
             )
             be
             applyed
             to
             the
             whole
             length
             of
             the
             Beam
             A
             B
             ;
             and
             producing
             C
             B
             howsoever
             to
             E
             ,
             let
             the
             Triangle
             A
             B
             E
             be
             described
             .
             Let
             now
             any
             part
             of
             the
             Parallelogram
             ,
             as
             A
             F
             ,
             be
             cut
             off
             by
             the
             plain
             F
             G
             ,
             parallel
             to
             the
             base
             C
             B
             ;
             and
             let
             F
             G
             be
             produced
             to
             A
             E
             in
             the
             point
             H.
             I
             say
             the
             Moment
             of
             the
             whole
             A
             B
             C
             D
             to
             the
             Moment
             of
             its
             part
             A
             F
             ,
             is
             as
             the
             Triangle
             A
             B
             E
             to
             the
             Triangle
             A
             G
             H
             ,
             that
             is
             ,
             in
             proportion
             duplicate
             to
             that
             of
             the
             distances
             from
             the
             center
             of
             the
             Scale
             .
          
           
             For
             ,
             the
             Parallelogram
             A
             B
             C
             D
             being
             divided
             into
             equal
             parts
             
             infinite
             in
             number
             ,
             by
             straight
             lines
             drawn
             parallel
             to
             the
             base
             ;
             and
             supposing
             the
             Moment
             of
             the
             straight
             line
             C
             B
             to
             be
             B
             E
             ;
             the
             Moment
             of
             the
             straight
             line
             F
             G
             ,
             will
             (
             by
             the
             7th
             Article
             )
             be
             G
             H
             ;
             and
             the
             Moments
             of
             all
             the
             straight
             lines
             of
             that
             Parallelogram
             ,
             will
             be
             so
             many
             straight
             lines
             in
             the
             Triangle
             A
             B
             E
             drawn
             parallel
             to
             the
             base
             B
             E
             ;
             all
             which
             parallels
             together
             taken
             are
             the
             Moment
             of
             the
             whole
             Parallelogram
             A
             B
             C
             D
             ;
             and
             the
             same
             parallels
             do
             also
             constitute
             the
             superficies
             of
             the
             Triangle
             A
             B
             E.
             Wherefore
             the
             Moment
             of
             the
             Parallelogram
             A
             B
             C
             D
             ,
             is
             the
             Triangle
             A
             B
             E
             ;
             and
             for
             the
             same
             reason
             ,
             the
             Moment
             of
             the
             Parallelogram
             A
             F
             ,
             is
             the
             Triangle
             A
             G
             H
             ;
             and
             therefore
             the
             Moment
             of
             the
             whole
             Parallelogram
             ,
             to
             the
             Moment
             of
             a
             Parallelogram
             which
             is
             part
             of
             the
             same
             ,
             is
             as
             the
             Triangle
             A
             B
             E
             ,
             to
             the
             Triangle
             A
             G
             H
             ,
             or
             in
             proportion
             duplicate
             to
             that
             of
             the
             Beams
             to
             which
             they
             are
             applyed
             .
             And
             what
             is
             here
             demonstrated
             in
             the
             case
             of
             a
             Parallelogram
             ,
             may
             be
             understood
             to
             serve
             for
             that
             of
             a
             Cylinder
             ,
             and
             of
             a
             Prisma
             ,
             and
             their
             Superficies
             ;
             as
             also
             for
             the
             Superficies
             of
             a
             Sphere
             ,
             of
             an
             Hemisphere
             ,
             or
             any
             portion
             of
             a
             Sphere
             ,
             (
             for
             the
             parts
             of
             the
             Superficies
             of
             a
             Sphere
             ,
             have
             the
             same
             proportion
             with
             that
             of
             the
             parts
             of
             the
             Axis
             cut
             off
             by
             the
             same
             parallels
             by
             which
             the
             parts
             of
             the
             Superficies
             are
             cut
             off
             ,
             as
             Archimedes
             has
             demonstrated
             )
             ;
             and
             therefore
             when
             the
             parts
             of
             any
             of
             these
             figures
             are
             equal
             and
             at
             equal
             distances
             from
             the
             Center
             of
             the
             Scale
             ,
             their
             Moments
             also
             are
             equal
             ,
             in
             the
             same
             manner
             as
             they
             are
             in
             Parallelograms
             .
          
           
             Secondly
             ,
             let
             the
             Parallelogram
             A
             K
             I
             B
             not
             be
             rectangled
             ;
             the
             straight
             line
             I
             B
             wil
             nevertheless
             press
             the
             point
             B
             perpendicularly
             in
             the
             straight
             line
             B
             E
             ;
             &
             the
             straight
             line
             L
             G
             wil
             press
             the
             point
             G
             perpendicularly
             in
             the
             straight
             line
             G
             H
             ;
             and
             all
             the
             rest
             of
             the
             straight
             lines
             which
             are
             parallel
             to
             I
             B
             will
             do
             the
             like
             .
             Whatsoever
             therefore
             the
             Moment
             be
             which
             is
             assigned
             to
             the
             straight
             line
             I
             B
             ,
             as
             here
             (
             for
             example
             )
             it
             is
             supposed
             to
             be
             B
             E
             ,
             if
             A
             E
             be
             drawn
             ,
             the
             Moment
             of
             the
             whole
             Parallelogram
             A
             I
             will
             be
             the
             Triangle
             A
             B
             E
             ;
             and
             the
             Moment
             of
             the
             part
             A
             L
             will
             be
             the
             Triangle
             A
             G
             H.
             Wherefore
             the
             Moment
             of
             any
             Ponderant
             ,
             which
             has
             its
             sides
             equally
             applyed
             to
             the
             Beam
             ,
             (
             whether
             they
             be
             applyed
             perpendicularly
             or
             obliquely
             )
             will
             be
             always
             to
             the
             Moment
             of
             a
             part
             of
             the
             same
             ,
             in
             such
             proportion
             ,
             as
             the
             whole
             Triangle
             has
             to
             a
             part
             
             of
             the
             same
             cut
             off
             by
             a
             plain
             which
             is
             parallel
             to
             the
             base
             .
          
           
             9
             The
             Center
             of
             Equiponderation
             of
             any
             figure
             which
             is
             deficient
             according
             to
             commensurable
             proportions
             of
             the
             altitude
             and
             base
             diminished
             ,
             and
             whose
             complete
             figure
             is
             either
             a
             Parallelogram
             ,
             or
             a
             Cylinder
             ,
             or
             a
             Parallelopipedum
             ,
             divides
             the
             Axis
             ,
             so
             ,
             that
             the
             part
             next
             the
             Vertex
             ,
             to
             the
             other
             part
             ,
             is
             as
             the
             complete
             figure
             to
             the
             deficient
             figure
             .
          
           
             For
             let
             C
             I
             A
             P
             E
             (
             in
             the
             5th
             figure
             )
             be
             a
             deficient
             figure
             ,
             whose
             Axis
             is
             A
             B
             ,
             and
             whose
             complete
             figure
             is
             C
             D
             F
             E
             ;
             and
             let
             the
             Axis
             A
             B
             be
             so
             divided
             in
             Z
             ,
             that
             A
             Z
             be
             to
             Z
             B
             as
             C
             D
             F
             E
             is
             to
             C
             I
             A
             P
             E.
             I
             say
             the
             center
             of
             Equiponderation
             of
             the
             figure
             C
             I
             A
             P
             E
             will
             be
             in
             the
             point
             Z.
             
          
           
             First
             ,
             that
             the
             Center
             of
             Equiponderation
             of
             the
             figure
             C
             I
             AP
             E
             is
             somewhere
             in
             the
             Axis
             A
             B
             ,
             is
             manifest
             of
             it self
             ;
             and
             therefore
             A
             B
             is
             a
             Diameter
             of
             Equiponderation
             .
             Let
             A
             E
             be
             drawn
             ,
             and
             let
             B
             E
             be
             put
             for
             the
             Moment
             of
             the
             straight
             line
             C
             E
             ;
             the
             Triangle
             A
             B
             E
             will
             therefore
             (
             by
             the
             3d
             Article
             )
             be
             the
             Moment
             of
             the
             complete
             figure
             C
             D
             F
             E.
             Let
             the
             Axis
             A
             B
             be
             equally
             divided
             in
             L
             ,
             and
             let
             G
             L
             H
             be
             drawn
             parallel
             and
             equal
             to
             the
             straight
             line
             C
             E
             ,
             cutting
             the
             crooked
             line
             C
             I
             A
             P
             E
             in
             I
             and
             P
             ,
             and
             the
             straight
             lines
             A
             C
             and
             A
             E
             in
             K
             and
             M.
             Moreover
             ,
             let
             Z
             O
             be
             drawn
             parallel
             to
             the
             same
             C
             E
             ;
             and
             let
             it
             be
             ,
             as
             L
             G
             to
             L
             I
             ,
             so
             L
             M
             to
             another
             L
             N
             ;
             and
             let
             the
             same
             be
             done
             in
             all
             the
             rest
             of
             the
             straight
             lines
             possible
             ,
             parallel
             to
             the
             base
             ;
             and
             through
             all
             the
             points
             N
             ,
             let
             the
             line
             A
             N
             E
             be
             drawn
             ;
             the
             three-sided
             figure
             A
             N
             E
             B
             will
             therefore
             be
             the
             Moment
             of
             the
             figure
             C
             I
             A
             P
             E.
             Now
             the
             Triangle
             A
             B
             E
             is
             (
             by
             the
             9th
             Article
             of
             the
             17th
             Chapter
             )
             to
             the
             three-sided
             figure
             A
             N
             E
             B
             ,
             as
             A
             B
             C
             D
             +
             A
             I
             C
             B
             is
             to
             A
             I
             C
             B
             twice
             taken
             ,
             that
             is
             ,
             as
             C
             D
             F
             E
             +
             C
             I
             A
             P
             E
             is
             to
             C
             I
             A
             P
             E
             twice
             taken
             .
             But
             as
             C
             I
             A
             P
             E
             is
             to
             C
             D
             F
             E
             ,
             that
             is
             ,
             as
             the
             waight
             of
             the
             deficient
             figure
             ,
             is
             to
             the
             waight
             of
             the
             complete
             figure
             ,
             so
             is
             C
             I
             A
             P
             E
             twice
             taken
             ,
             to
             C
             D
             F
             E
             twice
             taken
             .
             Wherefore
             ,
             setting
             in
             order
             C
             D
             F
             E
             +
             C
             I
             A
             P
             E.
             2
             C
             I
             A
             P
             E.
             2
             C
             D
             F
             E
             ;
             the
             proportion
             of
             C
             D
             F
             E
             +
             C
             I
             A
             P
             E
             to
             C
             D
             F
             E
             twice
             taken
             ,
             will
             be
             compounded
             of
             the
             proportion
             of
             C
             D
             F
             E
             +
             C
             I
             A
             P
             E
             to
             C
             I
             A
             P
             E
             twice
             taken
             ,
             that
             is
             ,
             of
             the
             proportion
             of
             the
             Triangle
             A
             B
             E
             to
             the
             threesided
             figure
             A
             N
             E
             B
             ,
             that
             is
             ,
             of
             the
             Moment
             of
             the
             complete
             figure
             to
             the
             Moment
             of
             
             the
             deficient
             figure
             ,
             and
             of
             the
             proportion
             of
             C
             I
             A
             P
             E
             twice
             taken
             ,
             to
             C
             D
             F
             E
             twice
             taken
             ,
             that
             is
             ,
             to
             the
             proportion
             reciprocally
             taken
             of
             the
             waight
             of
             the
             deficient
             figure
             to
             the
             waight
             of
             the
             complete
             figure
             .
          
           
             Again
             ,
             seeing
             by
             supposition
             A
             Z.
             Z
             B
             :
             :
             C
             D
             F
             E.
             C
             I
             A
             P
             E
             are
             proportionals
             ;
             A
             B.
             A
             Z
             :
             :
             C
             D
             F
             E
             +
             C
             I
             A
             P
             E.
             C
             D
             F
             E
             will
             also
             (
             by
             cōpounding
             )
             be
             proportionals
             .
             And
             seeing
             A
             L
             is
             the
             half
             of
             A
             B
             ,
             A
             L.
             A
             Z
             :
             :
             C
             D
             F
             E
             +
             C
             I
             A
             P
             E.
             2
             C
             D
             F
             E
             will
             also
             be
             proportionals
             .
             But
             the
             proportion
             of
             C
             D
             F
             E
             +
             C
             I
             A
             P
             E
             to
             2
             C
             D
             F
             E
             is
             compounded
             (
             as
             was
             but
             now
             shewn
             )
             of
             the
             proportions
             of
             Moment
             to
             Moment
             &c.
             and
             therefore
             the
             proportion
             of
             A
             L
             to
             A
             Z
             is
             compounded
             of
             the
             proportion
             of
             the
             Moment
             of
             the
             complete
             figure
             C
             D
             F
             E
             to
             the
             Moment
             of
             the
             deficient
             figure
             C
             I
             A
             P
             E
             ,
             and
             of
             the
             proportion
             of
             the
             waight
             of
             the
             deficient
             figure
             C
             I
             AP
             E
             ,
             to
             the
             waight
             of
             the
             complete
             figure
             C
             D
             F
             E
             ;
             But
             the
             proportion
             of
             A
             L
             to
             A
             Z
             is
             compounded
             of
             the
             proportions
             of
             A
             L
             to
             B
             Z
             and
             of
             B
             Z
             to
             A
             Z.
             Now
             the
             proportion
             of
             B
             Z
             to
             A
             Z
             is
             the
             proportion
             of
             the
             Waights
             reciprocally
             taken
             ,
             that
             is
             to
             say
             ,
             of
             the
             waight
             C
             I
             A
             P
             F
             to
             the
             waight
             C
             D
             F
             E.
             Therefore
             the
             remayning
             proportion
             of
             A
             L
             to
             B
             Z
             ,
             that
             is
             ,
             of
             L
             B
             to
             B
             Z
             is
             the
             proportion
             of
             the
             Moment
             of
             the
             waight
             C
             D
             F
             E
             to
             the
             Moment
             of
             the
             waight
             C
             I
             A
             P
             E.
             But
             the
             proportion
             of
             A
             L
             to
             B
             Z
             is
             compounded
             of
             the
             proportions
             of
             A
             L
             to
             A
             Z
             and
             of
             A
             Z
             to
             Z
             B
             ;
             of
             which
             proportions
             that
             of
             A
             Z
             to
             Z
             B
             is
             the
             proportion
             of
             the
             waight
             C
             D
             F
             E
             to
             the
             waight
             C
             I
             A
             P
             E.
             Wherefore
             (
             by
             the
             5th
             Article
             of
             this
             Chapter
             )
             the
             remayning
             proportion
             of
             A
             L
             to
             A
             Z
             is
             the
             proportion
             of
             the
             distances
             of
             the
             points
             Z
             and
             L
             from
             the
             center
             of
             the
             Scale
             ,
             which
             is
             A.
             And
             therefore
             (
             by
             the
             6th
             Article
             )
             the
             waight
             C
             I
             A
             P
             E
             shall
             hang
             from
             O
             in
             the
             straight
             line
             O
             Z.
             So
             that
             O
             Z
             is
             one
             Diameter
             of
             Equiponderation
             of
             the
             waight
             C
             I
             A
             P
             E.
             But
             the
             straight
             line
             A
             B
             is
             the
             other
             Diameter
             of
             Equiponderation
             of
             the
             same
             waight
             C
             I
             A
             P
             E.
             Wherefore
             (
             by
             the
             7th
             Definition
             )
             the
             point
             Z
             is
             the
             center
             of
             the
             same
             Equiponderation
             ;
             which
             point
             (
             by
             construction
             )
             divides
             the
             axis
             so
             ,
             that
             the
             part
             A
             Z
             which
             is
             the
             part
             next
             the
             vertex
             ,
             is
             to
             the
             other
             part
             Z
             B
             ,
             as
             the
             complete
             figure
             C
             D
             F
             E
             is
             to
             the
             deficient
             figure
             C
             I
             A
             P
             E
             ;
             which
             is
             that
             which
             was
             to
             be
             demonstrated
             .
          
           
             
             
               Corollary
               .
               The
               Center
               of
               Equiponderation
               of
               any
               of
               those
               plain
               three-sided
               figures
               ,
               which
               are
               compared
               with
               their
               complete
               figures
               in
               the
               Table
               of
               the
               third
               Article
               of
               the
               17th
               Chapter
               ,
               is
               to
               be
               found
               in
               the
               same
               Table
               ,
               by
               taking
               the
               Denominator
               of
               the
               fraction
               for
               the
               part
               of
               the
               axis
               cut
               off
               next
               the
               vertex
               ,
               and
               the
               Numerator
               for
               the
               other
               part
               next
               the
               base
               .
               For
               example
               ,
               if
               it
               be
               required
               to
               find
               the
               Center
               of
               Equiponderation
               of
               the
               second
               three-sided
               figure
               of
               foure
               Meanes
               ,
               there
               is
               in
               the
               concourse
               of
               the
               second
               columne
               with
               the
               row
               of
               three-sided
               figures
               of
               four
               Meanes
               this
               fraction
               ●
               /
               7
               ,
               which
               signifies
               that
               that
               figure
               is
               to
               its
               parallelogrā
               or
               compleat
               figure
               as
               5
               /
               7
               to
               Unity
               ,
               that
               is
               ,
               as
               5
               /
               7
               to
               7
               /
               7
               ,
               or
               as
               5
               to
               7
               ;
               and
               therefore
               the
               Center
               of
               Equiponderation
               of
               that
               figure
               ,
               divides
               the
               axis
               ,
               so
               ,
               that
               the
               part
               next
               the
               vertex
               is
               to
               the
               other
               part
               as
               7
               to
               5.
               
            
             
               2
               Corallary
               .
               The
               Center
               of
               Equiponderation
               of
               any
               of
               the
               Solids
               of
               those
               figures
               which
               are
               contained
               in
               the
               Table
               of
               the
               8th
               Article
               of
               the
               same
               17th
               Chapter
               ,
               is
               exhibited
               in
               the
               same
               Table
               .
               For
               example
               ,
               if
               the
               Center
               of
               Equiponderation
               of
               a
               Cone
               be
               sought
               for
               ;
               the
               Cone
               will
               be
               found
               to
               be
               ⅓
               of
               its
               Cylinder
               ;
               and
               therefore
               the
               Center
               of
               its
               Equiponderation
               will
               so
               divide
               the
               axis
               ,
               that
               the
               part
               next
               the
               vertex
               ,
               to
               the
               other
               part
               ,
               will
               be
               as
               3
               to
               1.
               
               Also
               the
               Solid
               of
               a
               three-sided
               figure
               of
               one
               Meane
               ,
               that
               is
               ,
               a
               parabolical
               Solid
               ,
               seeing
               it
               is
               2
               /
               4
               ,
               that
               is
               ½
               of
               its
               Cylinder
               ,
               will
               have
               its
               Center
               of
               Equiponderation
               in
               that
               point
               ,
               which
               divides
               the
               axis
               ,
               so
               ,
               that
               the
               part
               towards
               the
               vertex
               be
               double
               to
               the
               part
               towards
               the
               base
               .
            
          
           
             10
             The
             Diameter
             of
             Equiponderation
             of
             the
             Complement
             of
             the
             half
             of
             any
             of
             those
             figu●es
             which
             are
             contained
             in
             the
             Table
             of
             the
             3d
             article
             of
             the
             17th
             Chapter
             ,
             divides
             that
             line
             which
             is
             drawne
             through
             the
             Vertex
             parallel
             and
             equall
             to
             the
             base
             ,
             so
             ,
             that
             the
             part
             next
             the
             Vertex
             ,
             will
             be
             to
             the
             other
             part
             ,
             as
             the
             Complete
             figure
             to
             the
             Complement
             .
          
           
             For
             let
             A
             I
             C
             B
             (
             in
             the
             same
             5
             fig.
             )
             be
             the
             halfe
             of
             a
             Parabola
             ,
             or
             of
             any
             other
             of
             those
             three-sided
             figures
             which
             are
             in
             the
             Table
             of
             the
             3d
             article
             of
             the
             17th
             Chap
             whose
             Axis
             is
             A
             B
             ,
             and
             base
             B
             C
             ;
             having
             A
             D
             drawn
             from
             the
             Vertex
             ,
             equall
             and
             parallel
             to
             the
             base
             B
             C
             ;
             and
             whose
             complete
             figure
             is
             the
             parallelogramme
             A
             B
             C
             D.
             Let
             I
             Q
             be
             drawne
             ,
             at
             any
             distance
             from
             the
             side
             C
             D
             ,
             
             but
             parallel
             to
             it
             ;
             and
             let
             A
             D
             be
             the
             altitude
             of
             the
             Complement
             A
             I
             C
             D
             ,
             and
             Q
             I
             a
             line
             ordinately
             applyed
             in
             it
             .
             Wherefore
             the
             altitude
             A
             L
             in
             the
             deficient
             figure
             A
             I
             C
             B
             ,
             is
             equal
             to
             Q
             I
             the
             line
             ordinately
             applyed
             in
             its
             Complement
             ;
             and
             contrarily
             ,
             L
             I
             the
             line
             ordinately
             applyed
             in
             the
             figure
             A
             I
             C
             B
             ,
             is
             equall
             to
             the
             altitude
             A
             Q
             in
             its
             Complement
             ;
             and
             so
             in
             all
             the
             rest
             of
             the
             ordinate
             lines
             and
             altitudes
             ,
             the
             mutation
             is
             such
             ,
             that
             that
             line
             which
             is
             ordinately
             applyed
             in
             the
             figure
             ,
             is
             the
             altitude
             of
             its
             Complement
             .
             And
             therefore
             the
             proportion
             of
             the
             altitudes
             decreasing
             ,
             to
             that
             of
             the
             ordinate
             lines
             decreasing
             ,
             being
             multiplicate
             according
             to
             any
             number
             in
             the
             deficient
             figure
             ,
             is
             submultiplicate
             according
             to
             the
             same
             number
             in
             its
             Complement
             .
             For
             example
             ,
             if
             A
             I
             C
             B
             be
             a
             Parabola
             ,
             seeing
             the
             proportion
             of
             A
             B
             to
             A
             L
             is
             duplicate
             to
             that
             of
             B
             C
             to
             L
             I
             ,
             the
             proportion
             of
             A
             D
             to
             A
             Q
             in
             the
             Complement
             A
             I
             C
             D
             (
             which
             is
             the
             same
             with
             that
             of
             B
             C
             to
             L
             I
             )
             will
             be
             subduplicate
             to
             that
             of
             C
             D
             to
             Q
             I
             (
             which
             is
             the
             same
             with
             that
             of
             A
             B
             to
             A
             L
             )
             ;
             and
             consequently
             ,
             in
             a
             Parabola
             ,
             the
             Complement
             will
             be
             to
             the
             Parallelogramme
             as
             1
             to
             3
             ;
             in
             a
             three-sided
             figure
             of
             two
             Meanes
             ,
             as
             1
             to
             4
             ;
             in
             a
             three-sided
             figure
             of
             three
             Meanes
             ,
             as
             1
             to
             5
             ,
             &c.
             
             But
             all
             the
             ordinate
             lines
             together
             in
             A
             I
             C
             D
             are
             its
             moment
             ;
             and
             all
             the
             ordinate
             lines
             in
             A
             I
             C
             B
             are
             its
             moment
             .
             Wherefore
             the
             moments
             of
             the
             Complements
             of
             the
             halves
             of
             Deficient
             figures
             in
             the
             Table
             of
             the
             3d
             article
             of
             the
             17th
             Chap.
             being
             compared
             ,
             are
             as
             the
             Deficient
             figures
             themselves
             ;
             and
             therefore
             the
             Diameter
             of
             Equiponderation
             will
             divide
             the
             straight
             line
             A
             D
             in
             such
             proportion
             ,
             that
             the
             part
             next
             the
             Vertex
             be
             to
             the
             other
             part
             ,
             as
             the
             complete
             figure
             A
             B
             C
             D
             is
             to
             the
             Complement
             A
             I
             C
             D.
             
          
           
             Coroll
             .
             The
             diameter
             of
             Equiponderation
             of
             these
             halves
             ,
             may
             be
             found
             by
             the
             Table
             of
             the
             ●d
             article
             of
             the
             17th
             Chapter
             in
             this
             manner
             .
             Let
             there
             be
             propounded
             any
             deficient
             figure
             ,
             namely
             the
             second
             three-sided
             figure
             of
             two
             Meanes
             .
             This
             figure
             is
             to
             the
             complete
             figure
             as
             ⅗
             to
             1
             ,
             that
             is
             as
             3
             to
             5.
             
             Wherefore
             the
             Complement
             to
             the
             same
             complete
             figure
             is
             as
             2
             to
             5
             ;
             and
             therefore
             the
             diameter
             of
             Equiponderation
             of
             this
             Complement
             will
             cut
             the
             straight
             line
             drawne
             from
             the
             Vertex
             parallel
             to
             the
             base
             ,
             so
             ,
             that
             the
             part
             next
             the
             Vertex
             will
             be
             to
             the
             other
             part
             as
             5
             to
             2.
             
             
             And
             in
             like
             manner
             ,
             any
             other
             of
             the
             said
             three-sided
             figures
             being
             propounded
             ,
             if
             the
             numerator
             of
             its
             fraction
             (
             found
             out
             in
             the
             Table
             )
             be
             taken
             from
             the
             denominator
             ,
             the
             straight
             line
             drawn
             from
             the
             Vertex
             is
             to
             be
             divided
             ,
             so
             ,
             that
             the
             part
             next
             the
             Vertex
             be
             to
             the
             other
             part
             ,
             as
             the
             denominator
             is
             to
             the
             remainder
             which
             that
             substraction
             leaves
             .
          
           
             11
             The
             center
             of
             Equiponderation
             of
             the
             halfe
             of
             any
             of
             those
             crooked-lined
             figures
             which
             are
             in
             the
             first
             row
             of
             the
             Table
             of
             the
             3d
             article
             of
             the
             17th
             chapter
             ,
             is
             in
             that
             straight
             line
             ,
             which
             being
             parallel
             to
             the
             Axis
             ,
             divides
             the
             base
             according
             to
             the
             numbers
             of
             the
             fraction
             next
             below
             it
             in
             the
             second
             row
             ,
             so
             ,
             that
             the
             Numerator
             be
             answerable
             to
             that
             part
             which
             is
             towards
             the
             Axis
             .
          
           
             For
             example
             ,
             let
             the
             first
             figure
             of
             three
             Means
             be
             taken
             ,
             whose
             half
             is
             A
             B
             C
             D
             (
             in
             the
             6th
             figure
             )
             ,
             and
             let
             the
             rectangle
             A
             B
             E
             D
             be
             completed
             .
             The
             Complement
             therefore
             will
             be
             B
             C
             D
             E.
             And
             seeing
             A
             B
             E
             D
             is
             to
             the
             figure
             A
             B
             C
             D
             (
             by
             the
             Table
             )
             as
             5
             to
             4
             ,
             the
             same
             A
             B
             E
             D
             will
             be
             to
             the
             Complement
             B
             C
             D
             E
             as
             5
             to
             1.
             
             Wherefore
             if
             F
             G
             be
             drawn
             parallel
             to
             the
             base
             D
             A
             ,
             cutting
             the
             axis
             ,
             so
             ,
             that
             A
             G
             be
             to
             G
             B
             as
             4
             to
             5
             ,
             the
             center
             of
             Equiponderation
             of
             the
             figure
             A
             B
             C
             D
             ,
             will
             (
             by
             the
             precedent
             article
             )
             be
             somewhere
             in
             the
             same
             F
             G.
             Again
             ,
             seeing
             (
             by
             the
             same
             article
             )
             the
             complete
             figure
             A
             B
             E
             D
             ,
             is
             to
             the
             Complement
             B
             C
             D
             E
             as
             5
             to
             1
             ,
             therefore
             if
             B
             E
             and
             A
             D
             be
             divided
             in
             H
             and
             I
             as
             5
             to
             1
             ,
             the
             center
             of
             Equiponderation
             of
             the
             Complement
             B
             C
             D
             E
             will
             be
             somewhere
             in
             the
             straight
             line
             which
             connects
             H
             and
             I.
             Let
             now
             the
             straight
             line
             L
             K
             be
             drawn
             through
             M
             the
             center
             of
             the
             complete
             figure
             ,
             parallel
             to
             the
             base
             ;
             and
             the
             straight
             line
             N
             O
             ,
             through
             the
             same
             center
             M
             ,
             perpendicular
             to
             it
             ;
             and
             let
             the
             straight
             lines
             L
             K
             and
             F
             G
             cut
             the
             straight
             line
             H
             I
             in
             P
             and
             Q.
             Let
             P
             R
             be
             taken
             quadruple
             to
             P
             Q
             ;
             and
             let
             R
             M
             be
             drawn
             and
             produced
             to
             F
             G
             in
             S.
             R
             M
             therefore
             will
             be
             to
             M
             S
             as
             4
             to
             1
             ,
             that
             is
             ,
             as
             the
             figure
             A
             B
             C
             D
             to
             its
             Complement
             B
             C
             D
             E.
             Wherefore
             seeing
             M
             is
             the
             center
             of
             the
             Complete
             figure
             A
             B
             E
             D
             ,
             and
             the
             distances
             of
             R
             and
             S
             from
             the
             center
             M
             be
             in
             proportion
             reciprocall
             to
             that
             of
             the
             waight
             of
             the
             Complement
             B
             C
             D
             E
             to
             the
             waight
             of
             the
             figure
             A
             B
             C
             D
             ,
             R
             and
             S
             will
             either
             be
             the
             centers
             of
             Equiponderation
             of
             their
             own
             
             figures
             ,
             or
             those
             centers
             will
             be
             in
             some
             other
             points
             of
             the
             diameters
             of
             Equiponderation
             H
             I
             and
             F
             G.
             But
             this
             last
             is
             impossible
             .
             For
             no
             other
             straight
             line
             can
             be
             drawn
             through
             the
             point
             M
             terminating
             in
             the
             straight
             lines
             H
             I
             and
             F
             G
             ,
             and
             retaining
             the
             proportion
             of
             M
             R
             to
             M
             S
             ,
             that
             is
             ,
             of
             the
             figure
             A
             B
             C
             D
             to
             its
             complement
             B
             C
             D
             E.
             The
             center
             therefore
             of
             Equiponderation
             of
             the
             figure
             A
             B
             C
             D
             is
             in
             the
             point
             S.
             Now
             seeing
             P
             M
             hath
             the
             same
             proportion
             to
             Q
             S
             which
             R
             P
             hath
             to
             R
             Q
             ,
             Q
             S
             will
             be
             5
             of
             those
             parts
             of
             which
             P
             M
             is
             4
             ,
             that
             is
             ,
             of
             which
             I
             N
             is
             4.
             
             But
             I
             N
             or
             P
             M
             is
             2
             of
             those
             parts
             of
             which
             E
             B
             or
             F
             G
             is
             6
             ;
             and
             therefore
             if
             it
             be
             ,
             as
             4
             to
             5
             ,
             so
             2
             to
             a
             fourth
             ,
             that
             fourth
             will
             be
             2½
             .
             Wherefore
             Q
             S
             is
             2½
             of
             those
             parts
             of
             which
             F
             G
             is
             6.
             
             But
             F
             Q
             is
             1
             ;
             and
             therefore
             F
             S
             is
             3½
             .
             Wherefore
             the
             remayning
             part
             G
             S
             is
             2½
             .
             So
             that
             F
             G
             is
             so
             divided
             in
             S
             ,
             that
             the
             part
             towards
             the
             Axis
             ,
             is
             in
             proportion
             to
             the
             other
             part
             as
             2½
             to
             3½
             ,
             that
             is
             ,
             as
             5
             to
             7
             ;
             which
             answereth
             to
             the
             fraction
             5
             /
             7
             in
             the
             second
             row
             ,
             next
             under
             the
             fraction
             ⅘
             in
             the
             first
             row
             .
             Wherefore
             drawing
             S
             T
             parallel
             to
             the
             Axis
             ,
             the
             base
             wil
             be
             divided
             in
             like
             manner
             .
          
           
             By
             this
             Method
             it
             is
             manifest
             ,
             that
             the
             base
             of
             a
             Semiparabola
             will
             be
             divided
             into
             3
             and
             5
             ;
             and
             the
             base
             of
             the
             first
             three-sided
             figure
             of
             two
             Means
             ,
             into
             4
             and
             6
             ;
             and
             of
             the
             first
             three-sided
             figure
             of
             four
             Means
             ,
             into
             6
             and
             8.
             
             The
             fractions
             therefore
             of
             the
             second
             row
             denote
             the
             proportions
             into
             which
             the
             bases
             of
             the
             figures
             of
             the
             first
             row
             are
             divided
             by
             the
             diameters
             of
             Equiponderation
             .
             But
             the
             first
             row
             begins
             one
             place
             higher
             then
             the
             second
             row
             .
          
           
             12
             The
             center
             of
             Equiponderation
             of
             the
             half
             of
             any
             of
             the
             figures
             in
             the
             second
             row
             of
             the
             same
             Table
             of
             the
             3d
             article
             of
             the
             17th
             Chapter
             ,
             is
             in
             a
             straight
             line
             parallel
             to
             the
             Axis
             ,
             and
             dividing
             the
             base
             according
             to
             the
             nūbers
             of
             the
             fraction
             in
             the
             fourth
             row
             ,
             two
             places
             lower
             ,
             so
             ,
             as
             that
             the
             Numerator
             be
             answerable
             to
             that
             part
             which
             is
             next
             the
             Axis
             .
          
           
             Let
             the
             half
             of
             the
             second
             three-sided
             figure
             of
             two
             Means
             be
             taken
             ;
             and
             let
             it
             be
             A
             B
             C
             D
             (
             in
             the
             7th
             Figure
             )
             ;
             whose
             complement
             is
             B
             C
             D
             E
             ,
             and
             the
             rectangle
             completed
             A
             B
             E
             D.
             Let
             this
             rectangle
             be
             divided
             by
             the
             two
             straight
             lines
             L
             K
             &
             N
             O
             ,
             cutting
             one
             another
             in
             the
             center
             M
             at
             right
             angles
             ;
             and
             because
             A
             B
             E
             D
             
             is
             to
             A
             B
             C
             D
             as
             5
             to
             3
             ,
             let
             A
             B
             be
             divided
             in
             G
             ,
             so
             ,
             that
             A
             G
             to
             B
             G
             be
             as
             3
             to
             5
             ;
             and
             let
             F
             G
             be
             drawn
             parallel
             to
             the
             base
             .
             Also
             because
             A
             B
             E
             D
             is
             (
             by
             the
             9th
             article
             )
             to
             B
             C
             D
             E
             as
             5
             to
             2
             ,
             let
             B
             E
             be
             divided
             in
             the
             point
             I
             ,
             so
             ,
             that
             B
             I
             be
             to
             I
             E
             as
             5
             to
             2
             ;
             and
             let
             I
             H
             be
             drawn
             parallel
             to
             the
             Axis
             ,
             cutting
             L
             K
             and
             F
             G
             in
             P
             and
             Q.
             Let
             now
             P
             R
             be
             so
             taken
             ,
             that
             it
             be
             to
             P
             Q
             as
             3
             to
             2
             ,
             and
             let
             R
             M
             be
             drawn
             and
             produced
             to
             F
             G
             in
             S.
             Seeing
             therefore
             R
             P
             is
             to
             R
             Q
             ,
             that
             is
             ,
             R
             M
             to
             M
             S
             ,
             as
             A
             B
             C
             D
             is
             to
             its
             complement
             B
             C
             D
             E
             ,
             and
             the
             centers
             of
             Equiponderation
             of
             A
             B
             C
             D
             and
             B
             C
             D
             E
             are
             in
             the
             straight
             lines
             F
             G
             and
             H
             I
             ,
             and
             the
             center
             of
             Equiponderation
             of
             them
             both
             together
             in
             the
             point
             M
             ;
             R
             will
             be
             the
             center
             of
             the
             Complement
             B
             C
             D
             E
             ,
             and
             S
             the
             center
             of
             the
             Figure
             A
             B
             C
             D.
             And
             seeing
             P
             M
             ,
             that
             is
             I
             N
             ,
             is
             to
             Q
             S
             ,
             as
             R
             P
             is
             to
             R
             Q
             ;
             and
             I
             N
             ,
             or
             P
             M
             is
             3
             of
             those
             parts
             ,
             of
             which
             B
             E
             ,
             that
             is
             ,
             F
             G
             is
             14
             ;
             therefore
             Q
             S
             is
             5
             of
             the
             same
             parts
             ;
             and
             E
             I
             ,
             that
             is
             F
             Q
             ,
             4
             ;
             and
             F
             S
             ,
             9
             ;
             and
             G
             S
             ,
             5.
             
             Wherefore
             the
             straight
             line
             S
             T
             being
             drawn
             parallel
             to
             the
             Axis
             ,
             will
             divide
             the
             base
             A
             D
             into
             5
             and
             9.
             
             But
             the
             fraction
             5
             /
             9
             is
             found
             in
             the
             fourth
             row
             of
             the
             Table
             ,
             two
             places
             below
             the
             fraction
             9
             /
             5
             in
             the
             second
             row
             .
          
           
             By
             the
             same
             method
             ,
             if
             in
             the
             same
             second
             row
             ,
             there
             be
             taken
             the
             second
             three-sided
             Figure
             of
             three
             Meanes
             ,
             the
             center
             of
             Equiponderation
             of
             the
             half
             of
             it
             ,
             will
             be
             found
             to
             be
             in
             a
             straight
             line
             parallel
             to
             the
             Axis
             ,
             dividing
             the
             base
             according
             to
             the
             numbers
             of
             the
             fraction
             6
             /
             10
             ,
             two
             places
             below
             in
             the
             fourth
             row
             .
             And
             the
             same
             way
             serves
             for
             all
             the
             rest
             of
             the
             Figures
             in
             the
             second
             row
             .
             In
             like
             manner
             ,
             the
             center
             of
             Equiponderation
             of
             the
             third
             three-sided
             Figure
             of
             three
             Means
             ,
             will
             be
             found
             to
             be
             in
             a
             straight
             line
             parallel
             to
             the
             Axis
             ,
             dividing
             the
             base
             ,
             so
             ,
             that
             the
             part
             next
             the
             Axis
             ,
             be
             to
             the
             other
             part
             ,
             as
             7
             to
             13
             ,
             &c.
             
          
           
             Coroll
             .
             The
             Centers
             of
             Equiponderation
             of
             the
             halves
             of
             the
             said
             Figures
             are
             known
             ,
             seeing
             they
             are
             in
             the
             intersection
             of
             the
             straight
             lines
             S
             T
             and
             F
             G
             ,
             which
             are
             both
             known
             .
          
           
             13
             The
             center
             of
             Equiponderation
             of
             the
             half
             of
             any
             of
             the
             Figures
             ,
             which
             (
             in
             the
             Table
             of
             the
             3d
             Article
             of
             the
             17th
             Chap.
             )
             are
             compared
             with
             their
             Parallelograms
             ,
             being
             known
             ;
             the
             center
             of
             Equiponderation
             of
             the
             excess
             of
             the
             same
             Figure
             above
             its
             triangle
             ,
             is
             also
             known
             .
          
           
           
             For
             example
             ,
             let
             the
             Semiparabola
             A
             B
             C
             D
             (
             in
             the
             8th
             Fig.
             )
             be
             taken
             ;
             whose
             Axis
             is
             A
             B
             ;
             whose
             complete
             Figure
             is
             A
             B
             E
             D
             ;
             and
             whose
             excess
             above
             its
             triangle
             is
             B
             C
             D
             B.
             It
             s
             center
             of
             Equiponderation
             may
             be
             found
             out
             in
             this
             manner
             .
             Let
             F
             G
             be
             drawn
             parallel
             to
             the
             base
             ,
             so
             ,
             that
             A
             F
             be
             a
             third
             part
             of
             the
             Axis
             ;
             and
             let
             H
             I
             be
             drawn
             parallel
             to
             the
             Axis
             ,
             so
             ,
             that
             A
             H
             be
             a
             third
             part
             of
             the
             base
             .
             This
             being
             done
             ,
             the
             center
             of
             Equiponderation
             of
             the
             triangle
             A
             B
             D
             ,
             will
             be
             I.
             Again
             ,
             let
             K
             L
             be
             drawn
             parallel
             to
             the
             base
             ,
             so
             ,
             that
             A
             K
             be
             to
             A
             B
             as
             2
             to
             5
             ;
             and
             M
             N
             parallel
             to
             the
             Axis
             ,
             so
             ,
             that
             A
             M
             be
             to
             A
             D
             as
             3
             to
             8
             ;
             and
             let
             M
             N
             terminate
             in
             the
             straight
             line
             K
             L.
             The
             center
             therefore
             of
             Equiponderation
             of
             the
             Parabola
             A
             B
             C
             D
             is
             N
             ;
             and
             therefore
             we
             have
             the
             centers
             of
             Equiponderation
             of
             the
             Semiparabola
             A
             B
             C
             D
             ,
             and
             of
             its
             part
             the
             triangle
             A
             B
             D.
             That
             we
             may
             now
             finde
             the
             Center
             of
             Equiponderation
             of
             the
             remayning
             part
             B
             C
             D
             B
             ,
             let
             I
             N
             be
             drawn
             and
             produced
             to
             O
             ,
             so
             ,
             that
             N
             O
             be
             triple
             to
             I
             N
             ;
             and
             O
             will
             be
             the
             center
             sought
             for
             .
             For
             seeing
             the
             waight
             of
             A
             B
             D
             ,
             to
             the
             waight
             of
             B
             C
             D
             B
             is
             in
             proportion
             reciprocall
             to
             that
             of
             the
             straight
             line
             N
             O
             to
             the
             straight
             line
             I
             N
             ;
             and
             N
             is
             the
             center
             of
             the
             whole
             ,
             and
             I
             the
             center
             of
             the
             triangle
             A
             B
             D
             ;
             O
             will
             be
             the
             center
             of
             the
             remaining
             part
             ,
             namely
             ,
             of
             the
             figure
             B
             D
             C
             B
             ;
             which
             was
             to
             be
             found
             .
          
           
             Coroll
             .
             The
             Center
             of
             Equiponderation
             of
             the
             figure
             B
             D
             C
             B
             ,
             is
             in
             the
             concourse
             of
             two
             straight
             lines
             ,
             whereof
             one
             is
             parallel
             to
             the
             base
             ,
             and
             divides
             the
             Axis
             ,
             so
             ,
             that
             the
             part
             next
             the
             base
             be
             ⅖
             or
             6
             /
             15
             of
             the
             whole
             Axis
             ;
             the
             other
             is
             parallel
             to
             the
             Axis
             ,
             and
             so
             divides
             the
             base
             ,
             that
             the
             part
             towards
             the
             Axis
             be
             ½
             or
             12
             /
             24
             of
             the
             whole
             base
             .
             For
             drawing
             O
             P
             parallel
             to
             the
             base
             ,
             it
             will
             be
             as
             I
             N
             to
             N
             O
             ,
             so
             F
             K
             to
             K
             P
             ,
             that
             is
             ,
             so
             1
             to
             3
             ,
             or
             5
             to
             15.
             
             But
             A
             F
             is
             5
             /
             15
             or
             ⅓
             of
             the
             whole
             A
             B
             ;
             and
             A
             K
             is
             6
             /
             15
             or
             ⅖
             ;
             and
             F
             K
             ●
             /
             15
             ;
             and
             K
             P
             3
             /
             15
             ;
             and
             therefore
             A
             P
             is
             9
             /
             15
             of
             the
             Axis
             A
             B.
             Also
             A
             H
             is
             ⅓
             or
             8
             /
             24
             ;
             and
             A
             M
             ⅜
             or
             9
             /
             24
             of
             the
             whole
             base
             ;
             and
             therefore
             O
             Q
             being
             drawn
             parallel
             to
             the
             Axis
             ,
             M
             Q
             (
             which
             is
             triple
             to
             H
             M
             )
             will
             be
             3
             /
             24.
             
             Wherefore
             A
             Q
             is
             12
             /
             24
             or
             ½
             of
             the
             base
             A
             D.
             
          
           
             The
             excesses
             of
             the
             rest
             of
             the
             three-sided
             figures
             in
             the
             first
             row
             of
             the
             Table
             of
             the
             3d
             article
             of
             the
             17th
             Chapter
             ,
             have
             their
             centers
             of
             Equiponderation
             in
             two
             straight
             lines
             which
             divide
             the
             
             Axis
             and
             base
             according
             to
             those
             fractions
             ,
             which
             adde
             4
             to
             the
             numerators
             of
             the
             fractions
             of
             a
             Parabola
             9
             /
             15
             and
             12
             /
             24
             ;
             and
             6
             to
             the
             denominators
             ,
             in
             this
             manner
             ,
          
           
             
               In
               a
               Parabola
               ,
               The
               Axis
               9
               /
               15
               ,
               The
               Base
               12
               /
               24
            
             
               In
               the
               first
               three-sided
               figure
               ,
               The
               Axis
               13
               /
               21
               ,
               The
               Base
               16
               /
               30
            
             
               In
               the
               second
               three-sided
               figure
               ,
               The
               Axis
               17
               /
               27
               ,
               The
               Base
               20
               /
               36
               &c.
               
            
          
           
             And
             by
             the
             same
             method
             ,
             any
             man
             (
             if
             it
             be
             worth
             the
             paines
             )
             may
             find
             out
             the
             centers
             of
             Equiponderation
             of
             the
             excesses
             above
             their
             triangles
             of
             the
             rest
             of
             the
             figures
             in
             the
             second
             &
             third
             row
             ,
             &c.
             
          
           
             14
             The
             center
             of
             Equiponderation
             of
             the
             Sector
             of
             a
             Sphere
             (
             that
             is
             ,
             of
             a
             figure
             compounded
             of
             a
             right
             Cone
             whose
             Vertex
             is
             the
             center
             of
             the
             Sphere
             ,
             and
             the
             portion
             of
             the
             Sphere
             whose
             base
             is
             the
             same
             with
             that
             of
             the
             Cone
             )
             ,
             divides
             the
             straight
             line
             which
             is
             made
             of
             the
             Axis
             of
             the
             Cone
             and
             halfe
             the
             Axis
             of
             the
             portion
             together
             taken
             ,
             so
             ,
             that
             the
             part
             next
             the
             Vertex
             be
             triple
             to
             the
             other
             part
             ,
             or
             to
             the
             whole
             straight
             line
             ,
             as
             3
             to
             4.
             
          
           
             For
             let
             A
             B
             C
             (
             in
             the
             9th
             fig.
             )
             be
             the
             Sector
             of
             a
             Sphere
             ,
             whose
             Vertex
             is
             the
             ce●ter
             of
             the
             Sphere
             A
             ;
             whose
             Axis
             is
             A
             D
             ;
             and
             the
             circle
             upon
             B
             C
             is
             the
             common
             base
             of
             the
             portion
             of
             the
             Sphere
             and
             of
             the
             Cone
             whose
             Vertex
             is
             A
             ;
             the
             Axis
             of
             which
             portion
             is
             E
             D
             ,
             and
             the
             halfe
             thereof
             F
             D
             ;
             and
             the
             Axis
             of
             the
             Cone
             ,
             A
             E.
             Lastly
             let
             A
             G
             be
             ¾
             of
             the
             straight
             line
             A
             F.
             I
             say
             G
             is
             the
             center
             of
             Equiponderation
             of
             the
             Sector
             A
             B
             C.
             
          
           
             Let
             the
             straight
             line
             F
             H
             be
             drawne
             of
             any
             length
             ,
             making
             right
             angles
             with
             A
             F
             at
             F
             ;
             and
             drawing
             the
             straight
             line
             A
             H
             ,
             let
             the
             triangle
             A
             F
             H
             be
             made
             .
             Then
             upon
             the
             same
             center
             A
             let
             any
             arch
             I
             K
             be
             drawne
             ,
             cutting
             A
             D
             in
             L
             ;
             and
             its
             chord
             ,
             cutting
             A
             D
             in
             M
             ;
             and
             dividing
             M
             L
             equally
             in
             N
             ,
             let
             N
             O
             be
             drawne
             parallel
             to
             the
             straight
             line
             F
             H
             ,
             and
             meeting
             with
             the
             straight
             line
             A
             H
             in
             O.
             
          
           
             Seeing
             now
             B
             D
             C
             is
             the
             Spherical
             Superficies
             of
             the
             portion
             cut
             off
             with
             a
             plain
             passing
             through
             B
             C
             ,
             and
             cutting
             the
             Axis
             at
             right
             angles
             ;
             and
             seeing
             F
             H
             divides
             E
             D
             the
             Axis
             of
             the
             portion
             into
             two
             equal
             parts
             in
             F
             ;
             the
             center
             of
             Equiponderation
             of
             the
             Superficies
             
             B
             D
             C
             will
             be
             in
             F
             (
             by
             the
             8th
             article
             )
             ;
             and
             for
             the
             same
             reason
             the
             center
             of
             Equiponderation
             of
             the
             Superficies
             I
             L
             K
             (
             K
             being
             in
             the
             straight
             line
             A
             C
             )
             will
             be
             in
             N.
             And
             in
             like
             manner
             ,
             if
             there
             were
             drawne
             between
             the
             center
             of
             the
             Sphere
             A
             and
             the
             outermost
             Spherical
             Superficies
             of
             the
             Sector
             ,
             arches
             infinite
             in
             number
             ,
             the
             centers
             of
             Equiponderation
             of
             the
             Sphericall
             Superficies
             in
             which
             those
             arches
             are
             ,
             ,
             would
             be
             found
             to
             be
             in
             that
             part
             of
             the
             Axis
             ,
             which
             is
             intercepted
             between
             the
             Superficies
             it selfe
             and
             a
             plaine
             passing
             along
             by
             the
             chord
             of
             the
             arch
             ,
             and
             cutting
             the
             Axis
             in
             the
             middle
             at
             right
             angles
             .
          
           
             Let
             it
             now
             be
             supposed
             that
             the
             moment
             of
             the
             outermost
             sphericall
             Superficies
             B
             D
             C
             is
             F
             H.
             Seeing
             therefore
             the
             Superficies
             B
             D
             C
             is
             to
             the
             Superficies
             I
             L
             K
             in
             proportion
             duplicate
             to
             that
             of
             the
             arch
             B
             D
             C
             to
             the
             arch
             I
             L
             K
             ,
             that
             is
             ,
             of
             B
             E
             to
             I
             M
             ,
             that
             is
             ,
             of
             F
             H
             to
             N
             O
             ;
             let
             it
             be
             as
             F
             H
             to
             N
             O
             ,
             so
             N
             O
             to
             another
             N
             P
             ;
             and
             again
             ,
             as
             N
             O
             to
             N
             P
             ,
             so
             N
             P
             to
             another
             N
             Q
             ;
             and
             let
             this
             be
             done
             in
             all
             the
             straight
             lines
             parallel
             to
             the
             base
             F
             H
             that
             can
             possibly
             be
             drawn
             between
             the
             base
             and
             the
             vertex
             of
             the
             triangle
             A
             F
             H.
             If
             then
             through
             all
             the
             points
             Q
             there
             be
             drawn
             the
             crooked
             line
             A
             Q
             H
             ,
             the
             figure
             A
             F
             H
             Q
             A
             will
             be
             the
             complement
             of
             the
             first
             three-si●ed
             figure
             of
             two
             Meanes
             ;
             and
             the
             same
             will
             also
             be
             the
             moment
             of
             all
             the
             Sphericall
             Superficies
             of
             which
             the
             Solid
             Sector
             A
             B
             C
             D
             is
             compounded
             ;
             and
             by
             consequent
             ,
             the
             moment
             of
             the
             Sector
             it selfe
             .
             Let
             now
             F
             H
             be
             understood
             to
             be
             the
             semidiameter
             of
             the
             base
             of
             a
             right
             Cone
             ,
             whose
             side
             is
             A
             H
             ,
             and
             Axis
             A
             F.
             Wherfore
             seeing
             the
             bases
             of
             the
             Cones
             which
             passe
             through
             F
             and
             N
             and
             the
             rest
             of
             the
             points
             of
             the
             Axis
             ,
             are
             in
             proportion
             duplicate
             to
             that
             of
             the
             straight
             lines
             F
             H
             and
             N
             O
             ,
             &c.
             the
             moment
             of
             all
             the
             bases
             together
             ,
             that
             is
             ,
             of
             the
             whole
             Cone
             ,
             will
             be
             the
             figure
             it self
             A
             F
             H
             Q
             A
             ;
             and
             therefore
             the
             center
             of
             Equiponderation
             of
             the
             Cone
             A
             F
             H
             is
             the
             same
             with
             that
             of
             the
             solid
             Sector
             .
             Wherefore
             seeing
             A
             G
             is
             ¾
             of
             the
             Axis
             A
             F
             ,
             the
             center
             of
             Equiponderation
             of
             the
             Cone
             A
             F
             H
             is
             in
             G
             ;
             and
             therefore
             the
             center
             of
             the
             solid
             Sector
             is
             in
             G
             also
             ,
             and
             divides
             the
             part
             A
             F
             of
             the
             Axis
             ,
             so
             ,
             that
             A
             G
             is
             triple
             to
             G
             F
             ;
             that
             is
             ,
             A
             G
             is
             to
             A
             F
             as
             3
             to
             4
             ;
             which
             was
             to
             be
             demonstrated
             .
          
           
             Note
             ,
             that
             when
             the
             Sector
             is
             a
             Hemisphere
             ,
             the
             Axis
             of
             the
             
             Cone
             vanisheth
             into
             that
             point
             which
             is
             the
             center
             of
             the
             Sphere
             ;
             and
             therefore
             it
             addeth
             nothing
             to
             half
             the
             Axis
             of
             the
             portion
             .
             Wherefore
             ,
             if
             in
             the
             Axis
             of
             the
             Hemisphere
             ,
             there
             be
             taken
             from
             the
             center
             ,
             ¾
             of
             halfe
             the
             Axis
             ,
             that
             is
             ,
             3
             /
             ●
             of
             the
             Semidiameter
             of
             the
             Sphere
             ,
             there
             will
             be
             the
             center
             of
             Equiponderation
             of
             the
             Hemisphere
             .
          
        
         
           
           
           
           
             
               CHAP.
               XXIV
            
             .
             Of
             Refraction
             and
             Reflection
             .
          
           
             
               
                 1
                 Definitions
                 .
              
               
                 2
                 In
                 perpendicular
                 Motion
                 there
                 is
                 no
                 Refraction
                 .
              
               
                 3
                 Things
                 thrown
                 out
                 of
                 a
                 thinner
                 into
                 a
                 thicker
                 Medium
                 ,
                 are
                 so
                 refracted
                 ,
                 that
                 the
                 Angle
                 Refracted
                 is
                 greater
                 then
                 the
                 Angle
                 of
                 Inclination
                 .
              
               
                 4
                 Endeavour
                 which
                 from
                 one
                 point
                 tendeth
                 every
                 way
                 ,
                 will
                 be
                 so
                 Refracted
                 ,
                 at
                 that
                 the
                 sine
                 of
                 the
                 Angle
                 Refracted
                 ,
                 will
                 be
                 to
                 the
                 sine
                 of
                 the
                 Angle
                 of
                 Inclination
                 ,
                 as
                 the
                 Density
                 of
                 the
                 first
                 Medium
                 is
                 to
                 the
                 Density
                 of
                 the
                 second
                 Medium
                 ,
                 reciprocally
                 taken
                 .
              
               
                 5
                 The
                 sine
                 of
                 the
                 Refracted
                 Angle
                 in
                 one
                 Inclination
                 is
                 to
                 the
                 sine
                 of
                 the
                 Refracted
                 Angle
                 in
                 another
                 Inclination
                 ,
                 as
                 the
                 sine
                 of
                 the
                 Angle
                 of
                 that
                 Inclination
                 is
                 to
                 the
                 sine
                 of
                 the
                 Angle
                 of
                 this
                 Inclination
                 .
              
               
                 6
                 If
                 two
                 lines
                 of
                 Incidence
                 ,
                 having
                 equal
                 Inclination
                 ,
                 be
                 the
                 one
                 in
                 a
                 thinner
                 ,
                 the
                 other
                 in
                 a
                 thicker
                 Medium
                 ,
                 the
                 sine
                 of
                 the
                 angle
                 of
                 Inclination
                 will
                 be
                 a
                 Mean
                 proportional
                 between
                 the
                 two
                 sines
                 of
                 the
                 Refracted
                 angles
                 .
              
               
                 7
                 If
                 the
                 angle
                 of
                 Inclination
                 be
                 semirect
                 ,
                 and
                 the
                 line
                 of
                 Inclination
                 be
                 in
                 the
                 thicker
                 Medium
                 ,
                 and
                 the
                 proportion
                 of
                 their
                 Densities
                 be
                 the
                 same
                 with
                 that
                 of
                 the
                 Diagonal
                 to
                 the
                 side
                 of
                 a
                 Square
                 ,
                 and
                 the
                 separating
                 Superficies
                 be
                 plain
                 ,
                 the
                 Refracted
                 line
                 will
                 be
                 in
                 the
                 separating
                 Superficies
                 .
              
               
                 8
                 If
                 a
                 Body
                 be
                 carried
                 in
                 a
                 straight
                 line
                 upon
                 another
                 Body
                 ,
                 and
                 do
                 not
                 penetrate
                 the
                 same
                 ,
                 but
                 be
                 reflected
                 from
                 it
                 ,
                 the
                 angle
                 of
                 Reflexion
                 will
                 be
                 equal
                 to
                 the
                 Angle
                 of
                 Incidence
                 .
              
               
                 9
                 The
                 same
                 happens
                 in
                 the
                 generation
                 of
                 Motion
                 in
                 the
                 line
                 of
                 Incidence
                 .
              
            
          
           
             1
          
           
             
               Definitions
               .
            
             
               1
               REFRACTION
               ,
               is
               the
               breaking
               of
               that
               straight
               Line
               ,
               in
               which
               a
               Body
               is
               moved
               ,
               or
               its
               Action
               would
               proceed
               in
               one
               and
               the
               same
               Medium
               ,
               into
               two
               straight
               lines
               ,
               by
               reason
               of
               the
               different
               natures
               of
               the
               two
               Mediums
               .
            
             
               2
               The
               former
               of
               these
               is
               called
               the
               
                 Line
                 of
                 Incidence
              
               ;
               the
               later
               the
               
                 Refracted
                 Line
              
               .
            
             
               3
               The
               
                 Point
                 of
                 Refraction
              
               ,
               is
               the
               common
               point
               of
               the
               Line
               of
               Incidence
               and
               of
               the
               Refracted
               Line
               .
            
             
               
               4
               The
               
                 Refracting
                 Superficies
              
               ,
               which
               also
               is
               the
               
                 Separating
                 Superficies
              
               of
               the
               two
               Mediums
               ,
               is
               that
               in
               which
               is
               the
               point
               of
               Refraction
               .
            
             
               5
               The
               
                 Angle
                 Refracted
              
               ,
               is
               that
               which
               the
               Refracted
               Line
               makes
               in
               the
               point
               of
               Refraction
               ,
               with
               that
               Line
               which
               from
               the
               same
               point
               is
               drawn
               perpendicular
               to
               the
               separating
               Superficies
               in
               a
               different
               Medium
               .
            
             
               6
               The
               
                 Angle
                 of
                 Refraction
              
               ,
               is
               that
               which
               the
               Refracted
               line
               makes
               with
               the
               Line
               of
               Incidence
               produced
               .
            
             
               7
               The
               
                 Angle
                 of
                 Inclination
              
               ,
               is
               that
               which
               the
               Line
               of
               Incidence
               makes
               with
               that
               Line
               which
               from
               the
               point
               of
               Refraction
               is
               drawn
               perpendicular
               to
               the
               separating
               Superficies
               .
            
             
               
                 8
                 The
                 
                   Angle
                   of
                   Incidence
                
                 ,
                 is
                 the
                 Complement
                 to
                 a
                 right
                 Angle
                 of
                 the
                 Angle
                 of
                 Inclination
                 .
              
               
                 And
                 so
                 ,
                 (
                 in
                 the
                 first
                 Figure
                 )
                 the
                 Refraction
                 is
                 made
                 in
                 A
                 B
                 F.
                 The
                 
                   Refracted
                   Line
                
                 is
                 B
                 F.
                 The
                 
                   Line
                   of
                   Incidence
                
                 is
                 A
                 B.
                 The
                 
                   Point
                   of
                   Incidence
                   ,
                   and
                   of
                   Refraction
                
                 is
                 B.
                 The
                 Refracting
                 or
                 
                   Separating
                   Superficies
                
                 is
                 D
                 B
                 E.
                 The
                 
                   Line
                   of
                   Incidence
                   produced
                   directly
                
                 is
                 A
                 B
                 C
                 The
                 Perpendicular
                 to
                 the
                 separating
                 Superficies
                 is
                 B
                 H.
                 The
                 
                   Angle
                   of
                   Refraction
                
                 is
                 C
                 B
                 F.
                 The
                 
                   Angle
                   Refracted
                
                 is
                 H
                 B
                 F.
                 The
                 
                   Angle
                   of
                   Inclination
                
                 is
                 A
                 B
                 G
                 or
                 H
                 B
                 C.
                 The
                 
                   Angle
                   of
                   Incidence
                
                 is
                 A
                 B
                 D.
                 
              
            
             
               9
               Moreover
               the
               
                 Thinner
                 Medium
              
               ,
               is
               understood
               to
               be
               that
               in
               which
               there
               is
               less
               resistance
               to
               Motion
               or
               to
               the
               generation
               of
               Motion
               ;
               &
               the
               Thicker
               ,
               that
               wherin
               there
               is
               greater
               resistance
               .
            
             
               10
               And
               that
               Medium
               in
               which
               there
               is
               equal
               resistance
               every
               where
               ,
               is
               a
               
                 Homogeneous
                 Medium
              
               .
               All
               other
               Mediums
               are
               Heterogeneous
               .
            
          
           
             2
             If
             a
             Body
             pass
             ,
             or
             there
             be
             generation
             of
             Motion
             ,
             from
             one
             Medium
             to
             another
             of
             different
             Density
             ,
             in
             a
             line
             perpendicular
             to
             the
             Separating
             Superficies
             ;
             there
             will
             be
             no
             Refraction
             .
          
           
             For
             seeing
             on
             every
             side
             of
             the
             perdendicular
             all
             things
             in
             the
             Mediums
             are
             supposed
             to
             be
             like
             and
             equal
             ;
             if
             the
             Motion
             it self
             be
             supposed
             to
             be
             perpendicular
             ,
             the
             Inclinations
             also
             will
             be
             equal
             ,
             or
             rather
             none
             at
             all
             ;
             and
             therefore
             there
             can
             be
             no
             cause
             ,
             from
             which
             Refraction
             may
             be
             inferred
             to
             be
             on
             one
             side
             of
             the
             
             perpendicular
             ,
             which
             wil
             not
             cōclude
             the
             same
             Refraction
             to
             be
             on
             the
             other
             side
             .
             Which
             being
             so
             ,
             Refraction
             on
             one
             side
             will
             destroy
             Refraction
             on
             the
             other
             side
             ;
             and
             consequently
             ,
             either
             the
             Refracted
             line
             will
             be
             every
             where
             ,
             (
             which
             is
             absurd
             )
             ,
             or
             there
             will
             be
             no
             Refracted
             line
             at
             all
             ;
             which
             was
             to
             be
             demonstrated
             .
          
           
             Corol.
             It
             is
             manifest
             from
             hence
             ,
             that
             the
             cause
             of
             Refraction
             consisteth
             onely
             in
             the
             obliquity
             of
             the
             line
             of
             Incidence
             ,
             whether
             the
             Incident
             Body
             penetrate
             both
             the
             Mediums
             ,
             or
             without
             penetrating
             ,
             propagate
             motion
             by
             Pressure
             onely
             .
          
           
             3
             If
             a
             Body
             ,
             without
             any
             change
             of
             situation
             of
             its
             internal
             parts
             ,
             as
             a
             stone
             ,
             be
             moved
             obliquely
             out
             of
             the
             thinner
             Medium
             ,
             and
             proceed
             penetrating
             the
             thicker
             Medium
             ;
             and
             the
             thicker
             Medium
             be
             such
             ,
             as
             that
             its
             internal
             parts
             being
             moved
             ,
             restore
             themselves
             to
             their
             former
             situation
             ;
             the
             angle
             Refracted
             will
             be
             greater
             then
             the
             angle
             of
             Inclination
             .
          
           
             For
             let
             D
             B
             E
             (
             in
             the
             same
             first
             figure
             )
             be
             the
             separating
             Superficies
             of
             two
             Mediums
             ;
             and
             let
             a
             Body
             ,
             as
             a
             stone
             thrown
             ,
             be
             understood
             to
             be
             moved
             as
             is
             supposed
             in
             the
             straight
             line
             A
             B
             C
             ;
             and
             let
             A
             B
             be
             in
             the
             thinner
             Medium
             ,
             as
             in
             the
             Aire
             ;
             and
             B
             C
             in
             the
             thicker
             ,
             as
             in
             the
             Water
             .
             I
             say
             the
             stone
             ,
             w
             ch
             being
             thrown
             ,
             is
             moved
             in
             the
             line
             A
             B
             ,
             will
             not
             proceed
             in
             the
             line
             B
             C
             ,
             but
             in
             some
             other
             line
             ,
             namely
             that
             ,
             with
             which
             the
             perpendicular
             B
             H
             makes
             the
             Refracted
             angle
             H
             B
             F
             greater
             then
             the
             angle
             of
             Inclination
             H
             B
             C.
             
          
           
             For
             seeing
             the
             stone
             coming
             from
             A
             ,
             and
             falling
             upon
             B
             ,
             makes
             that
             which
             is
             at
             B
             proceed
             towards
             H
             ,
             and
             that
             the
             like
             is
             done
             in
             all
             the
             straight
             lines
             which
             are
             parallel
             to
             B
             H
             ;
             and
             seeing
             the
             parts
             moved
             restore
             themselves
             by
             contrary
             motion
             in
             the
             same
             line
             ;
             there
             will
             be
             contrary
             motion
             generated
             in
             H
             B
             ,
             and
             in
             all
             the
             straight
             lines
             which
             are
             parallel
             to
             it
             .
             Wherefore
             the
             motion
             of
             the
             stone
             will
             be
             made
             by
             the
             concourse
             of
             the
             motions
             in
             A
             G
             ,
             that
             is
             ,
             in
             D
             B
             ,
             and
             in
             G
             B
             ,
             that
             is
             ,
             in
             B
             H
             ,
             and
             lastly
             ,
             in
             H
             B
             ,
             that
             is
             ,
             by
             the
             concourse
             of
             three
             motions
             .
             But
             by
             the
             concourse
             of
             the
             motions
             in
             A
             G
             and
             B
             H
             ,
             the
             stone
             will
             be
             carried
             to
             C
             ;
             and
             therefore
             by
             adding
             the
             motion
             in
             H
             B
             ,
             it
             will
             be
             carried
             higher
             in
             some
             other
             line
             ,
             as
             in
             B
             F
             ,
             and
             make
             the
             angle
             H
             B
             F
             greater
             then
             the
             angle
             H
             B
             C.
             
          
           
             And
             from
             hence
             may
             be
             derived
             the
             cause
             ,
             why
             Bodies
             which
             
             are
             thrown
             in
             a
             very
             oblique
             line
             ,
             if
             either
             they
             be
             any
             thing
             flat
             ,
             or
             be
             thrown
             with
             great
             force
             ,
             will
             when
             they
             fall
             upon
             the
             water
             ,
             be
             cast
             up
             again
             from
             the
             water
             into
             the
             aire
             .
          
           
             For
             let
             A
             B
             (
             in
             the
             2d
             figure
             )
             be
             the
             superficies
             of
             the
             water
             ;
             into
             which
             from
             the
             point
             C
             ,
             let
             a
             stone
             be
             thrown
             in
             the
             straight
             line
             C
             A
             ,
             making
             with
             the
             line
             B
             A
             produced
             a
             very
             little
             angle
             C
             A
             D
             ;
             and
             producing
             B
             A
             indefinitely
             to
             D
             ,
             let
             C
             D
             be
             drawn
             perpendicular
             to
             it
             ,
             and
             A
             E
             parallel
             to
             C
             D.
             The
             stone
             therefore
             will
             be
             moved
             in
             C
             A
             by
             the
             concourse
             of
             two
             motions
             in
             C
             D
             and
             D
             A
             ,
             whose
             velocities
             are
             as
             the
             lines
             themselves
             C
             D
             and
             D
             A.
             And
             from
             the
             motion
             in
             C
             D
             and
             all
             its
             parallels
             downwards
             ,
             as
             soon
             as
             the
             stone
             falls
             upon
             A
             ,
             there
             will
             be
             Reaction
             upwards
             ,
             because
             the
             water
             restores
             it self
             to
             its
             former
             situation
             .
             If
             now
             the
             stone
             be
             thrown
             with
             sufficient
             obliquity
             ,
             that
             is
             ,
             if
             the
             straight
             line
             C
             D
             be
             short
             enough
             ,
             that
             is
             ,
             if
             the
             endeavour
             of
             the
             stone
             downwards
             be
             less
             then
             the
             Reaction
             of
             the
             water
             upwards
             ,
             that
             is
             ,
             less
             then
             the
             endeavour
             it
             hath
             from
             its
             own
             gravity
             ,
             (
             for
             that
             may
             be
             )
             ,
             the
             stone
             will
             (
             by
             reason
             of
             the
             excess
             of
             the
             endeavour
             which
             the
             water
             hath
             to
             restore
             it self
             ,
             above
             that
             which
             the
             stone
             hath
             downwards
             )
             be
             raised
             again
             above
             the
             Superficies
             A
             B
             ,
             and
             be
             carried
             higher
             ,
             being
             reflected
             in
             a
             line
             which
             goes
             higher
             ,
             as
             the
             line
             A
             G.
             
          
           
             4
             If
             from
             a
             point
             ,
             whatsoever
             the
             Medium
             be
             ,
             Endeavour
             be
             propagated
             every
             way
             into
             all
             the
             parts
             of
             that
             Medium
             ;
             and
             to
             the
             same
             Endeavour
             there
             be
             obliquely
             opposed
             another
             Medium
             of
             a
             different
             nature
             ,
             that
             is
             ,
             either
             thinner
             or
             thicker
             ;
             that
             Endeavour
             will
             be
             so
             refracted
             ,
             that
             the
             sine
             of
             the
             angle
             Refracted
             ,
             to
             the
             sine
             of
             the
             angle
             of
             Inclination
             ,
             will
             be
             as
             the
             density
             of
             the
             first
             Medium
             to
             the
             density
             of
             the
             second
             Medium
             ,
             reciprocally
             taken
             .
          
           
             First
             ,
             let
             a
             Body
             be
             in
             the
             thinner
             Medium
             in
             A
             (
             Figure
             3d.
             )
             ;
             and
             let
             it
             be
             understood
             to
             have
             endeavour
             every
             way
             ,
             and
             consequently
             that
             its
             endeavour
             proceed
             in
             the
             lines
             A
             B
             and
             A
             b
             ;
             to
             which
             let
             B
             b
             the
             superficies
             of
             the
             thicker
             Medium
             be
             obliquely
             opposed
             in
             B
             and
             b
             ,
             so
             that
             A
             B
             and
             A
             b
             be
             equal
             ;
             and
             let
             the
             straight
             line
             B
             b
             be
             produced
             both
             wayes
             .
             From
             the
             points
             B
             and
             b
             let
             the
             perpendiculars
             B
             C
             and
             
               b
               c
            
             be
             drawn
             ;
             and
             upon
             the
             centers
             
             B
             and
             b
             ,
             and
             at
             the
             equal
             distances
             B
             A
             and
             b
             A
             ,
             let
             the
             Circles
             A
             C
             and
             A
             c
             be
             described
             ,
             cutting
             B
             C
             and
             
               b
               c
            
             in
             C
             and
             c
             ,
             and
             the
             same
             C
             B
             and
             
               c
               b
            
             produced
             in
             D
             and
             d
             ,
             as
             also
             A
             B
             and
             A
             b
             produced
             in
             E
             and
             e.
             Then
             from
             the
             point
             A
             to
             the
             straight
             lines
             B
             C
             and
             
               b
               c
            
             let
             the
             perpendiculars
             A
             F
             and
             A
             f
             be
             drawn
             .
             A
             F
             therefore
             will
             be
             the
             sine
             of
             the
             angle
             of
             Inclination
             of
             the
             straight
             line
             A
             B
             ,
             and
             A
             f
             the
             sine
             of
             the
             angle
             of
             Inclination
             of
             the
             straight
             line
             A
             h
             ,
             which
             two
             Inclinations
             are
             by
             construction
             made
             equal
             .
             I
             say
             ,
             as
             the
             density
             of
             the
             Medium
             in
             which
             are
             B
             C
             and
             
               b
               c
            
             ,
             is
             to
             the
             density
             of
             the
             Medium
             in
             which
             are
             B
             D
             and
             
               b
               d
            
             ,
             so
             is
             the
             sine
             of
             the
             angle
             Refracted
             ,
             to
             the
             sine
             of
             the
             angle
             of
             Inclination
             .
          
           
             Let
             the
             straight
             line
             F
             G
             be
             drawn
             parallel
             to
             the
             straight
             line
             A
             B
             ,
             meeting
             with
             the
             straight
             line
             b
             B
             produced
             in
             G.
             
          
           
             Seeing
             therefore
             A
             F
             and
             B
             G
             are
             also
             parallels
             ,
             they
             will
             be
             equal
             ;
             and
             consequently
             ,
             the
             endeavour
             in
             A
             F
             is
             propagated
             in
             the
             same
             time
             ,
             in
             which
             the
             endeavour
             in
             B
             G
             would
             be
             propagated
             if
             the
             Medium
             were
             of
             the
             same
             density
             .
             But
             because
             B
             G
             is
             in
             a
             thicker
             Medium
             ,
             that
             is
             ,
             in
             a
             Medium
             which
             resists
             the
             endeavour
             more
             then
             the
             Medium
             in
             which
             A
             F
             is
             ,
             the
             endeavour
             will
             be
             propagated
             less
             in
             B
             G
             then
             in
             A
             F
             ,
             according
             to
             the
             proportion
             which
             the
             density
             of
             the
             Medium
             in
             which
             A
             F
             is
             ,
             hath
             to
             the
             density
             of
             the
             Medium
             in
             which
             B
             G
             is
             .
             Let
             therefore
             the
             density
             of
             the
             Medium
             in
             which
             B
             G
             is
             ,
             be
             to
             the
             density
             of
             the
             Medium
             in
             which
             A
             F
             is
             ,
             as
             B
             G
             is
             to
             B
             H
             ;
             and
             let
             the
             measure
             of
             the
             time
             be
             the
             Radius
             of
             the
             Circle
             .
             Let
             H
             I
             be
             drawn
             parallel
             to
             B
             D
             ,
             meeting
             with
             the
             circumference
             in
             I
             ;
             and
             from
             the
             point
             I
             let
             I
             K
             be
             drawn
             perpendicular
             to
             B
             D
             ;
             which
             being
             done
             ,
             B
             H
             and
             I
             K
             will
             be
             equal
             ;
             and
             I
             K
             will
             be
             to
             A
             F
             ,
             as
             the
             density
             of
             the
             Medium
             in
             which
             is
             A
             F
             ,
             is
             to
             the
             density
             of
             the
             Medium
             in
             which
             is
             I
             K.
             Seeing
             therefore
             in
             the
             time
             A
             B
             (
             which
             is
             the
             Radius
             of
             the
             Circle
             )
             the
             endeavour
             is
             propagated
             in
             A
             F
             in
             the
             thinner
             Medium
             ,
             it
             will
             be
             propagated
             in
             the
             same
             time
             ,
             that
             is
             ,
             in
             the
             time
             B
             I
             in
             the
             thicker
             Medium
             from
             K
             to
             I.
             Therefore
             B
             I
             is
             the
             Refracted
             line
             of
             the
             line
             of
             Incidence
             A
             B
             ;
             and
             I
             K
             is
             the
             sine
             of
             the
             angle
             Refracted
             ;
             and
             A
             F
             ,
             the
             sine
             of
             the
             angle
             of
             Inclination
             .
             Wherefore
             seeing
             I
             K
             is
             to
             A
             F
             ,
             as
             the
             density
             of
             the
             Medium
             in
             which
             is
             A
             F
             to
             the
             density
             of
             the
             Medium
             in
             which
             is
             I
             K
             ;
             it
             will
             be
             as
             the
             density
             of
             the
             Medium
             in
             which
             is
             A
             F
             ,
             (
             or
             
             B
             C
             )
             to
             the
             density
             of
             the
             Medium
             in
             which
             is
             I
             K
             (
             or
             B
             D
             )
             ,
             so
             the
             sine
             of
             the
             angle
             Refracted
             to
             the
             sine
             of
             the
             angle
             of
             Inclination
             .
             And
             by
             the
             same
             reason
             it
             may
             be
             shewn
             ,
             that
             as
             the
             density
             of
             the
             thinner
             Medium
             is
             to
             the
             density
             of
             the
             thicker
             Medium
             ,
             so
             will
             K
             I
             the
             sine
             of
             the
             angle
             Refracted
             be
             to
             A
             F
             the
             sine
             of
             the
             Angle
             of
             Inclination
             .
          
           
             Secondly
             ,
             let
             the
             Body
             which
             endeavoureth
             every
             way
             ,
             be
             in
             the
             thicker
             Medium
             at
             I.
             If
             therefore
             both
             the
             Mediums
             were
             of
             the
             same
             density
             ,
             the
             endeavour
             of
             the
             Body
             in
             I
             B
             would
             tend
             directly
             to
             L
             ;
             and
             the
             sine
             of
             the
             angle
             of
             Inclination
             L
             M
             would
             be
             equal
             to
             I
             K
             or
             B
             H.
             But
             because
             the
             density
             of
             the
             Medium
             in
             which
             is
             IK
             ,
             to
             the
             density
             of
             the
             Medium
             in
             which
             is
             L
             M
             ,
             is
             as
             BH
             to
             B
             G
             ,
             that
             is
             ,
             to
             A
             F
             ,
             the
             endeavour
             will
             be
             propagated
             further
             in
             the
             Medium
             in
             which
             L
             M
             is
             ,
             then
             in
             the
             Medium
             in
             which
             I
             K
             is
             ,
             in
             the
             proportion
             of
             density
             to
             density
             ,
             that
             is
             ,
             of
             M
             L
             to
             A
             F.
             Wherefore
             B
             A
             being
             drawn
             ,
             the
             angle
             Refracted
             will
             be
             C
             B
             A
             ,
             and
             its
             sine
             A
             F.
             But
             L
             M
             is
             the
             sine
             of
             the
             angle
             of
             Inclination
             ;
             and
             therefore
             again
             ,
             as
             the
             density
             of
             one
             Medium
             is
             to
             the
             density
             of
             the
             different
             Medium
             ,
             so
             reciprocally
             is
             the
             sine
             of
             the
             angle
             Refracted
             to
             the
             sine
             of
             the
             angle
             of
             Inclination
             ,
             which
             was
             to
             be
             demonstrated
             .
          
           
             In
             this
             Demonstration
             ,
             I
             have
             made
             the
             separating
             Superficies
             B
             b
             plain
             by
             construction
             .
             But
             though
             it
             were
             concave
             or
             convex
             ,
             the
             Theoreme
             would
             nevertheless
             be
             true
             .
             For
             the
             Refraction
             being
             made
             in
             the
             point
             B
             of
             the
             plain
             separating
             Superficies
             ,
             if
             a
             crooked
             line
             ,
             as
             P
             Q
             be
             drawn
             ,
             touching
             the
             separating
             line
             in
             the
             point
             B
             ;
             neither
             the
             Refracted
             line
             B
             I
             ,
             nor
             the
             perpendicular
             B
             D
             will
             be
             altered
             ;
             and
             the
             Refracted
             angle
             K
             B
             I
             ,
             as
             also
             its
             sine
             K
             I
             will
             be
             still
             the
             same
             they
             were
             .
          
           
             5
             The
             sine
             of
             the
             angle
             Refracted
             in
             one
             Inclination
             ,
             is
             to
             the
             sine
             of
             the
             angle
             Refracted
             in
             another
             Inclination
             ,
             as
             the
             sine
             of
             the
             angle
             of
             that
             Inclination
             to
             the
             sine
             of
             the
             angle
             of
             this
             Inclination
             .
          
           
             For
             seeing
             the
             sine
             of
             the
             Refracted
             angle
             is
             to
             the
             sine
             of
             the
             angle
             of
             Inclination
             ,
             (
             whatsoever
             that
             Inclination
             be
             )
             as
             the
             density
             of
             one
             Medium
             ,
             to
             the
             density
             of
             the
             other
             Medium
             ;
             the
             proportion
             of
             the
             sine
             of
             the
             Refracted
             angle
             ,
             to
             the
             sine
             of
             the
             angle
             of
             Inclination
             ,
             will
             be
             compounded
             of
             the
             proportions
             of
             density
             to
             density
             ,
             
             and
             of
             the
             sine
             of
             the
             angle
             of
             one
             Inclination
             to
             the
             sine
             of
             the
             angle
             of
             the
             other
             Inclination
             .
             But
             the
             proportions
             of
             the
             densities
             in
             the
             same
             Homogeneous
             Body
             ,
             are
             supposed
             to
             be
             the
             same
             .
             Wherefore
             Refracted
             angles
             in
             different
             Inclinations
             ,
             are
             as
             the
             sines
             of
             the
             angles
             of
             those
             Inclinations
             ;
             which
             was
             to
             be
             demonstrated
             .
          
           
             6
             If
             two
             lines
             of
             Incidence
             having
             equal
             inclination
             ,
             be
             the
             one
             in
             a
             thinner
             ,
             the
             other
             in
             a
             thicker
             Medium
             ;
             the
             sine
             of
             the
             angle
             of
             their
             Inclination
             ,
             will
             be
             a
             mean
             proportional
             between
             the
             two
             sines
             of
             their
             angles
             Refracted
             .
          
           
             For
             let
             the
             straight
             line
             AB
             (
             in
             the
             same
             3d
             figure
             )
             have
             its
             Inclination
             in
             the
             thinner
             Medium
             ,
             and
             be
             refracted
             in
             the
             thicker
             Medium
             in
             B
             I
             ;
             and
             let
             E
             B
             have
             as
             much
             Inclination
             in
             the
             thicker
             Medium
             ,
             and
             be
             refracted
             in
             the
             thinner
             Medium
             in
             B
             S
             ;
             and
             let
             R
             S
             the
             sine
             of
             the
             angle
             Refracted
             be
             drawn
             .
             I
             say
             the
             straight
             lines
             R
             S
             ,
             A
             F
             and
             I
             K
             are
             in
             continual
             proportion
             .
             For
             it
             is
             ,
             as
             the
             density
             of
             the
             thicker
             Medium
             to
             the
             density
             of
             the
             thinner
             Medium
             ,
             so
             R
             S
             to
             A
             F.
             But
             it
             is
             also
             ,
             as
             the
             density
             of
             the
             same
             thicker
             Medium
             ,
             to
             that
             of
             the
             same
             thinner
             Medium
             ,
             so
             AF
             to
             IK
             .
             Wherefore
             R
             S.
             A
             F
             :
             :
             A
             F.
             I
             K
             are
             propoortionals
             ;
             that
             is
             ,
             R
             S
             ,
             A
             F
             and
             I
             K
             are
             in
             continual
             proportion
             ,
             and
             A
             F
             is
             the
             Mean
             proportional
             ;
             which
             was
             to
             be
             proved
             .
          
           
             7
             If
             the
             angle
             of
             Inclination
             be
             semirect
             ,
             and
             the
             line
             of
             Inclination
             be
             in
             the
             thicker
             Medium
             ,
             and
             the
             proportion
             of
             the
             Densities
             be
             as
             that
             of
             a
             Diagonal
             to
             the
             side
             of
             its
             Square
             ,
             and
             the
             separating
             Superficies
             be
             plain
             ,
             the
             Refracted
             line
             will
             be
             in
             that
             separating
             Superficies
             .
          
           
             For
             in
             the
             Circle
             A
             C
             (
             in
             the
             4th
             figure
             )
             let
             the
             angle
             of
             Inclination
             A
             B
             C
             be
             an
             angle
             of
             45
             degrees
             .
             Let
             C
             B
             be
             produced
             to
             the
             Circumference
             in
             D
             ;
             &
             let
             C
             E
             (
             the
             sine
             of
             the
             angle
             ●
             B
             C
             )
             be
             drawn
             ▪
             to
             which
             ,
             let
             B
             F
             be
             taken
             equal
             in
             the
             separating
             line
             B
             G.
             B
             C
             E
             F
             will
             therefore
             be
             a
             Parallelogram
             ,
             &
             F
             E
             &
             B
             C
             ,
             that
             is
             ,
             F
             E
             and
             B
             G
             equal
             .
             Let
             AG
             be
             drawn
             ,
             namely
             ,
             the
             Diagonal
             of
             the
             Square
             whose
             side
             is
             B
             G
             ;
             and
             it
             will
             be
             ,
             as
             A
             G
             to
             E
             F
             ,
             so
             B
             G
             to
             B
             F
             ;
             &
             so
             (
             by
             supposition
             )
             the
             density
             of
             the
             Medium
             in
             which
             C
             is
             ,
             to
             the
             density
             of
             the
             Medium
             in
             which
             D
             is
             ;
             and
             so
             also
             the
             sine
             of
             the
             angle
             Refracted
             to
             the
             sine
             of
             the
             angle
             of
             Inclination
             .
             
             Drawing
             therefore
             F
             D
             ,
             &
             from
             D
             the
             line
             D
             H
             perpendicular
             to
             A
             B
             produced
             ,
             DH
             will
             be
             the
             sine
             of
             the
             angle
             of
             Inclination
             .
             And
             seeing
             the
             sine
             of
             the
             angle
             Refracted
             is
             to
             the
             sine
             of
             the
             angle
             of
             Inclination
             ,
             as
             the
             density
             of
             the
             Medium
             in
             which
             is
             C
             ,
             is
             to
             the
             density
             of
             the
             Medium
             in
             which
             is
             D
             ,
             that
             is
             ,
             (
             by
             supposition
             )
             as
             A
             G
             is
             to
             F
             E
             ,
             that
             is
             ,
             as
             D
             H
             is
             to
             B
             G
             ;
             and
             seeing
             D
             H
             is
             the
             sine
             of
             the
             angle
             of
             Inclination
             ,
             B
             G
             will
             therefore
             be
             the
             sine
             of
             the
             angle
             Refracted
             .
             Wherefore
             B
             G
             will
             be
             the
             Refracted
             line
             ,
             and
             lye
             in
             the
             plain
             separating
             Superficies
             ;
             which
             was
             to
             be
             demonstrated
             .
          
           
             Coroll
             .
             It
             is
             therefore
             manifest
             ,
             that
             when
             the
             Inclination
             is
             greater
             then
             45
             degrees
             ,
             as
             also
             when
             it
             is
             less
             ,
             provided
             the
             density
             be
             greater
             ,
             it
             may
             happen
             that
             the
             Refraction
             will
             not
             enter
             the
             thinner
             Medium
             at
             all
             .
          
           
             8
             If
             a
             Body
             fall
             in
             a
             straight
             line
             upon
             another
             Body
             ,
             and
             do
             not
             penetrate
             it
             ,
             but
             be
             reflected
             from
             it
             ,
             the
             angle
             of
             Reflexion
             will
             be
             equal
             to
             the
             angle
             of
             Incidence
             .
          
           
             Let
             there
             be
             a
             Body
             at
             A
             (
             in
             the
             5th
             figure
             )
             ,
             which
             falling
             with
             straight
             motion
             in
             the
             line
             A
             C
             upon
             another
             Body
             at
             C
             ,
             passeth
             no
             further
             ,
             but
             is
             reflected
             ;
             and
             let
             the
             angle
             of
             Incidence
             be
             any
             angle
             ,
             as
             A
             C
             D.
             Let
             the
             straight
             line
             C
             E
             be
             drawn
             ,
             making
             with
             D
             C
             produced
             the
             angle
             E
             C
             F
             equall
             to
             the
             angle
             A
             C
             D
             ;
             and
             let
             A
             D
             be
             drawn
             perpendicular
             to
             the
             straight
             line
             D
             F.
             Also
             in
             the
             same
             straight
             line
             D
             F
             let
             C
             G
             be
             taken
             equall
             to
             C
             D
             ;
             and
             let
             the
             perpendicular
             G
             E
             be
             raised
             ,
             cutting
             C
             E
             in
             E.
             This
             being
             done
             ,
             the
             triangles
             A
             C
             D
             and
             E
             C
             G
             will
             be
             equall
             and
             like
             .
             Let
             C
             H
             be
             drawn
             equal
             and
             parallel
             to
             the
             straight
             line
             A
             D
             ;
             and
             let
             H
             C
             be
             produced
             indefinitely
             to
             I.
             Lastly
             let
             E
             A
             be
             drawn
             ,
             which
             will
             passe
             through
             H
             ,
             and
             be
             parallel
             and
             equall
             to
             G
             D.
             I
             say
             the
             motion
             from
             A
             to
             C
             in
             the
             straight
             line
             of
             Incidence
             AC
             ,
             will
             be
             reflected
             in
             the
             straight
             line
             C
             E.
             
          
           
             For
             the
             motion
             from
             A
             to
             C
             is
             made
             by
             two
             coefficient
             or
             concurrent
             motions
             ,
             the
             one
             in
             A
             H
             parallel
             to
             D
             G
             ,
             the
             other
             in
             A
             D
             perpendicular
             to
             the
             same
             D
             G
             ;
             of
             which
             two
             motions
             ,
             that
             in
             A
             H
             workes
             nothing
             upon
             the
             Body
             A
             after
             it
             has
             been
             moved
             as
             farre
             as
             C
             ,
             because
             (
             by
             supposition
             )
             it
             doth
             not
             passe
             the
             straight
             line
             D
             G
             ;
             whereas
             the
             endeavour
             in
             A
             D
             ,
             that
             is
             in
             H
             C
             ,
             worketh
             further
             towards
             I.
             But
             seeing
             it
             doth
             onely
             presse
             and
             not
             penetrate
             ,
             
             there
             will
             be
             reaction
             in
             H
             ,
             which
             causeth
             motion
             from
             C
             towards
             H
             ;
             and
             in
             the
             mean
             time
             the
             motion
             in
             H
             E
             remaines
             the
             same
             it
             was
             in
             A
             H
             ;
             and
             therefore
             the
             Body
             will
             now
             be
             moved
             by
             the
             concourse
             of
             two
             motions
             in
             C
             H
             and
             H
             E
             ,
             which
             are
             equall
             to
             the
             two
             motions
             it
             had
             formerly
             in
             A
             H
             and
             H
             C.
             Wherefore
             it
             will
             be
             carried
             on
             in
             C
             E.
             The
             angle
             therefore
             of
             Reflection
             will
             be
             E
             C
             G
             ,
             equall
             (
             by
             construction
             )
             to
             the
             angle
             A
             C
             D
             ;
             which
             was
             to
             be
             demonstrated
             .
          
           
             Now
             when
             the
             Body
             is
             considered
             but
             as
             a
             point
             ,
             it
             is
             all
             one
             ,
             whether
             the
             Superficies
             or
             line
             in
             which
             the
             Reflection
             is
             made
             ,
             be
             straight
             or
             crooked
             ;
             for
             the
             point
             of
             Incidence
             and
             Reflexion
             C
             ,
             is
             as
             well
             in
             the
             crooked
             line
             which
             toucheth
             D
             G
             in
             C
             ,
             as
             in
             D
             G
             it selfe
             .
          
           
             9
             But
             if
             we
             suppose
             that
             not
             a
             Body
             be
             moved
             ,
             but
             some
             Endeavour
             onely
             be
             propagated
             from
             A
             to
             C
             ,
             the
             Demonstration
             will
             neverthelesse
             be
             the
             same
             .
             For
             all
             Endeavour
             is
             motion
             ;
             and
             when
             it
             hath
             reached
             the
             Solid
             Body
             in
             C
             ,
             it
             presseth
             it
             ,
             and
             endeavoureth
             further
             in
             C
             I.
             Wherefore
             the
             reaction
             will
             proceed
             in
             C
             H
             ;
             and
             the
             endeavour
             in
             C
             H
             concurring
             with
             the
             endeavour
             in
             H
             E
             ,
             will
             generate
             the
             endeavour
             in
             C
             E
             ,
             in
             the
             same
             manner
             as
             in
             the
             repercussion
             of
             Bodies
             moved
             .
          
           
             If
             therefore
             Endeavour
             be
             propagated
             from
             any
             point
             to
             the
             concave
             Superficies
             of
             a
             Spherical
             Body
             ,
             the
             Reflected
             line
             with
             the
             circumference
             of
             a
             great
             circle
             in
             the
             same
             Sphere
             ,
             will
             make
             an
             angle
             equall
             to
             the
             angle
             of
             Incidence
             .
          
           
             For
             if
             Endeavour
             be
             propagated
             from
             A
             (
             in
             the
             6
             fig.
             )
             to
             the
             circumference
             in
             B
             ,
             and
             the
             center
             of
             the
             Sphere
             be
             C
             ,
             and
             the
             line
             C
             B
             be
             drawne
             ,
             as
             also
             the
             Tangent
             D
             B
             E
             ;
             and
             lastly
             if
             the
             angle
             F
             B
             D
             be
             made
             equall
             to
             the
             angle
             A
             B
             E
             ,
             the
             Reflexion
             will
             be
             made
             in
             the
             line
             B
             F
             ,
             as
             hath
             been
             newly
             shewn
             .
             Wherefore
             the
             angles
             which
             the
             straight
             lines
             A
             B
             and
             F
             B
             make
             with
             the
             circumference
             ,
             will
             also
             be
             equall
             .
             But
             it
             is
             here
             to
             be
             noted
             that
             if
             C
             B
             be
             produced
             howsoever
             to
             G
             ,
             the
             endeavour
             in
             the
             line
             G
             B
             C
             will
             proceed
             onely
             from
             the
             perpendicular
             reaction
             in
             G
             B
             ;
             and
             that
             therefore
             there
             will
             be
             no
             other
             endeavour
             in
             the
             point
             B
             towards
             the
             parts
             which
             are
             within
             the
             Sphere
             ,
             besides
             that
             which
             tends
             towards
             the
             center
             .
          
           
           
             And
             here
             I
             put
             an
             end
             to
             the
             third
             part
             of
             this
             Discourse
             ;
             in
             which
             I
             have
             considered
             Motion
             and
             Magnitude
             by
             themselves
             in
             the
             abstract
             .
             The
             fourth
             and
             last
             part
             ,
             concerning
             the
             
               Phaenomena
               of
               Nature
            
             ,
             that
             is
             to
             say
             ,
             concerning
             the
             Motions
             and
             Magnitudes
             of
             the
             Bodies
             which
             are
             parts
             of
             the
             World
             ,
             reall
             and
             existent
             ,
             is
             that
             which
             followes
             .
          
        
      
       
         
         
         
         
           PHYSIQVES
           ▪
           or
           the
           PHAENOMENA
           of
           NATVRE
           .
        
         
           
             
               CHAP.
               XXV
            
             .
             Of
             Sense
             and
             
               Animal
               Motion
            
             .
          
           
             
               
                 1
                 The
                 connexion
                 of
                 what
                 hath
                 been
                 said
                 with
                 that
                 which
                 followeth
                 .
              
               
                 2
                 The
                 investigation
                 of
                 the
                 nature
                 of
                 Sense
                 ,
                 and
                 the
                 Definition
                 of
                 Sense
                 .
              
               
                 3
                 The
                 Subject
                 and
                 Object
                 of
                 Sense
                 .
              
               
                 4
                 The
                 Organ
                 of
                 Sense
                 .
              
               
                 5
                 All
                 Bodies
                 are
                 not
                 indued
                 with
                 Sense
                 .
              
               
                 6
                 But
                 one
                 Phantasme
                 at
                 one
                 and
                 the
                 same
                 time
                 .
              
               
                 7
                 Imagination
                 ,
                 the
                 Remayns
                 of
                 past
                 Sense
                 ,
                 (
                 which
                 also
                 is
                 Memory
                 )
                 .
                 Of
                 Sleep
                 .
              
               
                 8
                 How
                 Phantasmes
                 succeed
                 one
                 another
                 .
              
               
                 9
                 Dreames
                 ,
                 whence
                 they
                 proceed
                 .
              
               
                 10
                 Of
                 the
                 Senses
                 ,
                 their
                 kindes
                 ,
                 their
                 Organs
                 ,
                 and
                 Phantasmes
                 proper
                 and
                 common
                 .
              
               
                 11
                 The
                 Magnitude
                 of
                 Images
                 ,
                 how
                 and
                 by
                 what
                 it
                 is
                 determined
                 .
              
               
                 12
                 Pleasure
                 ,
                 Pain
                 ,
                 Appetite
                 ,
                 and
                 Aversion
                 ,
                 what
                 they
                 are
                 .
              
               
                 13
                 Deliberation
                 and
                 Will
                 ,
                 what
                 .
              
            
          
           
             1
             I
             Have
             (
             in
             the
             first
             Chapter
             )
             defined
             Philosophy
             to
             be
             
               Knowledge
               of
               Effects
               acquired
               by
               true
               Ratiocination
               ,
               from
               knowledge
               first
               had
               of
               their
               Causes
               and
               Generation
               ;
               and
               of
               such
               Causes
               or
               Generations
               as
               may
               be
               ,
               from
               former
               knowledge
               of
               their
               Effects
               or
               Appearances
               .
            
             There
             are
             therefore
             two
             Methods
             of
             Philosophy
             ,
             One
             from
             the
             Generation
             of
             things
             to
             their
             possible
             Effects
             ,
             and
             the
             other
             from
             their
             Effects
             or
             Appearances
             to
             some
             possible
             Generation
             of
             the
             same
             .
             In
             the
             former
             of
             these
             ,
             the
             Truth
             of
             the
             first
             Principles
             of
             our
             ratiocination
             (
             namely
             Definitions
             )
             is
             made
             and
             constituted
             by
             our selves
             ,
             whilest
             we
             consent
             and
             agree
             about
             the
             Appellations
             of
             things
             .
             And
             this
             part
             I
             have
             finished
             in
             the
             foregoing
             Chapters
             ;
             in
             which
             (
             if
             I
             am
             not
             deceived
             )
             I
             have
             affirmed
             nothing
             (
             saving
             
             the
             Definitions
             themselves
             )
             which
             hath
             not
             good
             coherence
             with
             the
             Definitions
             I
             have
             given
             ;
             that
             is
             to
             say
             ,
             which
             is
             not
             sufficiently
             demonstrated
             to
             all
             those
             that
             agree
             with
             me
             in
             the
             use
             of
             Words
             and
             Appellations
             ;
             for
             whose
             sake
             onely
             I
             have
             written
             the
             same
             .
             I
             now
             enter
             upon
             the
             other
             part
             ;
             which
             is
             the
             finding
             out
             by
             the
             Appearances
             or
             Effects
             of
             Nature
             which
             we
             know
             by
             Sense
             ,
             some
             wayes
             and
             means
             by
             which
             they
             may
             be
             (
             I
             do
             not
             say
             ,
             they
             are
             )
             generated
             .
             The
             Principles
             therefore
             ,
             upon
             which
             the
             following
             discourse
             depends
             ,
             are
             not
             such
             as
             we
             our selves
             make
             and
             pronounce
             in
             general
             terms
             ,
             as
             Definitions
             ;
             but
             such
             ,
             as
             being
             placed
             in
             the
             things
             themselves
             by
             the
             Authour
             of
             Nature
             ,
             are
             by
             us
             observed
             in
             them
             ;
             and
             we
             make
             use
             of
             them
             in
             single
             and
             particular
             ,
             not
             universal
             propositions
             .
             Nor
             do
             they
             impose
             upon
             us
             any
             necessity
             of
             constituting
             Theoremes
             ;
             their
             use
             being
             onely
             (
             though
             not
             without
             such
             general
             Propositions
             as
             have
             been
             already
             demonstrated
             )
             to
             shew
             us
             the
             possibility
             of
             some
             production
             or
             generation
             .
             Seeing
             therefore
             the
             Science
             which
             is
             here
             taught
             ,
             hath
             its
             Principles
             in
             the
             Appearances
             of
             Nature
             ,
             and
             endeth
             in
             the
             attayning
             of
             some
             knowledge
             of
             Natural
             causes
             ,
             I
             have
             given
             to
             this
             Part
             ,
             the
             title
             of
             PHYSIQUES
             ,
             or
             the
             PHAENOMENA
             of
             NATURE
             .
             Now
             such
             things
             as
             appear
             ,
             or
             are
             shewn
             to
             us
             by
             Nature
             ,
             we
             call
             Phaenomena
             or
             Appearances
             .
          
           
             Of
             all
             the
             Phaenomena
             ,
             or
             Appearances
             which
             are
             neer
             us
             ,
             the
             most
             admirable
             is
             Apparition
             it self
             ,
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             ;
             namely
             ,
             that
             some
             Natural
             Bodies
             have
             in
             themselves
             the
             patterns
             almost
             of
             all
             things
             ,
             and
             others
             of
             none
             at
             all
             .
             So
             that
             if
             the
             Appearances
             be
             the
             Principles
             by
             which
             we
             know
             all
             other
             things
             ,
             we
             must
             needs
             acknowledge
             Sense
             to
             be
             the
             Principle
             by
             which
             we
             know
             those
             Principles
             ;
             &
             that
             all
             the
             knowledge
             we
             have
             is
             derived
             from
             it
             .
             And
             as
             for
             the
             causes
             of
             Sense
             ,
             we
             cannot
             begin
             our
             search
             of
             them
             from
             any
             other
             Phaenomenon
             then
             that
             of
             Sense
             it self
             .
             But
             you
             will
             say
             ,
             by
             what
             Sense
             shall
             we
             take
             notice
             of
             Sense
             ?
             I
             answer
             ,
             by
             Sense
             it self
             ,
             namely
             ,
             by
             the
             Memory
             which
             for
             some
             time
             remains
             in
             us
             of
             things
             sensible
             ,
             though
             they
             themselves
             pass
             away
             .
             For
             he
             that
             perceiues
             that
             he
             hath
             perceived
             ,
             remembers
             .
          
           
             In
             the
             first
             place
             therefore
             the
             causes
             of
             our
             Perception
             ,
             that
             is
             ,
             the
             causes
             of
             those
             Ideas
             and
             Phantasmes
             which
             are
             perpetually
             
             generated
             within
             us
             whilest
             we
             make
             use
             of
             our
             Senses
             ,
             are
             to
             be
             enquired
             into
             ;
             and
             in
             what
             manner
             their
             generation
             proceeds
             .
             To
             help
             which
             Inquisition
             ,
             we
             may
             observe
             first
             of
             all
             ,
             that
             our
             Phantasmes
             or
             Ideas
             are
             not
             alwayes
             the
             same
             ;
             but
             that
             new
             ones
             appear
             to
             us
             ,
             and
             old
             ones
             vanish
             ,
             according
             as
             we
             apply
             our
             Organs
             of
             Sense
             ,
             now
             to
             one
             Object
             ,
             now
             to
             another
             .
             Wherefore
             they
             are
             generated
             ,
             and
             perish
             .
             And
             from
             hence
             it
             is
             manifest
             ,
             that
             they
             are
             some
             change
             or
             mutation
             in
             the
             Sentient
             .
          
           
             2
             Now
             that
             all
             Mutation
             or
             Alteration
             is
             Motion
             ,
             or
             Endeavour
             (
             and
             Endeavour
             also
             is
             Motion
             )
             in
             the
             internal
             parts
             of
             the
             thing
             that
             is
             altered
             ,
             hath
             been
             proved
             (
             in
             the
             9th
             Article
             of
             the
             8th
             Chapter
             )
             from
             this
             ,
             that
             whilest
             even
             the
             least
             parts
             of
             any
             Body
             remain
             in
             the
             same
             situation
             in
             respect
             of
             one
             another
             ,
             it
             cannot
             be
             said
             that
             any
             alteration
             (
             unless
             perhaps
             that
             the
             whole
             Body
             together
             hath
             been
             moved
             )
             hath
             hapned
             to
             it
             ;
             but
             that
             it
             both
             appeareth
             and
             is
             the
             same
             it
             appeared
             &
             was
             before
             .
             Sense
             therefore
             in
             the
             Sentient
             ,
             can
             be
             nothing
             else
             but
             motion
             in
             some
             of
             the
             internal
             parts
             of
             the
             Sentient
             ;
             and
             the
             parts
             so
             moved
             ,
             are
             parts
             of
             the
             Organs
             of
             Sense
             .
             For
             the
             parts
             of
             our
             Body
             by
             which
             we
             perceive
             any
             thing
             ,
             are
             those
             we
             commonly
             call
             the
             
               Organs
               of
               Sense
            
             .
             And
             so
             we
             find
             what
             is
             the
             Subject
             of
             our
             Sense
             ,
             namely
             ,
             that
             in
             which
             are
             the
             Phantasmes
             ;
             and
             partly
             also
             we
             have
             discovered
             the
             nature
             of
             Sense
             ,
             namely
             ,
             that
             it
             is
             some
             internal
             Motion
             in
             the
             Sentient
             .
          
           
             I
             have
             shewn
             besides
             (
             in
             the
             8th
             Chap.
             at
             the
             7th
             Article
             )
             that
             no
             Motion
             is
             generated
             but
             by
             a
             Body
             contiguous
             and
             Moved
             .
             From
             whence
             it
             is
             manifest
             ,
             that
             the
             immediate
             cause
             of
             Sense
             or
             Perception
             consists
             in
             this
             ,
             that
             the
             first
             Organ
             of
             Sense
             is
             touched
             and
             pressed
             .
             For
             when
             the
             uttermost
             part
             of
             the
             Organ
             is
             pressed
             ,
             it
             no
             sooner
             yeilds
             ,
             but
             the
             part
             next
             within
             it
             ,
             is
             pressed
             also
             ;
             and
             in
             this
             manner
             ,
             the
             pressure
             or
             Motion
             is
             propagated
             through
             all
             the
             parts
             of
             the
             Organ
             to
             the
             innermost
             .
             And
             thus
             also
             the
             pressure
             of
             the
             uttermost
             part
             proceeds
             from
             the
             pressure
             of
             some
             more
             remote
             Body
             ,
             and
             so
             continually
             ,
             till
             we
             come
             to
             that
             from
             which
             ,
             as
             from
             its
             fountain
             we
             derive
             the
             Phantasme
             or
             Idea
             that
             is
             made
             in
             us
             by
             our
             Sense
             .
             And
             this
             ,
             whatsoever
             it
             be
             ,
             is
             that
             we
             commonly
             call
             the
             Object
             .
             Sense
             therefore
             is
             some
             internal
             Motion
             
             in
             the
             Sentient
             ,
             generated
             by
             some
             internal
             Motion
             of
             the
             parts
             of
             the
             Object
             ,
             and
             propagated
             through
             all
             the
             Media
             to
             the
             innermost
             part
             of
             the
             Organ
             .
             By
             which
             words
             I
             have
             almost
             defined
             what
             Sense
             is
             .
          
           
             Moreover
             ,
             I
             have
             shewn
             (
             in
             the
             2d
             Article
             of
             the
             15
             Chapter
             )
             that
             all
             Resistance
             is
             Endeavour
             opposite
             to
             another
             Endeavour
             ,
             that
             is
             to
             say
             ,
             Reaction
             .
             Seeing
             therefore
             there
             is
             in
             the
             whole
             Organ
             by
             reason
             of
             its
             own
             internal
             natural
             Motion
             ;
             some
             Resistance
             or
             Reaction
             against
             the
             Motion
             which
             is
             propagated
             from
             the
             Object
             to
             the
             innermost
             part
             of
             the
             Organ
             ,
             there
             is
             also
             in
             the
             same
             Organ
             an
             Endeavour
             opposite
             to
             the
             Endeavour
             which
             proceeds
             from
             the
             Object
             ;
             so
             that
             when
             that
             Endeavour
             inwards
             is
             the
             last
             action
             in
             the
             act
             of
             Sense
             ,
             then
             from
             the
             Reaction
             ,
             how
             little
             soever
             the
             duration
             of
             it
             be
             ,
             a
             Phantasme
             or
             Idea
             hath
             its
             being
             ;
             which
             by
             reason
             the
             Endeavour
             is
             now
             outwards
             ,
             doth
             alwayes
             appear
             as
             something
             situate
             without
             the
             Organ
             .
             So
             that
             now
             I
             shall
             give
             you
             the
             whole
             Definition
             of
             Sense
             ,
             as
             it
             is
             drawn
             from
             the
             explication
             of
             the
             causes
             thereof
             ,
             and
             the
             order
             of
             its
             generation
             ,
             thus
             ;
             SENSE
             
               is
               a
               Phantasme
               ,
               made
               by
               the
               Reaction
               and
               endeavour
               outwards
               in
               the
               Organ
               of
               Sense
               ,
               caused
               by
               an
               Endeavour
               inwards
               from
               the
               Object
               ,
               remayning
               for
               some
               time
               more
               or
               less
               .
            
          
           
             3
             The
             Subject
             of
             Sense
             ,
             is
             the
             Sentient
             it self
             ,
             namely
             ,
             some
             living
             Creature
             ;
             and
             we
             speak
             more
             correctly
             ,
             when
             we
             say
             a
             
               Living
               Creature
            
             seeth
             ,
             then
             when
             we
             say
             the
             Eye
             seeth
             .
             The
             Object
             ,
             is
             the
             thing
             Perceived
             ;
             and
             it
             is
             more
             accurately
             said
             ,
             that
             we
             see
             the
             Sun
             ,
             then
             that
             we
             see
             the
             Light.
             For
             Light
             &
             Colour
             &
             Heat
             &
             Sound
             ,
             and
             other
             qualities
             which
             are
             commonly
             called
             Sensible
             ,
             are
             not
             Objects
             ,
             but
             Phantasms
             in
             the
             Sentients
             .
             For
             a
             Phantasm
             is
             the
             act
             of
             Sense
             ,
             and
             differs
             no
             otherwise
             from
             Sense
             then
             fieri
             (
             that
             is
             ,
             Being
             a
             doing
             )
             differs
             from
             
               Factum
               esse
            
             ,
             (
             that
             is
             ,
             Being
             done
             ;
             )
             which
             difference
             ,
             in
             things
             that
             are
             done
             in
             an
             Instant
             ,
             is
             none
             at
             all
             ;
             and
             a
             Phantasme
             is
             made
             in
             an
             Instant
             .
             For
             in
             all
             Motion
             which
             proceeds
             by
             perpetual
             propagation
             ,
             the
             first
             part
             being
             moved
             moves
             the
             second
             ,
             the
             second
             the
             third
             ,
             and
             so
             on
             to
             the
             last
             ,
             and
             that
             to
             any
             distance
             how
             great
             soever
             .
             And
             in
             what
             point
             of
             time
             the
             first
             or
             formost
             part
             proceeded
             to
             the
             place
             of
             the
             second
             ,
             which
             is
             thrust
             on
             ;
             in
             the
             same
             point
             of
             time
             the
             last
             save
             
             one
             proceeded
             into
             the
             place
             of
             the
             last
             yeilding
             part
             ;
             which
             by
             reaction
             ,
             in
             the
             same
             instant
             ,
             if
             the
             reaction
             be
             strong
             enough
             ,
             makes
             a
             Phantasme
             ;
             and
             a
             Phantasme
             being
             made
             ,
             Perception
             is
             made
             together
             with
             it
             ,
          
           
             4
             The
             Organs
             of
             Sense
             ,
             which
             are
             in
             the
             Sentient
             ,
             are
             such
             parts
             thereof
             ,
             that
             if
             they
             be
             hurt
             ,
             the
             very
             generation
             of
             Phantasmes
             is
             thereby
             destroyed
             ,
             though
             all
             the
             rest
             of
             the
             parts
             remain
             intire
             .
             Now
             these
             parts
             in
             the
             most
             of
             Living
             Creatures
             are
             found
             to
             be
             certain
             Spirits
             and
             Membranes
             ,
             which
             proceeding
             from
             the
             
               Pia
               Mater
            
             ,
             involve
             the
             Brain
             and
             all
             the
             Nerves
             ;
             also
             the
             Brain
             it self
             ,
             and
             the
             Arteries
             which
             are
             in
             the
             Brain
             ;
             and
             such
             other
             parts
             ,
             as
             being
             stirred
             ,
             the
             Hart
             also
             ,
             which
             is
             the
             fountain
             of
             all
             Sense
             is
             stirred
             together
             with
             them
             .
             For
             whensoever
             the
             action
             of
             the
             Object
             reacheth
             the
             Body
             of
             the
             Sentient
             ,
             that
             action
             is
             by
             some
             Nerve
             propagated
             to
             the
             Brain
             ;
             and
             if
             the
             Nerve
             leading
             thither
             be
             so
             hurt
             or
             obstructed
             ,
             that
             the
             Motion
             can
             be
             propagated
             no
             further
             ,
             no
             Sense
             follows
             .
             Also
             ▪
             if
             the
             motion
             be
             intercepted
             between
             the
             Brain
             and
             the
             Heart
             by
             the
             defect
             of
             the
             Organ
             by
             which
             the
             action
             is
             propagated
             ,
             there
             will
             be
             no
             perception
             of
             the
             Object
             .
          
           
             5
             But
             though
             all
             Sense
             ,
             as
             I
             have
             said
             ,
             be
             made
             by
             Reaction
             ,
             nevertheless
             it
             is
             not
             necessary
             that
             every
             thing
             that
             Reacteth
             should
             have
             Sense
             .
             I
             know
             there
             have
             been
             Philosophers
             ,
             &
             those
             learned
             men
             ,
             who
             have
             maintained
             that
             all
             Bodies
             are
             endued
             with
             Sense
             .
             Nor
             do
             I
             see
             how
             they
             can
             be
             refuted
             ,
             if
             the
             nature
             of
             Sense
             be
             placed
             in
             Reaction
             onely
             .
             And
             ,
             though
             by
             the
             Reaction
             of
             Bodies
             inanimate
             a
             Phantasme
             might
             be
             made
             ,
             it
             would
             nevertheless
             cease
             ,
             as
             soon
             as
             ever
             the
             Object
             were
             removed
             .
             For
             unless
             those
             Bodies
             had
             Organs
             ,
             (
             as
             living
             Creatures
             have
             )
             fit
             for
             the
             retaining
             of
             such
             Motion
             as
             is
             made
             in
             them
             ,
             their
             Sense
             would
             be
             such
             ,
             as
             that
             they
             should
             never
             remēber
             the
             same
             .
             And
             therefore
             this
             hath
             nothing
             to
             do
             with
             that
             Sense
             which
             is
             the
             subject
             of
             my
             discourse
             .
             For
             by
             Sense
             we
             commonly
             understand
             the
             judgement
             we
             make
             of
             Objects
             by
             their
             Phantasmes
             ;
             namely
             ,
             by
             comparing
             and
             distinguishing
             those
             Phantasmes
             ;
             which
             we
             could
             never
             do
             ,
             if
             that
             motion
             in
             the
             Organ
             ,
             by
             which
             the
             Phantasme
             is
             made
             ,
             did
             not
             remain
             there
             for
             some
             time
             ,
             and
             make
             the
             same
             Phantasme
             return
             .
             
             Wherefore
             Sense
             ,
             as
             I
             here
             understand
             it
             ,
             and
             which
             is
             commonly
             so
             called
             ,
             hath
             necessarily
             some
             memory
             adhering
             to
             it
             ,
             by
             which
             former
             and
             later
             Phantasmes
             may
             be
             compared
             together
             ,
             and
             distinguished
             from
             one
             another
             .
          
           
             Sense
             therefore
             properly
             so
             called
             ,
             must
             necessarily
             have
             in
             it
             a
             perpetual
             variety
             of
             Phantasmes
             ,
             that
             they
             may
             be
             discerned
             one
             from
             another
             .
             For
             if
             we
             should
             suppose
             a
             man
             to
             be
             made
             with
             cleer
             Eyes
             ,
             and
             all
             the
             rest
             of
             his
             Organs
             of
             Sight
             well
             disposed
             ,
             but
             endued
             with
             no
             other
             Sense
             ;
             and
             that
             he
             should
             look
             onely
             upon
             one
             thing
             ,
             which
             is
             alwayes
             of
             the
             same
             colour
             and
             figure
             without
             the
             least
             appearance
             of
             variety
             ,
             he
             would
             seem
             to
             me
             ,
             whatsoever
             others
             may
             say
             ,
             to
             see
             ,
             no
             more
             then
             I
             seem
             to
             my self
             to
             feel
             the
             Bones
             of
             my
             own
             Limbs
             by
             my
             Organs
             of
             Feeling
             ;
             and
             yet
             those
             Bones
             are
             alwayes
             ,
             and
             on
             all
             sides
             touched
             by
             a
             most
             sensible
             Membrane
             .
             I
             might
             perhaps
             say
             he
             were
             astonished
             ,
             and
             looked
             upon
             it
             ;
             but
             I
             should
             not
             say
             he
             saw
             it
             ;
             it
             being
             almost
             all
             one
             for
             a
             man
             to
             be
             alwayes
             sensible
             of
             one
             and
             the
             same
             thing
             ,
             and
             not
             to
             be
             sensible
             at
             all
             of
             any
             thing
             .
          
           
             6
             And
             yet
             such
             is
             the
             nature
             of
             Sense
             ,
             that
             it
             does
             not
             permit
             a
             man
             to
             discern
             many
             things
             at
             once
             .
             For
             seeing
             the
             nature
             of
             Sense
             consists
             in
             Motion
             ;
             as
             long
             as
             the
             Organs
             are
             employed
             about
             one
             Object
             ,
             they
             cannot
             be
             so
             Moved
             by
             another
             at
             the
             same
             time
             ,
             as
             to
             make
             by
             both
             their
             Motions
             one
             sincere
             Phantasme
             of
             each
             of
             them
             at
             once
             .
             And
             therefore
             two
             several
             Phantasmes
             will
             not
             be
             made
             by
             two
             Objects
             working
             together
             ,
             but
             onely
             one
             Phantasme
             compounded
             from
             the
             action
             of
             both
             .
          
           
             Besides
             ,
             as
             when
             we
             divide
             a
             Body
             ,
             we
             divide
             its
             place
             ;
             and
             when
             we
             reckon
             many
             Bodies
             ,
             we
             must
             necessarily
             reckon
             as
             many
             places
             ;
             and
             contrarily
             ,
             as
             I
             have
             shewn
             in
             the
             first
             Article
             of
             the
             7th
             Chapter
             ;
             so
             what
             number
             soever
             we
             say
             there
             be
             of
             Times
             ,
             we
             must
             understand
             the
             same
             number
             of
             Motions
             also
             ;
             and
             as
             oft
             as
             we
             count
             many
             Motions
             ,
             so
             oft
             we
             reckon
             many
             times
             .
             For
             though
             the
             object
             we
             looke
             upon
             be
             of
             diverse
             colours
             ,
             yet
             with
             those
             diverse
             colours
             it
             is
             but
             one
             varied
             Object
             ,
             and
             not
             variety
             of
             Objects
             .
          
           
             Moreover
             ,
             whilest
             those
             Organs
             which
             are
             common
             to
             all
             the
             Senses
             (
             such
             as
             are
             those
             parts
             of
             every
             Organ
             which
             proceed
             (
             in
             
             Men
             )
             from
             the
             root
             of
             the
             Nerves
             ,
             to
             the
             Hart
             )
             are
             vehemently
             stirred
             by
             a
             strong
             action
             from
             some
             one
             Object
             ,
             they
             are
             (
             by
             reaof
             the
             contumacy
             which
             the
             motion
             they
             have
             already
             gives
             them
             against
             the
             reception
             of
             all
             other
             motion
             )
             made
             the
             lesse
             fit
             to
             receive
             any
             other
             impression
             from
             whatsoever
             other
             Objects
             ,
             to
             what
             sense
             soever
             those
             Objects
             belong
             .
             And
             hence
             it
             is
             ,
             that
             an
             earnest
             studying
             of
             one
             Object
             ,
             takes
             away
             the
             Sense
             of
             all
             other
             Objects
             for
             the
             present
             .
             For
             Study
             is
             nothing
             else
             but
             a
             possession
             of
             the
             mind
             ,
             that
             is
             to
             say
             ,
             a
             vehement
             motion
             made
             by
             some
             one
             Object
             in
             the
             Organs
             of
             Sense
             ,
             which
             are
             stupid
             to
             all
             other
             motions
             as
             long
             as
             this
             lasteth
             ;
             according
             to
             what
             was
             said
             by
             
               Terence
               ,
               Populus
               studio
               stupidus
               in
               funambulo
               animum
               occuparat
               .
            
             For
             what
             is
             Stupor
             but
             that
             which
             the
             Greekes
             call
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             ,
             that
             is
             ,
             a
             cessation
             from
             the
             Sense
             of
             other
             things
             ?
             Wherefore
             at
             one
             and
             the
             same
             time
             ,
             we
             cannot
             by
             Sense
             perceive
             more
             then
             one
             single
             Object
             ;
             as
             in
             reading
             ,
             we
             see
             the
             letters
             successively
             one
             by
             one
             ,
             and
             not
             all
             together
             ,
             though
             the
             whole
             Page
             be
             presented
             to
             our
             eye
             ;
             and
             though
             every
             severall
             letter
             be
             distinctly
             written
             there
             ,
             yet
             when
             we
             looke
             upon
             the
             whole
             page
             at
             once
             ,
             we
             read
             nothing
             .
          
           
             From
             hence
             it
             is
             manifest
             ,
             that
             every
             endeavour
             of
             the
             Organ
             ●utwards
             ,
             is
             not
             to
             be
             called
             Sense
             ,
             but
             that
             onely
             which
             at
             severall
             times
             is
             by
             Vehemence
             made
             stronger
             and
             more
             praedominant
             than
             the
             rest
             ;
             which
             deprives
             us
             of
             the
             Sense
             of
             other
             Phantasmes
             ,
             no
             otherwise
             then
             the
             Sun
             deprives
             the
             rest
             of
             the
             starres
             of
             light
             ,
             not
             by
             hindering
             their
             action
             ,
             but
             by
             obscuring
             and
             hiding
             them
             with
             his
             excesse
             of
             brightnesse
             .
          
           
             7.
             
             But
             the
             motion
             of
             the
             Organ
             ,
             by
             which
             a
             Phantasme
             is
             made
             ,
             is
             not
             commonly
             called
             Sense
             ,
             except
             the
             Object
             be
             present
             .
             And
             the
             Phantasme
             remaining
             after
             the
             Object
             is
             removed
             or
             past
             by
             ,
             is
             called
             Fancy
             ,
             and
             in
             latine
             Imaginatio
             ;
             which
             word
             (
             because
             all
             Phantasmes
             are
             not
             Images
             )
             doth
             not
             fully
             answer
             the
             signification
             of
             the
             word
             Fancy
             in
             its
             generall
             acceptation
             .
             Neverthelesse
             I
             may
             use
             it
             safely
             enough
             ,
             by
             understanding
             it
             for
             the
             Greek
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             .
          
           
             IMAGINATION
             therefore
             is
             nothing
             else
             but
             
               Sense
               decaying
            
             ,
             or
             weakned
             ,
             by
             the
             absence
             of
             the
             Object
             .
             But
             what
             may
             be
             the
             cause
             of
             this
             decay
             or
             weakning
             ?
             Is
             the
             Motion
             the
             weaker
             because
             the
             
             Object
             is
             taken
             away
             ?
             If
             it
             were
             ,
             then
             Phantasmes
             would
             alwayes
             and
             necessarily
             be
             less
             cleare
             in
             the
             Imagination
             ,
             then
             they
             are
             in
             Sense
             ;
             which
             is
             not
             true
             .
             For
             in
             Dreams
             (
             which
             are
             the
             Imaginations
             of
             those
             that
             sleep
             )
             they
             are
             no
             less
             clear
             then
             in
             Sense
             it self
             .
             But
             the
             reason
             why
             in
             men
             Waking
             the
             Phantasms
             of
             things
             past
             are
             more
             obscure
             then
             those
             of
             things
             present
             ,
             is
             this
             ,
             that
             their
             Organs
             being
             at
             the
             same
             time
             moved
             by
             other
             present
             Objects
             ,
             those
             Phantasmes
             are
             the
             lesse
             praedominant
             .
             Whereas
             in
             Sleep
             ,
             the
             passages
             being
             shut
             up
             ,
             externall
             action
             doth
             not
             at
             all
             disturbe
             or
             hinder
             internall
             motion
             .
          
           
             If
             this
             be
             true
             ,
             the
             next
             thing
             to
             be
             considered
             ,
             will
             be
             ,
             whether
             any
             cause
             may
             be
             found
             out
             ,
             from
             the
             supposition
             whereof
             it
             will
             follow
             ,
             that
             the
             passage
             is
             shut
             up
             from
             the
             externall
             Objects
             of
             Sense
             to
             the
             internall
             Organ
             .
             I
             suppose
             therefore
             ,
             that
             by
             the
             continuall
             action
             of
             Objects
             ,
             (
             to
             which
             a
             Reaction
             of
             the
             Organ
             ,
             and
             more
             esqecially
             of
             the
             Spirits
             ,
             is
             necessarily
             consequent
             )
             the
             Organ
             is
             wearied
             ,
             that
             is
             ,
             its
             parts
             are
             no
             longer
             moved
             by
             the
             Spirits
             without
             some
             pain
             ;
             and
             consequently
             the
             Nerves
             being
             abandoned
             and
             grown
             slack
             ,
             they
             retire
             to
             their
             fountain
             which
             is
             the
             cavity
             either
             of
             the
             Brain
             or
             of
             the
             Heart
             ;
             by
             which
             means
             the
             action
             which
             proceeded
             by
             the
             Nerves
             is
             necessarily
             intercepted
             ▪
             For
             Action
             ,
             upon
             a
             Patient
             that
             retires
             from
             it
             ,
             makes
             but
             little
             Impression
             at
             the
             first
             ;
             and
             at
             last
             ,
             when
             the
             Nerves
             are
             by
             little
             and
             little
             slack●ed
             ,
             none
             at
             all
             .
             And
             therefore
             there
             is
             no
             more
             Reaction
             ,
             that
             is
             ,
             no
             more
             Sense
             ,
             till
             the
             Organ
             being
             refreshed
             by
             Rest
             ,
             and
             by
             a
             supply
             of
             new
             Spirits
             recovering
             strength
             and
             motion
             ,
             the
             Sentient
             awaketh
             .
             And
             thus
             it
             seems
             to
             be
             alwayes
             ,
             unless
             some
             other
             praeternatural
             cause
             intervene
             ;
             as
             Heat
             in
             the
             internal
             parts
             from
             lassitude
             ,
             or
             from
             some
             disease
             stirring
             the
             Spirits
             and
             other
             parts
             of
             the
             Organ
             in
             some
             extraordinary
             manner
             .
          
           
             8
             Now
             it
             is
             not
             without
             cause
             ,
             nor
             so
             casual
             a
             thing
             as
             many
             perhaps
             think
             it
             ,
             that
             Phantasmes
             in
             this
             their
             great
             variety
             ,
             proceed
             from
             one
             another
             ;
             and
             that
             the
             same
             Phantasmes
             sometimes
             bring
             into
             the
             mind
             other
             Phantasmes
             like
             themselves
             ,
             and
             at
             other
             times
             extreamly
             unlike
             .
             For
             in
             the
             motion
             of
             any
             continued
             Body
             ,
             one
             part
             followes
             another
             by
             cohaesion
             ;
             and
             therefore
             ,
             whilst
             we
             turne
             our
             Eies
             and
             other
             Organs
             successively
             to
             many
             
             Objects
             ,
             the
             motion
             which
             was
             made
             by
             every
             one
             of
             them
             remayning
             ,
             the
             Phantasmes
             are
             renewed
             as
             often
             as
             any
             of
             those
             motions
             comes
             to
             be
             praedominant
             above
             the
             rest
             ;
             and
             they
             become
             praedominant
             in
             the
             same
             order
             ,
             in
             which
             at
             any
             time
             formerly
             they
             were
             generated
             by
             Sense
             .
             So
             that
             when
             by
             length
             of
             time
             very
             many
             Phantasmes
             have
             been
             generated
             within
             us
             by
             Sense
             ,
             then
             allmost
             any
             thought
             may
             arise
             from
             any
             other
             thought
             ;
             in
             so
             much
             that
             it
             may
             seeme
             to
             be
             a
             thing
             indifferent
             and
             casuall
             ,
             which
             thought
             shall
             follow
             which
             .
             But
             for
             the
             most
             part
             this
             is
             not
             so
             uncertain
             a
             thing
             to
             waking
             as
             to
             sleeping
             men
             .
             For
             the
             thought
             or
             Phantasme
             of
             the
             desired
             End
             ,
             brings
             in
             all
             the
             Phantasmes
             that
             are
             meanes
             conducing
             to
             that
             end
             ,
             and
             that
             in
             order
             backewards
             from
             the
             last
             to
             the
             first
             ,
             and
             againe
             forwards
             from
             the
             beginning
             to
             the
             End
             :
             But
             this
             supposes
             both
             Appetite
             ,
             and
             Judgement
             to
             discerne
             what
             meanes
             conduce
             to
             the
             end
             ;
             which
             is
             gotten
             by
             Experience
             ;
             and
             Experience
             is
             store
             of
             Phantasmes
             ,
             arising
             from
             the
             sense
             of
             very
             many
             things
             .
             For
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             ▪
             and
             
               Meminisse
               ,
               Fancy
            
             and
             Memory
             differ
             onely
             in
             this
             ,
             that
             Memory
             supposeth
             the
             time
             past
             ,
             which
             Fancy
             doth
             not
             .
             In
             Memory
             ,
             the
             Phantasmes
             we
             consider
             are
             as
             if
             they
             were
             worne
             out
             with
             time
             ;
             but
             in
             our
             Fancy
             we
             consider
             them
             as
             they
             are
             ;
             which
             distinction
             is
             not
             of
             the
             things
             themselves
             ,
             but
             of
             the
             considerations
             of
             the
             Sentient
             .
             For
             there
             is
             in
             Memory
             ,
             something
             like
             that
             which
             happens
             in
             looking
             upon
             things
             at
             a
             great
             distance
             ;
             in
             which
             ,
             as
             the
             small
             parts
             of
             the
             Object
             are
             not
             discerned
             by
             reason
             of
             their
             remotenesse
             ;
             so
             in
             Memory
             ,
             many
             accidents
             and
             places
             and
             parts
             of
             things
             ,
             which
             were
             formerly
             perceived
             by
             Sense
             ,
             are
             by
             length
             of
             time
             decayed
             and
             lost
             .
          
           
             The
             perpetuall
             arising
             of
             Phantasmes
             ,
             both
             in
             Sense
             and
             Imagination
             ,
             is
             that
             which
             we
             commonly
             call
             
               Discourse
               of
               the
               Mind
            
             ,
             and
             is
             common
             to
             men
             with
             other
             living
             Creatures
             .
             For
             he
             thta
             thinketh
             ,
             compareth
             the
             Phantasmes
             that
             passe
             ,
             that
             is
             ,
             taketh
             notice
             of
             their
             likenesse
             or
             unlikenesse
             to
             one
             another
             .
             And
             as
             he
             that
             observes
             readily
             the
             likenesses
             of
             things
             of
             different
             natures
             ,
             or
             that
             are
             very
             remote
             from
             one
             another
             ,
             is
             said
             to
             have
             a
             good
             Fancy
             ;
             so
             he
             is
             said
             to
             have
             a
             good
             Judgement
             ,
             that
             finds
             out
             the
             unlikenesses
             or
             differences
             of
             things
             that
             are
             like
             one
             another
             .
             
             Now
             this
             observation
             of
             differences
             ,
             is
             not
             perception
             made
             by
             a
             common
             Organ
             of
             Sense
             ,
             distinct
             from
             Sense
             or
             Perception
             properly
             so
             called
             ;
             but
             is
             Memory
             of
             the
             differences
             of
             particular
             Phantasmes
             remaining
             for
             some
             time
             ;
             as
             the
             distinction
             between
             Hot
             and
             Lucid
             ,
             is
             nothing
             else
             but
             the
             Memory
             both
             of
             a
             Heating
             ,
             and
             of
             an
             Enlightning
             Object
             .
          
           
             9
             The
             Phantasmes
             of
             men
             that
             sleep
             ,
             are
             DREAMS
             .
             Concerning
             which
             we
             are
             taught
             by
             experience
             these
             five
             things
             .
             First
             ,
             that
             for
             the
             most
             part
             there
             is
             neither
             order
             nor
             coherence
             in
             them
             .
             Secondly
             ,
             that
             we
             dream
             of
             nothing
             but
             what
             is
             compounded
             and
             made
             up
             of
             the
             Phantasmes
             of
             Sense
             past
             .
             Thirdly
             ,
             that
             somtimes
             they
             proceed
             (
             as
             in
             those
             that
             are
             drowsy
             )
             from
             the
             interruption
             of
             their
             Phantasmes
             by
             little
             and
             little
             broken
             and
             altered
             through
             sleepiness
             ;
             and
             sometimes
             also
             they
             begin
             in
             the
             midst
             of
             sleep
             .
             Fourthly
             ,
             that
             they
             are
             clearer
             then
             the
             Imaginations
             of
             waking
             men
             ,
             except
             such
             as
             are
             made
             by
             Sense
             itself
             ,
             to
             which
             they
             are
             equal
             in
             clearness
             .
             Fifthly
             ,
             that
             when
             we
             dream
             ,
             we
             admire
             neither
             the
             places
             nor
             the
             looks
             of
             the
             things
             that
             appear
             to
             us
             .
             Now
             from
             what
             hath
             been
             said
             ,
             it
             is
             not
             hard
             to
             shew
             what
             may
             be
             the
             causes
             of
             these
             Phaenomena
             .
             For
             as
             for
             the
             first
             ,
             seeing
             all
             Order
             and
             Coherence
             proceeds
             from
             frequent
             looking
             back
             to
             the
             End
             ,
             that
             is
             ,
             from
             Consultation
             ;
             it
             must
             needs
             be
             ,
             that
             seeing
             in
             sleep
             we
             lose
             all
             thought
             of
             the
             End
             ,
             our
             Phantasmes
             succeed
             one
             another
             ,
             not
             in
             that
             order
             which
             tends
             to
             any
             End
             ,
             but
             as
             it
             hapneth
             ,
             and
             in
             such
             manner
             ,
             as
             Objects
             present
             themselves
             to
             our
             Eyes
             when
             we
             look
             indifferently
             upon
             all
             things
             before
             us
             ,
             and
             see
             them
             ,
             not
             because
             we
             would
             see
             them
             ,
             but
             because
             we
             do
             not
             shut
             our
             Eyes
             ;
             for
             then
             they
             appear
             to
             us
             without
             any
             order
             at
             all
             .
             The
             second
             proceeds
             from
             this
             ,
             that
             in
             the
             silence
             of
             Sense
             ,
             there
             is
             no
             new
             motion
             from
             the
             Objects
             ,
             and
             therefore
             no
             new
             Phantasme
             ,
             unless
             we
             call
             that
             new
             ,
             which
             is
             compounded
             of
             old
             ones
             ,
             as
             a
             Chimaera
             ,
             a
             golden
             Mountain
             ,
             and
             the
             like
             .
             As
             for
             the
             third
             ,
             why
             a
             Dream
             is
             sometimes
             as
             it
             were
             the
             continuation
             of
             Sense
             ,
             made
             up
             of
             broken
             Phantasmes
             ,
             as
             in
             men
             distempered
             with
             sickness
             ,
             the
             reason
             is
             manifestly
             this
             ,
             that
             in
             some
             of
             the
             Organs
             Sense
             remains
             ,
             and
             in
             others
             it
             faileth
             .
             But
             how
             some
             Phantasmes
             may
             be
             revived
             ,
             when
             all
             the
             exteriour
             Organs
             are
             benummed
             with
             sleep
             ,
             is
             not
             so
             easily
             shewn
             .
             Nevertheless
             ,
             that
             which
             
             hath
             already
             been
             said
             ,
             contains
             the
             reason
             of
             this
             also
             .
             For
             whatsoever
             strikes
             the
             
               Pia
               Mater
            
             ,
             reviveth
             some
             of
             those
             Phantasmes
             that
             are
             still
             in
             motion
             in
             the
             Brain
             ;
             and
             when
             any
             internal
             motion
             of
             the
             Heart
             reacheth
             that
             Membrane
             ,
             then
             the
             praedominant
             motion
             in
             the
             Brain
             makes
             the
             Phantasme
             .
             Now
             the
             Motions
             of
             the
             Heart
             are
             Appetites
             and
             Aversions
             ,
             of
             which
             I
             shall
             presently
             speak
             further
             .
             And
             as
             Appetites
             and
             Aversions
             are
             generated
             by
             Phantasmes
             ,
             so
             reciprocally
             Phantasmes
             are
             generated
             by
             Appetites
             and
             Aversions
             .
             For
             example
             ,
             Heat
             in
             the
             Heart
             proceeds
             from
             Anger
             and
             Fighting
             ;
             and
             again
             from
             Heat
             in
             the
             Heart
             ,
             (
             whatsoever
             be
             the
             cause
             of
             it
             )
             is
             generated
             Anger
             ,
             and
             the
             Image
             of
             an
             Enemy
             in
             Sleep
             .
             And
             as
             Love
             and
             Beauty
             stirre
             up
             heat
             in
             certain
             Organs
             ;
             so
             Heat
             in
             the
             same
             Organs
             ,
             from
             whatsoever
             it
             proceeds
             ,
             often
             causeth
             Desire
             ,
             and
             the
             Image
             of
             an
             unresisting
             Beauty
             .
             Lastly
             ,
             Cold
             doth
             in
             the
             same
             manner
             generate
             Feare
             in
             those
             that
             sleep
             ,
             and
             causeth
             them
             to
             dream
             of
             Ghosts
             ,
             and
             to
             have
             Phantasmes
             of
             horrour
             and
             danger
             ;
             as
             Fear
             also
             causeth
             Cold
             in
             those
             that
             wake
             ;
             so
             reciprocal
             are
             the
             motions
             of
             the
             Heart
             and
             Brain
             .
             The
             fourth
             ,
             namely
             ,
             that
             the
             things
             we
             seem
             to
             see
             and
             feel
             in
             sleep
             ,
             are
             as
             clear
             as
             in
             sense
             it self
             ,
             proceeds
             from
             two
             causes
             ;
             one
             ,
             that
             having
             then
             no
             sense
             of
             things
             without
             us
             ,
             that
             internal
             motion
             which
             makes
             the
             Phantasme
             ,
             in
             the
             absence
             of
             all
             other
             impressions
             ,
             is
             praedominant
             ;
             and
             the
             other
             ,
             that
             the
             parts
             of
             our
             Phantasms
             which
             are
             decayed
             and
             worn
             out
             by
             time
             ,
             are
             made
             up
             with
             other
             fictitious
             parts
             .
             To
             conclude
             ,
             when
             we
             dream
             ,
             we
             do
             not
             wonder
             at
             strange
             places
             ,
             and
             the
             appearances
             of
             things
             unknown
             to
             us
             ,
             because
             Admiration
             requires
             that
             the
             things
             appearing
             be
             new
             and
             unusual
             ,
             which
             can
             happen
             to
             none
             but
             those
             that
             remember
             former
             appearances
             ;
             whereas
             in
             sleep
             ,
             all
             things
             appear
             as
             present
             .
          
           
             But
             it
             is
             here
             to
             be
             observed
             ,
             that
             certain
             Dreams
             ,
             especially
             such
             as
             some
             men
             have
             when
             they
             are
             between
             sleeping
             and
             waking
             ,
             and
             such
             as
             happen
             to
             those
             that
             have
             no
             knowledge
             of
             the
             nature
             of
             Dreams
             ,
             and
             are
             with
             all
             superstitious
             ,
             were
             not
             heretofore
             ,
             nor
             are
             now
             accounted
             Dreams
             .
             For
             the
             Apparitions
             men
             thought
             they
             saw
             ,
             and
             the
             Voices
             they
             thought
             they
             heard
             in
             sleep
             ,
             were
             not
             believed
             to
             be
             Phantasmes
             ,
             but
             things
             subsisting
             of
             themselves
             ,
             
             and
             Objects
             without
             those
             that
             dreamed
             .
             For
             to
             some
             men
             ,
             as
             well
             sleeping
             as
             waking
             ,
             but
             especially
             to
             guilty
             men
             ,
             and
             in
             the
             night
             ,
             and
             in
             hallowed
             places
             ,
             Feare
             alone
             ,
             helped
             a
             little
             with
             the
             stories
             of
             such
             Apparitions
             ,
             hath
             raised
             in
             their
             minds
             terrible
             Phantasmes
             ,
             which
             have
             been
             ,
             and
             are
             still
             deceiptfully
             received
             for
             things
             really
             true
             ,
             under
             the
             names
             of
             Ghosts
             and
             
               Incorporeal
               Substances
            
             .
          
           
             10
             In
             most
             living
             Creatures
             there
             are
             observed
             five
             kinds
             of
             Senses
             ,
             which
             are
             distinguished
             by
             their
             Organs
             ,
             and
             by
             their
             different
             kinds
             of
             Phantasmes
             ;
             namely
             ,
             
               Sight
               ,
               Hearing
               ,
               Smell
               ,
               Tast
            
             and
             Touch
             ;
             and
             these
             have
             their
             Organs
             partly
             peculiar
             to
             each
             of
             them
             severally
             ,
             and
             partly
             common
             to
             them
             all
             .
             The
             Organ
             of
             Sight
             is
             partly
             animate
             ,
             and
             partly
             inanimate
             .
             The
             inanimate
             parts
             ,
             are
             the
             three
             Humours
             ;
             namely
             ,
             the
             
               Watry
               Humour
            
             ,
             which
             by
             the
             interposition
             of
             the
             Membrane
             called
             U●ea
             ,
             (
             the
             perforation
             whereof
             is
             called
             the
             Apple
             of
             the
             Eye
             )
             is
             contained
             on
             one
             side
             by
             the
             first
             concave
             superficies
             of
             the
             Eye
             ,
             and
             on
             the
             other
             side
             by
             the
             
               Ciliary
               processes
            
             and
             the
             Coat
             of
             the
             
               Cristalline
               humour
            
             ;
             the
             Cristalline
             ,
             which
             (
             hanging
             in
             the
             midst
             between
             the
             Ciliary
             processes
             ,
             and
             being
             almost
             of
             Spherical
             figure
             ,
             and
             of
             a
             thick
             consistence
             )
             is
             inclosed
             on
             all
             sides
             with
             its
             own
             transparent
             Coat
             ;
             and
             the
             Vitreous
             or
             
               Glassie
               Humour
            
             ,
             which
             filleth
             all
             the
             rest
             of
             the
             Cavity
             of
             the
             Eye
             ,
             and
             is
             somewhat
             thicker
             then
             the
             Watry
             Humour
             ,
             but
             thinner
             then
             the
             Cristalline
             .
             The
             animate
             part
             of
             the
             Organ
             is
             ,
             first
             ,
             the
             Membrane
             Choroeides
             ,
             which
             is
             a
             part
             of
             the
             
               Pia
               Mater
            
             ,
             saving
             that
             it
             is
             covered
             with
             a
             Coat
             derived
             from
             the
             marrow
             of
             the
             Optique
             Nerve
             ,
             which
             is
             called
             the
             Retina
             ;
             and
             this
             Choroeides
             ,
             seeing
             it
             is
             part
             of
             the
             
               Pia
               Mater
            
             ,
             is
             continued
             to
             the
             beginning
             of
             the
             
               Medulla
               Spinalis
            
             within
             the
             Scull
             ,
             in
             which
             all
             the
             Nerves
             which
             are
             within
             the
             Head
             have
             their
             roots
             .
             Wherefore
             all
             the
             
               Animal
               Spirits
            
             that
             the
             Nerves
             receive
             ,
             enter
             into
             them
             there
             ,
             for
             it
             is
             not
             imaginable
             that
             they
             can
             enter
             into
             them
             any
             where
             else
             .
             Seeing
             therefore
             Sense
             is
             nothing
             else
             but
             the
             action
             of
             Objects
             propagated
             to
             the
             furthest
             part
             of
             the
             Organ
             ;
             and
             seeing
             also
             that
             
               Animal
               Spirits
            
             are
             nothing
             but
             
               Vital
               Spirits
            
             purified
             by
             the
             Hart
             and
             carried
             from
             it
             by
             the
             Arteries
             ;
             it
             follows
             necessarily
             ,
             that
             the
             action
             is
             derived
             from
             the
             Heart
             by
             some
             of
             the
             Arteries
             to
             the
             roots
             of
             the
             Nerves
             
             which
             are
             in
             the
             Head
             ,
             whether
             those
             Arteries
             be
             the
             
               Plexus
               Retiformis
            
             ,
             or
             whether
             they
             be
             other
             Arteries
             which
             are
             inserted
             into
             the
             substance
             of
             the
             Brain
             .
             And
             therefore
             those
             Arteries
             are
             the
             Complement
             ,
             or
             the
             remaying
             part
             of
             the
             whole
             Organ
             of
             Sight
             .
             And
             this
             last
             part
             is
             a
             common
             Organ
             to
             all
             the
             Senses
             ;
             wheras
             that
             which
             reacheth
             from
             the
             Eie
             to
             the
             roots
             of
             the
             Nerves
             is
             proper
             onely
             to
             Sight
             .
             The
             proper
             Organ
             of
             Hearing
             is
             the
             Tympanum
             of
             the
             Eare
             ,
             and
             its
             own
             Nerve
             ;
             from
             which
             to
             the
             Heart
             the
             Organ
             is
             Common
             .
             So
             the
             proper
             Organs
             of
             Smel
             &
             Tast
             are
             Nervous
             Membranes
             ,
             in
             the
             Palate
             and
             Tongue
             for
             the
             Taste
             ,
             and
             in
             the
             Nostrils
             for
             the
             Smell
             ;
             and
             from
             the
             roots
             of
             those
             Nerves
             to
             the
             Heart
             all
             is
             common
             .
             Lastly
             ,
             ●he
             proper
             Organ
             of
             Touch
             are
             Nerves
             and
             Membranes
             dispersed
             through
             the
             whole
             Body
             ;
             which
             Membranes
             are
             derived
             from
             the
             root
             of
             the
             Nerves
             .
             And
             all
             things
             else
             belonging
             alike
             to
             all
             the
             Senses
             seem
             to
             be
             administred
             by
             the
             Arteries
             ,
             and
             not
             by
             the
             Nerves
             .
          
           
             The
             proper
             Phantasme
             of
             Sight
             is
             Light
             ;
             and
             under
             this
             name
             of
             Light
             ,
             Colour
             also
             (
             which
             is
             nothing
             but
             perturbed
             Light
             )
             is
             comprehended
             .
             Wherefore
             the
             Phantasme
             of
             a
             Lucid
             Body
             ,
             is
             Light
             ;
             and
             of
             a
             coloured
             Body
             ,
             Colour
             .
             But
             the
             Object
             of
             Sight
             ,
             properly
             so
             called
             is
             neither
             Light
             nor
             Colour
             ,
             but
             the
             Body
             itself
             which
             is
             lucid
             ,
             or
             enlightned
             ,
             or
             coloured
             .
             For
             Light
             and
             Colour
             being
             Phantasmes
             of
             the
             Sentient
             ,
             cannot
             be
             Accidents
             of
             the
             Object
             .
             Which
             is
             manifest
             enough
             from
             this
             ,
             that
             Visible
             things
             appear
             oftentimes
             in
             places
             ,
             in
             which
             we
             know
             assuredly
             they
             are
             not
             ,
             and
             that
             in
             different
             places
             they
             are
             of
             different
             colours
             ,
             and
             may
             at
             one
             and
             the
             same
             time
             appear
             in
             divers
             places
             .
             Motion
             ,
             Rest
             ,
             Magnitude
             and
             Figure
             are
             common
             both
             to
             the
             Sight
             and
             Touch
             ;
             and
             the
             whole
             appearance
             together
             of
             
               Figure
               ,
               and
               Light
               or
               Colour
               ,
            
             is
             by
             the
             Greeks
             commonly
             called
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             and
             by
             the
             Latines
             ,
             Species
             and
             Imago
             ;
             all
             which
             names
             signifie
             no
             more
             but
             Appearance
             .
          
           
             The
             phantasme
             which
             is
             made
             by
             Hearing
             ,
             is
             Sound
             ;
             by
             Smell
             ,
             Odour
             ;
             by
             Tast
             ,
             Savour
             ;
             and
             by
             Touch
             ,
             Hardness
             and
             
               Softness
               ,
               Heat
            
             and
             
               Cold
               ,
               Wetness
               ,
               Oiliness
            
             ,
             and
             many
             more
             ,
             which
             are
             easier
             to
             be
             distinguished
             by
             sense
             then
             words
             .
             
               Smoothness
               ,
               Roughness
               ,
               Rarity
            
             and
             Density
             refer
             to
             Figure
             ,
             and
             are
             therefore
             common
             both
             to
             
             Touch
             and
             Sight
             .
             And
             as
             for
             the
             Objects
             of
             Hearing
             ,
             Smel
             ,
             Tast
             ,
             and
             Touch
             ,
             they
             are
             not
             Sound
             ,
             Odour
             ,
             Savour
             ,
             Hardness
             ,
             &c.
             but
             the
             Bodies
             themselves
             from
             which
             Sound
             ,
             Odour
             ,
             Savour
             ,
             Hardness
             ,
             &c.
             proceed
             ;
             Of
             the
             causes
             of
             which
             ,
             and
             of
             the
             manner
             how
             they
             are
             produced
             ,
             I
             shall
             speak
             hereafter
             .
          
           
             But
             these
             Phantasmes
             ,
             though
             they
             be
             effects
             in
             the
             Sentient
             ,
             (
             as
             Subject
             )
             produced
             by
             Objects
             working
             upon
             the
             Organs
             ;
             yet
             there
             are
             also
             other
             effects
             besides
             these
             ,
             produced
             by
             the
             same
             Objects
             ,
             in
             the
             same
             Organs
             ;
             namely
             ,
             certain
             Motions
             proceeding
             from
             Sense
             ,
             which
             are
             called
             
               Animal
               Motions
            
             .
             For
             seeing
             in
             all
             Sense
             of
             external
             things
             ,
             there
             is
             mutual
             Action
             and
             Reaction
             ,
             that
             is
             ,
             two
             Endeavours
             opposing
             one
             another
             ,
             it
             is
             manifest
             ,
             that
             the
             motion
             of
             both
             of
             them
             together
             will
             be
             continued
             every
             way
             ,
             especially
             to
             the
             confines
             of
             both
             the
             Bodies
             .
             And
             when
             this
             happens
             in
             the
             internal
             Organ
             ,
             the
             Endeavour
             outwards
             will
             proceed
             in
             a
             solid
             Angle
             ,
             which
             will
             be
             greater
             ,
             and
             consequently
             the
             Idea
             greater
             ,
             then
             it
             would
             have
             been
             if
             the
             impression
             had
             been
             weaker
             .
          
           
             11
             From
             hence
             the
             Natural
             cause
             is
             manifest
             ,
             First
             ,
             why
             those
             things
             seem
             to
             be
             greater
             ,
             which
             (
             
               caeteris
               paribus
            
             )
             are
             seen
             in
             a
             greater
             Angle
             .
             Secondly
             ,
             why
             in
             a
             serene
             cold
             night
             when
             the
             Moon
             doth
             not
             shine
             ,
             more
             of
             the
             fixed
             Stars
             appear
             ,
             then
             at
             another
             time
             .
             For
             their
             action
             is
             less
             hindred
             by
             the
             serenity
             of
             the
             Aire
             ,
             and
             not
             obscured
             by
             the
             greater
             Light
             of
             the
             Moon
             ,
             which
             is
             then
             absent
             ;
             and
             the
             Cold
             making
             the
             Air
             more
             pressing
             ,
             helpeth
             or
             strengtheneth
             the
             action
             of
             the
             Stars
             upon
             our
             Eies
             ,
             in
             so
             much
             as
             Stars
             may
             then
             be
             seen
             which
             are
             seen
             at
             no
             other
             time
             .
             And
             this
             may
             suffice
             to
             be
             said
             in
             general
             concerning
             Sense
             made
             by
             the
             Reaction
             of
             the
             Organ
             .
             For
             ,
             as
             for
             the
             place
             of
             the
             Image
             ,
             the
             deceptions
             of
             Sight
             ,
             and
             other
             things
             of
             which
             we
             have
             experience
             in
             our selves
             by
             Sense
             ,
             being
             they
             depend
             for
             the
             most
             part
             upon
             the
             Fabrick
             it self
             of
             the
             Eie
             of
             Man
             ,
             I
             shall
             speak
             of
             them
             then
             when
             I
             come
             to
             speak
             of
             Man.
             
          
           
             12
             But
             ,
             there
             is
             another
             kind
             of
             Sense
             ,
             of
             which
             I
             will
             say
             somthing
             in
             this
             place
             ,
             namely
             ,
             the
             Sense
             of
             Pleasure
             and
             Pain
             ,
             proceeding
             ,
             not
             from
             the
             Reaction
             of
             the
             Heart
             outwards
             ,
             but
             from
             continual
             action
             from
             the
             outermost
             part
             of
             the
             Organ
             towards
             the
             
             Heart
             .
             For
             the
             original
             of
             Life
             being
             in
             the
             Heart
             ,
             that
             motion
             in
             the
             Sentient
             which
             is
             propagated
             to
             the
             Heart
             ,
             must
             necessarily
             make
             some
             alteration
             or
             diversion
             of
             Vital
             Motion
             ,
             namely
             ,
             by
             quickning
             or
             slackening
             ,
             helping
             or
             hindering
             the
             same
             .
             Now
             when
             it
             helpeth
             ,
             it
             is
             Pleasure
             ,
             and
             when
             it
             hindereth
             ,
             it
             is
             Pain
             ,
             Trouble
             ,
             Grief
             ,
             &c.
             
             And
             as
             Phantasmes
             seem
             to
             be
             without
             ,
             by
             reason
             of
             the
             Endeavour
             outwards
             ;
             so
             Pleasure
             and
             Pain
             by
             reason
             of
             the
             Endeavour
             of
             the
             Organ
             inwards
             seem
             to
             be
             within
             ,
             namely
             ,
             there
             where
             the
             first
             Cause
             of
             the
             Pleasure
             or
             Pain
             is
             ;
             as
             when
             the
             Pain
             proceeds
             from
             a
             Wound
             ,
             we
             think
             the
             Pain
             and
             the
             Wound
             are
             both
             in
             the
             same
             place
             .
          
           
             Now
             Vital
             Motion
             ,
             is
             the
             Motion
             of
             the
             Bloud
             ,
             perpetually
             circulating
             (
             as
             hath
             been
             shewn
             from
             many
             infallible
             signes
             and
             marks
             by
             Doctor
             Harvey
             ,
             the
             first
             Observer
             of
             it
             )
             in
             the
             Veins
             and
             Arteries
             .
             Which
             Motion
             ,
             when
             it
             is
             hindered
             by
             some
             other
             Motion
             made
             by
             the
             action
             of
             sensible
             Objects
             ,
             may
             be
             restored
             again
             either
             by
             bending
             or
             setting
             straight
             the
             parts
             of
             the
             Body
             ;
             which
             is
             done
             when
             the
             Spirits
             are
             carried
             now
             into
             these
             ,
             now
             into
             other
             Nerves
             ,
             till
             the
             Pain
             (
             as
             farre
             as
             is
             possible
             )
             be
             quite
             taken
             away
             .
             But
             if
             Vital
             Motion
             be
             helped
             by
             Motion
             made
             by
             Sense
             ,
             then
             the
             parts
             of
             the
             Organ
             will
             be
             disposed
             to
             guide
             the
             Spirits
             in
             such
             manner
             ,
             as
             conduceth
             most
             to
             the
             preservation
             and
             augmentation
             of
             that
             motion
             by
             the
             help
             of
             the
             Nerves
             .
             And
             in
             animal
             motion
             this
             is
             the
             very
             first
             Endeavour
             ,
             and
             found
             even
             in
             the
             Embrio
             ;
             which
             while
             it
             is
             in
             the
             wombe
             ,
             moveth
             its
             limbes
             with
             voluntary
             motion
             ,
             for
             the
             avoiding
             of
             whatsoever
             troubleth
             it
             ,
             or
             for
             the
             pursuing
             of
             what
             pleaseth
             it
             .
             And
             this
             first
             Endeavour
             ,
             when
             it
             tends
             towards
             such
             things
             as
             are
             known
             by
             experience
             to
             be
             pleasant
             ,
             is
             called
             APPETITE
             ,
             (
             that
             is
             ,
             an
             Approaching
             ;
             )
             and
             when
             it
             shuns
             what
             is
             troublesome
             ,
             AVERSION
             ,
             or
             Flying
             from
             it
             .
             And
             little
             Infants
             ,
             at
             the
             beginning
             ,
             and
             as
             soon
             as
             they
             are
             born
             ,
             have
             appetite
             to
             very
             few
             things
             ,
             as
             also
             they
             avoid
             very
             few
             ,
             by
             reason
             of
             their
             want
             of
             Experience
             and
             Memory
             ;
             &
             therefore
             they
             have
             not
             so
             great
             a
             variety
             of
             animal
             Motion
             as
             we
             see
             in
             those
             that
             are
             more
             grown
             .
             For
             it
             is
             not
             possible
             ,
             without
             such
             knowledge
             as
             is
             derived
             from
             Sense
             ,
             that
             is
             ,
             without
             Experience
             and
             Memory
             ,
             to
             know
             what
             will
             prove
             pleasant
             ,
             or
             hurtful
             ;
             onely
             there
             is
             some
             
             place
             for
             conjecture
             from
             the
             looks
             or
             aspects
             of
             things
             .
             And
             hence
             it
             is
             ,
             that
             though
             they
             do
             not
             know
             what
             may
             do
             them
             good
             or
             harm
             ,
             yet
             sometimes
             they
             approach
             ,
             and
             sometimes
             retire
             from
             the
             same
             thing
             as
             their
             doubt
             prompts
             them
             .
             But
             afterwards
             by
             accustoming
             themselves
             by
             little
             and
             little
             ,
             they
             come
             to
             know
             readily
             what
             is
             to
             be
             pursued
             ,
             and
             what
             to
             be
             avoided
             ;
             and
             also
             to
             have
             a
             ready
             use
             of
             their
             Nerves
             and
             other
             Organs
             in
             the
             pursuing
             and
             avoiding
             of
             good
             and
             bad
             .
             Wherefore
             Appetite
             and
             Aversion
             are
             the
             first
             Endeavours
             of
             
               Animal
               Motion
            
             .
          
           
             Consequent
             to
             this
             first
             Endeavour
             ,
             is
             the
             Impulsion
             into
             the
             Nerves
             ,
             and
             Retraction
             again
             of
             Animal
             Spirits
             ,
             of
             which
             it
             is
             necessary
             there
             be
             some
             Receptacle
             on
             place
             neer
             the
             original
             of
             the
             Nerves
             ;
             and
             this
             Motion
             or
             Endeavour
             is
             followed
             by
             a
             swelling
             and
             Relaxation
             of
             the
             Muscles
             ;
             and
             lastly
             ,
             these
             are
             followed
             by
             Contraction
             and
             Extension
             of
             the
             limbes
             ,
             which
             is
             Animal
             Motion
             .
          
           
             13
             The
             Considerations
             of
             Appetites
             and
             Aversions
             are
             divers
             .
             For
             seeing
             Living
             Creatures
             have
             sometimes
             Appetite
             ,
             and
             sometimes
             Aversion
             to
             the
             same
             thing
             ,
             as
             they
             think
             it
             will
             either
             be
             for
             their
             good
             or
             their
             hurt
             ,
             while
             that
             vicissitude
             of
             Appetites
             and
             Aversions
             remains
             in
             them
             ▪
             they
             have
             that
             series
             of
             Thoughts
             which
             is
             called
             DELIBERATION
             ;
             which
             lasteth
             as
             long
             as
             they
             have
             it
             in
             their
             power
             to
             obtain
             that
             which
             pleaseth
             ,
             or
             to
             avoid
             that
             which
             displeaseth
             them
             .
             Appetite
             therefore
             and
             Aversion
             are
             simply
             so
             called
             as
             long
             as
             they
             follow
             not
             Deliberation
             .
             But
             if
             Deliberation
             have
             gone
             before
             ,
             then
             the
             last
             act
             of
             it
             ,
             if
             it
             be
             Appetite
             ,
             is
             called
             WILL
             ;
             if
             Aversion
             ,
             UNWILLINGNESSE
             ;
             so
             that
             the
             same
             thing
             is
             called
             both
             Will
             and
             Appetite
             ;
             but
             the
             consideration
             of
             them
             (
             namely
             ,
             before
             and
             after
             Deliberation
             )
             is
             divers
             .
             Nor
             is
             that
             which
             is
             done
             within
             a
             Man
             whilest
             he
             Willeth
             any
             thing
             ,
             different
             from
             that
             which
             is
             done
             in
             other
             living
             Creatures
             ,
             whilest
             (
             Deliberation
             having
             preceded
             )
             they
             have
             Appetite
             .
          
           
             Neither
             is
             the
             freedome
             of
             Willing
             ,
             or
             not
             willing
             ,
             greater
             in
             Man
             ,
             then
             in
             other
             living
             Creatures
             .
             For
             where
             there
             is
             Appetite
             ,
             the
             entire
             cause
             of
             Appetite
             hath
             preceded
             ;
             and
             consequently
             the
             act
             of
             Appetite
             could
             not
             choose
             but
             follow
             ,
             that
             is
             ,
             hath
             of
             necessity
             followed
             ,
             (
             as
             is
             shewn
             Chapt.
             9th
             Article
             5.
             )
             And
             therefore
             
             such
             a
             Liberty
             as
             is
             free
             from
             Necessity
             ,
             is
             not
             to
             be
             found
             in
             the
             Will
             either
             of
             Men
             or
             Beasts
             .
             But
             if
             by
             Liberty
             we
             understand
             the
             faculty
             or
             power
             ,
             not
             of
             Willing
             ,
             but
             of
             Doing
             what
             they
             Will
             ,
             then
             certainly
             that
             Liberty
             is
             to
             be
             allowed
             to
             both
             ;
             and
             both
             may
             equally
             have
             it
             ,
             whensover
             it
             is
             to
             be
             had
             .
          
           
             Again
             ,
             when
             Appetite
             and
             Aversion
             do
             with
             celerity
             succeed
             one
             another
             ,
             the
             whole
             series
             made
             by
             them
             ,
             hath
             its
             name
             sometimes
             from
             one
             ,
             sometimes
             from
             the
             other
             .
             For
             the
             same
             Deliberation
             (
             whilest
             it
             inclines
             sometimes
             to
             one
             ,
             sometimes
             to
             the
             other
             )
             is
             from
             Appetite
             called
             HOPE
             ,
             and
             from
             Aversion
             FEAR
             .
             For
             where
             there
             is
             no
             Hope
             ,
             it
             is
             not
             to
             be
             called
             Fear
             ,
             but
             HATE
             ;
             and
             where
             no
             Fear
             ,
             not
             Hope
             ,
             but
             DESIRE
             .
             To
             conclude
             ,
             all
             the
             Passions
             ,
             called
             
               Passions
               of
               the
               Minde
            
             ,
             consist
             of
             Appetite
             and
             Aversion
             (
             except
             pure
             Pleasure
             and
             Pain
             ,
             which
             are
             a
             certain
             Fruition
             of
             good
             or
             Evil
             ;
             )
             as
             Anger
             ,
             is
             Aversion
             from
             some
             imminent
             evil
             ,
             but
             such
             as
             is
             joyned
             with
             Appetite
             of
             avoiding
             that
             evil
             by
             force
             .
             But
             because
             the
             Passions
             and
             Perturbations
             of
             the
             Minde
             are
             innumerable
             ;
             and
             many
             of
             them
             not
             to
             be
             discerned
             in
             any
             Creatures
             besides
             Men
             ;
             I
             will
             speak
             of
             them
             more
             at
             large
             in
             that
             Section
             which
             is
             concerning
             Man.
             As
             for
             those
             Objects
             (
             if
             there
             be
             any
             such
             )
             which
             do
             not
             at
             all
             stir
             the
             Mind
             ,
             we
             are
             said
             to
             Contemn
             them
             .
          
           
             And
             thus
             much
             of
             Sense
             in
             general
             .
             In
             the
             next
             place
             I
             shall
             speak
             of
             Sensible
             Objects
             .
          
        
         
           
           
             
               CHAP.
               XXVI
            
             .
             Of
             the
             World
             ,
             and
             of
             the
             Starres
             .
          
           
             
               
                 1
                 The
                 Magnitude
                 and
                 Duration
                 of
                 the
                 World
                 ,
                 inscrutable
                 .
              
               
                 2
                 No
                 place
                 in
                 the
                 World
                 Empty
                 .
              
               
                 3
                 The
                 arguments
                 of
                 Lucretius
                 for
                 Vacuum
                 ,
                 invalid
                 .
              
               
                 4
                 Other
                 arguments
                 for
                 the
                 establishing
                 of
                 Vacuum
                 ,
                 invalid
                 .
              
               
                 5
                 Six
                 suppositions
                 for
                 the
                 salving
                 of
                 the
                 Phaenomena
                 of
                 Nature
                 .
              
               
                 6
                 Possible
                 causes
                 of
                 the
                 Motions
                 Annual
                 and
                 Diurnal
                 ;
                 and
                 of
                 the
                 apparent
                 Direction
                 ,
                 Station
                 and
                 Retrogradation
                 of
                 the
                 Planets
                 .
              
               
                 7
                 The
                 supposition
                 of
                 Simple
                 Motion
                 ,
                 why
                 likely
                 .
              
               
                 8
                 The
                 cause
                 of
                 the
                 Excentricity
                 of
                 the
                 annual
                 motion
                 of
                 the
                 Earth
                 .
              
               
                 9
                 The
                 cause
                 why
                 the
                 Moon
                 hath
                 alwayes
                 one
                 and
                 the
                 same
                 face
                 turned
                 towards
                 the
                 Earth
                 .
              
               
                 10
                 The
                 cause
                 of
                 the
                 Tides
                 of
                 the
                 Ocean
                 .
              
               
                 11
                 The
                 cause
                 of
                 the
                 Praecession
                 of
                 the
                 Equinoxes
                 .
              
            
          
           
             1
             COnsequent
             to
             the
             Contemplation
             of
             Sense
             is
             the
             contemplation
             of
             Bodies
             ,
             which
             are
             the
             efficient
             causes
             or
             Objects
             of
             Sense
             .
             Now
             every
             Object
             is
             either
             a
             part
             of
             the
             whole
             World
             ,
             or
             an
             Aggregate
             of
             parts
             .
             The
             greatest
             of
             all
             Bodies
             ,
             or
             sensible
             Objects
             ,
             is
             the
             World
             it self
             ,
             which
             we
             behold
             when
             we
             look
             round
             about
             us
             from
             this
             point
             of
             the
             same
             which
             we
             call
             the
             Earth
             .
             Concerning
             the
             World
             ,
             as
             it
             is
             one
             Aggregate
             of
             many
             parts
             ,
             the
             things
             that
             fall
             under
             inquiry
             are
             but
             few
             ;
             and
             those
             we
             can
             determine
             ,
             none
             .
             Of
             the
             whole
             World
             we
             may
             inquire
             what
             is
             its
             Magnitude
             ,
             what
             its
             Duration
             ,
             and
             how
             many
             there
             be
             ;
             but
             nothing
             else
             .
             For
             as
             for
             Place
             and
             Time
             ,
             that
             is
             to
             say
             ,
             Magnitude
             and
             Duration
             ,
             they
             are
             only
             our
             own
             fancies
             of
             a
             Body
             simply
             so
             called
             ,
             that
             is
             to
             say
             of
             a
             Body
             indefinitely
             taken
             ,
             as
             I
             have
             shewne
             before
             in
             the
             7
             chapter
             .
             All
             other
             Phantasmes
             ,
             are
             of
             Bodies
             or
             Objects
             as
             they
             are
             distinguished
             from
             one
             another
             ;
             as
             Colour
             ,
             the
             Phantasme
             of
             coloured
             Bodies
             ;
             Sound
             ,
             of
             Bodies
             that
             move
             the
             Sense
             of
             Hearing
             ,
             &c.
             
             The
             questions
             concerning
             the
             Magnitude
             of
             the
             World
             ,
             are
             ,
             whether
             it
             be
             Finite
             or
             
               Infinite
               ,
               Full
            
             or
             
               not
               Full
            
             ;
             Concerning
             its
             Duration
             ,
             whether
             it
             had
             a
             Beginning
             ,
             or
             be
             Eternall
             ;
             and
             concerning
             the
             
             number
             ,
             whether
             there
             be
             One
             or
             Many
             (
             though
             as
             concerning
             the
             Number
             ,
             if
             it
             were
             of
             infinite
             Magnitude
             ,
             there
             could
             be
             no
             controversy
             at
             all
             .
             )
             Also
             if
             it
             had
             a
             beginning
             ,
             then
             by
             what
             Cause
             and
             of
             what
             Matter
             it
             was
             made
             ;
             and
             againe
             ,
             from
             whence
             that
             Cause
             and
             that
             Matter
             had
             their
             being
             ,
             will
             be
             new
             questions
             ;
             till
             at
             last
             we
             come
             to
             one
             or
             many
             eternall
             Cause
             or
             Causes
             .
             And
             the
             determination
             of
             all
             these
             things
             belongeth
             to
             him
             that
             professeth
             the
             universal
             doctrine
             of
             Philosophy
             ,
             in
             case
             as
             much
             could
             be
             known
             as
             can
             be
             sought
             .
             But
             the
             knowledge
             of
             what
             is
             Infinite
             can
             never
             be
             attained
             by
             a
             finite
             Inquirer
             .
             Whatsoever
             we
             know
             that
             are
             Men
             ,
             we
             learn
             it
             from
             our
             Phantasmes
             ;
             and
             of
             Infinite
             (
             whether
             Magnitude
             or
             Time
             )
             there
             is
             no
             Phantasme
             at
             all
             ;
             so
             that
             it
             is
             impossible
             either
             for
             a
             man
             ,
             or
             any
             other
             creature
             to
             have
             any
             conception
             of
             Infinite
             .
             And
             though
             a
             man
             may
             from
             some
             Effect
             proceed
             to
             the
             immediate
             Cause
             thereof
             ,
             &
             frō
             that
             to
             a
             more
             remote
             Cause
             ,
             and
             so
             ascend
             continually
             by
             right
             ratiocination
             from
             Cause
             to
             Cause
             ;
             yet
             he
             will
             not
             be
             able
             to
             proceed
             eternally
             ;
             but
             wearied
             ,
             will
             at
             last
             give
             over
             ,
             without
             knowing
             whether
             it
             were
             possible
             for
             him
             to
             proceed
             to
             an
             end
             ,
             or
             not
             .
             But
             whether
             we
             suppose
             the
             World
             to
             be
             Finite
             ,
             or
             Infinite
             ,
             no
             absurdity
             will
             follow
             .
             For
             the
             same
             things
             which
             now
             appear
             ,
             might
             appear
             ,
             whether
             the
             Creator
             had
             pleased
             it
             should
             be
             Finite
             or
             Infinite
             .
             Besides
             ,
             though
             from
             this
             ,
             that
             nothing
             can
             move
             it self
             ,
             it
             may
             rightly
             be
             inferred
             that
             there
             was
             some
             first
             eternal
             Movent
             ;
             yet
             it
             can
             never
             be
             inferred
             (
             though
             some
             use
             to
             make
             such
             inference
             )
             that
             that
             Movent
             was
             eternally
             Immoveable
             ,
             but
             rather
             eternally
             Moved
             .
             For
             as
             it
             is
             true
             ,
             that
             nothing
             is
             moved
             by
             it self
             ;
             so
             it
             is
             true
             also
             that
             nothing
             is
             moved
             but
             by
             that
             which
             is
             already
             moved
             .
             The
             questions
             therefore
             about
             the
             Magnitude
             and
             Beginning
             of
             the
             World
             ,
             are
             not
             to
             be
             determined
             by
             Philosophers
             ,
             but
             by
             those
             that
             are
             lawfully
             authorised
             to
             order
             the
             Worship
             of
             God.
             For
             as
             Almighty
             God
             ,
             when
             he
             had
             brought
             his
             People
             into
             Judaea
             ,
             allowed
             the
             Priests
             the
             first
             fruits
             reserved
             to
             himself
             ;
             so
             when
             he
             had
             delivered
             up
             the
             World
             to
             the
             disputations
             of
             Men
             ,
             it
             was
             his
             pleasure
             ,
             that
             all
             Opinions
             concerning
             the
             nature
             of
             Infinite
             and
             Eternal
             ,
             known
             onely
             to
             himself
             ,
             should
             (
             as
             the
             first-fruits
             of
             Wisdom
             )
             be
             judged
             by
             those
             whose
             Ministery
             he
             meant
             to
             use
             in
             the
             ordering
             of
             Religion
             .
             I
             cannot
             therefore
             
             commend
             those
             that
             boast
             they
             have
             demonstrated
             by
             reasons
             drawn
             from
             natural
             things
             ,
             that
             the
             World
             had
             a
             Beginning
             .
             They
             are
             contemned
             by
             Idiots
             ,
             because
             they
             understand
             them
             not
             ;
             and
             by
             the
             Learned
             ,
             because
             they
             understand
             them
             ;
             by
             both
             deservedly
             .
             For
             who
             can
             commend
             him
             that
             demonstrates
             thus
             ?
             If
             the
             World
             be
             Eternal
             ,
             then
             an
             infinite
             number
             of
             dayes
             (
             or
             other
             measures
             of
             Time
             )
             preceded
             the
             birth
             of
             Abraham
             .
             But
             the
             birth
             of
             Abraham
             preceded
             the
             birth
             of
             Isaac
             ;
             and
             therefore
             one
             Infinite
             is
             greater
             then
             another
             Infinite
             ,
             or
             one
             Eternal
             then
             another
             Eternal
             ;
             which
             (
             he
             sayes
             )
             is
             absurd
             .
             This
             Demonstration
             is
             like
             his
             ,
             who
             from
             this
             ,
             that
             the
             number
             of
             even
             Numbers
             is
             infinite
             ,
             would
             conclude
             ,
             that
             there
             are
             as
             many
             even
             Numbers
             ,
             as
             there
             are
             Numbers
             simply
             ,
             that
             is
             to
             say
             ,
             the
             even
             Numbers
             are
             as
             many
             ,
             as
             all
             the
             even
             and
             od
             together
             .
             They
             which
             in
             this
             manner
             take
             away
             Eternity
             from
             the
             World
             ,
             do
             they
             not
             by
             the
             same
             means
             take
             away
             Eternity
             from
             the
             Creator
             of
             the
             Wo●ld
             ?
             From
             this
             absurdity
             therefore
             they
             run
             into
             another
             ,
             being
             forced
             to
             call
             Eternity
             
               Nunc
               stans
            
             ,
             a
             standing
             still
             of
             the
             present
             Time
             ,
             or
             
               an
               abiding
               Now
            
             ;
             and
             (
             which
             is
             much
             more
             absurd
             )
             to
             give
             to
             the
             infinite
             number
             of
             Numbers
             ,
             the
             name
             of
             Unity
             .
             But
             why
             should
             Eternity
             be
             called
             an
             
               Abiding
               Now
            
             ,
             rather
             then
             an
             
               Abiding
               Then
            
             ?
             Wherefore
             there
             must
             either
             be
             many
             Eternities
             ,
             or
             Now
             and
             Then
             must
             signifie
             the
             same
             .
             With
             such
             Demonstrators
             as
             these
             ,
             that
             speak
             in
             another
             language
             ,
             it
             is
             impossible
             to
             enter
             into
             disputation
             .
             And
             the
             men
             that
             reason
             thus
             absurdly
             ,
             are
             not
             Idiots
             ,
             but
             (
             which
             makes
             the
             absurdity
             unpardonable
             )
             Geometricians
             ,
             and
             such
             as
             take
             upon
             them
             to
             be
             Judges
             (
             impertinent
             ,
             but
             severe
             Judges
             )
             of
             other
             mens
             Demonstrations
             .
             The
             reason
             is
             this
             ,
             that
             as
             soon
             as
             they
             are
             entangled
             in
             the
             Words
             Infinite
             and
             Eternal
             ,
             (
             of
             which
             we
             have
             in
             our
             mind
             no
             Idea
             ,
             but
             that
             of
             our
             own
             insufficiency
             to
             comprehend
             them
             )
             they
             are
             forced
             either
             to
             speak
             something
             absurd
             ,
             or
             (
             which
             they
             love
             worse
             )
             to
             hold
             their
             peace
             .
             For
             Geometry
             hath
             in
             it
             somewhat
             like
             Wine
             ;
             which
             when
             new
             ,
             is
             windy
             ;
             but
             afterwards
             though
             less
             pleasant
             ,
             yet
             more
             wholsome
             .
             Whatsoever
             therefore
             is
             true
             ,
             young
             Geometricians
             think
             Demonstrable
             ;
             but
             elder
             not
             .
             Wherefore
             I
             purposely
             pass
             over
             the
             Questions
             of
             Infinite
             and
             Eternal
             ;
             contenting
             my self
             with
             that
             Doctrine
             
             concerning
             the
             Beginning
             and
             Magnitude
             of
             the
             World
             ,
             which
             I
             have
             been
             perswaded
             to
             by
             the
             holy
             Scriptures
             ,
             and
             fame
             of
             the
             Miracles
             which
             confirm
             them
             ;
             and
             by
             the
             Custome
             of
             my
             Country
             ,
             and
             reverence
             due
             to
             the
             Laws
             .
             And
             so
             I
             pass
             on
             to
             such
             things
             as
             it
             is
             not
             unlawful
             to
             dispute
             of
             .
          
           
             2
             Concerning
             the
             World
             it
             is
             further
             questioned
             ,
             whether
             the
             parts
             thereof
             be
             contiguous
             to
             one
             another
             ,
             in
             such
             manner
             ,
             as
             not
             to
             admit
             of
             the
             least
             empty
             space
             between
             ;
             and
             the
             disputation
             both
             for
             &
             against
             it
             ,
             is
             carried
             on
             with
             probability
             enough
             .
             For
             the
             taking
             away
             of
             Vacuum
             ,
             I
             will
             instance
             in
             onely
             one
             experiment
             ,
             a
             common
             one
             ,
             but
             (
             I
             think
             )
             unanswerable
             .
          
           
             Let
             AB
             (
             in
             the
             first
             fig.
             )
             represent
             a
             vessel
             (
             such
             as
             Gardiners
             use
             to
             water
             their
             Gardens
             withal
             ;
             )
             whose
             bottom
             B
             is
             ful
             of
             litle
             holes
             ;
             &
             whose
             mouth
             A
             may
             be
             stopt
             with
             ones
             finger
             when
             there
             shall
             be
             need
             .
             If
             now
             this
             vessel
             be
             filled
             with
             water
             ,
             the
             hole
             at
             the
             top
             A
             being
             stopt
             ,
             the
             water
             will
             not
             flow
             out
             at
             any
             of
             the
             holes
             in
             the
             bottom
             B.
             But
             if
             the
             finger
             be
             removed
             to
             let
             in
             the
             air
             above
             ,
             it
             will
             run
             out
             at
             them
             all
             ;
             and
             as
             soon
             as
             the
             finger
             is
             applyed
             to
             it
             again
             ,
             the
             water
             wil
             suddenly
             &
             totally
             be
             stayed
             again
             from
             running
             out
             .
             The
             cause
             whereof
             seems
             to
             be
             no
             other
             but
             this
             ,
             that
             the
             Water
             cannot
             by
             its
             natural
             endeavour
             to
             descend
             ,
             drive
             down
             the
             aire
             below
             it
             ,
             because
             there
             is
             no
             place
             for
             it
             to
             go
             into
             ,
             unless
             either
             by
             thrusting
             away
             the
             next
             contiguous
             aire
             it
             proceed
             by
             continual
             endeavour
             to
             the
             hole
             A
             ,
             where
             it
             may
             enter
             and
             succeed
             into
             the
             place
             of
             the
             water
             that
             floweth
             out
             ;
             or
             else
             by
             resisting
             the
             endeavour
             of
             the
             water
             Downwards
             ,
             penetrate
             the
             same
             ,
             and
             pass
             up
             through
             it
             .
             By
             the
             first
             of
             these
             wayes
             (
             while
             the
             hole
             at
             A
             remains
             stopped
             )
             there
             is
             no
             possible
             passage
             ;
             nor
             by
             the
             second
             ,
             unless
             the
             holes
             be
             so
             great
             ,
             that
             the
             water
             flowing
             out
             at
             them
             ,
             can
             by
             its
             own
             waight
             force
             the
             Air
             at
             the
             same
             time
             to
             ascend
             into
             the
             vessel
             by
             the
             same
             holes
             ;
             as
             we
             see
             it
             does
             in
             a
             vessel
             whose
             mouth
             is
             wide
             enough
             ,
             when
             we
             turn
             suddenly
             the
             bottom
             upwards
             to
             poure
             out
             the
             water
             ;
             for
             then
             the
             Aire
             being
             forced
             by
             the
             waight
             of
             the
             water
             ,
             enters
             (
             as
             is
             evident
             by
             the
             sobbing
             and
             resistance
             of
             the
             water
             )
             at
             the
             sides
             or
             circumference
             of
             the
             orifice
             .
             And
             this
             I
             take
             for
             a
             sign
             that
             all
             Space
             is
             full
             ;
             for
             without
             this
             the
             naturall
             motion
             of
             the
             water
             (
             which
             is
             a
             heavy
             Body
             )
             downwards
             ,
             would
             not
             be
             hindered
             .
          
           
           
             3
             On
             the
             contrary
             ,
             for
             the
             establishing
             of
             Vacuum
             ,
             many
             &
             specious
             arguments
             and
             experiments
             have
             been
             brought
             .
             Neverthelesse
             there
             seemes
             to
             be
             something
             wanting
             in
             all
             of
             them
             to
             conclude
             it
             firmely
             .
             These
             arguments
             for
             Vacuum
             are
             partly
             made
             by
             the
             followers
             of
             the
             doctrine
             of
             Epicurus
             ;
             who
             taught
             that
             the
             World
             consists
             of
             very
             small
             Spaces
             not
             filled
             by
             any
             Bodie
             ,
             and
             of
             very
             small
             Bodies
             that
             have
             within
             them
             no
             empty
             Space
             ,
             (
             which
             by
             reason
             of
             their
             hardnesse
             he
             calls
             Atomes
             )
             ;
             and
             that
             these
             small
             Bodies
             and
             Spaces
             are
             every
             where
             intermingled
             .
             Their
             arguments
             are
             thus
             delivered
             by
             Lucretius
             .
          
           
             And
             first
             he
             sayes
             that
             unlesse
             it
             were
             so
             ,
             there
             could
             be
             no
             motion
             ;
             For
             the
             office
             and
             property
             of
             Bodies
             is
             to
             withstand
             and
             hinder
             motion
             .
             If
             therfore
             the
             Universe
             were
             filled
             with
             Body
             ,
             motion
             would
             every
             where
             be
             hindered
             ,
             so
             ,
             as
             to
             have
             no
             beginning
             any
             where
             ;
             &
             consequently
             there
             would
             be
             no
             motion
             at
             all
             .
             It
             is
             true
             that
             in
             whatsoever
             is
             full
             ,
             and
             at
             rest
             in
             all
             its
             parts
             ,
             it
             is
             not
             possible
             motion
             should
             have
             beginning
             .
             But
             nothing
             is
             drawn
             from
             hence
             for
             the
             proving
             of
             Vacuum
             .
             For
             though
             it
             should
             be
             granted
             that
             there
             is
             Vacuum
             ;
             yet
             if
             the
             Bodies
             which
             are
             intermingled
             with
             it
             ,
             should
             all
             at
             once
             and
             together
             be
             at
             rest
             ,
             they
             would
             never
             be
             moved
             again
             .
             For
             it
             has
             been
             demonstrated
             above
             (
             in
             the
             9th
             Chapter
             7th
             Article
             )
             that
             nothing
             can
             be
             moved
             but
             by
             that
             which
             is
             contiguous
             and
             already
             moved
             .
             But
             supposing
             that
             all
             things
             are
             at
             rest
             together
             ,
             there
             can
             be
             nothing
             contiguous
             and
             moved
             ;
             and
             therefore
             no
             beginning
             of
             motion
             .
             Now
             the
             denying
             of
             the
             beginning
             of
             motion
             ,
             doth
             not
             take
             away
             present
             motion
             ,
             unless
             beginning
             be
             taken
             away
             from
             Body
             also
             .
             For
             motion
             may
             be
             either
             coeternal
             ,
             or
             concreated
             with
             Body
             .
             Nor
             doth
             it
             seem
             more
             necessary
             that
             Bodies
             were
             first
             at
             rest
             ,
             and
             afterwards
             moved
             ,
             then
             that
             they
             were
             first
             moved
             ,
             and
             rested
             (
             if
             ever
             they
             rested
             at
             all
             )
             afterwards
             .
             Neither
             doth
             there
             appear
             any
             cause
             ,
             why
             the
             matter
             of
             the
             World
             should
             for
             the
             admission
             of
             motion
             ,
             be
             intermingled
             with
             empty
             spaces
             ,
             rather
             then
             full
             ;
             I
             say
             full
             ,
             but
             withall
             fluid
             .
             Nor
             lastly
             ,
             is
             there
             any
             reason
             why
             those
             hard
             Atomes
             may
             not
             also
             by
             the
             motion
             of
             intermingled
             fluid
             matter
             be
             congregated
             &
             brought
             together
             into
             compounded
             Bodies
             of
             such
             bigness
             as
             we
             see
             .
             Wherefore
             nothing
             can
             by
             this
             argument
             be
             
             concluded
             ,
             but
             that
             motion
             was
             either
             coeternal
             ,
             or
             of
             the
             same
             duration
             with
             that
             which
             is
             moved
             ;
             neither
             of
             which
             conclusions
             consisteth
             with
             the
             doctrine
             of
             Epicurus
             ,
             who
             allows
             neither
             to
             the
             World
             nor
             to
             Motion
             any
             Beginning
             at
             all
             .
             The
             necessity
             therefore
             of
             Vacuum
             is
             not
             hitherto
             demonstrated
             .
             And
             the
             cause
             (
             as
             far
             as
             I
             understand
             from
             them
             that
             have
             discoursed
             with
             me
             of
             Vacuum
             )
             is
             this
             ,
             that
             whilest
             they
             contemplate
             the
             nature
             of
             Fluid
             ,
             they
             conceive
             it
             to
             consist
             as
             it
             were
             of
             small
             grains
             of
             hard
             matter
             ,
             in
             such
             manner
             as
             meal
             is
             fluid
             ,
             made
             so
             by
             grinding
             of
             the
             Corn
             ;
             when
             nevertheless
             it
             is
             possible
             to
             conceive
             Fluid
             to
             be
             of
             its
             own
             nature
             as
             homogeneous
             ,
             as
             either
             an
             Atome
             ,
             or
             as
             Vacuum
             ,
             it self
             .
          
           
             The
             second
             of
             their
             arguments
             is
             taken
             from
             waight
             ,
             and
             is
             contained
             in
             these
             Verses
             of
             Lucretius
             ,
          
           
             
               Corporis
               officium
               est
               quoniam
               premere
               omnia
               deorsum
               ;
            
             
               Contrà
               autem
               natura
               manet
               sine
               Pondere
               Inanis
               ;
            
             
               Ergo
               quod
               magnum
               est
               aeque
               ,
               Leviusque
               videtur
               ,
            
             
               Nimirum
               plus
               esse
               sibi
               declarat
               Inanis
               .
            
          
           
             That
             is
             to
             say
             ,
             
               Seeing
               the
               office
               and
               property
               of
               Body
               is
               to
               press
               all
               things
               downwards
               ;
               and
               on
               the
               contrary
               ,
               seeing
               the
               nature
               of
            
             Vacuum
             
               is
               to
               have
               no
               waight
               at
               all
               ;
               Therefore
               when
               of
               two
               Bodies
               of
               equal
               magnitude
               ,
               one
               is
               lighter
               then
               the
               other
               ,
               it
               is
               manifest
               that
               the
               lighter
               Body
               hath
               in
               it
               more
            
             Vacuum
             
               then
               the
               other
            
             .
          
           
             To
             say
             nothing
             of
             the
             Assumption
             concerning
             the
             endeavour
             of
             Bodies
             downwards
             ,
             which
             is
             not
             rightly
             assumed
             ,
             because
             the
             World
             hath
             nothing
             to
             do
             with
             Downwards
             ,
             which
             is
             a
             mere
             fiction
             of
             ours
             ;
             Nor
             of
             this
             ,
             that
             if
             al
             things
             tended
             to
             the
             same
             lowest
             part
             of
             the
             World
             ,
             either
             there
             would
             be
             no
             coalescence
             at
             all
             of
             Bodies
             ,
             or
             they
             would
             all
             be
             gathered
             together
             into
             the
             same
             place
             .
             This
             onely
             is
             sufficient
             to
             take
             away
             the
             force
             of
             the
             argument
             ,
             that
             Aire
             intermingled
             with
             those
             his
             Atomes
             ,
             had
             served
             as
             well
             for
             his
             purpose
             ,
             as
             his
             intermingled
             Vacuum
             .
          
           
             The
             third
             argument
             is
             drawn
             from
             this
             ,
             That
             Lightning
             ,
             Sound
             ,
             Heat
             and
             Cold
             do
             penetrate
             all
             Bodies
             (
             except
             Atomes
             )
             how
             solid
             soever
             they
             be
             .
             But
             this
             reason
             ,
             except
             it
             be
             first
             demonstrated
             that
             the
             same
             things
             cannot
             happen
             (
             without
             Vacuum
             )
             by
             perpetual
             
             generation
             of
             Motion
             ,
             is
             altogether
             invalid
             .
             But
             that
             all
             the
             same
             things
             may
             so
             happen
             ,
             shall
             in
             due
             place
             be
             demonstrated
             .
          
           
             Lastly
             ,
             the
             fourth
             argument
             is
             set
             down
             by
             the
             same
             Lucretius
             in
             these
             Verses
             .
          
           
             
               Duo
               de
               concursu
               corpora
               lata
            
             
               Si
               citò
               dissiliant
               ,
               nempe
               aer
               omne
               necesse
               est
            
             
               Inter
               corpora
               quod
               fuerat
               ,
               possidat
               Inane
               .
            
             
               Is
               porro
               quamvis
               circum
               celerantibus
               auris
            
             
               Confluat
               ,
               haud
               poterit
               tamen
               uno
               tempore
               totum
            
             
               Compleri
               spatium
               ;
               nam
               primum
               quemque
               necesse
               est
            
             
               Occupet
               ille
               locum
               ,
               deinde
               omnia
               possideantur
               .
            
          
           
             That
             is
             ,
             
               If
               two
               flat
               Bodies
               be
               suddenly
               pulled
               asunder
               ,
               of
               necessity
               the
               Air
               must
               come
               between
               them
               to
               fill
               all
               the
               space
               they
               left
               empty
               .
               But
               with
               what
               celerity
               soever
               the
               Air
               flow
               in
               ,
               yet
               it
               cannot
               in
               one
               instant
               of
               time
               fill
               the
               whole
               space
               ,
               but
               first
               one
               part
               of
               it
               ,
               then
               successively
               all
               .
            
          
           
             Which
             nevertheless
             is
             more
             repugnant
             to
             the
             opinion
             of
             Epicurus
             ,
             then
             of
             those
             that
             deny
             Vacuum
             .
             For
             though
             it
             be
             true
             ,
             that
             if
             two
             Bodies
             were
             of
             infinite
             hardness
             ,
             and
             were
             joyned
             together
             by
             their
             Superficies
             which
             were
             most
             exactly
             plain
             ,
             it
             would
             be
             impossible
             to
             pull
             them
             asunder
             ,
             in
             regard
             it
             could
             not
             be
             done
             but
             by
             Motion
             in
             an
             instant
             ;
             yet
             ,
             if
             as
             the
             greatest
             of
             all
             Magnitudes
             cannot
             be
             given
             ,
             nor
             the
             swiftest
             of
             all
             Motions
             ,
             so
             neither
             the
             hardest
             of
             all
             hard
             Bodies
             ;
             it
             might
             be
             ,
             that
             by
             the
             application
             of
             very
             great
             force
             ,
             there
             might
             be
             place
             made
             for
             a
             successive
             flowing
             in
             of
             the
             Aire
             ,
             namely
             by
             separating
             the
             parts
             of
             the
             joyned
             Bodies
             by
             succession
             ,
             beginning
             at
             the
             outermost
             and
             ending
             at
             the
             innermost
             part
             .
             He
             ought
             therefore
             first
             to
             have
             proved
             ,
             that
             there
             are
             some
             Bodies
             extreamly
             hard
             ,
             not
             relatively
             ,
             as
             compared
             with
             softer
             Bodies
             ,
             but
             absolutely
             ,
             that
             is
             to
             say
             ,
             infinitely
             hard
             ;
             which
             is
             not
             true
             .
             But
             if
             we
             suppose
             (
             as
             Epicurus
             doth
             )
             that
             Atomes
             are
             indivisible
             ,
             and
             yet
             have
             small
             superficies
             of
             their
             own
             ;
             then
             if
             two
             Bodies
             should
             be
             joyned
             together
             by
             many
             ,
             or
             but
             one
             onely
             small
             superficies
             of
             either
             of
             them
             ,
             then
             I
             say
             this
             argument
             of
             Lucretius
             would
             be
             a
             firme
             demonstration
             ,
             that
             no
             two
             Bodies
             made
             up
             of
             Atomes
             (
             as
             he
             supposes
             )
             could
             ever
             possibly
             be
             pulled
             asunder
             by
             
             any
             force
             whatsoever
             .
             But
             this
             is
             repugnant
             to
             daily
             experience
             .
          
           
             And
             thus
             much
             of
             the
             arguments
             of
             Lucretius
             .
             Let
             us
             now
             consider
             the
             arguments
             which
             are
             drawn
             from
             the
             experiments
             of
             later
             Writers
             .
          
           
             4
             The
             first
             experiment
             is
             this
             ,
             That
             if
             a
             hollow
             vessel
             be
             thrust
             into
             water
             with
             the
             bottom
             upwards
             ,
             the
             water
             will
             ascend
             into
             it
             ;
             which
             they
             say
             it
             could
             not
             do
             ,
             unless
             the
             Aire
             within
             were
             thrust
             together
             into
             a
             narrower
             place
             ;
             and
             that
             this
             were
             also
             impossible
             except
             there
             were
             little
             empty
             places
             in
             the
             Aire
             .
             Also
             that
             when
             the
             Aire
             is
             compressed
             to
             a
             certain
             degree
             ,
             it
             can
             receive
             no
             further
             compression
             ,
             its
             small
             particles
             not
             suffering
             themselves
             to
             be
             pent
             into
             less
             room
             .
             This
             reason
             ,
             if
             the
             Aire
             could
             not
             pass
             through
             the
             Water
             as
             it
             ascends
             within
             the
             vessel
             ,
             might
             seem
             valid
             .
             But
             it
             is
             sufficiently
             known
             ,
             that
             Aire
             will
             penetrate
             Water
             by
             the
             application
             of
             a
             force
             equal
             to
             the
             gravity
             of
             the
             Water
             .
             If
             therefore
             the
             force
             by
             which
             the
             Vessel
             is
             thrust
             down
             ,
             be
             greater
             ,
             or
             equal
             to
             the
             endeavour
             by
             which
             the
             water
             naturally
             tendeth
             downwards
             ,
             the
             Aire
             will
             go
             out
             that
             way
             where
             the
             resistance
             is
             made
             ,
             namely
             ,
             towards
             the
             edges
             of
             the
             Vessel
             .
             For
             ,
             by
             how
             much
             the
             deeper
             is
             the
             water
             which
             is
             to
             be
             penetrated
             ,
             so
             much
             greater
             must
             be
             the
             depressing
             force
             .
             But
             after
             the
             Vessel
             is
             quite
             under
             water
             ,
             the
             force
             by
             which
             it
             is
             depressed
             ,
             that
             is
             to
             say
             ,
             the
             force
             by
             which
             the
             water
             riseth
             up
             ,
             is
             no
             longer
             encreased
             .
             There
             is
             therefore
             such
             an
             equilibration
             between
             them
             ,
             as
             that
             the
             natural
             endeavour
             of
             the
             water
             downwards
             ,
             is
             equal
             to
             the
             endeavour
             by
             which
             the
             same
             water
             is
             to
             be
             penetrated
             to
             the
             encreased
             depth
             .
          
           
             The
             second
             experiment
             is
             ,
             That
             if
             a
             concave
             Cylinder
             of
             sufficient
             length
             (
             made
             of
             Glass
             ,
             that
             the
             experiment
             may
             be
             the
             better
             seen
             )
             having
             one
             end
             open
             ,
             and
             the
             other
             close
             shut
             ,
             be
             filled
             with
             Quicksilver
             ,
             and
             the
             open
             end
             being
             stopped
             with
             ones
             finger
             ,
             be
             together
             with
             the
             finger
             dipped
             into
             a
             dish
             or
             other
             vessel
             in
             which
             also
             there
             is
             Quicksilver
             ,
             and
             the
             Cylinder
             be
             set
             upright
             ,
             we
             shall
             ,
             the
             finger
             being
             taken
             away
             (
             to
             make
             way
             for
             the
             descent
             of
             the
             Quicksilver
             )
             see
             it
             descend
             into
             the
             Vessel
             under
             it
             ,
             till
             there
             be
             onely
             so
             much
             remayning
             within
             the
             Cylinder
             as
             may
             fill
             about
             26
             Inches
             of
             the
             same
             ;
             and
             thus
             it
             will
             alwayes
             happen
             whatsoever
             be
             the
             Cylinder
             ,
             provided
             that
             the
             length
             be
             
             not
             less
             then
             26
             Inches
             .
             From
             whence
             they
             conclude
             that
             the
             cavity
             of
             the
             Cylinder
             above
             the
             Quicksilver
             remayns
             empty
             of
             all
             Body
             .
             But
             in
             this
             experiment
             I
             finde
             no
             necessity
             at
             all
             of
             Vacuum
             .
             For
             when
             the
             Quicksilver
             which
             is
             in
             the
             Cylinder
             descends
             ,
             the
             Vessel
             under
             it
             must
             needs
             be
             filled
             to
             a
             greater
             height
             ,
             and
             consequently
             so
             much
             of
             the
             contiguous
             Air
             must
             be
             thrust
             away
             as
             may
             make
             place
             for
             the
             Quicksilver
             which
             is
             descended
             .
             Now
             if
             it
             be
             asked
             whether
             that
             Aire
             goes
             ,
             what
             can
             be
             answered
             but
             this
             ,
             that
             it
             thrusteth
             away
             the
             next
             Aire
             ,
             &
             that
             the
             next
             ,
             &
             so
             successively
             ,
             till
             there
             be
             a
             return
             to
             the
             place
             where
             the
             propulsion
             first
             began
             ?
             and
             there
             ,
             the
             last
             Aire
             thus
             thrust
             on
             will
             press
             the
             Quicksilver
             in
             the
             Vessel
             with
             the
             same
             force
             with
             which
             the
             first
             Aire
             was
             thrust
             away
             ;
             and
             if
             the
             force
             with
             which
             the
             Quicksilver
             descends
             be
             great
             enough
             (
             which
             is
             greater
             or
             less
             ,
             as
             it
             descends
             from
             a
             place
             of
             greater
             or
             less
             height
             )
             it
             will
             make
             the
             Aire
             penetrate
             the
             Quicksilver
             in
             the
             vessel
             ,
             and
             go
             up
             into
             the
             Cylinder
             to
             fill
             the
             place
             which
             they
             thought
             was
             left
             empty
             .
             But
             because
             the
             Quicksilver
             hath
             not
             in
             every
             degree
             of
             height
             force
             enough
             to
             cause
             such
             penetration
             ,
             therefore
             in
             descending
             it
             must
             of
             necessity
             stay
             somewhere
             ,
             namely
             there
             ,
             where
             its
             endeavour
             downwards
             ,
             and
             the
             resistance
             of
             the
             same
             to
             the
             penetration
             of
             the
             Aire
             come
             to
             an
             aequilibrium
             .
             And
             by
             this
             experiment
             it
             is
             manifest
             ,
             that
             this
             aequilibrium
             will
             be
             at
             the
             height
             of
             26
             Inches
             ,
             or
             thereabouts
             .
          
           
             The
             third
             experiment
             is
             ,
             That
             when
             a
             Vessel
             hath
             as
             much
             Air
             in
             it
             as
             it
             can
             naturally
             contain
             ,
             there
             may
             nevertheless
             be
             forced
             into
             it
             as
             much
             water
             as
             will
             fill
             three
             quarters
             of
             the
             same
             Vessel
             .
             And
             the
             experiment
             is
             made
             in
             this
             manner
             .
             Into
             the
             glass
             bottle
             ,
             represented
             (
             in
             the
             2d
             figure
             )
             by
             the
             Sphere
             F
             G
             ,
             whose
             center
             is
             A
             ,
             let
             the
             pipe
             B
             A
             C
             be
             so
             fitted
             ,
             that
             it
             may
             precisely
             fill
             the
             mouth
             of
             the
             bottle
             ;
             and
             let
             the
             end
             B
             ,
             be
             so
             neer
             the
             bottom
             ,
             that
             there
             may
             be
             onely
             space
             enough
             left
             for
             the
             free
             passage
             of
             the
             water
             which
             is
             thrust
             in
             above
             .
             Let
             the
             upper
             end
             of
             this
             pipe
             have
             a
             Cover
             at
             D
             ,
             with
             a
             spout
             at
             E
             ,
             by
             which
             the
             water
             (
             when
             it
             ascends
             in
             the
             pipe
             )
             may
             run
             out
             .
             Also
             let
             H
             C
             be
             a
             Cock
             ,
             for
             the
             opening
             or
             shutting
             of
             the
             passage
             of
             the
             water
             between
             B
             and
             D
             ,
             as
             there
             shall
             be
             occasion
             .
             Let
             the
             Cover
             D
             E
             be
             taken
             off
             ;
             and
             the
             Cock
             H
             C
             being
             opened
             ,
             let
             a
             Syringe
             fall
             of
             water
             be
             forced
             
             in
             ;
             and
             before
             the
             Syringe
             be
             taken
             away
             ,
             let
             the
             Cock
             be
             turned
             to
             hinder
             the
             going
             out
             of
             the
             Aire
             .
             And
             in
             this
             manner
             let
             the
             injection
             of
             water
             be
             repeated
             as
             often
             as
             it
             shall
             be
             requisite
             ,
             till
             the
             water
             rise
             within
             the
             bottle
             ,
             for
             example
             ,
             to
             G
             F.
             Lastly
             ,
             the
             Cover
             being
             fastened
             on
             again
             ,
             and
             the
             Cock
             H
             C
             opened
             ,
             the
             water
             will
             run
             swiftly
             out
             at
             E
             ,
             and
             sink
             by
             little
             and
             little
             from
             G
             F
             to
             the
             bottom
             of
             the
             pipe
             B.
             
          
           
             From
             this
             Phaenomenon
             they
             argue
             for
             the
             necessity
             of
             Vacuum
             in
             this
             manner
             .
             The
             Bottle
             from
             the
             beginning
             was
             full
             of
             Aire
             ;
             which
             Aire
             could
             neither
             go
             out
             by
             penetrating
             so
             great
             a
             length
             of
             water
             as
             was
             injected
             by
             the
             pipe
             ,
             nor
             by
             any
             other
             way
             .
             Of
             necessity
             therefore
             all
             the
             water
             as
             high
             as
             F
             G
             ,
             as
             also
             all
             the
             Aire
             that
             was
             in
             the
             bottle
             before
             the
             water
             was
             forced
             in
             ,
             must
             now
             be
             in
             the
             same
             place
             ,
             which
             at
             first
             was
             filled
             by
             the
             Aire
             alone
             ;
             which
             were
             impossible
             ,
             if
             all
             the
             space
             within
             the
             bottle
             were
             formerly
             filled
             with
             Aire
             precisely
             ,
             that
             is
             ,
             without
             any
             Vacuum
             .
             Besides
             ,
             though
             some
             man
             perhaps
             may
             think
             the
             Air
             ,
             being
             a
             thinne
             Body
             ,
             may
             pass
             through
             the
             Body
             of
             the
             water
             contained
             in
             the
             pipe
             ,
             yet
             from
             that
             other
             Phaenomenon
             ,
             (
             namely
             ,
             that
             all
             the
             water
             which
             is
             in
             the
             space
             B
             F
             G
             ,
             is
             cast
             out
             again
             by
             the
             spout
             at
             E
             ,
             for
             which
             it
             seems
             impossible
             that
             any
             other
             reason
             can
             be
             given
             besides
             the
             force
             by
             which
             the
             Aire
             frees
             it self
             from
             compression
             )
             it
             follows
             ,
             that
             either
             there
             was
             in
             the
             bottle
             some
             space
             empty
             ,
             or
             that
             many
             Bodies
             may
             be
             together
             in
             the
             same
             place
             .
             But
             this
             last
             is
             absurd
             ;
             and
             therefore
             the
             former
             is
             true
             ,
             namely
             ,
             that
             there
             was
             Vacuum
             .
          
           
             This
             argument
             is
             infirm
             in
             two
             places
             .
             For
             first
             that
             is
             assumed
             which
             is
             not
             to
             be
             granted
             ;
             and
             in
             the
             second
             place
             an
             experiment
             is
             brought
             ,
             which
             I
             think
             is
             repugnant
             to
             Vacuum
             .
             That
             which
             is
             assumed
             is
             ,
             that
             the
             Aire
             can
             have
             no
             passage
             out
             through
             the
             pipe
             .
             Nevertheless
             we
             see
             daily
             that
             Aire
             easily
             ascends
             from
             the
             bottom
             to
             the
             Superficies
             of
             a
             River
             (
             as
             is
             manifest
             by
             the
             bubbles
             that
             rise
             )
             ;
             nor
             doth
             it
             need
             any
             other
             cause
             to
             give
             it
             this
             motion
             ,
             then
             the
             natural
             endeavour
             downwards
             of
             the
             Water
             .
             Why
             therefore
             may
             not
             the
             endeavour
             upwards
             of
             the
             same
             Water
             acquired
             by
             the
             injection
             (
             which
             endeavour
             upwards
             is
             greater
             then
             the
             natural
             endeavour
             of
             the
             water
             downwards
             )
             cause
             the
             aire
             in
             the
             bottle
             to
             penetrate
             in
             like
             manner
             the
             water
             that
             presseth
             
             it
             downwards
             ;
             especially
             seeing
             the
             water
             as
             it
             riseth
             in
             the
             bottle
             ,
             doth
             so
             press
             the
             Aire
             that
             is
             above
             it
             ,
             as
             that
             it
             generateth
             in
             every
             part
             thereof
             an
             endeavour
             towards
             the
             external
             superficies
             of
             the
             pipe
             ,
             and
             consequently
             maketh
             all
             the
             parts
             of
             the
             enclosed
             aire
             to
             tend
             directly
             towards
             the
             passage
             at
             B
             ?
             I
             say
             this
             is
             no
             less
             manifest
             ,
             then
             that
             he
             aire
             which
             riseth
             up
             from
             the
             bottom
             of
             a
             River
             should
             penetrate
             the
             water
             ,
             how
             deep
             soever
             it
             be
             .
             Werefore
             I
             do
             not
             yet
             see
             any
             cause
             why
             the
             force
             by
             which
             the
             water
             is
             injected
             ,
             should
             not
             at
             the
             same
             time
             eject
             the
             aire
             .
          
           
             And
             as
             for
             their
             arguing
             the
             necessity
             of
             Vacuum
             from
             the
             rejection
             of
             the
             water
             ;
             In
             the
             first
             place
             ,
             supposing
             there
             is
             Vacuum
             ,
             I
             demand
             by
             what
             principle
             of
             motion
             that
             ejection
             is
             made
             .
             Certainly
             ,
             seeing
             this
             motion
             is
             from
             within
             outwards
             ,
             it
             must
             needs
             be
             caused
             by
             some
             Agent
             within
             the
             bottle
             ;
             that
             is
             to
             say
             ,
             by
             the
             aire
             it self
             .
             Now
             the
             motion
             of
             that
             aire
             ,
             being
             caused
             by
             the
             rising
             of
             the
             water
             ,
             begins
             at
             the
             bottom
             ,
             and
             tends
             upwards
             ;
             whereas
             the
             motion
             by
             which
             it
             ejecteth
             the
             water
             ought
             to
             begin
             above
             ,
             and
             tend
             downwards
             .
             From
             whence
             therefore
             hath
             the
             enclosed
             aire
             this
             endeavour
             towards
             the
             bottom
             .
             To
             this
             question
             I
             know
             not
             what
             answer
             can
             be
             given
             ,
             unless
             it
             be
             said
             ,
             that
             the
             aire
             descends
             of
             its
             own
             accord
             to
             expel
             the
             water
             .
             Which
             because
             it
             is
             absurd
             ,
             and
             that
             the
             aire
             after
             the
             water
             is
             forced
             in
             ,
             hath
             as
             much
             room
             as
             its
             magnitude
             requires
             ,
             there
             will
             remain
             no
             cause
             at
             all
             why
             the
             water
             should
             be
             forced
             out
             .
             Wherefore
             the
             assertion
             of
             Vacuum
             is
             repugnant
             to
             the
             very
             experiment
             which
             is
             here
             brought
             to
             establish
             it
             ,
          
           
             Many
             other
             Phaenomena
             are
             usually
             brought
             for
             Vacuum
             ,
             as
             those
             of
             
               Weather-glasses
               ,
               Aeolipiles
               ,
               Wind-guns
            
             ,
             &c.
             
             Which
             would
             all
             be
             very
             hard
             to
             be
             salved
             ,
             unless
             water
             be
             penetrable
             by
             aire
             ,
             without
             the
             intermixture
             of
             empty
             space
             .
             But
             now
             ,
             seeing
             aire
             may
             with
             no
             great
             endeavour
             pass
             through
             ,
             not
             onely
             water
             ,
             but
             any
             other
             fluid
             Body
             ,
             though
             never
             so
             stubborn
             ,
             as
             Quicksilver
             ,
             these
             Phaenomena
             prove
             nothing
             .
             Nevertheless
             ,
             it
             might
             in
             reason
             be
             expected
             ,
             that
             he
             that
             would
             take
             away
             Vacuum
             ,
             should
             without
             Vacuum
             shew
             us
             such
             causes
             of
             these
             Phaenomena
             ,
             as
             should
             be
             at
             least
             of
             equal
             ,
             if
             not
             greater
             probability
             .
             This
             therefore
             shall
             be
             done
             in
             the
             following
             discourse
             ,
             when
             I
             come
             to
             speak
             of
             these
             Phaenomena
             in
             their
             
             proper
             places
             .
             But
             first
             the
             most
             general
             Hypotheses
             of
             natural
             Philosophy
             are
             to
             be
             premised
             .
          
           
             And
             seeing
             that
             Suppositions
             are
             put
             for
             the
             true
             Causes
             of
             apparent
             Effects
             ,
             every
             Supposition
             ,
             except
             such
             as
             be
             absurd
             ,
             must
             of
             necessity
             consist
             of
             some
             supposed
             possible
             Motion
             (
             for
             Rest
             can
             never
             be
             the
             Essicient
             Cause
             of
             any
             thing
             )
             ,
             &
             Motion
             supposeth
             Bodies
             Moveable
             ;
             of
             which
             there
             are
             three
             kinds
             ,
             
               Fluid
               ,
               Consistent
            
             ,
             and
             
               mixt
               of
               both
               .
               Fluid
            
             are
             those
             ,
             whose
             parts
             may
             by
             very
             weak
             endeavonr
             be
             separated
             from
             one
             another
             ;
             and
             Consistent
             those
             ,
             for
             the
             separation
             of
             whose
             parts
             greater
             force
             is
             to
             be
             applyed
             .
             There
             are
             therefore
             degrees
             of
             Consistency
             ;
             which
             degrees
             ,
             by
             comparison
             with
             more
             or
             less
             Consistent
             ,
             have
             the
             names
             of
             Hardness
             ,
             or
             Softness
             .
             Wherefore
             a
             Fluid
             Body
             is
             alwayes
             divisible
             into
             Bodies
             equally
             Fluid
             ,
             as
             Quantity
             into
             Quantities
             ;
             and
             Soft
             Bodies
             ,
             of
             whatsoever
             degree
             of
             Softness
             ,
             into
             Soft
             Bodies
             of
             the
             same
             degree
             .
             And
             though
             many
             men
             seem
             to
             conceive
             no
             other
             difference
             of
             Fluidity
             ,
             but
             such
             as
             ariseth
             from
             the
             different
             magnitudes
             of
             the
             parts
             (
             in
             which
             Sense
             ,
             Dust
             ,
             though
             of
             Diamonds
             ,
             may
             be
             called
             Fluid
             )
             ;
             Yet
             I
             understand
             by
             Fluidity
             ,
             that
             which
             is
             made
             such
             by
             Nature
             equally
             in
             every
             part
             of
             the
             Fluid
             Body
             ;
             not
             as
             Dust
             is
             Fluid
             ,
             for
             so
             a
             House
             which
             is
             falling
             in
             pieces
             may
             be
             called
             Fluid
             ;
             but
             in
             such
             manner
             as
             Water
             seems
             Fluid
             ,
             and
             to
             divide
             it self
             into
             parts
             perpetually
             Fluid
             .
             And
             this
             being
             well
             understood
             ,
             I
             come
             to
             my
             Suppositions
             .
          
           
             5
             First
             ,
             therefore
             I
             suppose
             ,
             That
             the
             Immense
             Space
             which
             we
             call
             the
             World
             ,
             is
             the
             Aggregate
             of
             all
             Bodies
             ;
             which
             are
             either
             Consistent
             &
             Visible
             ,
             as
             the
             Earth
             and
             the
             Starres
             ;
             or
             Invisible
             ,
             as
             the
             small
             Atomes
             which
             are
             disseminated
             through
             the
             whole
             space
             between
             the
             Earth
             and
             the
             Stars
             ;
             and
             lastly
             ,
             that
             most
             Fluid
             Aether
             ,
             which
             so
             fils
             all
             the
             rest
             of
             the
             Universe
             ,
             as
             that
             it
             leaves
             in
             it
             no
             empty
             place
             at
             all
             .
          
           
             Secondly
             ,
             I
             suppose
             with
             Copernicus
             ,
             That
             the
             greater
             Bodies
             of
             the
             World
             ,
             which
             are
             both
             consistent
             and
             permanent
             ,
             have
             such
             order
             amongst
             themselves
             ,
             as
             that
             the
             Sunne
             hath
             the
             first
             place
             ,
             Mercury
             the
             second
             ,
             Venus
             the
             third
             ,
             
               The
               Earth
            
             (
             with
             the
             Moon
             going
             about
             it
             )
             the
             fourth
             ,
             Mars
             the
             fifth
             ,
             Jupiter
             (
             with
             his
             Attendants
             )
             the
             sixth
             ,
             Saturne
             the
             seventh
             ,
             and
             after
             these
             the
             
               Fixed
               Starres
            
             have
             their
             several
             distances
             from
             the
             Sunne
             .
          
           
           
             Thirdly
             ,
             I
             suppose
             ,
             That
             in
             the
             Sunne
             &
             the
             rest
             of
             the
             Planets
             ,
             there
             is
             and
             alwayes
             has
             been
             a
             Simple
             Circular
             Motion
             .
          
           
             Fourthly
             ,
             I
             suppose
             ,
             That
             in
             the
             Body
             of
             the
             Aire
             there
             are
             certain
             other
             Bodies
             intermingled
             ,
             which
             are
             not
             Fluid
             ;
             but
             withal
             that
             they
             are
             so
             small
             ,
             that
             they
             are
             not
             preceptible
             by
             Sense
             ,
             and
             that
             these
             also
             have
             their
             proper
             Simple
             Motion
             ;
             and
             are
             some
             of
             them
             more
             ,
             some
             less
             hard
             or
             consistent
             .
          
           
             Fifthly
             ,
             I
             suppose
             with
             Kepler
             ,
             That
             as
             the
             distance
             between
             the
             Sunne
             and
             the
             Earth
             ,
             is
             to
             the
             distance
             between
             the
             Moon
             and
             the
             Earth
             ;
             so
             the
             distance
             between
             the
             Moon
             and
             the
             Earth
             ,
             is
             to
             the
             Semidiameter
             of
             the
             Earth
             .
          
           
             As
             for
             the
             Magnitude
             of
             the
             Circles
             ,
             and
             the
             Times
             in
             which
             they
             are
             described
             by
             the
             Bodies
             which
             are
             in
             them
             ,
             I
             will
             suppose
             them
             to
             be
             such
             as
             shall
             seem
             most
             agreeable
             to
             the
             Phaenomena
             in
             question
             ,
          
           
             6
             The
             causes
             of
             the
             different
             Seasons
             of
             the
             Year
             ,
             and
             of
             the
             several
             variations
             of
             Dayes
             and
             Nights
             in
             all
             the
             parts
             of
             the
             superficies
             of
             the
             Earth
             ,
             have
             been
             demonstrated
             first
             by
             Copernicus
             ,
             and
             since
             by
             
               Kepler
               ,
               Galilaeus
            
             and
             others
             ,
             from
             the
             supposition
             of
             the
             Earths
             diurnal
             revolution
             about
             its
             own
             Axis
             ,
             together
             with
             its
             Annual
             motion
             about
             the
             Sunne
             ,
             in
             the
             Ecliptick
             according
             to
             the
             order
             of
             the
             Signes
             ;
             and
             thirdly
             ,
             by
             the
             annual
             revolution
             of
             the
             same
             Earth
             about
             its
             own
             center
             contrary
             to
             the
             order
             of
             the
             Signs
             .
             I
             suppose
             with
             Copernicus
             ,
             That
             the
             diurnal
             revolution
             is
             from
             the
             motion
             of
             the
             Earth
             ,
             by
             which
             the
             Aequinoctial
             Circle
             is
             described
             about
             it
             .
             And
             as
             for
             the
             other
             two
             annual
             motions
             ,
             they
             are
             the
             efficient
             cause
             of
             the
             Earths
             being
             carried
             about
             in
             the
             Ecliptick
             in
             such
             manner
             ,
             as
             that
             its
             Axis
             is
             alwayes
             kept
             parallel
             to
             it self
             .
             Which
             parallelisme
             was
             for
             this
             reason
             introduced
             ,
             lest
             by
             the
             Earths
             annual
             revolution
             ,
             its
             Poles
             should
             seem
             to
             be
             necessarily
             carried
             about
             the
             Sunne
             ,
             contrary
             to
             experience
             .
             I
             have
             (
             in
             the
             10th
             Artic.
             of
             the
             ●●th
             Chap.
             )
             demonstrated
             from
             the
             supposition
             of
             Simple
             Circular
             Motion
             in
             the
             Sun
             ,
             that
             the
             Earth
             is
             so
             carried
             about
             the
             Sunne
             ,
             as
             that
             its
             Axis
             is
             thereby
             kept
             always
             parallel
             to
             it self
             .
             Wherefore
             ,
             from
             these
             two
             supposed
             motions
             in
             the
             Sunne
             ,
             the
             one
             Simple
             Circular
             Motion
             ;
             the
             other
             Circular
             Motion
             about
             its
             owns
             Center
             ,
             it
             may
             be
             demonstrated
             ,
             that
             the
             Year
             hath
             both
             the
             same
             variations
             of
             Dayes
             and
             Nights
             ,
             as
             have
             been
             demonstrated
             by
             Copernicus
             .
          
           
           
             For
             if
             the
             Circle
             abcd
             (
             in
             the
             3d
             Figure
             )
             be
             the
             Ecliptick
             ,
             whose
             Center
             is
             e
             ,
             and
             Diameter
             aec
             ;
             and
             the
             Earth
             be
             placed
             in
             a
             ,
             &
             the
             Sunne
             be
             moved
             in
             the
             little
             Circle
             fghi
             ,
             namely
             ,
             according
             to
             the
             order
             
               f
               ,
               g
               ,
               h
            
             &
             i
             ,
             it
             hath
             been
             demonstrated
             ,
             that
             a
             Body
             placed
             in
             a
             ,
             will
             be
             moved
             in
             the
             same
             order
             through
             the
             points
             of
             the
             Ecliptick
             
               a
               ,
               b
               ,
               c
            
             &
             d
             ,
             and
             will
             alwayes
             keep
             its
             Axis
             parallel
             to
             its
             self
             .
          
           
             But
             if
             (
             as
             I
             have
             supposed
             )
             the
             Earth
             also
             be
             moved
             with
             Simple
             Circular
             Motion
             in
             a
             plain
             that
             passeth
             through
             a
             ,
             cutting
             the
             plain
             of
             the
             Ecliptick
             so
             ,
             as
             that
             the
             common
             section
             of
             both
             the
             plains
             be
             in
             ac
             ,
             thus
             also
             the
             Axis
             of
             the
             Earth
             will
             be
             kept
             alwayes
             parallel
             to
             it self
             .
             For
             let
             the
             Center
             of
             the
             Earth
             be
             moved
             about
             in
             the
             Circumference
             of
             the
             Epicycle
             whose
             Diameter
             is
             lak
             ,
             which
             is
             a
             part
             of
             the
             straight
             line
             lac
             .
             Therefore
             lak
             the
             Diameter
             of
             the
             Epicycle
             ,
             passing
             through
             the
             Center
             of
             the
             Earth
             ,
             will
             be
             in
             the
             plain
             of
             the
             Ecliptick
             .
             Wherefore
             seeing
             that
             by
             reason
             of
             the
             Earths
             Simple
             Motion
             both
             in
             the
             Ecliptick
             and
             in
             its
             Epicycle
             ,
             the
             straight
             line
             lak
             is
             kept
             alwayes
             parallel
             to
             it self
             ,
             every
             other
             straight
             line
             also
             taken
             in
             the
             Body
             of
             the
             Earth
             ,
             and
             consequently
             its
             Axis
             ,
             will
             in
             like
             manner
             be
             kept
             alwayes
             parallel
             to
             it self
             ;
             so
             that
             in
             what
             part
             soever
             of
             the
             Ecliptick
             the
             Center
             of
             the
             Epicycle
             be
             found
             ,
             and
             in
             what
             part
             soever
             of
             the
             Epicycle
             the
             Center
             of
             the
             Earth
             be
             found
             at
             the
             same
             time
             ,
             the
             Axis
             of
             the
             Earth
             will
             be
             parallel
             to
             the
             place
             where
             the
             same
             Axis
             would
             have
             been
             ,
             if
             the
             Center
             of
             the
             Earth
             had
             never
             gone
             out
             of
             the
             Ecliptick
             .
          
           
             Now
             as
             I
             have
             demonstrated
             the
             simple
             annual
             motion
             of
             the
             Earth
             from
             the
             supposition
             of
             Simple
             Motion
             in
             the
             Sunne
             ;
             so
             from
             the
             supposition
             of
             Simple
             Motion
             in
             the
             Earth
             may
             be
             demonstrated
             the
             monethly
             Simple
             Motion
             of
             the
             Moon
             .
             For
             if
             the
             names
             be
             but
             changed
             the
             Demonstration
             will
             be
             the
             same
             ,
             and
             therefore
             need
             not
             be
             repeated
             .
          
           
             7
             That
             which
             makes
             this
             supposition
             of
             the
             Sunnes
             Simple
             Motion
             in
             the
             Epicycle
             fghi
             probable
             ,
             is
             First
             ,
             that
             the
             Periods
             of
             all
             the
             Planets
             are
             not
             onely
             described
             about
             the
             Sunne
             ,
             but
             so
             described
             ,
             as
             that
             they
             are
             al
             contained
             within
             the
             Zodiack
             ,
             that
             is
             to
             say
             ,
             within
             the
             latitude
             of
             about
             16
             degrees
             ;
             for
             the
             cause
             of
             this
             
             seems
             to
             depend
             upon
             some
             power
             in
             the
             Sunne
             ,
             especially
             in
             that
             part
             of
             the
             Sunne
             which
             respects
             the
             Zodiack
             .
             Secondly
             ,
             that
             in
             the
             whole
             co●passe
             of
             the
             heavens
             there
             appears
             no
             other
             Body
             ,
             from
             which
             the
             cause
             of
             this
             Phaenomenon
             can
             in
             probability
             be
             derived
             .
             Besides
             ,
             I
             could
             not
             imagine
             ,
             that
             so
             many
             and
             such
             various
             motions
             of
             the
             Planets
             should
             have
             no
             dependance
             at
             all
             upon
             one
             another
             .
             But
             by
             supposing
             motive
             power
             in
             the
             Sunne
             ,
             we
             suppose
             motion
             also
             ;
             for
             power
             to
             move
             ,
             without
             motion
             ,
             is
             no
             power
             at
             all
             .
             I
             have
             therefore
             supposed
             that
             there
             is
             in
             the
             Sunne
             for
             the
             governing
             of
             the
             primary
             Planets
             ,
             and
             in
             the
             Earth
             for
             the
             governing
             of
             the
             Moon
             ,
             such
             motion
             ,
             as
             being
             received
             by
             the
             primary
             Planets
             and
             by
             the
             Moon
             ,
             makes
             them
             necessarily
             appear
             to
             us
             in
             such
             manner
             as
             we
             see
             them
             ;
             Whereas
             ,
             that
             circular
             motion
             (
             which
             is
             commonly
             attributed
             to
             them
             )
             about
             a
             fixed
             Axis
             ,
             (
             which
             is
             called
             Conversion
             )
             being
             a
             motion
             of
             their
             parts
             onely
             ,
             and
             not
             of
             their
             whole
             Bodies
             ,
             is
             insufficient
             to
             salve
             their
             Appearances
             .
             For
             seeing
             whatsoever
             is
             so
             moved
             ,
             hath
             no
             endeavour
             at
             all
             towards
             those
             parts
             which
             are
             without
             the
             circle
             ,
             they
             have
             no
             power
             to
             propagate
             any
             endeavour
             to
             such
             Bodies
             as
             are
             placed
             without
             it
             .
             And
             as
             for
             them
             that
             suppose
             this
             may
             be
             done
             by
             Magnetical
             Virtue
             ,
             or
             by
             incorporeall
             and
             immateriall
             Species
             ,
             they
             suppose
             no
             naturall
             cause
             ;
             nay
             no
             cause
             at
             all
             .
             For
             there
             is
             no
             such
             thing
             as
             an
             Incorporeal
             Movent
             ;
             and
             Magnetical
             Virtue
             is
             a
             thing
             altogether
             unknown
             ;
             and
             whensoever
             it
             shall
             be
             known
             ,
             it
             will
             be
             found
             to
             be
             a
             motion
             of
             Body
             .
             It
             remaines
             therefore
             ,
             that
             if
             the
             primary
             Planets
             be
             carried
             about
             by
             the
             Sunne
             ,
             and
             the
             Moon
             by
             the
             Earth
             ,
             they
             have
             the
             simple
             circular
             motions
             of
             the
             Sunne
             and
             the
             Earth
             for
             the
             causes
             of
             their
             circulations
             .
             Otherwise
             ,
             if
             they
             be
             not
             carried
             about
             by
             the
             Sunne
             and
             the
             Earth
             ,
             but
             that
             every
             Planet
             hath
             been
             moved
             as
             it
             is
             now
             moved
             ever
             since
             it
             was
             made
             ,
             there
             will
             be
             of
             their
             motions
             no
             cause
             naturall
             .
             For
             either
             these
             motions
             were
             concreated
             with
             their
             Bodies
             ,
             and
             their
             cause
             is
             supernatural
             ;
             or
             they
             are
             coeternal
             with
             them
             ,
             and
             so
             they
             have
             no
             cause
             at
             all
             .
             For
             whatsoever
             is
             Eternall
             was
             never
             generated
             .
          
           
             I
             may
             add
             besides
             ,
             to
             confirme
             the
             probability
             of
             this
             simple
             motion
             ,
             that
             allmost
             all
             learned
             men
             are
             now
             of
             the
             same
             opinion
             
             with
             Copernicus
             concerning
             the
             parallelisme
             of
             the
             Axis
             of
             the
             Earth
             ,
             it
             seemed
             to
             me
             to
             be
             more
             agreeable
             to
             truth
             ,
             or
             at
             least
             more
             handsome
             ,
             that
             it
             should
             be
             caused
             by
             simple
             Circular
             Motion
             alone
             ,
             than
             by
             two
             motions
             ,
             one
             in
             the
             Ecliptick
             ,
             and
             the
             other
             about
             the
             Earths
             own
             Axis
             the
             contrary
             way
             ,
             neither
             of
             them
             Simple
             ,
             nor
             either
             of
             them
             such
             as
             might
             be
             produced
             by
             any
             motion
             of
             the
             Sunne
             .
             I
             thought
             best
             therefore
             to
             retain
             this
             Hypothesis
             of
             Simple
             Motion
             ;
             and
             from
             it
             to
             derive
             the
             causes
             of
             as
             many
             of
             the
             Phaenomena
             as
             I
             could
             ,
             and
             to
             let
             such
             alone
             as
             I
             could
             not
             deduce
             frm
             thence
             .
          
           
             It
             will
             perhaps
             be
             objected
             ,
             that
             although
             by
             this
             supposition
             the
             reason
             may
             be
             given
             of
             the
             Parallelisme
             of
             the
             Axis
             of
             the
             Earth
             ,
             and
             of
             many
             other
             Appearances
             ;
             nevertheless
             ,
             seeing
             it
             is
             done
             by
             placing
             the
             Body
             of
             the
             Sunne
             in
             the
             Center
             of
             that
             Orbe
             which
             the
             Earth
             describes
             with
             its
             annual
             motion
             ,
             the
             supposition
             it self
             is
             false
             ,
             because
             this
             annual
             Orbe
             is
             excentrique
             to
             the
             Sunne
             .
             In
             the
             first
             place
             therefore
             let
             us
             examine
             what
             that
             Excentricity
             is
             ,
             and
             whence
             it
             proceeds
             .
          
           
             8
             Let
             the
             annual
             Circle
             of
             the
             Earth
             abcd
             (
             in
             the
             same
             3d
             figure
             )
             be
             divided
             into
             four
             equal
             parts
             by
             the
             straight
             lines
             ac
             &
             bd
             ,
             cutting
             one
             another
             in
             the
             Center
             e
             ;
             and
             let
             a
             be
             the
             beginning
             of
             
               Libra
               ,
               b
            
             of
             
               Capricorn
               ,
               c
            
             of
             Aries
             ,
             and
             d
             of
             Cancer
             ;
             and
             let
             the
             whole
             Orbe
             abcd
             be
             understood
             (
             according
             to
             Copernicus
             )
             to
             have
             every
             way
             so
             great
             distance
             from
             the
             Zodiack
             of
             the
             fixed
             Starres
             ,
             that
             it
             be
             in
             comparison
             with
             it
             but
             as
             a
             point
             .
             Let
             the
             Earth
             be
             now
             supposed
             to
             be
             in
             the
             beginning
             of
             Libra
             at
             a.
             The
             Sunne
             therefore
             will
             appear
             in
             the
             beginning
             of
             Aries
             at
             c.
             Wherefore
             if
             the
             Earth
             be
             moved
             from
             a
             to
             b
             ,
             the
             apparent
             motion
             of
             the
             Sunne
             will
             be
             from
             c
             to
             the
             beginning
             of
             Cancer
             in
             d
             ;
             and
             the
             Earth
             being
             moved
             forwards
             from
             b
             to
             c
             ,
             the
             Sunne
             also
             will
             appear
             to
             be
             moved
             forwards
             to
             the
             beginning
             of
             Libra
             in
             a
             ;
             Wherefore
             cda
             will
             be
             the
             Summer
             Arch
             ,
             and
             the
             Winter
             Arch
             will
             be
             abc
             .
             Now
             in
             the
             time
             of
             the
             Suns
             apparent
             motion
             in
             the
             Summer
             Arch
             ,
             there
             are
             numbred
             186¾
             dayes
             ;
             and
             consequently
             the
             Earth
             makes
             in
             the
             same
             time
             the
             same
             number
             of
             diurnal
             conversions
             in
             the
             Arch
             abc
             ;
             and
             therefore
             the
             Earth
             in
             its
             motion
             through
             the
             Arch
             cda
             will
             make
             onely
             178½
             diurnal
             conversions
             ,
             Wherefore
             
             the
             Arch
             
               a
               b
               c
            
             ought
             to
             be
             greater
             then
             the
             Arch
             
               c
               d
               a
            
             by
             8¼
             dayes
             ,
             that
             is
             to
             say
             ,
             by
             almost
             so
             many
             degrees
             .
             Let
             the
             Arch
             
               a
               r
            
             ,
             as
             also
             
               c
               s
            
             ,
             be
             each
             of
             them
             an
             Arch
             of
             two
             degrees
             and
             1
             /
             16.
             
             Wherefore
             the
             Arch
             
               r
               b
               s
            
             will
             be
             greater
             then
             the
             Semicircle
             
               a
               b
               c
            
             by
             4
             degrees
             and
             ⅛
             ,
             and
             greater
             then
             the
             Arch
             
               s
               d
               r
            
             by
             8
             degrees
             and
             ¼
             .
             The
             Equinoxes
             therefore
             will
             be
             in
             the
             points
             r
             &
             s
             ;
             and
             therefore
             also
             when
             the
             Earth
             is
             in
             r
             ,
             the
             Sunne
             will
             appear
             in
             s.
             Wherefore
             the
             true
             place
             of
             the
             Sunne
             will
             be
             in
             t
             ,
             that
             is
             to
             say
             ,
             without
             the
             Center
             of
             the
             Earths
             annual
             motion
             by
             the
             quantity
             of
             the
             Sine
             of
             the
             Arch
             
               a
               r
            
             ,
             or
             the
             Sine
             of
             two
             degrees
             and
             16
             minutes
             .
             Now
             this
             Sine
             ,
             putting
             100000
             for
             the
             Radius
             ,
             will
             be
             neer
             3580
             parts
             thereof
             .
             And
             so
             munh
             is
             the
             Excentricity
             of
             the
             Earths
             annual
             motion
             ,
             provided
             that
             that
             motion
             be
             in
             a
             perfect
             circle
             ;
             and
             s
             &
             r
             are
             the
             Equinoctial
             points
             ;
             and
             the
             straight
             lines
             
               s
               r
            
             &
             
               c
               a
            
             produced
             both
             wayes
             till
             they
             reach
             the
             Zodiack
             of
             the
             fixed
             Starres
             ,
             wil
             fall
             stil
             upon
             the
             same
             fixed
             Starres
             ,
             because
             the
             whole
             Orbe
             
               a
               b
               c
               d
            
             is
             supposed
             to
             have
             no
             magnitude
             at
             all
             in
             respect
             of
             the
             great
             distance
             of
             the
             fixed
             Starres
             .
          
           
             Supposing
             now
             the
             Sun
             to
             be
             in
             c
             ,
             it
             remains
             that
             I
             shew
             the
             cause
             why
             the
             Earth
             is
             neerer
             to
             the
             Sunne
             ,
             when
             in
             its
             annual
             motion
             it
             is
             found
             to
             be
             in
             d
             ,
             then
             when
             it
             is
             in
             b.
             And
             I
             take
             the
             cause
             to
             be
             this
             .
             When
             the
             Earth
             is
             in
             the
             beginning
             of
             Capricorn
             at
             b
             ,
             the
             Sunne
             appears
             in
             the
             beginning
             of
             Cancer
             at
             d
             ;
             &
             then
             is
             the
             midst
             of
             Summer
             .
             But
             in
             the
             midst
             of
             Summer
             ,
             the
             Northern
             parts
             of
             the
             Earth
             are
             towards
             the
             Sunne
             ,
             which
             is
             almost
             all
             dry
             land
             ,
             containing
             all
             Europe
             ,
             and
             much
             the
             greatest
             part
             of
             Asia
             and
             America
             .
             But
             when
             the
             Earth
             is
             in
             the
             beginning
             of
             Cancer
             at
             d
             ,
             it
             is
             the
             midst
             of
             Winter
             ,
             and
             that
             part
             of
             the
             Earth
             is
             towards
             the
             Sunne
             ,
             which
             contains
             those
             great
             Seas
             called
             the
             South
             Sea
             and
             the
             Indian
             Sea
             ,
             which
             are
             of
             farre
             greater
             extent
             then
             all
             the
             dry
             Land
             in
             that
             Hemisphere
             .
             Wherefore
             (
             by
             the
             last
             Article
             of
             the
             21
             Chapter
             )
             when
             the
             Earth
             is
             in
             d
             ,
             it
             will
             come
             neerer
             to
             its
             first
             Movent
             ,
             that
             is
             ,
             to
             the
             Sunne
             which
             is
             in
             t
             ;
             that
             is
             to
             say
             ,
             the
             Earth
             is
             neerer
             to
             the
             Sunne
             in
             the
             midst
             of
             Winter
             when
             it
             is
             in
             d
             ,
             then
             in
             the
             midst
             of
             Summer
             when
             it
             is
             b
             ;
             and
             therefore
             during
             the
             Winter
             the
             Sunne
             is
             in
             its
             Perigaeum
             ,
             and
             in
             its
             Apogaeum
             during
             the
             Summer
             .
             And
             thus
             I
             have
             shewn
             a
             possible
             cause
             of
             the
             
             Excentricity
             of
             the
             Earth
             ;
             which
             was
             to
             be
             done
             .
          
           
             I
             am
             therefore
             of
             Keplers
             opinion
             ,
             in
             this
             ,
             that
             he
             attributes
             the
             Excentricity
             of
             the
             Earth
             to
             the
             difference
             of
             the
             parts
             thereof
             ,
             and
             supposes
             one
             part
             to
             be
             affected
             ,
             and
             another
             disaffected
             to
             the
             Sunne
             .
             And
             I
             dissent
             from
             him
             in
             this
             ,
             that
             he
             thinks
             it
             to
             be
             by
             Magnetick
             virtue
             ,
             and
             that
             this
             Magnetick
             virtue
             ,
             or
             attraction
             and
             thrusting
             back
             of
             the
             Earth
             is
             wrought
             by
             immateriate
             Species
             ;
             which
             cannot
             be
             ;
             because
             nothing
             can
             give
             motion
             ,
             but
             a
             Body
             moved
             and
             contiguous
             .
             For
             if
             those
             Bodies
             be
             not
             moved
             which
             are
             contiguous
             to
             a
             Body
             unmoved
             ,
             how
             this
             Body
             should
             begin
             to
             be
             moved
             is
             not
             imaginable
             ;
             as
             has
             been
             demonstrated
             in
             the
             7th
             Article
             of
             the
             9th
             Chapter
             ,
             and
             often
             inculcated
             in
             other
             places
             ,
             to
             the
             end
             that
             Philosophers
             might
             at
             last
             abstain
             from
             the
             use
             of
             such
             unconceiveable
             connexions
             of
             words
             .
             I
             dissent
             also
             from
             him
             in
             this
             ,
             that
             he
             says
             the
             similitude
             of
             Bodies
             is
             the
             cause
             of
             their
             mutual
             attraction
             ,
             For
             if
             it
             were
             so
             ,
             I
             see
             no
             reason
             why
             one
             Egg
             should
             not
             be
             attracted
             by
             another
             .
             If
             therefore
             one
             part
             of
             the
             Earth
             be
             more
             affected
             by
             the
             Sunne
             ,
             then
             another
             part
             ,
             it
             proceeds
             from
             this
             ,
             that
             one
             part
             hath
             more
             water
             ,
             the
             other
             more
             dry
             land
             .
             And
             from
             hence
             it
             is
             ,
             (
             as
             I
             shewed
             above
             )
             that
             the
             Earth
             comes
             neerer
             to
             the
             Sunne
             when
             it
             shines
             upon
             that
             part
             where
             there
             is
             more
             water
             ,
             then
             when
             it
             shines
             upon
             that
             where
             there
             is
             more
             dry
             Land.
             
          
           
             9
             This
             Excentricity
             of
             the
             Earth
             is
             the
             cause
             why
             the
             way
             of
             its
             annual
             motion
             is
             not
             a
             perfect
             Circle
             ,
             but
             either
             an
             Elliptical
             ,
             or
             almost
             an
             Elliptical
             line
             ;
             as
             also
             why
             the
             Axis
             of
             the
             Earth
             is
             not
             kept
             exactly
             parallel
             to
             it self
             in
             all
             places
             ,
             but
             onely
             in
             the
             Equinoctial
             points
             .
          
           
             Now
             seeing
             I
             have
             said
             that
             the
             Moon
             is
             carried
             about
             by
             the
             Earth
             ,
             in
             the
             same
             manner
             that
             the
             Earth
             is
             by
             the
             Sunne
             ;
             and
             that
             the
             Earth
             goeth
             about
             the
             Sunne
             in
             such
             manner
             as
             that
             it
             shews
             sometimes
             one
             Hemisphere
             ,
             sometimes
             the
             other
             to
             the
             Sunne
             ;
             it
             remains
             to
             be
             enquired
             ,
             why
             the
             Moon
             has
             alwayes
             one
             and
             the
             same
             face
             turned
             towards
             the
             Earth
             .
          
           
             Suppose
             therefore
             the
             Sunne
             to
             be
             moved
             with
             Simple
             Motion
             in
             the
             little
             Circle
             
               f
               g
               h
               i
            
             ,
             (
             in
             the
             fourth
             figure
             )
             whose
             Center
             is
             t
             ;
             and
             let
             ♈
             ♋
             ♎
             ♑
             be
             the
             annuall
             Circle
             of
             the
             Earth
             ;
             and
             a
             the
             beginning
             ,
             
             of
             Libra
             .
             About
             the
             point
             a
             let
             the
             little
             Circle
             
               l
               k
            
             be
             described
             ;
             and
             in
             it
             let
             the
             Center
             of
             the
             Earth
             be
             understood
             to
             be
             moved
             with
             Simple
             motion
             ;
             and
             both
             the
             Sunne
             &
             the
             Earth
             to
             be
             moved
             according
             to
             the
             order
             of
             the
             Signes
             .
             Upon
             the
             Center
             a
             let
             the
             way
             of
             the
             Moon
             
               m
               n
               o
               p
            
             be
             described
             ;
             and
             let
             
               q
               r
            
             be
             the
             Diameter
             of
             a
             Circle
             cutting
             the
             Globe
             of
             the
             Moon
             into
             two
             Hemispheres
             ,
             whereof
             one
             is
             seen
             by
             us
             when
             the
             Moon
             is
             at
             the
             full
             ,
             and
             the
             other
             is
             turned
             from
             us
             .
          
           
             The
             Diameter
             therefore
             of
             the
             Moon
             
               q
               o
               r
            
             will
             be
             perpendicular
             to
             the
             Straight
             Line
             
               t
               a.
            
             Wherefore
             the
             Moon
             is
             carried
             by
             reason
             of
             the
             Motion
             of
             the
             Earth
             from
             o
             towards
             p.
             But
             by
             reason
             of
             the
             motion
             of
             the
             Sunne
             ,
             if
             it
             were
             in
             p
             it
             would
             at
             the
             same
             time
             be
             carried
             from
             p
             towards
             o
             ;
             and
             by
             these
             two
             contrary
             Movents
             the
             straight
             line
             
               q
               r
            
             will
             be
             turned
             about
             ;
             and
             in
             a
             Quadrant
             of
             the
             Circle
             
               m
               n
               o
               p
            
             it
             will
             be
             turned
             so
             much
             as
             makes
             the
             fourth
             part
             of
             its
             whole
             conversion
             .
             Wherefore
             when
             the
             Moon
             is
             in
             
               p
               ,
               q
               r
            
             will
             be
             parallel
             to
             the
             straight
             line
             
               m
               o.
            
             Secondly
             ,
             when
             the
             Moon
             is
             in
             m
             ,
             the
             straight
             line
             
               q
               r
            
             will
             by
             reason
             of
             the
             motion
             of
             the
             Earth
             be
             in
             
               m
               o.
            
             But
             by
             the
             working
             of
             the
             Suns
             motion
             upon
             it
             in
             the
             quadrant
             
               p
               m
            
             ,
             to●
             same
             
               q
               r
            
             will
             be
             turned
             so
             much
             as
             makes
             another
             quarter
             of
             its
             whole
             conversion
             .
             When
             therefore
             the
             Moon
             is
             in
             
               m
               ,
               q
               r
            
             will
             be
             perpendicular
             to
             the
             straight
             line
             
               o
               m.
            
             By
             the
             same
             reason
             ,
             when
             the
             Moon
             is
             in
             
               n
               ,
               q
               r
            
             will
             be
             parallel
             to
             the
             straight
             line
             
               m
               o
            
             ;
             and
             the
             Moon
             returning
             to
             o
             ,
             the
             same
             
               q
               r
            
             will
             return
             to
             its
             first
             place
             ;
             and
             the
             Body
             of
             the
             Moon
             will
             in
             one
             entire
             period
             make
             also
             one
             entire
             conversion
             upon
             her
             own
             Axis
             .
             In
             the
             making
             of
             which
             ,
             it
             is
             manifest
             ,
             that
             one
             and
             the
             same
             face
             of
             the
             Moon
             is
             always
             turned
             towards
             the
             Earth
             .
             And
             if
             any
             Diameter
             were
             taken
             in
             that
             little
             Circle
             ,
             in
             which
             the
             Moon
             were
             supposed
             to
             be
             carried
             about
             with
             simple
             motion
             ,
             the
             same
             effect
             would
             follow
             ;
             for
             if
             there
             were
             no
             action
             from
             the
             Sun
             ,
             every
             Diameter
             of
             the
             Moon
             would
             be
             carried
             about
             always
             parallel
             to
             it self
             .
             Wherefore
             I
             have
             given
             a
             possible
             cause
             ,
             why
             one
             and
             the
             same
             face
             of
             the
             Moon
             is
             alwayes
             turned
             towards
             the
             Earth
             .
          
           
             But
             it
             is
             to
             be
             noted
             ,
             that
             when
             the
             Moon
             is
             without
             the
             Ecliptick
             ,
             we
             do
             not
             alwayes
             see
             the
             same
             face
             precisely
             .
             For
             we
             see
             onely
             that
             part
             which
             is
             illuminated
             .
             But
             when
             the
             Moon
             is
             without
             
             the
             Ecliptick
             ,
             that
             part
             which
             is
             towards
             us
             ,
             is
             not
             exactly
             the
             same
             with
             that
             which
             is
             illuminated
             .
          
           
             10
             To
             these
             three
             simple
             motions
             ,
             one
             of
             the
             Sunne
             ,
             another
             of
             the
             Moon
             ,
             and
             the
             third
             of
             the
             Earth
             ,
             in
             their
             own
             little
             Circles
             
               f
               g
               h
               i
               ,
               l
               k
               ,
            
             &
             
               q
               r
            
             ,
             together
             with
             the
             Diurnal
             conversion
             of
             the
             Earth
             (
             by
             which
             conversion
             all
             things
             that
             adhere
             to
             its
             superficies
             are
             necessarily
             carried
             about
             with
             it
             )
             may
             be
             referred
             the
             three
             Phaenomena
             concerning
             the
             Tides
             of
             the
             Ocean
             .
             Whereof
             the
             first
             ,
             is
             the
             alternate
             elevation
             and
             depression
             of
             the
             Water
             at
             the
             Shores
             ,
             twice
             in
             the
             space
             of
             24
             houres
             and
             neer
             upon
             52
             minutes
             ;
             for
             so
             it
             has
             constantly
             continued
             in
             all
             ages
             .
             The
             second
             ,
             that
             at
             the
             New
             and
             Full
             Moons
             the
             elevations
             of
             the
             Water
             are
             greater
             ,
             then
             at
             other
             times
             between
             .
             And
             the
             third
             ,
             that
             when
             the
             Sunne
             is
             in
             the
             Equinoctial
             ,
             they
             are
             yet
             greater
             then
             at
             any
             other
             time
             .
             For
             the
             salving
             of
             which
             Phaenomena
             ,
             we
             have
             already
             the
             foure
             above-mentioned
             Motions
             ;
             to
             which
             I
             assume
             also
             this
             ,
             that
             the
             part
             of
             the
             Earth
             which
             is
             called
             America
             ,
             being
             higher
             then
             the
             Water
             ,
             and
             extended
             almost
             the
             space
             of
             a
             whole
             Semicircle
             from
             North
             to
             South
             ,
             gives
             a
             stop
             to
             the
             motion
             of
             the
             Water
             .
             This
             being
             granted
             ,
          
           
             In
             the
             same
             4th
             figure
             ,
             where
             
               l
               b
               k
               c
            
             is
             supposed
             to
             be
             in
             the
             plain
             of
             the
             Moons
             monethly
             motion
             ,
             let
             the
             little
             Circle
             
               l
               d
               k
               e
            
             be
             described
             about
             the
             same
             Center
             a
             in
             the
             plain
             of
             the
             Equinoctial
             .
             This
             Circle
             therefore
             will
             decline
             from
             the
             Circle
             
               l
               b
               k
               c
            
             in
             an
             angle
             of
             almost
             28
             degrees
             and
             ½
             (
             for
             the
             greatest
             declination
             of
             the
             Ecliptick
             is
             23½
             ,
             to
             which
             adding
             5
             for
             the
             greatest
             declination
             of
             the
             Moon
             from
             the
             Ecliptick
             ,
             the
             summe
             wil
             be
             28
             degrees
             and
             ½
             )
             .
             Seeing
             now
             the
             Waters
             which
             are
             under
             the
             Circle
             of
             the
             Moons
             course
             ,
             are
             (
             by
             reason
             of
             the
             Earths
             Simple
             Motion
             in
             the
             plain
             of
             the
             same
             Circle
             )
             moved
             together
             with
             the
             Earth
             ,
             (
             that
             is
             to
             say
             ,
             together
             with
             their
             own
             bottoms
             )
             neither
             out-going
             nor
             out-gone
             ;
             if
             we
             add
             the
             Diurnal
             motion
             ,
             by
             which
             the
             other
             Waters
             which
             are
             under
             the
             Equinoctial
             are
             moved
             in
             the
             same
             order
             ;
             and
             consider
             withall
             that
             the
             Circles
             of
             the
             Moon
             and
             of
             the
             Equinoctial
             intersect
             one
             another
             ;
             it
             will
             be
             manifest
             ,
             that
             both
             those
             Waters
             ,
             which
             are
             under
             the
             Circle
             of
             the
             Moon
             ,
             and
             under
             the
             Equinoctiall
             ,
             will
             runne
             together
             under
             the
             Equinoctial
             ;
             and
             consequently
             ,
             
             that
             their
             Motion
             will
             not
             onely
             be
             swifter
             then
             the
             ground
             that
             carries
             them
             ;
             but
             also
             that
             the
             waters
             themselves
             will
             have
             greater
             elevation
             whensoever
             the
             Earth
             is
             in
             the
             Equinoctial
             .
             Wherefore
             ,
             whatsoever
             the
             cause
             of
             the
             Tides
             may
             be
             ,
             this
             may
             be
             the
             cause
             of
             their
             augmentation
             at
             that
             time
             .
          
           
             Againe
             ,
             seeing
             I
             have
             supposed
             the
             Moon
             to
             be
             carried
             about
             by
             the
             simple
             motion
             of
             the
             Earth
             in
             the
             little
             circle
             lbkc
             ;
             and
             demonstrated
             (
             at
             the
             4
             article
             of
             the
             21
             chapter
             )
             that
             whatsoever
             is
             moved
             by
             a
             Movent
             that
             hath
             simple
             motion
             ,
             will
             be
             moved
             allwayes
             with
             the
             same
             velocity
             ;
             it
             follows
             ,
             that
             the
             center
             of
             the
             Earth
             will
             be
             carried
             in
             the
             circumference
             lbkc
             with
             the
             same
             velocity
             ,
             with
             which
             the
             Moon
             is
             carried
             in
             the
             circumference
             mnop
             .
             Wherefore
             the
             time
             in
             which
             the
             Moon
             is
             carried
             about
             in
             mnop
             ,
             is
             to
             the
             time
             in
             which
             the
             Earth
             is
             carried
             about
             in
             lbkc
             ,
             as
             one
             circumference
             to
             the
             other
             ,
             that
             is
             ,
             as
             ao
             to
             ak
             .
             But
             ao
             is
             observed
             to
             be
             to
             the
             Semidiameter
             of
             the
             Earth
             as
             59
             to
             1
             ;
             and
             therefore
             the
             Earth
             (
             if
             ak
             be
             put
             for
             its
             Semidiameter
             )
             will
             make
             59
             revolutions
             in
             lbkc
             ,
             in
             the
             time
             that
             the
             Moon
             makes
             one
             monthly
             circuit
             in
             mnop
             .
             But
             the
             Moon
             makes
             her
             monthly
             circuit
             in
             little
             more
             then
             29
             dayes
             .
             Wherefore
             the
             Earth
             shal
             make
             its
             circuit
             in
             the
             circumference
             lbkc
             in
             12
             hours
             and
             a
             little
             more
             ,
             namely
             about
             26
             minutes
             more
             ;
             that
             is
             to
             say
             ,
             it
             shall
             make
             two
             circuits
             in
             24
             hours
             and
             allmost
             52
             minutes
             ;
             which
             is
             observed
             to
             be
             the
             time
             between
             the
             high
             water
             of
             one
             day
             and
             the
             high
             water
             of
             the
             day
             following
             .
             Now
             the
             course
             of
             the
             waters
             being
             hindered
             by
             the
             southern
             part
             of
             America
             ,
             their
             motion
             will
             be
             interrupted
             there
             ;
             and
             consequently
             ,
             they
             will
             be
             elevated
             in
             those
             places
             ,
             and
             sink
             down
             again
             by
             their
             own
             waight
             ,
             twice
             in
             the
             space
             of
             24
             hours
             and
             52
             minutes
             .
             And
             thus
             I
             have
             given
             a
             possible
             cause
             of
             the
             diurnall
             reciprocation
             of
             the
             Ocean
             .
          
           
             Now
             from
             this
             swelling
             of
             the
             Ocean
             in
             those
             parts
             of
             the
             Earth
             ,
             proceed
             the
             Flowings
             and
             Ebbings
             in
             the
             Atlantick
             ,
             Spanish
             ,
             Brittish
             and
             German
             Seas
             ;
             which
             though
             they
             have
             their
             set
             times
             ,
             yet
             upon
             severall
             Shores
             they
             happen
             at
             severall
             hours
             of
             the
             day
             ;
             and
             they
             receive
             some
             augmentation
             from
             the
             North
             ,
             by
             reason
             that
             the
             shores
             of
             China
             and
             Tartaria
             ,
             hindering
             
             the
             generall
             course
             of
             the
             waters
             ,
             makes
             them
             swell
             there
             ▪
             and
             discharge
             themselves
             in
             part
             through
             the
             straight
             of
             Anian
             into
             the
             Northern
             Ocean
             ,
             and
             so
             into
             the
             German
             Sea.
             
          
           
             As
             for
             the
             Spring
             Tides
             which
             happen
             at
             the
             time
             of
             the
             New
             &
             Full
             Moons
             ,
             they
             are
             caused
             by
             that
             simple
             motion
             which
             at
             the
             beginning
             I
             supposed
             to
             be
             allwayes
             in
             the
             Moone
             .
             For
             as
             ,
             when
             I
             shewed
             the
             cause
             of
             the
             Excentricity
             of
             the
             Earth
             ,
             I
             derived
             the
             elevation
             of
             the
             waters
             from
             the
             simple
             motion
             of
             the
             Sunne
             ;
             so
             the
             same
             may
             here
             be
             derived
             from
             the
             simple
             motion
             of
             the
             Moon
             .
             For
             though
             from
             the
             generation
             of
             Clouds
             ,
             there
             appeare
             in
             the
             Sunne
             a
             more
             manifest
             power
             of
             elevating
             the
             waters
             ,
             then
             in
             the
             Moon
             ;
             yet
             the
             power
             of
             encreasing
             moisture
             in
             Vegetables
             and
             living
             creatures
             appears
             more
             manifestly
             in
             the
             Moon
             then
             in
             the
             Sunne
             ;
             which
             may
             perhaps
             proceed
             from
             this
             ,
             that
             the
             Sunne
             raiseth
             up
             greater
             ,
             and
             the
             Moon
             lesser
             drops
             of
             water
             .
             Neverthelesse
             ,
             it
             is
             more
             likely
             ,
             and
             more
             agreeable
             to
             common
             observation
             ,
             that
             Raine
             is
             raised
             not
             only
             by
             the
             Sunne
             but
             also
             by
             the
             Moon
             ;
             for
             allmost
             all
             men
             expect
             change
             of
             weather
             at
             the
             time
             of
             the
             Conjunctions
             of
             the
             Sunne
             and
             Moon
             with
             one
             another
             ,
             and
             with
             the
             Earth
             ,
             more
             then
             in
             the
             time
             of
             their
             Quarters
             .
          
           
             In
             the
             last
             place
             ,
             the
             cause
             why
             the
             Spring
             Tides
             are
             greater
             at
             the
             time
             of
             the
             Aequinoxes
             ,
             hath
             been
             already
             sufficiently
             declared
             in
             this
             Article
             ,
             where
             I
             have
             demonstrated
             ,
             that
             the
             two
             Motions
             of
             the
             Earth
             ,
             namely
             ,
             its
             Simple
             Motion
             in
             the
             little
             Circle
             lbkc
             ,
             and
             its
             Diurnal
             motion
             in
             ldke
             ,
             cause
             necessarily
             a
             greater
             elevation
             of
             Waters
             when
             the
             Sunne
             is
             about
             the
             Aequinoxes
             ,
             then
             when
             he
             is
             in
             other
             places
             .
             I
             have
             therefore
             given
             possible
             causes
             of
             the
             Phaenomenon
             of
             the
             Flowing
             and
             Ebbing
             of
             the
             Ocean
             .
          
           
             11
             As
             for
             the
             explication
             of
             the
             yearly
             
               Praecession
               of
               the
               Aequinoctial
               points
            
             ,
             we
             must
             remember
             ,
             that
             (
             as
             I
             have
             already
             shewn
             )
             the
             annual
             motion
             of
             the
             Earth
             is
             not
             in
             the
             Circumference
             of
             a
             Circle
             ,
             but
             of
             an
             Ellipsis
             ,
             or
             a
             line
             not
             considerably
             different
             from
             that
             of
             an
             Ellipsis
             .
             In
             the
             first
             place
             therefore
             this
             Elliptical
             line
             is
             to
             be
             described
             .
          
           
             Let
             the
             Ecliptick
             ♎
             ♑
             ♈
             ♋
             (
             in
             the
             5th
             figure
             )
             be
             divided
             into
             four
             equal
             parts
             by
             the
             two
             straight
             lines
             ab
             and
             ♑
             ♋
             ,
             cutting
             one
             
             another
             at
             right
             angles
             in
             the
             Center
             c
             ;
             and
             taking
             the
             Arch
             bd
             of
             2
             degrees
             and
             16
             minutes
             ,
             let
             the
             straight
             line
             de
             be
             drawn
             parallel
             to
             ab
             and
             cutting
             ♑
             ♋
             in
             f
             ;
             which
             being
             done
             ,
             the
             Excentricity
             of
             the
             Earth
             will
             be
             cf.
             
             Seeing
             therefore
             the
             annual
             motion
             of
             the
             Earth
             is
             in
             the
             Circumference
             of
             an
             Ellipsis
             ,
             of
             which
             ♑
             ♋
             is
             the
             greater
             Axis
             ,
             ab
             cannot
             be
             the
             lesser
             Axis
             ;
             for
             ab
             and
             ♑
             ♋
             are
             equal
             .
             Wherefore
             the
             Earth
             passing
             through
             a
             &
             b
             ,
             will
             either
             pass
             above
             ♑
             ,
             as
             through
             g
             ,
             or
             passing
             through
             ♑
             ,
             will
             fall
             between
             c
             and
             a
             ;
             it
             is
             no
             matter
             which
             .
             Let
             it
             pass
             therefore
             through
             g
             ;
             and
             let
             gl
             be
             taken
             equal
             to
             the
             straight
             line
             ♑
             ♋
             ;
             and
             dividing
             gl
             equally
             in
             
               i
               ,
               gi
            
             will
             be
             equal
             to
             ♑
             ♋
             ,
             &
             il
             equal
             to
             f
             ♋
             ;
             and
             consequently
             the
             point
             i
             will
             cut
             the
             Excentricity
             cf
             into
             two
             equal
             parts
             ;
             and
             taking
             ih
             equal
             to
             
               if
               ,
               hi
            
             will
             be
             the
             whole
             Excentricity
             .
             If
             now
             a
             straight
             line
             (
             namely
             ,
             the
             line
             ♎
             i
             ♈
             )
             be
             drawn
             through
             i
             parallel
             to
             the
             straight
             lines
             ab
             and
             ed
             ,
             the
             way
             of
             the
             Sunne
             in
             Summer
             (
             namely
             ,
             the
             Arch
             ♎
             g
             ♈
             )
             will
             be
             greater
             then
             his
             way
             in
             Winter
             by
             8
             degrees
             and
             ¼
             .
             Wherefore
             the
             true
             Aequinoxes
             wil
             be
             in
             the
             straight
             line
             ♎
             i
             ♈
             ;
             and
             therefore
             the
             Ellipsis
             of
             the
             Earths
             annual
             motion
             will
             not
             pass
             through
             
               a
               ,
               g
               ,
               b
               &
               l
            
             ;
             but
             through
             ♎
             ,
             g
             ,
             ♈
             &
             l.
             Wherfore
             the
             annual
             motion
             of
             the
             Earth
             is
             in
             the
             Ellipsis
             ♎
             g
             ♈
             l
             ,
             and
             cannot
             be
             (
             the
             Excentricity
             being
             salved
             )
             in
             any
             other
             line
             .
             And
             this
             perhaps
             is
             the
             reason
             ,
             why
             Kepler
             ,
             against
             the
             opinion
             of
             all
             the
             Astronomers
             of
             former
             time
             ,
             thought
             fit
             to
             bisect
             the
             Excentricity
             of
             the
             Earth
             ,
             or
             (
             according
             to
             the
             Ancients
             )
             of
             the
             Sunne
             ,
             not
             by
             diminishing
             the
             quantity
             of
             the
             same
             Excentricity
             ,
             because
             the
             true
             measure
             of
             that
             quantity
             ,
             is
             the
             difference
             by
             which
             the
             Summer
             Arch
             exceeds
             the
             Winter
             Arch
             ,
             but
             by
             taking
             for
             the
             Center
             of
             the
             Ecliptick
             of
             the
             great
             Orbe
             the
             point
             c
             neerer
             to
             f
             ,
             &
             so
             placing
             the
             whole
             great
             Orbe
             as
             much
             neerer
             to
             the
             Ecliptick
             of
             the
             fixed
             Stars
             towards
             ♋
             ,
             as
             is
             the
             distance
             between
             c
             &
             i.
             For
             seeing
             the
             whole
             great
             Orbe
             is
             but
             as
             a
             point
             in
             respect
             of
             the
             immense
             distance
             of
             the
             fixed
             Starres
             ,
             the
             two
             straight
             lines
             ♎
             ♈
             and
             ab
             being
             produced
             both
             wayes
             to
             the
             beginnings
             of
             Aries
             and
             Libra
             ,
             will
             fall
             upon
             the
             same
             points
             of
             the
             Sphere
             of
             the
             fixed
             Stars
             .
             Let
             therefore
             the
             Diameter
             of
             the
             Earth
             mn
             be
             in
             the
             plain
             of
             the
             Earths
             annual
             motion
             .
             If
             now
             the
             Earth
             be
             moved
             by
             the
             Sunnes
             simple
             motion
             in
             the
             Circumference
             of
             
             the
             Ecliptick
             about
             the
             Center
             i
             ,
             this
             Diameter
             will
             bee
             kept
             alwayes
             parallel
             to
             itself
             and
             to
             the
             straight
             line
             gl
             .
             But
             seeing
             the
             Earth
             is
             moved
             in
             the
             Circumference
             of
             an
             Ellipsis
             without
             the
             Ecliptick
             ,
             the
             point
             n
             ,
             whilst
             it
             passeth
             through
             ♎
             ♑
             ♈
             will
             go
             in
             a
             lesser
             Circumference
             then
             the
             point
             m
             ;
             and
             consequently
             as
             soon
             as
             ever
             it
             begins
             to
             be
             moved
             ,
             it
             will
             lose
             its
             parallelisme
             with
             the
             straight
             line
             ♑
             ♋
             ▪
             so
             that
             mn
             produced
             will
             at
             last
             cut
             the
             straight
             line
             gl
             produced
             .
             And
             contrarily
             ,
             as
             soon
             as
             mn
             is
             past
             ♈
             ,
             (
             the
             Earth
             making
             its
             way
             in
             the
             internal
             Ellipticall
             line
             ♈
             l
             ♎
             )
             the
             same
             mn
             produced
             towards
             m
             ,
             will
             cut
             lg
             produced
             .
             And
             when
             the
             Earth
             hath
             allmost
             finished
             its
             whole
             circumference
             ,
             the
             same
             ▪
             mn
             shall
             againe
             make
             a
             right
             angle
             with
             a
             line
             drawn
             from
             the
             center
             i
             ,
             a
             little
             short
             of
             the
             point
             from
             which
             the
             Earth
             began
             its
             motion
             .
             And
             there
             the
             next
             yeare
             shall
             be
             one
             of
             the
             Aequinoctial
             points
             ,
             namely
             ,
             neer
             the
             end
             of
             ♍
             ;
             the
             other
             shall
             be
             opposite
             to
             it
             neer
             the
             end
             of
             ♓
             .
             And
             thus
             the
             points
             in
             which
             the
             Days
             and
             Nights
             are
             made
             equall
             ,
             doe
             every
             year
             fall
             back
             ;
             but
             with
             so
             slow
             a
             motion
             ,
             that
             in
             a
             whole
             year
             ,
             it
             makes
             but
             51
             first
             minutes
             .
             And
             this
             relapse
             being
             contrary
             to
             the
             order
             of
             the
             Signes
             ,
             is
             commonly
             called
             the
             
               Praecession
               of
               the
               Aequinoxes
            
             .
             Of
             which
             I
             have
             from
             my
             former
             Suppositions
             deduced
             a
             possible
             cause
             ;
             which
             was
             to
             be
             done
             .
          
           
             According
             to
             what
             I
             have
             said
             concerning
             the
             cause
             of
             the
             Excentricity
             of
             the
             Earth
             ;
             and
             according
             to
             Kepler
             ,
             who
             for
             the
             cause
             thereof
             supposeth
             one
             part
             of
             the
             Earth
             to
             be
             affected
             to
             the
             Sunne
             the
             other
             part
             to
             be
             disaffected
             ,
             the
             Apogaeum
             &
             Perigaeum
             of
             the
             Sunne
             should
             be
             moved
             every
             year
             in
             the
             same
             order
             ,
             and
             with
             the
             same
             velocity
             ,
             with
             which
             the
             Aequinoctiall
             points
             are
             moved
             ;
             and
             their
             distance
             from
             them
             should
             allwayes
             be
             the
             quadrant
             of
             a
             circle
             ;
             which
             seems
             to
             be
             otherwise
             .
             For
             Astronomers
             say
             ,
             that
             the
             Aequinoxes
             are
             now
             ,
             the
             one
             about
             28
             degrees
             gone
             back
             from
             the
             first
             Star
             of
             Aries
             ,
             the
             other
             as
             much
             from
             the
             beginning
             of
             Libra
             .
             So
             that
             the
             Apogaeum
             of
             the
             Sunne
             ,
             or
             the
             Aphelium
             of
             the
             Earth
             ought
             to
             be
             about
             the
             28th
             degree
             of
             Cancer
             ;
             but
             it
             is
             reckoned
             to
             be
             in
             the
             7th
             degree
             .
             Seeing
             therefore
             we
             have
             not
             sufficient
             evidence
             of
             the
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             (
             
               that
               so
               it
               is
            
             )
             ,
             it
             is
             in
             vaine
             to
             seek
             for
             the
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             (
             
               why
               it
               is
               so
            
             )
             .
             Wherefore
             ,
             as
             long
             as
             the
             motion
             of
             the
             
             Apogaeum
             is
             not
             observable
             by
             reason
             of
             the
             slownesse
             thereof
             ;
             and
             as
             long
             as
             it
             remaiues
             doubtful
             whether
             their
             distance
             from
             the
             Aequinoctiall
             points
             be
             more
             or
             lesse
             then
             a
             quadrant
             precisely
             ;
             so
             long
             it
             may
             be
             lawfull
             for
             me
             to
             thinke
             they
             proceed
             both
             of
             them
             with
             equall
             velocity
             .
          
           
             Also
             ,
             I
             doe
             not
             at
             all
             meddle
             with
             the
             causes
             of
             the
             Excentricities
             of
             
               Saturne
               ,
               Jupiter
               ,
               Mars
            
             ,
             and
             Mercury
             .
             Neverthelesse
             ,
             seeing
             the
             Excentricity
             of
             the
             Earth
             may
             (
             as
             I
             have
             shewne
             )
             be
             caused
             by
             the
             unlike
             constitution
             of
             the
             several
             parts
             of
             the
             Earth
             which
             are
             alternately
             turned
             towards
             the
             Sunne
             ,
             it
             is
             credible
             also
             that
             like
             effects
             may
             be
             produced
             in
             these
             other
             Planets
             from
             their
             having
             their
             Superficies
             of
             unlike
             parts
             .
          
           
             And
             this
             is
             all
             I
             shall
             say
             concerning
             
               Sidereal
               Philosophy
            
             .
             And
             though
             the
             causes
             I
             have
             here
             supposed
             be
             not
             the
             true
             causes
             of
             these
             Phaenomena
             ,
             yet
             I
             have
             demonstrated
             that
             they
             are
             sufficient
             to
             produce
             them
             ,
             according
             to
             what
             I
             at
             first
             propounded
             .
          
        
         
           
           
           
           
             
               CHAP.
               XXVII
            
             .
             Of
             
               Light
               ,
               Heat
            
             ,
             and
             of
             Colours
             .
          
           
             
               
                 1
                 Of
                 the
                 immense
                 Magnitude
                 of
                 some
                 Bodies
                 ,
                 and
                 the
                 unspeakable
                 Littleness
                 of
                 others
                 .
              
               
                 2
                 Of
                 the
                 cause
                 of
                 the
                 Light
                 of
                 the
                 Sun.
                 
              
               
                 3
                 How
                 Light
                 heateth
                 .
              
               
                 4
                 The
                 generation
                 of
                 Fire
                 from
                 the
                 Sunne
                 .
              
               
                 5
                 The
                 generation
                 of
                 Fire
                 from
                 Collision
                 .
              
               
                 6
                 The
                 cause
                 of
                 Light
                 in
                 Glow-wormes
                 ,
                 Rotten
                 Wood
                 ,
                 and
                 the
                 Bolonian
                 Stone
                 .
              
               
                 7
                 The
                 cause
                 of
                 Light
                 in
                 the
                 concussion
                 of
                 Sea-water
                 .
              
               
                 8
                 The
                 cause
                 of
                 Flame
                 ,
                 Sparks
                 ,
                 and
                 Colliquation
                 .
              
               
                 9
                 The
                 cause
                 why
                 wet
                 Hay
                 sometimes
                 burns
                 of
                 its
                 own
                 accord
                 .
                 Also
                 the
                 cause
                 of
                 Lightning
                 .
              
               
                 10
                 The
                 cause
                 of
                 the
                 force
                 of
                 Gunpowder
                 ;
                 and
                 what
                 is
                 to
                 be
                 ascribed
                 to
                 the
                 Coals
                 ,
                 what
                 to
                 the
                 Brimstone
                 ,
                 and
                 what
                 to
                 the
                 Nitre
                 .
              
               
                 11
                 How
                 Heat
                 is
                 caused
                 by
                 Attrition
                 .
              
               
                 12
                 The
                 distinction
                 of
                 Light
                 into
                 First
                 ,
                 Second
                 ,
                 &c.
                 
              
               
                 13
                 The
                 causes
                 of
                 the
                 Colours
                 we
                 see
                 in
                 looking
                 through
                 a
                 Prisma
                 of
                 Glass
                 ,
                 namely
                 ,
                 of
                 Red
                 ,
                 Yellow
                 ,
                 Blue
                 and
                 Violet
                 Colour
                 .
              
               
                 14
                 Why
                 the
                 Moon
                 and
                 the
                 Starres
                 appear
                 redder
                 in
                 the
                 Horizon
                 then
                 in
                 the
                 midst
                 of
                 the
                 Heaven
                 .
              
               
                 15
                 The
                 cause
                 of
                 Whiteness
                 .
              
               
                 16
                 The
                 cause
                 of
                 Blackness
                 .
              
            
          
           
             1
             BEsides
             the
             Starres
             (
             of
             which
             I
             have
             spoken
             in
             the
             last
             Chapt.
             )
             ,
             whatsoever
             other
             Bodies
             there
             be
             in
             the
             World
             ,
             they
             may
             be
             all
             comprehended
             under
             the
             name
             of
             Intersidereal
             Bodies
             .
             And
             these
             I
             have
             already
             supposed
             to
             be
             either
             the
             most
             fluid
             Aether
             ,
             or
             such
             Bodies
             whose
             parts
             have
             some
             degree
             of
             cohaesion
             .
             Now
             these
             differ
             from
             one
             another
             in
             their
             several
             
               Consistencies
               ,
               Magnitudes
               ,
               Motions
            
             and
             Figures
             .
             In
             Consistency
             ▪
             I
             suppose
             some
             Bodies
             to
             be
             Harder
             ,
             others
             Softer
             through
             all
             the
             several
             degrees
             of
             Tenacity
             .
             In
             Magnitude
             ,
             some
             to
             be
             Greater
             ,
             others
             Less
             ,
             and
             many
             unspeakably
             Little.
             For
             we
             must
             remember
             that
             by
             the
             Understanding
             ,
             Quantity
             is
             divisible
             into
             divisibles
             perpetually
             .
             And
             therefore
             if
             a
             man
             could
             do
             as
             much
             with
             his
             hands
             as
             he
             can
             with
             his
             Understanding
             ,
             he
             would
             be
             able
             to
             take
             from
             any
             given
             magnitude
             ,
             a
             part
             which
             should
             be
             less
             then
             any
             other
             magnitude
             given
             .
             But
             the
             Omnipotent
             Creator
             of
             the
             World
             can
             actually
             from
             a
             part
             of
             any
             thing
             take
             another
             par●
             ,
             as
             farre
             as
             we
             
             by
             our
             Understanding
             can
             conceive
             the
             same
             to
             be
             divisible
             .
             Wherefore
             there
             is
             no
             impossible
             Smalness
             of
             Bodies
             .
             And
             what
             hinders
             but
             that
             we
             may
             think
             this
             likely
             ?
             For
             we
             know
             there
             are
             some
             living
             Creatures
             so
             small
             ▪
             that
             we
             can
             scarce
             see
             their
             whole
             Bodies
             .
             Yet
             even
             these
             have
             their
             young
             ones
             ,
             their
             little
             Veins
             ,
             and
             other
             Vessels
             ,
             and
             their
             Eyes
             so
             smal
             ,
             as
             that
             no
             Microscope
             can
             make
             them
             visible
             .
             So
             that
             we
             cannot
             suppose
             any
             magnitude
             so
             little
             ,
             but
             that
             our
             very
             supposition
             is
             actually
             exceeded
             by
             Nature
             .
             Besides
             ,
             there
             are
             now
             such
             Microscopes
             commonly
             made
             ,
             that
             the
             things
             we
             see
             with
             them
             appear
             a
             hundred
             thousand
             times
             bigger
             ,
             then
             they
             would
             do
             if
             we
             looked
             upon
             them
             with
             our
             bare
             Eyes
             .
             Nor
             is
             there
             any
             doubt
             but
             that
             by
             augmenting
             the
             power
             of
             these
             Microscopes
             (
             for
             it
             may
             be
             augmented
             as
             long
             as
             neither
             Matter
             nor
             the
             hands
             of
             Workmen
             are
             wanting
             )
             every
             one
             of
             those
             hundred
             thousandth
             parts
             might
             yet
             appear
             a
             hundred
             thousand
             times
             geater
             then
             they
             did
             before
             .
             Neither
             is
             the
             Smalness
             of
             some
             Bodies
             to
             be
             more
             admired
             ,
             then
             the
             vast
             Greatness
             of
             others
             .
             For
             it
             belongs
             to
             the
             same
             infinite
             Power
             ,
             as
             well
             to
             augment
             infinitely
             ,
             as
             infinitely
             to
             diminish
             ;
             To
             make
             the
             great
             Orbe
             ,
             (
             namely
             ,
             that
             whose
             Radius
             reacheth
             from
             the
             Earth
             to
             the
             Sunne
             )
             but
             as
             a
             point
             in
             respect
             of
             the
             distance
             between
             the
             Sunne
             and
             the
             fixed
             Starres
             ;
             and
             on
             the
             contrary
             ,
             to
             make
             a
             Body
             so
             little
             ,
             as
             to
             be
             in
             the
             same
             proportion
             less
             then
             any
             other
             visible
             Body
             ,
             proceeds
             equally
             from
             one
             and
             the
             same
             Authour
             of
             Nature
             .
             But
             this
             of
             the
             immense
             distance
             of
             the
             fixed
             Starres
             (
             which
             for
             a
             long
             time
             was
             accounted
             an
             incredible
             thing
             )
             is
             now
             believed
             by
             almost
             all
             the
             Learned
             .
             Why
             then
             should
             not
             that
             other
             of
             the
             smalness
             of
             some
             Bodies
             ,
             become
             credible
             at
             some
             time
             or
             other
             ?
             For
             the
             Majesty
             of
             God
             appears
             no
             less
             in
             small
             things
             then
             in
             great
             ;
             and
             as
             it
             exceedeth
             humane
             sense
             in
             the
             immense
             greatness
             of
             the
             Universe
             ;
             so
             also
             it
             doth
             in
             the
             smalness
             of
             the
             parts
             thereof
             .
             Nor
             are
             the
             first
             Elements
             of
             Compositions
             ,
             nor
             the
             first
             Beginnings
             of
             Actions
             ,
             nor
             the
             first
             Moments
             of
             Times
             more
             credible
             ,
             then
             that
             which
             is
             now
             believed
             of
             the
             vast
             distance
             of
             the
             fixed
             Starres
             .
          
           
             Some
             things
             are
             acknowledged
             by
             mortal
             men
             to
             be
             very
             Great
             ,
             though
             Finite
             ,
             as
             seeing
             them
             to
             be
             such
             .
             They
             acknowledge
             also
             ,
             that
             some
             things
             which
             they
             do
             not
             see
             ,
             may
             be
             of
             infinite
             magnitude
             .
             
             But
             they
             are
             not
             presently
             ,
             nor
             without
             great
             study
             perswaded
             that
             there
             is
             any
             Mean
             between
             Infinite
             &
             the
             Greatest
             of
             those
             things
             which
             either
             they
             see
             or
             imagine
             .
             Nevertheless
             ,
             when
             after
             meditation
             &
             contemplation
             ,
             many
             things
             which
             we
             wondred
             at
             before
             are
             now
             grown
             more
             familiar
             to
             us
             ,
             we
             then
             believe
             them
             ,
             and
             transferre
             our
             admiration
             from
             the
             Creatures
             to
             the
             Creator
             .
             But
             how
             little
             soever
             some
             Bodies
             may
             be
             ;
             yet
             I
             will
             not
             suppose
             their
             quantity
             to
             be
             less
             ,
             then
             is
             requisite
             for
             the
             salving
             of
             the
             Phaenomena
             .
             And
             in
             like
             manner
             I
             shall
             suppose
             their
             motion
             ,
             namely
             ,
             their
             Velocity
             and
             Slowness
             ,
             and
             the
             Variety
             of
             their
             Figures
             ,
             to
             be
             onely
             such
             as
             the
             explication
             of
             their
             natural
             causes
             requires
             .
             And
             lastly
             ,
             I
             suppose
             ,
             that
             the
             parts
             of
             the
             pure
             Aether
             (
             as
             if
             it
             were
             the
             First
             Matter
             )
             have
             no
             motion
             at
             all
             ,
             but
             what
             they
             receive
             from
             Bodies
             which
             float
             in
             them
             ,
             and
             are
             not
             themselves
             fluid
             .
          
           
             2
             Having
             laid
             these
             Grounds
             ,
             let
             us
             come
             to
             speake
             of
             Causes
             ,
             and
             in
             the
             first
             place
             let
             us
             inquire
             what
             may
             be
             the
             cause
             of
             the
             Light
             of
             the
             Sunne
             .
             Seeing
             therefore
             the
             Body
             of
             the
             Sunne
             doth
             by
             its
             simple
             circular
             motion
             thrust
             away
             the
             ambient
             aethereall
             substance
             sometimes
             one
             way
             sometimes
             another
             ,
             so
             that
             those
             parts
             which
             are
             next
             the
             Sunne
             being
             moved
             by
             it
             ,
             doe
             propagate
             that
             motion
             to
             the
             next
             remote
             parts
             ,
             and
             these
             to
             the
             next
             ,
             and
             so
             on
             continually
             ;
             it
             must
             needs
             be
             ,
             that
             notwithstanding
             any
             distance
             ,
             the
             foremost
             part
             of
             the
             Eie
             will
             at
             last
             be
             pressed
             ;
             and
             by
             the
             pressure
             of
             that
             part
             ,
             the
             motion
             will
             be
             propagated
             to
             the
             innermost
             part
             of
             the
             Organ
             of
             Sight
             ,
             namely
             to
             the
             Heart
             ;
             and
             from
             the
             reaction
             of
             the
             Heart
             ,
             there
             will
             proceed
             an
             endeavour
             back
             by
             the
             same
             way
             ,
             ending
             in
             the
             endeavour
             outwards
             of
             the
             Coat
             of
             the
             Eie
             called
             the
             Retina
             .
             But
             this
             endeavour
             outwards
             (
             as
             has
             been
             defined
             in
             the
             25
             chapter
             )
             is
             the
             thing
             which
             is
             called
             Light
             ,
             or
             the
             Phantasme
             of
             a
             Lucid
             Body
             .
             For
             it
             is
             by
             reason
             of
             this
             Phantasme
             that
             an
             Object
             is
             called
             Lucid.
             Wherefore
             we
             have
             a
             possible
             cause
             of
             the
             Light
             of
             the
             Sunne
             ;
             which
             I
             undertook
             to
             find
             .
          
           
             3
             The
             generation
             of
             the
             Light
             of
             the
             Sunne
             is
             accompanied
             with
             the
             generation
             of
             Heat
             .
             Now
             every
             man
             knowes
             what
             Heat
             is
             in
             himselfe
             ,
             by
             feeling
             it
             when
             he
             growes
             Hot
             ;
             but
             what
             it
             is
             in
             other
             things
             he
             knowes
             onely
             by
             ratiocination
             .
             For
             it
             is
             one
             thing
             to
             
             grow
             Hot
             ,
             and
             another
             thing
             to
             Heat
             ,
             or
             make
             Hot.
             And
             therefore
             though
             we
             perceive
             that
             the
             Fire
             or
             the
             Sunne
             Heateth
             ,
             yet
             we
             doe
             not
             perceive
             that
             it
             is
             it selfe
             Hot.
             That
             other
             living
             creatures
             whilest
             they
             make
             other
             things
             Hot
             ,
             are
             Hot
             themselves
             ,
             we
             inferre
             by
             reasoning
             from
             the
             like
             sense
             in
             our selves
             .
             But
             this
             is
             not
             a
             necessary
             inference
             .
             For
             though
             it
             may
             truly
             be
             said
             of
             living
             Creatures
             ,
             that
             
               They
               Heat
               ,
               therefore
               they
               are
               themselves
               Hot
            
             ;
             yet
             it
             cannot
             from
             hence
             be
             truly
             inferred
             ,
             that
             
               Fire
               Heateth
               ,
               therefore
               it
               is
               it selfe
               Hot
            
             ;
             no
             more
             then
             this
             ,
             
               Fire
               causeth
               Pain
               ,
               therefore
               it
               is
               it self
               in
               Pain
               .
            
             Wherefore
             that
             is
             onely
             and
             properly
             called
             Hot
             ,
             which
             when
             we
             feel
             ,
             we
             are
             necessarily
             Hot.
             
          
           
             Now
             when
             we
             grow
             Hot
             ,
             we
             find
             that
             our
             Spirits
             and
             Blood
             ,
             and
             whatsoever
             is
             fluid
             within
             us
             ,
             is
             called
             out
             from
             the
             internall
             to
             the
             externall
             parts
             of
             our
             Bodies
             ,
             more
             or
             lesse
             ,
             according
             to
             the
             degree
             of
             the
             Heat
             ;
             and
             that
             our
             Skin
             swelleth
             .
             He
             therefore
             that
             can
             give
             a
             possible
             cause
             of
             this
             Evocation
             and
             Swelling
             ,
             and
             such
             as
             agreeth
             with
             the
             rest
             of
             the
             Phaenomena
             of
             Heat
             ;
             may
             be
             thought
             to
             have
             given
             the
             cause
             of
             the
             Heat
             of
             the
             Sunne
             .
          
           
             It
             hath
             been
             shewn
             (
             in
             the
             5
             article
             of
             the
             21
             chapter
             )
             that
             the
             fluid
             Medium
             which
             we
             call
             the
             Aire
             ,
             is
             so
             moved
             by
             the
             simple
             circular
             motion
             of
             the
             Sunne
             ,
             as
             that
             all
             its
             parts
             ,
             even
             the
             least
             ,
             do
             perpetually
             change
             places
             with
             one
             another
             ;
             which
             change
             of
             places
             is
             that
             which
             there
             I
             called
             Fermentation
             .
             From
             this
             Fermentation
             of
             the
             Aire
             ,
             I
             have
             (
             in
             the
             8
             article
             of
             the
             last
             chapter
             )
             demonstrated
             ,
             that
             the
             water
             may
             be
             drawn
             up
             into
             the
             clouds
             .
          
           
             And
             I
             shall
             now
             shew
             ,
             that
             the
             fluid
             parts
             may
             in
             like
             manner
             by
             the
             same
             Fermentation
             be
             drawn
             out
             from
             the
             internall
             to
             the
             externall
             parts
             of
             our
             Bodies
             .
             For
             seeing
             that
             wheresoever
             the
             fluid
             Medium
             is
             contiguous
             to
             the
             Body
             of
             any
             living
             creature
             ,
             there
             the
             parts
             of
             that
             Medium
             are
             by
             perpetuall
             change
             of
             place
             separated
             from
             one
             another
             ;
             the
             contiguous
             parts
             of
             the
             living
             creature
             must
             of
             necessity
             endeavour
             to
             enter
             into
             the
             spaces
             of
             the
             separated
             parts
             .
             For
             otherwise
             those
             parts
             (
             supposing
             there
             is
             no
             Vacuum
             )
             would
             have
             no
             place
             to
             go
             into
             .
             And
             therefore
             that
             which
             is
             most
             fluid
             and
             separable
             in
             the
             parts
             of
             the
             living
             creature
             which
             are
             contiguous
             to
             the
             Medium
             ,
             will
             go
             first
             out
             ;
             and
             into
             the
             place
             thereof
             will
             succeed
             such
             other
             parts
             as
             can
             most
             
             easily
             transpire
             through
             the
             po●es
             of
             the
             skin
             .
             And
             from
             hence
             it
             is
             necessary
             ,
             that
             the
             rest
             of
             the
             parts
             which
             are
             not
             separated
             ,
             must
             all
             together
             be
             moved
             outwards
             ,
             for
             the
             keeping
             of
             all
             places
             full
             .
             But
             this
             motion
             outwards
             of
             all
             parts
             together
             must
             of
             necessity
             press
             those
             parts
             of
             the
             ambient
             Aire
             which
             are
             ready
             to
             leave
             their
             places
             ;
             and
             therefore
             all
             the
             parts
             of
             the
             Body
             endeavouring
             at
             once
             that
             way
             ,
             makes
             the
             Body
             swell
             .
             Wherefore
             a
             possible
             cause
             is
             given
             of
             Heat
             from
             the
             Sunne
             ;
             which
             was
             to
             be
             done
             .
          
           
             4
             We
             have
             now
             seen
             how
             Light
             and
             Heat
             are
             generated
             ;
             Heat
             by
             the
             simple
             motion
             of
             the
             Medium
             ,
             making
             the
             parts
             perpetually
             change
             places
             with
             one
             another
             ;
             and
             Light
             by
             this
             ,
             that
             by
             the
             same
             simple
             motion
             Action
             is
             propagated
             in
             a
             straight
             line
             .
             But
             when
             a
             Body
             hath
             its
             parts
             so
             moved
             ,
             that
             it
             sensibly
             both
             Heats
             and
             Shines
             at
             the
             same
             time
             ,
             then
             it
             is
             ,
             that
             we
             say
             Fire
             is
             generated
             .
          
           
             Now
             by
             Fire
             I
             do
             not
             understand
             a
             Body
             distinct
             from
             matter
             combustible
             or
             glowing
             ,
             as
             Wood
             or
             Iron
             ,
             but
             the
             matter
             it self
             ,
             not
             simply
             and
             always
             ,
             but
             then
             onely
             when
             it
             shineth
             and
             heateth
             .
             He
             therefore
             that
             renders
             a
             cause
             possible
             ,
             and
             agreeable
             to
             the
             rest
             of
             the
             Phaenomena
             ,
             namely
             ,
             whence
             and
             from
             what
             action
             both
             the
             Shining
             and
             Heating
             proceed
             ,
             may
             be
             thought
             to
             have
             given
             a
             possible
             cause
             of
             the
             generation
             of
             Fire
             .
          
           
             Let
             therefore
             ABC
             (
             in
             the
             first
             Figure
             )
             be
             a
             Sphere
             ,
             or
             the
             portion
             of
             a
             Sphere
             ,
             whose
             Center
             is
             D
             ;
             and
             let
             it
             be
             transparent
             and
             homogeneous
             ,
             as
             Cristal
             ,
             Glass
             ,
             or
             Water
             ,
             and
             objected
             to
             the
             Sunne
             .
             Wherefore
             the
             foremost
             part
             ABC
             ,
             will
             by
             the
             simple
             motion
             of
             the
             Sunne
             ,
             by
             which
             it
             thrusts
             forwards
             the
             Medium
             ▪
             be
             wrought
             upon
             by
             the
             Sun-beams
             in
             the
             straight
             lines
             EA
             ,
             FB
             and
             GC
             ;
             which
             straight
             lines
             may
             in
             respect
             of
             the
             great
             distance
             of
             the
             Sunne
             be
             taken
             for
             parallels
             .
             And
             seeing
             the
             Medium
             within
             the
             Sphere
             is
             thicker
             then
             the
             Medium
             without
             it
             ,
             those
             Beams
             will
             be
             refracted
             towards
             their
             perpendiculars
             .
             Let
             the
             straight
             lines
             EA
             and
             GC
             be
             produced
             till
             they
             cut
             the
             Sphere
             in
             H
             and
             I
             ;
             and
             drawing
             the
             perpendiculars
             AD
             and
             CD
             ,
             the
             refracted
             Beams
             EA
             and
             GC
             will
             of
             necessity
             fall
             ,
             the
             one
             between
             AH
             and
             AD
             ,
             the
             other
             between
             CI
             and
             CD
             .
             Let
             those
             refracted
             
             Beams
             be
             AK
             and
             CL.
             And
             again
             ,
             let
             the
             lines
             DKM
             &
             DLN
             be
             drawn
             per●endicular
             to
             the
             Sphere
             ;
             and
             let
             AK
             and
             CL
             be
             produced
             till
             they
             meet
             with
             the
             straight
             line
             BD
             produced
             in
             O.
             Seeing
             therefore
             the
             Medium
             within
             the
             Sphere
             is
             thicker
             then
             that
             without
             it
             ,
             the
             refracted
             line
             AK
             will
             recede
             further
             from
             the
             perpendicular
             KM
             ,
             then
             KO
             will
             recede
             from
             the
             same
             .
             Wherefore
             KO
             will
             fall
             between
             the
             refracted
             line
             and
             the
             perpendicular
             .
             Let
             therefore
             the
             refracted
             line
             be
             KP
             ,
             cutting
             FO
             in
             P
             ,
             and
             for
             the
             same
             reason
             the
             straight
             line
             LP
             will
             be
             the
             refracted
             line
             of
             the
             straight
             line
             CL.
             Wherfore
             ,
             seeing
             the
             Beams
             are
             nothing
             else
             but
             the
             Wayes
             in
             which
             the
             motion
             is
             propagated
             ,
             the
             motion
             about
             P
             will
             be
             so
             much
             more
             vehement
             then
             the
             motion
             about
             ABC
             ,
             by
             how
             much
             the
             base
             of
             the
             portion
             ABC
             is
             greater
             then
             the
             base
             of
             a
             like
             portion
             in
             the
             Sphere
             whose
             Center
             is
             P
             ,
             and
             whose
             magnitude
             is
             equal
             to
             that
             of
             the
             little
             Circle
             about
             P
             ,
             which
             comprehendeth
             all
             the
             Beams
             that
             are
             propagated
             from
             ABC
             ;
             and
             this
             Sphere
             being
             much
             less
             then
             the
             Sphere
             ABC
             ,
             the
             parts
             of
             the
             Medium
             ,
             that
             is
             ,
             of
             the
             Aire
             about
             P
             ,
             will
             change
             places
             with
             one
             another
             with
             much
             greater
             celerity
             ,
             then
             those
             about
             ABC
             .
             If
             therefore
             any
             matter
             Combustible
             ,
             that
             is
             to
             say
             ,
             such
             as
             may
             be
             easily
             dissipated
             ,
             be
             placed
             in
             P
             ,
             the
             parts
             of
             that
             matter
             (
             if
             the
             proportion
             be
             great
             enough
             between
             AC
             and
             a
             like
             portion
             of
             the
             little
             circle
             about
             P
             )
             wil
             be
             freed
             from
             their
             mutual
             cohaesion
             ,
             and
             being
             separated
             will
             acquire
             simple
             motion
             .
             But
             vehement
             simple
             motion
             generates
             in
             the
             beholder
             a
             Phantasm
             of
             Lucid
             and
             Hot
             ,
             as
             I
             have
             before
             de●onstrated
             of
             the
             simple
             motion
             of
             the
             Sunne
             ;
             and
             therefore
             the
             combustible
             matter
             which
             is
             placed
             in
             P
             will
             be
             made
             Lucid
             and
             Hot
             ,
             that
             is
             to
             say
             ,
             will
             be
             Fire
             .
             Wherefore
             I
             have
             rendered
             a
             possible
             cause
             of
             Fire
             ;
             which
             was
             to
             be
             done
             .
          
           
             5
             From
             the
             manner
             by
             which
             the
             Sunne
             generateth
             Fire
             ,
             it
             is
             easy
             to
             explaine
             the
             manner
             by
             which
             Fire
             may
             be
             generated
             by
             the
             collision
             of
             two
             Flints
             .
             For
             by
             that
             Collision
             ,
             some
             of
             those
             particles
             of
             which
             the
             stone
             is
             compacted
             ,
             are
             violently
             separated
             and
             thrown
             off
             ;
             and
             being
             withall
             swiftly
             turned
             round
             ,
             the
             Eie
             is
             moved
             by
             them
             ,
             as
             it
             is
             in
             the
             generation
             of
             Light
             by
             the
             Sunne
             .
             Wherefore
             they
             shine
             ;
             and
             falling
             upon
             matter
             which
             
             is
             already
             halfe
             dissipated
             ,
             such
             as
             is
             Tinder
             ,
             they
             throughly
             dissipate
             the
             parts
             thereof
             ,
             and
             make
             them
             turn
             round
             .
             From
             whence
             (
             as
             I
             have
             newly
             shewn
             )
             Light
             and
             Heat
             ,
             that
             is
             to
             say
             ,
             Fire
             is
             generated
             .
          
           
             6
             The
             shining
             of
             Glow-worms
             ,
             some
             kinds
             of
             Rotten
             Wood
             ,
             and
             of
             a
             kinde
             of
             stone
             made
             at
             Bolognia
             ,
             may
             have
             one
             common
             cause
             ,
             namely
             the
             exposing
             of
             them
             to
             the
             hot
             Sunne
             .
             We
             finde
             by
             experience
             that
             the
             Bolonian
             stone
             shines
             not
             unless
             it
             be
             so
             exposed
             ;
             and
             after
             it
             has
             been
             exposed
             it
             shines
             but
             for
             a
             little
             time
             ,
             namely
             ,
             as
             long
             as
             it
             retains
             a
             certain
             degree
             of
             heat
             .
             And
             the
             cause
             may
             be
             ,
             that
             the
             parts
             of
             which
             it
             is
             made
             ,
             may
             together
             with
             heat
             have
             Simple
             Motion
             imprinted
             in
             them
             by
             the
             Sunne
             .
             Which
             if
             it
             be
             so
             ,
             it
             is
             necessary
             ,
             that
             it
             shine
             in
             the
             dark
             ,
             as
             long
             as
             there
             is
             sufficient
             heat
             in
             it
             ;
             but
             this
             ceasing
             ,
             it
             will
             shine
             no
             longer
             .
             Also
             we
             find
             by
             experience
             ,
             that
             in
             the
             Glow-worm
             there
             is
             a
             certain
             thick
             humour
             ,
             like
             the
             Cristalline
             humour
             of
             the
             Eie
             ;
             which
             if
             it
             be
             taken
             out
             ,
             and
             held
             long
             enough
             in
             ones
             fingers
             ,
             and
             then
             be
             carried
             into
             the
             dark
             ,
             it
             will
             shine
             by
             reason
             of
             the
             warmth
             it
             received
             from
             the
             fingers
             ;
             but
             as
             soon
             as
             it
             is
             cold
             ,
             it
             will
             cease
             shining
             .
             From
             whence
             therefore
             can
             these
             creatures
             have
             their
             Light
             ,
             but
             from
             lying
             all
             day
             in
             the
             Sun-shine
             ,
             in
             the
             hottest
             time
             of
             Summer
             ?
             In
             the
             same
             manner
             ,
             Rotten
             Wood
             ,
             except
             it
             grow
             rotten
             in
             the
             Sun-shine
             ,
             or
             be
             afterwards
             long
             enough
             exposed
             to
             the
             Sunne
             ,
             will
             not
             shine
             .
             That
             this
             doth
             not
             happen
             in
             every
             Worm
             ,
             nor
             in
             all
             kinds
             of
             Rotten
             Wood
             ,
             nor
             in
             all
             Calcined
             Stones
             ,
             the
             cause
             may
             be
             ,
             that
             the
             parts
             of
             which
             those
             Bodies
             are
             made
             ,
             are
             different
             both
             in
             motion
             and
             figure
             from
             the
             parts
             of
             Bodies
             of
             other
             kinds
             .
          
           
             7
             Also
             the
             Sea-water
             shineth
             when
             it
             is
             either
             dashed
             with
             the
             strokes
             of
             Oares
             ,
             or
             when
             a
             Ship
             in
             its
             course
             breaks
             strongly
             through
             it
             ;
             but
             more
             or
             less
             according
             as
             the
             Winde
             blows
             from
             different
             points
             .
             The
             cause
             whereof
             may
             be
             this
             ,
             that
             the
             particles
             of
             salt
             (
             though
             they
             never
             shine
             in
             the
             Salt-pits
             ,
             where
             they
             are
             but
             slowly
             drawn
             up
             by
             the
             Sunne
             )
             being
             here
             beaten
             up
             into
             the
             aire
             in
             greater
             quantities
             ,
             and
             with
             more
             force
             ,
             are
             withall
             made
             to
             turn
             round
             ,
             and
             consequently
             to
             shine
             ,
             though
             weakly
             .
             I
             have
             therefore
             given
             a
             possible
             cause
             of
             this
             Phaenomenon
             .
          
           
             8
             If
             such
             matter
             as
             is
             compounded
             of
             hard
             little
             Bodies
             ,
             be
             set
             
             on
             fire
             ,
             it
             must
             needs
             be
             ,
             that
             as
             they
             flye
             out
             in
             greater
             or
             lesse
             quantities
             ,
             the
             Flame
             which
             is
             made
             by
             them
             ,
             will
             be
             greater
             or
             less
             .
             And
             if
             the
             aethereal
             or
             fluid
             part
             of
             that
             matter
             fly
             out
             together
             with
             them
             ,
             their
             motion
             will
             be
             the
             Swifter
             ,
             as
             it
             is
             in
             Wood
             ,
             and
             other
             things
             which
             flame
             with
             a
             manifest
             mixture
             of
             Winde
             .
             When
             therefore
             these
             hard
             particles
             by
             their
             flying
             out
             ,
             move
             the
             Eye
             strongly
             ,
             they
             shine
             bright
             ;
             and
             a
             great
             quantity
             of
             them
             flying
             out
             together
             ,
             they
             make
             a
             great
             shining
             Body
             .
             For
             Flame
             being
             nothing
             but
             an
             aggregate
             of
             shining
             particles
             ,
             the
             greater
             the
             aggregate
             is
             ,
             the
             greater
             and
             more
             manifest
             will
             be
             the
             Flame
             .
             I
             have
             therefore
             shewn
             a
             possible
             cause
             of
             Flame
             .
             And
             from
             hence
             the
             cause
             appears
             evidently
             ,
             why
             Glass
             is
             so
             easily
             and
             quickly
             melted
             by
             the
             small
             Flame
             of
             a
             candle
             blown
             ,
             which
             will
             not
             be
             melted
             without
             blowing
             ,
             but
             by
             a
             very
             strong
             Fire
             .
          
           
             Now
             if
             from
             the
             same
             matter
             ,
             there
             be
             a
             part
             broken
             off
             (
             namely
             such
             a
             part
             as
             consisteth
             of
             many
             of
             the
             small
             particles
             )
             ,
             of
             this
             is
             made
             a
             Spark
             .
             For
             from
             the
             breaking
             off
             ,
             it
             hath
             a
             violent
             turning
             round
             ;
             and
             from
             hence
             it
             shines
             .
             But
             though
             from
             this
             matter
             ,
             there
             fly
             neither
             Flame
             nor
             Sparks
             ;
             yet
             some
             of
             the
             smallest
             parts
             of
             it
             may
             be
             carried
             out
             as
             farre
             as
             to
             the
             Superficies
             ,
             and
             remain
             there
             ,
             as
             Ashes
             ;
             the
             parts
             whereof
             are
             so
             extremely
             small
             ,
             that
             it
             cannot
             any
             longer
             be
             doubted
             how
             farre
             Nature
             may
             proceed
             in
             Dividing
             .
          
           
             Lastly
             ,
             though
             by
             the
             application
             of
             fire
             to
             this
             matter
             ,
             there
             fly
             little
             or
             nothing
             from
             it
             ,
             yet
             there
             will
             be
             in
             the
             parts
             an
             endeavour
             to
             Simple
             motion
             ;
             by
             which
             the
             whole
             Body
             will
             either
             be
             Melted
             ,
             or
             (
             which
             is
             a
             degree
             of
             Melting
             )
             Softned
             .
             For
             all
             Motion
             has
             some
             effect
             upon
             all
             Matter
             whatsoever
             (
             as
             has
             been
             shewn
             at
             the
             3d
             Article
             of
             the
             16th
             Chapter
             )
             .
             Now
             if
             it
             be
             softned
             to
             such
             a
             degree
             ,
             as
             that
             the
             stubborness
             of
             the
             parts
             be
             exceeded
             by
             their
             gravity
             ,
             then
             we
             say
             it
             is
             Melted
             ;
             otherwise
             ,
             Softned
             ,
             and
             made
             Pliant
             and
             Ductile
             .
          
           
             Again
             ,
             the
             matter
             having
             in
             it
             some
             particles
             hard
             ,
             others
             aethereal
             or
             watery
             ,
             if
             by
             the
             application
             of
             fire
             these
             later
             be
             called
             out
             ,
             the
             former
             will
             thereby
             come
             to
             a
             more
             full
             contact
             with
             one
             another
             ;
             and
             consequently
             will
             not
             be
             so
             easily
             separated
             ;
             that
             is
             to
             say
             ,
             the
             whole
             Body
             will
             be
             made
             Harder
             .
             And
             this
             may
             
             be
             the
             cause
             why
             the
             same
             Fire
             makes
             some
             things
             Soft
             ,
             others
             Hard.
             
          
           
             9
             It
             is
             known
             by
             experience
             ,
             that
             if
             Hay
             be
             laid
             wet
             together
             in
             a
             heap
             ,
             it
             will
             after
             a
             time
             begin
             to
             smoke
             ,
             and
             then
             burn
             as
             it
             were
             of
             it self
             .
             The
             cause
             whereof
             seems
             to
             be
             this
             ,
             that
             in
             the
             Aire
             which
             is
             enclosed
             within
             the
             Hay
             ,
             there
             are
             those
             little
             Bodies
             ,
             which
             (
             as
             I
             have
             supposed
             )
             are
             moved
             freely
             with
             simple
             Motion
             .
             But
             this
             Motion
             being
             by
             degrees
             hindred
             more
             and
             more
             by
             the
             descending
             moisture
             ,
             which
             at
             the
             last
             fils
             and
             stops
             all
             the
             passages
             ,
             the
             thinner
             parts
             of
             the
             Aire
             ascend
             by
             penetrating
             the
             water
             ;
             and
             those
             hard
             little
             Bod●●
             being
             so
             thrust
             together
             that
             they
             touch
             and
             press
             one
             another
             ,
             acquire
             stronger
             motion
             ,
             till
             at
             last
             by
             the
             increased
             strength
             of
             this
             motion
             the
             watery
             parts
             are
             first
             driven
             outwards
             ,
             from
             whence
             appears
             Vapour
             ;
             and
             by
             the
             continued
             increase
             of
             this
             motion
             ,
             the
             smallest
             particles
             of
             the
             dryed
             Hay
             are
             forced
             out
             ,
             and
             recovering
             their
             natural
             simple
             Motion
             ,
             they
             grow
             Hot
             and
             Shine
             ,
             that
             is
             to
             say
             ,
             they
             are
             set
             on
             Fire
             .
          
           
             The
             same
             also
             may
             be
             the
             cause
             of
             Lightning
             ;
             which
             happens
             in
             the
             hottest
             time
             of
             the
             yeare
             ,
             when
             the
             water
             is
             raised
             up
             in
             greatest
             quantity
             ,
             and
             carried
             highest
             .
             For
             after
             the
             first
             Clouds
             are
             raised
             ,
             others
             after
             others
             follow
             them
             ;
             and
             being
             congeled
             above
             ,
             they
             happen
             (
             whilest
             some
             of
             them
             ascend
             and
             others
             descend
             )
             to
             fall
             upon
             another
             in
             such
             manner
             ,
             as
             that
             in
             some
             places
             all
             their
             parts
             are
             joyned
             together
             ,
             in
             others
             they
             leave
             hollow
             Spaces
             between
             them
             ;
             aud
             into
             these
             spaces
             (
             the
             aethereall
             parts
             being
             forced
             out
             by
             the
             compressure
             of
             the
             Clouds
             )
             many
             of
             the
             harder
             little
             Bodies
             are
             so
             pent
             together
             ,
             as
             that
             they
             have
             not
             the
             liberty
             of
             such
             motion
             as
             is
             naturall
             to
             the
             Aire
             .
             Wherefore
             their
             endeavour
             growes
             more
             vehement
             ,
             till
             at
             last
             they
             force
             their
             way
             through
             the
             Clouds
             ,
             sometimes
             in
             one
             place
             ,
             sometimes
             in
             another
             ;
             and
             breaking
             through
             with
             great
             noise
             ,
             they
             move
             the
             aire
             violently
             ,
             &
             striking
             our
             Eies
             generate
             Light
             ,
             that
             is
             to
             say
             ,
             they
             Shine
             .
             And
             this
             Shining
             is
             that
             we
             call
             Lightning
             .
          
           
             10
             The
             most
             common
             Phaenomenon
             proceeding
             from
             Fire
             ,
             and
             yet
             the
             most
             admirable
             of
             all
             others
             ,
             is
             the
             force
             of
             Gunpowder
             fired
             ;
             which
             being
             compounded
             of
             Niter
             ,
             Brimstone
             and
             Coles
             beaten
             small
             ,
             hath
             from
             the
             Coles
             its
             first
             taking
             fire
             ;
             from
             the
             
             Brimstone
             its
             nourishment
             and
             flame
             ,
             that
             is
             to
             say
             ,
             Light
             and
             motion
             ;
             and
             from
             the
             Niter
             the
             vehemence
             of
             both
             .
             Now
             if
             a
             piece
             of
             Niter
             ,
             (
             before
             it
             is
             beaten
             )
             be
             laid
             upon
             a
             burning
             Cole
             ,
             first
             it
             melts
             ,
             and
             like
             water
             quencheth
             that
             part
             of
             the
             Cole
             it
             toucheth
             .
             Then
             Vapor
             or
             Aire
             flying
             out
             where
             the
             Cole
             and
             Niter
             joyne
             ,
             bloweth
             the
             Cole
             with
             great
             swiftnesse
             and
             vehemence
             on
             all
             sides
             .
             And
             from
             hence
             it
             comes
             to
             passe
             ,
             that
             by
             two
             contrary
             motions
             (
             the
             one
             ,
             of
             the
             particles
             which
             go
             out
             of
             the
             burning
             Cole
             ,
             the
             other
             ,
             of
             those
             of
             the
             aethereall
             and
             watery
             substance
             of
             the
             Niter
             )
             is
             generated
             that
             vehement
             Motion
             and
             Inflammation
             .
             And
             lastly
             ,
             when
             there
             is
             no
             more
             action
             from
             the
             Niter
             (
             that
             is
             to
             say
             ,
             when
             the
             volatile
             parts
             of
             the
             Niter
             are
             flown
             out
             )
             there
             is
             found
             about
             the
             sides
             a
             certain
             white
             substance
             ,
             which
             being
             thrown
             again
             into
             the
             fire
             ,
             will
             grow
             red
             hot
             again
             ,
             but
             will
             not
             be
             dissipated
             ,
             at
             least
             ,
             unlesse
             the
             fire
             be
             augmented
             .
             If
             now
             a
             possible
             cause
             of
             this
             be
             found
             out
             ,
             the
             same
             will
             also
             be
             a
             possible
             cause
             why
             a
             grain
             of
             Gunpowder
             set
             on
             fire
             doth
             expand
             it selfe
             with
             such
             vehement
             motion
             ,
             and
             Shine
             .
             And
             it
             may
             be
             caused
             in
             this
             manner
             .
          
           
             Let
             the
             particles
             of
             which
             Niter
             consisteth
             ,
             be
             supposed
             to
             be
             some
             of
             them
             hard
             ,
             others
             watery
             ,
             and
             the
             rest
             aethereall
             .
             Also
             let
             the
             hard
             particles
             be
             supposed
             to
             be
             spherically
             hollow
             ,
             like
             small
             bubbles
             ,
             so
             that
             many
             of
             them
             growing
             together
             may
             constitute
             a
             Body
             whose
             little
             cavernes
             are
             filled
             with
             a
             substance
             which
             is
             either
             watery
             ,
             or
             aethereal
             ,
             or
             both
             .
             As
             soon
             therefore
             as
             the
             hard
             particles
             are
             dissipated
             ,
             the
             watery
             and
             aethereal
             particles
             will
             necessarily
             fly
             out
             ;
             and
             as
             they
             fly
             ,
             of
             necessity
             blow
             strongly
             the
             burning
             Coles
             and
             Brimstone
             which
             are
             mingled
             together
             ;
             whereupon
             there
             will
             follow
             a
             great
             expansion
             of
             Light
             ,
             with
             vehement
             flame
             ,
             and
             a
             violent
             dissipation
             of
             the
             particles
             of
             the
             Niter
             ,
             the
             Brimstone
             and
             the
             Coles
             .
             Wherefore
             I
             have
             given
             a
             possible
             cause
             of
             the
             force
             of
             fired
             Gunpowder
             .
          
           
             It
             is
             manifest
             from
             hence
             ,
             that
             for
             the
             rendering
             of
             the
             cause
             why
             a
             bullet
             of
             lead
             or
             iron
             shot
             from
             a
             peece
             of
             Ordnance
             flies
             with
             so
             great
             velocity
             ,
             there
             is
             no
             necessity
             to
             introduce
             such
             Rarefaction
             ,
             as
             (
             by
             the
             common
             definition
             of
             it
             )
             makes
             the
             same
             Matter
             to
             have
             sometimes
             more
             ,
             sometimes
             lesse
             Quantity
             ;
             which
             is
             
             unconceiveable
             .
             For
             every
             thing
             is
             said
             to
             be
             greater
             or
             lesse
             ,
             as
             it
             hath
             more
             or
             lesse
             Quantity
             .
             The
             violence
             with
             which
             a
             bullet
             is
             thrust
             out
             of
             a
             Gun
             ,
             proceeds
             from
             the
             swiftnesse
             of
             the
             small
             particles
             of
             the
             fired
             Powder
             ;
             at
             least
             it
             may
             proceed
             from
             that
             cause
             ▪
             without
             the
             supposition
             of
             any
             Empty
             Space
             .
          
           
             11
             Besides
             ,
             by
             the
             attrition
             or
             rubbing
             of
             one
             Body
             against
             another
             ,
             as
             of
             Wood
             against
             Wood
             ,
             we
             find
             that
             not
             only
             a
             certaine
             degree
             of
             Heat
             ,
             but
             Fire
             it selfe
             is
             sometimes
             generated
             .
             For
             such
             motion
             ,
             is
             the
             reciprocation
             of
             pressure
             ,
             sometimes
             one
             way
             sometimes
             the
             other
             ;
             and
             by
             this
             reciprocation
             ,
             whatsoever
             is
             fluid
             in
             both
             the
             peeces
             of
             Wood
             ,
             is
             forced
             hither
             and
             thither
             ;
             and
             consequently
             ,
             to
             an
             endeavour
             of
             getting
             out
             ;
             and
             at
             last
             by
             breaking
             out
             makes
             Fire
             .
          
           
             12
             Now
             Light
             is
             distinguished
             into
             ,
             First
             ,
             Second
             ,
             Third
             ,
             and
             so
             on
             infinitely
             .
             And
             we
             call
             that
             First
             Light
             ,
             which
             is
             in
             the
             first
             Lucid
             Bodie
             ;
             as
             the
             Sunne
             ,
             Fire
             ,
             &c.
             
             Second
             ,
             that
             which
             is
             in
             such
             Bodies
             as
             being
             not
             transparent
             are
             illuminated
             by
             the
             Sunne
             ;
             as
             the
             Moon
             ,
             a
             Wall
             ,
             &c.
             and
             Third
             ,
             that
             which
             is
             in
             Bodies
             not
             transparent
             but
             illuminated
             by
             Second
             Light
             ,
             &c.
             
          
           
             13
             Colour
             is
             Light
             ,
             but
             troubled
             Light
             ;
             namely
             ,
             such
             as
             is
             generated
             by
             perturbed
             motion
             ;
             as
             shall
             be
             made
             manifest
             by
             the
             Red
             ,
             Yellow
             ,
             Blew
             and
             Purple
             which
             are
             generated
             by
             the
             interposition
             of
             a
             
               Diaphanous
               Prisma
            
             (
             whose
             opposite
             bases
             are
             triangular
             )
             between
             the
             Light
             and
             that
             which
             is
             enlightened
             .
          
           
             For
             let
             there
             be
             a
             Prisma
             of
             Glasse
             ,
             or
             of
             any
             other
             transparent
             matter
             which
             is
             of
             greater
             density
             then
             Aire
             ;
             and
             let
             the
             triangle
             ABC
             be
             the
             base
             of
             this
             Prisma
             .
             Also
             let
             the
             straight
             line
             DE
             be
             the
             diameter
             of
             the
             Sunnes
             Body
             ,
             having
             oblique
             position
             to
             the
             straight
             line
             AB
             ;
             and
             let
             the
             Sunne-beames
             passe
             in
             the
             lines
             DA
             and
             EBC
             .
             And
             lastly
             let
             the
             straight
             lines
             DA
             and
             EC
             be
             produced
             indefinitely
             to
             F
             and
             G.
             Seeing
             therefore
             the
             straight
             line
             DA
             ,
             by
             reason
             of
             the
             density
             of
             the
             Glasse
             is
             refracted
             towards
             the
             perpendicular
             ;
             let
             the
             line
             refracted
             at
             the
             point
             A
             be
             the
             straight
             line
             AH
             .
             And
             againe
             ,
             seeing
             the
             Medium
             below
             AC
             is
             thinner
             then
             that
             above
             it
             ,
             the
             other
             refraction
             which
             will
             be
             made
             there
             ,
             will
             diverge
             from
             the
             perpendicular
             .
             Let
             therefore
             this
             second
             refracted
             line
             be
             AI.
             Also
             let
             the
             same
             
             be
             done
             at
             the
             point
             C
             ,
             by
             making
             the
             first
             refracted
             line
             to
             be
             CK
             ,
             and
             the
             second
             CL.
             Seeing
             therefore
             the
             cause
             of
             the
             refraction
             in
             the
             point
             A
             of
             the
             straight
             line
             of
             AB
             ,
             is
             the
             excess
             of
             the
             resistance
             of
             the
             Medium
             in
             AB
             above
             the
             resistance
             of
             the
             Aire
             ,
             there
             must
             of
             necessity
             be
             reaction
             from
             the
             point
             A
             towards
             the
             point
             B
             ;
             and
             consequently
             the
             Medium
             at
             A
             within
             the
             triangle
             ABC
             ,
             will
             have
             its
             motion
             troubled
             ;
             that
             is
             to
             say
             ,
             the
             straight
             motion
             in
             AF
             and
             AH
             ,
             will
             be
             mixed
             with
             the
             transverse
             motion
             between
             the
             same
             AF
             and
             AH
             ,
             represented
             by
             the
             short
             transverse
             lines
             in
             the
             triangle
             AFH
             .
             Againe
             ,
             seeing
             at
             the
             point
             A
             of
             the
             straight
             line
             AC
             ,
             there
             is
             a
             second
             refraction
             from
             AH
             in
             AI
             ,
             the
             motion
             of
             the
             Medium
             will
             againe
             be
             perturbed
             by
             reason
             of
             the
             transverse
             reaction
             from
             A
             towards
             C
             ,
             represented
             likewise
             by
             the
             short
             transverse
             lines
             in
             the
             triangle
             AHI
             .
             And
             in
             the
             same
             manner
             there
             is
             a
             double
             perturbation
             represented
             by
             the
             transverse
             lines
             in
             the
             triangles
             CGK
             and
             CKL
             .
             But
             as
             for
             the
             light
             between
             AI
             and
             CG
             ,
             it
             will
             not
             be
             perturbed
             ;
             because
             if
             there
             were
             in
             all
             the
             points
             of
             the
             straight
             lines
             AB
             and
             AC
             ,
             the
             same
             action
             which
             is
             in
             the
             points
             A
             and
             C
             ,
             then
             the
             plaine
             of
             the
             triangle
             CGK
             would
             be
             every
             where
             coincident
             with
             the
             plaine
             of
             the
             triangle
             AFH
             ;
             by
             which
             meanes
             all
             would
             appear
             alike
             between
             A
             and
             C.
             Besides
             ,
             it
             is
             to
             be
             observed
             ,
             that
             all
             the
             reaction
             at
             A
             ,
             tends
             towards
             the
             illuminated
             parts
             which
             are
             between
             A
             and
             C
             ,
             and
             consequently
             perturbeth
             the
             First
             Light.
             And
             on
             the
             contrary
             ,
             that
             all
             the
             reaction
             at
             C
             tends
             towards
             the
             parts
             without
             the
             triangle
             ,
             or
             without
             the
             Prisma
             ABC
             ,
             where
             there
             is
             none
             but
             Second
             Light
             ;
             and
             that
             the
             triangle
             AFH
             shewes
             that
             perturbation
             of
             Light
             which
             is
             made
             in
             the
             Glasse
             it selfe
             ;
             as
             the
             triangle
             AHI
             shewes
             that
             perturbation
             of
             Light
             which
             is
             made
             below
             the
             Glasse
             .
             In
             like
             manner
             ,
             that
             CGK
             shewes
             the
             perturbation
             of
             Light
             within
             the
             Glasse
             ;
             and
             CKL
             that
             which
             is
             below
             the
             Glasse
             .
             From
             whence
             there
             are
             four
             divers
             motions
             ,
             or
             four
             different
             illuminations
             or
             Colours
             ;
             whose
             differences
             appear
             most
             manifestly
             to
             the
             Sense
             in
             a
             Prisma
             (
             whose
             base
             is
             an
             equilaterall
             triangle
             )
             when
             the
             Sunne-beames
             that
             passe
             through
             it
             are
             received
             upon
             a
             white
             paper
             .
             For
             the
             triangle
             AFH
             appears
             Red
             to
             the
             Sense
             ;
             the
             triangle
             AHI
             Yellow
             ;
             the
             
             triangle
             CGK
             Green
             ,
             and
             approaching
             to
             Blew
             ;
             and
             lastly
             the
             triangle
             CKL
             appears
             Purple
             .
             It
             is
             therefore
             evident
             ,
             that
             when
             weak
             but
             First
             Light
             passeth
             through
             a
             more
             resisting
             diaphanous
             Body
             ,
             as
             Glasse
             ,
             the
             beames
             which
             fall
             upon
             it
             tranversly
             ,
             make
             Rednesse
             ;
             and
             when
             the
             same
             First
             Light
             is
             stronger
             ,
             as
             it
             is
             in
             the
             thinner
             Medium
             below
             the
             straight
             line
             AC
             ,
             the
             transverse
             beames
             make
             Yellownesse
             .
             Also
             when
             Second
             Light
             is
             strong
             ,
             as
             it
             is
             in
             the
             triangle
             CGK
             (
             which
             is
             neerest
             to
             the
             First
             Light
             )
             the
             transverse
             beames
             make
             Greenesse
             ;
             and
             when
             the
             same
             Second
             Light
             is
             weaker
             ,
             as
             in
             the
             triangle
             CKL
             ,
             they
             make
             a
             Purple
             colour
             .
          
           
             14
             From
             hence
             may
             be
             deduced
             a
             cause
             why
             the
             Moon
             and
             Starres
             appear
             bigger
             and
             redder
             neer
             the
             Horizon
             then
             in
             the
             Mid-heaven
             .
             For
             between
             the
             Eie
             and
             the
             apparent
             Horizon
             ,
             there
             is
             more
             impure
             aire
             ,
             such
             as
             is
             mingled
             with
             Watery
             and
             Earthy
             little
             Bodies
             ,
             then
             is
             between
             the
             same
             Eie
             and
             the
             more
             elevated
             part
             of
             Heaven
             .
             But
             Vision
             is
             made
             by
             Beames
             which
             constitute
             a
             Cone
             ,
             whose
             base
             ,
             if
             we
             look
             upon
             the
             Moon
             ,
             is
             the
             Moons
             Face
             ,
             and
             whose
             vertex
             is
             in
             the
             Eie
             ;
             and
             therefore
             many
             beams
             from
             the
             Moon
             must
             needs
             fall
             upon
             little
             Bodies
             that
             are
             without
             the
             Visual
             Cone
             ,
             and
             be
             by
             them
             reflected
             to
             the
             Eie
             .
             But
             these
             reflected
             beams
             tend
             all
             in
             lines
             which
             are
             transverse
             to
             the
             Visual
             Cone
             ,
             and
             make
             at
             the
             Eie
             an
             angle
             which
             is
             greater
             then
             the
             angle
             of
             the
             Cone
             .
             Wherefore
             the
             Moon
             appeares
             greater
             in
             the
             Horizon
             ,
             then
             when
             she
             is
             more
             elevated
             .
             And
             because
             those
             reflected
             beames
             go
             transversely
             ,
             there
             will
             be
             generated
             (
             by
             the
             last
             article
             )
             Rednesse
             .
             A
             possible
             cause
             therefore
             is
             shewne
             ,
             why
             the
             Moon
             ,
             as
             also
             the
             Starres
             appear
             Greater
             and
             Redder
             in
             the
             Horizon
             ,
             then
             in
             the
             midst
             of
             heaven
             .
             The
             same
             also
             may
             be
             the
             cause
             why
             the
             Sunne
             appears
             in
             the
             Horizon
             ,
             Greater
             ,
             and
             of
             a
             colour
             more
             degenerating
             to
             Yellow
             ,
             then
             when
             he
             is
             higher
             elevated
             .
             For
             the
             reflection
             from
             the
             little
             Bodies
             between
             ,
             and
             the
             transverse
             motion
             of
             the
             Medium
             are
             still
             the
             same
             .
             But
             the
             Light
             of
             the
             Sunne
             is
             much
             stronger
             then
             that
             of
             the
             Moon
             ;
             and
             therefore
             (
             by
             the
             last
             article
             )
             his
             Splendor
             must
             needs
             by
             this
             perturbation
             degenerate
             into
             Yellownesse
             .
          
           
             But
             for
             the
             generation
             of
             these
             four
             colours
             ,
             it
             is
             not
             necessary
             
             that
             the
             figure
             of
             the
             Glass
             be
             a
             Prisma
             ;
             for
             if
             it
             were
             Spherical
             it
             would
             doe
             the
             same
             .
             For
             in
             a
             Sphere
             the
             Sunne-beames
             are
             twice
             refracted
             and
             twice
             reflected
             .
             And
             this
             being
             observed
             by
             
               Des
               Cartes
            
             ;
             and
             with
             all
             that
             a
             Rainebow
             never
             appeares
             but
             when
             it
             rains
             ;
             as
             also
             ,
             that
             the
             drops
             of
             raine
             have
             their
             figures
             almost
             Spherical
             ;
             he
             hath
             shewne
             from
             thence
             the
             cause
             of
             the
             colours
             in
             the
             Rainbow
             ;
             which
             therefore
             need
             not
             be
             repeated
             .
          
           
             15
             Whiteness
             is
             Light
             ;
             but
             Light
             perturbed
             by
             the
             reflexions
             of
             many
             beams
             of
             Light
             comming
             to
             the
             Eye
             together
             within
             a
             little
             space
             .
             For
             if
             Glass
             ,
             or
             any
             other
             Diaphanous
             Body
             be
             reduced
             to
             very
             small
             parts
             by
             contusion
             or
             concussion
             ;
             every
             one
             of
             those
             parts
             (
             if
             the
             Beams
             of
             a
             lucid
             Body
             be
             from
             any
             one
             point
             of
             the
             same
             reflected
             to
             the
             Eye
             )
             will
             represent
             to
             the
             beholder
             an
             Idea
             or
             Image
             of
             the
             whole
             lucid
             Body
             ,
             that
             is
             to
             say
             ,
             a
             Phantasme
             of
             White
             .
             For
             the
             strongest
             Light
             is
             the
             most
             White
             ;
             and
             therefore
             many
             such
             parts
             will
             make
             many
             such
             Images
             .
             Wherefore
             if
             those
             parts
             lie
             thick
             and
             close
             together
             ,
             those
             many
             Images
             will
             appear
             confusedly
             ,
             and
             will
             by
             reason
             of
             the
             confused
             Light
             represent
             a
             White
             Colour
             .
             So
             that
             from
             hence
             may
             be
             deduced
             a
             possible
             cause
             ,
             why
             Glass
             beaten
             ,
             that
             is
             ,
             reduced
             to
             powder
             ,
             looks
             White
             .
             Also
             why
             Water
             and
             Snow
             are
             White
             ;
             they
             being
             nothing
             but
             a
             heap
             of
             very
             small
             Diaphanous
             Bodies
             ,
             namely
             ,
             of
             little
             Bubbles
             ,
             from
             whose
             several
             convex
             Superficies
             ,
             there
             are
             by
             reflexion
             made
             several
             confused
             Phantasmes
             of
             the
             whole
             lucid
             Body
             ;
             that
             is
             to
             say
             ,
             Whiteness
             .
             For
             the
             same
             reason
             ,
             Salt
             and
             Nitre
             are
             White
             ;
             as
             consisting
             of
             small
             Bubbles
             which
             contain
             within
             them
             Water
             and
             Aire
             ;
             as
             is
             manifest
             in
             Nitre
             ,
             from
             this
             ,
             that
             being
             thrown
             into
             the
             fire
             ,
             it
             violently
             blowes
             the
             same
             ;
             which
             Salt
             also
             doth
             ,
             but
             with
             less
             violence
             .
             But
             if
             a
             White
             Body
             be
             exposed
             ,
             not
             to
             the
             Light
             of
             the
             Day
             ,
             but
             to
             that
             of
             the
             Fire
             ,
             or
             of
             a
             Candle
             ,
             it
             will
             not
             at
             the
             first
             sight
             be
             easily
             judged
             whether
             it
             be
             White
             or
             Yellow
             ;
             the
             cause
             whereof
             may
             be
             this
             ,
             that
             the
             light
             of
             those
             things
             which
             burn
             and
             flame
             ,
             is
             almost
             of
             a
             middle
             Colour
             between
             Whiteness
             and
             Yellowness
             .
          
           
             16
             As
             Whiteness
             is
             Light
             ,
             so
             Blackness
             is
             the
             privation
             of
             Light
             ,
             or
             Darkness
             .
             And
             from
             hence
             it
             is
             ;
             First
             ,
             that
             all
             Holes
             ,
             from
             which
             no
             light
             can
             be
             reflected
             to
             the
             Eie
             ,
             appear
             Black.
             
             Secondly
             ,
             that
             when
             a
             Body
             hath
             little
             eminent
             particles
             erected
             straight
             up
             from
             the
             Superficies
             (
             so
             that
             the
             Beams
             of
             Light
             which
             fall
             upon
             them
             are
             reflected
             ,
             not
             to
             the
             Eie
             ,
             but
             to
             the
             Body
             it self
             )
             that
             Superficies
             appears
             Black
             ,
             in
             the
             same
             manner
             as
             the
             Sea
             appears
             Black
             ,
             when
             ruffled
             by
             the
             Wind.
             Thirdly
             ,
             that
             any
             combustible
             matter
             is
             by
             the
             fire
             made
             to
             look
             Black
             before
             it
             shines
             .
             For
             the
             endeavour
             of
             the
             fire
             being
             to
             dissipate
             the
             smallest
             parts
             of
             such
             Bodies
             as
             are
             thrown
             into
             it
             ,
             it
             must
             first
             raise
             and
             erect
             those
             parts
             ,
             before
             it
             can
             work
             their
             dissipation
             .
             If
             therefore
             the
             fire
             be
             put
             out
             before
             the
             parts
             be
             totally
             dissipated
             ,
             the
             Cole
             will
             appear
             Black
             ;
             for
             the
             parts
             being
             onely
             erected
             ,
             the
             Beams
             of
             Light
             falling
             upon
             them
             will
             not
             be
             reflected
             to
             the
             Eie
             ,
             but
             to
             the
             Cole
             it self
             .
             Fourthly
             ,
             that
             Burning
             Glasses
             do
             more
             easily
             burn
             Black
             things
             then
             White
             .
             For
             in
             a
             White
             Superficies
             ,
             the
             eminent
             parts
             are
             convex
             ,
             like
             little
             bubbles
             ;
             and
             therefore
             the
             Beams
             of
             Light
             which
             fall
             upon
             them
             are
             reflected
             every
             way
             from
             the
             reflecting
             Body
             .
             But
             in
             a
             Black
             Superficies
             ,
             where
             the
             eminent
             particles
             are
             more
             erected
             ,
             the
             Beams
             of
             Light
             falling
             upon
             them
             ,
             are
             all
             necessarily
             reflected
             towards
             the
             Body
             it self
             ;
             and
             therefore
             Bodies
             that
             are
             Black
             are
             more
             easily
             set
             on
             fire
             by
             the
             Sun-beams
             ,
             then
             those
             that
             are
             White
             .
             Fifthly
             ,
             that
             all
             Colours
             that
             are
             made
             of
             the
             mixture
             of
             White
             and
             Black
             ,
             proceed
             from
             the
             different
             position
             of
             the
             particles
             that
             rise
             above
             the
             Superficies
             ,
             and
             their
             different
             forms
             of
             asperity
             .
             For
             according
             to
             these
             differences
             ,
             more
             or
             fewer
             Beams
             of
             Light
             are
             reflected
             from
             several
             Bodies
             to
             the
             Eie
             .
             But
             in
             regard
             those
             differences
             are
             innumerable
             ,
             and
             the
             Bodies
             themselves
             so
             small
             ,
             that
             we
             cannot
             perceive
             them
             ,
             the
             explication
             and
             precise
             determination
             of
             the
             Causes
             of
             all
             Colours
             is
             a
             thing
             of
             so
             great
             difficulty
             ,
             that
             I
             dare
             not
             undertake
             it
             .
          
        
         
           
           
           
             
               CHAP.
               XXVIII
            
             .
             Of
             
               Cold
               ,
               Wind
               ,
               Hard
               ,
               Ice
               ,
               Restitution
            
             of
             Bodies
             bent
             ,
             
               Diaphanous
               ,
               Lightning
            
             and
             Thunder
             ;
             and
             of
             the
             Heads
             of
             Rivers
             .
          
           
             
               
                 1
                 Why
                 Breath
                 from
                 the
                 same
                 mouth
                 sometimes
                 heats
                 ,
                 and
                 sometimes
                 cools
                 .
              
               
                 2
                 Wind
                 ,
                 and
                 the
                 Inconstancy
                 of
                 Winds
                 ,
                 whence
                 .
              
               
                 3
                 Why
                 there
                 is
                 a
                 constant
                 ,
                 though
                 not
                 a
                 great
                 Wind
                 from
                 East
                 to
                 West
                 neer
                 the
                 Equator
                 .
              
               
                 4
                 What
                 is
                 the
                 effect
                 of
                 Aire
                 pent
                 in
                 between
                 the
                 Clouds
                 .
              
               
                 5
                 No
                 change
                 from
                 Soft
                 to
                 Hard
                 ,
                 but
                 by
                 motion
                 .
              
               
                 6
                 What
                 is
                 the
                 cause
                 of
                 Cold
                 neer
                 the
                 Poles
                 .
              
               
                 7
                 The
                 cause
                 of
                 Ice
                 ;
                 and
                 why
                 the
                 Cold
                 is
                 more
                 remiss
                 in
                 rainy
                 then
                 in
                 clear
                 weather
                 .
                 Why
                 water
                 doth
                 not
                 freeze
                 in
                 deep
                 Wells
                 ,
                 as
                 it
                 doth
                 neer
                 the
                 Superficies
                 of
                 the
                 Earth
                 .
                 Why
                 Ice
                 is
                 not
                 so
                 heavy
                 as
                 Water
                 ;
                 and
                 why
                 Wine
                 is
                 not
                 so
                 easily
                 frozen
                 as
                 Water
                 .
              
               
                 8
                 Another
                 cause
                 of
                 Hardness
                 from
                 the
                 fuller
                 contact
                 of
                 Atomes
                 .
                 Also
                 how
                 Hard
                 things
                 are
                 broken
                 .
              
               
                 9
                 A
                 third
                 cause
                 of
                 Hardness
                 from
                 Heat
                 .
              
               
                 10
                 A
                 fourth
                 cause
                 of
                 Hardness
                 from
                 the
                 motion
                 of
                 Atomes
                 enclosed
                 in
                 a
                 narrow
                 space
                 .
              
               
                 11
                 How
                 Hard
                 things
                 are
                 Softned
                 .
              
               
                 12
                 Whence
                 proceeds
                 the
                 spontaneous
                 Restitution
                 of
                 things
                 Bent.
                 
              
               
                 13
                 Diaphanous
                 ,
                 and
                 Opacous
                 ,
                 what
                 they
                 are
                 ,
                 and
                 whence
                 .
              
               
                 14
                 The
                 cause
                 of
                 Lightning
                 and
                 Thunder
                 .
              
               
                 15
                 Whence
                 it
                 proceeds
                 that
                 Clouds
                 can
                 fall
                 again
                 ,
                 after
                 they
                 are
                 once
                 elevated
                 and
                 frozen
                 .
              
               
                 16
                 How
                 it
                 could
                 be
                 that
                 the
                 Moon
                 was
                 eclipsed
                 ,
                 when
                 she
                 was
                 not
                 diametrally
                 opposite
                 to
                 the
                 Sunne
                 .
              
               
                 17
                 By
                 what
                 means
                 many
                 Sunnes
                 may
                 appear
                 at
                 once
                 .
              
               
                 18
                 Of
                 the
                 Heads
                 of
                 Rivers
                 .
              
            
          
           
             1
             AS
             ,
             when
             the
             motion
             of
             the
             ambient
             aethereal
             substance
             makes
             the
             Spirits
             and
             fluid
             parts
             of
             our
             Bodies
             tend
             outwards
             ,
             we
             acknowledge
             Heat
             ;
             so
             ,
             by
             the
             endeavour
             inwards
             of
             the
             same
             spirits
             and
             humours
             ,
             we
             feel
             Cold.
             So
             that
             to
             Cool
             ,
             is
             to
             make
             the
             exterior
             parts
             of
             the
             Body
             endeavour
             inwards
             ,
             by
             a
             motion
             contrary
             to
             that
             of
             Calefaction
             ,
             by
             which
             the
             internal
             parts
             are
             called
             outwards
             .
             
             He
             therefore
             that
             would
             know
             the
             cause
             of
             Cold
             must
             find
             by
             what
             motion
             or
             motions
             ,
             the
             exterior
             parts
             of
             any
             Body
             endeavour
             to
             retire
             inwards
             .
             To
             begin
             with
             those
             Phaenomena
             which
             are
             the
             most
             familiar
             ;
             There
             is
             almost
             no
             man
             but
             knows
             ,
             that
             breath
             blown
             strongly
             ,
             and
             which
             comes
             from
             the
             mouth
             with
             violence
             ,
             that
             is
             to
             say
             ,
             the
             passage
             being
             straight
             ,
             will
             Cool
             the
             hand
             ;
             and
             that
             the
             same
             breath
             blown
             gently
             ,
             that
             is
             to
             say
             ,
             through
             a
             greater
             aperture
             ,
             wil
             warm
             the
             same
             The
             cause
             of
             which
             Phaenomenon
             may
             be
             this
             .
             The
             breath
             going
             out
             ,
             hath
             two
             motiōs
             ;
             the
             one
             ,
             of
             the
             whole
             and
             direct
             ,
             by
             which
             the
             formost
             parts
             of
             the
             hand
             are
             driven
             inwards
             ,
             the
             other
             ,
             simple
             motion
             of
             the
             small
             particles
             of
             the
             same
             breath
             ,
             which
             (
             as
             I
             have
             shewn
             in
             the
             3d
             Article
             of
             the
             last
             Chapter
             )
             causeth
             Heat
             .
             According
             therefore
             as
             either
             of
             these
             Motions
             is
             predominant
             ,
             so
             there
             is
             the
             sense
             sometimes
             of
             Cold
             ,
             sometimes
             of
             Heat
             .
             Wherefore
             ,
             when
             the
             breath
             is
             softly
             breathed
             out
             at
             a
             large
             passage
             ,
             that
             simple
             Motion
             which
             causeth
             Heat
             prevaileth
             ,
             and
             consequently
             Heat
             is
             felt
             ;
             and
             when
             by
             compressing
             the
             lips
             the
             breath
             is
             more
             strongly
             blown
             out
             ,
             then
             is
             the
             direct
             motion
             prevalent
             ,
             which
             makes
             us
             feel
             Cold.
             For
             the
             direct
             motion
             of
             the
             breath
             or
             aire
             ,
             is
             Wind
             ;
             and
             all
             Wind
             Cools
             ,
             or
             diminisheth
             former
             heat
             .
          
           
             2
             And
             seeing
             not
             onely
             great
             Wind
             ,
             but
             almost
             any
             Ventilation
             and
             stirring
             of
             the
             Aire
             ,
             doth
             refrigerate
             ;
             the
             reason
             of
             many
             experiments
             concerning
             Cold
             cannot
             well
             be
             given
             ,
             without
             finding
             first
             what
             are
             the
             causes
             of
             Wind.
             Now
             Wind
             is
             nothing
             else
             but
             the
             direct
             motion
             of
             the
             Aire
             thrust
             forwards
             ;
             which
             nevertheless
             ,
             when
             many
             Winds
             concurre
             may
             be
             circular
             ,
             or
             otherwise
             indirect
             ,
             as
             it
             is
             in
             Whirle-winds
             .
             Wherefore
             in
             the
             first
             place
             we
             are
             to
             enquire
             into
             the
             Causes
             of
             Winds
             .
             Wind
             is
             Aire
             moved
             in
             a
             considerable
             quantity
             ,
             and
             that
             either
             in
             the
             manner
             of
             Waves
             ,
             which
             is
             both
             forwards
             &
             also
             up
             &
             down
             ;
             or
             else
             forwards
             onely
             .
          
           
             Supposing
             therefore
             the
             Aire
             both
             cleer
             and
             calm
             ,
             for
             any
             time
             how
             little
             soever
             ;
             yet
             the
             greater
             Bodies
             of
             the
             World
             ,
             being
             so
             disposed
             and
             ordered
             as
             has
             been
             said
             ,
             it
             will
             be
             necessary
             that
             a
             Wind
             presently
             arise
             some
             where
             .
             For
             seeing
             that
             motion
             of
             the
             parts
             of
             the
             Aire
             which
             is
             made
             by
             the
             Simple
             Motion
             of
             the
             Sunne
             in
             his
             own
             Epicycle
             ,
             causeth
             an
             exhalation
             of
             the
             particles
             
             of
             water
             from
             the
             Seas
             and
             all
             other
             moist
             Bodies
             ,
             and
             those
             particles
             make
             Clouds
             ;
             it
             must
             needs
             follow
             ,
             that
             whilest
             the
             particles
             of
             water
             pass
             upwards
             ,
             the
             particles
             of
             Aire
             (
             for
             the
             keeping
             of
             all
             Spaces
             full
             )
             be
             justled
             out
             on
             every
             side
             ,
             and
             urge
             the
             next
             particles
             ,
             and
             these
             the
             next
             ,
             till
             having
             made
             their
             circuit
             ,
             there
             comes
             continually
             so
             much
             Aire
             to
             the
             hinder
             parts
             of
             the
             Earth
             ,
             as
             there
             went
             water
             from
             before
             it
             .
             Wherefore
             the
             ascending
             Vapours
             move
             the
             Aire
             towards
             the
             sides
             every
             way
             ;
             and
             all
             direct
             motion
             of
             the
             Aire
             being
             Wind
             ,
             they
             make
             a
             Wind.
             And
             if
             this
             Wind
             meet
             often
             with
             other
             Vapours
             which
             arise
             in
             other
             places
             ,
             it
             is
             manifest
             that
             the
             force
             thereof
             will
             be
             augmented
             ,
             &
             the
             way
             or
             course
             of
             it
             changed
             .
             Besides
             ,
             according
             as
             the
             Earth
             by
             its
             diurnal
             motion
             turns
             sometimes
             the
             drier
             ,
             sometimes
             the
             moister
             part
             towards
             the
             Sunne
             ,
             so
             sometimes
             a
             greater
             ,
             sometimes
             a
             less
             quantity
             of
             Vapours
             will
             be
             raised
             ,
             that
             is
             to
             say
             ,
             sometimes
             there
             will
             be
             a
             less
             ,
             sometimes
             a
             greater
             Wind.
             Wherefore
             I
             have
             rendred
             a
             possible
             cause
             of
             such
             Winds
             ,
             as
             are
             generated
             by
             Vapours
             ;
             and
             also
             of
             their
             Inconstancy
             .
          
           
             From
             hence
             it
             follows
             ,
             that
             these
             Winds
             cannot
             be
             made
             in
             any
             place
             which
             is
             higher
             then
             that
             to
             which
             Vapours
             may
             ascend
             .
             Nor
             is
             that
             incredible
             which
             is
             reported
             of
             the
             highest
             Mountains
             ,
             as
             the
             
               Pique
               of
               Tenariffe
            
             and
             the
             
               Andes
               of
               Peru
            
             ,
             namely
             ,
             that
             they
             are
             not
             at
             all
             troubled
             with
             these
             inconstant
             Winds
             .
             And
             if
             it
             were
             certain
             ,
             that
             neither
             Rain
             nor
             Snow
             were
             ever
             seen
             in
             the
             highest
             tops
             of
             those
             Mountains
             ,
             it
             could
             not
             be
             doubted
             but
             that
             they
             are
             higher
             then
             any
             place
             to
             which
             Vapours
             use
             to
             ascend
             .
          
           
             3
             Nevertheless
             ,
             there
             may
             be
             Wind
             there
             ,
             though
             not
             that
             which
             is
             made
             by
             the
             ascent
             of
             Vapours
             ,
             yet
             a
             less
             &
             more
             constant
             Wind
             (
             like
             the
             continued
             blast
             of
             a
             pair
             of
             bellows
             )
             blowing
             from
             the
             East
             .
             And
             this
             may
             have
             a
             double
             cause
             ;
             the
             one
             ,
             the
             diurnal
             mo
             tion
             of
             the
             Earth
             ;
             the
             other
             ,
             its
             simple
             motion
             in
             its
             own
             Epicycle
             .
             For
             these
             Mountains
             being
             (
             by
             reason
             of
             their
             height
             )
             more
             eminent
             then
             all
             the
             rest
             of
             the
             parts
             of
             the
             Earth
             ,
             do
             by
             both
             these
             Motions
             drive
             the
             Aire
             from
             the
             West
             Eastwards
             .
             To
             which
             though
             the
             diurnal
             Motion
             contribute
             but
             little
             ;
             yet
             seeing
             I
             have
             supposed
             that
             the
             simple
             Motion
             of
             the
             Earth
             in
             its
             own
             Epicycle
             ,
             makes
             two
             revolutions
             in
             the
             same
             time
             in
             which
             the
             diurnal
             Motion
             
             makes
             but
             one
             ;
             and
             that
             the
             Semidiameter
             of
             the
             Epicycle
             is
             double
             to
             the
             Semidiameter
             of
             the
             diurnal
             Conversion
             ,
             the
             Motion
             of
             every
             point
             of
             the
             Earth
             in
             its
             own
             Epicycle
             will
             have
             its
             velocity
             quadruple
             to
             that
             of
             the
             diurnal
             Motion
             ;
             so
             that
             by
             both
             these
             Motions
             together
             ,
             the
             tops
             of
             those
             Hils
             will
             sensibly
             be
             moved
             against
             the
             Aire
             ;
             and
             consequently
             a
             Wind
             will
             be
             felt
             .
             For
             whether
             the
             Air
             strike
             the
             Sentient
             ,
             or
             the
             Sentient
             the
             Air
             ,
             the
             perception
             of
             Motion
             will
             be
             the
             same
             .
             But
             this
             Wind
             ,
             seeing
             it
             is
             not
             caused
             by
             the
             ascent
             of
             Vapours
             ,
             must
             necessarily
             be
             very
             Constant.
             
          
           
             4
             When
             one
             Cloud
             is
             already
             ascended
             into
             the
             Aire
             ,
             if
             another
             Cloud
             ascend
             towards
             it
             ,
             that
             part
             of
             the
             Aire
             which
             is
             intercepted
             between
             them
             both
             ,
             must
             of
             necessity
             be
             pressed
             out
             every
             way
             .
             Also
             when
             both
             of
             them
             ,
             whilest
             the
             one
             ascends
             ,
             and
             the
             other
             either
             stayes
             ,
             or
             descends
             ,
             come
             to
             be
             joyned
             in
             such
             manner
             as
             that
             the
             aethereal
             substance
             be
             shut
             within
             them
             on
             every
             side
             ,
             it
             will
             by
             this
             compression
             also
             go
             out
             by
             penetrating
             the
             Water
             .
             But
             in
             the
             mean
             time
             ,
             the
             hard
             particles
             which
             are
             mingled
             with
             the
             Aire
             ,
             and
             are
             agitated
             (
             as
             I
             have
             supposed
             )
             with
             Simple
             Motion
             ,
             wil
             not
             pass
             through
             the
             water
             of
             the
             clouds
             ,
             but
             be
             more
             straightly
             compressed
             within
             their
             cavities
             .
             And
             this
             I
             have
             demonstrated
             at
             the
             4th
             and
             5th
             Articles
             of
             the
             22th
             Chapter
             .
             Besides
             ,
             seeing
             the
             Globe
             of
             the
             Earth
             floateth
             in
             the
             Aire
             which
             is
             agitated
             by
             the
             Sunnes
             Motion
             ,
             the
             parts
             of
             the
             Aire
             resisted
             by
             the
             Earth
             ,
             will
             spread
             themselves
             every
             way
             upon
             the
             Earths
             Superficies
             ;
             as
             I
             have
             shewn
             at
             the
             8th
             Article
             of
             the
             21th
             Chapter
             .
          
           
             5
             We
             perceive
             a
             Body
             to
             be
             Hard
             ,
             from
             this
             ,
             that
             when
             touching
             it
             we
             would
             thrust
             forwards
             that
             part
             of
             the
             same
             which
             we
             touch
             ,
             we
             cannot
             do
             it
             otherwise
             then
             by
             thrusting
             forwards
             the
             whole
             Body
             .
             We
             may
             indeed
             easily
             and
             sensibly
             thrust
             forwards
             any
             particle
             of
             the
             Aire
             or
             Water
             which
             we
             touch
             ,
             whilst
             yet
             the
             rest
             of
             its
             parts
             remain
             (
             to
             sense
             )
             unmoved
             .
             But
             we
             cannot
             do
             so
             to
             any
             part
             of
             a
             stone
             .
             Wherfore
             I
             define
             a
             Hard
             Body
             to
             be
             that
             ,
             whereof
             no
             part
             can
             be
             sensibly
             moved
             ,
             unless
             the
             whole
             be
             moved
             .
             Whatsoever
             therefore
             is
             Soft
             or
             Fluid
             ,
             the
             same
             can
             never
             be
             made
             Hard
             but
             by
             such
             motion
             ,
             as
             makes
             many
             of
             the
             parts
             together
             stop
             the
             motion
             of
             some
             one
             part
             ,
             by
             resisting
             the
             same
             .
          
           
             6
             These
             things
             premised
             ,
             I
             shall
             shew
             a
             possible
             cause
             why
             there
             
             is
             greater
             Cold
             neer
             the
             Poles
             of
             the
             Earth
             ,
             then
             further
             from
             them
             .
             The
             motion
             of
             the
             Sunne
             between
             the
             Tropicks
             ,
             driving
             the
             Aire
             towards
             that
             part
             of
             the
             Earths
             Superficies
             which
             is
             perpendicularly
             under
             it
             ,
             makes
             it
             spread
             it self
             every
             way
             ;
             and
             the
             velocity
             of
             this
             expansion
             of
             the
             Aire
             grows
             greater
             and
             greater
             ,
             as
             the
             Superficies
             of
             the
             Earth
             comes
             to
             be
             more
             and
             more
             straightned
             ,
             that
             is
             to
             say
             ,
             as
             the
             Circles
             which
             are
             parallel
             to
             the
             Aequator
             come
             to
             be
             less
             and
             less
             .
             Wherefore
             this
             expansive
             motion
             of
             the
             Aire
             ,
             drives
             before
             it
             the
             parts
             of
             the
             Aire
             which
             are
             in
             its
             way
             continually
             towards
             the
             Poles
             more
             and
             more
             strongly
             ,
             as
             its
             force
             comes
             to
             be
             more
             and
             more
             united
             ,
             that
             is
             to
             say
             ,
             as
             the
             Circles
             which
             are
             parallel
             to
             the
             Aequator
             are
             less
             and
             less
             ;
             that
             is
             ,
             so
             much
             the
             more
             ,
             by
             how
             much
             they
             are
             neerer
             to
             the
             Poles
             of
             the
             Earth
             .
             In
             those
             places
             therefore
             which
             are
             neerer
             to
             the
             Poles
             ,
             there
             is
             greater
             Cold
             ,
             then
             in
             those
             which
             are
             more
             remote
             from
             them
             .
             Now
             this
             expansion
             of
             the
             Aire
             upon
             the
             Superficies
             of
             the
             Earth
             from
             East
             to
             West
             ,
             doth
             by
             reason
             of
             the
             Sunnes
             perpetual
             accession
             to
             the
             places
             which
             are
             successively
             under
             it
             ,
             make
             it
             Cold
             at
             the
             time
             of
             the
             Sunnes
             Rising
             and
             Setting
             ;
             but
             as
             the
             Sunne
             comes
             to
             be
             continually
             more
             and
             more
             perpendicular
             to
             those
             cooled
             places
             ,
             so
             by
             the
             Heat
             which
             is
             generated
             by
             the
             supervening
             Simple
             Motion
             of
             the
             Sunn
             ,
             that
             Cold
             is
             again
             remitted
             ;
             and
             can
             never
             be
             great
             ,
             because
             the
             action
             by
             the
             which
             it
             was
             generated
             ,
             is
             not
             permanent
             .
             Wherefore
             I
             have
             rendred
             a
             possible
             cause
             of
             Cold
             in
             those
             places
             that
             are
             neer
             the
             Poles
             ,
             or
             where
             the
             obliquity
             of
             the
             Sunne
             is
             great
             .
          
           
             7
             How
             Water
             may
             be
             congeled
             by
             Cold
             ,
             may
             be
             explained
             in
             this
             manner
             .
             Let
             A
             (
             in
             the
             first
             figure
             )
             represent
             the
             Sunne
             ,
             and
             B
             the
             Earth
             .
             A
             will
             therefore
             be
             much
             greater
             then
             B.
             Let
             EF
             be
             in
             the
             plain
             of
             the
             Equinoctial
             ;
             to
             which
             let
             GH
             ,
             IK
             and
             LC
             be
             parallel
             .
             Lastly
             ,
             let
             C
             and
             D
             be
             the
             Poles
             of
             the
             Earth
             .
             The
             Aire
             therefore
             by
             its
             action
             in
             those
             parallels
             will
             rake
             the
             Superficies
             of
             the
             Earth
             ;
             and
             that
             with
             motion
             so
             much
             the
             stronger
             ,
             by
             how
             much
             the
             parallel
             Circles
             towards
             the
             Poles
             grow
             less
             and
             less
             .
             From
             whence
             must
             arise
             a
             Wind
             ,
             which
             will
             force
             together
             the
             uppermost
             parts
             of
             the
             water
             ,
             and
             withal
             raise
             them
             a
             little
             ,
             weakning
             their
             endeavour
             towards
             the
             Center
             of
             the
             Earth
             .
             And
             from
             
             their
             endeavour
             towards
             the
             center
             of
             the
             Earth
             ,
             joyned
             with
             the
             endeavour
             of
             the
             said
             Wind
             ,
             the
             uppermost
             parts
             of
             the
             water
             will
             be
             pressed
             together
             ,
             and
             coagulated
             ,
             that
             is
             to
             say
             ,
             the
             top
             of
             the
             water
             will
             be
             skinned
             over
             and
             hardned
             .
             And
             so
             againe
             the
             Water
             next
             the
             top
             will
             be
             hardned
             in
             the
             same
             manner
             ,
             till
             at
             length
             the
             Ice
             be
             thick
             .
             And
             this
             Ice
             being
             now
             compacted
             of
             little
             hard
             Bodies
             ,
             must
             also
             containe
             many
             particles
             of
             ayre
             received
             into
             it
             .
          
           
             As
             Rivers
             and
             Seas
             ,
             so
             also
             in
             the
             same
             manner
             may
             the
             Clouds
             be
             frozen
             .
             For
             when
             by
             the
             ascending
             and
             descending
             of
             severall
             Clouds
             at
             the
             same
             time
             ,
             the
             Air
             intercepted
             between
             them
             is
             by
             compression
             forced
             out
             ,
             it
             rakes
             ,
             &
             by
             little
             &
             little
             hardens
             them
             .
             And
             though
             those
             smal
             drops
             (
             which
             usually
             make
             Clouds
             )
             be
             not
             yet
             united
             into
             greater
             Bodies
             ,
             yet
             the
             same
             Wind
             will
             be
             made
             ;
             &
             by
             it
             ,
             as
             water
             is
             congeled
             into
             Ice
             ,
             so
             will
             Vapours
             in
             the
             same
             manner
             be
             congeled
             into
             Snow
             .
             From
             the
             same
             cause
             it
             is
             that
             Ice
             may
             be
             made
             by
             art
             ,
             and
             that
             not
             farre
             from
             the
             fire
             .
             For
             it
             is
             done
             by
             the
             mingling
             of
             Snow
             and
             Salt
             together
             ,
             and
             by
             burying
             in
             it
             a
             small
             vessell
             full
             of
             Water
             .
             Now
             while
             the
             Snow
             and
             Salt
             (
             which
             have
             in
             them
             a
             great
             deale
             of
             aire
             )
             are
             melting
             ,
             the
             aire
             which
             is
             pressed
             out
             every
             way
             in
             Wind
             ,
             rakes
             the
             sides
             of
             the
             Vessel
             ;
             and
             as
             the
             Wind
             by
             its
             motion
             rakes
             the
             Vessell
             ,
             so
             the
             Vessell
             by
             the
             same
             motion
             and
             action
             congeles
             the
             Water
             within
             it
             .
          
           
             We
             find
             by
             experience
             ,
             that
             Cold
             is
             allwayes
             more
             Remisse
             in
             places
             where
             it
             raynes
             ,
             or
             where
             the
             weather
             is
             cloudy
             (
             things
             being
             alike
             in
             all
             other
             respects
             )
             then
             where
             the
             aire
             is
             cleare
             .
             And
             this
             agreeth
             very
             well
             with
             what
             I
             have
             sayd
             before
             .
             For
             in
             cleare
             weather
             ,
             the
             course
             of
             the
             Wind
             which
             (
             as
             I
             sayd
             even
             now
             )
             rakes
             the
             Superficies
             of
             the
             Earth
             ,
             as
             it
             is
             free
             from
             all
             interruption
             ,
             so
             also
             it
             is
             very
             strong
             .
             But
             when
             small
             drops
             of
             water
             are
             either
             rising
             or
             falling
             ,
             that
             Wind
             is
             repelled
             ,
             broken
             and
             dissipated
             by
             them
             ;
             and
             the
             lesse
             the
             Wind
             is
             ,
             the
             lesse
             is
             the
             Cold.
             
          
           
             We
             find
             also
             by
             experience
             ,
             that
             in
             deep
             Wells
             the
             Water
             freezeth
             not
             so
             much
             ,
             as
             it
             doth
             upon
             the
             Superficies
             of
             the
             Earth
             .
             For
             the
             Wind
             by
             which
             Ice
             is
             made
             ,
             entring
             into
             the
             Earth
             (
             by
             reason
             of
             the
             laxity
             of
             its
             parts
             )
             more
             or
             lesse
             ,
             loseth
             some
             of
             its
             
             force
             ,
             though
             not
             much
             .
             So
             that
             if
             the
             Well
             be
             not
             deep
             ,
             it
             will
             freeze
             ;
             whereas
             if
             it
             be
             so
             deep
             ,
             as
             that
             the
             Wind
             which
             causeth
             cold
             cannot
             reach
             it
             ,
             it
             will
             not
             freeze
             .
          
           
             We
             find
             moreover
             by
             experience
             ,
             that
             Ice
             is
             lighter
             then
             Water
             .
             The
             cause
             whereof
             is
             manifest
             from
             that
             which
             I
             have
             already
             shewn
             ,
             namely
             ,
             that
             Aire
             is
             received
             in
             and
             mingled
             with
             the
             particles
             of
             the
             Water
             whilest
             it
             is
             in
             congeling
             .
          
           
             8
             We
             have
             seen
             one
             way
             of
             making
             things
             Hard
             ,
             namely
             ,
             by
             Congelation
             .
             Another
             way
             is
             thus
             .
             Having
             already
             supposed
             ,
             that
             innumerable
             Atomes
             ,
             some
             harder
             then
             others
             ,
             and
             that
             have
             several
             simple
             motions
             of
             their
             own
             ,
             are
             intermingled
             with
             the
             aethereal
             substance
             ;
             it
             follows
             necessarily
             from
             hence
             ,
             that
             by
             reason
             of
             the
             fermentation
             of
             the
             whole
             Aire
             (
             of
             which
             I
             have
             spoken
             in
             the
             21
             Chap.
             )
             some
             of
             those
             Atomes
             meeting
             with
             others
             ,
             will
             cleave
             together
             ,
             by
             applying
             themselvs
             to
             one
             another
             in
             such
             manner
             as
             is
             agreeable
             to
             their
             motions
             and
             mutual
             contacts
             ;
             and
             (
             seeing
             there
             is
             no
             Vacuum
             )
             cannot
             be
             pulled
             asunder
             ,
             but
             by
             so
             much
             force
             as
             is
             sufficient
             to
             overcome
             their
             Hardness
             .
          
           
             Now
             there
             are
             innumerable
             degrees
             of
             Hardness
             .
             As
             (
             for
             example
             )
             there
             is
             a
             degree
             of
             it
             in
             Water
             ,
             as
             is
             manifest
             from
             this
             ,
             that
             upon
             a
             plain
             it
             may
             be
             drawn
             any
             way
             at
             pleasure
             by
             ones
             finger
             .
             There
             is
             a
             greater
             degree
             of
             it
             in
             clammy
             liquors
             ;
             which
             when
             they
             are
             poured
             out
             ,
             doe
             in
             falling
             downwards
             dispose
             themselves
             into
             one
             continued
             thred
             ;
             which
             thred
             before
             it
             be
             broken
             will
             by
             little
             and
             little
             diminish
             its
             thickness
             ,
             till
             at
             last
             it
             be
             so
             small
             ,
             as
             that
             it
             seems
             to
             break
             onely
             in
             a
             point
             ;
             and
             in
             their
             separation
             the
             external
             parts
             break
             first
             from
             one
             another
             ,
             and
             then
             the
             more
             internal
             parts
             successively
             one
             after
             another
             .
             In
             Wax
             there
             is
             yet
             a
             greater
             degree
             of
             Hardness
             .
             For
             when
             we
             would
             pull
             one
             part
             of
             it
             from
             another
             ,
             we
             first
             make
             the
             whole
             mass
             slenderer
             ,
             before
             we
             can
             pull
             it
             asunder
             .
             And
             how
             much
             the
             harder
             anything
             is
             which
             we
             would
             break
             ,
             so
             much
             the
             more
             force
             we
             must
             apply
             to
             it
             .
             Wherefore
             ,
             if
             we
             go
             on
             to
             harder
             things
             ,
             as
             Ropes
             ,
             Wood
             ,
             Metals
             ,
             Stones
             ,
             &c.
             reason
             prompteth
             us
             to
             believe
             that
             the
             same
             (
             though
             not
             alwayes
             sensibly
             )
             will
             necessarily
             happen
             ;
             and
             that
             even
             the
             hardest
             things
             are
             broken
             asunder
             in
             the
             same
             manner
             ,
             namely
             ,
             by
             Solution
             of
             their
             continuity
             ,
             begun
             in
             the
             outermost
             Superficies
             ,
             
             and
             proceeding
             successively
             to
             the
             innermost
             parts
             .
             In
             like
             manner
             ,
             when
             the
             parts
             of
             Bodies
             are
             to
             be
             separated
             ,
             not
             by
             pulling
             them
             asunder
             ,
             but
             by
             breaking
             them
             ,
             the
             first
             separation
             will
             necessarily
             be
             in
             the
             convex
             Superficies
             of
             the
             bowed
             part
             of
             the
             Body
             ,
             and
             afterwards
             in
             the
             concave
             Superficies
             .
             For
             in
             all
             bowing
             ,
             there
             is
             in
             the
             convex
             Superficies
             an
             endeavour
             in
             the
             parts
             to
             go
             one
             from
             another
             ,
             and
             in
             the
             concave
             Superficies
             to
             penetrate
             one
             another
             .
          
           
             This
             being
             well
             understood
             ,
             a
             reason
             may
             be
             given
             ,
             how
             two
             Bodies
             which
             are
             contiguous
             in
             one
             common
             Superficies
             ,
             may
             by
             force
             be
             separated
             without
             the
             introduction
             of
             Vacuum
             ;
             though
             Lucretius
             thought
             otherwise
             ,
             believing
             that
             such
             separation
             was
             a
             strong
             establishment
             of
             Vacuum
             .
             For
             a
             Marble
             Pillar
             being
             made
             to
             hang
             by
             one
             of
             its
             bases
             ,
             if
             it
             be
             long
             enough
             it
             will
             by
             its
             own
             weight
             be
             broken
             asunder
             ,
             and
             yet
             it
             will
             not
             necessarily
             follow
             that
             there
             should
             be
             Vacuum
             ,
             seeing
             the
             solution
             of
             its
             continuity
             may
             begin
             in
             the
             Circumference
             ,
             and
             proceed
             successively
             to
             the
             midst
             thereof
             .
          
           
             Lastly
             ,
             Wine
             is
             not
             so
             easily
             congeled
             as
             Water
             ,
             because
             in
             Wine
             there
             are
             particles
             which
             being
             not
             fluid
             ,
             are
             moved
             very
             swiftly
             ,
             and
             by
             their
             motion
             congelation
             is
             retarded
             ;
             but
             if
             the
             Cold
             prevail
             against
             this
             motion
             ,
             then
             the
             outermost
             parts
             of
             the
             Wine
             will
             be
             first
             frozen
             ,
             and
             afterwards
             the
             inner
             parts
             ;
             whereof
             this
             is
             a
             signe
             ,
             that
             the
             Wine
             which
             remains
             unfrozen
             in
             the
             midst
             will
             be
             very
             strong
             .
          
           
             9
             Another
             cause
             of
             Hardness
             in
             some
             things
             may
             be
             in
             this
             manner
             .
             If
             a
             soft
             Body
             consist
             of
             many
             hard
             particles
             ,
             which
             by
             the
             intermixture
             of
             many
             other
             fluid
             particles
             cohaere
             but
             loosely
             together
             ,
             those
             fluid
             parts
             (
             as
             hath
             been
             shewn
             in
             the
             last
             Article
             of
             the
             21
             Chapter
             )
             will
             be
             exhaled
             ;
             by
             which
             means
             each
             hard
             particle
             will
             apply
             it self
             to
             the
             next
             to
             it
             according
             to
             a
             greater
             Superficies
             ;
             and
             consequently
             they
             will
             cohaere
             more
             closely
             to
             one
             another
             ;
             that
             is
             to
             say
             ,
             the
             whole
             mass
             will
             be
             made
             Harder
             .
          
           
             10
             Again
             ,
             in
             some
             things
             Hardness
             may
             be
             made
             to
             a
             certain
             degree
             ,
             in
             this
             manner
             .
             When
             any
             fluid
             substance
             hath
             in
             it
             certain
             very
             small
             Bodies
             intermingled
             ,
             which
             being
             moved
             with
             simple
             motion
             of
             their
             own
             ,
             contribute
             like
             motion
             to
             the
             parts
             of
             the
             
             fluid
             substance
             ,
             and
             this
             be
             done
             in
             a
             small
             enclosed
             space
             (
             as
             in
             the
             hollow
             of
             a
             little
             Sphere
             ,
             or
             a
             very
             slender
             Pipe
             )
             if
             the
             motion
             be
             vehement
             ,
             and
             there
             be
             a
             great
             number
             of
             these
             small
             enclosed
             Bodies
             ,
             two
             things
             will
             happen
             ;
             the
             one
             ,
             that
             the
             fluid
             substance
             will
             have
             an
             endeavour
             of
             dilating
             it self
             at
             once
             every
             way
             ;
             the
             other
             ,
             that
             if
             those
             smal
             Bodies
             can
             no
             where
             get
             out
             ,
             then
             from
             their
             reflexion
             it
             will
             follow
             ,
             that
             the
             motion
             of
             the
             parts
             of
             the
             enclosed
             fluid
             substance
             ,
             which
             was
             vehement
             before
             ,
             will
             now
             be
             much
             more
             vehement
             .
             Wherefore
             if
             any
             one
             particle
             of
             that
             fluid
             substance
             should
             be
             touched
             &
             pressed
             by
             some
             external
             Movent
             ,
             it
             could
             not
             yeild
             but
             by
             the
             application
             of
             very
             sensible
             force
             .
             Wherefore
             the
             fluid
             substance
             which
             is
             enclosed
             ,
             and
             so
             moved
             ,
             hath
             some
             degree
             of
             Hardness
             .
             Now
             greater
             and
             less
             degree
             of
             Hardness
             depends
             upon
             the
             quantity
             and
             velocity
             of
             those
             small
             Bodies
             and
             upon
             the
             narrowness
             of
             the
             place
             both
             together
             .
          
           
             11
             Such
             things
             as
             are
             made
             Hard
             by
             sudden
             heat
             ,
             namely
             ,
             such
             as
             are
             hardned
             by
             fire
             ,
             are
             commonly
             reduced
             to
             their
             former
             soft
             form
             by
             Maceration
             .
             For
             fire
             hardens
             by
             Evaporation
             ,
             and
             therefore
             if
             the
             evaporated
             moisture
             be
             restored
             again
             ,
             the
             former
             nature
             and
             form
             is
             restored
             together
             with
             it
             .
             And
             such
             things
             as
             are
             frozen
             with
             Cold
             ,
             if
             the
             Wind
             by
             which
             they
             were
             frozen
             change
             into
             the
             opposite
             quarter
             ,
             they
             will
             be
             unfrozen
             again
             (
             unless
             they
             have
             gotten
             a
             habit
             of
             new
             motion
             or
             endeavour
             by
             long
             continuance
             in
             that
             hardness
             )
             .
             Nor
             is
             it
             enough
             to
             cause
             thawing
             ,
             that
             there
             be
             a
             cessation
             of
             the
             freezing
             Wind
             (
             for
             the
             taking
             away
             of
             the
             Cause
             doth
             not
             destroy
             a
             produced
             effect
             )
             ;
             but
             the
             thawing
             also
             must
             have
             its
             proper
             cause
             ,
             namely
             ,
             a
             contrary
             Wind
             ,
             or
             at
             least
             a
             Wind
             opposite
             in
             some
             degree
             .
             And
             this
             we
             finde
             to
             be
             true
             by
             experience
             .
             For
             if
             Ice
             be
             laid
             in
             a
             place
             so
             well
             enclosed
             that
             the
             motion
             of
             the
             Aire
             cannot
             get
             to
             it
             ,
             that
             Ice
             will
             remain
             unchanged
             ,
             though
             the
             place
             be
             not
             sensibly
             cold
             .
          
           
             12
             Of
             Hard
             Bodies
             ,
             some
             may
             manifestly
             be
             bowed
             ;
             others
             not
             ,
             but
             are
             broken
             in
             the
             very
             first
             moment
             of
             their
             bending
             .
             And
             of
             such
             Bodies
             as
             may
             manifestly
             be
             bended
             ,
             some
             being
             bent
             ,
             do
             as
             soon
             as
             ever
             they
             are
             set
             at
             liberty
             ,
             Restore
             themselves
             to
             their
             former
             posture
             ,
             others
             remain
             still
             bent
             .
             Now
             if
             the
             cause
             of
             this
             Restitution
             be
             asked
             ,
             I
             say
             it
             may
             be
             in
             this
             manner
             ;
             namely
             ,
             that
             the
             particles
             of
             the
             bended
             Body
             ,
             whilest
             it
             is
             held
             bent
             ,
             do
             nevertheless
             
             retain
             their
             motion
             ;
             and
             by
             this
             motion
             they
             restore
             it
             as
             soon
             as
             the
             force
             is
             removed
             by
             which
             it
             was
             bent
             .
             For
             when
             any
             thing
             is
             bent
             (
             as
             a
             plate
             of
             steel
             )
             ,
             and
             as
             soon
             as
             the
             force
             is
             removed
             restores
             it self
             again
             ,
             it
             is
             evident
             that
             the
             cause
             of
             its
             restitution
             cannot
             be
             referred
             to
             the
             ambient
             aire
             ;
             nor
             can
             it
             be
             referred
             to
             the
             removal
             of
             the
             force
             by
             which
             it
             was
             bent
             ;
             for
             in
             things
             that
             are
             at
             rest
             ,
             the
             taking
             away
             of
             impediments
             ,
             is
             not
             a
             sufficient
             cause
             of
             their
             future
             Motion
             ;
             there
             being
             no
             other
             cause
             of
             Motion
             ,
             but
             Motion
             .
             The
             cause
             therefore
             of
             such
             Restitution
             is
             in
             the
             parts
             of
             the
             Steel
             it self
             .
             Wherefore
             whilest
             it
             remains
             bent
             ,
             there
             is
             in
             the
             parts
             of
             which
             it
             consisteth
             ,
             some
             motion
             ,
             though
             invisible
             ,
             that
             is
             to
             say
             ,
             some
             endeavour
             at
             least
             that
             way
             by
             which
             the
             restitution
             is
             to
             be
             made
             ;
             and
             therefore
             this
             endeavour
             of
             all
             the
             parts
             together
             is
             the
             first
             beginning
             of
             Restitution
             ;
             so
             that
             the
             impediment
             being
             removed
             ,
             that
             is
             to
             say
             ,
             the
             force
             by
             which
             it
             was
             held
             bent
             ,
             it
             will
             be
             restored
             again
             .
             Now
             the
             motion
             of
             the
             parts
             by
             which
             this
             is
             done
             ,
             is
             that
             which
             I
             called
             Simple
             Motion
             ,
             or
             Motion
             returning
             into
             it self
             .
             When
             therefore
             in
             the
             bending
             of
             a
             plate
             ,
             the
             ends
             are
             drawn
             together
             ,
             there
             is
             on
             one
             side
             a
             mutual
             compression
             of
             the
             parts
             ;
             which
             compression
             is
             one
             endeavour
             opposite
             to
             another
             endeavour
             ;
             and
             on
             the
             other
             side
             a
             divulsion
             of
             the
             parts
             .
             The
             endeavour
             therefore
             of
             the
             parts
             on
             one
             side
             tends
             to
             the
             restitution
             of
             the
             plate
             from
             the
             middle
             towards
             the
             ends
             ;
             and
             on
             the
             other
             side
             ,
             from
             the
             ends
             towards
             the
             middle
             .
             Wherefore
             the
             impediment
             being
             taken
             away
             ,
             this
             endeavour
             (
             which
             is
             the
             beginning
             of
             restitution
             )
             will
             restore
             the
             plate
             to
             its
             former
             posture
             .
             And
             thus
             I
             have
             given
             a
             possible
             cause
             why
             some
             Bodies
             when
             they
             are
             bent
             Restore
             themselves
             again
             ;
             which
             was
             to
             be
             done
             .
          
           
             As
             for
             Stones
             ,
             seeing
             they
             are
             made
             by
             the
             accretion
             of
             many
             very
             hard
             particles
             within
             the
             Earth
             ;
             which
             particles
             have
             no
             great
             coherence
             ,
             that
             is
             to
             say
             ,
             touch
             one
             another
             in
             small
             latitude
             ,
             and
             consequently
             admit
             many
             particles
             of
             aire
             ,
             it
             must
             needs
             be
             that
             in
             bending
             of
             them
             ,
             their
             internal
             parts
             will
             not
             easily
             be
             compressed
             by
             reason
             of
             their
             hardness
             .
             And
             because
             their
             coherence
             is
             not
             firm
             ,
             as
             soon
             as
             the
             external
             hard
             particles
             are
             disjoyned
             ,
             the
             aethereal
             parts
             will
             necessarily
             break
             out
             ,
             and
             so
             the
             Body
             will
             suddenly
             be
             broken
             .
          
           
           
             13
             Those
             Bodies
             are
             called
             Diaphanous
             ,
             upon
             which
             whilest
             the
             Beams
             of
             a
             lucid
             Body
             do
             work
             ,
             the
             action
             of
             every
             one
             of
             those
             Beams
             is
             propagated
             in
             them
             in
             such
             manner
             as
             that
             they
             still
             retain
             the
             same
             order
             amongst
             themselves
             ,
             or
             the
             inversion
             of
             that
             order
             ;
             and
             therefore
             Bodies
             which
             are
             perfectly
             Diaphanous
             ,
             are
             also
             perfectly
             homogeneous
             .
             On
             the
             contrary
             ,
             an
             Opacous
             Body
             is
             that
             ,
             which
             by
             reason
             of
             its
             heterogeneous
             nature
             ,
             doth
             by
             innumerable
             reflexions
             and
             refractions
             in
             particles
             of
             different
             figures
             and
             unequal
             hardness
             ,
             weaken
             the
             Beams
             that
             fall
             upon
             it
             before
             they
             reach
             the
             Eie
             .
             And
             of
             Diaphanous
             Bodies
             ,
             some
             are
             made
             such
             by
             Nature
             from
             the
             beginning
             ;
             as
             the
             substance
             of
             the
             Aire
             ,
             and
             of
             the
             Water
             ,
             and
             perhaps
             also
             some
             parts
             of
             Stones
             ,
             unless
             these
             also
             be
             Water
             that
             has
             been
             long
             congeled
             .
             Others
             are
             made
             so
             by
             the
             power
             of
             Heat
             ,
             which
             congregates
             homogeneons
             Bodies
             .
             But
             such
             as
             are
             made
             Diaphanous
             in
             this
             manner
             ,
             consist
             of
             parts
             which
             were
             formerly
             Diaphanous
             .
          
           
             14
             In
             what
             manner
             Clouds
             are
             made
             by
             the
             motion
             of
             the
             Sunne
             ,
             elevating
             the
             particles
             of
             Water
             from
             the
             Sea
             and
             other
             moist
             places
             ,
             hath
             been
             explained
             in
             the
             26th
             Chapter
             .
             Also
             how
             Clouds
             come
             to
             be
             frozen
             ,
             hath
             been
             shewn
             above
             at
             the
             7th
             Article
             .
             Now
             from
             this
             ,
             that
             Aire
             may
             be
             enclosed
             ,
             as
             it
             were
             in
             Caverns
             ,
             and
             pent
             together
             more
             and
             more
             by
             the
             meeting
             of
             ascending
             and
             descending
             Clouds
             ,
             may
             be
             deduced
             a
             possible
             Cause
             of
             Thunder
             and
             Lightening
             .
             For
             seeing
             the
             Aire
             consists
             of
             two
             parts
             ,
             the
             one
             Aethereal
             ,
             which
             has
             no
             proper
             motion
             of
             its
             own
             ,
             as
             being
             a
             thing
             divisible
             into
             the
             least
             parts
             ;
             the
             other
             Hard
             ,
             namely
             ,
             consisting
             of
             many
             hard
             Atomes
             which
             have
             every
             one
             of
             them
             a
             very
             swift
             simple
             motion
             of
             its
             own
             ;
             whilest
             the
             Clouds
             by
             their
             meeting
             do
             more
             and
             more
             straighten
             such
             Cavities
             as
             they
             intercept
             ,
             the
             Aethereal
             parts
             will
             penetrate
             and
             pass
             through
             their
             watry
             substance
             ;
             but
             the
             hard
             parts
             will
             in
             the
             mean
             time
             be
             the
             more
             thrust
             together
             ,
             and
             press
             one
             another
             ;
             and
             consequently
             (
             by
             reason
             of
             their
             vehement
             motions
             )
             they
             will
             have
             an
             endeavour
             to
             rebound
             from
             each
             other
             .
             Whensoever
             therefore
             the
             compression
             is
             great
             enough
             ,
             and
             the
             concave
             parts
             of
             the
             Clouds
             are
             (
             for
             the
             cause
             I
             have
             already
             given
             )
             congeled
             into
             Ice
             ,
             the
             Cloud
             wil
             necessarily
             be
             broken
             ;
             &
             this
             breaking
             of
             the
             Cloud
             produceth
             
             the
             first
             clap
             of
             Thunder
             .
             Afterwards
             ,
             the
             Aire
             which
             was
             pent
             in
             ,
             having
             now
             broken
             through
             ,
             makes
             a
             concussion
             of
             the
             Aire
             without
             ;
             and
             from
             hence
             proceeds
             the
             roaring
             and
             murmur
             which
             follows
             ;
             and
             both
             the
             first
             Clap
             and
             the
             Murmur
             that
             follows
             it
             ,
             make
             that
             noise
             which
             is
             called
             Thunder
             .
             Also
             from
             the
             same
             Aire
             breaking
             through
             the
             Clouds
             ,
             and
             with
             concussion
             falling
             upon
             the
             Eie
             ,
             proceeds
             that
             action
             upon
             our
             Eie
             ,
             which
             causeth
             in
             us
             a
             perception
             of
             that
             Light
             which
             we
             call
             Lightening
             .
             Wherefore
             I
             have
             given
             a
             possible
             cause
             of
             Thunder
             and
             Lightening
             .
          
           
             15
             But
             if
             the
             Vapours
             which
             are
             raised
             into
             Clouds
             ,
             do
             run
             together
             again
             into
             Water
             ,
             or
             be
             congeled
             into
             Ice
             ,
             from
             whence
             is
             it
             (
             seeing
             both
             Ice
             and
             Water
             are
             heavy
             )
             that
             they
             are
             sustained
             in
             the
             Aire
             ?
             Or
             rather
             ,
             what
             may
             the
             cause
             be
             ,
             that
             being
             once
             elevated
             ,
             they
             fall
             down
             again
             ?
             For
             there
             is
             no
             doubt
             but
             the
             same
             force
             which
             could
             carry
             up
             that
             Water
             ,
             could
             also
             sustain
             it
             there
             .
             Why
             therefore
             being
             once
             carried
             up
             ,
             doth
             it
             fall
             again
             ?
             I
             say
             it
             proceeds
             from
             the
             same
             Simple
             Motion
             of
             the
             Sunne
             ,
             both
             that
             Vapours
             are
             forced
             to
             ascend
             ,
             and
             that
             Water
             gathered
             into
             Clouds
             is
             forced
             to
             descend
             .
             For
             in
             the
             21th
             Chapter
             and
             11th
             Article
             I
             have
             shewn
             how
             Vapours
             are
             elevated
             ;
             and
             in
             the
             same
             Chapter
             and
             5th
             Article
             I
             have
             also
             shewn
             how
             by
             the
             same
             motion
             Homogeneous
             Bodies
             are
             congregated
             ,
             &
             Heterogeneous
             dissipated
             ;
             that
             is
             to
             say
             ,
             how
             such
             things
             as
             have
             a
             like
             nature
             to
             that
             of
             the
             Earth
             ,
             are
             driven
             towards
             the
             Earth
             ;
             that
             is
             to
             say
             ,
             what
             is
             the
             cause
             of
             the
             descent
             of
             Heavy
             Bodies
             .
             Now
             if
             the
             action
             of
             the
             Sun
             be
             hindered
             in
             the
             raising
             of
             vapours
             ,
             and
             be
             not
             at
             all
             hindered
             in
             the
             casting
             of
             them
             down
             ,
             the
             Water
             will
             descend
             .
             But
             a
             Cloud
             cannot
             hinder
             the
             action
             of
             the
             Sunne
             in
             making
             things
             of
             an
             earthly
             nature
             descend
             to
             the
             Earth
             ,
             though
             it
             may
             hinder
             it
             in
             making
             Vapours
             ascend
             For
             the
             lower
             part
             of
             a
             thick
             Cloud
             is
             so
             covered
             by
             its
             upper
             part
             ,
             as
             that
             it
             cannot
             receive
             that
             action
             of
             the
             Sunne
             by
             which
             Vapours
             are
             carried
             up
             ,
             because
             Vapours
             are
             raised
             by
             the
             perpetual
             fermentation
             of
             the
             Aire
             ,
             or
             by
             the
             separating
             of
             its
             smallest
             parts
             from
             one
             another
             ,
             which
             is
             much
             weaker
             when
             a
             thick
             Cloud
             is
             interposed
             ,
             then
             when
             the
             Skie
             is
             cleere
             .
             And
             therefore
             whensoever
             a
             Cloud
             is
             made
             thick
             enough
             ,
             the
             water
             which
             
             would
             not
             descend
             before
             ,
             will
             then
             descend
             ,
             unless
             it
             be
             kept
             up
             by
             the
             agitation
             of
             the
             Winde
             .
             Wherefore
             I
             have
             rendred
             a
             possible
             cause
             ,
             both
             why
             the
             Clouds
             may
             be
             sustained
             in
             the
             Aire
             ,
             and
             also
             why
             they
             may
             fall
             down
             again
             to
             the
             Earth
             ;
             which
             was
             propounded
             to
             be
             done
             .
          
           
             16
             Granting
             that
             the
             Clouds
             may
             be
             frozen
             ,
             it
             is
             no
             wonder
             if
             the
             Moon
             have
             been
             seen
             eclipsed
             at
             such
             time
             as
             she
             hath
             been
             almost
             two
             degrees
             above
             the
             Horizon
             ,
             the
             Sunne
             at
             the
             same
             time
             appearing
             in
             the
             Horizon
             ;
             for
             such
             an
             Eclipse
             was
             observed
             by
             Mestline
             at
             Tubing
             in
             the
             year
             1590.
             
             For
             it
             might
             happen
             that
             a
             frozen
             Cloud
             was
             then
             interposed
             between
             the
             Sunne
             and
             the
             Eie
             of
             the
             Observer
             .
             And
             if
             it
             were
             so
             ,
             the
             Sunne
             which
             was
             then
             almost
             two
             Degrees
             below
             the
             Horizon
             ,
             might
             appear
             to
             be
             in
             it
             ,
             by
             reason
             of
             the
             passing
             of
             his
             Beams
             through
             the
             Ice
             .
             And
             it
             is
             to
             be
             noted
             ,
             that
             those
             that
             attribute
             such
             refractions
             to
             the
             Atmosphere
             ,
             cannot
             attribute
             to
             it
             so
             great
             a
             refraction
             as
             this
             .
             Wherefore
             not
             the
             Atmosphere
             ,
             but
             either
             Water
             in
             a
             continued
             Body
             ,
             or
             else
             Ice
             must
             be
             the
             cause
             of
             that
             refraction
             .
          
           
             17
             Again
             ,
             granting
             that
             there
             may
             be
             Ice
             in
             the
             Clouds
             ,
             it
             will
             be
             no
             longer
             a
             wonder
             that
             many
             Sunnes
             have
             sometimes
             appeared
             at
             once
             .
             For
             Looking-glasses
             may
             be
             so
             placed
             ,
             as
             by
             reflections
             to
             shew
             the
             same
             object
             in
             many
             places
             .
             And
             may
             not
             so
             many
             frozen
             Clouds
             serve
             for
             so
             many
             Looking-glasses
             ?
             and
             may
             they
             not
             be
             fitly
             disposed
             for
             that
             purpose
             ?
             Besides
             ,
             the
             number
             of
             Appearances
             may
             be
             encreased
             by
             refractions
             also
             ;
             and
             therefore
             it
             would
             be
             a
             greater
             wonder
             to
             me
             ,
             if
             such
             Phaenomena
             as
             these
             should
             never
             happen
             .
          
           
             And
             were
             it
             not
             for
             that
             one
             Phaenomenon
             of
             the
             new
             Starre
             which
             was
             seen
             in
             Cassiopaea
             ,
             I
             should
             think
             Comets
             were
             made
             in
             the
             same
             manner
             ,
             namely
             ,
             by
             Vapours
             drawn
             not
             onely
             from
             the
             Earth
             ,
             but
             from
             the
             rest
             of
             the
             Planets
             also
             ,
             and
             congeled
             into
             one
             continued
             Body
             .
             For
             I
             could
             very
             well
             from
             hence
             give
             a
             reason
             both
             of
             their
             Haire
             ,
             and
             of
             their
             motions
             .
             But
             seeing
             that
             Starre
             remained
             sixteen
             whole
             moneths
             in
             the
             same
             place
             amongst
             the
             fixed
             Starres
             ,
             I
             cannot
             believe
             the
             matter
             of
             it
             was
             Ice
             .
             Wherefore
             I
             leave
             to
             others
             ,
             the
             disquisition
             of
             the
             cause
             of
             Comets
             ;
             concerning
             which
             ,
             nothing
             that
             hath
             hitherto
             been
             published
             ,
             (
             〈…〉
             
             the
             bare
             Histories
             of
             them
             )
             is
             worth
             considering
             .
          
           
             18
             The
             Heads
             of
             Rivers
             may
             be
             deduced
             from
             Rain-water
             ,
             or
             from
             melted
             Snowes
             ,
             very
             easily
             ;
             but
             from
             other
             causes
             ,
             very
             hardly
             ,
             or
             not
             at
             all
             .
             For
             both
             Rain-water
             ,
             and
             melted
             Snowes
             run
             down
             the
             descents
             of
             Mountains
             ;
             and
             if
             they
             descend
             onely
             by
             the
             outward
             Superficies
             ,
             the
             Showres
             or
             Snowes
             themselves
             may
             be
             accounted
             the
             Springs
             or
             Fountains
             ;
             but
             if
             they
             enter
             the
             Earth
             &
             descend
             within
             it
             ,
             then
             wheresoever
             they
             break
             out
             ,
             there
             are
             their
             Springs
             .
             And
             as
             these
             Spings
             make
             small
             streams
             ,
             so
             ,
             many
             small
             streams
             running
             together
             make
             Rivers
             .
             Now
             there
             was
             never
             any
             Spring
             foūd
             ,
             but
             where
             the
             Water
             w
             ch
             flowed
             to
             it
             ,
             was
             either
             further
             ,
             or
             at
             least
             as
             farre
             from
             the
             center
             of
             the
             Earth
             ,
             as
             the
             Spring
             it self
             .
             And
             whereas
             it
             has
             bin
             objected
             by
             a
             great
             Philosopher
             ,
             that
             in
             the
             top
             of
             Mount-Cenis
             (
             which
             parts
             Savoy
             from
             Piemont
             )
             there
             Springs
             a
             River
             which
             runs
             down
             by
             Susa
             ;
             it
             is
             not
             true
             .
             For
             there
             are
             above
             that
             River
             ,
             for
             two
             miles
             length
             ,
             very
             high
             hils
             on
             both
             sides
             ,
             which
             are
             almost
             perpetually
             covered
             with
             Snow
             ;
             from
             which
             ,
             innumerable
             little
             streams
             running
             down
             do
             manifestly
             supply
             that
             River
             with
             water
             sufficient
             for
             its
             magnitude
             .
          
        
         
           
           
           
           
             
               CHAP.
               XXIX
            
             .
             Of
             
               Sound
               ,
               Odour
               ,
               Savour
            
             ,
             and
             Touch
          
           
             
               
                 1
                 The
                 definition
                 of
                 Sound
                 ,
                 and
                 the
                 distinctions
                 of
                 Sounds
                 .
              
               
                 2
                 The
                 cause
                 of
                 the
                 degrees
                 of
                 Sounds
                 .
              
               
                 3
                 The
                 difference
                 between
                 Sounds
                 Acute
                 and
                 Grave
                 .
              
               
                 4
                 The
                 difference
                 between
                 Clear
                 and
                 Hoarse
                 Sounds
                 ,
                 whence
                 .
              
               
                 5
                 The
                 Sound
                 of
                 Thunder
                 and
                 of
                 a
                 Gunne
                 ,
                 whence
                 it
                 proceeds
                 .
              
               
                 6
                 Whence
                 it
                 is
                 ,
                 that
                 Pipes
                 by
                 blowing
                 into
                 them
                 have
                 a
                 clear
                 Sound
                 .
              
               
                 7
                 Of
                 Reflected
                 Sound
                 .
              
               
                 8
                 From
                 whence
                 it
                 is
                 that
                 Sound
                 is
                 Uniform
                 and
                 Lasting
                 .
              
               
                 9
                 How
                 Sound
                 may
                 be
                 helped
                 aud
                 hindered
                 by
                 the
                 Wind.
                 
              
               
                 10
                 Not
                 onely
                 Aire
                 ,
                 but
                 other
                 Bodies
                 how
                 hard
                 soever
                 they
                 be
                 ,
                 conveigh
                 Sound
                 .
              
               
                 11
                 The
                 causes
                 of
                 Grave
                 and
                 Acute
                 Sounds
                 ,
                 and
                 of
                 Concent
                 .
              
               
                 12
                 Phaenomena
                 for
                 Smelling
                 .
              
               
                 13
                 The
                 first
                 Organ
                 and
                 the
                 generation
                 of
                 Smelling
                 .
              
               
                 14
                 How
                 it
                 is
                 helped
                 by
                 Heat
                 and
                 by
                 Wind.
                 
              
               
                 15
                 Why
                 such
                 Bodies
                 are
                 least
                 smelt
                 ,
                 which
                 have
                 least
                 intermixture
                 of
                 Aire
                 in
                 them
                 .
              
               
                 16
                 Why
                 Odorous
                 things
                 become
                 more
                 Odorous
                 by
                 being
                 bruised
                 .
              
               
                 17
                 The
                 first
                 Organ
                 of
                 Tasting
                 ;
                 and
                 why
                 some
                 Savours
                 cause
                 Nauseousness
                 .
              
               
                 18
                 The
                 first
                 Organ
                 of
                 Feeling
                 ;
                 and
                 how
                 we
                 come
                 to
                 the
                 knowledge
                 of
                 such
                 Objects
                 as
                 are
                 common
                 to
                 the
                 Touch
                 and
                 other
                 Senses
                 .
              
            
          
           
             SOUND
             
               is
               Sense
               generated
               by
               the
               action
               of
               the
               Medium
               ,
               when
               its
               motion
               reacheth
               the
               Eare
               and
               the
               rest
               of
               the
               Organs
               of
               Sense
               .
            
             Now
             the
             motion
             of
             the
             Medium
             is
             not
             the
             Sound
             it self
             ,
             but
             the
             cause
             of
             it
             .
             For
             the
             Phantasme
             which
             is
             made
             in
             us
             ,
             that
             is
             to
             say
             ,
             the
             Reaction
             of
             the
             Organ
             is
             properly
             that
             which
             we
             call
             Sound
             .
          
           
             The
             principal
             distinctions
             of
             Sounds
             are
             these
             ;
             First
             ,
             that
             one
             Sound
             is
             stronger
             ,
             another
             Weaker
             .
             Secondly
             ,
             that
             one
             is
             more
             Grave
             ,
             another
             more
             Acute
             .
             Thirdly
             ,
             that
             one
             is
             Clear
             ,
             another
             Hoarse
             .
             Fourthly
             ,
             that
             one
             is
             Primary
             ,
             another
             Derivative
             .
             Fifthly
             ,
             that
             one
             is
             Uniform
             ,
             another
             not
             .
             Sixthly
             ,
             that
             one
             is
             more
             Durable
             ,
             another
             less
             Durable
             .
             Of
             all
             which
             distinctions
             the
             members
             may
             be
             subdistinguished
             into
             parts
             distinguishable
             almost
             infinitely
             .
             For
             the
             variety
             of
             Sounds
             seems
             to
             be
             not
             much
             less
             then
             that
             of
             Colours
             .
          
           
           
             As
             Vision
             ,
             so
             Hearing
             is
             generated
             by
             the
             motion
             of
             the
             Medium
             ,
             but
             not
             in
             the
             same
             manner
             .
             For
             Sight
             is
             from
             Pressure
             ,
             that
             is
             ,
             from
             an
             Endeavour
             ;
             in
             which
             there
             is
             no
             perceptible
             progression
             of
             any
             of
             the
             parts
             of
             the
             Medium
             ;
             but
             one
             part
             urging
             or
             thrusting
             on
             an
             other
             ,
             propagateth
             that
             action
             successively
             to
             any
             distance
             whatsoever
             ;
             whereas
             the
             motion
             of
             the
             Medium
             by
             which
             Sound
             is
             made
             ,
             is
             a
             Stroke
             .
             For
             when
             we
             Hear
             ,
             the
             Drumme
             of
             the
             Eare
             (
             which
             is
             the
             first
             Organ
             of
             Hearing
             )
             is
             stricken
             ;
             and
             the
             Drumme
             being
             stricken
             ,
             the
             
               Pia
               Mater
            
             is
             also
             shaken
             ,
             and
             with
             it
             the
             Arteries
             which
             are
             inserted
             into
             it
             ;
             by
             which
             the
             action
             being
             propagated
             to
             the
             Heart
             it self
             ,
             by
             the
             reaction
             of
             the
             Heart
             a
             Phantasm
             is
             made
             which
             we
             call
             Sound
             ;
             and
             (
             because
             the
             reaction
             tendeth
             outwards
             )
             we
             think
             it
             is
             without
             .
          
           
             2
             And
             seeing
             the
             effects
             produced
             by
             Motion
             ,
             are
             greater
             or
             lesse
             ,
             not
             onely
             when
             the
             Velocity
             is
             greater
             or
             less
             ,
             but
             also
             when
             the
             Body
             hath
             greater
             or
             less
             Magnitude
             though
             the
             Velocity
             be
             the
             same
             ;
             a
             Sound
             may
             be
             greater
             or
             lesse
             both
             these
             wayes
             .
             And
             because
             neither
             the
             greatest
             nor
             the
             least
             Magnitude
             or
             Velocity
             can
             be
             given
             ,
             it
             may
             happen
             ,
             that
             either
             the
             motion
             may
             be
             of
             so
             small
             velocity
             ,
             or
             the
             Body
             it self
             of
             so
             small
             magnitude
             ,
             as
             to
             produce
             no
             Sound
             at
             all
             ;
             or
             either
             of
             them
             may
             be
             so
             great
             ,
             as
             to
             take
             away
             the
             Faculty
             of
             Sense
             by
             hurting
             the
             Organ
             .
          
           
             From
             hence
             may
             be
             deduced
             possible
             causes
             of
             the
             strength
             and
             weakness
             of
             Sounds
             in
             the
             following
             Phaenomena
             .
          
           
             The
             first
             whereof
             is
             this
             ,
             That
             if
             a
             man
             speak
             through
             a
             Trunk
             which
             hath
             on
             end
             applyed
             to
             the
             mouth
             of
             the
             Speaker
             ,
             and
             the
             other
             to
             the
             eare
             of
             the
             Hearer
             ,
             the
             Sound
             will
             come
             stronger
             then
             it
             would
             do
             through
             the
             open
             Aire
             .
             And
             the
             cause
             (
             not
             onley
             the
             possible
             ,
             but
             the
             certain
             and
             manifest
             cause
             )
             is
             this
             ,
             that
             the
             Aire
             which
             is
             moved
             by
             the
             first
             breath
             ,
             and
             carried
             forwards
             in
             the
             Trunk
             ,
             is
             not
             diffused
             ,
             as
             it
             would
             be
             in
             the
             open
             Aire
             ,
             and
             is
             consequently
             brought
             to
             the
             eare
             almost
             with
             the
             same
             velocity
             with
             which
             it
             was
             first
             breathed
             out
             ;
             Whereas
             in
             the
             open
             Aire
             ,
             the
             first
             motion
             diffuseth
             it self
             every
             way
             into
             Circles
             ,
             such
             as
             are
             made
             by
             the
             throwing
             of
             a
             Stone
             into
             a
             standing
             water
             ,
             where
             the
             velocity
             grows
             less
             and
             less
             as
             the
             Undulation
             proceeds
             further
             and
             further
             from
             the
             beginning
             of
             its
             motion
             .
          
           
             The
             second
             is
             this
             ,
             That
             if
             the
             Trunk
             be
             short
             ,
             and
             the
             
             end
             which
             is
             applyed
             to
             the
             mouth
             be
             wider
             then
             that
             which
             is
             applyed
             to
             the
             eare
             ,
             thus
             also
             the
             Sound
             will
             be
             stronger
             then
             if
             it
             were
             made
             in
             the
             open
             aire
             .
             And
             the
             cause
             is
             the
             same
             ,
             namely
             ,
             that
             by
             how
             much
             the
             wider
             end
             of
             the
             Trunk
             is
             less
             distant
             from
             the
             beginning
             of
             the
             Sound
             ,
             by
             so
             much
             the
             less
             is
             the
             diffusion
             .
          
           
             The
             third
             ,
             That
             it
             is
             easier
             for
             one
             that
             is
             within
             a
             Chamber
             ,
             to
             heare
             what
             is
             spoken
             without
             ,
             then
             for
             him
             that
             stands
             without
             ,
             to
             hear
             what
             is
             spoken
             within
             .
             For
             the
             Windows
             and
             other
             inlets
             of
             the
             moved
             Aire
             ,
             are
             as
             the
             wide
             end
             of
             the
             Trunk
             .
             And
             for
             this
             reason
             some
             creatures
             seem
             to
             hear
             the
             better
             ,
             because
             Nature
             has
             bestowed
             upon
             them
             wide
             and
             capacious
             Ears
             .
          
           
             The
             fourth
             is
             this
             ,
             That
             though
             he
             which
             standeth
             upon
             the
             Sea
             shore
             ,
             cannot
             heare
             the
             Collision
             of
             the
             two
             neerest
             waves
             ,
             yet
             neverthess
             he
             hears
             the
             roaring
             of
             the
             whole
             Sea.
             And
             the
             cause
             seems
             to
             be
             this
             ,
             that
             though
             the
             several
             collisions
             move
             the
             Organ
             ,
             yet
             they
             are
             not
             severally
             great
             enough
             to
             cause
             Sense
             ;
             whereas
             nothing
             hinders
             but
             that
             all
             of
             them
             together
             may
             make
             Sound
             .
          
           
             3
             That
             Bodies
             when
             they
             are
             stricken
             do
             yeild
             some
             a
             more
             Grave
             ,
             others
             a
             more
             Acute
             Sound
             ,
             the
             cause
             may
             consist
             in
             the
             difference
             of
             the
             times
             in
             which
             the
             parts
             stricken
             and
             forced
             out
             of
             their
             places
             ,
             return
             to
             the
             same
             places
             again
             .
             For
             in
             some
             Bodies
             ,
             the
             restitution
             of
             the
             moved
             parts
             is
             quick
             ,
             in
             others
             slow
             .
             And
             this
             also
             may
             be
             the
             cause
             why
             the
             parts
             of
             the
             Organ
             which
             are
             moved
             by
             the
             Medium
             ,
             return
             to
             their
             rest
             again
             ,
             sometimes
             sooner
             ,
             sometimes
             later
             .
             Now
             by
             how
             much
             the
             Vibrations
             ,
             or
             the
             reciprocal
             motions
             of
             the
             parts
             are
             more
             frequent
             ,
             by
             so
             much
             doth
             the
             whole
             Sound
             made
             (
             at
             the
             same
             time
             )
             by
             one
             stroke
             ,
             consist
             of
             more
             ,
             and
             consequently
             of
             smaller
             parts
             .
             For
             what
             is
             Acute
             in
             Sound
             ,
             the
             same
             is
             Subtle
             in
             Matter
             ;
             and
             both
             of
             them
             ,
             namely
             ,
             Acute
             Sound
             ,
             and
             Subtle
             Matter
             consist
             of
             very
             small
             parts
             ,
             that
             of
             Time
             ,
             and
             this
             of
             the
             Matter
             it self
             .
          
           
             The
             third
             distinction
             of
             Sounds
             cannot
             be
             conceived
             clearly
             enough
             by
             the
             names
             I
             have
             used
             of
             Clear
             and
             Hoarse
             ,
             nor
             by
             any
             other
             that
             I
             know
             ;
             and
             therefore
             it
             is
             needful
             to
             explain
             them
             by
             examples
             .
             When
             I
             say
             Hoarse
             ,
             I
             understand
             Whispering
             and
             Hissing
             ,
             and
             whatsoever
             is
             like
             to
             these
             ,
             by
             what
             appellation
             soever
             it
             be
             expressed
             .
             And
             Sounds
             of
             this
             kind
             seem
             to
             be
             made
             by
             the
             force
             of
             some
             strong
             Wind
             ,
             raking
             rather
             then
             striking
             such
             hard
             
             Bodies
             ,
             as
             it
             falls
             upon
             .
             On
             the
             contrary
             ,
             when
             I
             use
             the
             word
             Clear
             ,
             I
             do
             not
             understand
             such
             a
             Sound
             as
             may
             be
             easily
             and
             distinctly
             heard
             ,
             for
             so
             Whispers
             would
             be
             Clear
             ,
             but
             such
             as
             is
             made
             by
             somewhat
             that
             is
             Broken
             ;
             and
             such
             as
             is
             Clamor
             ,
             Tinkling
             ,
             the
             Sound
             of
             a
             Trumpet
             ,
             &c.
             and
             (
             to
             express
             it
             significantly
             in
             one
             word
             )
             ,
             Noise
             .
             And
             seeing
             no
             Sound
             is
             made
             but
             by
             the
             concourse
             of
             two
             Bodies
             at
             the
             least
             ,
             by
             which
             concourse
             it
             is
             necessary
             that
             there
             be
             as
             well
             Reaction
             as
             Action
             ,
             that
             is
             to
             say
             ,
             one
             motion
             opposite
             to
             another
             ;
             it
             follows
             ,
             that
             according
             as
             the
             proportion
             between
             those
             two
             opposite
             motions
             is
             diversified
             ,
             so
             the
             Sounds
             which
             are
             made
             will
             be
             different
             from
             one
             another
             .
             And
             whensoever
             the
             proportion
             between
             them
             is
             so
             great
             ,
             as
             that
             the
             motion
             of
             one
             of
             the
             Bodies
             be
             insensible
             if
             compared
             with
             the
             motion
             of
             the
             other
             ,
             then
             the
             Sound
             will
             not
             be
             of
             the
             same
             kind
             ;
             as
             when
             the
             Wind
             falls
             very
             obliquely
             upon
             a
             hard
             Body
             ,
             or
             when
             a
             hard
             Body
             is
             carried
             swiftly
             through
             the
             Aire
             ;
             for
             then
             there
             is
             made
             that
             Sound
             which
             I
             call
             a
             Hoarse
             Sound
             ,
             in
             Greek
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             .
             Therefore
             the
             breath
             blown
             with
             violence
             from
             the
             mouth
             makes
             a
             Hissing
             ,
             because
             in
             going
             out
             it
             rakes
             the
             Superficies
             of
             the
             Lips
             ,
             whose
             reaction
             against
             the
             force
             of
             the
             breath
             is
             not
             Sensible
             .
             And
             this
             is
             the
             cause
             why
             the
             Winds
             have
             that
             Hoarse
             Sound
             .
             Also
             if
             two
             Bodies
             how
             hard
             soever
             ,
             be
             rubbed
             together
             with
             no
             great
             pressure
             ,
             they
             make
             a
             Hoarse
             Sound
             .
             And
             this
             Hoarse
             Sound
             when
             it
             is
             made
             (
             as
             I
             have
             said
             )
             by
             the
             Aire
             raking
             the
             Superficies
             of
             a
             hard
             Body
             ,
             seemeth
             to
             be
             nothing
             but
             the
             dividing
             of
             the
             Aire
             into
             innumerable
             and
             very
             small
             Files
             .
             For
             the
             asperity
             of
             the
             Superficies
             doth
             by
             the
             eminencies
             of
             its
             innumerable
             parts
             divide
             or
             cut
             in
             pieces
             the
             Aire
             that
             slides
             upon
             it
             .
          
           
             4
             Noise
             ,
             or
             that
             which
             I
             call
             Clear
             Sound
             ,
             is
             made
             two
             wayes
             ;
             one
             ,
             by
             two
             Hoarse
             Sounds
             ,
             made
             by
             opposite
             motions
             ;
             the
             other
             ,
             by
             Collision
             ,
             or
             by
             the
             suddain
             pulling
             asunder
             of
             two
             Bodies
             ,
             whereby
             their
             small
             particles
             are
             put
             into
             commotion
             ,
             or
             being
             already
             in
             commotion
             ,
             suddenly
             restore
             themselves
             again
             ;
             which
             motion
             making
             impression
             upon
             the
             Medium
             ,
             is
             propagated
             to
             the
             Organ
             of
             Hearing
             .
             And
             seeing
             there
             is
             in
             this
             Collision
             ,
             or
             divulsion
             ,
             an
             endeavour
             in
             the
             particles
             of
             one
             Body
             ,
             opposite
             to
             the
             endeavour
             of
             the
             particles
             of
             the
             other
             Body
             ,
             there
             will
             also
             be
             made
             in
             the
             Organ
             of
             Hearing
             a
             like
             opposition
             of
             endeavours
             ,
             that
             is
             to
             
             say
             ,
             of
             motions
             ;
             and
             consequently
             the
             Sound
             arising
             from
             thence
             ,
             will
             be
             made
             by
             two
             opposite
             motions
             ,
             that
             is
             to
             say
             ,
             by
             two
             opposite
             Hoarse
             Sounds
             in
             one
             and
             the
             same
             part
             of
             the
             Organ
             .
             For
             (
             as
             I
             have
             already
             said
             )
             a
             Hoarse
             Sound
             supposeth
             the
             sensible
             motion
             of
             but
             one
             of
             the
             Bodies
             .
             And
             this
             opposition
             of
             motions
             in
             the
             Organ
             is
             the
             cause
             why
             two
             Bodies
             make
             a
             Noyse
             ,
             when
             they
             are
             either
             suddenly
             stricken
             against
             one
             another
             ,
             or
             suddenly
             broken
             asunder
             .
          
           
             5
             This
             being
             granted
             ;
             and
             seeing
             withall
             ,
             that
             Thunder
             is
             made
             by
             the
             vehement
             eruption
             of
             the
             Aire
             out
             of
             the
             cavities
             of
             congeled
             Clouds
             ;
             the
             cause
             of
             the
             great
             Noyse
             or
             Clap
             ,
             may
             be
             the
             suddain
             breaking
             asunder
             of
             the
             Ice
             .
             For
             in
             this
             action
             it
             is
             necessary
             ,
             that
             there
             be
             not
             onely
             a
             great
             concussion
             of
             the
             small
             particles
             of
             the
             broken
             parts
             ,
             but
             also
             that
             this
             Concussion
             (
             by
             being
             communicated
             to
             the
             Aire
             )
             be
             carried
             to
             the
             Organ
             of
             Hearing
             ,
             &
             make
             impression
             upon
             it
             .
             And
             then
             ,
             from
             the
             first
             reaction
             of
             the
             Organ
             proceeds
             that
             first
             and
             greatest
             Sound
             ,
             which
             is
             made
             by
             the
             collision
             of
             the
             parts
             whilst
             they
             restore
             themselves
             .
             And
             seeing
             there
             is
             in
             all
             Concussion
             a
             reciprocation
             of
             Motion
             forwards
             and
             backwards
             in
             the
             parts
             stricken
             ,
             (
             for
             opposite
             motions
             cannot
             extinguish
             one
             another
             in
             an
             instant
             ,
             as
             I
             have
             shewn
             in
             the
             11th
             Art.
             of
             the
             8th
             Chap.
             )
             it
             follows
             necessarily
             ,
             that
             the
             Sound
             will
             both
             continue
             ,
             and
             grow
             weaker
             and
             weaker
             ,
             till
             at
             last
             the
             action
             of
             the
             reciprocating
             aire
             grow
             so
             weak
             ,
             as
             to
             be
             unperceptible
             .
             Wherefore
             a
             possible
             cause
             is
             given
             both
             of
             the
             first
             fierce
             Noyse
             of
             the
             Thunder
             ;
             and
             also
             of
             the
             Murmur
             that
             follows
             it
             .
          
           
             The
             cause
             of
             the
             great
             Sound
             from
             a
             discharged
             piece
             of
             Ordnance
             ,
             is
             like
             that
             of
             a
             Clap
             of
             Thunder
             .
             For
             the
             Gunpowder
             being
             fired
             ,
             doth
             in
             its
             endeavour
             to
             go
             out
             ,
             attempt
             every
             way
             the
             sides
             of
             the
             metal
             in
             such
             manner
             ,
             as
             that
             it
             enlargeth
             the
             Circumference
             all
             along
             ,
             and
             withall
             shortneth
             the
             axis
             ;
             so
             that
             whilest
             the
             peece
             of
             Ordnance
             is
             in
             discharging
             ,
             it
             is
             made
             both
             wider
             and
             shorter
             then
             it
             was
             before
             ;
             and
             therefore
             also
             presently
             after
             it
             is
             discharged
             its
             wideness
             will
             be
             diminished
             ,
             and
             its
             length
             encreased
             again
             by
             the
             restitution
             of
             all
             the
             particles
             of
             the
             matter
             of
             which
             it
             consisteth
             to
             their
             former
             position
             .
             And
             this
             is
             done
             with
             such
             motion
             of
             the
             parts
             ,
             as
             are
             not
             onely
             very
             vehement
             ,
             but
             also
             opposite
             to
             one
             another
             ;
             which
             motions
             
             being
             communicated
             to
             the
             Aire
             ,
             make
             impression
             upon
             the
             Organ
             ,
             and
             by
             the
             reaction
             of
             the
             Organ
             create
             a
             Sound
             ;
             which
             lasteth
             for
             some
             time
             ,
             as
             I
             have
             already
             shewn
             in
             this
             Article
             .
          
           
             I
             note
             by
             the
             way
             (
             as
             not
             belonging
             to
             this
             place
             )
             that
             the
             possible
             cause
             why
             a
             Gun
             recoyles
             when
             it
             is
             shot
             off
             ,
             may
             be
             this
             ;
             That
             being
             first
             swoln
             by
             the
             force
             of
             the
             fire
             ,
             and
             afterwards
             restoring
             it self
             ;
             from
             this
             restitution
             there
             proceeds
             an
             endeavour
             from
             all
             the
             sides
             towards
             the
             cavity
             ;
             and
             consequently
             this
             endeavour
             is
             in
             those
             parts
             which
             are
             next
             the
             breech
             ;
             which
             being
             not
             hollow
             ,
             but
             solid
             ,
             the
             effect
             of
             the
             restitution
             is
             by
             it
             hindered
             and
             diverted
             into
             the
             length
             ;
             and
             by
             this
             means
             both
             the
             breech
             and
             the
             whole
             Gun
             is
             thrust
             backwards
             ;
             and
             the
             more
             forcibly
             by
             how
             much
             the
             force
             is
             greater
             by
             which
             the
             part
             next
             the
             breech
             is
             restored
             to
             its
             former
             posture
             ;
             that
             is
             to
             say
             ,
             by
             how
             much
             the
             thiner
             is
             that
             part
             .
             The
             cause
             therefore
             why
             Gunnes
             recoyle
             ,
             some
             more
             ,
             some
             less
             ,
             is
             the
             difference
             of
             their
             thickness
             towards
             the
             breech
             ;
             &
             the
             greater
             that
             thickness
             is
             ,
             the
             less
             they
             recoyl
             ;
             and
             contrarily
             .
          
           
             6
             Also
             the
             cause
             why
             the
             Sound
             of
             a
             Pipe
             ,
             which
             is
             made
             by
             blowing
             into
             it
             ,
             is
             nevertheless
             Clear
             ,
             is
             the
             same
             with
             that
             of
             the
             Sound
             which
             is
             made
             by
             collision
             .
             For
             if
             the
             breath
             when
             it
             is
             blown
             into
             a
             Pipe
             doe
             onely
             rake
             its
             concave
             Superficies
             ,
             or
             fall
             upon
             it
             with
             a
             very
             sharp
             angle
             of
             incidence
             ,
             the
             Sound
             will
             not
             be
             Clear
             ,
             but
             Hoarse
             .
             But
             if
             the
             angle
             be
             great
             enough
             ,
             the
             percussion
             which
             is
             made
             against
             one
             of
             the
             hollow
             sides
             ,
             will
             be
             reverberated
             to
             the
             opposite
             side
             ;
             and
             so
             successive
             repercussions
             will
             be
             made
             from
             side
             to
             side
             ,
             till
             at
             last
             the
             whole
             concave
             Superficies
             of
             the
             Pipe
             be
             put
             into
             motion
             ;
             which
             motion
             will
             be
             reciprocated
             ,
             as
             it
             is
             in
             Collision
             ;
             and
             this
             reciprocation
             being
             propagated
             to
             the
             Organ
             ,
             from
             the
             reaction
             of
             the
             Organ
             will
             arise
             a
             Cleare
             Sound
             ,
             such
             as
             is
             made
             by
             Collision
             ,
             or
             by
             breaking
             asunder
             of
             hard
             Bodies
             .
          
           
             In
             the
             same
             manner
             it
             is
             with
             the
             Sound
             of
             a
             Mans
             voice
             .
             For
             when
             the
             breath
             passeth
             out
             ,
             without
             interruption
             ,
             and
             doth
             but
             lightly
             touch
             the
             cavities
             through
             which
             it
             is
             sent
             ,
             the
             Sound
             it
             maketh
             is
             a
             Hoarse
             Sound
             .
             But
             if
             in
             going
             out
             it
             strike
             strongly
             upon
             the
             Larinx
             ,
             then
             a
             Clear
             Sound
             is
             made
             ,
             as
             in
             a
             Pipe.
             And
             the
             same
             breath
             ,
             as
             it
             comes
             in
             divers
             manners
             to
             the
             Palate
             ,
             the
             Tongue
             ,
             the
             Lips
             ,
             the
             Teeth
             ,
             and
             other
             Organs
             of
             Speech
             ,
             so
             the
             
             Sounds
             into
             which
             it
             is
             articulated
             become
             different
             from
             one
             another
             .
          
           
             7
             I
             call
             that
             
               Primary
               Sound
            
             ,
             which
             is
             generated
             by
             motion
             from
             the
             sounding
             Body
             to
             the
             Organ
             in
             a
             straight
             line
             without
             reflexion
             ;
             and
             I
             call
             that
             
               Reflected
               Sound
            
             ,
             which
             is
             generated
             by
             one
             or
             more
             reflexions
             ;
             being
             the
             same
             with
             that
             we
             call
             Echo
             ;
             and
             is
             iterated
             as
             often
             as
             there
             are
             reflexions
             made
             from
             the
             Object
             to
             the
             Eare.
             And
             these
             reflexions
             are
             made
             by
             Hils
             ,
             Wals
             ,
             and
             other
             resisting
             Bodies
             ,
             so
             placed
             ,
             as
             that
             they
             make
             more
             or
             fewer
             reflexions
             of
             the
             motion
             ,
             according
             as
             they
             are
             themselves
             more
             or
             fewer
             in
             number
             ;
             and
             they
             make
             them
             more
             or
             less
             frequently
             ,
             according
             as
             they
             are
             more
             or
             less
             distant
             from
             one
             another
             .
             Now
             the
             cause
             of
             both
             these
             things
             is
             to
             be
             sought
             for
             in
             the
             situation
             of
             the
             reflecting
             Bodies
             ,
             as
             is
             usually
             done
             in
             Sight
             .
             For
             the
             Lawes
             of
             Reflexion
             are
             the
             same
             in
             both
             ,
             namely
             ,
             that
             the
             Angles
             of
             Incidence
             and
             Reflexion
             be
             equal
             to
             one
             another
             .
             If
             therefore
             in
             a
             hollow
             Elliptique
             Body
             whose
             inside
             is
             well
             polished
             ,
             or
             in
             two
             right
             Parabolical
             Solids
             which
             are
             joyned
             together
             by
             one
             common
             base
             ,
             there
             be
             placed
             a
             Sounding
             Body
             in
             one
             of
             the
             Burning
             Points
             ,
             &
             the
             Ear
             in
             the
             other
             ,
             there
             will
             be
             heard
             a
             Sound
             by
             many
             degrees
             greater
             then
             in
             the
             open
             Aire
             ;
             and
             both
             this
             ,
             and
             the
             burning
             of
             such
             combustible
             things
             ,
             as
             being
             put
             in
             the
             same
             places
             are
             set
             on
             fire
             by
             the
             Sun-beams
             ,
             are
             effects
             of
             one
             and
             the
             same
             cause
             .
             But
             as
             when
             the
             visible
             Object
             is
             placed
             in
             one
             of
             the
             Burning
             Points
             ,
             it
             is
             not
             distinctly
             seen
             in
             the
             other
             ,
             because
             every
             part
             of
             the
             Object
             being
             seen
             in
             every
             line
             which
             is
             reflected
             from
             the
             Concave
             Superficies
             to
             the
             Eie
             ,
             makes
             a
             confusion
             in
             the
             Sight
             ;
             so
             neither
             is
             Sound
             heard
             articulately
             and
             distinctly
             when
             it
             comes
             to
             the
             Eare
             in
             all
             those
             reflected
             lines
             .
             And
             this
             may
             be
             the
             reason
             ,
             why
             in
             Churches
             which
             have
             arched
             rooffs
             ,
             though
             they
             be
             neither
             Elliptical
             nor
             Parabolical
             ;
             yet
             because
             their
             figure
             is
             not
             much
             different
             from
             these
             ,
             the
             voice
             from
             the
             Pulpit
             will
             not
             be
             heard
             so
             articulately
             as
             it
             would
             be
             if
             there
             were
             no
             vaulting
             at
             all
             .
          
           
             8
             Concerning
             the
             Uniformity
             and
             Duration
             of
             Sounds
             ,
             both
             which
             have
             one
             common
             cause
             ,
             we
             may
             observe
             ,
             that
             such
             Bodies
             as
             being
             stricken
             yeild
             an
             unequal
             or
             harsh
             Sound
             ,
             are
             very
             heterogeous
             ,
             that
             is
             to
             say
             ,
             they
             consist
             of
             parts
             which
             are
             very
             unlike
             both
             in
             figure
             and
             hardness
             ,
             such
             as
             are
             Wood
             ,
             Stones
             ,
             and
             others
             not
             a
             
             few
             .
             When
             these
             are
             stricken
             ,
             there
             follows
             a
             concussion
             of
             their
             internal
             particles
             ,
             and
             a
             restitution
             of
             them
             again
             .
             But
             they
             are
             neither
             moved
             alike
             ,
             nor
             have
             they
             the
             same
             action
             upon
             one
             another
             ;
             some
             of
             them
             recoyling
             from
             the
             stroke
             whilest
             others
             which
             have
             already
             finished
             their
             recoylings
             are
             now
             returning
             ;
             by
             which
             means
             they
             hinder
             and
             stop
             on
             another
             .
             And
             from
             hence
             it
             is
             that
             their
             motions
             are
             not
             only
             unequal
             and
             harsh
             ,
             but
             also
             that
             their
             reciprocations
             come
             to
             be
             quickly
             extinguished
             .
             Whensoever
             therfore
             this
             motion
             is
             propagated
             to
             the
             Eare
             ,
             the
             Sound
             it
             makes
             is
             Unequal
             and
             of
             small
             Duration
             .
             On
             the
             contrary
             ,
             if
             a
             Body
             that
             is
             stricken
             ,
             be
             not
             onely
             sufficiently
             hard
             ,
             but
             have
             also
             the
             particles
             of
             which
             it
             consisteth
             like
             to
             one
             another
             both
             in
             hardness
             and
             figure
             ,
             (
             such
             as
             are
             the
             particles
             of
             Glass
             and
             Metals
             ,
             which
             being
             first
             melted
             do
             afterwards
             settle
             and
             harden
             )
             the
             Sound
             it
             yeildeth
             ,
             will
             (
             because
             the
             motions
             of
             its
             parts
             and
             their
             reciprocations
             are
             like
             and
             Uniform
             )
             be
             Uniform
             and
             pleasant
             ,
             and
             be
             more
             or
             less
             Lasting
             according
             as
             the
             Body
             stricken
             hath
             gteater
             or
             less
             magnitude
             .
             The
             possible
             cause
             therefore
             of
             Sounds
             Uniform
             ,
             and
             Harsh
             ,
             and
             of
             their
             longer
             or
             shorter
             Duration
             ,
             may
             be
             one
             and
             the
             same
             likeness
             and
             unlikeness
             of
             the
             internal
             parts
             of
             the
             Sounding
             Body
             ,
             in
             respect
             both
             of
             their
             figure
             and
             hardness
             .
          
           
             Besides
             ,
             if
             two
             plain
             Bodies
             of
             the
             same
             matter
             ,
             and
             of
             equal
             thickness
             ,
             do
             both
             yeild
             an
             Uniform
             Sound
             ,
             the
             Sound
             of
             that
             Body
             which
             hath
             the
             greatest
             extent
             of
             length
             will
             be
             the
             longest
             heard
             .
             For
             the
             motion
             which
             in
             both
             of
             them
             hath
             its
             beginning
             from
             the
             point
             of
             percussion
             ,
             is
             to
             be
             propagated
             in
             the
             greater
             Body
             through
             a
             greater
             Space
             ,
             and
             consequently
             that
             propagation
             will
             require
             more
             time
             ;
             and
             therefore
             also
             the
             parts
             which
             are
             moved
             will
             require
             more
             time
             for
             their
             return
             .
             Wherefore
             all
             the
             reciprocations
             cannot
             be
             finished
             but
             in
             longer
             time
             ;
             and
             being
             carried
             to
             the
             Eare
             ,
             will
             make
             the
             Sound
             last
             the
             longer
             .
             And
             from
             hence
             it
             is
             manifest
             ,
             that
             of
             hard
             Bodies
             which
             yeild
             an
             Uniform
             Sound
             ,
             the
             Sound
             lasteth
             longer
             which
             comes
             from
             those
             that
             are
             round
             and
             hollow
             ,
             then
             from
             those
             that
             are
             plain
             ,
             if
             they
             be
             like
             in
             all
             other
             respects
             .
             For
             in
             circular
             lines
             the
             action
             which
             begins
             at
             any
             point
             ,
             hath
             not
             frō
             the
             figure
             any
             end
             of
             its
             propagation
             ,
             because
             the
             line
             in
             which
             it
             is
             propagated
             returns
             again
             to
             its
             beginning
             ;
             so
             that
             the
             figure
             hinders
             not
             but
             that
             the
             motion
             may
             
             have
             infinite
             progression
             ;
             whereas
             in
             a
             plain
             ,
             every
             line
             hath
             its
             magnitude
             finite
             ,
             beyond
             which
             the
             action
             cannot
             proceed
             .
             If
             therefore
             the
             matter
             be
             the
             same
             ,
             the
             motion
             of
             the
             parts
             of
             that
             Body
             whose
             figure
             is
             round
             and
             hollow
             ,
             wil
             last
             longer
             ,
             then
             of
             that
             which
             is
             plain
             .
          
           
             Also
             ,
             if
             a
             string
             which
             is
             stretched
             ,
             be
             fastned
             at
             both
             ends
             to
             a
             hollow
             Body
             ,
             and
             be
             stricken
             ,
             the
             Sound
             will
             last
             longer
             then
             if
             it
             were
             not
             so
             fastned
             ;
             because
             the
             trembling
             or
             reciprocation
             which
             it
             receives
             from
             the
             stroke
             ,
             is
             by
             reason
             of
             the
             connexion
             communicated
             to
             the
             hollow
             Body
             ;
             and
             this
             trembling
             ,
             if
             the
             hollow
             Body
             be
             great
             ,
             will
             last
             the
             longer
             by
             reason
             of
             that
             greatness
             .
             Wherefore
             also
             (
             for
             the
             reason
             above
             mentioned
             )
             the
             Sound
             will
             last
             the
             longer
             .
          
           
             9
             In
             Hearing
             ,
             it
             happens
             (
             otherwise
             then
             in
             Seeing
             )
             that
             the
             action
             of
             the
             Medium
             is
             made
             stronger
             by
             the
             Wind
             when
             it
             blows
             the
             same
             way
             ,
             and
             weaker
             when
             it
             blows
             the
             contrary
             way
             .
             The
             cause
             whereof
             cannot
             proceed
             from
             any
             thing
             but
             the
             different
             generation
             of
             Sound
             and
             Light.
             For
             in
             the
             generation
             of
             Light
             ,
             none
             of
             the
             parts
             of
             the
             Medium
             between
             the
             object
             and
             the
             Eie
             are
             moved
             from
             their
             own
             places
             to
             other
             places
             sensibly
             distant
             ;
             but
             the
             action
             is
             propagated
             in
             spaces
             imperceptible
             ;
             so
             that
             no
             contrary
             Wind
             can
             diminish
             ,
             nor
             favourable
             Winde
             encrease
             the
             Light
             ,
             unless
             it
             be
             so
             strong
             as
             to
             remove
             the
             Object
             further
             off
             ,
             or
             bring
             it
             nearer
             to
             the
             Eie
             .
             For
             the
             Wind
             ,
             that
             is
             to
             say
             ,
             the
             aire
             moved
             ,
             doth
             not
             by
             its
             interposition
             between
             the
             object
             and
             the
             eie
             ,
             worke
             otherwise
             then
             it
             would
             doe
             if
             it
             were
             stil
             and
             calme
             .
             For
             where
             the
             pressure
             is
             perpetuall
             ,
             one
             part
             of
             the
             aire
             is
             no
             sooner
             carried
             away
             ,
             but
             another
             by
             succeeding
             it
             receives
             the
             same
             impression
             which
             the
             part
             carried
             away
             had
             received
             before
             .
             But
             in
             the
             generation
             of
             Sound
             ,
             the
             first
             collision
             or
             breaking
             asunder
             ,
             beateth
             away
             &
             driveth
             out
             of
             its
             place
             the
             nearest
             part
             of
             the
             aire
             ,
             and
             that
             to
             a
             considerable
             distance
             ,
             and
             with
             considerable
             velocity
             ;
             and
             as
             the
             circles
             grow
             (
             by
             their
             remotenesse
             )
             wider
             and
             wider
             ,
             so
             the
             aire
             being
             more
             &
             more
             dissipated
             ,
             hath
             also
             its
             motion
             more
             &
             more
             weakned
             .
             Whensoever
             therfore
             the
             air
             is
             so
             stricken
             as
             to
             cause
             Sound
             ,
             if
             the
             Wind
             fall
             upon
             it
             ,
             it
             will
             move
             it
             all
             ,
             neerer
             to
             the
             Eare
             if
             it
             blow
             that
             way
             ,
             and
             further
             from
             it
             if
             it
             blow
             the
             contrary
             way
             ;
             so
             that
             according
             
             as
             it
             blowes
             from
             or
             towards
             the
             Object
             ,
             so
             the
             Sound
             which
             is
             heard
             will
             seeme
             to
             come
             from
             a
             neerer
             or
             remoter
             place
             ;
             and
             the
             action
             by
             reason
             of
             the
             unequall
             distances
             be
             strengthened
             or
             debilitated
             .
          
           
             From
             hence
             may
             be
             understood
             the
             reason
             ,
             why
             the
             voice
             of
             such
             as
             are
             said
             to
             speake
             in
             their
             bellies
             ,
             though
             it
             be
             uttered
             neer
             hand
             ,
             is
             neverthelesse
             heard
             by
             those
             that
             suspect
             nothing
             ,
             as
             if
             it
             were
             a
             great
             way
             off
             .
             For
             having
             no
             former
             thought
             of
             any
             determined
             place
             from
             which
             the
             voice
             should
             proceed
             ,
             and
             judging
             according
             to
             the
             greatesse
             ,
             if
             it
             be
             weake
             they
             thinke
             it
             a
             great
             way
             off
             ,
             if
             strong
             neer
             .
             These
             Ventriloqui
             therefore
             by
             forming
             their
             voice
             ,
             not
             (
             as
             others
             )
             by
             the
             emission
             of
             their
             breath
             ,
             but
             by
             drawing
             it
             inwards
             ,
             doe
             make
             the
             same
             appear
             small
             and
             weake
             ;
             which
             weaknesse
             of
             the
             voice
             deceives
             those
             that
             neither
             suspect
             the
             artifice
             ,
             nor
             observe
             the
             endeavour
             which
             they
             use
             in
             speaking
             ;
             and
             so
             instead
             of
             thinking
             it
             weake
             they
             thinke
             it
             farre
             off
             .
          
           
             10
             As
             for
             the
             Medium
             which
             conveighs
             Sound
             ,
             it
             is
             not
             Aire
             onely
             .
             For
             Water
             ,
             or
             any
             other
             Body
             how
             hard
             soever
             may
             be
             that
             Medium
             .
             For
             the
             Motion
             may
             be
             propagated
             perpetually
             in
             any
             hard
             continuous
             Body
             ;
             but
             by
             reason
             of
             the
             difficulty
             with
             which
             the
             parts
             of
             hard
             Bodies
             are
             moved
             ,
             the
             motion
             in
             going
             out
             of
             hard
             matter
             makes
             but
             a
             weak
             impression
             upon
             the
             Aire
             .
             Nevertheless
             if
             one
             end
             of
             a
             very
             long
             and
             hard
             beam
             be
             stricken
             ,
             &
             the
             eare
             be
             applyed
             at
             the
             same
             time
             to
             the
             other
             end
             ,
             so
             that
             when
             the
             action
             goeth
             out
             of
             the
             beam
             ,
             the
             aire
             which
             it
             striketh
             may
             immediately
             be
             received
             by
             the
             eare
             ,
             and
             be
             carried
             to
             the
             Tympanum
             ,
             the
             Sound
             will
             be
             considerably
             strong
             .
          
           
             In
             like
             manner
             ,
             if
             in
             the
             night
             (
             when
             all
             other
             noyse
             ,
             which
             may
             hinder
             Sound
             ,
             ceaseth
             )
             a
             man
             lay
             his
             eare
             to
             the
             ground
             ,
             he
             will
             hear
             the
             Sound
             of
             the
             steps
             of
             Passengers
             ,
             though
             at
             a
             great
             distance
             ;
             because
             the
             motion
             which
             by
             their
             treading
             they
             communicate
             to
             the
             earth
             ,
             is
             propagated
             to
             the
             eare
             by
             the
             uppermost
             parts
             of
             the
             earth
             which
             receiveth
             it
             from
             their
             feet
             .
          
           
             11
             I
             have
             shewn
             above
             ,
             that
             the
             difference
             between
             Grave
             and
             Acute
             Sounds
             consisteth
             in
             this
             ,
             that
             by
             how
             much
             the
             shorter
             the
             time
             is
             ,
             in
             which
             the
             reciprocations
             of
             the
             parts
             of
             a
             Body
             stricken
             are
             made
             ,
             by
             so
             much
             the
             more
             Acute
             will
             be
             the
             Sound
             .
             
             Now
             by
             how
             much
             a
             Body
             of
             the
             same
             bigness
             ,
             is
             either
             more
             heavy
             ,
             or
             less
             stretched
             ,
             by
             so
             much
             the
             longer
             will
             the
             reciprocations
             last
             ;
             and
             therefore
             heavier
             ,
             and
             less
             stretched
             Bodies
             (
             if
             they
             be
             like
             in
             all
             other
             respects
             )
             will
             yeild
             a
             Graver
             Sound
             then
             such
             as
             be
             lighter
             and
             more
             stretched
             .
          
           
           
             12
             For
             the
             finding
             out
             of
             the
             cause
             of
             Smels
             ,
             I
             shall
             make
             use
             of
             the
             evidence
             of
             these
             following
             Phaenomena
             .
             First
             ,
             that
             Smelling
             is
             hindred
             by
             Cold
             ,
             and
             helped
             by
             Heat
             .
             Secondly
             ,
             that
             when
             the
             Wind
             bloweth
             from
             the
             Object
             ,
             the
             Smel
             is
             the
             stronger
             ;
             and
             contrarily
             when
             it
             bloweth
             from
             the
             Sentient
             towards
             the
             Object
             ,
             the
             weaker
             ;
             both
             which
             Phaenomena
             are
             by
             experience
             manifestly
             found
             to
             be
             true
             in
             Doggs
             which
             follow
             the
             track
             of
             Beasts
             by
             the
             Sent.
             Thirdly
             ,
             that
             such
             Bodies
             as
             are
             less
             pervious
             to
             the
             fluid
             Medium
             ,
             yeild
             less
             Smell
             then
             such
             as
             are
             more
             pervious
             ;
             as
             may
             be
             seen
             in
             Stones
             and
             Metals
             ,
             which
             compared
             with
             Plants
             and
             Living
             Creatures
             ,
             and
             their
             Parts
             ,
             Fruits
             and
             Excretions
             ,
             have
             very
             little
             or
             no
             Smell
             at
             all
             .
             Fourthly
             ,
             that
             such
             Bodies
             as
             are
             of
             their
             own
             nature
             Odorous
             ,
             become
             yet
             more
             Odorous
             when
             they
             are
             bruised
             .
             Fifthly
             ,
             that
             when
             the
             breath
             is
             stopped
             (
             at
             least
             in
             Men
             )
             nothing
             can
             be
             Smelt
             .
             Sixthly
             ,
             that
             the
             sense
             of
             Smelling
             is
             also
             taken
             
             away
             by
             the
             stopping
             of
             the
             Nostrils
             ,
             though
             the
             mouth
             be
             left
             open
             .
          
           
             13
             By
             the
             fourth
             and
             fifth
             Phaenomenon
             it
             is
             manifest
             ,
             that
             the
             first
             and
             immediate
             Organ
             of
             Smelling
             is
             the
             innermost
             cuticle
             of
             the
             Nostrils
             ,
             and
             that
             part
             of
             it
             which
             is
             below
             the
             passage
             common
             to
             the
             Nostrils
             and
             the
             Palate
             .
             For
             when
             we
             draw
             breath
             by
             the
             Nostrils
             ,
             we
             draw
             it
             into
             the
             Lungs
             .
             That
             breath
             therefore
             which
             conveighs
             Smels
             ,
             is
             in
             the
             way
             which
             passeth
             to
             the
             Lungs
             ,
             that
             is
             to
             say
             ,
             in
             that
             part
             of
             the
             Nostrils
             which
             is
             below
             the
             passage
             through
             which
             the
             breath
             goeth
             .
             For
             nothing
             is
             Smelt
             ,
             neither
             beyond
             the
             passage
             of
             the
             breath
             within
             ,
             nor
             at
             all
             without
             the
             Nostrils
             .
          
           
             And
             seeing
             that
             from
             different
             Smels
             there
             must
             necessarily
             proceed
             some
             mutation
             in
             the
             Organ
             ,
             and
             all
             mutation
             is
             motion
             ;
             it
             is
             therefore
             also
             necessary
             that
             in
             Smelling
             ,
             the
             parts
             of
             the
             Organ
             ,
             that
             is
             to
             say
             ,
             of
             that
             internal
             cuticle
             ,
             and
             the
             nerves
             that
             are
             inserted
             into
             it
             ,
             must
             be
             diversly
             moved
             by
             different
             Smels
             .
             And
             seeing
             also
             that
             it
             hath
             been
             demonstrated
             ,
             that
             nothing
             can
             be
             moved
             but
             by
             a
             Body
             that
             is
             already
             moved
             and
             contiguous
             ;
             and
             that
             there
             is
             no
             other
             Body
             contiguous
             to
             the
             internal
             membrane
             of
             the
             nostrils
             ,
             but
             breath
             ,
             that
             is
             to
             say
             ,
             attracted
             aire
             ,
             and
             such
             little
             solid
             invisible
             Bodies
             (
             if
             there
             be
             any
             such
             )
             as
             are
             intermingled
             with
             the
             aire
             ;
             it
             follows
             necessarily
             ,
             that
             the
             cause
             of
             Smelling
             is
             either
             the
             motion
             of
             that
             pure
             aire
             or
             aethereal
             Substance
             ,
             or
             the
             motion
             of
             those
             small
             Bodies
             .
             But
             this
             motion
             is
             an
             effect
             proceding
             from
             the
             Object
             of
             Smell
             ,
             and
             therefore
             either
             the
             whole
             Object
             it self
             ,
             or
             its
             several
             parts
             must
             necessarily
             be
             moved
             .
             Now
             we
             know
             ,
             that
             Odorous
             Bodies
             make
             Odour
             though
             their
             whole
             bulk
             be
             not
             moved
             .
             Wherefore
             the
             cause
             of
             Odour
             is
             the
             motion
             of
             the
             invisible
             parts
             of
             the
             Odorous
             Body
             .
             And
             these
             invisible
             parts
             do
             either
             go
             out
             of
             the
             Object
             ,
             or
             else
             retaining
             their
             former
             situation
             with
             the
             rest
             of
             the
             parts
             ,
             are
             moved
             together
             with
             them
             ,
             that
             is
             to
             say
             ,
             they
             have
             simple
             and
             invisible
             motion
             .
             They
             that
             say
             there
             goes
             something
             out
             of
             the
             Odorous
             Body
             ,
             call
             it
             an
             Effluvium
             ;
             which
             Effluvium
             is
             either
             of
             the
             aethereal
             substance
             ,
             or
             of
             the
             small
             Bodies
             ,
             that
             are
             intermingled
             with
             it
             .
             But
             that
             all
             variety
             of
             Odours
             should
             proceed
             from
             the
             
             Effluviums
             of
             those
             small
             Bodies
             that
             are
             intermingled
             with
             the
             aethereal
             substance
             ,
             is
             altogether
             incredible
             ,
             for
             these
             considerations
             ;
             First
             ,
             that
             certain
             Unguents
             ,
             though
             very
             little
             in
             quantity
             ,
             do
             nevertheless
             send
             forth
             very
             strong
             Odours
             ,
             not
             onely
             to
             a
             great
             distance
             of
             place
             ,
             but
             also
             to
             a
             great
             continuance
             of
             time
             ,
             and
             are
             to
             be
             Smelt
             in
             every
             point
             both
             of
             that
             place
             and
             time
             ;
             so
             that
             the
             parts
             issued
             out
             are
             sufficient
             to
             fil
             ten
             thousand
             times
             more
             space
             ,
             then
             the
             whole
             Odorous
             Body
             is
             able
             to
             fill
             ;
             which
             is
             impossible
             .
             Secondly
             ,
             that
             whether
             that
             issuing
             out
             be
             with
             straight
             or
             with
             crooked
             motion
             ,
             if
             the
             same
             quantity
             should
             flow
             from
             any
             other
             Odorous
             Body
             with
             the
             same
             motion
             ,
             it
             would
             follow
             ,
             that
             all
             Odorous
             Bodies
             would
             yeild
             the
             same
             Smell
             .
             Thirdly
             ,
             that
             seeing
             those
             Effluviums
             have
             great
             Velocity
             of
             motion
             (
             as
             is
             manifest
             from
             this
             ,
             that
             noysome
             Odours
             proceeding
             from
             caverns
             are
             presently
             Smelt
             at
             a
             great
             distance
             )
             it
             would
             follow
             ,
             that
             by
             reason
             there
             is
             nothing
             to
             hinder
             the
             passage
             of
             those
             Effluviums
             to
             the
             Organ
             ,
             such
             motion
             alone
             were
             sufficient
             to
             cause
             Smelling
             .
             Which
             is
             not
             so
             ;
             for
             we
             cannot
             Smell
             at
             all
             unless
             we
             draw
             in
             our
             breath
             through
             our
             Nostrils
             .
             Smelling
             therefore
             is
             not
             caused
             by
             the
             Effluvium
             of
             Atomes
             ;
             nor
             ,
             for
             the
             same
             reason
             is
             it
             caused
             by
             the
             Effluvium
             of
             aethereal
             substance
             ;
             for
             so
             also
             we
             should
             Smell
             without
             the
             drawing
             in
             of
             our
             breath
             .
             Besides
             the
             aethereal
             substance
             being
             the
             same
             in
             all
             Odorous
             Bodies
             ,
             they
             would
             always
             affect
             the
             Organ
             in
             the
             same
             manner
             ,
             and
             consequently
             the
             Odours
             of
             all
             things
             would
             be
             like
             .
          
           
             It
             remains
             therefore
             ,
             that
             the
             cause
             of
             Smelling
             must
             consist
             in
             the
             Simple
             motion
             of
             the
             parts
             of
             Odorous
             Bodies
             ,
             without
             any
             efflux
             or
             diminution
             of
             their
             whole
             substance
             .
             And
             by
             this
             motion
             ,
             there
             is
             propagated
             to
             the
             Organ
             by
             the
             intermediate
             aire
             ,
             the
             like
             motion
             ,
             but
             not
             strong
             enough
             to
             excite
             Sense
             of
             it self
             without
             the
             attraction
             of
             aire
             by
             respiration
             .
             And
             this
             is
             a
             possible
             cause
             of
             Smelling
             .
          
           
             14
             The
             cause
             why
             Smelling
             is
             hindred
             by
             Cold
             ,
             and
             helped
             by
             Heat
             ,
             may
             be
             this
             ,
             that
             Heat
             (
             as
             hath
             been
             shewn
             in
             the
             21
             Chapter
             )
             generateth
             Simple
             motion
             ;
             and
             therefore
             also
             wheresoever
             it
             is
             already
             ,
             there
             it
             will
             encrease
             it
             ;
             and
             the
             cause
             of
             Smelling
             being
             encreased
             the
             Smell
             it self
             will
             also
             be
             encreased
             .
             As
             for
             the
             cause
             
             why
             the
             Wind
             blowing
             from
             the
             Object
             makes
             the
             Smell
             the
             stronger
             ,
             it
             is
             all
             one
             with
             that
             for
             which
             the
             attraction
             of
             aire
             in
             respiration
             doth
             the
             same
             .
             For
             he
             that
             draws
             in
             the
             aire
             next
             to
             him
             ,
             draws
             with
             it
             by
             succession
             that
             aire
             in
             which
             is
             the
             Object
             .
             Now
             this
             motion
             of
             the
             aire
             is
             Wind
             ,
             and
             when
             another
             Wind
             bloweth
             from
             the
             Object
             ,
             will
             be
             encreased
             by
             it
             .
          
           
             15
             That
             Bodies
             which
             cōtain
             the
             least
             quantity
             of
             air
             ,
             as
             Stones
             and
             Metals
             ,
             yeild
             less
             Smell
             then
             Plants
             and
             Living
             Creatures
             ,
             the
             cause
             may
             be
             ,
             that
             the
             motion
             which
             causeth
             Smelling
             ,
             is
             a
             motion
             of
             the
             fluid
             parts
             onely
             ;
             which
             parts
             ,
             if
             they
             have
             any
             motion
             from
             the
             hard
             parts
             in
             which
             they
             are
             contained
             ,
             they
             communicate
             the
             same
             to
             the
             open
             aire
             ,
             by
             which
             it
             is
             propagated
             to
             the
             Organ
             .
             Where
             therefore
             there
             are
             no
             fluid
             parts
             ,
             as
             in
             Metals
             ;
             or
             where
             the
             fluid
             parts
             receive
             no
             motion
             from
             the
             hard
             parts
             ,
             as
             in
             Stones
             ,
             which
             are
             made
             hard
             by
             accretion
             ,
             there
             can
             be
             no
             Smell
             .
             And
             therefore
             also
             the
             Water
             ,
             whose
             parts
             have
             little
             or
             no
             motion
             ,
             yeildeth
             no
             Smell
             .
             But
             if
             the
             same
             Water
             ,
             by
             Seeds
             ,
             and
             the
             heat
             of
             the
             Sunne
             ,
             be
             together
             with
             particles
             of
             Earth
             raised
             into
             a
             Plant
             ,
             and
             be
             afterwards
             pressed
             out
             again
             ,
             it
             will
             be
             Odorous
             ,
             as
             Wine
             from
             the
             Vine
             .
             And
             as
             Water
             passing
             through
             plants
             is
             by
             the
             motion
             of
             the
             parts
             of
             those
             plants
             made
             an
             Odorous
             liquour
             ;
             so
             also
             of
             aire
             passing
             through
             the
             same
             plants
             whilest
             they
             are
             growing
             ,
             are
             made
             Odorous
             aires
             .
             And
             thus
             also
             it
             is
             with
             the
             Juices
             and
             Spirits
             which
             are
             bred
             in
             Living
             Creatures
             .
          
           
             16
             That
             Odorous
             Bodies
             may
             be
             made
             more
             Odorous
             by
             Contrition
             ,
             proceeds
             from
             this
             ,
             that
             being
             broken
             into
             many
             parts
             which
             are
             all
             Odorous
             ,
             the
             aire
             which
             by
             respiration
             is
             drawn
             from
             the
             Object
             towards
             the
             Organ
             ,
             doth
             in
             its
             passage
             touch
             upon
             all
             those
             parts
             ,
             and
             receives
             their
             motion
             .
             Now
             the
             aire
             toucheth
             the
             superficies
             onely
             ;
             and
             a
             Body
             having
             less
             superficies
             whilest
             it
             is
             whole
             ,
             then
             all
             its
             parts
             together
             have
             after
             it
             is
             reduced
             to
             powder
             ,
             it
             follows
             that
             the
             same
             Odorous
             Body
             yeildeth
             less
             Smell
             whilest
             it
             is
             whole
             ,
             then
             it
             will
             do
             after
             it
             is
             broken
             into
             smaller
             parts
             .
             And
             thus
             much
             of
             Smels
             .
          
           
             17
             The
             Tast
             follows
             ;
             whose
             generation
             hath
             this
             difference
             from
             that
             of
             the
             Sight
             ,
             Hearing
             and
             Smelling
             ,
             that
             by
             these
             we
             have
             Sense
             of
             remote
             Objects
             ;
             whereas
             we
             Tast
             nothing
             but
             what
             is
             contiguous
             ,
             and
             doth
             immediately
             touch
             either
             the
             Tongue
             or
             
             Palate
             ,
             or
             both
             .
             From
             whence
             it
             is
             evident
             ,
             that
             the
             cuticles
             of
             the
             Tongue
             and
             Palate
             ,
             and
             the
             Nerves
             inserted
             into
             them
             are
             the
             first
             Organ
             of
             Tast
             ;
             and
             (
             because
             from
             the
             concussion
             of
             the
             parts
             of
             these
             ,
             there
             followeth
             necessarily
             a
             concussion
             of
             the
             
               Pia
               Mater
            
             )
             that
             the
             action
             communicated
             to
             these
             ,
             is
             propagated
             to
             the
             Brain
             ,
             and
             from
             thence
             to
             the
             farthest
             Organ
             ,
             namely
             ,
             the
             Heart
             ;
             in
             whose
             reaction
             consisteth
             the
             nature
             of
             Sense
             .
          
           
             Now
             that
             Savours
             (
             as
             well
             as
             Odours
             )
             doe
             not
             onely
             move
             the
             Brain
             ,
             but
             the
             Stomack
             also
             ,
             as
             is
             manifest
             by
             the
             loathings
             that
             are
             caused
             by
             them
             both
             ,
             they
             that
             consider
             the
             Organ
             of
             both
             these
             Senses
             will
             not
             wonder
             at
             all
             ;
             seeing
             the
             Tongue
             ,
             the
             Palate
             &
             the
             Nostrils
             have
             one
             and
             the
             same
             continued
             cuticle
             ,
             derived
             from
             the
             
               Dura
               Mater
            
             .
          
           
             And
             that
             Effluviums
             have
             nothing
             to
             doe
             in
             the
             Sense
             of
             Tasting
             ,
             is
             manifest
             from
             this
             ,
             that
             there
             is
             no
             Tast
             where
             the
             Organ
             and
             the
             Object
             are
             not
             contiguous
             .
          
           
             By
             what
             variety
             of
             motions
             the
             different
             kinds
             of
             Tasts
             (
             which
             are
             innumerable
             )
             may
             be
             distinguished
             ,
             I
             know
             not
             .
             I
             might
             (
             with
             others
             )
             derive
             them
             from
             the
             divers
             figures
             of
             those
             Atomes
             ,
             of
             which
             whatsoever
             may
             be
             Tasted
             consisteth
             ;
             or
             from
             the
             diverse
             motions
             which
             I
             might
             (
             by
             way
             of
             Supposition
             )
             attribute
             to
             those
             Atomes
             ;
             conjecturing
             (
             not
             without
             some
             likelyhood
             of
             truth
             )
             that
             such
             things
             as
             tast
             Sweet
             ,
             have
             their
             particles
             moved
             with
             slow
             circular
             motion
             ,
             and
             their
             figures
             Spherical
             ,
             which
             makes
             them
             smooth
             and
             pleasing
             to
             the
             Organ
             ;
             that
             Bitter
             things
             have
             circular
             motion
             ,
             but
             vehement
             ,
             and
             their
             figures
             full
             of
             Angles
             ,
             by
             which
             they
             trouble
             the
             Organ
             ;
             and
             that
             Sowre
             things
             have
             straight
             and
             reciprocal
             motion
             ,
             and
             their
             figures
             long
             and
             small
             ,
             so
             that
             they
             cut
             and
             wound
             the
             Organ
             .
             And
             in
             like
             manner
             I
             might
             assigne
             for
             the
             causes
             of
             other
             Tasts
             such
             several
             motions
             and
             figures
             of
             Atomes
             as
             might
             in
             probability
             seem
             to
             be
             the
             true
             causes
             .
             But
             this
             would
             be
             to
             revolt
             from
             Philosophy
             to
             Divination
             .
          
           
             18
             By
             the
             Touch
             ,
             we
             feel
             what
             Bodies
             are
             Cold
             or
             Hot
             ,
             though
             they
             be
             distant
             from
             us
             .
             Others
             ,
             as
             Hard
             ,
             Soft
             ,
             Rough
             and
             Smooth
             ,
             we
             cannot
             feel
             ,
             unless
             they
             be
             contiguous
             .
             The
             Organ
             of
             Touch
             ,
             is
             every
             one
             of
             those
             membranes
             ,
             which
             being
             continued
             from
             the
             
               Pia
               Mater
            
             ,
             are
             so
             diffused
             throughout
             the
             whole
             Body
             ,
             as
             
             that
             no
             part
             of
             it
             can
             be
             pressed
             ,
             but
             the
             
               Pia
               Mater
            
             is
             pressed
             together
             with
             it
             .
             Whatsoever
             therefore
             presseth
             it
             ,
             is
             felt
             as
             Hard
             or
             Soft
             ,
             that
             is
             to
             say
             ,
             as
             more
             or
             less
             Hard.
             And
             as
             for
             the
             Sense
             of
             Rough
             ,
             it
             is
             nothing
             else
             but
             innumerable
             perceptions
             of
             Hard
             and
             Hard
             succeeding
             one
             another
             by
             short
             intervals
             both
             of
             time
             and
             place
             .
             For
             we
             take
             notice
             of
             Rough
             and
             Smooth
             ,
             as
             also
             of
             Magnitude
             and
             Figure
             ,
             not
             onely
             by
             the
             Touch
             ,
             but
             also
             by
             Memory
             .
             For
             though
             some
             things
             are
             touched
             in
             one
             Point
             ;
             yet
             Rough
             and
             Smooth
             ,
             like
             Quantity
             and
             Figure
             ,
             are
             not
             perceived
             but
             by
             the
             Flux
             of
             a
             Point
             ,
             that
             is
             to
             say
             ,
             we
             have
             no
             Sense
             of
             them
             without
             Time
             ;
             and
             we
             can
             have
             no
             Sense
             of
             Time
             ,
             without
             Memory
             .
          
        
         
           
           
             
               CHAP.
               XXX
            
             .
             Of
             Gravity
             .
          
           
             
               
                 1
                 A
                 Thick
                 Body
                 doth
                 not
                 contain
                 more
                 Matter
                 (
                 unless
                 also
                 more
                 Place
                 )
                 then
                 a
                 Thinne
                 .
              
               
                 2
                 That
                 the
                 Descent
                 of
                 Heavy
                 Bodies
                 proceeds
                 not
                 from
                 their
                 own
                 Appetite
                 ;
                 but
                 from
                 some
                 Power
                 of
                 the
                 Earth
                 .
              
               
                 3
                 The
                 difference
                 of
                 Gravities
                 proceedeth
                 from
                 the
                 difference
                 of
                 the
                 Impetus
                 with
                 which
                 the
                 Elements
                 whereof
                 Heavy
                 Bodies
                 are
                 made
                 do
                 fall
                 vpon
                 the
                 Earth
                 .
              
               
                 4
                 The
                 cause
                 of
                 the
                 Descent
                 of
                 Heavy
                 Bodies
                 .
              
               
                 5
                 In
                 what
                 proportion
                 the
                 Descent
                 of
                 Heavy
                 Bodies
                 is
                 accelerated
                 .
              
               
                 6
                 Why
                 those
                 that
                 Dive
                 do
                 not
                 when
                 they
                 are
                 under
                 Water
                 ,
                 feel
                 the
                 waight
                 of
                 the
                 Water
                 above
                 them
                 .
              
               
                 7
                 The
                 Waight
                 of
                 a
                 Body
                 that
                 floateth
                 ,
                 is
                 equal
                 to
                 the
                 Waight
                 of
                 so
                 much
                 Water
                 as
                 would
                 fill
                 the
                 space
                 which
                 the
                 immersed
                 part
                 of
                 the
                 Body
                 takes
                 up
                 within
                 the
                 Water
                 .
              
               
                 8
                 If
                 a
                 Body
                 be
                 Lighter
                 then
                 Water
                 ,
                 then
                 how
                 big
                 soever
                 that
                 Body
                 be
                 ,
                 it
                 will
                 float
                 upon
                 any
                 quantity
                 of
                 Water
                 ,
                 how
                 little
                 soever
                 .
              
               
                 9
                 How
                 Water
                 may
                 be
                 lifted
                 up
                 and
                 forced
                 out
                 of
                 a
                 Vessel
                 by
                 Air.
                 
              
               
                 10
                 Why
                 a
                 Bladder
                 is
                 Heavier
                 when
                 blown
                 full
                 of
                 aire
                 ,
                 then
                 when
                 it
                 is
                 empty
                 .
              
               
                 11
                 The
                 cause
                 of
                 the
                 ejection
                 upwards
                 of
                 Heavy
                 Bodies
                 from
                 a
                 Wind-Gun
                 .
              
               
                 12
                 The
                 cause
                 of
                 the
                 ascent
                 of
                 Water
                 in
                 a
                 Weather-glass
                 .
              
               
                 13
                 The
                 cause
                 of
                 motion
                 upwards
                 in
                 Living
                 Creatures
                 .
              
               
                 14
                 That
                 there
                 is
                 in
                 Nature
                 a
                 kind
                 of
                 Body
                 Heavier
                 then
                 Aire
                 ,
                 which
                 nevertheless
                 is
                 not
                 by
                 Sense
                 distinguishable
                 from
                 it
                 .
              
               
                 15
                 Of
                 the
                 cause
                 of
                 Magnetical
                 vertue
                 .
              
            
          
           
             1
             IN
             the
             21
             Chapter
             I
             have
             defined
             Thick
             and
             Thinne
             (
             as
             that
             place
             required
             )
             so
             ,
             as
             that
             by
             Thick
             was
             signified
             a
             more
             Resisting
             Body
             ,
             and
             by
             Thinne
             a
             Body
             less
             Resisting
             ;
             following
             the
             custome
             of
             those
             that
             have
             before
             me
             discoursed
             of
             Refraction
             .
             Now
             if
             we
             consider
             the
             true
             and
             vulgar
             signification
             of
             those
             words
             ,
             we
             shall
             find
             them
             to
             be
             Names
             Collective
             ,
             that
             is
             to
             say
             ,
             Names
             of
             Multitude
             ;
             as
             Thick
             to
             be
             that
             which
             takes
             up
             more
             parts
             of
             a
             space
             given
             ,
             &
             Thinne
             that
             which
             contains
             fewer
             parts
             of
             the
             same
             magnitude
             ,
             in
             the
             same
             space
             ,
             or
             in
             a
             space
             equal
             to
             it
             .
             Thick
             therefore
             is
             the
             same
             with
             Frequent
             ,
             as
             a
             Thick
             Troop
             ;
             And
             Thinne
             the
             same
             with
             Unfrequent
             ,
             as
             a
             Thinne
             Rank
             ,
             Thinne
             of
             Houses
             ;
             not
             that
             there
             is
             more
             matter
             in
             one
             place
             then
             in
             another
             equal
             
             place
             ,
             but
             a
             greater
             quantity
             of
             some
             named
             Body
             .
             For
             there
             is
             not
             less
             matter
             or
             Body
             indefinitely
             taken
             ,
             in
             a
             Desert
             ,
             then
             there
             is
             in
             a
             City
             ;
             but
             fewer
             Houses
             ,
             or
             fewer
             Men.
             Nor
             is
             there
             in
             a
             Thick
             Rank
             a
             greater
             quantity
             of
             Body
             ,
             but
             a
             greater
             number
             of
             Souldiers
             ,
             then
             in
             a
             Thinne
             .
             Wherefore
             the
             multitude
             &
             paucity
             of
             the
             parts
             contained
             within
             the
             same
             space
             ,
             do
             constitute
             Density
             and
             Rarity
             ,
             whether
             those
             parts
             be
             separated
             by
             Vacuum
             ,
             or
             by
             Aire
             .
             But
             the
             consideration
             of
             this
             is
             not
             of
             any
             great
             moment
             in
             Philosophy
             ;
             and
             therefore
             I
             let
             it
             alone
             ,
             and
             pass
             on
             to
             the
             search
             of
             the
             causes
             of
             Gravity
             .
          
           
             2
             Now
             we
             call
             those
             Bodies
             Heavy
             ,
             which
             (
             unless
             they
             be
             hindred
             by
             some
             force
             )
             are
             carried
             towards
             the
             center
             of
             the
             Earth
             ,
             and
             that
             by
             their
             own
             accord
             ,
             for
             ought
             we
             can
             by
             Sense
             perceive
             to
             the
             contrary
             .
             Some
             Philosophers
             therefore
             have
             been
             of
             opinion
             ,
             that
             the
             Descent
             of
             Heavy
             Bodies
             proceeded
             from
             some
             internal
             Appetite
             ,
             by
             which
             when
             they
             were
             cast
             upwards
             ,
             they
             descended
             again
             ,
             as
             moved
             by
             themselves
             ,
             to
             such
             place
             as
             was
             agreeable
             to
             their
             nature
             .
             Others
             thought
             they
             were
             attracted
             by
             the
             Earth
             .
             To
             the
             former
             I
             cannot
             assent
             ,
             because
             I
             think
             I
             have
             already
             clearly
             enough
             demonstrated
             ,
             that
             there
             can
             be
             no
             beginning
             of
             motion
             ,
             but
             from
             an
             external
             &
             moved
             Body
             ;
             and
             consequently
             ,
             that
             whatsoever
             hath
             motion
             or
             endeavour
             towards
             any
             place
             ,
             will
             alwayes
             move
             or
             endeavour
             towards
             that
             same
             place
             ,
             unless
             it
             be
             hindered
             by
             the
             reaction
             of
             some
             external
             Body
             .
             Heavy
             Bodies
             therefore
             being
             once
             cast
             upwards
             ,
             cannot
             be
             cast
             down
             again
             but
             by
             external
             motion
             .
             Besides
             ,
             seeing
             inanimate
             Bodies
             have
             no
             Appetite
             at
             all
             ,
             it
             is
             ridiculous
             to
             think
             that
             by
             their
             own
             innate
             Appetite
             they
             should
             to
             preserve
             themselves
             (
             not
             understanding
             what
             preserves
             them
             )
             forsake
             the
             place
             they
             are
             in
             ,
             and
             transferre
             themselves
             to
             another
             place
             ;
             whereas
             Man
             (
             who
             hath
             both
             Appetite
             and
             understanding
             )
             cannot
             for
             the
             preservation
             of
             his
             own
             life
             ,
             raise
             himselfe
             by
             leaping
             above
             three
             or
             four
             feet
             from
             the
             ground
             .
             Lastly
             ,
             to
             attribute
             to
             created
             Bodies
             the
             power
             to
             move
             themselves
             ,
             what
             is
             it
             else
             then
             to
             say
             that
             there
             be
             creatures
             which
             have
             no
             dependance
             upon
             the
             Creator
             ?
             To
             the
             later
             ,
             who
             attribute
             the
             Descent
             of
             Heavy
             Bodies
             to
             the
             attraction
             of
             the
             Earth
             ,
             I
             assent
             .
             But
             by
             what
             motion
             this
             is
             done
             ,
             hath
             not
             as
             yet
             been
             explained
             by
             any
             man.
             I
             shall
             therefore
             in
             this
             place
             say
             somewhat
             
             of
             the
             manner
             ,
             and
             of
             the
             way
             by
             which
             the
             Earth
             by
             its
             action
             attracteth
             Heavy
             Bodies
             .
          
           
             3
             That
             by
             the
             supposition
             of
             simple
             motion
             in
             the
             Sunne
             ,
             homogeneous
             Bodies
             are
             congregated
             ,
             and
             heterogeneous
             dissipated
             ,
             has
             already
             been
             demonstrated
             in
             the
             5th
             Article
             of
             the
             21
             Chapter
             .
             I
             have
             also
             supposed
             ,
             that
             there
             are
             intermingled
             with
             the
             pure
             Air
             ,
             certain
             little
             Bodies
             ,
             or
             (
             as
             others
             call
             them
             )
             Atomes
             ,
             which
             by
             reason
             of
             their
             extreme
             smalness
             are
             invisible
             ,
             and
             differing
             from
             one
             another
             in
             
               Consistence
               ,
               Figure
               ,
               Motion
               &
               Magnitude
            
             ;
             from
             whence
             it
             comes
             to
             pass
             ,
             that
             some
             of
             them
             are
             congregated
             to
             the
             Earth
             ,
             others
             to
             other
             Planets
             ,
             and
             others
             are
             carried
             up
             and
             down
             in
             the
             spaces
             between
             .
             And
             seeing
             those
             which
             are
             carried
             to
             the
             Earth
             ,
             differ
             from
             one
             another
             in
             Figure
             ,
             Motion
             and
             Magnitude
             ,
             they
             will
             fall
             upon
             the
             Earth
             ,
             some
             with
             greater
             ,
             others
             with
             less
             Impetus
             .
             And
             seeing
             also
             that
             we
             compute
             the
             several
             degrees
             of
             Gravity
             no
             otherwise
             then
             by
             this
             their
             falling
             upon
             the
             Earth
             with
             greater
             or
             less
             Impetus
             ;
             it
             follows
             ,
             that
             we
             conclude
             those
             to
             be
             the
             more
             Heavy
             that
             have
             the
             greater
             Impetus
             ,
             and
             those
             to
             be
             less
             Heavy
             that
             have
             the
             less
             Impetus
             .
             Our
             enquiry
             therefore
             must
             be
             ,
             by
             what
             means
             it
             may
             come
             to
             pass
             ,
             that
             of
             Bodies
             which
             descend
             from
             above
             to
             the
             Earth
             ,
             some
             are
             carried
             with
             greater
             ,
             others
             with
             less
             Impetus
             ;
             that
             is
             to
             say
             ,
             some
             are
             more
             Heavy
             then
             others
             .
             We
             must
             also
             enquire
             ,
             by
             what
             means
             such
             Bodies
             as
             settle
             upon
             the
             Earth
             ,
             may
             by
             the
             Earth
             it self
             be
             forced
             to
             ascend
             .
          
           
             4
             Let
             the
             Circle
             made
             upon
             the
             center
             C
             (
             in
             the
             2d
             figure
             )
             be
             a
             great
             Circle
             in
             the
             Superficies
             of
             the
             Earth
             ,
             passing
             through
             the
             points
             A
             and
             B.
             Also
             let
             any
             Heavy
             Body
             ,
             as
             the
             stone
             A
             D
             be
             placed
             any
             where
             in
             the
             plain
             of
             the
             Aequator
             ;
             and
             let
             it
             be
             conceived
             to
             be
             cast
             up
             from
             A
             D
             perpendicularly
             ,
             or
             to
             be
             carried
             in
             any
             other
             line
             to
             E
             ,
             and
             supposed
             to
             rest
             there
             .
             Therefore
             how
             much
             space
             soever
             the
             stone
             took
             up
             in
             A
             D
             ,
             so
             much
             space
             it
             takes
             up
             now
             in
             E.
             And
             because
             all
             place
             is
             supposed
             to
             be
             full
             ,
             the
             space
             A
             D
             will
             be
             filled
             by
             the
             aire
             which
             flows
             into
             it
             first
             from
             the
             neerest
             places
             of
             the
             Earth
             ,
             and
             afterwards
             successively
             from
             more
             remote
             places
             .
             Upon
             the
             center
             C
             let
             a
             Circle
             be
             understood
             to
             be
             drawn
             through
             E
             ;
             and
             let
             the
             plain
             space
             which
             is
             between
             the
             Superficies
             of
             the
             Earth
             and
             that
             Circle
             ,
             be
             divided
             into
             plain
             Orbs
             equal
             and
             concentrique
             ;
             of
             which
             ,
             let
             that
             be
             the
             first
             which
             is
             
             contained
             by
             the
             two
             perimeters
             that
             pass
             through
             A
             &
             D.
             Whilst
             therefore
             the
             aire
             which
             is
             in
             the
             first
             Orbe
             ,
             filleth
             the
             place
             A
             D
             ,
             the
             Orbe
             it self
             is
             made
             so
             much
             less
             ,
             and
             consequently
             its
             latitude
             is
             less
             then
             the
             straight
             line
             A
             D.
             Wherefore
             there
             will
             necessarily
             descend
             so
             much
             aire
             from
             the
             Orbe
             next
             abvoe
             .
             In
             like
             manner
             ,
             for
             the
             same
             cause
             ,
             there
             will
             also
             be
             a
             descent
             of
             aire
             from
             the
             Orbe
             next
             above
             that
             ;
             and
             so
             by
             Succession
             from
             the
             Orbe
             in
             which
             the
             Stone
             is
             at
             rest
             in
             E.
             Either
             therefore
             the
             Stone
             it self
             ,
             or
             so
             much
             aire
             will
             descend
             .
             And
             seeing
             aire
             is
             by
             the
             diurnal
             revolution
             of
             the
             Earth
             more
             easily
             thrust
             away
             ,
             then
             the
             Stone
             ,
             the
             aire
             which
             is
             in
             the
             Orbe
             that
             contains
             the
             Stone
             will
             be
             forced
             further
             upwards
             then
             the
             Stone
             .
             But
             this
             ,
             without
             the
             admission
             of
             Vacuum
             ,
             cannot
             be
             ,
             unless
             so
             much
             aire
             descend
             to
             E
             from
             the
             place
             next
             above
             ;
             which
             being
             done
             ,
             the
             Stone
             will
             be
             thrust
             downwards
             .
             By
             this
             means
             therefore
             the
             Stone
             now
             receives
             the
             beginning
             of
             its
             Descent
             ,
             that
             is
             to
             say
             ,
             of
             its
             Gravity
             .
             Furthermore
             ,
             whatsoever
             is
             once
             moved
             ,
             will
             be
             moved
             continually
             (
             as
             hath
             been
             shewn
             in
             the
             19th
             Article
             of
             the
             8th
             Chapter
             )
             in
             the
             same
             way
             ,
             and
             with
             the
             same
             celerity
             ,
             except
             it
             be
             retarded
             or
             accelerated
             by
             some
             external
             Movent
             .
             Now
             the
             aire
             (
             which
             is
             the
             onely
             Body
             that
             is
             interposed
             between
             the
             Earth
             A
             and
             the
             stone
             above
             it
             E
             )
             will
             have
             the
             same
             action
             in
             every
             point
             of
             the
             straight
             line
             E
             A
             ,
             which
             it
             hath
             in
             E.
             But
             it
             depressed
             the
             stone
             in
             E
             ;
             and
             therefore
             also
             it
             will
             depress
             it
             equally
             in
             every
             point
             of
             the
             straight
             line
             E
             A.
             Wherefore
             the
             stone
             will
             descend
             from
             E
             to
             A
             with
             accelerated
             motion
             .
             The
             possible
             cause
             therefore
             of
             the
             Descent
             of
             Heavy
             Bodies
             under
             the
             Aequator
             ,
             is
             the
             Diurnal
             motion
             of
             the
             Earth
             .
             And
             the
             same
             demonstration
             will
             serve
             ,
             if
             the
             stone
             be
             placed
             in
             the
             plain
             of
             any
             other
             Circle
             parallel
             to
             the
             Aequator
             .
             But
             because
             this
             motion
             hath
             by
             reason
             of
             its
             greater
             slowness
             ,
             less
             force
             to
             thrust
             off
             the
             aire
             in
             the
             parallel
             Circles
             then
             in
             the
             Aequator
             ,
             and
             no
             force
             at
             all
             at
             the
             Poles
             ,
             it
             may
             well
             be
             thought
             (
             for
             it
             is
             a
             certain
             consequent
             )
             that
             Heavy
             Bodies
             descend
             with
             less
             and
             less
             velocity
             ,
             as
             they
             are
             more
             &
             more
             remote
             from
             the
             Aequator
             ;
             &
             that
             at
             the
             Poles
             themselves
             they
             wil
             either
             not
             descend
             at
             all
             ,
             or
             not
             descend
             by
             the
             Axis
             ;
             which
             whether
             it
             be
             true
             or
             false
             ,
             Experience
             ▪
             must
             determine
             .
             But
             it
             is
             hard
             to
             make
             the
             experiment
             ,
             both
             because
             the
             times
             of
             their
             descents
             
             cannot
             be
             easily
             measured
             with
             sufficient
             exactness
             ,
             and
             also
             because
             the
             places
             neer
             the
             Poles
             are
             inaccessible
             .
             Nevertheless
             ,
             this
             we
             know
             ,
             that
             by
             how
             much
             the
             neerer
             we
             come
             to
             the
             Poles
             ,
             by
             so
             much
             the
             greater
             are
             the
             Flakes
             of
             the
             Snow
             that
             falls
             ;
             and
             by
             how
             much
             the
             more
             swiftly
             such
             Bodies
             descend
             as
             are
             fluid
             and
             dissipable
             ,
             by
             so
             much
             the
             smaller
             are
             the
             particles
             into
             which
             they
             are
             dissipated
             .
          
           
             5
             Supposing
             therefore
             this
             to
             be
             the
             cause
             of
             the
             Descent
             of
             Heavy
             Bodies
             ;
             it
             will
             follow
             ,
             that
             their
             motion
             will
             be
             accelerated
             in
             such
             manner
             ,
             as
             that
             the
             spaces
             which
             are
             transmitted
             by
             them
             in
             the
             several
             times
             ,
             will
             have
             to
             one
             another
             the
             same
             proportion
             which
             the
             odd
             numbers
             have
             in
             succession
             from
             Unity
             .
             For
             if
             the
             straight
             line
             EA
             be
             divided
             into
             any
             number
             of
             equal
             parts
             ,
             the
             Heavy
             Body
             descending
             ,
             will
             (
             by
             reason
             of
             the
             perpetual
             action
             of
             the
             Diurnal
             motion
             )
             receive
             from
             the
             aire
             in
             every
             one
             of
             those
             times
             ,
             in
             every
             several
             point
             of
             the
             streight
             line
             EA
             ,
             a
             several
             new
             and
             equal
             impulsion
             ;
             and
             therefore
             also
             in
             every
             one
             of
             those
             times
             it
             will
             acquire
             a
             several
             and
             equal
             degree
             of
             celerity
             .
             And
             from
             hence
             it
             follows
             ,
             by
             that
             which
             Galilaeus
             hath
             in
             his
             Dialogues
             of
             Motion
             demonstrated
             ,
             that
             Heavy
             Bodies
             descend
             in
             the
             several
             times
             with
             such
             differences
             of
             transmitted
             spaces
             as
             are
             equal
             to
             the
             differences
             of
             the
             square
             numbers
             that
             succeed
             one
             another
             from
             Unity
             ;
             which
             square
             numbers
             being
             1
             ,
             4
             ,
             9
             ,
             16
             ,
             &c.
             their
             differences
             are
             3
             ,
             5
             ,
             7
             ,
             that
             is
             to
             say
             ,
             the
             odd
             numbers
             which
             succeed
             another
             from
             Unity
             .
             Against
             this
             cause
             of
             Gravity
             which
             I
             have
             given
             ,
             it
             will
             perhaps
             be
             objected
             ,
             that
             if
             a
             Heavy
             Body
             be
             placed
             in
             the
             bottom
             of
             some
             hollow
             Cylinder
             of
             Iron
             or
             Adamant
             ,
             and
             the
             bottom
             be
             turned
             upwards
             ,
             the
             Body
             will
             descend
             ,
             though
             the
             aire
             above
             cannot
             depress
             it
             ,
             much
             less
             accelerate
             its
             motion
             .
             But
             it
             is
             to
             be
             considered
             ,
             that
             there
             can
             be
             no
             Cylinder
             or
             Cavern
             ,
             but
             such
             as
             is
             supported
             by
             the
             Earth
             ,
             and
             being
             so
             supported
             ,
             is
             together
             with
             the
             Earth
             carried
             about
             by
             its
             diurnal
             Motion
             .
             For
             by
             this
             means
             the
             bottom
             of
             the
             Cylinder
             will
             be
             as
             the
             Superficies
             of
             the
             Earth
             ;
             and
             by
             thrusting
             off
             the
             next
             and
             lowest
             aire
             ,
             will
             make
             the
             uppermost
             aire
             depress
             the
             Heavy
             Body
             which
             is
             at
             the
             top
             of
             the
             Cylinder
             ,
             in
             such
             manner
             as
             is
             above
             explicated
             .
          
           
             6
             The
             Gravity
             of
             Water
             being
             so
             great
             as
             by
             experience
             wee
             find
             it
             is
             ,
             the
             reason
             is
             demanded
             by
             many
             ,
             why
             those
             that
             Dive
             ,
             
             how
             deep
             soever
             they
             go
             under
             water
             ,
             do
             not
             at
             all
             feel
             the
             weight
             of
             the
             water
             which
             lyes
             upon
             them
             .
             And
             the
             cause
             seems
             to
             be
             this
             ,
             that
             all
             Bodies
             by
             how
             much
             the
             Heavier
             they
             are
             ,
             by
             so
             much
             the
             greater
             is
             the
             endeavour
             by
             which
             they
             tend
             downwards
             .
             But
             the
             Body
             of
             a
             Man
             is
             Heavier
             then
             so
             much
             water
             as
             is
             equal
             to
             it
             in
             magnitude
             ,
             and
             therefore
             the
             endeavour
             downwards
             of
             a
             Mans
             Body
             is
             greater
             then
             that
             of
             water
             .
             And
             seeing
             all
             endeavour
             is
             motion
             ,
             the
             Body
             also
             of
             a
             Man
             will
             be
             carried
             towards
             the
             bottom
             with
             greater
             Velocity
             then
             so
             much
             water
             .
             Wherefore
             there
             is
             greater
             Reaction
             from
             the
             bottom
             ;
             and
             the
             Endeavour
             upwards
             is
             equal
             to
             the
             endeavour
             downwards
             ,
             whether
             the
             water
             be
             pressed
             by
             water
             ,
             or
             by
             another
             Body
             which
             is
             Heavier
             then
             water
             .
             And
             therefore
             by
             these
             two
             opposite
             equal
             endeavours
             ,
             the
             endeavour
             both
             ways
             in
             the
             water
             is
             taken
             away
             ;
             and
             consequently
             ,
             those
             that
             Dive
             are
             not
             at
             all
             pressed
             by
             it
             .
          
           
             Coroll
             .
             From
             hence
             also
             it
             is
             manifest
             ,
             that
             water
             in
             water
             hath
             no
             Waight
             at
             all
             ,
             because
             all
             the
             parts
             of
             water
             (
             both
             the
             parts
             above
             ,
             and
             the
             parts
             that
             are
             directly
             under
             )
             tend
             towards
             the
             bottom
             with
             equal
             endeavour
             and
             in
             the
             same
             straight
             lines
             .
          
           
             7
             If
             a
             Body
             float
             upon
             the
             water
             ,
             the
             waight
             of
             that
             Body
             is
             equal
             to
             the
             waight
             of
             so
             much
             water
             as
             would
             fill
             the
             place
             which
             the
             immersed
             part
             of
             the
             Body
             takes
             up
             within
             the
             water
             .
          
           
             Let
             EF
             (
             in
             the
             3d
             figure
             )
             be
             a
             Body
             floating
             in
             the
             water
             ABCD
             ;
             and
             let
             the
             part
             E
             be
             above
             ,
             and
             the
             other
             part
             F
             under
             the
             water
             .
             I
             say
             the
             waight
             of
             the
             whole
             Body
             EF
             is
             equal
             to
             the
             waight
             of
             so
             much
             water
             as
             the
             Space
             F
             will
             receive
             .
             For
             seeing
             the
             waight
             of
             the
             Body
             EF
             forceth
             the
             water
             out
             of
             the
             space
             F
             ,
             and
             placeth
             it
             upon
             the
             Superficies
             AB
             ,
             where
             it
             presseth
             downwards
             ;
             it
             follows
             ,
             that
             from
             the
             resistance
             of
             the
             bottom
             there
             will
             also
             be
             an
             endeavour
             upwards
             .
             And
             seeing
             again
             that
             by
             this
             endeavour
             of
             the
             water
             upwards
             ,
             the
             Body
             EF
             is
             lifted
             up
             ;
             it
             follows
             ,
             that
             if
             the
             endeavour
             of
             the
             Body
             downwards
             be
             not
             equal
             to
             the
             endeavour
             of
             the
             water
             upwards
             ,
             either
             the
             whole
             Body
             EF
             will
             (
             by
             reason
             of
             that
             inequality
             of
             their
             endeavours
             or
             moments
             )
             be
             raised
             out
             of
             the
             water
             ,
             or
             else
             it
             will
             descend
             to
             the
             bottom
             .
             But
             it
             is
             supposed
             to
             stand
             so
             ,
             as
             neither
             to
             ascend
             nor
             descend
             .
             Wherefore
             there
             is
             an
             Aequilibrium
             between
             the
             two
             endeavours
             ;
             that
             is
             to
             say
             ,
             the
             waight
             of
             the
             Body
             EF
             is
             equal
             to
             the
             
             waight
             of
             so
             much
             water
             as
             the
             Space
             F
             will
             receive
             ;
             Which
             was
             to
             be
             proved
             .
          
           
             8
             From
             hence
             it
             follows
             ,
             that
             any
             Body
             of
             how
             great
             magnitude
             soever
             ,
             provided
             it
             consist
             of
             matter
             less
             Heavy
             then
             water
             ,
             may
             nevertheless
             float
             upon
             any
             quantity
             of
             water
             how
             little
             soever
             .
          
           
             Let
             ABCD
             (
             in
             the
             4th
             figure
             )
             be
             a
             vessel
             ;
             and
             in
             it
             let
             EFGH
             be
             a
             Body
             consisting
             of
             matter
             which
             is
             less
             Heavy
             then
             water
             ;
             and
             let
             the
             space
             AGCF
             be
             filled
             with
             water
             .
             I
             say
             the
             Body
             EFGH
             will
             not
             sink
             to
             the
             bottom
             DC
             .
             For
             seeing
             the
             matter
             of
             the
             Body
             EFGH
             is
             less
             Heavy
             then
             Water
             ,
             if
             the
             whole
             space
             without
             ABCD
             were
             full
             of
             Water
             ,
             yet
             some
             part
             of
             the
             Body
             EFGH
             ,
             as
             EFIK
             would
             be
             above
             the
             Water
             ;
             and
             the
             waight
             of
             so
             much
             water
             as
             would
             fill
             the
             space
             IGHK
             would
             be
             equal
             to
             the
             waight
             of
             the
             whole
             Body
             EFGH
             ;
             and
             consequently
             GH
             would
             not
             touch
             the
             bottom
             DC
             .
             As
             for
             the
             sides
             of
             the
             vessel
             ,
             it
             is
             no
             matter
             whether
             they
             be
             hard
             ,
             or
             fluid
             ;
             for
             they
             serve
             onely
             to
             terminate
             the
             Water
             ;
             which
             may
             be
             done
             as
             well
             by
             water
             as
             by
             any
             other
             matter
             how
             hard
             soever
             ;
             and
             the
             water
             without
             the
             Vessel
             is
             terminated
             somewhere
             ,
             so
             as
             that
             it
             can
             spread
             no
             further
             .
             The
             part
             therefore
             EFIG
             will
             be
             extant
             above
             the
             water
             AGCF
             which
             is
             contained
             in
             the
             vessel
             .
             Wherefore
             the
             Body
             EFGH
             will
             also
             float
             upon
             the
             water
             AGCF
             ,
             how
             little
             soever
             that
             water
             be
             ;
             which
             was
             to
             be
             demonstrated
             .
          
           
             9
             In
             the
             4th
             Article
             of
             the
             26th
             Chapter
             ,
             there
             is
             brought
             for
             the
             proving
             of
             Vacuum
             ,
             the
             experiment
             of
             water
             enclosed
             in
             a
             vessel
             ;
             which
             water
             ,
             the
             Orifice
             above
             being
             opened
             ,
             is
             ejected
             upwards
             by
             the
             impulsion
             of
             the
             aire
             .
             It
             is
             therefore
             demanded
             (
             seeing
             water
             is
             Heavier
             then
             aire
             )
             how
             that
             can
             be
             done
             .
             Let
             the
             2d
             .
             figure
             of
             the
             same
             26th
             Chap.
             be
             considered
             ,
             where
             the
             water
             is
             with
             great
             force
             injected
             by
             a
             Syringe
             into
             the
             space
             FGB
             .
             In
             that
             injection
             ,
             the
             aire
             (
             but
             pure
             aire
             )
             goeth
             with
             the
             same
             force
             out
             of
             the
             vessel
             through
             the
             injected
             water
             .
             But
             as
             for
             those
             small
             Bodies
             which
             formerly
             I
             supposed
             to
             be
             intermingled
             with
             aire
             ,
             &
             to
             be
             moved
             with
             simple
             motion
             ,
             they
             can
             not
             together
             with
             the
             oure
             air
             penetrate
             the
             water
             ;
             but
             remayning
             behind
             are
             necessarily
             thrust
             together
             into
             a
             narrower
             place
             ,
             namely
             into
             the
             space
             which
             is
             above
             the
             water
             FG.
             The
             motions
             therefore
             of
             those
             small
             Bodies
             will
             be
             less
             and
             less
             free
             ,
             by
             how
             much
             the
             quantity
             of
             the
             injected
             
             water
             is
             greater
             and
             greater
             ;
             so
             that
             by
             their
             motions
             falling
             upon
             one
             another
             the
             same
             small
             Bodies
             will
             mutually
             compress
             each
             other
             ,
             and
             have
             a
             perpetual
             endeavour
             of
             regayning
             their
             liberty
             ,
             and
             of
             depressing
             the
             water
             that
             hinders
             them
             .
             Wherefore
             ,
             as
             soone
             as
             the
             orifice
             above
             is
             opened
             ,
             the
             water
             which
             is
             next
             it
             will
             have
             an
             endeavour
             to
             ascend
             ;
             and
             will
             therefore
             necessarily
             go
             out
             .
             But
             it
             cannot
             go
             out
             ,
             unless
             at
             the
             same
             time
             there
             enter
             in
             as
             much
             aire
             ;
             and
             therefore
             both
             the
             water
             will
             go
             out
             ,
             and
             the
             aire
             enter
             in
             ,
             till
             those
             small
             Bodies
             which
             were
             left
             within
             the
             vessel
             have
             recovered
             their
             former
             liberty
             of
             motion
             ;
             that
             is
             to
             say
             ,
             till
             the
             vessel
             be
             again
             filled
             with
             aire
             ,
             and
             no
             water
             be
             left
             of
             sufficient
             height
             to
             stop
             the
             passage
             at
             B.
             Wherefore
             I
             have
             shewn
             a
             possible
             cause
             of
             this
             Phaenomenon
             ;
             namely
             ,
             the
             same
             with
             that
             of
             Thunder
             .
             For
             as
             in
             the
             generation
             of
             Thunder
             ,
             the
             small
             Bodies
             enclosed
             within
             the
             Clouds
             by
             being
             too
             closely
             pent
             together
             ,
             do
             by
             their
             motion
             break
             the
             Clouds
             ,
             and
             restore
             themselves
             to
             their
             natural
             liberty
             ;
             so
             here
             also
             the
             small
             Bodies
             enclosed
             within
             the
             space
             which
             is
             above
             the
             straight
             line
             FG
             ,
             do
             by
             their
             own
             motion
             expel
             the
             water
             as
             soon
             as
             the
             passage
             is
             opened
             above
             .
             And
             if
             the
             passage
             be
             kept
             stopped
             ,
             and
             these
             small
             Bodies
             be
             more
             vehemently
             compressed
             by
             the
             perpetual
             forcing
             in
             of
             more
             water
             ,
             they
             will
             at
             last
             break
             the
             vessel
             it self
             with
             great
             noise
             .
          
           
             10
             If
             Aire
             be
             blown
             into
             a
             hollow
             Cylinder
             ,
             or
             into
             a
             Bladder
             ,
             it
             will
             encrease
             the
             waight
             of
             either
             of
             them
             a
             little
             ,
             as
             many
             have
             found
             by
             experience
             ,
             who
             with
             great
             accurateness
             have
             tried
             the
             same
             .
             And
             it
             is
             no
             wonder
             ,
             seeing
             (
             as
             I
             have
             supposed
             )
             there
             are
             intermingled
             with
             the
             common
             aire
             a
             great
             number
             of
             small
             hard
             Bodies
             which
             are
             Heavier
             then
             the
             pure
             aire
             .
             For
             the
             aethereal
             substance
             ,
             being
             on
             all
             sides
             equally
             agitated
             by
             the
             motion
             of
             the
             Sunne
             ,
             hath
             an
             equal
             endeavour
             towards
             all
             the
             parts
             of
             the
             Universe
             ;
             and
             therefore
             it
             hath
             no
             Gravity
             at
             all
             .
          
           
             11
             We
             find
             also
             by
             experience
             ,
             that
             by
             the
             force
             of
             air
             enclosed
             in
             a
             hollow
             Canon
             ,
             a
             bullet
             of
             lead
             may
             with
             considerable
             violence
             be
             shot
             out
             of
             a
             Gunne
             of
             late
             invention
             ,
             called
             the
             Wind-Gun
             .
             In
             the
             end
             of
             this
             Canon
             there
             are
             two
             holes
             with
             their
             Valves
             on
             the
             inside
             ,
             to
             shut
             them
             close
             ;
             one
             of
             them
             serving
             for
             the
             admission
             of
             aire
             ,
             and
             the
             other
             for
             the
             letting
             of
             it
             out
             .
             Also
             to
             that
             
             end
             which
             serves
             for
             the
             receiving
             in
             of
             aire
             ,
             there
             is
             joyned
             another
             Canon
             of
             the
             same
             metal
             and
             bigness
             ,
             in
             which
             there
             is
             fitted
             a
             Rammer
             ,
             which
             is
             perforated
             ,
             and
             hath
             also
             a
             Valve
             opening
             towards
             the
             former
             Canon
             .
             By
             the
             help
             of
             this
             Valve
             the
             Rammer
             is
             easily
             drawn
             back
             ,
             and
             letteth
             in
             aire
             from
             without
             ;
             and
             being
             often
             drawn
             back
             and
             returned
             again
             with
             violent
             strokes
             ,
             it
             forceth
             some
             part
             of
             that
             aire
             into
             the
             former
             Canon
             ,
             so
             long
             ,
             till
             at
             last
             the
             resistance
             of
             the
             enclosed
             aire
             is
             greater
             then
             the
             force
             of
             the
             stroke
             .
             And
             by
             this
             means
             men
             think
             there
             is
             now
             a
             greater
             quantity
             of
             aire
             in
             the
             Canon
             then
             there
             was
             formerly
             ,
             though
             it
             were
             full
             before
             .
             Also
             the
             aire
             thus
             forced
             in
             ,
             how
             much
             soever
             it
             be
             ,
             is
             hindered
             from
             getting
             out
             again
             by
             the
             foresaid
             Valves
             ,
             which
             the
             very
             endeavour
             of
             the
             aire
             to
             get
             out
             doth
             necessarily
             shut
             .
             Lastly
             ,
             that
             Valve
             being
             opened
             which
             was
             made
             for
             the
             letting
             out
             of
             the
             aire
             ,
             it
             presently
             breaketh
             out
             with
             violence
             ,
             &
             driveth
             the
             bullet
             before
             it
             with
             great
             force
             and
             velocity
             .
          
           
             As
             for
             the
             cause
             of
             this
             ,
             I
             could
             easily
             attribute
             it
             (
             as
             most
             men
             do
             )
             to
             Condensation
             ,
             and
             think
             that
             the
             aire
             ,
             which
             had
             at
             the
             first
             but
             its
             ordinary
             degree
             of
             Rarity
             ,
             was
             afterwards
             by
             the
             forcing
             in
             of
             more
             aire
             condensed
             ,
             and
             last
             of
             all
             rarified
             again
             by
             being
             let
             out
             and
             restored
             to
             its
             natural
             liberty
             .
             But
             I
             cannot
             imagine
             how
             the
             same
             place
             can
             be
             alwayes
             full
             ,
             and
             nevertheless
             contain
             sometimes
             a
             greater
             ,
             sometimes
             a
             less
             quantity
             of
             matter
             ;
             that
             is
             to
             say
             ,
             that
             it
             can
             be
             fuller
             then
             full
             .
             Nor
             can
             I
             conceive
             ,
             how
             Fulness
             can
             of
             it self
             be
             an
             efficient
             cause
             of
             motion
             .
             For
             both
             ●hese
             are
             impossible
             .
             Wherefore
             we
             must
             seek
             out
             some
             other
             possible
             cause
             of
             this
             Phaenomenon
             .
             Whilst
             therefore
             the
             Valve
             w
             ch
             serves
             for
             the
             letting
             in
             of
             aire
             is
             opened
             by
             the
             first
             stroke
             of
             the
             Rammer
             ,
             the
             aire
             within
             doth
             with
             equal
             force
             resist
             the
             entering
             of
             the
             aire
             from
             without
             ;
             so
             that
             the
             endeavours
             between
             the
             internal
             and
             external
             aire
             are
             opposite
             ,
             that
             is
             ,
             there
             are
             two
             opposite
             motions
             ,
             whilest
             the
             one
             goeth
             in
             and
             the
             other
             cometh
             out
             ;
             but
             no
             augmentation
             at
             all
             of
             aire
             within
             the
             Canon
             .
             For
             there
             is
             driven
             out
             by
             the
             stroke
             as
             much
             pure
             aire
             which
             passeth
             between
             the
             Rammer
             and
             the
             sides
             of
             the
             Canon
             ,
             as
             there
             is
             forced
             in
             of
             aire
             impure
             by
             the
             same
             stroke
             .
             And
             thus
             by
             many
             forcible
             strokes
             the
             quantity
             of
             small
             hard
             Bodies
             will
             be
             encreased
             
             within
             the
             Canon
             ,
             and
             their
             motions
             also
             will
             grow
             stronger
             and
             stronger
             as
             long
             as
             the
             matter
             of
             the
             Canon
             is
             able
             to
             endure
             their
             force
             ;
             by
             which
             if
             it
             be
             not
             broken
             ,
             it
             will
             at
             least
             be
             urged
             every
             way
             by
             their
             endeavour
             to
             free
             themselves
             ;
             and
             as
             soon
             as
             the
             Valve
             which
             serves
             to
             let
             them
             out
             is
             opened
             ,
             they
             will
             fly
             out
             with
             violent
             motion
             ,
             and
             carry
             with
             them
             the
             bullet
             which
             is
             in
             their
             way
             .
             Wherefore
             I
             have
             given
             a
             possible
             cause
             of
             this
             Phaenomenon
             .
          
           
             12
             Water
             ,
             contrary
             to
             the
             custome
             of
             Heavy
             Bodies
             ,
             ascendeth
             in
             the
             Weather-glasse
             ;
             but
             it
             doth
             it
             when
             the
             aire
             is
             cold
             ;
             for
             when
             it
             is
             warme
             it
             descendeth
             againe
             .
             And
             this
             Organ
             is
             called
             a
             Thermometer
             ,
             or
             Thermoscope
             ,
             because
             the
             degrees
             of
             Heat
             and
             Cold
             are
             measured
             and
             marked
             by
             it
             .
             It
             is
             made
             in
             this
             manner
             .
             Let
             A
             B
             C
             D
             (
             in
             the
             5th
             figure
             )
             be
             a
             vessel
             full
             of
             water
             ,
             and
             E
             F
             G
             a
             hollow
             Cylinder
             of
             glasse
             ,
             closed
             at
             E
             ,
             and
             open
             at
             G.
             Let
             it
             be
             heated
             ,
             and
             set
             upright
             within
             the
             water
             to
             F
             ;
             and
             let
             the
             open
             end
             reach
             to
             G.
             This
             being
             done
             ▪
             as
             the
             aire
             by
             little
             and
             little
             grows
             colder
             ,
             the
             water
             will
             ascend
             slowly
             within
             the
             Cylinder
             from
             F
             towards
             E
             ;
             till
             at
             last
             the
             externall
             and
             internall
             aire
             coming
             to
             be
             both
             of
             the
             same
             temper
             ,
             it
             will
             neither
             ascend
             higher
             ,
             nor
             descend
             lower
             ,
             till
             the
             temper
             of
             the
             aire
             be
             changed
             .
             Suppose
             it
             therefore
             to
             be
             setled
             any
             where
             ,
             as
             at
             H.
             If
             now
             the
             heat
             of
             the
             aire
             be
             augmented
             ,
             the
             water
             will
             descend
             below
             H
             ;
             and
             if
             the
             heat
             be
             diminished
             ,
             it
             will
             ascend
             above
             it
             .
             Which
             though
             it
             be
             certainely
             known
             to
             be
             true
             by
             experience
             ,
             the
             cause
             neverthelesse
             hath
             not
             as
             yet
             been
             discovered
             .
          
           
             In
             the
             6
             and
             7
             articles
             of
             the
             27th
             chapter
             (
             where
             I
             consider
             the
             cause
             of
             Cold
             )
             I
             have
             shewne
             ,
             that
             fluid
             Bodies
             are
             made
             colder
             by
             the
             pressure
             of
             the
             aire
             ,
             that
             is
             to
             say
             ,
             by
             a
             constant
             Wind
             that
             presseth
             them
             .
             For
             the
             same
             cause
             it
             is
             ,
             that
             the
             Superficies
             of
             the
             water
             is
             pressed
             at
             F
             ;
             and
             having
             no
             place
             to
             which
             it
             may
             retire
             from
             this
             pressure
             besides
             the
             cavity
             of
             the
             Cylinder
             between
             H
             and
             E
             ,
             it
             is
             therefore
             necessarily
             forced
             thither
             by
             the
             Cold
             ,
             and
             consequently
             it
             ascendeth
             more
             or
             lesse
             ,
             according
             as
             the
             Cold
             is
             more
             or
             lesse
             encreased
             .
             And
             againe
             ,
             as
             the
             Heat
             is
             more
             intense
             ,
             or
             the
             Cold
             more
             remisse
             ,
             the
             same
             water
             will
             be
             depressed
             more
             or
             lesse
             by
             its
             own
             Gravity
             ,
             that
             is
             to
             say
             ,
             by
             the
             cause
             of
             Gravity
             above
             explicated
             .
          
           
           
             13
             Also
             Living
             creatures
             ,
             though
             they
             be
             Heavy
             ,
             can
             by
             Leaping
             ,
             Swimming
             &
             Flying
             raise
             themselvs
             to
             a
             certain
             degree
             of
             height
             .
             But
             they
             cannot
             do
             this
             except
             they
             be
             supported
             by
             some
             resisting
             Body
             ,
             as
             the
             Earth
             ,
             the
             Water
             and
             the
             Aire
             .
             For
             these
             motions
             have
             their
             beginning
             from
             the
             contraction
             (
             by
             the
             helpe
             of
             the
             Muscles
             )
             of
             the
             Body
             animate
             .
             For
             to
             this
             contraction
             there
             succeedeth
             a
             distension
             of
             their
             whole
             Bodies
             ;
             by
             which
             distension
             ,
             the
             Earth
             ,
             the
             Water
             or
             the
             Aire
             which
             supporteth
             them
             ,
             is
             pressed
             ;
             and
             from
             hence
             ,
             by
             the
             reaction
             of
             those
             pressed
             Bodies
             ,
             Living
             Creatures
             acquire
             an
             endeavour
             upwards
             ,
             but
             such
             ,
             as
             by
             reason
             of
             the
             Gravity
             of
             their
             Bodies
             is
             presently
             lost
             againe
             .
             By
             this
             endeavour
             therefore
             it
             is
             ,
             that
             Living
             creatures
             rayse
             themselues
             up
             a
             little
             way
             by
             Leaping
             ,
             but
             to
             no
             great
             purpose
             ;
             but
             by
             Swimming
             &
             Flying
             they
             raise
             themselves
             to
             a
             greater
             height
             ;
             because
             before
             the
             effect
             of
             their
             endeavour
             is
             quite
             extinguished
             by
             the
             Gravity
             of
             their
             bodies
             ,
             they
             can
             renew
             the
             same
             endeavour
             againe
             .
          
           
             That
             by
             the
             power
             of
             the
             Soule
             ,
             without
             any
             antecedent
             contraction
             of
             the
             Muscles
             ,
             or
             the
             helpe
             of
             something
             to
             support
             him
             ,
             any
             man
             can
             be
             able
             to
             raise
             his
             Body
             upwards
             ,
             is
             a
             childish
             conceipt
             .
             For
             if
             it
             were
             true
             ,
             a
             man
             might
             raise
             himselfe
             to
             what
             height
             he
             pleased
             .
          
           
             14
             The
             diaphanous
             Medium
             which
             surrounds
             the
             Eie
             on
             all
             fides
             ,
             is
             invisible
             ;
             Nor
             is
             Aire
             to
             be
             seen
             in
             Aire
             ,
             nor
             Water
             in
             Water
             ,
             nor
             any
             thing
             but
             that
             which
             is
             more
             opacous
             .
             But
             in
             the
             confines
             of
             two
             diaphanous
             Bodies
             ,
             one
             of
             them
             may
             be
             distinguished
             from
             the
             other
             .
             It
             is
             not
             therefore
             a
             thing
             so
             very
             ridiculous
             for
             ordinary
             people
             to
             think
             all
             that
             Space
             empty
             ,
             in
             which
             we
             say
             is
             Aire
             ;
             it
             being
             the
             worke
             of
             Reason
             to
             make
             us
             conceive
             that
             the
             Aire
             is
             any
             thing
             .
             For
             by
             which
             of
             our
             Senses
             is
             it
             ,
             that
             we
             take
             notice
             of
             the
             Aire
             ,
             seeing
             we
             neither
             See
             ,
             nor
             Hear
             ,
             nor
             Tast
             ,
             nor
             Smell
             ,
             nor
             Feel
             it
             to
             be
             any
             thing
             ?
             When
             we
             feel
             Heat
             ,
             we
             do
             not
             impute
             it
             to
             the
             Air
             ,
             but
             to
             the
             Fire
             ;
             nor
             do
             we
             say
             the
             aire
             is
             Cold
             ,
             but
             we
             our selves
             are
             Cold
             ;
             and
             when
             we
             feel
             the
             Wind
             ,
             we
             rather
             think
             something
             is
             comming
             ,
             then
             that
             any
             thing
             is
             already
             come
             .
             Also
             we
             do
             not
             at
             al
             feel
             the
             waight
             of
             water
             in
             water
             ,
             much
             less
             of
             air
             in
             air
             .
             That
             we
             come
             to
             know
             that
             to
             be
             a
             Body
             which
             we
             call
             Aire
             ,
             it
             is
             by
             Reasoning
             ;
             but
             it
             is
             from
             one
             
             Reason
             onely
             ,
             namely
             ,
             because
             it
             is
             impossible
             for
             remote
             Bodies
             to
             work
             upon
             our
             Organs
             of
             Sense
             but
             by
             the
             help
             of
             Bodies
             intermediate
             ,
             without
             which
             we
             could
             have
             no
             sense
             of
             them
             ,
             till
             they
             came
             to
             be
             contiguous
             .
             Wherefore
             ,
             from
             the
             Senses
             alone
             ,
             without
             reasoning
             from
             effects
             ,
             we
             cannot
             have
             sufficient
             evidence
             of
             the
             nature
             of
             Bodies
             .
          
           
             For
             there
             is
             under-ground
             in
             some
             Mines
             of
             Coles
             ,
             a
             certain
             matter
             of
             a
             middle
             nature
             between
             Water
             and
             Aire
             ,
             which
             nevertheless
             cannot
             by
             Sense
             be
             distinguished
             from
             aire
             ;
             for
             it
             is
             as
             Diaphanous
             as
             the
             purest
             aire
             ;
             and
             as
             farre
             as
             Sense
             can
             judge
             ,
             equally
             penetrable
             .
             But
             if
             we
             look
             upon
             the
             effect
             ,
             it
             is
             like
             that
             of
             water
             .
             For
             when
             that
             matter
             breaks
             out
             of
             the
             Earth
             into
             one
             of
             those
             Pits
             ,
             it
             fils
             the
             same
             either
             totally
             ,
             or
             to
             some
             degree
             ;
             and
             if
             a
             Man
             ,
             or
             Fire
             be
             then
             let
             down
             into
             it
             ,
             it
             extinguishes
             them
             in
             almost
             as
             little
             time
             as
             water
             would
             do
             .
             But
             for
             the
             better
             understanding
             of
             this
             Phaenomenon
             ,
             I
             shall
             describe
             the
             6th
             figure
             .
             In
             which
             ,
             let
             A
             B
             represent
             the
             pit
             of
             the
             Mine
             ;
             and
             let
             part
             thereof
             ,
             namely
             ,
             C
             B
             ▪
             be
             supposed
             to
             be
             filled
             with
             that
             matter
             .
             If
             now
             a
             lighted
             Cādle
             be
             let
             down
             into
             it
             below
             C
             ,
             it
             wil
             as
             suddenly
             be
             extinguished
             ,
             as
             if
             it
             were
             thrust
             into
             water
             .
             Also
             if
             a
             grate
             filled
             with
             coles
             throughly
             kindled
             and
             burning
             never
             so
             brightly
             ,
             be
             let
             down
             ;
             as
             soon
             as
             ever
             it
             is
             below
             C
             ,
             the
             fire
             will
             begin
             to
             grow
             pale
             ,
             and
             shortly
             after
             (
             losing
             its
             light
             )
             be
             extinguished
             ,
             no
             otherwise
             then
             if
             it
             were
             quenched
             in
             water
             .
             But
             if
             the
             grate
             be
             drawn
             up
             again
             presently
             ,
             whilest
             the
             coles
             are
             still
             very
             hot
             ,
             the
             fire
             will
             by
             little
             and
             little
             be
             kindled
             again
             ,
             and
             shine
             as
             before
             .
             There
             is
             indeed
             between
             this
             matter
             &
             water
             this
             considerable
             difference
             ,
             that
             it
             neither
             wetteth
             ,
             nor
             sticketh
             to
             such
             things
             as
             are
             put
             down
             into
             it
             ,
             as
             water
             doth
             ;
             which
             by
             the
             moisture
             it
             leaveth
             ,
             hindereth
             the
             kindling
             again
             of
             the
             matter
             once
             extinguished
             .
             In
             like
             manner
             ,
             if
             a
             Man
             be
             let
             down
             below
             C
             ,
             he
             will
             presently
             fall
             into
             a
             great
             difficulty
             of
             breathing
             ,
             and
             immediately
             after
             into
             a
             swoun
             ,
             and
             die
             ,
             unless
             he
             be
             suddenly
             drawn
             up
             again
             .
             They
             therefore
             that
             go
             down
             into
             these
             pits
             ,
             have
             this
             custome
             ,
             that
             as
             soon
             as
             ever
             they
             feel
             themselves
             sick
             ,
             they
             shake
             the
             rope
             by
             which
             they
             were
             let
             down
             ,
             to
             signifie
             they
             are
             not
             well
             ,
             and
             to
             the
             end
             that
             they
             may
             speedily
             be
             pulled
             up
             again
             .
             For
             if
             a
             man
             be
             drawn
             out
             too
             late
             ,
             
             void
             of
             sense
             and
             motion
             ,
             they
             digg
             up
             a
             Turff
             ,
             and
             put
             his
             face
             and
             mouth
             into
             the
             fresh
             earth
             ;
             by
             which
             means
             (
             unless
             he
             be
             quite
             dead
             )
             he
             comes
             to
             himself
             again
             by
             little
             and
             little
             ,
             and
             recovers
             life
             by
             the
             breathing
             out
             (
             as
             it
             were
             )
             of
             that
             suffocating
             matter
             which
             he
             had
             sucked
             in
             whilest
             he
             was
             in
             the
             pit
             ;
             almost
             in
             the
             same
             manner
             as
             they
             that
             are
             drowned
             come
             to
             themselves
             again
             by
             vomiting
             up
             the
             water
             .
             But
             this
             doth
             not
             happen
             in
             all
             Mines
             ,
             but
             in
             some
             onely
             ;
             and
             in
             those
             not
             alwayes
             ,
             but
             often
             .
             In
             such
             Pits
             as
             are
             subject
             to
             it
             ,
             they
             use
             this
             remedy
             .
             They
             dig
             another
             pit
             ,
             as
             DE
             ,
             close
             by
             it
             ,
             of
             equal
             depth
             ;
             and
             joyning
             them
             both
             together
             with
             one
             common
             channel
             EF
             ,
             they
             make
             a
             Fire
             in
             the
             bottom
             E
             ,
             which
             carries
             out
             at
             D
             the
             aire
             contained
             in
             the
             pit
             DE
             ;
             and
             this
             draws
             with
             it
             the
             aire
             contained
             in
             the
             channel
             EF
             ;
             which
             in
             like
             manner
             is
             followed
             by
             the
             noxious
             matter
             contained
             in
             CB
             ;
             &
             by
             this
             means
             the
             pit
             is
             for
             that
             time
             made
             healthful
             .
             Out
             of
             this
             History
             (
             which
             I
             write
             onely
             to
             such
             as
             have
             had
             experience
             of
             the
             truth
             of
             it
             ,
             without
             any
             designe
             to
             support
             my
             Philosophy
             with
             Stories
             of
             doubtful
             credit
             )
             may
             be
             collected
             the
             following
             possible
             cause
             of
             this
             Phaenomenon
             ;
             namely
             ,
             that
             there
             is
             a
             certain
             matter
             ,
             fluid
             ,
             &
             most
             transparent
             ,
             and
             not
             much
             lighter
             then
             water
             ;
             which
             breaking
             out
             of
             the
             Earth
             fills
             the
             Pit
             to
             C
             ;
             and
             that
             in
             this
             matter
             ,
             as
             in
             water
             ,
             both
             Fire
             and
             Living
             creatures
             are
             extinguished
             .
          
           
             15
             About
             the
             nature
             of
             Heavy
             Bodies
             ,
             the
             greatest
             difficulty
             ariseth
             from
             the
             contemplation
             of
             those
             things
             which
             make
             other
             Heavy
             Bodies
             ascend
             to
             them
             ;
             such
             are
             
               Jet
               ,
               Amber
            
             ,
             and
             the
             Loadstone
             .
             But
             that
             which
             troubles
             men
             most
             is
             the
             Loadstone
             ,
             which
             is
             also
             called
             
               Lapis
               Herculeus
            
             ;
             a
             stone
             ,
             though
             otherwise
             despicable
             ,
             yet
             of
             so
             great
             power
             ,
             that
             it
             taketh
             up
             Iron
             from
             the
             Earth
             ,
             and
             holds
             it
             suspended
             in
             the
             aire
             ,
             as
             Hercules
             did
             Antaeus
             .
             Nevertheless
             ,
             we
             wonder
             at
             it
             somewhat
             the
             less
             ,
             because
             we
             see
             Jet
             draw
             up
             Straws
             ,
             which
             are
             Heavy
             Bodies
             ,
             though
             not
             so
             Heavy
             as
             Iron
             .
             But
             as
             for
             Jet
             ,
             it
             must
             first
             be
             excited
             by
             rubbing
             ,
             that
             is
             to
             say
             ,
             by
             motion
             to
             and
             fro
             ;
             whereas
             the
             Loadstone
             hath
             sufficient
             excitation
             from
             its
             own
             nature
             ,
             that
             is
             to
             say
             ,
             from
             some
             internal
             principle
             of
             motion
             peculiar
             to
             it self
             .
             Now
             whatsoever
             is
             moved
             ,
             is
             moved
             by
             some
             contiguous
             and
             moved
             Body
             ,
             as
             hath
             been
             formerly
             demonstrated
             .
             And
             from
             hence
             it
             follows
             evidently
             ,
             that
             the
             first
             endeavour
             
             which
             Iron
             hath
             towards
             the
             Loadstone
             ,
             is
             caused
             by
             the
             motion
             of
             that
             aire
             which
             is
             contiguous
             to
             the
             Iron
             .
             Also
             that
             this
             motion
             is
             generated
             by
             the
             motion
             of
             the
             next
             aire
             ,
             and
             so
             on
             successively
             ,
             till
             by
             this
             succession
             we
             find
             that
             the
             motion
             of
             all
             the
             intermediate
             air
             taketh
             its
             beginning
             from
             some
             motion
             which
             is
             in
             the
             Loadstone
             it self
             ;
             which
             motion
             (
             because
             the
             Loadstone
             seems
             to
             be
             at
             rest
             )
             is
             invisible
             .
             It
             is
             therefore
             certain
             ,
             that
             the
             attractive
             power
             of
             the
             Loadstone
             is
             nothing
             else
             but
             some
             motiō
             of
             the
             smallest
             particles
             thereof
             .
             Supposing
             therefore
             that
             those
             small
             Bodies
             of
             which
             the
             Loadstone
             is
             (
             in
             the
             bowels
             of
             the
             Earth
             )
             composed
             ,
             have
             by
             nature
             such
             motion
             or
             endeavour
             as
             was
             above
             attributed
             to
             Jet
             ,
             namely
             a
             reciprocal
             motiō
             in
             a
             line
             too
             short
             to
             be
             seen
             ,
             both
             those
             stones
             wil
             have
             one
             &
             the
             same
             cause
             of
             attraction
             .
             Now
             in
             what
             manner
             ,
             and
             in
             what
             order
             of
             working
             this
             cause
             produceth
             the
             effect
             of
             attraction
             ,
             is
             the
             thing
             to
             be
             enquired
             .
             And
             first
             we
             know
             ,
             that
             when
             the
             string
             of
             a
             Lute
             or
             Viol
             is
             stricken
             ,
             the
             Vibration
             ,
             that
             is
             ,
             the
             reciprocal
             motion
             of
             that
             string
             in
             the
             same
             straight
             Line
             ,
             causeth
             like
             Vibration
             in
             another
             string
             which
             has
             like
             tension
             .
             We
             know
             also
             ,
             that
             the
             dregs
             or
             small
             sands
             which
             sink
             to
             the
             bottom
             of
             a
             Vessel
             ,
             will
             be
             raised
             up
             from
             the
             bottom
             by
             any
             strong
             and
             reciprocal
             agitation
             of
             the
             water
             stirred
             with
             the
             hand
             or
             with
             a
             staff
             .
             Why
             therefore
             should
             not
             reciprocal
             motion
             of
             the
             parts
             of
             the
             Loadstone
             contribute
             as
             much
             towards
             the
             moving
             of
             Iron
             ?
             For
             if
             in
             the
             Loadstone
             there
             be
             supposed
             such
             reciprocal
             motion
             ,
             or
             motion
             of
             the
             parts
             forwards
             and
             backwards
             ,
             it
             will
             follow
             ,
             that
             the
             like
             motion
             will
             be
             propagated
             by
             the
             aire
             to
             the
             Iron
             ,
             and
             consequently
             that
             there
             will
             be
             in
             all
             the
             parts
             of
             the
             Iron
             the
             same
             reciprocations
             or
             motions
             forwards
             and
             backwards
             .
             And
             from
             hence
             also
             it
             will
             follow
             ,
             that
             the
             intermediate
             aire
             between
             the
             Stone
             and
             the
             Iron
             will
             by
             little
             and
             little
             be
             thrust
             away
             ;
             and
             the
             aire
             being
             thrust
             away
             ,
             the
             Bodies
             of
             the
             Loadstone
             and
             the
             Iron
             will
             necessarily
             come
             together
             .
             The
             possible
             cause
             therefore
             why
             the
             Loadstone
             and
             Jet
             draw
             to
             them
             ,
             the
             one
             Iron
             ,
             the
             other
             Strawes
             ,
             may
             be
             this
             ,
             that
             those
             attracting
             Bodies
             have
             reciprocal
             motion
             either
             in
             a
             straight
             line
             ,
             or
             in
             an
             Elliptical
             line
             ,
             when
             there
             is
             nothing
             in
             the
             nature
             of
             the
             attracted
             Bodies
             which
             is
             repugnant
             to
             such
             a
             motion
             .
          
           
           
             But
             why
             the
             Loadstone
             (
             if
             with
             the
             help
             of
             Cork
             it
             float
             at
             liberty
             upon
             the
             top
             of
             the
             water
             )
             should
             from
             any
             position
             whatsoever
             so
             place
             it self
             in
             the
             plain
             of
             the
             Meridian
             ,
             as
             that
             the
             same
             points
             which
             at
             one
             time
             of
             its
             being
             at
             rest
             respect
             the
             Poles
             of
             the
             Earth
             ,
             should
             at
             all
             other
             times
             respect
             the
             same
             Poles
             ,
             the
             cause
             may
             be
             this
             ,
             That
             the
             reciprocal
             motion
             which
             I
             supposed
             to
             be
             in
             the
             parts
             of
             the
             Stone
             ,
             is
             made
             in
             a
             line
             parallel
             to
             the
             Axis
             of
             the
             Earth
             ,
             and
             has
             been
             in
             those
             parts
             ever
             since
             the
             Stone
             was
             generated
             .
             Seeing
             therefore
             the
             Stone
             whilest
             it
             remains
             in
             the
             Mine
             ,
             and
             is
             carried
             about
             together
             with
             the
             Earth
             by
             its
             diurnal
             motion
             ,
             doth
             by
             length
             of
             time
             get
             a
             habit
             of
             being
             moved
             in
             a
             line
             which
             is
             perpendicular
             to
             the
             line
             of
             its
             reciprocal
             motion
             ,
             it
             will
             afterwards
             ,
             though
             its
             axis
             be
             removed
             from
             the
             parallel
             situation
             it
             had
             with
             the
             axis
             of
             the
             Earth
             ,
             retain
             its
             endeavour
             of
             returning
             to
             that
             situation
             again
             ;
             and
             all
             endeavour
             being
             the
             beginning
             of
             motion
             ,
             and
             nothing
             intervening
             that
             may
             hinder
             the
             same
             ,
             the
             Loadstone
             will
             therefore
             return
             to
             its
             former
             situation
             .
             For
             any
             piece
             of
             Iron
             that
             has
             for
             a
             long
             time
             rested
             in
             the
             plain
             of
             the
             Meridian
             ,
             whensoever
             it
             is
             forced
             from
             that
             situation
             ,
             and
             afterwards
             left
             to
             its
             own
             liberty
             again
             ,
             will
             of
             it self
             return
             to
             lie
             in
             the
             Meridian
             again
             ;
             which
             return
             is
             caused
             by
             the
             endeavour
             it
             acquired
             from
             the
             diurnal
             motion
             of
             the
             Earth
             in
             the
             parallel
             circles
             which
             are
             perpendicular
             to
             the
             Meridians
             .
          
           
             If
             Iron
             be
             rubbed
             by
             the
             Loadstone
             drawn
             from
             one
             Pole
             to
             the
             other
             ,
             two
             things
             will
             happen
             ;
             one
             ,
             that
             the
             Iron
             will
             acquire
             the
             same
             direction
             with
             the
             Loadstone
             ,
             that
             is
             to
             say
             ,
             that
             it
             will
             lie
             in
             the
             Meridian
             ,
             and
             have
             its
             Axis
             and
             Poles
             in
             the
             same
             position
             with
             those
             of
             the
             Stone
             ;
             the
             other
             ,
             that
             the
             like
             Poles
             of
             the
             Stone
             and
             of
             the
             Iron
             will
             avoid
             one
             another
             ,
             and
             the
             unlike
             Poles
             approach
             one
             another
             .
             And
             the
             cause
             of
             the
             former
             may
             be
             this
             ,
             that
             Iron
             being
             touched
             by
             motion
             which
             is
             not
             reciprocal
             ,
             but
             drawn
             the
             same
             way
             from
             Pole
             to
             Pole
             ,
             there
             will
             be
             imprinted
             in
             the
             Iron
             also
             an
             endeavour
             from
             the
             same
             Pole
             to
             the
             same
             Pole.
             For
             seeing
             the
             Loadstone
             differs
             from
             Iron
             no
             otherwise
             then
             as
             Ore
             from
             Metal
             ,
             there
             will
             be
             no
             repugnance
             at
             all
             in
             the
             Iron
             to
             receive
             the
             same
             motion
             which
             is
             in
             the
             Stone
             .
             From
             whence
             it
             follows
             ,
             that
             seeing
             they
             are
             both
             affected
             alike
             by
             the
             diurnal
             motion
             
             of
             the
             Earth
             ,
             they
             will
             both
             equally
             return
             to
             their
             situation
             in
             the
             Meridian
             whensoever
             they
             are
             put
             frō
             the
             same
             ▪
             Also
             of
             the
             later
             this
             may
             be
             the
             cause
             ,
             that
             as
             the
             Loadstone
             in
             touching
             the
             Irō
             doth
             by
             its
             action
             imprint
             in
             the
             Iron
             an
             endeavour
             towards
             one
             of
             the
             Poles
             ,
             suppose
             towards
             the
             North
             Pole
             ;
             so
             reciprocally
             ,
             the
             Iron
             by
             its
             action
             upon
             the
             Loadstone
             doth
             imprint
             in
             it
             an
             endeavour
             towards
             the
             other
             Pole
             ,
             namely
             towards
             the
             South
             Pole.
             It
             happens
             therefore
             in
             these
             reciprocations
             or
             motions
             forwards
             and
             backwards
             of
             the
             particles
             of
             the
             Stone
             and
             of
             the
             Iron
             betwixt
             the
             North
             &
             the
             South
             ,
             that
             whilest
             in
             one
             of
             them
             the
             motion
             is
             from
             North
             to
             South
             ,
             and
             the
             return
             from
             South
             to
             North
             ,
             in
             the
             other
             the
             motion
             wil
             be
             from
             South
             to
             North
             ,
             &
             the
             return
             frō
             North
             to
             South
             ;
             which
             motions
             being
             opposite
             to
             one
             another
             ,
             and
             communicated
             to
             the
             Air
             ,
             the
             North
             Pole
             of
             the
             Iron
             (
             whilest
             the
             attraction
             is
             working
             )
             will
             be
             depressed
             towards
             the
             South
             Pole
             of
             the
             Loadstone
             ;
             or
             contrarily
             the
             North
             Pole
             of
             the
             Loadstone
             will
             be
             depressed
             towards
             the
             South
             Pole
             of
             the
             Iron
             ;
             and
             the
             Axes
             both
             of
             the
             Loadstone
             and
             of
             the
             Iron
             will
             be
             situate
             in
             the
             same
             straight
             line
             .
             The
             truth
             whereof
             is
             taught
             us
             by
             experience
             .
          
           
             As
             for
             the
             propagation
             of
             this
             Magnetical
             vertue
             ,
             not
             onely
             through
             the
             Aire
             ,
             but
             through
             any
             other
             Bodies
             how
             hard
             so
             ever
             ,
             it
             is
             not
             to
             be
             wondred
             at
             ,
             seeing
             no
             motion
             can
             be
             so
             weak
             ,
             but
             that
             it
             may
             be
             propagated
             infinitely
             through
             a
             space
             filled
             with
             Body
             of
             any
             hardness
             whatsoever
             .
             For
             in
             a
             full
             Medium
             ,
             there
             can
             be
             no
             motion
             which
             doth
             not
             make
             the
             next
             part
             yeild
             ,
             and
             that
             the
             next
             ,
             and
             so
             successively
             without
             end
             ;
             so
             that
             there
             is
             no
             effect
             whatsoever
             but
             to
             the
             production
             thereof
             something
             is
             necessarily
             contributed
             by
             the
             several
             motions
             of
             all
             the
             several
             things
             that
             are
             in
             the
             World.
             
          
           
             And
             thus
             much
             concerning
             the
             nature
             of
             Body
             in
             general
             ;
             with
             which
             I
             conclude
             this
             my
             first
             Section
             of
             the
             Elements
             of
             Philosophy
             .
             In
             the
             first
             ,
             second
             and
             third
             Parts
             ,
             where
             the
             Principles
             of
             Ratiocination
             consist
             in
             our
             own
             Understanding
             ,
             that
             is
             to
             say
             ,
             in
             the
             legitimate
             use
             of
             such
             Words
             as
             we
             our selves
             constitute
             ,
             all
             the
             Theoremes
             (
             if
             I
             be
             not
             deceived
             )
             are
             rightly
             demonstrated
             .
             The
             fourth
             Part
             depends
             upon
             Hypotheses
             ;
             which
             unless
             we
             know
             them
             to
             be
             true
             ,
             it
             is
             impossible
             for
             us
             to
             demonstrate
             that
             those
             
             Causes
             which
             I
             have
             there
             explicated
             ,
             are
             the
             true
             Causes
             of
             the
             things
             whose
             productions
             I
             have
             derived
             from
             them
             .
          
           
             Nevertheless
             ,
             seeing
             I
             have
             assumed
             no
             Hypothesis
             ,
             which
             is
             not
             both
             possible
             and
             easie
             to
             be
             comprehended
             ;
             and
             seeing
             also
             that
             I
             have
             reasoned
             aright
             from
             those
             Assumptions
             ,
             I
             have
             withall
             sufficiently
             demonstrated
             that
             they
             may
             be
             the
             true
             Causes
             ;
             w
             ch
             is
             the
             end
             of
             Physical
             Contemplation
             .
             If
             any
             other
             man
             from
             other
             Hypotheses
             shall
             demonstrate
             the
             same
             ,
             or
             greater
             things
             ,
             there
             wil
             be
             greater
             praise
             and
             thanks
             due
             to
             him
             then
             I
             demand
             for
             my self
             ,
             provided
             his
             Hypotheses
             be
             such
             as
             are
             conceivable
             .
             For
             as
             for
             those
             that
             say
             any
             thing
             may
             be
             moved
             or
             produced
             by
             it Self
             ,
             by
             Species
             ,
             by
             
               its
               own
               Power
            
             ,
             by
             
               Substantial
               Forms
            
             ,
             by
             
               Incorporeal
               Substances
            
             ,
             by
             Instinct
             ,
             by
             Anteperistasis
             ,
             by
             
               Antipathy
               ,
               Sympathy
               ,
               Occult
               Quality
               ,
            
             and
             other
             empty
             words
             of
             Schoolmen
             ,
             their
             saying
             so
             is
             to
             no
             purpose
             .
          
           
             And
             now
             I
             proceed
             to
             the
             Phaenomena
             of
             Mans
             Body
             ;
             Where
             I
             shall
             speak
             of
             the
             Opticks
             ,
             and
             of
             the
             
               Dispositions
               ,
               Affections
               ,
               and
               Manners
            
             of
             Men
             (
             if
             it
             shall
             please
             God
             to
             give
             me
             life
             )
             ,
             and
             shew
             their
             Causes
             .
          
        
      
       
         
           
           
           
             AD
             CAP.
             XIV
          
           
             
               
                 fig.
                 1.
                 
              
            
             
               
                 fig.
                 2.
                 
              
            
             
               
                 fig.
                 3.
                 
              
            
             
               
                 fig.
                 4.
                 
              
            
             
               
                 fig.
                 5.
                 
              
            
             
               
                 fig
                 :
                 6.
                 
              
            
             
               
                 fig
                 :
                 7.
                 
              
            
             
               
                 fig
                 :
                 8.
                 
              
            
             
               
                 fig
                 :
                 9.
                 
              
            
             
               
                 fig
                 :
                 10.
                 
              
            
          
        
         
           
           
             AD
             CAP.
             XVI
          
           
             
               
                 fig
                 :
                 1
              
            
             
               
                 fig
                 :
                 2
              
            
             
               
                 fig
                 :
                 3
              
            
             
               
                 fig
                 :
                 4
              
            
             
               
                 fig
                 :
                 5
              
            
             
               
                 fig
                 :
                 6
              
            
             
               
                 fig
                 :
                 7
              
            
             
               
                 fig
                 :
                 8
              
            
             
               
                 fig
                 :
                 9
              
            
             
               
                 fig
                 :
                 10
              
            
             
               
                 fig
                 :
                 11
              
            
          
        
         
           
           
             AD
             CAP.
             XVII
          
           
             
               
                 fig.
                 1.
                 
              
            
             
               
                 fig.
                 2.
                 
              
            
             
               
                 fig.
                 3.
                 
              
            
             
               
                 fig.
                 4.
                 
              
            
             
               
                 fig.
                 5.
                 
              
            
             
               
                 fig.
                 6.
                 
              
            
             
               
                 fig.
                 7.
                 
              
            
             
               
                 fig.
                 8.
                 
              
            
          
        
         
           
           
             Cap
             :
             XVIII
          
           
             
               
                 fig.
                 1.
                 
              
            
             
               
                 fig.
                 2.
                 
              
            
          
        
         
           
           
             AD
             CAP.
             XIX
          
           
             
               
                 fig
                 :
                 1
              
            
             
               
                 fig
                 :
                 2
              
            
             
               
                 fig
                 :
                 3
              
            
             
               
                 fig
                 :
                 4
              
            
             
               
                 fig
                 ▪
                 5
              
            
             
               
                 fig
                 :
                 6
              
            
             
               
                 fig
                 :
                 7
              
            
             
               
                 fig
                 :
                 8
              
            
             
               
                 fig
                 :
                 9
              
            
             
               
                 fig
                 :
                 10
              
            
          
        
         
           
           
             Cap
             :
             XX.
             
          
           
             
               
                 Fig
                 ▪
                 1
              
            
             
               
                 fig
                 ▪
                 2
              
            
             
               
                 fig
                 ▪
                 3
              
            
             
               
                 fig
                 ▪
                 4
              
            
             
               
                 fig
                 ▪
                 5
              
            
          
        
         
           
           
             AD
             CAP
             XXI
          
           
             
               
                 Fig
                 ▪
                 1
              
            
             
               
                 fig
                 :
                 2
              
            
             
               
                 fig
                 :
                 3
              
            
             
               
                 fig
                 ▪
                 4
              
            
             
               
                 fig
                 :
                 5
              
            
          
        
         
           
           
             AD
             CAP
             ▪
             XXII
          
           
             
               
                 fig
                 ▪
                 1
              
            
             
               
                 fig
                 :
                 2
              
            
             
               
                 fig
                 :
                 3
              
            
          
        
         
           
           
             AD
             CAP
             ▪
             XXIII
          
           
             
               
                 fig
                 :
                 1
              
            
             
               
                 fig
                 ▪
                 2
              
            
             
               
                 fig
                 :
                 3
              
            
             
               
                 fig
                 :
                 4
              
            
             
               
                 fig
                 :
                 5
              
            
             
               
                 fig
                 :
                 6
              
            
             
               
                 fig
                 :
                 7
              
            
             
               
                 fig
                 :
                 8
              
            
             
               
                 fig
                 :
                 9
              
            
          
        
         
           
           
             AD
             CAP.
             XXIIII
          
           
             
               
                 fig
                 :
                 1
              
            
             
               
                 fig
                 :
                 2
              
            
             
               
                 fig
                 :
                 3
              
            
             
               
                 fig
                 :
                 4
              
            
             
               
                 fig
                 :
                 5
              
            
             
               
                 fig
                 :
                 6
              
            
          
        
         
           
           
             AD
             Cap
             XXVI
          
           
             
               
                 fig
                 ▪
                 1
              
            
             
               
                 Fig
                 ▪
                 2
              
            
             
               
                 fig
                 ▪
                 3
              
            
             
               
                 fig
                 :
                 4.
                 
              
            
             
               
                 fig
                 :
                 5.
                 
              
            
          
        
         
           
           
             AD
             Cap
             XXVII
          
           
             
               
                 fig
                 :
                 1.
                 
              
            
             
               
                 fig
                 ▪
                 2
                 ▪
              
            
          
        
         
           
           
             AD
             Cap
             XXVIII
             et
             XXX
          
           
             
               
                 fig
                 ▪
                 1.
                 
              
            
             
               
                 fig
                 :
                 2.
                 
              
            
             
               
                 fig
                 ▪
                 3
                 ▪
              
            
             
               
                 fig
                 ▪
                 4
                 ▪
              
            
             
               
                 fig
                 ▪
                 5.
                 
              
            
             
               
                 fig
                 :
                 6.