The artificial clock-maker a treatise of watch, and clock-work, wherein the art of calculating numbers for most sorts of movements is explained to the capacity of the unlearned : also, the history of clock-work, both ancient and modern, with other useful matters, never before published / by W.D.
         Derham, W. (William), 1657-1735.
      
       
         
           1696
        
      
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         ESTC R24292
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         ocm 08118750
         40878
         
           
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             The artificial clock-maker a treatise of watch, and clock-work, wherein the art of calculating numbers for most sorts of movements is explained to the capacity of the unlearned : also, the history of clock-work, both ancient and modern, with other useful matters, never before published / by W.D.
             Derham, W. (William), 1657-1735.
          
           [10], 132 p., [1] leaf of plates : ill.
           
             Printed for James Knapton,
             London :
             1696.
          
           
             Reproduction of original in the Bodleian Library.
          
        
      
    
     
       
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         eng
      
       
         
           Clock and watch making.
        
      
    
     
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           THE
           ARTIFICIAL
           Clock-maker
           .
           A
           Treatise
           of
           Watch
           ,
           and
           Clock-work
           :
           Wherein
           the
           Art
           of
           Calculating
           Numbers
           For
           most
           sorts
           of
           MOVEMENTS
           Is
           explained
           to
           the
           capacity
           of
           the
           Unlearned
           .
           ALSO
           THE
           History
           of
           Clock-work
           ,
           Both
           Ancient
           and
           Modern
           .
           With
           other
           useful
           matters
           never
           before
           Published
           .
           By
           
             W.
             D.
          
           M.
           A.
           
        
         
           LONDON
           ,
           Printed
           for
           
             James
             Knapton
          
           ,
           at
           the
           Crown
           in
           St.
           Pauls
           Church-yard
           ,
           1696.
           
        
      
       
         
         
         
           The
           Preface
           .
        
         
           THe
           following
           Book
           was
           at
           first
           drawn
           up
           in
           a
           rude
           manner
           ,
           only
           to
           please
           my self
           ,
           and
           divert
           the
           vacant
           hours
           of
           a
           Solitary
           Country
           Life
           .
           But
           it
           is
           now
           published
           ,
           purely
           in
           hopes
           of
           its
           doing
           some
           good
           in
           the
           World
           ,
           among
           such
           ,
           whose
           Genius
           and
           Leisure
           lead
           them
           to
           Mechanical
           Studies
           ,
           or
           those
           whose
           business
           and
           livelihood
           it
           is
           .
        
         
           Many
           there
           are
           ,
           whose
           fault
           ,
           or
           calamity
           it
           is
           ,
           to
           have
           time
           lying
           upon
           their
           hands
           ;
           and
           for
           want
           of
           innocent
           ,
           do
           betake
           themselves
           to
           hurtful
           pleasures
           .
           This
           is
           the
           too
           common
           misfortune
           of
           Persons
           of
           Quality
           .
           Among
           some
           of
           the
           courser
           sort
           of
           these
           ,
           if
           this
           Book
           shall
           find
           some
           acceptance
           ,
           it
           may
           be
           a
           means
           to
           compose
           their
           loose
           Spirits
           ;
           and
           by
           an
           innocent
           guile
           ,
           initiate
           them
           in
           other
           Studies
           ,
           of
           greater
           use
           to
           themselves
           ,
           their
           family
           ,
           and
           country
           .
           However
           it
           may
           hinder
           their
           commission
           of
           many
           sins
           ,
           which
           are
           the
           effects
           of
           idleness
           .
        
         
           If
           there
           be
           any
           one
           person
           ,
           in
           whom
           these
           good
           effects
           are
           produced
           ,
           I
           shall
           think
           my
           idle
           hours
           well
           bestowed
           ,
           and
           bless
           God
           for
           it
           .
           However
           upon
           the
           account
           of
           the
           innocence
           of
           my
           end
           in
           publishing
           this
           Book
           ,
           and
           that
           it
           was
           written
           only
           as
           the
           ha●●less
           (
           I
           
           may
           add
           also
           the
           vertuous
           )
           sport
           of
           leisure
           hours
           ;
           I
           think
           my self
           excusable
           to
           God
           and
           the
           World
           ,
           for
           the
           expence
           of
           so
           much
           time
           ,
           in
           a
           subject
           different
           from
           my
           Profession
           .
        
         
           But
           besides
           ,
           I
           think
           my self
           under
           some
           little
           obligations
           of
           Justice
           and
           Charity
           ,
           to
           publish
           the
           ensuing
           papers
           for
           the
           sake
           of
           those
           ,
           whose
           business
           the
           Mechanick
           part
           is
           .
           I
           take
           it
           to
           be
           a
           Charity
           to
           the
           Trade
           ;
           because
           there
           are
           many
           (
           altho
           excellent
           in
           the
           working
           part
           )
           who
           are
           utterly
           unskilled
           in
           the
           artificial
           part
           of
           it
           .
           And
           then
           ,
           it
           is
           a
           debt
           I
           pay
           :
           because
           I
           owe
           somewhat
           of
           health
           ,
           as
           well
           as
           diversion
           to
           the
           Study
           ,
           and
           practice
           of
           these
           sort
           of
           Mechanicks
           .
           And
           the
           best
           requital
           I
           can
           make
           for
           my
           trespass
           ,
           is
           to
           publish
           what
           I
           have
           had
           better
           opportunities
           perhaps
           of
           Learning
           ,
           than
           many
           Workmen
           have
           .
        
         
           And
           further
           yet
           ,
           there
           is
           another
           rea&
           ;
           son
           ,
           which
           much
           prevailed
           with
           me
           to
           publish
           this
           Book
           ,
           viz.
           Because
           no
           body
           ,
           that
           I
           know
           of
           ,
           hath
           prevented
           me
           ,
           by
           treating
           so
           plainly
           and
           intelligibly
           of
           this
           subject
           ,
           as
           to
           be
           understood
           by
           a
           vulgar
           Workman
           .
           I
           have
           often
           wondered
           at
           it
           ,
           that
           so
           useful
           and
           delightful
           a
           part
           of
           Mechanical
           Mathematicks
           should
           lie
           in
           any
           obscurity
           ,
           in
           an
           age
           wherein
           such
           vast
           improvements
           have
           been
           made
           therein
           ,
           and
           when
           many
           Books
           are
           daily
           published
           upon
           every
           subject
           .
           I
           speak
           here
           of
           this
           Art
           remaining
           in
           obscurity
           ;
           not
           as
           if
           nothing
           was
           ever
           written
           of
           it
           ,
           and
           I
           the
           
           in
           venter
           of
           Automatical
           Computation
           .
        
         
           But
           altho
           I
           cannot
           assume
           the
           glory
           of
           being
           the
           first
           Writer
           upon
           this
           subject
           ,
           yet
           very
           few
           have
           as
           yet
           done
           it
           ;
           of
           which
           I
           shall
           next
           give
           some
           account
           .
        
         
           
             Cardan
             ,
             Kircher
          
           ,
           and
           Scottus
           promised
           it
           ;
           but
           I
           do
           not
           find
           they
           ever
           published
           any
           thing
           to
           the
           purpose
           of
           it
           .
           Our
           great
           Mr.
           Oughtred
           I
           take
           to
           be
           the
           first
           that
           ever
           wrote
           to
           any
           purpose
           about
           the
           Calculation
           of
           Automata
           :
           And
           I
           believe
           he
           was
           the
           first
           that
           brought
           that
           Art
           under
           Rules
           ,
           in
           his
           little
           treatise
           called
           Automtaa
           .
           This
           Book
           was
           first
           surreptitiously
           published
           in
           English
           in
           a
           little
           Book
           ,
           called
           
             Horolog
             .
             Dialogues
          
           ,
           in
           the
           year
           1675
           ;
           and
           afterwards
           far
           more
           compleatly
           in
           Latin
           ,
           at
           the
           Theatre
           in
           Oxon
           ,
           among
           Mr.
           Oughtred
           s
           
             Opusc
             .
             Mathem
          
           .
           in
           the
           year
           1677.
           
           This
           last
           edition
           it
           was
           my
           misfortune
           not
           to
           meet
           with
           ,
           until
           it
           was
           too
           late
           ,
           and
           therefore
           I
           have
           been
           forced
           to
           quote
           the
           first
           ,
           and
           worst
           in
           my
           Book
           .
        
         
           What
           Mr.
           Oughtred
           had
           wrapt
           up
           in
           ●his
           Algebraick
           obscure
           Characters
           ,
           was
           afterwards
           put
           into
           plainer
           Language
           ▪
           by
           that
           excellent
           Mathematician
           Sir
           
             Jon.
             Moor
          
           ,
           with
           some
           additions
           of
           his
           own
           ;
           which
           you
           have
           in
           his
           
             Math.
             Compend
          
           .
           and
           since
           him
           ,
           by
           Mr.
           Leyborne
           ,
           in
           his
           
             Pleasure
             with
             Profit
          
           .
        
         
           I
           hope
           I
           shall
           not
           be
           judged
           to
           have
           transgressed
           the
           Rules
           of
           Modesty
           ,
           in
           coming
           af●r
           three
           such
           famous
           men
           ;
           neither
           should
           I
           
           venture
           that
           censure
           ,
           but
           for
           two
           reasons
           .
           One
           is
           ,
           I
           find
           by
           experience
           ,
           that
           what
           they
           have
           written
           ,
           is
           understood
           by
           very
           few
           Workmen
           .
           And
           therefore
           I
           have
           endeavoured
           ,
           with
           all
           industry
           ,
           to
           make
           the
           matter
           as
           plain
           as
           I
           could
           for
           such
           .
           For
           which
           reason
           ,
           I
           hope
           the
           more
           learned
           Reader
           will
           excuse
           my
           using
           many
           words
           ,
           when
           fewer
           would
           have
           served
           his
           turn
           ;
           and
           that
           I
           have
           condescended
           to
           low
           things
           ,
           (
           and
           to
           him
           needless
           )
           as
           teaching
           the
           Golden-rule
           ,
           &c.
           
           The
           other
           reason
           is
           ,
           that
           what
           those
           three
           have
           written
           ,
           relates
           only
           ,
           or
           chiefly
           to
           the
           Watch-part
           .
           To
           which
           I
           have
           added
           several
           other
           things
           of
           my
           own
           :
           particularly
           the
           Calculation
           of
           the
           Clock-part
           ,
           &c.
           
           I
           have
           been
           forced
           to
           reduce
           to
           Rules
           my self
           ,
           and
           to
           name
           no
           more
           ,
           the
           Historical
           part
           hath
           not
           been
           so
           much
           as
           attempted
           before
           ,
           that
           I
           know
           of
           .
        
         
           These
           Reasons
           will
           ,
           I
           hope
           ,
           excuse
           me
           with
           the
           most
           censorious
           Reader
           ,
           not
           only
           for
           presuming
           to
           write
           after
           so
           accurate
           a
           ▪
           piece
           ,
           as
           Mr.
           Oughtred
           s
           is
           ,
           but
           also
           the
           Novelty
           of
           the
           subject
           ,
           will
           I
           hope
           procure
           for
           me
           a
           candid
           interpretation
           of
           the
           faults
           and
           blunders
           ,
           that
           I
           may
           have
           unwittingly
           committed
           .
        
         
           To
           the
           preceeding
           account
           of
           what
           others
           have
           written
           (
           which
           shews
           what
           help
           I
           have
           had
           from
           printed
           Books
           )
           I
           shall
           subjoyn
           my
           acknowledgments
           ,
           and
           thanks
           to
           the
           principal
           of
           my
           friends
           ,
           who
           have
           given
           me
           thei●
           assistance
           in
           compiling
           this
           Book
           .
           But
           thei●
           
           names
           I
           shall
           not
           make
           more
           publick
           than
           mine
           own
           ,
           being
           unwilling
           to
           be
           discovered
           my self
           .
           In
           the
           Chap.
           of
           the
           Terms
           of
           Art
           ,
           I
           owe
           much
           to
           the
           assistance
           of
           
             L.
             Br
             ....
          
           a
           judicious
           Workman
           in
           White-chappel
           ,
           who
           drew
           me
           up
           a
           Scheme
           of
           the
           Clock-maker
           ▪
           s
           Language
           .
           In
           the
           History
           of
           the
           Modern
           Inventions
           ,
           I
           have
           had
           (
           among
           some
           others
           )
           the
           assistance
           chiefly
           of
           the
           ingenious
           
             Dr.
             H
             ....
          
           and
           
             Mr.
             T
             ....
             ▪
          
           The
           former
           being
           the
           Author
           of
           some
           ,
           and
           well
           acquainted
           with
           others
           ,
           of
           the
           Mechanical
           Inventions
           of
           that
           fertile
           Reign
           of
           King
           Charles
           the
           II.
           and
           the
           latter
           actually
           concerned
           in
           all
           ,
           or
           most
           of
           the
           late
           inventions
           in
           Clock-work
           ,
           by
           means
           of
           his
           famed
           skill
           in
           that
           ,
           and
           other
           Mechanick
           operations
           .
        
         
           There
           are
           some
           other
           contrivances
           of
           this
           last
           age
           (
           besides
           those
           I
           have
           mentioned
           )
           which
           I
           have
           passed
           over
           in
           silence
           ;
           because
           either
           they
           are
           only
           branches
           ,
           or
           improvements
           of
           the
           inventions
           I
           have
           taken
           notice
           of
           ,
           (
           such
           as
           several
           ways
           of
           repeating
           work
           ,
           &c.
           )
           or
           else
           ,
           they
           only
           collaterally
           relate
           to
           Watch-work
           (
           as
           the
           inventions
           of
           Cutting-Engines
           ,
           Fusy-Engines
           ,
           &c.
           )
           To
           treat
           of
           all
           these
           ,
           would
           swell
           my
           Book
           far
           beyond
           its
           intended
           bounds
           ;
           which
           I
           have
           already
           somewhat
           exceeded
           .
           I
           shall
           therefore
           commit
           this
           task
           to
           some
           better
           Pen
           ,
           hoping
           that
           no
           person
           will
           take
           it
           amiss
           ,
           that
           I
           have
           not
           mentioned
           what
           I
           have
           been
           beholding
           to
           him
           for
           the
           relation
           of
           .
        
         
         
           For
           the
           resons
           last
           mentioned
           ,
           I
           have
           also
           left
           out
           of
           my
           Book
           ,
           a
           Chapter
           of
           the
           Art
           of
           making
           ,
           and
           using
           many
           sorts
           of
           Sodders
           ,
           the
           way
           of
           colouring
           Metals
           ,
           &c.
           useful
           in
           the
           practice
           of
           Clock-work
           .
           This
           I
           had
           prepared
           for
           the
           sake
           of
           Mercurial
           Gentlemen
           ,
           but
           omitted
           printing
           it
           ,
           and
           some
           other
           things
           ,
           out
           of
           Charity
           to
           poor
           Apprentices
           and
           other
           Workmen
           ,
           whose
           purses
           I
           am
           unwilling
           my
           volume
           should
           too
           much
           exceed
           .
        
         
           If
           I
           have
           at
           any
           time
           invaded
           the
           Workman's
           province
           ,
           it
           was
           not
           because
           I
           pretend
           to
           teach
           him
           his
           Trade
           ;
           but
           either
           for
           Gentlemen's
           sakes
           ,
           or
           when
           the
           matter
           led
           me
           necessarily
           to
           it
           .
        
         
           I
           have
           nothing
           more
           to
           add
           ,
           but
           that
           I
           would
           have
           this
           little
           Treatise
           looked
           upon
           only
           as
           an
           Essay
           ,
           which
           I
           hope
           will
           prompt
           some
           abler
           pen
           to
           perform
           the
           task
           better
           ,
           especially
           in
           the
           Historical
           part
           .
           For
           since
           Watch-work
           oweth
           so
           much
           to
           our
           Age
           ,
           and
           Country
           ,
           t
           is
           pity
           that
           it
           should
           not
           be
           remembred
           :
           especially
           when
           we
           cannot
           but
           lament
           the
           great
           defect
           of
           History
           ,
           about
           the
           beginning
           and
           improvements
           of
           this
           ingenious
           and
           useful
           Art.
           
        
      
       
         
         
           THE
           CONTENTS
           .
        
         
           
             
               CHap.
               I.
               Of
               the
               Terms
               of
               Art.
               
            
             
               
                 The
                 more
                 general
                 Terms
                 .
                 
                   p.
                   2.
                
                 
              
               
                 Names
                 belonging
                 properly
                 to
                 the
                 Watch-part
                 .
                 
                   p.
                   8.
                
                 
              
               
                 Names
                 of
                 the
                 Clock-part
                 .
                 
                   p.
                   5.
                
                 
              
            
          
           
             
               Chap.
               II.
               The
               Art
               of
               Calculation
               .
            
             
               
                 
                   Sect.
                   1.
                   
                   Preliminary
                   Rules
                   .
                
                 
                   
                     To
                     find
                     the
                     turns
                     of
                     a
                     Wheel
                     or
                     Pinion
                     ,
                     8.
                     
                  
                   
                     The
                     way
                     of
                     writing
                     down
                     the
                     Numbers
                     ,
                     9.
                     
                  
                   
                     To
                     find
                     the
                     turns
                     of
                     any
                     ,
                     or
                     all
                     the
                     Wheels
                     in
                     the
                     Movement
                     ,
                     10.
                     
                  
                   
                     To
                     find
                     the
                     Beats
                     of
                     the
                     Ballance
                     in
                     all
                     the
                     Watches
                     going
                     ,
                     or
                     in
                     one
                     turn
                     of
                     any
                     Wheel
                     ,
                     11.
                     
                  
                   
                     Two
                     strokes
                     to
                     every
                     tooth
                     of
                     the
                     Crown
                     ▪
                     wheel
                     ,
                     14.
                     
                  
                
              
               
                 
                   Sect.
                   2.
                   
                   Calculation
                   of
                   the
                   Watch-part
                   .
                
                 
                   
                     Several
                     ways
                     of
                     performing
                     one
                     and
                     the
                     same
                     motion
                     ,
                     15.
                     
                  
                   
                     A
                     Rule
                     to
                     vary
                     Numbers
                     ,
                     16.
                     
                  
                   
                     The
                     way
                     of
                     working
                     the
                     Golden
                     Rule
                     ,
                     17.
                     
                  
                   
                     A
                     very
                     useful
                     Rule
                     to
                     vary
                     inconvenient
                     Numbers
                     ,
                     18.
                     
                  
                   
                     Rules
                     of
                     perpetual
                     use
                     in
                     proportioning
                     the
                     parts
                     of
                     a
                     Watch
                     ,
                     19.
                     
                  
                   
                     Examples
                     of
                     contriving
                     a
                     piece
                     of
                     ordinary
                     Watch-work
                     ,
                     22.
                     
                  
                   
                     Examples
                     thereof
                     for
                     Minutes
                     ,
                     and
                     Seconds
                     ,
                     29.
                     
                  
                
              
               
                 
                   Sect.
                   3.
                   
                   Calculation
                   of
                   the
                   Striking-part
                   .
                
                 
                   
                     
                     General
                     Observations
                     and
                     Rules
                     relating
                     to
                     the
                     Wheel-work
                     of
                     a
                     Clock
                     ,
                     
                       p.
                       33.
                    
                     
                  
                   
                     Rules
                     of
                     perpetual
                     use
                     in
                     proportioning
                     the
                     parts
                     of
                     a
                     Clock
                     ,
                     35.
                     
                  
                   
                     Examples
                     of
                     Calculating
                     the
                     Numbers
                     of
                     a
                     small
                     Clock
                     ,
                     38.
                     
                  
                   
                     Examples
                     of
                     Clocks
                     of
                     longer
                     continuance
                     ,
                     39.
                     
                  
                   
                     An
                     useful
                     Rule
                     to
                     find
                     the
                     number
                     of
                     Strokes
                     in
                     one
                     turn
                     of
                     the
                     Fusy
                     ,
                     33.
                     
                  
                   
                     Examples
                     of
                     fixing
                     the
                     Pinion
                     of
                     Report
                     ,
                     34.
                     
                  
                
              
               
                 
                   Sect.
                   4.
                   
                   Of
                   Quarters
                   and
                   Chimes
                   .
                
                 
                   
                     Notes
                     concerning
                     the
                     Quarters
                     ,
                     45.
                     
                  
                   
                     Of
                     making
                     the
                     Chime-barrel
                     ,
                     46.
                     
                  
                   
                     Of
                     dividing
                     it
                     ,
                     and
                     setting
                     on
                     the
                     Chime-pins
                     ,
                     47.
                     
                  
                   
                     Chimes
                     of
                     
                       Psal
                       .
                       100
                    
                     ,
                     and
                     of
                     a
                     Song-tune
                     ,
                     50.
                     
                  
                   
                     Another
                     way
                     of
                     setting
                     Chimes
                     on
                     the
                     Barrel
                     ,
                     52.
                     
                  
                
              
               
                 
                   Sect.
                   5.
                   
                   To
                   calculate
                   Numbers
                   to
                   represent
                   the
                   Celestial
                   Motions
                   .
                
                 
                   
                     Contrivance
                     of
                     Movements
                     only
                     to
                     shew
                     these
                     Motions
                     ,
                     53.
                     
                  
                   
                     To
                     add
                     it
                     to
                     a
                     Watch
                     that
                     shews
                     the
                     hour
                     of
                     the
                     day
                     ,
                     55.
                     
                  
                   
                     A
                     motion
                     to
                     shew
                     the
                     day
                     of
                     the
                     month
                     ,
                     56.
                     
                  
                   
                     To
                     shew
                     the
                     Age
                     of
                     the
                     Moon
                     ,
                     57.
                     
                  
                   
                     To
                     shew
                     the
                     day
                     of
                     the
                     Year
                     ,
                     and
                     Sun's
                     place
                     in
                     the
                     Ecliptick
                     ,
                     his
                     Rising
                     or
                     Setting
                     ,
                     &c.
                     
                     58.
                     
                  
                   
                     To
                     shew
                     the
                     Tydes
                     ,
                     ib.
                     To
                     represent
                     the
                     motion
                     of
                     the
                     Planets
                     ,
                     fixed
                     Stars
                     ,
                     &c.
                     
                     60.
                     
                  
                
              
            
          
           
             
               Chap.
               
                 III.
                 To
                 alter
                 Clock-work
                 ,
                 p.
                 62.
                 
              
            
             
               
                 Example
                 of
                 converting
                 a
                 12
                 hour
                 Ballance-clock
                 into
                 a
                 Pendulum
                 ,
                 63.
                 
              
               
                 To
                 make
                 it
                 go
                 30
                 hours
                 ,
                 65.
                 
              
               
                 To
                 change
                 the
                 Clock-part
                 ,
                 67.
                 
              
            
          
           
             
               Chap.
               IV.
               To
               size
               Wheels
               and
               Pinions
               .
            
             
               
                 To
                 do
                 it
                 Arithmetically
                 ,
                 69.
                 
              
               
                 Mechanically
                 ,
                 70.
                 
              
            
          
           
             
               Chap.
               V.
               Of
               Pendulums
               .
            
             
               
                 Irregularities
                 of
                 Pendular
                 motions
                 remedied
                 ,
                 71.
                 
              
               
                 Cause
                 of
                 the
                 difference
                 of
                 the
                 motion
                 of
                 the
                 same
                 
                 Pendulum
                 ,
                 72.
                 
              
               
                 True
                 length
                 of
                 a
                 Pendulum
                 that
                 vibrateth
                 Seconds
                 ,
                 73.
                 
              
               
                 To
                 find
                 the
                 Center
                 of
                 Oscillation
                 ,
                 74.
                 
              
               
                 To
                 calculate
                 the
                 Lengths
                 ,
                 or
                 Vibrations
                 of
                 Pendulums
                 ,
                 75.
                 
              
               
                 A
                 Table
                 of
                 Lengths
                 and
                 Swings
                 ,
                 78.
                 
              
               
                 To
                 correct
                 the
                 motion
                 of
                 a
                 Pendulum
                 ,
                 79.
                 
              
            
          
           
             
               Chap.
               VI.
               The
               Antiquity
               ,
               and
               general
               History
               of
               Watch-work
               .
            
             
               
                 The
                 ancientest
                 Time-engine
                 ,
                 82.
                 
              
               
                 The
                 Grecian
                 and
                 Roman
                 ways
                 of
                 measuring
                 Time
                 ,
                 83.
                 
              
               
                 Some
                 horological
                 Instruments
                 mentioned
                 by
                 ancient
                 Authors
                 ,
                 84.
                 
              
               
                 Watch
                 ,
                 or
                 Clock-work
                 ,
                 no
                 new
                 German
                 Invention
                 ,
                 86.
                 
              
               
                 The
                 Sphere
                 of
                 
                   Archimedes
                   ,
                   87.
                
                 
              
               
                 Of
                 
                   Po●idonius
                   ,
                   89.
                
                 
              
               
                 The
                 beginning
                 of
                 our
                 present
                 Clock-work
                 ,
                 91.
                 
              
               
                 Clocks
                 that
                 perform
                 strange
                 feats
                 ,
                 92.
                 
              
            
          
           
             
               Chap.
               VII
               .
               The
               Invention
               of
               Pendulum
               Watches
               .
            
             
               
                 Mr.
                 Hugens
                 the
                 Inventer
                 ,
                 
                   p.
                   93.
                
                 
              
               
                 Others
                 claiming
                 it
                 ,
                 94.
                 
              
               
                 Their
                 beginning
                 in
                 
                   England
                   ,
                   95.
                
                 
              
               
                 The
                 contriver
                 of
                 their
                 carrying
                 a
                 heavy
                 Ball
                 ,
                 &c.
                 
                 96.
                 
              
               
                 Their
                 use
                 ,
                 ibid.
                 
              
               
                 The
                 Circular
                 Pendulum
                 ,
                 97.
                 
              
            
          
           
             
               Chap.
               VIII
               .
               Of
               the
               Invention
               of
               Pocket
               Pendulum
               Watches
               .
            
             
               
                 Inventer
                 ,
                 
                   p.
                   99.
                
                 
              
               
                 Several
                 ways
                 of
                 them
                 ,
                 ib.
                 
              
               
                 The
                 time
                 when
                 invented
                 ,
                 103.
                 
              
               
                 Mr.
                 
                 Hugens's
                 Watch
                 ,
                 104.
                 
              
            
          
           
             
               Chap.
               IX
               .
               The
               Invention
               of
               Repeating
               Clocks
               .
            
             
               
                 The
                 Inventer
                 ,
                 
                   p.
                   106.
                
                 
              
               
                 When
                 and
                 by
                 whom
                 first
                 used
                 in
                 Pocket
                 Clocks
                 ,
                 107.
                 
              
            
          
           
             
               Chap.
               XI
               .
               Numbers
               for
               various
               Movements
               .
            
             
               
                 The
                 way
                 of
                 Watch-makers
                 writing
                 down
                 their
                 
                 Numbers
                 ,
                 109.
                 
              
               
                 Numbers
                 of
                 an
                 8
                 day
                 Piece
                 ,
                 110.
                 
              
               
                 A
                 Month
                 Piece
                 ,
                 112.
                 
              
               
                 A
                 Two
                 Month
                 Piece
                 ,
                 113.
                 
              
               
                 A
                 Quarter
                 of
                 Year
                 piece
                 ,
                 114.
                 
              
               
                 An
                 Half
                 Year
                 Piece
                 ,
                 ib.
                 
              
               
                 A
                 Year
                 Piece
                 ,
                 115.
                 
              
               
                 A
                 lesser
                 30
                 hours
                 Piece
                 ,
                 ib.
                 
              
               
                 A
                 small
                 Week
                 Piece
                 ,
                 ib.
                 
              
               
                 A
                 small
                 Month
                 Piece
                 ,
                 116.
                 
              
               
                 A
                 small
                 Year
                 Piece
                 ,
                 ib.
                 
              
               
                 An
                 8
                 day
                 Piece
                 Pend.
                 3
                 inches
                 ,
                 117.
                 
              
               
                 Numbers
                 representing
                 the
                 Motion
                 of
                 the
                 Planet
                 
                   Saturn
                   ,
                   118.
                
                 
              
               
                 Of
                 
                   Jupiter
                   ,
                   ib.
                
                 
              
               
                 Monsieur
                 
                 Romer's
                 Instrument
                 for
                 
                 Jupiter's
                 Satellites
                 ,
                 119.
                 
              
               
                 Numbers
                 for
                 
                   Mars
                   ,
                   Venus
                
                 ,
                 and
                 
                   Mercury
                   ,
                   120.
                
                 
              
               
                 For
                 the
                 Dragons
                 Head
                 and
                 Tail
                 ,
                 121.
                 
              
               
                 Numbers
                 for
                 Pocket
                 Watches
                 of
                 8
                 days
                 ,
                 ib.
                 
              
               
                 Of
                 30
                 hours
                 ,
                 
                   122
                   ,
                   123.
                
                 
              
               
                 The
                 way
                 to
                 amend
                 the
                 Numbers
                 ,
                 123.
                 
              
            
          
           
             
               Chap.
               XI
               .
               Tables
               of
               Time.
               
            
             
               
                 A
                 Table
                 for
                 ready
                 casting
                 up
                 the
                 parts
                 of
                 Time
                 ,
                 124.
                 
              
               
                 A
                 Table
                 to
                 set
                 a
                 Watch
                 by
                 the
                 Fixed
                 Stars
                 ,
                 125.
                 
              
               
                 A
                 Table
                 of
                 the
                 Variations
                 of
                 the
                 Hour
                 by
                 the
                 Sun's
                 Refraction
                 ,
                 117.
                 
              
               
                 Observations
                 concerning
                 Refractions
                 ,
                 and
                 the
                 Variations
                 of
                 the
                 Hour
                 ,
                 
                   128
                   ▪
                
              
            
          
        
      
    
     
       
         
         The
         Artificial
         CLOCK-MAKER
         .
      
       
         
           CHAP.
           I.
           Of
           the
           Terms
           of
           Art
           ,
           or
           Names
           by
           which
           the
           parts
           of
           an
           Automaton
           are
           called
           .
        
         
           IT
           is
           necessary
           that
           I
           should
           shew
           the
           meaning
           of
           those
           Terms
           which
           Clock-makers
           use
           ,
           that
           Gentlemen
           and
           others
           ,
           unskilful
           in
           the
           Art
           ,
           may
           know
           how
           to
           express
           themselves
           properly
           ,
           in
           speaking
           ;
           and
           also
           understand
           what
           I
           shall
           say
           in
           the
           following
           Book
           .
        
         
           I
           shall
           not
           trouble
           the
           Reader
           with
           a
           recital
           of
           every
           name
           that
           doth
           occur
           ,
           but
           only
           such
           as
           I
           shall
           have
           occasion
           to
           
           use
           in
           the
           following
           discourse
           ,
           and
           some
           few
           others
           that
           offer
           themselves
           ,
           upon
           a
           transient
           view
           of
           a
           piece
           of
           work
           .
        
         
           I
           begin
           with
           the
           more
           general
           Terms
           :
           as
           ,
           the
           Frame
           ;
           which
           is
           that
           which
           contains
           the
           Wheels
           ,
           and
           the
           rest
           of
           the
           work
           .
           The
           Pillars
           ,
           and
           Plates
           ,
           are
           what
           it
           chiefly
           consists
           of
           .
        
         
           Next
           for
           the
           Spring
           ,
           and
           its
           appurtenances
           .
           That
           which
           the
           Spring
           lies
           in
           ,
           is
           the
           Spring-box
           ;
           that
           which
           the
           Spring
           laps
           about
           ,
           in
           the
           middle
           of
           the
           Spring-box
           ,
           is
           the
           Spring-Arbor
           ;
           to
           which
           the
           Spring
           is
           hooked
           at
           one
           end
           .
           At
           the
           top
           of
           the
           Spring-Arbor
           ,
           is
           the
           Endless-Screw
           ,
           and
           its
           Wheel
           .
        
         
           That
           which
           the
           Spring
           draweth
           ,
           and
           about
           which
           the
           Chain
           or
           String
           is
           wrapped
           ,
           and
           which
           is
           commonly
           taper
           ,
           is
           the
           Fusy
           .
           In
           larger
           work
           ,
           going
           with
           weights
           ,
           where
           it
           is
           cylindrical
           ,
           it
           is
           called
           the
           Barrel
           .
           The
           small
           Teeth
           at
           the
           bottom
           of
           the
           Fusy
           ,
           or
           Barrel
           ,
           that
           stop
           it
           in
           winding
           up
           ,
           is
           the
           Ratchet
           .
           That
           which
           stops
           it
           when
           wound
           up
           ,
           and
           is
           for
           that
           end
           driven
           up
           by
           the
           String
           ,
           is
           the
           Garde-caut
           ,
           or
           Guard-Cock
           ,
           as
           others
           ;
           and
           Garde-du-Cord
           ,
           and
           Gard-du-Gut
           ,
           as
           others
           call
           it
           .
        
         
         
           The
           parts
           of
           a
           Wheel
           are
           ,
           the
           Hoop
           ,
           or
           Rim
           :
           the
           Teeth
           :
           the
           Cross
           :
           and
           the
           Collet
           ,
           or
           piece
           of
           Brass
           ,
           soddered
           on
           the
           Arbor
           ,
           or
           Spindle
           ,
           on
           which
           the
           Wheel
           is
           rivetted
           .
        
         
           A
           Pinion
           is
           that
           little
           Wheel
           ,
           which
           plays
           in
           the
           teeth
           of
           the
           Wheel
           .
           Its
           teeth
           (
           which
           are
           commonly
           4
           ,
           5
           ,
           6
           ,
           8
           ,
           &c.
           )
           are
           called
           Leves
           ,
           not
           Teeth
           .
        
         
           The
           ends
           of
           the
           Spindle
           ,
           are
           called
           Pevetts
           :
           the
           holes
           in
           which
           they
           run
           ,
           Pevet-holes
           .
        
         
           The
           guttered
           Wheel
           ,
           with
           Iron
           spikes
           at
           the
           bottom
           ,
           in
           which
           the
           line
           of
           ordinary
           House-Clocks
           doth
           run
           ,
           is
           called
           the
           Pully
           .
        
         
           I
           need
           not
           speak
           of
           the
           Dial-plate
           ,
           the
           
             Hand
             ,
             Screws
             ,
             Wedges
             ,
             Stops
             ,
          
           &c.
           
        
         
           Thus
           much
           for
           general
           Names
           ,
           which
           are
           common
           to
           all
           parts
           of
           a
           Movement
           .
        
         
           The
           parts
           of
           a
           Movement
           ,
           which
           I
           shall
           consider
           ,
           are
           the
           Watch
           ,
           and
           Clock
           .
        
         
           The
           Watch-part
           of
           a
           Movement
           is
           that
           which
           serveth
           to
           the
           measuring
           the
           hours
           .
           In
           which
           the
           first
           thing
           I
           shall
           consider
           is
           the
           Ballance
           :
           whose
           parts
           are
           ,
           the
           Rim
           ,
           which
           is
           the
           circular
           part
           of
           it
           :
           the
           Verge
           ,
           is
           its
           Spindle
           :
           to
           which
           belong
           the
           two
           
           Pallets
           ,
           or
           Nuts
           ,
           which
           play
           in
           the
           fangs
           of
           the
           Crown
           ▪
           Wheel
           :
           in
           Pocket-Watches
           ,
           that
           strong
           Stud
           in
           which
           the
           lower
           Pevet
           of
           the
           Verge
           plays
           ,
           and
           in
           the
           middle
           of
           which
           one
           Pevet
           of
           the
           Crown-Wheel
           runs
           ,
           is
           called
           the
           Pottans
           :
           the
           wrought
           piece
           which
           covers
           the
           Ballance
           ,
           and
           in
           which
           the
           upper
           Pevet
           of
           the
           Ballance
           plays
           ,
           is
           the
           Cock.
           The
           small
           Spring
           in
           the
           new
           Pocket-Watches
           is
           the
           Regulator
           .
        
         
           The
           parts
           of
           a
           Pendulum
           are
           ,
           the
           
             Verge
             ▪
             Pallets
          
           and
           Cocks
           ,
           as
           before
           .
           The
           Ball
           in
           long
           Pendulums
           ,
           the
           Bob
           in
           short
           ones
           ,
           is
           the
           Weight
           at
           the
           bottom
           .
           The
           Rod
           ,
           or
           Wire
           is
           plain
           .
           The
           terms
           peculiar
           to
           the
           
             Royal
             Swing
          
           ,
           are
           the
           Pads
           ,
           which
           are
           the
           Pallets
           in
           others
           ,
           and
           are
           fixed
           on
           the
           Spindle
           .
           The
           Fork
           is
           also
           fixed
           on
           the
           Spindle
           ,
           and
           about
           6
           inches
           below
           ,
           catcheth
           hold
           on
           the
           Rod
           ,
           at
           a
           flat
           piece
           of
           Brass
           ,
           called
           the
           Flatt
           ,
           in
           which
           the
           lower
           end
           of
           the
           Spring
           is
           fastened
           .
        
         
           The
           names
           of
           the
           Wheels
           next
           follow
           .
           The
           Crown-Wheel
           in
           Small
           pieces
           ,
           and
           Swing-Wheel
           in
           Royal
           Pendulums
           ,
           is
           that
           Wheel
           which
           drives
           the
           ▪
           Ballance
           ,
           or
           Pendulum
           .
        
         
         
           The
           Contrate-Wheel
           ,
           is
           that
           Wheel
           in
           Pocket-Watches
           ,
           which
           is
           next
           to
           the
           Crown-Wheel
           ,
           whose
           Teeth
           and
           Hoop
           lye
           contrary
           to
           those
           of
           other
           Wheels
           .
        
         
           The
           Great-Wheel
           ,
           or
           First-Wheel
           ,
           is
           that
           which
           the
           Fusy
           ,
           &c.
           immediately
           driveth
           .
           Next
           it
           ,
           are
           the
           
             Second-Wheel
             ,
             Third-Wheel
          
           ,
           &c.
           
        
         
           Next
           followeth
           the
           Work
           between
           the
           Frame
           and
           Dial-Plate
           .
           And
           first
           ,
           is
           the
           Pinion
           of
           Report
           ;
           which
           is
           that
           Pinion
           which
           is
           commonly
           fixed
           on
           the
           Arbor
           of
           the
           Great-Wheel
           ,
           and
           in
           old
           Watches
           used
           to
           have
           commonly
           but
           four
           Leaves
           ;
           which
           driveth
           the
           Dial-Wheel
           ,
           and
           this
           carrieth
           about
           the
           Hand
           .
        
         
           The
           last
           Part
           which
           I
           shall
           speak
           of
           ,
           is
           the
           Clock
           ,
           which
           is
           that
           part
           which
           serveth
           to
           strike
           the
           Hours
           :
           In
           which
           I
           shall
        
         
           First
           speak
           of
           the
           
             Great
             ▪
          
           or
           First-Wheel
           ;
           which
           is
           that
           which
           the
           Weight
           or
           Spring
           first
           drives
           .
           In
           16
           or
           30
           hour
           Clocks
           ,
           this
           is
           commonly
           the
           Pin-Wheel
           ;
           in
           8
           Day
           pieces
           ▪
           the
           Second-Wheel
           is
           commonly
           the
           ▪
           〈◊〉
           This
           Wheel
           with
           Pins
           is
           sometimes
           called
           the
           Striking-Wheel
           ,
           or
           Pin-Wheel
           .
        
         
         
           Next
           to
           this
           Striking-Wheel
           ,
           followeth
           the
           Detent-Wheel
           ,
           or
           Hoop-Wheel
           ,
           having
           a
           Hoop
           almost
           round
           it
           ,
           in
           whic●
           is
           a
           vacancy
           ,
           at
           which
           the
           Cloc●
           locks
           .
        
         
           The
           next
           is
           the
           Third
           ,
           or
           
             Fourth
             ▪
             Wheel
          
           (
           according
           as
           it
           is
           distant
           fro●
           the
           First-Wheel
           )
           called
           also
           the
           
             Warning
             ▪
             Wheel
          
           .
        
         
           And
           lastly
           is
           the
           Flying-Pinion
           ,
           with
           a
           Fly
           or
           Fan
           to
           gather
           Air
           ,
           and
           so
           bridle
           the
           rapidity
           of
           the
           Clock's
           motion
           .
        
         
           Besides
           these
           ,
           there
           are
           the
           Pinion
           o●
           Report
           ,
           of
           which
           before
           ;
           which
           driveth
           round
           the
           Locking-Wheel
           ,
           called
           also
           the
           Count-Wheel
           ,
           with
           11
           Notches
           in
           it
           commonly
           ,
           unequally
           distant
           from
           one
           another
           ,
           to
           make
           the
           Clock
           strike
           the
           hour●
           of
           1
           ,
           2
           ,
           3
           ,
           &c.
           
        
         
           Thus
           much
           for
           the
           Wheels
           of
           the
           Clock
           part
           .
        
         
           Besides
           which
           there
           are
           the
           Rash
           ,
           or
           Ratch
           ;
           which
           is
           that
           sort
           of
           Wheel
           ,
           of
           twelve
           large
           Fangs
           ,
           that
           runneth
           concentrical
           to
           the
           Dial-Wheel
           ,
           and
           serveth
           to
           lift
           up
           the
           Detents
           every
           hour
           ,
           and
           make
           the
           Clock
           strike
           .
        
         
           The
           Detents
           are
           those
           Stops
           ,
           which
           
           by
           being
           lifted
           up
           ,
           or
           let
           ●all
           down
           ,
           do
           lock
           and
           unlock
           the
           Clock
           in
           striking
           .
        
         
           The
           Hammers
           s●rike
           the
           Bell
           :
           The
           Hammer-tails
           are
           what
           the
           Striking-pins
           draw
           back
           the
           Hammers
           by
           .
        
         
           Latches
           are
           what
           li●t
           up
           ,
           and
           unlock
           the
           Work.
           
        
         
           Catches
           are
           what
           hold
           by
           hooking
           ,
           or
           catching
           hold
           of
           .
        
         
           The
           Lifting-pieces
           do
           lift
           up
           ,
           and
           unlock
           the
           Detents
           ,
           in
           the
           Clock
           part
           .
        
      
       
         
           CHAP.
           II.
           The
           Art
           of
           Calculation
           .
        
         
           
             SECT
             .
             I.
             General
             preliminary
             Rules
             and
             Directions
             for
             Calculation
             .
          
           
             §
             1.
             
             FOR
             the
             more
             clear
             understanding
             this
             Chapter
             it
             must
             be
             observed
             ,
             that
             those
             Automata
             (
             whose
             Calculation
             I
             chiefly
             intend
             )
             do
             by
             little
             Interstices
             ,
             or
             Strokes
             ,
             measure
             out
             longer
             
             portions
             of
             Time.
             Thus
             the
             strokes
             o●
             the
             Balance
             of
             a
             Watch
             ,
             do
             measure
             ou●
             Minutes
             ,
             Hours
             ,
             Days
             ,
             &c.
             
          
           
             Now
             to
             scatter
             those
             strokes
             among
             Wheels
             and
             Pinions
             ,
             and
             to
             proportionat●
             them
             ,
             ●so
             as
             to
             measure
             Time
             regularly
             is
             the
             design
             of
             Calculation
             .
             For
             th●
             clearer
             discovery
             of
             which
             ,
             it
             will
             be
             necessary
             to
             proceed
             leisurely
             ,
             and
             gradually
             .
          
           
             
             §
             2.
             
             And
             in
             the
             first
             place
             ,
             you
             are
             to
             know
             ,
             that
             any
             Wheel
             being
             divided
             by
             its
             Pinion
             ,
             shews
             how
             many
             turns
             that
             Pinion
             hath
             to
             one
             turn
             of
             that
             Wheel
             .
             Thus
             a
             Wheel
             of
             60
             teeth
             driving
             a
             Pinion
             of
             6
             ,
             will
             turn
             round
             the
             Pinion
             10
             times
             in
             going
             round
             once
             .
          
           
             From
             the
             Fusy
             to
             the
             Ballance
             the
             Wheels
             drive
             the
             Pinions
             ;
             and
             consequently
             the
             Pinions
             run
             faster
             ,
             or
             go
             more
             turns
             ,
             than
             the
             Wheels
             they
             run
             in
             .
             But
             it
             is
             contrary
             from
             the
             Great-Wheel
             to
             the
             Dial
             Wheel
             .
             Thus
             in
             the
             last
             Example
             ,
             The
             Wheel
             drives
             round
             the
             Pinion
             10
             times
             :
             but
             if
             the
             Pinion
             drove
             the
             Wheel
             ,
             it
             must
             turn
             10
             times
             to
             drive
             the
             Wheel
             round
             once
             .
          
           
             §
             3.
             
             Before
             I
             proceed
             further
             ,
             I
             must
             
             shew
             how
             to
             
               write
               down
            
             the
             Wheels
             and
             Pinions
             .
             Which
             may
             be
             done
             ,
             either
             as
             Vulgar
             Fractions
             ,
             or
             in
             the
             way
             of
             Division
             in
             Vulgar
             Arithmetick
             .
             E.
             C.
             A
             Wheel
             of
             60
             moving
             a
             Pinion
             of
             5
             ,
             may
             be
             set
             down
             thus
             ,
             60
             /
             3
             :
             or
             rather
             thus
             ,
             5
             )
             60
             :
             where
             the
             first
             figure
             is
             the
             Pinion
             ,
             the
             next
             without
             the
             hook
             ,
             is
             the
             Wheel
             .
          
           
             The
             number
             of
             Turns
             ,
             which
             the
             Pinion
             hath
             in
             one
             turn
             of
             the
             Wheel
             ,
             is
             set
             without
             a
             hook
             on
             the
             right
             hand
             :
             as
             5
             )
             60
             (
             12
             ,
             
               i.
               e.
            
             a
             Pinion
             5
             playing
             in
             a
             Wheel
             of
             60
             ,
             moveth
             round
             12
             times
             ,
             in
             one
             turn
             of
             the
             Wheel
             .
          
           
             
             A
             whole
             Movement
             ma●
             be
             noted
             thus
             ,
             4
             /
             ●●
             55
             /
             5
             45
             /
             5
             40
             /
             5
             17
             Notches
             in
             the
             Crown
             ▪
             Wheel
             .
             Or
             rather
             as
             you
             see
             here
             in
             the
             Margin
             :
             where
             the
             uppermost
             number
             ,
             above
             the
             line
             ,
             is
             the
             Pinion
             of
             Report
             4
             ,
             the
             Dial-wheel
             36
             ,
             and
             9
             turns
             of
             the
             Pin.
             of
             Report
             .
             The
             second
             number
             (
             under
             the
             line
             )
             is
             5
             the
             Pinion
             ,
             55
             is
             the
             Great-wheel
             ,
             and
             11
             turns
             of
             the
             Pinion
             it
             driveth
             .
             The
             third
             numbers
             ,
             are
             the
             Second-wheel
             ,
             &c.
             
             The
             fourth
             the
             Contrate-wheel
             ,
             &c.
             
             An●
             the
             single
             number
             17
             under
             all
             ,
             is
             th●
             Crown-wheel
             .
          
           
             §
             4
             By
             the
             §
             2.
             before
             ,
             knowing
             th●
             number
             of
             turns
             ,
             which
             any
             Pinion
             hath
             in
             one
             turn
             of
             the
             Wheel
             it
             worketh
             in
             ▪
             you
             may
             also
             find
             out
             how
             many
             turns
             a●
             Wheel
             or
             Pinion
             hath
             ,
             at
             a
             greater
             distance
             ;
             as
             the
             Contrate-wheel
             ,
             Crown
             ▪
             wheel
             ,
             or
             &c.
             
             For
             it
             is
             but
             multiplying
             
             together
             the
             Quotients
             ,
             and
             the
             number
             produced
             ,
             is
             the
             number
             of
             Turns
             .
             An
             Example
             will
             make
             what
             I
             say
             plain
             :
             
             let
             us
             chuse
             these
             3
             numbers
             here
             set
             down
             ;
             the
             first
             of
             which
             hath
             11
             turns
             ,
             the
             next
             9
             ▪
             and
             the
             last
             8.
             
             If
             you
             multiply
             11
             and
             9
             it
             produceth
             99
             ,
             for
             9
             times
             11
             is
             99
             ,
             that
             is
             ,
             in
             one
             turn
             of
             the
             Whee●
             55
             ,
             there
             are
             99
             turns
             of
             the
             second
             Pinion
             5
             ,
             or
             of
             the
             Wheel
             40.
             
             If
             you
             multiply
             99
             by
             the
             last
             Quotient
             8
             (
             that
             is
             ,
             8
             times
             99
             is
             792
             )
             it
             shews
             the
             number
             of
             turns
             ,
             which
             the
             third
             and
             last
             Pinion
             5
             hath
             .
             So
             that
             this
             third
             ,
             and
             last
             Pinion
             turns
             792
             times
             in
             
             one
             turn
             of
             the
             first
             Wheel
             55.
             
             
             Another
             Example
             will
             make
             it
             still
             more
             plain
             The
             Example
             is
             in
             the
             Margin
             .
             The
             turns
             are
             10
             ,
             9
             and
             8.
             
             These
             multiplied
             as
             before
             run
             thus
             ,
             viz.
             10
             times
             9
             is
             90
             ,
             that
             is
             ,
             the
             Pinion
             6
             (
             which
             is
             the
             Pin.
             of
             the
             third
             Wheel
             40
             )
             turns
             90
             times
             in
             one
             turn
             of
             the
             First
             ▪
             wheel
             80.
             
             This
             last
             product
             90
             being
             multiplied
             by
             8
             ,
             produces
             720
             ▪
             that
             is
             ,
             the
             Pinion
             5
             (
             which
             is
             the
             Pin.
             of
             the
             Crown-wheel
             15
             )
             turns
             720
             times
             in
             one
             turn
             of
             the
             First-wheel
             ,
             of
             80
             teeth
             .
          
           
             §
             5.
             
             We
             may
             now
             proceed
             to
             that
             ,
             which
             is
             the
             very
             groundwork
             of
             all
             ;
             which
             is
             ,
             not
             only
             to
             find
             out
             the
             turns
             ,
             but
             the
             Beats
             also
             of
             the
             Ballance
             in
             those
             turns
             of
             the
             Wheels
             .
             By
             the
             last
             Paragraph
             ,
             having
             found
             out
             the
             number
             of
             turns
             ,
             which
             the
             Crown-wheel
             hath
             in
             one
             turn
             of
             the
             Wheel
             you
             seek
             for
             ,
             you
             must
             then
             multiply
             those
             turns
             of
             the
             Crown-wheel
             by
             its
             number
             of
             Notches
             ,
             and
             this
             will
             give
             you
             half
             the
             number
             of
             Beats
             ,
             in
             that
             one
             turn
             of
             the
             Wheel
             .
             Half
             the
             number
             ,
             I
             say
             ,
             for
             the
             reasons
             in
             the
             following
             
             6
             §
             .
             For
             the
             Explication
             of
             what
             hath
             been
             said
             ,
             we
             will
             take
             the
             example
             in
             the
             last
             §
             :
             the
             Crown-wheel
             there
             ,
             has
             720
             turns
             in
             one
             turn
             of
             the
             first
             Wheel
             〈◊〉
             This
             number
             multiplied
             by
             15
             ,
             the
             Notches
             in
             the
             Crown-wheel
             ,
             produceth
             10800
             ,
             which
             are
             half
             the
             number
             o●
             strokes
             of
             the
             Ballance
             ,
             in
             one
             turn
             of
             the
             first
             wheel
             80.
             
             The
             like
             may
             be
             done
             for
             any
             of
             the
             other
             Wheels
             ;
             as
             the
             Wheel
             54
             ,
             or
             40
             :
             but
             I
             shall
             not
             insist
             upon
             these
             ,
             having
             said
             enough
             .
          
           
             I
             shall
             give
             but
             one
             Example
             more
             which
             will
             fully
             ,
             and
             very
             plainly
             illustrate
             the
             whole
             matter
             .
             
             The
             example
             is
             in
             the
             margin
             ,
             and
             't
             is
             of
             a
             16
             hour
             Watch
             ,
             wherein
             the
             Pinion
             of
             Report
             is
             4
             ,
             the
             Dial-wheel
             32
             ,
             the
             Great-wheel
             i●
             55
             ,
             the
             Pinion
             of
             the
             secon●-Wheel
             is
             5
             ,
             &c.
             the
             numbe●
             of
             Notches
             in
             the
             Crown
             ▪
             wheel
             are
             17
             :
             the
             quotients
             or
             number
             of
             turns
             in
             each
             ,
             are
             8
             ,
             11
             9
             ,
             8.
             
             All
             which
             being
             multiplied
             as
             be
             ▪
             fore
             ,
             make
             6336
             :
             this
             number
             multiplied
             by
             17
             ,
             produceth
             107712
             ;
             which
             la●summ
             is
             half
             the
             number
             of
             Beats
             in
             on●
             
             turn
             of
             the
             Dial-wheel
             .
             The
             half
             number
             of
             Beats
             in
             one
             turn
             of
             the
             Great-wheel
             ,
             you
             will
             find
             to
             be
             13464
             :
             For
             8
             times
             17
             is
             136
             ,
             which
             is
             the
             half
             number
             of
             Beats
             in
             one
             turn
             of
             the
             Contrate-wheel
             40
             :
             and
             9
             times
             136
             ,
             is
             1224
             ,
             the
             half
             beats
             in
             one
             turn
             of
             the
             Second-wheel
             :
             and
             11
             times
             1224
             ,
             is
             13464
             ,
             the
             half
             beats
             in
             one
             turn
             of
             the
             Great-wheel
             55.
             
             And
             8
             times
             this
             last
             ,
             is
             107712
             before
             named
             .
             If
             you
             multiply
             this
             by
             the
             two
             Pallets
             ,
             that
             is
             ,
             double
             it
             ,
             it
             is
             215424
             ,
             which
             is
             the
             number
             of
             Beats
             in
             one
             turn
             of
             the
             Dial-wheel
             ,
             or
             12
             hours
             .
             If
             you
             would
             know
             how
             many
             beats
             this
             Watch
             hath
             in
             an
             hour
             ,
             't
             is
             but
             dividing
             the
             beats
             in
             12
             hours
             ,
             into
             12
             parts
             ,
             and
             it
             gives
             17952
             ,
             the
             Train
             of
             the
             
             Watch
             ,
             or
             beats
             in
             an
             hour
             .
             If
             you
             divide
             this
             into
             60
             parts
             ,
             it
             gives
             299
             and
             a
             little
             more
             ,
             for
             the
             beats
             in
             a
             minute
             .
             And
             so
             you
             may
             go
             on
             to
             seconds
             and
             thirds
             ,
             if
             you
             please
             .
          
           
             Thus
             I
             have
             delivered
             my
             thoughts
             as
             plainly
             as
             I
             can
             ,
             that
             I
             may
             be
             well
             understood
             ;
             this
             being
             the
             very
             foundation
             of
             all
             the
             artificial
             part
             of
             Clock-work
             .
             And
             therefore
             let
             the
             young
             practiser
             exercise
             
             himself
             thorowly
             in
             it
             ,
             in
             more
             than
             one
             example
             .
          
           
             If
             I
             have
             offended
             the
             more
             learned
             ,
             quick-sighted
             Reader
             ,
             by
             using
             m●●y
             words
             ;
             my
             desire
             to
             instruct
             the
             most
             ignorant
             Artist
             ▪
             must
             plead
             my
             excuse
             .
          
           
             
             §
             6.
             
             The
             Ballance
             or
             Swing
             hath
             two
             strokes
             to
             every
             tooth
             of
             the
             Crown-wheel
             .
             For
             each
             of
             the
             two
             Pallets
             hath
             its
             blow
             against
             each
             tooth
             of
             the
             Crown-wheel
             ▪
             Wherefore
             a
             Pendulum
             that
             swings
             Seconds
             ,
             hath
             its
             Crown-wheel
             but
             30.
             
          
        
         
           
             SECT
             .
             II.
             The
             way
             to
             Calculate
             ,
             or
             contrive
             the
             Numbers
             of
             a
             piece
             of
             Watch
             work
             .
          
           
             HAving
             in
             the
             last
             Section
             led
             on
             the
             Reader
             to
             a
             general
             knowledge
             of
             Calculation
             ;
             I
             may
             now
             venture
             him
             further
             into
             the
             more
             obscure
             ,
             and
             useful
             parts
             of
             that
             Art
             :
             Which
             I
             shall
             explain
             with
             all
             possible
             plainness
             ,
             tho
             less
             brevity
             ,
             than
             I
             could
             wish
             .
          
           
             
             §
             1.
             
             The
             same
             motion
             may
             be
             performed
             either
             with
             one
             Wheel
             and
             one
             
             or
             by
             many
             Wheels
             and
             many
             Pinions
             :
             provided
             that
             the
             number
             of
             turns
             of
             all
             those
             Wheels
             bear
             the
             same
             proportion
             to
             all
             those
             Pinions
             ,
             which
             that
             one
             Wheel
             bears
             to
             its
             Pinion
             .
             Or
             (
             which
             is
             the
             same
             thing
             )
             that
             the
             number
             produced
             by
             multiplying
             all
             the
             Wheels
             together
             ,
             be
             to
             the
             number
             produced
             ●y
             multiplying
             all
             the
             Pinions
             together
             ;
             as
             that
             one
             Wheel
             is
             to
             that
             one
             Pinion
             .
             
             Thus
             suppose
             you
             had
             use
             for
             a
             Wheel
             of
             1440
             teeth
             ,
             with
             a
             Pin.
             of
             28
             leaves
             ,
             you
             may
             make
             it
             into
             3
             Wheels
             and
             Pinions
             ,
             viz.
             4
             ●
             36
             ,
             7
             )
             8
             ,
             and
             1
             )
             5.
             
             For
             if
             you
             multiply
             the
             three
             Wheels
             together
             ,
             viz.
             36
             ,
             8
             and
             5
             ;
             and
             the
             three
             Pinions
             together
             by
             themselves
             ,
             viz
             4
             ,
             7
             and
             1
             ,
             you
             will
             find
             1440
             to
             arise
             for
             the
             Wheels
             ,
             and
             28
             for
             the
             Pinions
             .
             Or
             if
             you
             try
             the
             example
             by
             the
             number
             of
             turns
             ,
             it
             will
             be
             the
             same
             .
             For
             28
             )
             1440
             (
             51
             3
             /
             7.
             
             And
             the
             quotients
             and
             turns
             of
             the
             3
             Wheels
             and
             Pinions
             multiplied
             together
             ,
             are
             51
             3
             /
             7
             also
             ,
             as
             in
             the
             last
             example
             .
          
           
             It
             matters
             not
             in
             what
             order
             the
             Wheels
             and
             Pinions
             are
             set
             ,
             or
             which
             Pinion
             runs
             
             in
             which
             Wheel
             :
             Only
             for
             convenien●
             sake
             ,
             they
             commonly
             set
             the
             biggest
             numbers
             to
             drive
             the
             rest
             .
          
           
             §
             2.
             
             Two
             Wheels
             and
             Pinions
             of
             diff●rent
             
             numbers
             may
             perform
             the
             same
             m●tion
             .
             As
             ,
             a
             Wheel
             of
             36
             drives
             a
             Pinio●
             of
             4
             ,
             all
             one
             as
             a
             Wheel
             of
             45
             drives
             Pin.
             of
             5
             ;
             or
             as
             a
             Wheel
             of
             90
             drives
             Pin.
             of
             10.
             
             The
             turns
             of
             each
             are
             9.
             
          
           
             §
             3.
             
             If
             in
             breaking
             your
             Train
             int●
             parcels
             (
             of
             which
             by
             and
             by
             )
             any
             of
             you●
             Quotients
             should
             not
             please
             you
             ;
             or
             
             you
             would
             alter
             any
             other
             two
             number
             which
             are
             to
             be
             multiplied
             together
             ,
             yo●
             may
             vary
             them
             by
             this
             Rule
             :
             Divid●
             your
             two
             numbers
             by
             any
             two
             oth●
             numbers
             which
             will
             measure
             them
             ;
             th●
             multiply
             the
             Quotients
             by
             the
             alternat●
             divisors
             ,
             the
             product
             of
             these
             two
             la●
             numbers
             found
             ,
             shall
             be
             equal
             to
             the
             product
             of
             the
             two
             numbers
             first
             give●
             Thus
             if
             you
             would
             vary
             36
             times
             8
             ,
             d●vide
             these
             by
             any
             two
             numbers
             that
             wi●
             evenly
             measure
             them
             ,
             as
             36
             by
             4
             ,
             and
             by
             1.
             
             The
             fourth
             part
             of
             36
             is
             9
             ,
             and
             divided
             by
             1
             gives
             8.
             
             Multiply
             9
             by
             〈◊〉
             the
             product
             is
             9
             ;
             and
             8
             multiplied
             by
             〈◊〉
             produceth
             32.
             
             So
             that
             for
             36
             times
             8●
             
             you
             shall
             have
             found
             32
             times
             9.
             
             
             The
             operation
             is
             in
             the
             Margin
             ,
             that
             you
             may
             see
             ,
             and
             apprehend
             it
             the
             better
             .
             These
             numbers
             are
             equal
             ,
             viz.
             36
             times
             8
             is
             equal
             to
             32
             times
             9
             ;
             both
             producing
             288.
             
             If
             you
             divide
             36
             by
             6
             ,
             and
             8
             by
             2
             ,
             and
             multiply
             as
             before
             is
             said
             ,
             you
             will
             have
             for
             36
             times
             8
             ,
             24
             times
             12
             ,
             equal
             to
             288
             also
             .
          
           
             If
             this
             Rule
             seem
             to
             the
             unskilful
             Reader
             hard
             to
             be
             understood
             ,
             let
             him
             not
             be
             discouraged
             ,
             because
             he
             may
             do
             without
             it
             ,
             altho
             it
             may
             be
             of
             good
             use
             to
             him
             that
             would
             be
             a
             more
             compleat
             Artist
             .
          
           
             §
             4.
             
             Because
             in
             the
             following
             Paragraphs
             ,
             I
             shall
             have
             frequent
             occasion
             to
             use
             the
             
               Rule
               of
               Three
            
             ,
             or
             
               Rule
               of
               Proportion
            
             ,
             it
             will
             be
             necessary
             to
             shew
             the
             unskilful
             Reader
             ,
             how
             to
             work
             this
             noble
             Rule
             .
          
           
             If
             you
             find
             3
             or
             4
             numbers
             thus
             set
             ,
             with
             four
             spots
             after
             the
             second
             of
             them
             ,
             't
             is
             the
             Rule
             of
             Proportion
             ;
             as
             in
             this
             example
             ,
             2.
             4
             
             :
             :
             3.
             6.
             
               i.
               e.
            
             As
             2
             is
             to
             4
             :
             :
             So
             is
             3
             to
             6.
             
          
           
             The
             way
             to
             work
             this
             Rule
             ,
             viz.
             by
             the
             3
             first
             number
             to
             find
             a
             fourth
             ,
             is
             ,
             
             To
             multiply
             the
             second
             number
             and
             the
             third
             together
             ,
             and
             divide
             their
             product
             by
             the
             first
             .
             Thus
             4
             times
             3
             is
             12
             ,
             which
             12
             divided
             by
             2
             ,
             gives
             6
             ;
             which
             is
             the
             number
             sought
             for
             ,
             and
             stands
             in
             the
             fourth
             place
             .
          
           
             You
             will
             find
             the
             great
             use
             of
             this
             Rule
             hereafter
             ;
             only
             take
             care
             to
             bear
             it
             in
             mind
             all
             along
             .
          
           
             §
             5.
             
             To
             proceed
             .
             If
             in
             seeking
             for
             your
             Pinion
             of
             Report
             ,
             or
             by
             any
             other
             means
             ,
             you
             happen
             to
             have
             a
             Wheel
             and
             Pinion
             fall
             out
             with
             cross
             numbers
             ,
             too
             big
             to
             be
             cut
             in
             Wheels
             ,
             and
             yet
             not
             to
             be
             altered
             by
             the
             former
             Rules
             ,
             you
             may
             find
             out
             two
             numbers
             of
             the
             same
             ,
             or
             a
             near
             proportion
             ,
             by
             this
             following
             Rule
             ,
             viz.
             As
             either
             of
             the
             two
             numbers
             given
             ,
             is
             to
             
             the
             other
             :
             :
             So
             is
             360
             to
             a
             fourth
             :
             Divide
             that
             fourth
             number
             ,
             as
             also
             360
             by
             4.
             5.
             6.
             8.
             9.
             10.
             12.
             15.
             
             (
             each
             of
             which
             numbers
             doth
             exactly
             measure
             360
             )
             o●
             by
             any
             one
             of
             those
             numbers
             that
             bringeth
             a
             quotient
             nearest
             to
             an
             integer
             (
             or
             whole
             number
             .
             )
             Thus
             if
             you
             had
             these
             two
             numbers
             ,
             147
             the
             Wheel
             ,
             and
             170
             the
             Pinion
             ,
             which
             are
             too
             great
             to
             be
             cut
             in
             small
             Wheels
             ,
             and
             yet
             can't
             be
             reduced
             
             into
             le●s
             ,
             because
             they
             have
             no
             other
             common
             measure
             ,
             but
             unity
             :
             say
             therefore
             according
             to
             the
             last
             paragraph
             ,
             As
             170
             is
             to
             147
             ;
             or
             as
             147
             is
             to
             170
             :
             :
             So
             is
             360
             to
             a
             fourth
             number
             sought
             .
             In
             numbers
             thus
             ,
             170
             147
             :
             :
             360.
             311.
             or
             147.
             170
             
             :
             :
             360.
             416.
             
             Divide
             the
             fourth
             number
             and
             360
             by
             one
             of
             the
             foregoing
             numbers
             ;
             as
             311
             and
             360
             by
             6
             ,
             it
             gives
             52
             and
             60.
             
             In
             numbers
             't
             is
             thus
             ,
             
               
                 
                   6
                   )
                
                 
                   311
                
                 
                   (
                   52
                
              
               
                 
                   360
                
                 
                   (
                   60
                
              
            
             Divide
             by
             8
             't
             is
             thus
             ,
             
               
                 
                   8
                   )
                
                 
                   311
                
                 
                   (
                   39
                
              
               
                 
                   360
                
                 
                   (
                   45
                
              
            
             If
             you
             divide
             360
             and
             416
             by
             8
             ,
             it
             will
             fall
             out
             exactly
             to
             be
             45
             and
             52
             
               
                 
                   8
                   )
                
                 
                   360
                
                 
                   (
                   45
                
              
               
                 
                   416
                
                 
                   (
                   52
                
              
            
             Wherefore
             for
             the
             two
             numbers
             147
             and
             170
             ,
             you
             may
             take
             52
             and
             60
             ;
             or
             39
             and
             45
             ;
             or
             45
             and
             52
             ,
             o●
             &c.
             
          
           
             §
             6.
             
             I
             shall
             add
             but
             one
             Rule
             more
             ,
             before
             I
             come
             to
             the
             practice
             of
             what
             hath
             been
             laid
             down
             ;
             which
             Rule
             will
             be
             of
             perpetual
             use
             ,
             and
             consists
             of
             these
             five
             particulars
             .
          
           
             
             1.
             
             To
             find
             what
             number
             of
             turns
             the
             Fusy
             will
             have
             ,
             thus
             ,
             As
             the
             Beats
             of
             the
             Ballance
             in
             one
             turn
             of
             the
             Great-Wheel
             or
             Fusy
             (
             suppose
             26928
             )
             To
             the
             Beats
             of
             the
             Ballance
             in
             one
             hour
             (
             suppose
             20196
             )
             
             :
             :
             So
             is
             the
             continuance
             of
             the
             Watches
             going
             in
             hours
             (
             suppose
             16
             )
             To
             the
             number
             of
             the
             turns
             of
             the
             Fusy
             12.
             
             In
             numbers
             't
             will
             stand
             thus
             ,
             26928.
             20196
             
             :
             ▪
             16.
             12.
             
             By
             §
             4.
             you
             may
             remember
             tha●
             you
             are
             to
             multiply
             20196
             by
             16
             ,
             the
             product
             is
             323136.
             
             Divide
             this
             by
             26928
             ,
             and
             there
             will
             arise
             12
             in
             the
             Quotient
             ,
             which
             must
             be
             placed
             in
             the
             fourth
             place
             ▪
             and
             is
             the
             number
             of
             turns
             which
             the
             Fusy
             hath
             .
          
           
             2.
             
             By
             the
             Beats
             and
             turns
             of
             the
             Fusy
             ▪
             to
             find
             how
             many
             hours
             the
             Watch
             will
             go
             ,
             thus
             ,
          
           
             As
             the
             Beats
             of
             the
             Ballance
             in
             one
             hour
             ,
             are
             to
             the
             Beats
             in
             one
             turn
             of
             the
             Fusy
             :
             :
             So
             is
             the
             number
             of
             the
             turns
             o●
             the
             Fusy
             ,
             to
             the
             continuance
             of
             the
             Watches
             going
             .
             In
             num●ers
             thus
             ,
          
           
             2196.
             26928
             
             :
             :
             12
             ▪
             16.
             
          
           
             3.
             
             To
             find
             the
             strokes
             of
             the
             Ballance
             in
             one
             turn
             of
             the
             Fusy
             ,
             say
             ,
             As
             the
             number
             of
             turns
             of
             the
             Fusy
             ,
             to
             the
             continuance
             of
             the
             Watch's
             going
             in
             hours
             :
             :
             S●
             are
             the
             Beats
             in
             one
             hour
             ,
             to
             the
             Beats
             o●
             one
             turn
             of
             the
             Fusy
             .
             In
             numbers
             it
             〈◊〉
             thus
             ,
          
           
             12.
             16
             
             :
             :
             20196
             ▪
             26928.
             
          
           
           
             4.
             
             To
             find
             the
             Beats
             of
             the
             Ballance
             in
             an
             hour
             ,
             say
             thus
             ,
             As
             the
             hours
             of
             the
             Watch's
             going
             ,
             To
             the
             number
             of
             turns
             of
             the
             Fusy
             :
             :
             So
             are
             the
             Beats
             in
             one
             turn
             of
             the
             Fusy
             ,
             To
             the
             Beats
             in
             an
             hour
             .
             In
             numbers
             thus
             ,
          
           
             16.
             12
             
             :
             :
             26928.
             20196.
             
          
           
             5.
             
             To
             find
             what
             Quotient
             is
             to
             be
             laid
             upon
             the
             Pinion
             of
             Report
             ,
             say
             thus
             ,
             As
             the
             beats
             in
             one
             turn
             of
             the
             Great-wheel
             ,
             To
             the
             beats
             in
             an
             hour
             :
             :
             So
             are
             the
             hours
             of
             the
             Face
             of
             the
             Clock
             
               (
               viz.
            
             12
             or
             24
             )
             To
             the
             Quotient
             of
             the
             Hour-Wheel
             divided
             by
             the
             Pinion
             of
             Report
             ,
             
               i.
               e.
            
             the
             number
             of
             turns
             ,
             which
             the
             Pinion
             of
             Report
             hath
             in
             one
             turn
             of
             the
             Dial-Wheel
             .
             In
             numbers
             thus
             ,
          
           
             26928.
             20196
             
             :
             :
             12.
             9.
             
          
           
             Or
             rather
             (
             to
             avoid
             trouble
             )
             say
             thus
             ,
             As
             the
             hours
             of
             the
             Watch's
             going
             ,
             Are
             to
             the
             numbers
             of
             the
             turns
             of
             the
             Fusy
             :
             :
             So
             are
             the
             hours
             of
             the
             Face
             ,
             To
             the
             Quotient
             of
             the
             Pinion
             of
             Report
             ▪
             In
             numbers
             thus
             ,
             16.
             12
             
             :
             :
             12.
             9.
             
             If
             the
             hours
             of
             the
             Face
             be
             24
             ,
             the
             Quotient
             will
             be
             18
             ;
             thus
             ,
             16.
             12
             
             :
             :
             24.
             18.
             
          
           
             §
             7.
             
             Having
             given
             a
             full
             account
             of
             all
             things
             necessary
             to
             the
             understanding
             the
             
             Art
             of
             Calculation
             ,
             I
             shall
             now
             reduc●
             what
             hath
             been
             said
             into
             practice
             ,
             by
             shewing
             how
             to
             proceed
             ,
             in
             Calculating
             a
             piece
             of
             Watch-work
             .
          
           
             The
             first
             thing
             you
             are
             to
             do
             ,
             is
             to
             pitch
             upon
             your
             Train
             ,
             or
             beats
             of
             the
             Ballance
             in
             an
             hour
             :
             as
             ,
             whither
             a
             swi●●
             Train
             ,
             about
             20000
             beats
             (
             which
             is
             the
             usual
             Train
             of
             a
             common
             30
             hour
             Pocket
             ▪
             Watch
             )
             or
             a
             slower
             Train
             of
             about
             16000
             (
             the
             Train
             of
             the
             new
             Pendulum
             Pocket
             ▪
             Watches
             ;
             )
             or
             any
             other
             Train
             .
          
           
             Having
             thus
             pitched
             upon
             your
             Train
             ,
             you
             must
             next
             resolve
             upon
             the
             number
             of
             turns
             you
             intend
             your
             Fusy
             shall
             have
             ;
             and
             also
             upon
             the
             number
             of
             Hours
             ,
             you
             would
             have
             your
             Piece
             to
             go
             :
             As
             suppose
             12
             turns
             ;
             and
             to
             go
             30
             hours
             ,
             or
             192
             hours
             (
             which
             is
             8
             days
             )
             or
             &c.
             
          
           
             These
             things
             being
             all
             soon
             determined
             ;
             you
             next
             proceed
             to
             find
             out
             the
             beats
             of
             the
             Ballance
             ,
             or
             Pendulum
             ,
             in
             one
             turn
             of
             the
             Fusy
             ,
             by
             the
             last
             §
             6.
             part
             3.
             viz.
             As
             the
             turns
             of
             the
             Fusy
             ,
             To
             the
             hours
             of
             the
             Watch's
             going
             :
             :
             So
             is
             the
             Train
             ,
             To
             the
             number
             of
             beats
             in
             one
             turn
             of
             the
             Fusy
             .
             In
             numbers
             thus
             ,
             12.
             16
             
             :
             :
             20000.
             26666.
             
             Which
             last
             
             number
             are
             the
             beats
             in
             one
             turn
             of
             the
             Fusy
             ,
             or
             Great-Wheel
             ;
             and
             (
             by
             Sect.
             I.
             §
             5.
             of
             this
             Chap.
             )
             are
             equal
             to
             the
             Quotients
             of
             all
             the
             Wheels
             unto
             the
             ballance
             ,
             multiplied
             together
             .
             This
             number
             therefore
             is
             to
             be
             broken
             into
             a
             convenient
             parcel
             of
             Quotients
             :
             which
             you
             are
             to
             do
             after
             this
             manner
             .
             First
             ,
             half
             your
             number
             of
             beats
             ,
             viz.
             26666
             ,
             for
             the
             reasons
             in
             Sect.
             I.
             §
             6.
             of
             this
             Chap.
             the
             half
             whereof
             is
             13333.
             
             Next
             you
             are
             to
             pitch
             upon
             the
             number
             of
             your
             Crown-wheel
             ,
             as
             suppose
             17.
             
             Divide
             13333
             by
             17
             ,
             the
             Quotient
             will
             be
             784
             (
             or
             to
             speak
             in
             the
             language
             of
             one
             that
             understands
             not
             Arithmetick
             ,
             divide
             13333
             into
             17
             parts
             ,
             and
             784
             will
             be
             one
             of
             them
             .
             )
             This
             784
             is
             the
             number
             left
             for
             the
             Quotients
             (
             or
             turns
             )
             of
             the
             rest
             of
             the
             Wheels
             and
             Pinions
             :
             which
             being
             too
             big
             for
             one
             or
             two
             Quotients
             ,
             may
             be
             best
             broken
             into
             three
             .
             Chuse
             therefore
             3
             numbers
             ,
             which
             when
             multiplied
             all
             together
             continually
             will
             come
             nearest
             784.
             
             As
             suppose
             you
             take
             10
             ,
             9
             ,
             and
             9.
             
             Now
             10
             times
             9
             is
             90
             ;
             and
             9
             times
             90
             is
             810
             ,
             which
             is
             somewhat
             too
             much
             .
             You
             may
             therefore
             try
             again
             other
             numbers
             ▪
             as
             suppose
             11
             ,
             9
             ,
             and
             8.
             
             
             These
             multiplied
             as
             the
             last
             ,
             produce
             792
             ▪
             which
             is
             as
             near
             as
             can
             be
             ,
             and
             convenient
             Quotients
             .
          
           
             Thus
             you
             have
             contrived
             your
             Piece
             ▪
             from
             the
             Great-Wheel
             to
             the
             Ballance
             ▪
             But
             the
             numbers
             not
             falling
             out
             exactly
             according
             as
             you
             at
             first
             proposed
             ;
             you
             must
             correct
             your
             work
             thus
             .
             First
             to
             find
             out
             the
             true
             number
             of
             beats
             ,
             in
             one
             turn
             of
             the
             Fusy
             ,
             you
             must
             multiply
             792
             ▪
             aforesaid
             ,
             which
             is
             the
             true
             product
             of
             al●
             the
             Quotients
             ,
             by
             17
             ,
             the
             notches
             of
             the
             Crown-wheel
             ;
             the
             product
             of
             this
             i●
             13464
             ,
             which
             is
             half
             the
             number
             of
             true
             beats
             in
             one
             turn
             of
             the
             Fusy
             ,
             by
             Sect.
             ●
             §
             5.
             of
             this
             Chap.
             Then
             to
             find
             the
             tru●
             number
             of
             beats
             in
             an
             hour
             ,
             say
             by
             §
             6
             ▪
             part
             4.
             of
             this
             Section
             ,
             as
             the
             hours
             o●
             the
             Watch's
             going
             ,
             viz.
             16
             ,
             to
             the
             1●
             turns
             of
             the
             Fusy
             :
             :
             So
             is
             13464
             the
             ha●●
             beats
             in
             one
             turn
             of
             the
             Fusy
             ,
             to
             1009●
             the
             half
             beats
             in
             an
             hour
             :
             the
             numbe●
             will
             stand
             thus
             16.
             12
             
             :
             :
             13464.
             1009●
          
           
             Then
             to
             know
             what
             Quotient
             is
             to
             b●
             laid
             upon
             the
             Pinion
             of
             Report
             ,
             say
             by
             〈◊〉
             6.
             part
             5.
             of
             this
             Sect
             :
             As
             the
             hours
             〈◊〉
             the
             Watch's
             going
             ,
             viz.
             16
             ,
             to
             the
             tur●
             of
             the
             Fusy
             ,
             viz.
             12
             :
             :
             So
             are
             the
             hou●●
             
             of
             the
             Dial-plate
             ,
             viz.
             12
             ,
             To
             the
             Quotient
             of
             the
             Pinion
             of
             Report
             .
             In
             numbers
             thus
             ,
             16.
             12
             
             :
             :
             12.
             9.
             
          
           
             
             Having
             thus
             found
             out
             all
             your
             Quotients
             ,
             't
             is
             easie
             to
             determine
             what
             numbers
             your
             Wheels
             shall
             have
             :
             for
             chuse
             what
             numbers
             your
             Pinions
             shall
             have
             ,
             and
             multiply
             the
             Pinions
             by
             their
             Quotients
             ,
             and
             that
             produceth
             the
             numbers
             for
             your
             Wheels
             ,
             as
             you
             see
             in
             the
             Margin
             .
             Thus
             4
             is
             the
             number
             of
             your
             Pinion
             of
             Report
             ,
             and
             9
             its
             quotient
             ;
             therefore
             4
             times
             9
             ,
             which
             makes
             36
             ,
             is
             the
             number
             for
             the
             Dial-wheel
             .
             So
             the
             next
             Pinion
             being
             5
             ,
             and
             its
             quotient
             11
             ,
             this
             multiplied
             produces
             55
             for
             the
             Great-wheel
             .
             And
             the
             like
             of
             the
             rest
             of
             the
             following
             numbers
             .
          
           
             Thus
             ,
             as
             plain
             as
             words
             can
             express
             it
             ,
             I
             have
             shewed
             how
             to
             Calculate
             the
             numbers
             of
             a
             16
             hour
             Watch.
             
          
           
             
             §
             8.
             
             This
             Watch
             may
             be
             made
             to
             go
             a
             longer
             time
             ,
             by
             lessening
             the
             Train
             ,
             and
             altering
             the
             Pinion
             of
             Report
             .
             Suppose
             you
             could
             conveniently
             slacken
             the
             Train
             to
             16000
             ,
             the
             half
             of
             which
             is
             8000.
             
             
             
             Then
             say
             (
             by
             §
             6
             part
             2.
             of
             this
             Sect.
             )
             As
             the
             halfed
             Train
             ,
             or
             Beats
             in
             an
             hour
             ,
             viz.
             8000
             ,
             To
             the
             halfed
             beats
             in
             one
             turn
             of
             the
             Fusy
             ,
             viz.
             13464
             :
             :
             So
             are
             the
             turns
             of
             the
             Fusy
             ,
             viz.
             12
             ,
             To
             the
             hours
             of
             the
             Watch's
             going
             :
             in
             numbers
             thus
             ,
             8000.
             13464
             
             :
             :
             12.
             20
             
             So
             that
             this
             Watch
             will
             go
             20
             hours
             .
          
           
             Then
             for
             the
             Pinion
             of
             Report
             ,
             say
             ,
             by
             the
             same
             §
             part
             5
             ,
             As
             20
             the
             Continuance
             ;
             To
             12
             the
             turns
             of
             the
             Fusy
             :
             :
             So
             are
             12
             the
             hours
             of
             the
             Face
             ,
             To
             7
             the
             quotient
             of
             the
             Pinion
             of
             Report
             .
             In
             numbers
             thus
             ,
             20.
             12
             
             :
             :
             12.
             7.
             
          
           
             
             The
             work
             is
             the
             same
             as
             before
             ,
             as
             to
             the
             numbers
             ;
             only
             the
             Dial-wheel
             is
             but
             28
             ,
             because
             its
             quotient
             is
             altered
             to
             7
             ;
             as
             appears
             in
             the
             Margin
             ,
             by
             the
             Scheme
             of
             the
             work
             .
          
           
             §
             9.
             
             I
             shall
             give
             the
             Reader
             one
             example
             more
             ,
             for
             the
             sake
             of
             shewing
             him
             the
             use
             of
             some
             of
             the
             foregoing
             Rules
             ,
             not
             yet
             taken
             notice
             of
             in
             the
             former
             operations
             .
             Suppose
             you
             
             would
             give
             numbers
             to
             a
             Watch
             of
             about
             10000
             beats
             in
             an
             hour
             ,
             to
             have
             12
             turns
             
             of
             the
             Fusy
             ,
             to
             go
             170
             hours
             ,
             and
             17
             notches
             in
             the
             Crown-wheel
             .
          
           
             This
             work
             is
             the
             same
             as
             in
             the
             last
             Example
             §
             7.
             
             In
             short
             therefore
             thus
             ,
             As
             the
             turns
             12
             :
             are
             To
             the
             Continuance
             170
             :
             :
             So
             is
             the
             Train
             10000
             ,
             To
             141666
             ,
             which
             are
             the
             beats
             in
             one
             turn
             of
             the
             Fusy
             .
             The
             numbers
             will
             stand
             thus
             ,
             12.
             170
             
             :
             :
             10000
             14●666
             .
             Half
             this
             last
             is
             70●33
             .
             Divide
             this
             half
             into
             17
             parts
             ,
             and
             4167
             will
             be
             for
             the
             quotients
             .
             And
             because
             this
             number
             is
             too
             big
             for
             3
             quotients
             ,
             therefore
             chuse
             4
             :
             as
             suppose
             10
             ,
             8
             ,
             8
             ,
             and
             6
             ⅗
             
               (
               i.
               e.
            
             6
             and
             3
             fifths
             )
             These
             multiplied
             together
             as
             before
             ,
             and
             with
             17
             ,
             maketh
             71808
             ,
             which
             are
             half
             the
             true
             beats
             in
             one
             turn
             of
             the
             Fusy
             .
             By
             this
             you
             are
             to
             find
             out
             your
             true
             Train
             first
             ▪
             saying
             as
             in
             the
             former
             example
             ,
             As
             170
             to
             12
             :
             :
             So
             71808
             ,
             to
             5069
             ;
             which
             last
             is
             the
             half
             of
             the
             true
             Train
             of
             your
             Watch.
             Then
             for
             the
             Pinion
             of
             Report
             ,
             say
             ,
             as
             170
             ,
             to
             12
             :
             :
             So
             12
             ,
             to
             1
             /
             ●4
             /
             74
             /
             ●
             .
             Which
             Fraction
             ari●eth
             thus
             :
             If
             you
             multiply
             12
             by
             12
             it
             makes
             144
             ;
             and
             divide
             144
             by
             170
             ,
             you
             cannot
             ;
             but
             setting
             the
             144
             (
             the
             dividend
             )
             over
             170
             (
             the
             Divisor
             )
             and
             there
             ari●eth
             this
             fraction
             ●
             /
             24
             /
             ●4
             /
             ●
             ,
             
             which
             is
             a
             Wheel
             and
             Pinion
             ;
             the
             lower
             is
             the
             Pinion
             of
             Report
             ,
             and
             the
             upper
             is
             the
             Dial-wheel
             ,
             according
             to
             Sect.
             I.
             §
             3.
             of
             this
             Chapter
             .
             Or
             (
             which
             perhaps
             will
             be
             more
             plain
             to
             the
             unlearned
             Reader
             )
             you
             may
             leave
             those
             two
             numbers
             ,
             in
             their
             Divisional
             posture
             thus
             ,
             170
             )
             144
             ,
             which
             does
             express
             the
             Pinion
             and
             Wheel
             ,
             in
             the
             way
             I
             have
             hitherto
             made
             use
             of
             .
             But
             to
             proceed
             .
             These
             numbers
             being
             too
             big
             to
             be
             cut
             in
             small
             
             Wheels
             ,
             may
             be
             varied
             ,
             as
             you
             see
             a
             like
             Example
             is
             §
             5.
             of
             this
             Section
             :
             viz.
             Say
             ,
             as
             144
             ▪
             is
             To
             170
             :
             :
             So
             is
             360
             ,
             To
             425.
             
             Or
             ,
             as
             170
             ,
             to
             144
             :
             :
             So
             is
             360
             ,
             To
             305.
             
             In
             number
             thus
             ,
             144.
             170
             
             :
             :
             360.
             425.
             
             Or
             170.
             144
             
             :
             :
             360.
             305.
             
             Divide
             360
             ,
             and
             either
             of
             these
             two
             fourth
             and
             last
             numbers
             by
             4
             ,
             5
             ,
             6
             ,
             8
             ,
             &c.
             (
             as
             is
             directed
             in
             the
             Rule
             last
             cited
             .
             )
             If
             you
             divide
             by
             8
             ,
             you
             will
             have
             for
             your
             numbers
             1
             /
             14
             /
             74
             /
             ●
             4
             /
             ●5
             /
             ●
             or
             3
             /
             48
             /
             5.
             
             If
             you
             divide
             by
             15
             (
             which
             will
             not
             bring
             it
             so
             near
             an
             integer
             )
             you
             will
             have
             2
             /
             24
             /
             8
             or
             2
             /
             ●0
             /
             4
             :
             which
             last
             are
             the
             numbers
             set
             down
             in
             the
             Margin
             ▪
             
             where
             the
             numbers
             of
             the
             whole
             Movement
             are
             set
             down
             .
          
           
             §
             10.
             
             Having
             said
             enough
             ,
             I
             think
             ,
             concerning
             the
             Calculation
             of
             ordinary
             Watches
             ,
             to
             shew
             the
             hour
             of
             the
             day
             :
             I
             shall
             next
             proceed
             to
             such
             as
             shew
             minutes
             and
             seconds
             .
             The
             process
             whereof
             is
             thus
             :
             First
             ,
             having
             resolved
             upon
             your
             beats
             in
             an
             hour
             ,
             you
             are
             next
             to
             find
             how
             many
             beats
             there
             will
             be
             in
             a
             minute
             ,
             by
             dividing
             your
             designed
             Train
             into
             60
             parts
             .
             And
             accordingly
             you
             are
             to
             find
             out
             such
             proper
             numbers
             for
             your
             Crown-wheel
             ,
             and
             quotients
             ,
             as
             that
             the
             Minute-wheel
             shall
             go
             round
             once
             in
             an
             hour
             ,
             and
             the
             Second-wheel
             once
             in
             a
             minute
             .
          
           
             An
             Example
             will
             make
             all
             plain
             .
             Let
             us
             chuse
             a
             Pendulum
             of
             6
             inches
             to
             go
             8
             days
             ,
             with
             16
             turns
             of
             the
             Fusy
             .
             By
             
             Mr
             
             Smith's
             Tables
             ,
             a
             Pendulum
             of
             6
             inches
             vibrates
             9368
             in
             an
             hour
             .
             This
             divided
             by
             60
             gives
             156
             beats
             for
             a
             minute
             .
             Half
             these
             summs
             are
             4684
             and
             
             78.
             
             Now
             the
             first
             work
             is
             to
             break
             this
             78
             into
             good
             proportions
             ;
             which
             will
             fall
             into
             one
             quotient
             ,
             and
             the
             Crown-wheel
             .
             First
             ,
             for
             the
             Crown-wheel
             ;
             let
             
             it
             have
             15
             notches
             .
             Divide
             78
             afore●
             by
             this
             15
             ,
             the
             quotient
             will
             be
             5.
             
             A●
             so
             this
             first
             work
             is
             done
             :
             for
             a
             Crow●
             wheel
             of
             15
             ,
             and
             a
             Wheel
             a●
             Pinion
             ,
             whose
             quotient
             is
             5
             (
             ●
             in
             the
             Margin
             )
             
             will
             go
             rou●
             in
             a
             minute
             ,
             to
             carry
             a
             Ha●
             to
             shew
             Seconds
             .
          
           
             Next
             for
             a
             Hand
             to
             go
             round
             in
             ●
             hour
             ,
             to
             shew
             Minutes
             .
             Now
             becau●
             there
             are
             60
             minutes
             in
             an
             hour
             ,
             't
             is
             b●
             breaking
             60
             into
             two
             goo●
             
             quotients
             (
             which
             may
             be
             ●
             and
             6
             ,
             or
             8
             and
             7½
             ,
             or
             &c.
             and
             the
             work
             is
             done
             .
          
           
             Thus
             your
             number
             4684
             ,
             broken
             ,
             as
             near
             as
             can
             be
             ,
             int●
             proper
             numbers
             .
          
           
             But
             because
             it
             does
             not
             fall
             out
             exact
             into
             the
             above-mentioned
             numbers
             ,
             yo●
             must
             Correct
             (
             as
             you
             were
             directed
             before
             )
             and
             find
             out
             the
             true
             number
             ●
             beats
             in
             an
             hour
             ,
             by
             multiplying
             15
             by
             5
             ,
             which
             makes
             75
             ;
             and
             this
             by
             6●
             makes
             4500
             ,
             which
             is
             the
             half
             of
             the
             tru●
             Train
             .
             Then
             to
             find
             out
             the
             beats
             in
             on
             
             turn
             of
             thy
             Fusy
             ,
             operate
             as
             before
             ,
             vi●
             As
             the
             number
             of
             turns
             ,
             16
             ,
             To
             the
             co●tinuan●
             
             192
             :
             :
             So
             is
             4500
             to
             54000
             ,
             which
             are
             half
             the
             beats
             in
             one
             turn
             of
             ●he
             Fusy
             .
             In
             numbers
             thus
             ,
             16.
             192
             
             :
             :
             4500.
             54000.
             
             This
             54000
             must
             be
             di●ided
             by
             4500
             ,
             which
             are
             the
             true
             ●umbers
             already
             pi●ched
             upon
             ,
             or
             beats
             ▪
             ●n
             an
             hour
             .
             The
             quotient
             of
             this
             division
             ●s
             12
             ,
             which
             being
             not
             too
             big
             for
             one
             single
             quotient
             ,
             needs
             not
             be
             divided
             into
             more
             .
             The
             work
             will
             stand
             ,
             as
             you
             see
             in
             the
             Margin
             .
             
          
           
             As
             to
             the
             Hour-hand
             ,
             the
             Great-Wheel
             ,
             which
             performs
             only
             one
             revolution
             in
             12
             turns
             of
             the
             Minute-wheel
             ,
             will
             shew
             the
             hour
             .
             Or
             rather
             you
             may
             order
             it
             to
             be
             done
             by
             the
             Minute-wheel
             ,
             ●s
             shall
             be
             shew'd
             hereafter
             .
          
           
             §
             11.
             
             I
             shall
             add
             but
             one
             Example
             more
             ,
             and
             so
             conclude
             this
             Section
             ;
             and
             ●hat
             is
             ,
             To
             calculate
             the
             numbers
             of
             a
             ●iece
             whose
             Pendulum
             swings
             ▪
             Seconds
             ,
             ●o
             shew
             the
             hour
             ,
             minutes
             ,
             and
             seconds
             ,
             ●nd
             to
             go
             8
             days
             ;
             which
             is
             the
             usual
             per●ormance
             of
             those
             Movements
             called
             
             ●oyal
             Pendulums
             at
             this
             day
             .
             First
             ,
             cast
             ●p
             the
             number
             of
             seconds
             in
             12
             hours
             
             (
             which
             are
             the
             beats
             in
             one
             turn
             of
             〈◊〉
             Great-wheel
             )
             These
             are
             12
             times
             〈◊〉
             minutes
             ,
             and
             60
             times
             that
             ,
             gives
             432●
             which
             are
             the
             seconds
             in
             12
             hours
             .
             H●
             
             this
             number
             (
             for
             the
             reasons
             before
             )
             21600.
             
             The
             Swing-wheel
             must
             ne●
             be
             30
             ,
             to
             swing
             60
             seconds
             in
             one
             of
             〈◊〉
             revolutions
             .
             Divide
             21600
             by
             it
             ,
             a●
             720
             is
             the
             quotient
             ,
             or
             number
             left
             to
             〈◊〉
             broken
             into
             quotients
             .
             Of
             these
             quo●ents
             ,
             the
             first
             must
             needs
             be
             12
             for
             〈◊〉
             Great-wheel
             ,
             which
             moves
             round
             on●
             in
             12
             hours
             .
             Divide
             720
             by
             12
             ,
             〈◊〉
             quotient
             is
             60
             ;
             which
             may
             be
             conve●
             ▪
             ently
             broken
             into
             two
             quotients
             ,
             as
             〈◊〉
             and
             6
             ,
             or
             5
             and
             12
             ,
             or
             8
             and
             7
             ½
             ,
             whi●
             last
             is
             most
             convenient
             .
             A●
             if
             you
             take
             all
             the
             Pinions
             the
             work
             will
             stand
             as
             in
             〈◊〉
             Margin
             .
             
          
           
             According
             to
             this
             compu●
             ▪
             tion
             ,
             the
             Great-wheel
             will
             〈◊〉
             about
             once
             in
             12
             hours
             ,
             shew
             the
             hour
             ,
             if
             you
             please
             :
             the
             Seco●
             wheel
             once
             in
             an
             hour
             ,
             to
             shew
             the
             〈◊〉
             nutes
             ;
             and
             the
             Swing-wheel
             once
             in
             a
             〈◊〉
             nute
             ,
             to
             shew
             the
             seconds
             .
          
           
           
             Thus
             I
             have
             endeavour'd
             with
             all
             possible
             plainness
             ,
             to
             unravel
             this
             most
             mysterious
             ,
             as
             well
             as
             useful
             part
             of
             Watch-work
             .
             In
             which
             ,
             if
             I
             have
             offended
             the
             more
             learned
             Reader
             ,
             by
             unartificial
             terms
             ,
             or
             multitude
             of
             words
             ,
             I
             desire
             the
             fault
             may
             be
             laid
             upon
             my
             earnest
             intent
             to
             condescend
             to
             the
             meanest
             capacity
             .
          
        
         
           
             SECT
             .
             III.
             To
             Calculate
             the
             Striking
             part
             of
             a
             Clock
             .
          
           
             §
             1.
             
             ALtho
             this
             part
             consists
             of
             many
             Wheels
             and
             Pinions
             ,
             yet
             respect
             needs
             to
             be
             had
             only
             to
             the
             
               Count-wheel
               ,
               Striking-wheel
            
             ,
             and
             Detent-wheel
             :
             which
             move
             round
             in
             this
             proportion
             ;
             The
             Count-wheel
             moveth
             round
             commonly
             ●nce
             in
             12
             ,
             or
             24
             hours
             .
             The
             Detent-wheel
             moves
             round
             every
             stroke
             the
             Clock
             striketh
             ,
             sometimes
             but
             once
             in
             two
             strokes
             .
             From
             whence
             it
             follows
             ,
          
           
             1.
             
             That
             as
             many
             Pins
             as
             are
             in
             the
             Pin-wheel
             ,
             so
             many
             turns
             hath
             the
             Detent-wheel
             ,
             in
             one
             turn
             of
             the
             Pin-wheel
             .
             Or
             (
             which
             is
             the
             same
             )
             the
             Pins
             of
             th●
             
             Pin-wheel
             are
             the
             Quotient
             of
             that
             Wheel
             ,
             divided
             by
             the
             Pinion
             of
             the
             Deten●-wheel
             .
             But
             if
             the
             Detent-wheel
             moveth
             but
             once
             round
             in
             two
             strokes
             o●
             the
             Clock
             ,
             then
             the
             said
             Quotient
             is
             bu●
             half
             the
             number
             of
             Pins
             .
          
           
             2.
             
             As
             many
             turns
             of
             the
             Pin-wheel
             a●
             are
             required
             to
             perform
             the
             strokes
             of
             1●
             hours
             (
             which
             are
             78
             )
             So
             many
             tur●●
             must
             the
             Pinion
             of
             Report
             have
             ,
             to
             turn
             round
             the
             Count-wheel
             once
             .
             Or
             thus
             ▪
             Divide
             78
             by
             the
             number
             of
             Striking
             pins
             ,
             and
             the
             Quotient
             thereof
             shall
             b●
             the
             Quotient
             of
             the
             Pinion
             of
             Report
             .
             Al●
             this
             is
             ,
             in
             case
             the
             Pinion
             of
             Report
             b●
             fixed
             to
             the
             arbor
             of
             the
             Pin-wheel
             ,
             as
             i●
             very
             commonly
             done
             .
          
           
             All
             this
             I
             take
             to
             be
             very
             plain
             :
             or
             〈◊〉
             it
             be
             not
             ,
             the
             example
             in
             the
             Margin
             wil●
             clear
             all
             difficulties
             .
             
             Her●
             the
             Locking-wheel
             is
             48●
             the
             Pinion
             of
             Report
             is
             8●
             the
             Pin-wheel
             is
             78
             ,
             th●
             Striking-pins
             are
             13.
             
             An●
             so
             of
             the
             rest
             .
             I
             need
             onl●
             to
             remark
             hero
             ,
             that
             7●
             being
             divided
             by
             the
             13
             pins
             ,
             gives
             6●
             which
             is
             the
             Quotient
             of●
             ▪
             the
             Pinion●
             
             of
             Report
             :
             as
             was
             before
             hinted
             .
          
           
             As
             for
             the
             Warning-wheel
             ,
             and
             Flying-Pinion
             ,
             it
             matters
             little
             what
             numbers
             they
             have
             ,
             their
             use
             being
             only
             to
             bridle
             the
             rapidity
             of
             the
             motion
             of
             the
             other
             Wheels
             .
          
           
             Besides
             the
             last
             observation
             ,
             there
             are
             other
             ways
             to
             find
             out
             the
             Pinion
             of
             Report
             ,
             which
             will
             fall
             under
             the
             next
             §
             .
          
           
             §
             2.
             
             These
             following
             Rules
             will
             be
             of
             great
             use
             in
             this
             part
             of
             Calculation
             ,
             viz.
             
          
           
             Rule
             1.
             
             As
             the
             number
             of
             turns
             of
             the
             Great-wheel
             ,
             or
             Fusy
             ;
          
           
             .
             To
             the
             days
             of
             the
             Clock's
             continuauc●
             :
          
           
             :
             :
             So
             is
             the
             number
             of
             strokes
             in
             24
             hours
             ,
             viz.
             156
             ▪
          
           
             .
             To
             the
             strokes
             in
             one
             turn
             of
             the
             Fusy
             ,
             or
             Great-wheel
             .
          
           
             Rule
             2.
             
             As
             the
             number
             of
             strokes
             in
             24
             hours
             ,
             which
             are
             156
             ,
          
           
             .
             To
             the
             strokes
             in
             one
             turn
             of
             the
             Fusy
             ,
             or
             Great-wheel
             ,
          
           
             :
             :
             So
             are
             the
             turns
             of
             the
             Fusy
             ,
             or
             Great-wheel
             ,
          
           
             .
             To
             the
             days
             of
             the
             Clock's
             continuance
             ,
             or
             going
             .
          
           
             Rule
             3.
             
             As
             the
             strokes
             in
             one
             turn
             of
             the
             Fu●y
             ,
          
           
           
             .
             To
             the
             strokes
             of
             24
             hours
             ,
             viz.
             156.
             
          
           
             :
             :
             So
             is
             the
             Clock's
             continuance
             ,
          
           
             .
             To
             the
             number
             of
             turns
             of
             the
             Fusy
             ,
             or
             Great-wheel
             .
          
           
             These
             two
             last
             Rules
             are
             of
             no
             great
             use
             (
             as
             the
             first
             is
             )
             but
             may
             serve
             to
             correct
             your
             work
             ,
             if
             need
             be
             ,
             when
             in
             breaking
             your
             Strokes
             into
             Quotients
             (
             of
             which
             presently
             )
             you
             cannot
             come
             near
             the
             true
             number
             ,
             but
             a
             good
             many
             strokes
             are
             left
             remaining
             .
             In
             this
             case
             ,
             by
             Rule
             2.
             you
             may
             find
             whether
             the
             continuance
             of
             your
             Clock
             be
             to
             your
             mind
             ▪
             And
             by
             Rule
             3
             ,
             you
             may
             enlarge
             or
             diminish
             the
             number
             of
             turns
             for
             this
             purpose
             .
             The
             praxis
             hereof
             will
             follow
             by
             and
             by
             .
          
           
             The
             2
             following
             Rules
             are
             to
             find
             fit
             numbers
             for
             the
             Pinion
             of
             Report
             ,
             and
             the
             Locking-wheel
             ,
             besides
             what
             is
             said
             before
             §
             1.
             
             Inference
             2.
             
          
           
             Rule
             4.
             
             As
             the
             number
             of
             Strokes
             in
             the
             Clock's
             continuance
             ,
             or
             in
             all
             it●
             turns
             of
             the
             Fusy
             ,
          
           
             .
             To
             the
             turns
             of
             the
             Fusy
             ,
          
           
             :
             :
             So
             are
             the
             Strokes
             in
             12
             hours
             ,
             which
             are
             78
             ,
          
           
             .
             To
             the
             Quotient
             of
             the
             Pinion
             of
             Report
             ,
             fixed
             upon
             the
             arbor
             of
             the
             Great-wheel
             .
          
           
           
             But
             if
             you
             would
             fix
             it
             to
             any
             other
             Wheel
             ,
             you
             may
             do
             it
             thus
             ,
             as
             is
             before
             
             hinted
             ,
             viz.
             
          
           
             Rule
             5.
             
             First
             ,
             find
             out
             the
             number
             of
             Strokes
             ,
             in
             one
             turn
             of
             the
             Wheel
             you
             intend
             to
             fix
             your
             Pinion
             of
             Report
             upon
             (
             which
             I
             shall
             shew
             you
             how
             to
             do
             in
             the
             following
             §
             .
             )
             Divide
             78
             by
             this
             number
             ,
             and
             the
             number
             arising
             in
             the
             Quotient
             ,
             is
             the
             Quotient
             of
             the
             Pinion
             of
             Report
             .
          
           
             Or
             thus
             .
             Take
             the
             number
             of
             Strokes
             in●●e
             turn
             of
             the
             Wheel
             ,
             for
             the
             number
             of
             the
             Pinion
             of
             Report
             ,
             and
             78
             for
             the
             Count
             (
             or
             Locking
             )
             wheel
             ,
             and
             vary
             them
             to
             lesser
             numbers
             ,
             by
             Sect.
             2.
             
             §
             5.
             of
             this
             Chapter
             .
          
           
             Rule
             6.
             
             The
             foregoing
             Rules
             are
             of
             greatest
             use
             ,
             in
             Clocks
             of
             a
             larger
             continuance
             ;
             altho
             ,
             where
             they
             can
             be
             applied
             ,
             they
             will
             indifferently
             serve
             all
             .
             But
             this
             Rule
             (
             which
             will
             serve
             larger
             Clocks
             too
             )
             I
             add
             chiefly
             for
             the
             use
             of
             lesser
             Pieces
             ,
             whose
             continuance
             is
             accounted
             by
             hours
             .
          
           
             The
             Rule
             is
             to
             find
             the
             Strokes
             in
             the
             Clock's
             continuance
             ,
             viz.
             As
             12
             ,
             is
             to
             78
             :
             :
             So
             are
             the
             hours
             of
             the
             Clocks
             continuance
             ,
             
             To
             the
             number
             of
             Strokes
             in
             that
             time
             .
          
           
             This
             Rule
             (
             I
             said
             )
             may
             be
             made
             use
             of
             for
             the
             largest
             Clock
             ;
             but
             then
             you
             must
             be
             at
             the
             trouble
             of
             reducing
             the
             Days
             into
             Hours
             Whereas
             the
             shortest
             way
             is
             to
             Multiply
             the
             strokes
             in
             one
             turn
             of
             the
             Great-wheel
             ,
             by
             the
             number
             of
             Turns
             .
             Thus
             in
             an
             8
             day
             piece
             the
             Strokes
             in
             one
             turn
             are
             78.
             
             These
             multiplied
             by
             16
             ,
             the
             turns
             ,
             produce
             1248
             ;
             which
             are
             the
             Strokes
             in
             the
             Clock's
             continuance
             .
             If
             you
             work
             by
             the
             foregoing
             Ruled
             the
             hours
             of
             8
             days
             are
             192.
             
             Then
             say
             ,
             12
             ▪
             78
             :
             :
             192.
             1248.
             
          
           
             §
             3.
             
             In
             this
             Paragraph
             ,
             I
             shall
             shew
             the
             use
             of
             the
             preceding
             Rules
             ,
             and
             by
             examples
             make
             all
             plain
             that
             might
             seem
             obscure
             in
             them
             .
          
           
             I
             begin
             with
             small
             Pieces
             :
             of
             which
             but
             briefly
             .
             And
             first
             ,
             having
             pitched
             upon
             the
             number
             of
             turns
             ,
             and
             the
             continuance
             ,
             you
             must
             find
             ,
             by
             the
             last
             Rule
             ,
             how
             many
             Strokes
             are
             in
             its
             continuance
             .
             Then
             divide
             these
             Strokes
             by
             the
             number
             of
             turns
             ,
             and
             you
             have
             the
             number
             of
             Striking-pins
             .
             Or
             divide
             by
             the
             number
             of
             Pins
             ,
             and
             you
             have
             the
             number
             of
             Turns
             .
          
           
           
             Thus
             a
             Clock
             of
             30
             hours
             ,
             with
             15
             turns
             of
             the
             Great-wheel
             ,
             hath
             195
             strokes
             .
             For
             by
             the
             last
             Rule
             ,
             12.
             78
             
             :
             :
             30.
             195.
             
             Divide
             195
             by
             15
             ,
             it
             gives
             13
             for
             the
             Striking-pins
             .
             Or
             if
             you
             chuse
             13
             for
             your
             number
             of
             Pins
             ,
             and
             divide
             195
             by
             it
             ,
             it
             gives
             15
             ,
             for
             the
             number
             of
             ●urns
             ,
             as
             you
             see
             in
             the
             Margin
             .
             
          
           
             As
             for
             the
             Pinion
             of
             Report
             ,
             and
             the
             rest
             of
             the
             Wheels
             ,
             enough
             is
             said
             in
             the
             §
             1.
             
          
           
             But
             suppose
             you
             would
             calculate
             the
             numbers
             of
             a
             Clock
             of
             much
             longer
             continuance
             ,
             which
             will
             necessitate
             you
             to
             make
             your
             Pin-wheel
             further
             distant
             from
             the
             Great-wheel
             ,
             you
             are
             to
             proceed
             thus
             :
             Having
             re●olved
             upon
             your
             turns
             ,
             you
             must
             find
             out
             the
             number
             of
             strokes
             in
             one
             turn
             of
             the
             Great-wheel
             ,
             or
             Fusy
             ,
             by
             §
             2
             ▪
             Rule
             1.
             
             Thus
             in
             an
             8
             day
             piece
             ,
             of
             16
             turns
             ,
             16.
             8
             
             :
             :
             156.
             78.
             
             So
             in
             a
             piece
             of
             32
             days
             ,
             and
             16
             turns
             ,
             16
             ▪
             32
             :
             :
             156.
             312.
             
             These
             strokes
             so
             found
             out
             ,
             are
             the
             number
             which
             is
             to
             be
             broken
             into
             a
             convenient
             parcel
             of
             Quotients
             ,
             thus
             ;
          
           
             First
             resolve
             upon
             your
             number
             of
             Striking-pins
             :
             
             divide
             the
             last
             named
             number
             by
             it
             :
             The
             quotient
             arising
             hence
             ,
             is
             to
             be
             one
             ,
             or
             more
             quotients
             ,
             for
             the
             Wheels
             and
             Pinions
             .
             As
             in
             the
             last
             examples
             ▪
             Divide
             78
             by
             8
             (
             the
             usual
             pins
             in
             an
             8
             day
             piece
             )
             and
             the
             quotient
             is
             9●
             ;
             which
             is
             a
             quotient
             little
             enough
             .
             So
             in
             the
             Month-piece
             :
             if
             you
             take
             your
             Pins
             8
             ▪
             divide
             312
             by
             it
             ,
             the
             quotient
             is
             39.
             
             Which
             being
             too
             big
             for
             one
             ,
             must
             b●
             broken
             into
             two
             quotients
             ,
             
             for
             Wheels
             and
             Pinions
             ,
             or
             as
             near
             〈◊〉
             can
             be
             :
             which
             may
             b●
             7
             and
             5
             ,
             or
             6
             and
             6½
             .
             Th●
             latter
             is
             exactly
             39
             ,
             and
             may
             there●o●
             stand
             :
             as
             you
             see
             is
             done
             in
             the
             Margin
             .
          
           
             The
             quotients
             being
             thus
             determined
             and
             accordingly
             the
             Wheels
             and
             Pinio●●
             as
             you
             see
             ;
             the
             next
             work
             is
             to
             find
             〈◊〉
             quotient
             for
             the
             Pinion
             of
             Report
             ,
             to
             ●●ry
             round
             the
             Count
             (
             or
             Locking
             )
             wh●●
             once
             in
             12
             hours
             ,
             or
             as
             you
             please
             .
             ●
             you
             fix
             your
             Pinion
             of
             Report
             on
             th●
             Great-wheel
             arbor
             ,
             you
             must
             operate
             〈◊〉
             the
             Rule
             4.
             of
             the
             last
             paragraph
             .
             As
             〈◊〉
             the
             last
             example
             in
             the
             Month-piece
             :
             〈◊〉
             Rule
             6.
             before
             ,
             the
             strokes
             in
             the
             conti●●ance
             
             are
             4992.
             
             Then
             by
             Rule
             4
             say
             ,
             4992.
             16
             
             :
             :
             78.
             499●
             /
             124●
             or
             thus
             ,
             4992
             )
             1248.
             
             The
             first
             of
             which
             two
             numbers
             is
             the
             Pinion
             ,
             the
             next
             is
             the
             Wheel
             .
             Which
             being
             too
             large
             ,
             may
             be
             varied
             to
             ●●
             /
             9
             or
             
             36
             )
             9
             ;
             or
             to
             ●4
             /
             6
             or
             24
             )
             6
             ,
             by
             Sect.
             2
             §
             5.
             before
             .
          
           
             These
             numbers
             being
             not
             the
             usual
             numbers
             of
             a
             Month-piece
             ,
             but
             only
             made
             use
             of
             by
             me
             ,
             as
             better
             illustrating
             the
             foregoing
             Rules
             ▪
             I
             shall
             therefore
             ,
             for
             the
             fuller
             explication
             of
             what
             has
             been
             said
             ,
             briefly
             touch
             upon
             the
             calculation
             of
             the
             more
             usual
             numbers
             .
             They
             commonly
             encrease
             the
             number
             of
             Striking-pins
             ,
             and
             so
             make
             the
             Second-wheel
             the
             Striking-wheel
             ▪
             Suppose
             you
             take
             24
             Pins
             ;
             Divide
             312
             by
             it
             ,
             and
             the
             Quotient
             is
             13.
             
             Which
             is
             little
             enough
             
             for
             one
             Quotient
             ;
             and
             may
             therefore
             stand
             as
             you
             see
             is
             done
             in
             the
             Margin
             :
             where
             the
             Quotient
             of
             the
             first
             Wheel
             is
             13.
             
             In
             the
             second
             Wheel
             of
             72
             teeth
             ,
             are
             the
             24
             pins
             ,
             altho
             its
             quotient
             is
             but
             12
             ,
             because
             the
             Hoop-wheel
             is
             double
             ,
             and
             goes
             round
             but
             once
             in
             two
             strokes
             of
             the
             Pin-wheel
             .
          
           
           
             The
             Pinion
             of
             Report
             here
             ,
             is
             the
             same
             with
             the
             last
             ,
             if
             fixed
             upon
             the
             arbor
             of
             the
             Great
             ▪
             wheel
             .
             But
             if
             you
             fix
             it
             on
             the
             arbor
             of
             the
             Second
             ,
             or
             Pin-wheel
             ,
             its
             quotient
             then
             is
             found
             by
             §
             1.
             
             Infer
             .
             2.
             or
             by
             §
             2.
             
             Rule
             5.
             viz.
             Divide
             78
             by
             24
             ,
             and
             the
             number
             arising
             in
             the
             quotient
             ,
             is
             the
             quotient
             of
             the
             Pinion
             of
             
             Report
             ,
             which
             is
             3
             ¼
             .
             The
             Pinion
             of
             Report
             then
             being
             12
             ,
             the
             Count-wheel
             will
             be
             39
             ,
             as
             in
             the
             Margin
             .
          
           
             To
             perfect
             the
             Reader
             in
             this
             part
             of
             Calculation
             ,
             I
             will
             finish
             this
             Section
             with
             the
             calculation
             of
             a
             Year-piece
             of
             Clock-work
             .
             The
             Process
             whereof
             is
             the
             same
             with
             the
             last
             ,
             and
             therefore
             I
             may
             be
             more
             brief
             with
             this
             ,
             except
             where
             I
             have
             not
             touched
             upon
             the
             foregoing
             Rules
             .
          
           
             We
             will
             chuse
             a
             piece
             to
             go
             395
             days
             with
             16
             turns
             ,
             and
             26
             Striking-pins
             .
             By
             §
             2.
             
             Rule
             1.
             there
             are
             3851
             strokes
             in
             one
             turn
             of
             the
             Great-wheel
             .
             For
             16.
             395
             
             :
             :
             156.
             3851.
             
             This
             last
             number
             divided
             by
             the
             26
             Pins
             ,
             leaves
             148
             in
             the
             quotient
             ,
             to
             be
             broken
             into
             two
             or
             more
             quotients
             ,
             for
             Wheels
             and
             Pinions
             .
             These
             quotients
             may
             be
             12
             and
             12
             ;
             which
             multiplied
             ,
             
             makes
             144
             ,
             which
             is
             
             as
             near
             as
             can
             well
             be
             ,
             to
             148.
             
             The
             work
             thus
             far
             contrived
             ,
             will
             stand
             as
             you
             see
             in
             the
             Margin
             .
          
           
             Before
             you
             go
             any
             further
             ,
             you
             may
             correct
             your
             work
             ,
             and
             see
             how
             near
             your
             numbers
             come
             to
             what
             you
             proposed
             at
             first
             ,
             because
             they
             did
             not
             fall
             out
             exact
             .
             And
             first
             ,
             for
             the
             true
             continuance
             of
             your
             Clock
             ▪
             
               If
               you
               multiply
            
             12
             ,
             12
             ,
             and
             26
             (
             i.
             e.
             
               the
               Quotients
               un
               ▪
               o
               the
               Stri●ing-pins
               ,
               and
               those
               Pins
               )
               you
               have
               the
               true
               number
               of
               Strokes
               ,
               in
               one
               turn
               of
               the
               Great-wheel
               :
            
             Which
             ,
             in
             this
             example
             ,
             make
             3744.
             
             For
             12
             times
             12
             ,
             is
             144
             ;
             and
             26
             times
             that
             ,
             is
             3744.
             
             (
             This
             Direction
             I
             would
             have
             noted
             ,
             and
             remembered
             ,
             as
             a
             Rule
             useful
             at
             any
             time
             to
             discover
             the
             nature
             of
             any
             piece
             of
             Clock-work
             .
             )
             Having
             thus
             the
             true
             number
             of
             Strokes
             desired
             ,
             by
             §
             2.
             
             Rule
             2.
             you
             may
             find
             the
             true
             Continuance
             to
             be
             only
             384
             days
             .
             For
             156.
             3744
             
             :
             :
             16.
             384.
             
             If
             this
             Continuance
             doth
             not
             please
             you
             ,
             you
             may
             come
             nearer
             to
             your
             first
             proposed
             number
             ,
             of
             395
             days
             ,
             by
             a
             small
             encrease
             of
             
             the
             number
             o●
             Turns
             ;
             according
             to
             §
             2.
             
             Rule
             3.
             viz.
             by
             making
             your
             turns
             almost
             16½
             .
             For
             3744.
             156
             
             :
             :
             395.
             16½
             almost
             .
          
           
             Lastiy
             ,
             For
             the
             Pinion
             of
             Report
             ,
             if
             you
             fix
             it
             upon
             the
             Great-wheel
             ,
             it
             will
             require
             an
             excessive
             number
             :
             if
             you
             fix
             it
             upon
             the
             Pin-wheel
             (
             which
             is
             usual
             )
             then
             by
             §
             2.
             
             Rule
             5
             ,
             the
             quotient
             
             is
             3
             ;
             and
             the
             Pinion
             of
             Report
             being
             13
             ,
             the
             Count-wheel
             will
             be
             39
             ;
             as
             you
             see
             in
             the
             Margin
             .
          
           
             But
             for
             the
             better
             exercising
             the
             Reader
             ,
             let
             us
             fix
             it
             upon
             the
             Spindle
             of
             the
             Second-wheel
             96.
             
             It
             s
             quotient
             is
             12
             ;
             which
             multiplied
             by
             26
             (
             the
             pins
             )
             produceth
             312
             ;
             which
             are
             the
             Strokes
             in
             one
             turn
             of
             that
             Second-wheel
             .
             Then
             by
             §
             2
             ▪
             Rule
             5
             ,
             Divide
             78
             by
             312
             ,
             
               i.
               e.
            
             Set
             them
             as
             a
             Wheel
             and
             Pinion
             thus
             ,
             312
             )
             78
             ,
             and
             vary
             them
             to
             lesser
             numbers
             (
             by
             Sect.
             ●
             §
             5.
             )
             viz.
             36
             ▪
             9
             ,
             or
             to
             24
             )
             6
             ,
             or
             th●
             like
             .
          
           
             I
             think
             it
             needless
             to
             say
             any
             thing
             o●
             Pocket-clocks
             ,
             whose
             calculation
             is
             the
             very
             same
             ,
             with
             what
             goes
             before
             .
          
           
           
             That
             the
             unlearned
             Reader
             may
             not
             think
             any
             thing
             going
             before
             difficult
             ,
             I
             need
             only
             to
             advise
             him
             ,
             to
             look
             over
             the
             working
             of
             the
             Rule
             of
             Proportion
             ,
             in
             Sect.
             2.
             
             §
             4.
             
             For
             I
             think
             all
             will
             be
             plain
             ,
             if
             that
             be
             well
             understood
             .
          
        
         
           
             SECT
             .
             4.
             
             Of
             Quarters
             and
             Chimes
             .
          
           
             THe
             Reader
             will
             expect
             that
             I
             should
             say
             somewhat
             concerning
             Quarters
             and
             Chimes
             :
             but
             because
             there
             is
             little
             ,
             but
             what
             is
             purely
             mechanical
             in
             it
             ,
             I
             shall
             say
             the
             less
             ,
             and
             leave
             the
             Reader
             to
             his
             own
             invention
             .
          
           
             §
             1.
             
             The
             Quarters
             are
             generally
             a
             distinct
             part
             from
             the
             Clock-part
             ,
             which
             striketh
             the
             Hour
             .
          
           
             The
             Striking-wheel
             may
             be
             the
             First
             ,
             Second
             ,
             or
             &c.
             
             Wheel
             ,
             according
             to
             your
             Clock's
             continuance
             .
             Unto
             which
             Wheel
             you
             may
             fix
             the
             Pinion
             of
             Report
             .
          
           
             The
             Locking-wheel
             must
             be
             divided
             (
             as
             other
             Locking-wheels
             )
             into
             4
             ,
             8
             ,
             or
             more
             unequal
             parts
             ,
             ●o
             as
             to
             strike
             the
             Quarter
             ,
             and
             lock
             at
             the
             first
             Notch
             ;
             the
             half-hour
             ,
             and
             
             lock
             at
             the
             second
             Notch
             ,
             &c.
             
             And
             in
             doing
             this
             ,
             you
             may
             make
             it
             to
             chime
             the
             Quarters
             ,
             or
             strike
             them
             upon
             two
             Bells
             ,
             or
             more
             .
          
           
             'T
             is
             usual
             for
             the
             Pin-wheel
             ▪
             or
             the
             Locking-wheel
             ,
             to
             unlock
             the
             Hour-part
             in
             these
             Clocks
             ;
             which
             is
             easily
             done
             by
             some
             jogg
             or
             Latch
             ,
             at
             the
             end
             of
             the
             last
             Quarter
             ,
             to
             lift
             up
             the
             Detents
             of
             the
             Hour-part
             .
          
           
             If
             you
             would
             have
             your
             Clock
             strike
             the
             Hour
             ,
             at
             the
             Half-hour
             ,
             as
             well
             as
             whole
             Hour
             ,
             you
             must
             make
             the
             Locking-wheel
             of
             the
             Hour-part
             double
             :
             
               i.
               e.
            
             it
             must
             have
             two
             Notches
             of
             a
             sort
             ,
             to
             strike
             1
             ,
             2
             ,
             3
             ,
             4
             ,
             &c.
             twice
             apiece
             .
          
           
             §
             2.
             
             As
             for
             Chimes
             ,
             I
             need
             say
             nothing
             of
             the
             Lifting-pieces
             and
             Detents
             ,
             to
             lock
             and
             unlock
             ;
             nor
             of
             the
             Wheels
             to
             bridle
             the
             motion
             of
             the
             Barrel
             .
             Only
             you
             are
             to
             observe
             ,
             that
             the
             Barrel
             must
             be
             as
             long
             in
             turning
             round
             ,
             as
             you
             are
             in
             Singing
             the
             Tune
             it
             is
             to
             play
             .
             As
             for
             the
             Chime-Barrel
             ,
             it
             may
             be
             made
             up
             of
             certain
             Barrs
             ,
             that
             run
             athwart
             it
             ,
             with
             a
             convenient
             number
             of
             holes
             punched
             in
             them
             ,
             to
             put
             in
             the
             Pins
             ,
             that
             are
             to
             draw
             each
             Hammer
             .
             By
             this
             means
             ,
             you
             may
             change
             
             the
             Tune
             ,
             without
             changing
             the
             Barrel
             .
             This
             is
             the
             way
             of
             the
             
               Royal
               Exchange
            
             Clock
             in
             London
             ,
             and
             of
             others
             .
             In
             this
             case
             ,
             the
             Pins
             or
             Nuts
             ,
             which
             draw
             the
             Hammers
             ,
             must
             hang
             down
             from
             the
             Barr
             ,
             some
             more
             ,
             some
             less
             ,
             and
             some
             stand
             upright
             in
             the
             Barr
             :
             the
             reason
             whereof
             is
             ,
             to
             play
             the
             Time
             of
             the
             Tune
             rightly
             .
             For
             the
             distance
             of
             each
             of
             these
             Barrs
             ,
             may
             be
             a
             Semi-brief
             ,
             or
             &c.
             of
             which
             hereafter
             .
          
           
             But
             the
             most
             usual
             way
             is
             ,
             to
             have
             the
             Pins
             that
             draw
             the
             Hammers
             ,
             fixed
             on
             the
             Barrel
             .
             For
             the
             placing
             of
             which
             Pins
             ,
             you
             may
             make
             use
             of
             the
             Musical
             Notes
             ,
             or
             proceed
             by
             the
             way
             of
             Changes
             on
             Bells
             ,
             viz.
             1
             ,
             2
             ,
             3
             ,
             4
             ,
             &c.
             
             The
             first
             being
             far
             the
             better
             way
             ,
             I
             shall
             speak
             of
             that
             chiefly
             ,
             especially
             because
             the
             latter
             will
             fall
             in
             to
             be
             explained
             with
             it
             .
          
           
             And
             first
             ,
             you
             are
             to
             observe
             what
             is
             the
             Compass
             of
             your
             Tune
             ,
             or
             how
             many
             Notes
             or
             Bells
             there
             are
             from
             the
             highest
             to
             the
             lowest
             :
             and
             accordingly
             you
             must
             divide
             your
             Barrel
             from
             end
             to
             end
             .
             Thus
             in
             the
             examples
             following
             ,
             each
             of
             those
             Tunes
             are
             8
             notes
             in
             compass
             ;
             and
             accordingly
             the
             Barrel
             is
             divided
             into
             8
             
             parts
             .
             These
             Divisions
             are
             struck
             round
             the
             Barrel
             ,
             opposite
             to
             which
             are
             the
             Hammer-tails
             .
          
           
             I
             speak
             here
             ,
             as
             if
             there
             was
             only
             one
             Hammer
             to
             each
             Bell
             ,
             that
             the
             Reader
             may
             more
             clearly
             apprehend
             what
             I
             am
             explaining
             .
             But
             when
             two
             Notes
             of
             the
             same
             sound
             come
             together
             in
             a
             Tune
             ,
             there
             must
             be
             two
             Hammers
             to
             that
             Bell
             ,
             to
             strike
             it
             .
             So
             that
             if
             in
             all
             the
             Tunes
             you
             intend
             to
             Chime
             ,
             of
             8
             notes
             compass
             ,
             there
             should
             happen
             to
             be
             such
             double
             Notes
             on
             every
             Bell
             ,
             instead
             of
             8
             ,
             you
             must
             have
             16
             Hammers
             :
             and
             accordingly
             you
             must
             divide
             your
             Barrel
             ,
             and
             strike
             16
             strokes
             round
             it
             opposite
             to
             each
             Hammer-tail
             .
             Thus
             much
             for
             dividing
             your
             Barrel
             from
             end
             to
             end
             .
          
           
             In
             the
             next
             place
             ,
             you
             are
             to
             divide
             i●
             (
             round
             about
             )
             into
             as
             many
             divisions
             ,
             as
             there
             are
             Musical
             Barrs
             ,
             Semibriefs
             ,
             Minums
             ,
             &c.
             in
             your
             Tune
             .
             Thus
             the
             100th
             Psalm-tune
             hath
             20
             Semibriefs
             ;
             the
             Song-tune
             following
             ,
             hath
             24
             Barrs
             of
             triple
             time
             :
             and
             accordingly
             their
             Barrels
             are
             divided
             .
             Each
             division
             therefore
             of
             the
             100th
             Psalm
             Barrel
             is
             a
             Semibrief
             ,
             ●nd
             of
             the
             Song-tune
             't
             is
             three
             crotchets
             ▪
             
             And
             therefore
             the
             intermediate
             Spaces
             serve
             for
             the
             shorter
             notes
             :
             as
             ,
             one
             third
             of
             a
             division
             ,
             is
             a
             Crotchet
             ,
             in
             the
             Song-tune
             .
             One
             half
             a
             division
             ,
             is
             a
             Minum
             ;
             and
             one
             quarter
             a
             Crotchet
             ,
             in
             the
             Psalm-tune
             .
             Thus
             the
             first
             note
             in
             the
             100th
             Psalm
             ,
             is
             a
             Semibrief
             ,
             and
             accordingly
             on
             the
             Barrel
             ,
             't
             is
             a
             whole
             division
             from
             5
             to
             5.
             
             The
             second
             is
             a
             Minum
             ,
             and
             therefore
             6
             is
             but
             half
             a
             division
             from
             5
             ;
             and
             so
             of
             the
             rest
             .
             And
             so
             also
             for
             the
             Song-tune
             ,
             which
             is
             shorter
             time
             :
             The
             two
             first
             notes
             being
             Quavers
             ,
             are
             distant
             from
             one
             another
             ,
             and
             from
             the
             third
             pin
             ,
             but
             half
             a
             third
             part
             of
             one
             of
             the
             divisions
             .
             But
             the
             two
             next
             pins
             (
             of
             the
             bell
             3
             ,
             3
             )
             being
             Crotchets
             ,
             are
             distant
             so
             many
             third
             parts
             of
             a
             division
             .
             And
             the
             next
             pin
             (
             of
             the
             bell
             1
             )
             being
             a
             Minum
             ,
             is
             distant
             from
             the
             following
             pin
             (
             4
             )
             two
             thirds
             of
             a
             division
             .
          
           
             
             
               A
               Table
               of
               Chimes
               to
               the
               100
               Psalm
               .
            
             
          
           
             
               The
               Musical
               Notes
               of
               Psalm
               100.
               
            
             
          
           
             
             
             
               
                 The
                 Musical
                 Notes
                 of
              
               ,
               Such
               Command
               o're
               my
               Fate
               ,
               &c
               A
               Song
            
             
          
           
             
               The
               Chimes
               of
               the
               Song
               ,
               
                 Such
                 Command
                 o're
                 my
                 Fate
              
               ,
               &c.
               
            
             
          
           
           
             Pins
             ,
             to
             be
             set
             on
             the
             Barrel
             .
          
           
             You
             may
             observe
             in
             the
             Tables
             ,
             that
             from
             the
             end
             of
             each
             Table
             to
             the
             beginning
             ,
             is
             the
             distance
             of
             two
             ,
             or
             near
             two
             divisions
             :
             which
             is
             for
             a
             Pause
             ,
             between
             the
             end
             of
             the
             Tune
             ,
             and
             its
             beginning
             to
             Chime
             again
             .
          
           
             I
             need
             not
             say
             ,
             that
             the
             Dotts
             running
             about
             the
             Tables
             ,
             are
             the
             places
             of
             the
             Pins
             that
             play
             the
             Tune
             .
          
           
             If
             you
             would
             have
             your
             Chimes
             compleat
             indeed
             ,
             you
             ought
             to
             have
             a
             set
             of
             Bells
             ,
             to
             the
             Gamut
             notes
             ;
             so
             as
             that
             each
             Bell
             having
             the
             true
             sound
             of
             Sol
             ,
             La
             ,
             Mi
             ,
             Fa
             ,
             you
             may
             play
             any
             Tune
             ,
             with
             its
             Flats
             and
             Sharps
             .
             Nay
             ,
             you
             may
             by
             these
             means
             ,
             play
             both
             the
             Bass
             and
             Treble
             ,
             with
             one
             Barrel
             .
          
           
             If
             any
             thing
             going
             before
             appears
             gibberish
             ,
             I
             can't
             help
             it
             ,
             unless
             I
             should
             here
             teach
             the
             skill
             of
             Musick
             too
             .
          
           
             As
             to
             setting
             a
             Tune
             upon
             the
             Chime-barrel
             from
             the
             number
             of
             Bells
             ,
             viz.
             1
             ,
             2
             ,
             3
             ,
             4
             ,
             I
             shall
             here
             give
             you
             a
             specimen
             thereof
             .
          
           
             Such
             Command
             o're
             my
             Fate
             ,
             
               in
               numbers
            
             .
          
           
           
             775
             ,
             3
             ,
             3
             ,
             1.
             4
             ,
             5
             ,
             6
             ,
             4.
             4
             ,
             2.
             
             4
             ,
             3
             ,
             2
             ,
             3
             ,
             4
             ,
             6
             ,
             3
             ,
             5
             ,
             7
             ,
             7
             ,
             7.
             
             ‖
             
             5
             ,
             6
             ,
             8
             ,
             8
             ,
             4.
             4
             ,
             4
             ;
             3
             ,
             5
             ,
             4.
             
             6
             ,
             5
             ,
             7
             ,
             5
             ,
             3
             ;
             41
             ,
             3
             ,
             5
             ,
             5
             ,
             5.
             
             3
             ,
             3
             ,
             1
             ,
             3
             ,
             5.
             554
             ,
             2
             ,
             4
             ,
             6.
             
             4
             ,
             3
             ;
             23
             ,
             3
             ;
             53
             ,
             5
             ,
             7
             ,
             7
             ,
             7.
             
          
           
             Note
             ,
             In
             these
             numbers
             ,
             a
             Comma
             [
             ,
             ]
             signifies
             the
             note
             before
             it
             ,
             to
             be
             a
             Crotchet
             .
             A
             prick'd
             Comma
             ,
             or
             Semi-colon
             [
             ;
             ]
             denoteth
             a
             prick'd
             Crotchet
             .
             And
             a
             Period
             [
             .
             ]
             is
             a
             Minum
             .
             Where
             no
             punctation
             is
             ,
             those
             Notes
             are
             Quavers
             .
          
           
             I
             shall
             only
             add
             further
             ,
             that
             by
             setting
             the
             Names
             of
             your
             Bells
             at
             the
             head
             of
             any
             Tune
             (
             as
             is
             done
             in
             the
             Tables
             before
             )
             you
             may
             easily
             transfer
             that
             Tune
             ,
             to
             your
             Chime-barrel
             ,
             without
             any
             great
             skill
             in
             Musick
             .
             But
             observe
             ,
             that
             each
             line
             in
             the
             Musick
             ,
             is
             three
             notes
             distant
             ;
             
               i.
               e.
            
             there
             is
             a
             Note
             between
             each
             line
             ,
             as
             well
             as
             upon
             it
             :
             as
             is
             manifest
             by
             inspecting
             the
             Tables
             .
          
        
         
           
           
             SECT
             .
             5.
             
             To
             Calculate
             any
             of
             the
             Celestial
             Motions
             .
          
           
             The
             Motions
             I
             here
             chiefly
             intend
             ,
             are
             the
             Day
             of
             the
             Month
             ,
             the
             Moons
             age
             ,
             the
             Day
             of
             the
             Year
             ,
             the
             Tides
             ,
             and
             (
             if
             you
             please
             )
             the
             slow
             motion
             of
             the
             Suns
             Apogaeum
             ,
             of
             the
             Fixed
             Stars
             ,
             the
             motion
             of
             the
             Planets
             ,
             &c.
             
          
           
             §
             1.
             
             For
             the
             effecting
             these
             Motions
             ,
             you
             may
             make
             them
             to
             depend
             upon
             the
             Work
             already
             in
             the
             Movement
             ;
             or
             else
             measure
             them
             by
             the
             beats
             of
             a
             Ballance
             ,
             or
             Pendulum
             .
          
           
             If
             the
             latter
             way
             ,
             you
             must
             however
             contrive
             a
             Piece
             (
             as
             before
             in
             Watch-work
             )
             to
             go
             a
             certain
             time
             ,
             with
             a
             certain
             number
             of
             turns
             .
          
           
             But
             then
             to
             Specificate
             ,
             or
             determine
             the
             Motion
             intended
             ,
             you
             must
             proceed
             one
             of
             these
             two
             ways
             :
             either
             ,
          
           
             1.
             
             Find
             how
             many
             beats
             are
             in
             the
             Revolution
             .
             Divide
             these
             beats
             by
             the
             beats
             in
             one
             turn
             of
             the
             Wheel
             ,
             or
             Pinion
             ,
             which
             you
             intend
             shall
             drive
             the
             intended
             Revolution
             ;
             and
             the
             Quotient
             shall
             be
             
             the
             number
             to
             perform
             the
             same
             .
             Which
             ,
             if
             too
             big
             for
             one
             ,
             may
             be
             broken
             into
             more
             Quotients
             .
             Thus
             ,
             if
             you
             would
             represent
             the
             Synodical
             Revolution
             of
             the
             Moon
             ,
             which
             is
             29
             days
             ,
             12
             ¾
             hours
             )
             with
             a
             Pendulum
             that
             swings
             Seconds
             ,
             the
             Movement
             to
             go
             8
             days
             ,
             with
             16
             turns
             of
             the
             Fusy
             ,
             and
             the
             Great-wheel
             to
             drive
             the
             Revolution
             .
             Divide
             2551500
             (
             the
             Beats
             in
             29
             days
             12
             ¾
             hours
             )
             by
             43200
             (
             the
             Beats
             in
             one
             turn
             of
             the
             Great-wheel
             )
             and
             you
             will
             have
             59
             in
             the
             Quotient
             :
             which
             being
             too
             big
             for
             one
             ,
             may
             be
             put
             into
             two
             Quotients
             .
             Or
          
           
             2.
             
             You
             may
             proceed
             as
             is
             directed
             before
             ,
             
             in
             the
             Section
             of
             Calculating
             Watch-work
             ,
             viz.
             Chuse
             your
             Train
             ,
             turns
             of
             the
             Fusy
             ,
             Continuance
             ,
             &c.
             
             And
             then
             instead
             of
             finding
             a
             Quotient
             for
             the
             Pinion
             of
             Report
             ,
             find
             a
             number
             (
             which
             is
             all
             one
             as
             a
             Pin.
             of
             Report
             )
             to
             Specificate
             your
             Revolution
             ,
             by
             this
             following
             Rule
             .
          
           
             Rule
             .
             As
             the
             Beats
             in
             one
             turn
             of
             the
             Great-wheel
             .
             To
             the
             Train
             :
             :
             So
             are
             the
             Hours
             of
             the
             Revolution
             ,
             To
             the
             Quotient
             of
             the
             Revolution
             .
          
           
             Thus
             to
             perform
             the
             Revolution
             of
             Saturn
             (
             which
             is
             29
             years
             ,
             183
             days
             )
             with
             a
             
             16
             hour
             Watch
             ,
             of
             26928
             Beats
             in
             one
             turn
             of
             the
             Fusy
             ,
             and
             20196
             ,
             the
             Train
             :
             the
             quotient
             of
             the
             Revolution
             ,
             will
             be
             193824.
             
             For
             ,
             As
             26928
             ,
             To
             20196
             :
             :
             So
             258432
             (
             the
             Hours
             in
             29
             y.
             and
             183
             d.
             )
             To
             193824.
             
             Note
             here
             ,
             That
             the
             Great-wheel
             Pinion
             is
             to
             drive
             the
             Revolution
             work
             .
          
           
             But
             if
             you
             would
             have
             the
             Revolution
             to
             be
             driven
             by
             the
             Dial-wheel
             ,
             and
             the
             Work
             already
             in
             the
             Movement
             (
             which
             in
             great
             Revolutions
             ,
             is
             for
             the
             most
             part
             ,
             as
             nice
             as
             the
             last
             way
             ,
             and
             in
             which
             I
             intend
             to
             treat
             of
             the
             particular
             Motions
             )
             in
             this
             case
             ,
             I
             say
             ,
             you
             must
             first
             know
             the
             Days
             of
             the
             Revolution
             .
             And
             because
             the
             Dial-wheel
             goeth
             round
             twice
             in
             a
             day
             ,
             therefore
             double
             the
             number
             of
             the
             days
             in
             the
             Revolution
             ,
             and
             you
             have
             the
             number
             of
             turns
             of
             the
             Dial-wheel
             in
             that
             time
             .
             This
             number
             of
             turns
             is
             what
             you
             are
             to
             break
             into
             a
             convenient
             number
             of
             quotients
             ,
             for
             the
             Wheels
             and
             Pinions
             ▪
             as
             shall
             be
             shewed
             in
             the
             following
             examples
             .
          
           
             §
             2.
             
             A
             Motion
             to
             shew
             the
             
               Day
               of
               the
               Month.
            
             
          
           
           
             
             The
             days
             in
             the
             largest
             Month
             are
             31.
             
             These
             doubled
             are
             62
             ,
             which
             are
             the
             turns
             of
             the
             Dial-wheel
             ,
             which
             may
             be
             broken
             into
             these
             two
             quotients
             15
             ½
             and
             4
             ;
             which
             multiplied
             together
             make
             62.
             
             Therefore
             chusing
             your
             Wheels
             and
             Pinions
             ,
             as
             hath
             been
             directed
             in
             the
             former
             Sections
             ,
             your
             work
             is
             done
             .
             The
             Wheels
             
             and
             Pinions
             may
             be
             ,
             as
             you
             see
             done
             in
             ▪
             the
             Margin
             .
             Or
             if
             a
             larger
             Pinion
             than
             one
             of
             5
             be
             necessary
             ,
             by
             reason
             it
             is
             concentrick
             to
             a
             Wheel
             ,
             you
             
             may
             take
             10
             for
             the
             Pinion
             ,
             and
             40
             for
             the
             Wheel
             ,
             as
             in
             the
             Margin
             .
          
           
             The
             work
             will
             lye
             thus
             in
             the
             Movement
             ,
             viz.
             Fix
             your
             Pinion
             10
             ,
             concentrical
             to
             the
             Dial-wheel
             (
             or
             to
             turn
             round
             with
             it
             upon
             the
             same
             Spindle
             .
             )
             This
             Pinion
             10
             drives
             the
             Wheel
             40
             :
             which
             Wheel
             has
             the
             Pinion
             4
             in
             its
             center
             ,
             which
             carrieth
             about
             a
             Ring
             of
             62
             teeth
             ,
             divided
             on
             the
             upper
             side
             into
             31
             days
             .
          
           
             Or
             ,
             you
             may
             ,
             without
             the
             trouble
             of
             many
             Wheels
             ,
             effect
             this
             motion
             ;
             vi●
             .
             By
             a
             Ring
             divided
             into
             30
             or
             31
             days
             ,
             and
             as
             many
             Fangs
             or
             Teeth
             ,
             like
             a
             Crown
             ▪
             
             wheel
             teeth
             ,
             which
             are
             caught
             and
             pushed
             forward
             once
             in
             24
             hours
             ,
             by
             a
             pin
             in
             a
             Wheel
             ,
             that
             goeth
             round
             in
             that
             time
             .
             This
             is
             the
             usual
             way
             in
             the
             Royal
             Pendulums
             ,
             and
             many
             other
             Clocks
             ;
             and
             therefore
             being
             common
             ,
             I
             shall
             say
             no
             more
             of
             it
             .
          
           
             §
             3.
             
             A
             Motion
             to
             shew
             the
             age
             of
             the
             
             Moon
             .
          
           
             The
             Moon
             finisheth
             her
             course
             ▪
             so
             as
             to
             overtake
             the
             Sun
             ,
             in
             29
             days
             ,
             and
             a
             little
             above
             an
             half
             .
             This
             29
             ½
             days
             (
             not
             regarding
             the
             small
             excess
             )
             makes
             59
             twelve
             hours
             ,
             or
             turns
             of
             the
             Dial-wheel
             ,
             which
             is
             to
             be
             broken
             into
             convenient
             quotients
             :
             which
             
             may
             be
             5
             ,
             9
             and
             10
             as
             in
             the
             first
             example
             ;
             or
             14¾
             and
             4
             ,
             as
             in
             the
             second
             example
             in
             the
             Margin
             .
             So
             that
             if
             you
             fix
             a
             Pinion
             of
             10
             concentrical
             with
             your
             Dial-wheel
             ,
             to
             drive
             a
             Wheel
             of
             40
             (
             according
             to
             the
             last
             example
             )
             which
             Wheel
             40
             drives
             a
             Pinion
             4
             ,
             which
             carries
             about
             a
             Ring
             ,
             or
             Wheel
             of
             59
             teeth
             ,
             divided
             on
             the
             upper
             side
             into
             29
             ½
             't
             will
             shew
             the
             Moons
             age
             .
          
           
           
             
             §
             4.
             
             A
             Motion
             to
             shew
             the
             day
             of
             the
             Year
             ,
             the
             Sun's
             place
             in
             the
             Ecliptick
             ,
             Sun's
             Rising
             or
             Setting
             ,
             or
             any
             other
             annual
             motion
             of
             365
             days
             .
          
           
             The
             double
             of
             365
             is
             730
             ,
             the
             turns
             of
             the
             Dial-wheel
             in
             an
             year
             :
             which
             may
             be
             broken
             into
             
             these
             quotients
             ,
             viz.
             18
             ¼
             ,
             and
             10
             ,
             and
             4
             ,
             according
             to
             the
             first
             example
             ;
             or
             18
             ¼
             ,
             8
             ,
             and
             5
             ,
             according
             to
             the
             second
             .
             So
             that
             a
             Pinion
             of
             5
             is
             to
             lead
             a
             Wheel
             of
             20
             ;
             which
             again
             by
             a
             Pinion
             of
             4
             ,
             leadeth
             a
             Wheel
             of
             40
             ;
             which
             thirdly
             ,
             by
             a
             Pinion
             of
             4
             ,
             carrieth
             about
             a
             Wheel
             ,
             or
             Ring
             of
             73
             ,
             divided
             into
             the
             12
             months
             ,
             and
             their
             days
             ;
             or
             into
             the
             12
             signs
             ,
             and
             their
             degrees
             ;
             or
             into
             the
             Sun's
             Rising
             and
             Setting
             ,
             &c.
             
             For
             the
             setting
             on
             of
             which
             last
             ,
             you
             have
             a
             Table
             in
             Mr.
             
             Oughtred's
             Opuscula
             .
          
           
             
             §
             5.
             
             To
             shew
             the
             Tides
             at
             any
             Port.
             
          
           
             This
             is
             done
             without
             any
             other
             trouble
             ,
             than
             the
             Moon
             's
             Ring
             (
             before
             mentioned
             §
             3.
             )
             to
             move
             round
             a
             fixed
             circle
             ,
             divided
             into
             twice
             12
             hours
             ,
             and
             numbered
             the
             contrary
             way
             to
             the
             age
             of
             the
             Moon
             .
          
           
           
             To
             set
             this
             to
             go
             right
             ,
             you
             must
             find
             out
             at
             what
             Point
             of
             the
             Compass
             the
             Moon
             makes
             full
             Sea
             ,
             at
             the
             place
             you
             would
             have
             your
             Watch
             serve
             to
             .
             Convert
             that
             point
             into
             hours
             ,
             allowing
             for
             every
             point
             North
             or
             S.
             lost
             45′
             of
             an
             hour
             .
             Thus
             at
             London-bridge
             't
             is
             vulgarly
             thought
             to
             be
             high
             Tide
             ,
             the
             Moon
             at
             N.
             E.
             and
             S.
             W
             ,
             which
             are
             4
             Points
             from
             the
             N.
             and
             S.
             Or
             you
             may
             do
             thus
             :
             by
             Tide-tables
             learn
             how
             many
             hours
             from
             the
             Moon
             's
             Southing
             ,
             't
             is
             High-water
             .
             Or
             thus
             ;
             find
             at
             what
             hour
             it
             is
             High-water
             ,
             at
             the
             Full
             or
             Change
             of
             the
             M●on
             :
             as
             at
             London-bridge
             ,
             the
             full
             Tide
             is
             commonly
             reckoned
             to
             be
             3
             hours
             from
             the
             Moon
             's
             Southing
             ;
             or
             at
             3
             of
             clock
             at
             the
             Full
             and
             Change.
             The
             day
             of
             Conjunction
             ,
             or
             New-Moon
             ,
             with
             a
             little
             stud
             to
             point
             ,
             being
             set
             to
             the
             hour
             so
             found
             ,
             will
             afterwards
             point
             to
             the
             hour
             of
             full
             Tide
             .
          
           
             This
             is
             the
             usual
             way
             ;
             but
             it
             being
             always
             in
             motion
             ,
             as
             the
             Tides
             are
             not
             ,
             a
             better
             way
             may
             be
             found
             out
             ,
             viz.
             By
             causing
             a
             Wheel
             ,
             or
             Ring
             to
             be
             moved
             forward
             ,
             only
             twice
             a
             day
             ,
             and
             to
             keep
             time
             (
             as
             near
             as
             can
             be
             )
             with
             Mr.
             
             Flamsteed's
             most
             correct
             Tables
             .
             But
             this
             I
             
             shall
             commit
             to
             the
             Readers
             contrivance
             ,
             it
             being
             easie
             ,
             and
             more
             of
             curiosity
             than
             use
             .
          
           
             §
             6.
             
             To
             Calculate
             Numbers
             ,
             to
             shew
             the
             Motion
             of
             the
             Planets
             ,
             the
             Slow
             Motion
             of
             the
             Fixed
             Stars
             ,
             and
             of
             the
             Sun's
             Apogeum
             ,
             &c.
             
          
           
             Having
             said
             enough
             before
             that
             may
             be
             applied
             here
             ,
             and
             they
             being
             only
             curiosities
             ,
             seldom
             put
             in
             practice
             ,
             I
             shall
             not
             therefore
             trouble
             the
             Reader
             ,
             or
             swell
             my
             Book
             with
             so
             many
             words
             ,
             as
             would
             be
             required
             to
             treat
             of
             these
             Motions
             distinctly
             ,
             and
             compleatly
             .
          
           
             Only
             thus
             much
             in
             general
             .
             Knowing
             the
             years
             of
             any
             of
             these
             Revolutions
             ,
             you
             may
             break
             this
             number
             into
             quotients
             ;
             if
             you
             will
             make
             the
             Revolution
             to
             depend
             upon
             the
             year's
             Motion
             ;
             which
             is
             already
             in
             the
             Movement
             ,
             and
             described
             §
             4.
             before
             .
             Or
             if
             you
             would
             have
             it
             depend
             upon
             the
             Dial-wheel
             ,
             or
             upon
             the
             Beats
             of
             a
             Pendulu●
             ,
             enough
             is
             said
             before
             to
             direct
             in
             mis
             matter
             .
          
           
             In
             all
             these
             Slow
             motions
             ,
             you
             may
             somewhat
             ●●●●ten
             your
             labour
             ,
             by
             endless
             Screws
             to
             serve
             for
             Pinions
             ,
             which
             are
             but
             as
             a
             Pinion
             of
             one
             tooth
             .
          
           
           
             
             Sir
             Jonas
             Moor's
             account
             of
             his
             large
             ●phere
             going
             by
             Clock-work
             ,
             will
             suffi●ently
             illustrate
             this
             paragraph
             .
             In
             this
             ●phere
             ,
             is
             a
             Motion
             of
             17100
             years
             ,
             for
             ●he
             Sun's
             Apogeum
             ,
             performed
             by
             six
             ●heels
             ,
             thus
             ,
             as
             Sir
             Jonas
             relates
             it
             ;
             For
             the
             Great-wheel
             fixed
             is
             96
             ,
             a
             Spindle-wheel
             of
             12
             bars
             turns
             round
             it
             8
             times
             in
             24
             hours
             ,
             that
             is
             ,
             in
             3
             hours
             ;
             after
             these
             ,
             there
             are
             four
             Wheels
             ,
             20
             ,
             73
             ,
             24
             ,
             and
             75
             ,
             wrought
             by
             endless
             Screws
             that
             are
             in
             value
             but
             one
             :
             therefore
             3
             ,
             20
             ,
             73
             ,
             24
             ,
             and
             75
             multiplied
             together
             continually
             ,
             produceth
             7884000
             
             hours
             ,
             which
             divided
             ,
             by
             24
             gives
             3285000
             days
             ,
             equal
             to
             900
             years
             .
             Now
             on
             the
             last
             wheel
             75
             is
             a
             pinion
             of
             6
             ,
             turning
             a
             great
             Wheel
             ,
             that
             carrieth
             the
             Apogeum
             number
             114
             :
             and
             114
             divided
             by
             6
             ,
             gives
             19
             the
             quotient
             :
             and
             900
             times
             19
             is
             17100
             years
             .
          
           
             Thus
             I
             have
             ,
             with
             all
             the
             perspicuity
             I
             ●ould
             ,
             led
             my
             Reader
             through
             the
             whole
             ●rt
             of
             Calculation
             ,
             so
             much
             of
             it
             at
             least
             ,
             ●at
             I
             hope
             he
             will
             be
             master
             of
             it
             all
             ;
             not
             ●ly
             of
             those
             motions
             ,
             which
             I
             have
             par●cularly
             treated
             about
             ,
             but
             of
             any
             other
             ●t
             mentioned
             :
             Such
             as
             the
             Revolution
             
             of
             the
             Dragons
             Head
             and
             Tail
             ,
             whereby
             the
             Eclipses
             of
             the
             Sun
             and
             Moon
             are
             found
             ,
             the
             Revolution
             of
             the
             several
             Orbs
             ,
             according
             to
             the
             Ptolemaick
             System
             ,
             or
             of
             the
             celestial
             bodies
             themselves
             ,
             according
             to
             better
             Systems
             ,
             with
             many
             other
             such
             curious
             performances
             ,
             which
             have
             made
             the
             Sphere
             of
             Archimedes
             of
             old
             famous
             :
             and
             since
             him
             ,
             that
             of
             William
             of
             Zeland
             ,
             
             and
             another
             of
             
               Janellus
               Turrianus
            
             of
             Cremona
             ,
             mentioned
             by
             Cardan
             :
             and
             of
             late
             ,
             that
             elaborate
             piece
             of
             Mr.
             Watson
             ,
             late
             of
             Coventry
             ,
             now
             of
             London
             ,
             in
             her
             late
             Majesties
             Closet
             .
          
        
      
       
         
           CHAP.
           III.
           To
           alter
           Clock-work
           ,
           or
           convert
           one
           Movement
           into
           another
           .
        
         
           THis
           Chapter
           I
           design
           for
           the
           use
           of
           such
           ,
           as
           would
           convert
           old
           Ballance
           Clocks
           into
           Pendulums
           ,
           or
           would
           make
           any
           old
           work
           serve
           for
           the
           tryal
           of
           new
           motions
           ,
           or
           would
           apply
           it
           to
           any
           other
           such
           like
           use
           .
        
         
         
           §
           1.
           
           To
           do
           this
           ,
           you
           may
           draw
           a
           Scheme
           of
           your
           old
           work
           :
           and
           so
           you
           will
           see
           what
           Quotients
           you
           have
           ,
           and
           what
           you
           will
           want
           .
           To
           do
           all
           which
           ,
           there
           are
           sufficient
           instructions
           in
           the
           preceding
           Chapter
           .
           A
           few
           instances
           will
           make
           all
           plain
           .
        
         
           §
           2.
           
           Let
           us
           chuse
           for
           instance
           an
           old
           Ballance
           clock
           to
           be
           turned
           into
           a
           Pendulum
           of
           6
           inches
           .
           The
           old
           work
           is
           ,
           The
           Great-wheel
           56
           ,
           the
           Pinion
           7
           ;
           the
           next
           Wheel
           54
           ,
           the
           Pinion
           6
           ;
           the
           Crown-wheel
           19
           ,
           &c.
           
           The
           Scheme
           
           of
           this
           work
           is
           in
           the
           Margin
           .
           The
           Quotients
           and
           Crown-wheel
           and
           2
           Pallets
           multiplied
           together
           continually
           ,
           produce
           2736
           ,
           which
           are
           the
           Strokes
           of
           the
           Ballance
           ,
           in
           one
           turn
           of
           the
           Great-wheel
           ,
           by
           Sect.
           l.
           §
           4
           ,
           5.
           of
           the
           last
           Chapter
           .
           And
           by
           the
           Quotient
           of
           the
           Dial-wheel
           (
           which
           is
           12
           )
           it
           appears
           ,
           that
           the
           Great-wheel
           goeth
           round
           once
           in
           an
           hour
           .
           Or
           you
           may
           find
           the
           Beats
           in
           an
           hour
           ,
           by
           §
           5.
           last
           cited
           .
           Having
           thus
           found
           the
           Beats
           in
           an
           hour
           ,
           of
           the
           old
           work
           ,
           you
           must
           next
           find
           the
           Beats
           in
           an
           hour
           ,
           of
           a
           6
           inches
           Pendulum
           ;
           which
           you
           may
           do
           by
           
           
           Chap.
           5.
           
           §
           4.
           following
           ;
           or
           by
           Mr.
           
           Smith's
           Tables
           ,
           according
           to
           whom
           the
           number
           is
           ▪
           9368.
           
           Divide
           this
           by
           2736
           ,
           and
           you
           have
           the
           Quotient
           ,
           which
           
           is
           to
           be
           added
           to
           the
           Scheme
           of
           the
           old
           work
           .
           This
           Quotient
           is
           3
           and
           near
           ½
           ▪
           as
           you
           see
           in
           the
           Margin
           .
        
         
           The
           work
           thus
           altered
           ,
           will
           
           stand
           as
           you
           see
           in
           the
           Margin
           ,
           viz.
           a
           Pinion
           6
           ,
           and
           a
           Contrate-wheel
           21
           ,
           must
           be
           added
           .
        
         
           According
           to
           this
           way
           ,
           the
           old
           work
           will
           stand
           as
           before
           ,
           only
           the
           Crown-wheel
           must
           be
           inverted
           .
        
         
           §
           3.
           
           But
           because
           the
           Crown-wheel
           is
           too
           big
           for
           the
           Contrate-wheel
           (
           which
           is
           unseemly
           )
           therefore
           it
           will
           be
           best
           ,
           ▪
           to
           make
           both
           the
           Contrate
           ,
           and
           Crown-wheels
           new
           ;
           and
           encrease
           the
           number
           of
           the
           Contrate-wheel
           ,
           but
           diminish
           that
           of
           the
           Crown-wheel
           .
           To
           do
           which
           ,
           pitch
           upon
           some
           convenient
           number
           for
           the
           Crown-wheel
           .
           Multiply
           all
           the
           Quotients
           ,
           and
           this
           new
           Crown-wheel
           number
           ,
           as
           before
           ;
           and
           divide
           9368
           by
           it
           .
           As
           ,
           
           Suppose
           you
           pitch
           upon
           11
           for
           the
           Crown-wheel
           :
           if
           you
           multiply
           8
           ,
           9
           and
           11
           ▪
           the
           Product
           is
           792
           ;
           which
           multiplied
           by
           the
           2
           Pallets
           ,
           makes
           1584
           ,
           which
           are
           the
           Beats
           in
           one
           turn
           of
           the
           Great-wheel
           ,
           or
           
           in
           an
           hour
           .
           Divide
           9368
           by
           it
           ,
           and
           you
           have
           near
           6
           for
           the
           Quotient
           
           of
           your
           Contrate-wheel
           .
           The
           work
           thus
           ordered
           ,
           will
           stand
           as
           in
           the
           Margin
           .
        
         
           If
           you
           would
           correct
           your
           work
           ,
           to
           find
           the
           true
           number
           of
           Beats
           in
           an
           hour
           ,
           &c.
           you
           must
           proceed
           ,
           as
           is
           shewn
           Sect.
           2.
           §
           6
           ,
           and
           latter
           end
           of
           §
           7.
           of
           the
           last
           Chapter
           .
        
         
           §
           4.
           
           But
           suppose
           you
           have
           a
           mind
           to
           change
           the
           former
           old
           Watch
           ,
           into
           a
           30
           hour
           piece
           ,
           and
           to
           retain
           the
           old
           Ballance-wheel
           (
           which
           may
           be
           often
           done
           :
           )
           in
           this
           case
           ,
           you
           must
           add
           a
           Contrate-wheel
           ,
           and
           alter
           the
           Pinion
           of
           report
           .
           For
           the
           Contrate-wheel
           ,
           chuse
           such
           a
           Quotient
           ,
           as
           will
           best
           suit
           with
           the
           rest
           of
           your
           work
           ;
           and
           then
           multiply
           all
           your
           Quotients
           ,
           Crown-wheel
           and
           ●
           Pallets
           together
           ,
           and
           so
           find
           the
           number
           of
           turns
           in
           the
           Great-wheel
           ,
           as
           before
           .
           Then
           say
           by
           
           Sect.
           2.
           
           §
           6.
           part
           5.
           before
           ,
           As
           the
           Beats
           in
           one
           turn
           of
           the
           Great-wheel
           ,
           To
           the
           Beats
           in
           an
           hour
           :
           :
           So
           are
           the
           hours
           of
           the
           Dial
           ,
           To
           the
           quotient
           of
           the
           Pinion
           of
           Report
           .
        
         
           Thus
           in
           the
           old
           work
           before
           ;
           to
           the
           old
           quotients
           8
           and
           9
           ,
           you
           may
           add
           another
           of
           8
           ,
           for
           the
           Contrate-wheel
           .
           Those
           multiplied
           ,
           as
           was
           now
           directed
           ,
           make
           21888
           ,
           for
           the
           Beats
           in
           one
           turn
           of
           the
           Great-wheel
           .
           And
           then
           for
           the
           quotient
           of
           the
           Pinion
           of
           Report
           ,
           say
           in
           numbers
           thus
           ,
           21888.
           9368
           
           :
           :
           12.
           5.
           
           
           The
           quotient
           for
           the
           Pinion
           of
           Report
           is
           somewhat
           more
           than
           5
           ,
           which
           overplus
           may
           be
           neglected
           ,
           as
           you
           see
           by
           the
           Scheme
           of
           the
           whole
           work
           in
           the
           Margin
           .
        
         
           If
           you
           desire
           to
           know
           what
           number
           of
           turns
           ,
           the
           Fusy
           must
           have
           in
           this
           work
           ;
           Say
           by
           the
           last
           quoted
           §
           ▪
           part
           1
           ,
           in
           numbers
           thus
           ,
           21888.
           9368
           
           :
           :
           30.
           13
           almost
           .
           So
           that
           near
           13
           turns
           will
           do
           .
        
         
           If
           you
           would
           correct
           your
           work
           ,
           to
           know
           the
           exact
           Beats
           ,
           &c.
           you
           are
           referred
           to
           directions
           in
           the
           end
           of
           the
           last
           paragraph
           .
        
         
         
           §
           5.
           
           I
           shall
           add
           but
           one
           thing
           more
           ,
           to
           what
           hath
           been
           said
           in
           this
           Chapter
           ,
           and
           that
           is
           ,
           to
           change
           the
           Striking
           part
           of
           this
           old
           Movement
           ,
           into
           a
           30
           hour
           piece
           .
        
         
           A
           Scheme
           of
           the
           old
           
           work
           is
           in
           the
           Margin
           .
        
         
           And
           to
           alter
           it
           ,
           the
           best
           way
           is
           ,
           to
           double
           the
           number
           of
           Striking
           pins
           ,
           making
           the
           8
           ,
           sixteen
           pins
           :
           and
           the
           Hoop
           of
           the
           Detent-wheel
           double
           ,
           that
           the
           Pin-wheel
           may
           strike
           two
           strokes
           ,
           in
           its
           going
           round
           once
           .
        
         
           The
           greatest
           inconvenience
           here
           ,
           will
           be
           to
           bridle
           the
           rapidity
           of
           the
           Strokes
           ;
           because
           a
           quotient
           of
           (
           2
           only
           ,
           added
           to
           the
           old
           work
           ,
           will
           be
           sufficient
           for
           this
           purpose
           :
           which
           being
           an
           inconvenient
           number
           ,
           't
           will
           be
           necessary
           to
           be
           content
           with
           the
           old
           numbers
           ,
           or
           make
           more
           Wheels
           and
           Pinions
           new
           ,
           than
           may
           be
           thought
           worth
           the
           while
           .
        
         
           If
           you
           would
           find
           what
           number
           of
           turns
           ,
           the
           Fusy
           will
           require
           ;
           you
           must
           find
           how
           many
           Strokes
           are
           in
           30
           hours
           ,
           by
           Sect.
           3.
           
           §
           2.
           
           R.
           6.
           before
           .
           These
           are
           195
           ;
           which
           divided
           by
           the
           16
           Pins
           ,
           
           gives
           somewhat
           more
           than
           12
           turns
           of
           the
           Fusy
           .
        
         
           Lastly
           ,
           for
           the
           Pinion
           of
           Report
           ,
           you
           must
           pursue
           the
           directions
           in
           the
           last
           quoted
           place
           ,
           R.
           5.
           
        
         
           The
           work
           thus
           altered
           ,
           
           will
           stand
           as
           in
           the
           Margin
           .
        
      
       
         
           CHAP.
           IV.
           To
           size
           the
           Wheels
           and
           Pinions
           ,
           or
           proportion
           them
           to
           each
           other
           ,
           both
           Arithmetically
           and
           Mechanically
           .
        
         
           §
           1.
           
           FOr
           the
           exact
           and
           easie
           moving
           of
           the
           Wheels
           and
           Pinions
           together
           ,
           it
           is
           necessary
           that
           they
           should
           fit
           each
           other
           ,
           by
           having
           their
           teeth
           and
           leaves
           of
           the
           same
           wideness
           ,
           or
           near
           of
           the
           same
           wideness
           .
           For
           many
           do
           make
           
           the
           Leaves
           of
           the
           Pinion
           narrower
           than
           the
           Teeth
           of
           its
           Wheel
           ,
           by
           reason
           of
           their
           running
           deep
           in
           each
           other
           ;
           which
           is
           as
           if
           the
           Diameters
           of
           the
           Wheel
           and
           Pinion
           were
           less
           .
           But
           this
           I
           leave
           to
           those
           ,
           whose
           practice
           and
           observations
           are
           greater
           than
           mine
           in
           these
           matters
           .
        
         
           §
           2.
           
           To
           make
           the
           Teeth
           of
           a
           Wheel
           and
           Pinion
           alike
           ,
           the
           way
           Arithmetically
           is
           thus
           ,
           First
           you
           must
           find
           the
           Circumference
           of
           your
           Wheel
           and
           Pinion
           ;
           which
           you
           may
           best
           do
           by
           the
           Rule
           of
           Three
           (
           so
           often
           made
           use
           of
           before
           )
           the
           Rule
           is
           thus
           ,
           as
           7
           is
           to
           22
           :
           :
           so
           is
           the
           Diameter
           to
           the
           Circumference
           .
           Or
           more
           exactly
           thus
           ,
           as
           1
           ,
           is
           to
           3
           ,
           1416
           :
           :
           So
           Diam
           .
           to
           Circum
           .
        
         
           Suppose
           you
           have
           a
           Wheel
           of
           2
           inches
           diameter
           ,
           and
           60
           Teeth
           ,
           and
           would
           fit
           to
           it
           a
           Pinion
           of
           6
           Leaves
           .
           First
           7
           22
           :
           :
           2.
           6
           ,
           3.
           
           The
           circumference
           of
           the
           Wheel
           ,
           is
           then
           6
           inches
           ,
           and
           3
           tenths
           of
           an
           inch
           .
           Then
           say
           ,
           as
           the
           Teeth
           of
           the
           Wheel
           ,
           to
           the
           circumference
           
           of
           it
           :
           :
           So
           are
           the
           Leaves
           of
           the
           Pinion
           ,
           to
           the
           circumference
           thereof
           .
           In
           numbers
           thus
           60
           ▪
           6
           ,
           3
           :
           :
           6
           ▪
           ,
           63.
           
           The
           Pinion
           then
           is
           63
           hundredth
           parts
           of
           an
           inch
           round
           .
        
         
         
           Now
           to
           find
           the
           Diameter
           ,
           't
           is
           but
           the
           reverse
           of
           the
           former
           Rule
           ,
           viz.
           As
           22.
           to
           7
           :
           :
           So
           the
           Circumference
           to
           the
           Diameter
           .
           In
           numbers
           thus
           ,
           for
           the
           foregoing
           Pinion
           ,
           22.
           7
           
           :
           :
           ,
           63
           ▪
           2
           ,
           The
           Diameter
           then
           of
           the
           Pinion
           must
           be
           two
           tenths
           of
           an
           inch
           ,
           to
           fit
           the
           aforesaid
           Wheel
           of
           2
           inches
           diameter
           .
        
         
           sect
           3.
           
           But
           because
           this
           way
           may
           be
           difficult
           to
           persons
           unacquainted
           with
           Decimal
           Arithmetick
           ,
           which
           is
           very
           necessary
           here
           ;
           therefore
           I
           shall
           set
           down
           a
           way
           to
           do
           it
           mechanically
           .
           Having
           drawn
           a
           Circle
           ,
           divide
           it
           into
           as
           many
           parts
           ,
           as
           you
           intend
           leaves
           in
           the
           Pinion
           you
           would
           size
           .
           From
           two
           of
           these
           points
           in
           the
           Circle
           ,
           draw
           two
           lines
           to
           the
           Center
           :
           to
           which
           apply
           two
           of
           the
           Teeth
           of
           your
           Wheel
           ,
           guiding
           them
           up
           and
           down
           until
           they
           touch
           at
           the
           same
           width
           on
           these
           Radii
           .
           Mark
           where
           this
           agreement
           is
           ,
           and
           a
           small
           circle
           drawn
           there
           ,
           will
           represent
           the
           circumference
           of
           the
           Pinion
           sought
           after
           .
        
      
       
         
         
           CHAP.
           V.
           Of
           Pendulums
           .
        
         
           sect
           1.
           
           AMong
           all
           known
           Motions
           ,
           none
           measureth
           Time
           so
           regularly
           ,
           as
           that
           of
           a
           Pendulum
           .
           But
           yet
           Watches
           governed
           hereby
           are
           not
           so
           persect
           ,
           but
           that
           they
           are
           subject
           to
           the
           variations
           of
           weather
           ,
           foulness
           ,
           &c.
           
           And
           the
           shorter
           ,
           and
           lesser
           the
           Pendulum
           is
           ,
           so
           much
           the
           more
           subject
           such
           Watches
           are
           to
           these
           annoyances
           .
        
         
           There
           are
           two
           ways
           to
           obviate
           these
           inconveniences
           in
           some
           measure
           .
           One
           way
           is
           ,
           to
           make
           the
           Pendulum
           long
           ,
           the
           Bob
           heavy
           ,
           and
           to
           vibrate
           but
           a
           little
           way
           from
           its
           settlement
           .
           Which
           is
           now
           the
           most
           usual
           way
           in
           England
           .
           The
           other
           is
           the
           contrivance
           of
           the
           ingenious
           Mr.
           Hu●ens
           ,
           which
           is
           ,
           to
           make
           the
           upper
           part
           of
           the
           rod
           ,
           play
           between
           two
           cheek
           parts
           of
           ●
           Cycloid
           .
           Sir
           
             Jonas
             Moor
          
           says
           ,
           that
           af●er
           some
           time
           ,
           and
           charge
           of
           Experiments
           ,
           
           
           he
           believes
           this
           latter
           to
           be
           the
           better
           way
           .
           And
           Mr
           Hugens
           calls
           it
           admirable
           .
        
         
           If
           any
           desire
           to
           know
           how
           to
           make
           those
           Cycloidal
           Cheeks
           ,
           fit
           to
           all
           Pendulums
           ,
           I
           refer
           him
           to
           the
           aforesaid
           Mr.
           
           Zulichem's
           
           Book
           ,
           because
           I
           can't
           shew
           how
           to
           do
           it
           ,
           without
           the
           trouble
           of
           Figures
           ;
           and
           this
           way
           is
           much
           ceased
           ,
           since
           the
           Crown-wheel
           method
           (
           to
           which
           it
           is
           chiefly
           proper
           )
           is
           swallowed
           up
           by
           the
           Royal
           Pendulums
           .
        
         
           sect
           2.
           
           Another
           thing
           to
           be
           remark'd
           in
           Pendulums
           is
           ,
           That
           the
           longer
           the
           Vibration
           is
           ,
           the
           ●lower
           it
           is
           .
           For
           if
           two
           isochrone
           Pendulums
           do
           move
           ,
           one
           the
           quadrant
           of
           a
           circle
           ,
           the
           other
           not
           above
           3
           or
           4
           degrees
           ,
           this
           latter
           shall
           move
           some-what
           quicker
           than
           the
           former
           .
           Which
           is
           the
           true
           reason
           ,
           why
           small
           Crown-wheel
           Pendulums
           go
           faster
           in
           cold
           weather
           ,
           or
           when
           soul
           ,
           than
           at
           other
           times
           .
           Yea
           ,
           in
           the
           best
           Royal
           Pendulum
           ,
           if
           you
           put
           a
           divided
           plate
           behind
           the
           Ball
           ,
           and
           observe
           its
           swings
           ,
           you
           may
           perceive
           the
           Vibrations
           to
           be
           sometimes
           shorter
           ;
           and
           that
           then
           the
           Watch
           doth
           gain
           too
           much
           ▪
           Somewhat
           also
           may
           perhaps
           be
           attributed
           to
           the
           rarity
           or
           density
           of
           the
           air
           ;
           which
           
           I
           have
           not
           yet
           had
           an
           opportunity
           of
           observing
           ,
           by
           comparing
           with
           a
           good
           Baroscope
           ,
           the
           various
           vibrations
           of
           a
           good
           Royal
           Pendulum
           .
           But
           Mr.
           Boyl
           says
           ,
           that
           a
           Pendulum
           moveth
           as
           long
           ,
           and
           as
           
           fast
           in
           a
           thick
           medium
           ,
           as
           a
           thin
           one
           ;
           contrary
           to
           the
           opinion
           of
           some
           Naturalists
           ,
           who
           think
           the
           contrary
           .
           His
           opinion
           is
           grounded
           upon
           the
           experiment
           of
           a
           Pendulum
           vibrating
           in
           his
           air-pump
           ,
           the
           air
           sucked
           out
           ,
           and
           in
           the
           open
           air
           ;
           wherein
           was
           no
           alteration
           .
        
         
           sect
           3.
           
           For
           the
           calculation
           of
           all
           Pendulums
           ,
           't
           is
           necessary
           to
           fix
           upon
           some
           one
           ,
           to
           be
           as
           a
           Standard
           to
           the
           rest
           .
           I
           pitch
           upon
           a
           Pend
           ▪
           to
           vibrate
           Seconds
           each
           stroke
           .
        
         
           Mr.
           Hugens
           lays
           down
           the
           length
           of
           a
           Pend.
           to
           swing
           Seconds
           to
           be
           3
           feet
           ,
           3
           inches
           ,
           and
           2
           tenths
           of
           an
           inch
           (
           according
           to
           Sir
           J.
           Moor's
           reduction
           of
           it
           to
           English
           measure
           .
           
           )
        
         
           
             
             The
             Honourable
             Lord
             Bru●cker
             (
             saith
             Sir
             Jonas
             )
             and
             Mr.
             Rook
             ,
             found
             the
             length
             to
             be
             39
             ,
             25
             inches
             ,
             which
             a
             little
             exceeds
             the
             other
             :
             and
             may
             be
             ,
             was
             justened
             by
             Mr.
             
             Hugens's
             Rule
             for
             the
             Center
             of
             Oscillation
             .
             For
             
             Mounton's
             Pendulum
             ,
             that
             ▪
             
             vibrate
             132
             times
             in
             a
             minute
             ,
             it
             will
             be
             found
             likewise
             8
             ,
             1
             inches
             ,
             agreeing
             to
             39
             ,
             2
             inches
             English
             .
             Therefore
             for
             certain
             39
             ,
             2
             inches
             may
             be
             called
             the
             
               Vniversal
               measure
            
             ,
             and
             relied
             on
             ,
             to
             be
             the
             near
             length
             of
             a
             Pend.
             that
             shall
             swing
             Seconds
             each
             vibration
             .
          
        
         
           But
           forasmuch
           as
           the
           different
           size
           of
           the
           Ball
           ,
           will
           make
           some
           difference
           in
           the
           length
           of
           this
           Standard
           Pend.
           ,
           therefore
           to
           make
           this
           Pend.
           an
           
             Vniversal
             measure
          
           ,
           to
           fit
           all
           Places
           and
           Ages
           ,
           you
           must
           measure
           from
           the
           Point
           of
           Suspension
           ,
           to
           the
           Center
           of
           Oscillation
           .
           Which
           Center
           is
           found
           by
           this
           Rule
           ,
           As
           the
           length
           of
           the
           
           String
           from
           the
           point
           of
           ▪
           Suspension
           to
           the
           center
           of
           a
           round
           Ball
           :
           is
           to
           the
           Semidiameter
           of
           a
           round
           Ball
           :
           is
           to
           the
           Semidiameter
           ,
           to
           a
           fourth
           number
           .
           Add
           two
           fifths
           of
           that
           fourth
           number
           ,
           to
           the
           former
           length
           ,
           and
           you
           have
           the
           center
           of
           Oscillation
           ;
           and
           thereby
           the
           true
           length
           of
           this
           
             Standard
             Pendulum
          
           .
        
         
           If
           it
           be
           desired
           to
           fit
           a
           Ball
           of
           a
           triangular
           ,
           quadrangular
           ,
           or
           any
           other
           form
           to
           this
           Pend
           ,
           the
           center
           of
           Oscillation
           in
           any
           of
           these
           bodies
           ,
           may
           be
           found
           in
           the
           last
           cited
           book
           of
           Mr
           Zulichem
           .
        
         
         
           If
           it
           be
           asked
           ,
           What
           is
           the
           meaning
           of
           the
           Center
           of
           Oscillation
           ?
           The
           most
           intelligible
           answer
           (
           altho
           not
           perfectly
           true
           )
           is
           ,
           That
           it
           is
           that
           point
           of
           the
           Ball
           ,
           at
           which
           if
           you
           imagine
           it
           divided
           into
           two
           parts
           ,
           by
           a
           circle
           ,
           whose
           center
           is
           in
           the
           point
           of
           Suspension
           ,
           the
           lower
           part
           of
           the
           Ball
           shall
           be
           of
           the
           same
           weight
           (
           or
           near
           so
           )
           with
           the
           upper
           .
        
         
           §
           4.
           
           Having
           thus
           fixed
           a
           Standard
           ,
           I
           shall
           next
           shew
           how
           from
           thence
           to
           find
           the
           Vibrations
           ,
           or
           Lengths
           of
           all
           other
           Pendulums
           .
           Which
           is
           done
           by
           this
           Rule
           ,
           
           
             The
             squares
             of
             the
             Vibrations
             ,
             bear
             the
             same
             Proportion
             to
             each
             other
             ,
             as
             their
             Length●
             do
             .
          
           And
           so
           contrary
           wise
           .
           Wherefore
           to
           find
           the
           length
           of
           a
           Pend
           ▪
           say
           ▪
           As
           the
           Square
           of
           the
           Vibrations
           given
           :
           To
           the
           Square
           of
           60
           (
           the
           Standard
           )
           :
           :
           So
           is
           the
           length
           of
           the
           Standard
           
             (
             viz.
          
           39
           ,
           2
           )
           To
           the
           length
           of
           the
           Pend.
           sought
           .
        
         
           If
           by
           the
           length
           ,
           you
           would
           find
           the
           Vibrations
           ,
           't
           is
           the
           reverse
           of
           the
           last
           Rule
           ,
           viz.
           As
           the
           length
           proposed
           :
           To
           the
           Standard
           (
           39
           ,
           2
           )
           :
           :
           So
           is
           the
           Square
           of
           60
           (
           the
           vibrations
           of
           the
           Standard
           )
           :
           To
           the
           Vibrations
           sought
           .
        
         
         
           Suppose
           for
           example
           ,
           you
           would
           know
           what
           length
           a
           Pend.
           is
           ,
           that
           vibrates
           153
           strokes
           in
           a
           minute
           .
           The
           Square
           of
           153
           
             (
             i.
             e.
          
           153
           times
           153
           )
           is
           23409.
           
           Say
           ,
           23409.
           3600
           
           :
           :
           39
           ,
           2.
           6.
           
           A
           Pend.
           then
           that
           vibrates
           153
           in
           a
           minute
           ,
           is
           about
           6
           inches
           long
           .
        
         
           On
           the
           other
           hand
           ,
           if
           you
           would
           know
           how
           many
           strokes
           a
           Pend.
           of
           6
           inches
           hath
           in
           a
           minute
           ;
           Say
           ,
           6.
           39
           ,
           2
           :
           :
           3600.
           23520.
           
           The
           square
           root
           whereof
           is
           153
           ,
           and
           somewhat
           more
           .
        
         
           Note
           ,
           Because
           141120
           is
           always
           the
           Product
           of
           the
           two
           middle
           terms
           multiplied
           together
           ,
           therefore
           you
           need
           only
           to
           divide
           this
           number
           by
           the
           Square
           of
           the
           Vibrations
           ,
           it
           gives
           the
           length
           sought
           :
           by
           the
           length
           ,
           it
           gives
           the
           square
           of
           the
           Vibrations
           .
        
         
           If
           you
           operate
           by
           the
           Logarithms
           ,
           you
           will
           much
           contract
           your
           labour
           .
           For
           if
           you
           seek
           the
           length
           ,
           't
           is
           but
           Substracting
           the
           Logarithm
           of
           the
           Square
           of
           the
           Vibrations
           ,
           out
           of
           the
           Logarithm
           of
           141120
           ,
           which
           is
           5.
           149588
           ,
           and
           the
           Remainder
           is
           the
           Logarithm
           of
           the
           length
           sought
           .
        
         
           If
           you
           seek
           the
           Vibrations
           ,
           it
           is
           but
           Substracting
           out
           of
           the
           aforesaid
           Logarithm
           
           5.
           149588
           ,
           the
           Logarithm
           of
           the
           length
           given
           ,
           and
           half
           the
           Residue
           is
           the
           Logarithm
           of
           the
           Vibrations
           required
           .
           The
           following
           examples
           will
           illustrate
           each
           particular
           .
        
         
           
             
               To
               find
               the
               Length
               .
            
             
               
                  
              
               
                 Logarithms
                 .
              
            
             
               
                 141120
              
               
                 5.
                 149588
              
            
             
               
                 153
                 Squared
                 is
                 23409
              
               
                 4.
                 369382
              
            
             
               
                 Length
                 is
                 more
                 than
                 6.
                 
              
               
                 0.
                 780206
              
            
          
           
             
               To
               find
               the
               Vibrations
               .
            
             
               
                  
              
               
                 Logarithms
                 .
              
            
             
               
                 141120
              
               
                 5.
                 149588
              
            
             
               
                 6
                 inches
                 long
              
               
                 0.
                 778151
              
            
             
               
                 Square
                 of
                 the
                 Vibr.
                 
              
               
                 4.
                 371437
              
            
             
               
                 Square-toor
                 ,
                 or
                 numb
                 .
                 of
                 Vibr.
                 is
                 153
                 ,
                 and
                 somewhat
                 more
                 .
              
               
                 2.
                 185718
              
            
          
        
         
           According
           to
           the
           foregoing
           Directions
           ▪
           I
           have
           calculated
           the
           following
           Table
           ▪
           to
           Pendulums
           of
           various
           lengths
           :
           and
           have
           therein
           shewed
           the
           Vibrations
           in
           a
           minute
           ,
           
           and
           an
           hour
           ,
           from
           1
           to
           100
           inches
           .
           If
           any
           desire
           a
           more
           minute
           account
           ,
           I
           refer
           
           him
           to
           Mr
           
           Smith's
           Tables
           in
           his
           late
           Book
           .
           The
           reason
           why
           his
           calculation
           and
           mine
           differ
           ,
           is
           because
           he
           measureth
           the
           length
           of
           the
           Pend.
           from
           the
           point
           of
           Suspension
           ,
           to
           the
           lower
           part
           of
           the
           Bob
           ;
           and
           I
           only
           to
           the
           center
           of
           the
           Bob.
           His
           Standards
           are
           6½
           inches
           ,
           and
           41
           inches
           ;
           and
           mine
           is
           39
           ,
           2
           ,
           for
           the
           reasons
           aforegoing
           .
        
         
           
             
               A
               Table
               of
               Swings
               in
               a
               Minute
               ,
               and
               in
               an
               hour
               ,
               to
               Pendulums
               of
               several
               lengths
               .
            
             
               
                 Pend.
                 length
                 in
                 inches
              
               
                 Vibrat
                 .
                 in
                 a
                 Minute
                 .
              
               
                 Vibrat
                 .
                 in
                 an
                 Hour
                 .
              
            
             
               
                 1
              
               
                 375,7
              
               
                 22542
              
            
             
               
                 2
              
               
                 265,6
              
               
                 15936
              
            
             
               
                 3
              
               
                 216,9
              
               
                 13014
              
            
             
               
                 4
              
               
                 187,8
              
               
                 11268
              
            
             
               
                 5
              
               
                 168,0
              
               
                 10080
              
            
             
               
                 6
              
               
                 153,3
              
               
                 9204
              
            
             
               
                 7
              
               
                 142,0
              
               
                 8520
              
            
             
               
                 8
              
               
                 132,8
              
               
                 7968
              
            
             
               
                 9
              
               
                 125,2
              
               
                 7512
              
            
             
               
                 10
              
               
                 118,8
              
               
                 7128
              
            
             
               
                 20
              
               
                 84,0
              
               
                 5040
              
            
             
               
                 30
              
               
                 68,6
              
               
                 4116
              
            
             
               
                 39,2
              
               
                 60.0
              
               
                 3600
              
            
             
               
                 40
              
               
                 59,4
              
               
                 3264
              
            
             
               
                 50
              
               
                 53,1
              
               
                 3186
              
            
             
               
                 60
              
               
                 48,5
              
               
                 2910
              
            
             
               
                 70
              
               
                 44,9
              
               
                 2694
              
            
             
               
                 80
              
               
                 42,0
              
               
                 2520
              
            
             
               
                 90
              
               
                 39,6
              
               
                 2376
              
            
             
               
                 100
              
               
                 37,5
              
               
                 2250
              
            
          
        
         
         
           The
           use
           of
           this
           Table
           is
           manifest
           ,
           and
           needs
           no
           explication
           .
           As
           to
           the
           Decimals
           in
           the
           column
           of
           Minute-Swings
           ,
           I
           have
           ●dded
           them
           for
           the
           sake
           of
           calculating
           the
           column
           of
           Hour-Swings
           ;
           which
           would
           have
           been
           judged
           false
           without
           them
           ,
           and
           would
           not
           have
           been
           exactly
           true
           without
           them
           .
        
         
           §
           5
           ▪
           I
           have
           but
           one
           thing
           more
           to
           add
           to
           this
           Chap.
           of
           Pendulums
           ,
           and
           that
           is
           ,
           
             ●o
             Correct
             their
             Motion
          
           .
        
         
           The
           usual
           way
           is
           ,
           to
           screw
           up
           ,
           or
           let
           down
           the
           Ball.
           In
           doing
           of
           which
           ,
           a
           small
           alteration
           will
           make
           a
           considerable
           ●ariation
           of
           Time
           :
           as
           you
           will
           find
           by
           calculation
           ,
           according
           to
           the
           last
           paragraph
           .
           To
           prevent
           the
           inconvenience
           of
           ●crewing
           the
           Ball
           too
           high
           ,
           or
           low
           ,
           Mr
           Smith
           hath
           contriv'd
           a
           very
           pretty
           Table
           
           ●or
           dividing
           the
           Nut
           of
           a
           Pendulum
           Screw
           ,
           ●o
           as
           to
           alter
           your
           Clock
           but
           a
           Second
           in
           ●day
           .
           But
           by
           reason
           no
           Screw
           and
           Nut
           can
           be
           so
           made
           ,
           as
           to
           be
           most
           exactly
           strait
           ●nd
           true
           ,
           therefore
           it
           may
           happen
           ,
           that
           instead
           of
           altering
           your
           Watch
           to
           your
           mind
           ,
           you
           may
           do
           quite
           contrary
           ;
           as
           ●nstead
           of
           letting
           the
           Ball
           down
           ,
           you
           may
           raise
           it
           higher
           ,
           by
           the
           false
           running
           
           of
           the
           Nut
           upon
           the
           Screw
           .
        
         
           Considering
           this
           irremediable
           inconvenience
           ,
           I
           am
           of
           opinion
           ,
           that
           Mr
           
           Hugens's
           
           way
           would
           do
           very
           well
           ,
           added
           to
           this
           .
           His
           way
           is
           ,
           To
           have
           a
           small
           Weight
           ,
           or
           Bob
           ,
           to
           slide
           up
           and
           down
           the
           Pend.
           rod
           ,
           above
           the
           Ball
           (
           which
           is
           immoveable
           .
           )
           But
           I
           would
           rather
           advise
           ,
           that
           the
           Ball
           be
           made
           to
           screw
           up
           and
           down
           ,
           to
           bring
           the
           Pend.
           pretty
           neer
           its
           gauge
           :
           and
           that
           this
           little
           Bob
           should
           serve
           only
           for
           more
           nice
           corrections
           ;
           as
           the
           alteration
           of
           a
           Second
           ,
           or
           &c.
           
           Which
           it
           will
           do
           ,
           better
           than
           the
           Great
           Ball.
           For
           a
           whole
           turn
           of
           this
           little
           Bob
           ,
           will
           not
           affect
           the
           motion
           of
           the
           Pend.
           near
           so
           much
           as
           a
           small
           alteration
           of
           the
           Great
           Ball.
           
        
         
           The
           Directions
           Mr
           Hugens
           gives
           ,
           about
           this
           little
           Corrector
           ,
           is
           ,
           That
           it
           should
           be
           equal
           to
           the
           weight
           of
           the
           Wire
           ,
           or
           Rod
           of
           the
           Pend.
           ,
           or
           about
           a
           50th
           part
           of
           the
           weight
           of
           the
           Great
           Ball
           ,
           which
           he
           appoints
           to
           be
           three
           pounds
           .
        
         
           Perhaps
           this
           Bob
           may
           do
           its
           office
           ,
           if
           it
           be
           made
           to
           screw
           only
           up
           and
           down
           the
           lower
           part
           of
           the
           Rod
           ,
           below
           the
           Ball.
           If
           not
           ,
           you
           must
           make
           it
           slide
           above
           the
           Ball
           ,
           or
           be
           screwed
           up
           and
           down
           there
           .
        
         
         
           Seeing
           this
           little
           Bob
           is
           not
           the
           only
           Corrector
           (
           as
           in
           Mr
           
           Zulichem's
           way
           )
           therefore
           it
           is
           not
           necessary
           to
           insert
           here
           ,
           that
           ingenious
           person's
           Table
           ,
           shewing
           what
           alterations
           of
           Time
           will
           be
           made
           by
           sliding
           the
           Bob
           up
           and
           down
           the
           rod.
           Only
           thus
           much
           may
           be
           observed
           in
           that
           Table
           of
           his
           ,
           viz.
           That
           a
           small
           alteration
           of
           the
           Corrector
           towards
           the
           lower
           end
           of
           the
           Pend
           ▪
           ,
           doth
           make
           as
           great
           an
           alteration
           of
           Time
           ,
           as
           a
           greater
           raising
           or
           falling
           of
           it
           ,
           doth
           make
           higher
           .
           Thus
           the
           little
           Bob
           raised
           7
           divisions
           of
           the
           Rod
           ,
           from
           the
           Center
           of
           Oscillation
           ,
           will
           alter
           the
           Watch
           15
           seconds
           ;
           raised
           15
           ,
           2
           't
           will
           alter
           it
           30″
           .
           But
           whereas
           ,
           if
           it
           be
           raised
           to
           154
           ▪
           3
           parts
           of
           the
           Rod
           ,
           it
           will
           make
           the
           Watch
           go
           faster
           3
           minutes
           ,
           15
           seconds
           ,
           the
           Watch
           shall
           be
           but
           3′
           .
           30″
           faster
           ,
           if
           the
           Bob
           be
           raised
           to
           192
           ▪
           6.
           
           So
           that
           here
           you
           have
           but
           15″
           variation
           ,
           by
           raising
           the
           Bob
           above
           38
           parts
           ;
           whereas
           lower
           ,
           you
           had
           the
           same
           variation
           ,
           when
           raised
           not
           above
           7
           or
           8
           parts
           .
        
         
           From
           what
           hath
           been
           said
           ,
           it
           appears
           ,
           that
           about
           half
           a
           turn
           of
           this
           little
           justening
           Bob
           ,
           will
           at
           no
           time
           alter
           the
           Watch
           ,
           above
           a
           second
           in
           24
           hours
           ▪
           and
           that
           
           above
           a
           whole
           turn
           ,
           will
           not
           alter
           it
           so
           much
           ,
           higher
           on
           the
           Rod
           ;
           supposing
           that
           the
           Bob
           at
           every
           turn
           ascended
           or
           descended
           a
           whole
           degree
           of
           the
           Rod
           ;
           which
           perhaps
           it
           will
           not
           do
           in
           20
           turns
           :
           and
           consequently
           ,
           it
           will
           require
           many
           turns
           ,
           to
           alter
           the
           Watch
           but
           one
           second
           .
        
      
       
         
           CHAP.
           VI.
           The
           Antiquity
           ,
           and
           general
           History
           of
           Watch
           ,
           or
           Clock-work
           .
        
         
           §
           1.
           
           IT
           is
           probable
           ,
           that
           in
           all
           Ages
           ,
           some
           Instruments
           or
           other
           have
           been
           used
           ,
           for
           the
           Measuring
           of
           Time.
           But
           the
           earliest
           we
           read
           of
           ,
           is
           the
           Dial
           of
           Ahaz
           .
           Concerning
           which
           ,
           little
           of
           certainty
           
           can
           be
           said
           .
           The
           Hebrew
           word
           Ma●aloth
           doth
           properly
           signifie
           Degrees
           ,
           Steps
           ,
           or
           Stairs
           ,
           by
           which
           we
           ascend
           to
           any
           place
           .
           And
           so
           this
           word
           Ma●aloth
           is
           rendered
           Ezek.
           40.
           26.
           
           And
           accordingly
           the
           LXXII
           translate
           the
           Ma●aloth
           of
           Ahaz
           ,
           by
           the
           words
           
             〈◊〉
             〈◊〉
             〈◊〉
             〈◊〉
             〈◊〉
          
           ,
           and
           '
           
             〈◊〉
             〈◊〉
             〈◊〉
             〈◊〉
             〈◊〉
          
           ,
           
           
             i.
             e.
             Steps
          
           or
           Ascents
           .
           The
           like
           doth
           the
           
             Syriack
             ,
             Arabick
          
           ,
           and
           other
           Versions
           .
        
         
           Some
           pretend
           to
           give
           a
           description
           of
           this
           Dial
           of
           Ahaz
           :
           but
           it
           being
           meer
           guessing
           ,
           and
           little
           to
           my
           purpose
           ,
           I
           shall
           not
           trouble
           the
           Reader
           with
           the
           various
           opinions
           about
           it
           .
        
         
           Among
           the
           Greeks
           and
           Romans
           ,
           there
           were
           two
           ways
           chiefly
           used
           to
           measure
           their
           Hours
           .
           One
           was
           by
           Clepsydrae
           ,
           or
           Hour
           ▪
           glasses
           .
           The
           other
           by
           the
           Solaria
           ,
           or
           Sun-dials
           .
           The
           
             〈◊〉
             〈◊〉
             〈◊〉
             〈◊〉
             〈◊〉
          
           ,
           says
           Suidas
           and
           Phavorinus
           ,
           was
           
             〈◊〉
             〈◊〉
             〈◊〉
             〈◊〉
             〈◊〉
          
           
           
             〈◊〉
             〈◊〉
             〈◊〉
             〈◊〉
             〈◊〉
          
           
             i.
             e.
             An
             Astronomical
             Instrument
             ,
             by
             which
             the
             Hours
             were
             measured
             .
          
           Also
           ,
           
             That
             it
             was
             a
             Vessel
             ,
             having
             a
             little
             hole
             in
             the
             bottom
             ,
             which
             was
             set
             in
             the
             Courts
             of
             Judicature
             ,
             full
             of
             water
             ;
             by
             which
             the
             Lawyers
             pleaded
             .
          
           This
           was
           ,
           says
           Phavorinus
           ,
           to
           prevent
           babbling
           ,
           that
           such
           as
           spake
           ,
           ought
           to
           brief
           in
           their
           Speeches
           .
        
         
           As
           to
           the
           Invention
           of
           those
           Waterwatches
           (
           which
           were
           ,
           no
           doubt
           ,
           of
           more
           common
           use
           ,
           than
           only
           in
           the
           Law-Courts
           )
           
           the
           Invention
           ,
           I
           say
           ,
           of
           them
           ,
           is
           attributed
           ,
           by
           Censorinus
           ,
           to
           
             P.
             Cornelius
             Nasica
          
           ,
           the
           
             Censor
             (
             Scipio
             Nasica
             ,
             Pliny
          
           calls
           him
           .
           )
        
         
         
           The
           other
           way
           of
           measuring
           the
           Hours
           ,
           
           with
           Sun-dials
           ,
           seems
           ,
           from
           Pliny
           and
           Censorinus
           ,
           to
           have
           been
           an
           earlier
           invention
           
           than
           the
           last
           .
           Pliny
           says
           ,
           that
           
             Anaximenes
             Milesius
          
           ,
           the
           Scholar
           of
           Anaximander
           ,
           invented
           Dialling
           ,
           and
           was
           the
           first
           that
           shewed
           a
           Sun-dial
           at
           
             Lacedoemon
             .
             Vitruvius
          
           
           calls
           him
           
             Milesius
             Anaximander
          
           .
           This
           Anaximander
           or
           Anaximenes
           was
           cotemporary
           with
           Pythagoras
           ,
           says
           Laertius
           ;
           and
           flourished
           about
           the
           time
           of
           the
           Prophet
           Daniel
           .
        
         
           But
           enough
           of
           these
           ancient
           Timeengines
           ,
           which
           are
           not
           very
           much
           to
           my
           purpose
           ,
           being
           not
           pieces
           of
           Watch-work
           .
        
         
           §
           I
           shall
           in
           the
           next
           place
           take
           notice
           of
           a
           few
           Horological
           Machines
           ,
           that
           I
           have
           met
           with
           ;
           which
           ,
           whether
           pieces
           of
           Clock-work
           ,
           or
           not
           ,
           I
           leave
           to
           the
           Readers
           judgment
           .
        
         
           
           The
           first
           is
           that
           of
           Dionysius
           ,
           which
           Plutarch
           commends
           for
           a
           very
           magnificent
           ,
           and
           illustrious
           Piece
           .
           But
           this
           might
           be
           only
           a
           well
           delineated
           Sun-dial
           .
        
         
           Another
           Piece
           ,
           is
           that
           of
           Sapor
           King
           of
           Persia
           .
           Whether
           that
           Sapor
           ,
           who
           was
           
           cotemporary
           with
           
             Constantine
             the
             Great
          
           ,
           I
           
           ●ow
           not
           .
           ●ardan
           saith
           it
           was
           made
           of
           
           Glass
           ;
           that
           the
           King
           could
           sit
           in
           the
           middle
           of
           it
           ,
           and
           see
           its
           Stars
           rise
           and
           set
           ▪
           But
           not
           finding
           whether
           this
           Sphere
           was
           moved
           by
           Clock-work
           ,
           or
           whether
           it
           ●ad
           any
           regular
           motion
           ,
           I
           shall
           say
           no
           ●ore
           concerning
           it
           .
        
         
           The
           last
           Machine
           I
           shall
           mention
           in
           ●his
           Paragraph
           ,
           is
           one
           I
           find
           des●ribed
           by
           
           ●itruvius
           .
           Which
           to
           me
           seems
           to
           be
           a
           piece
           of
           Watch-work
           ,
           moved
           by
           an
           equal
           ●nflux
           of
           Water
           .
        
         
           If
           the
           Reader
           will
           consult
           the
           
           French
           ●dition
           of
           Vitruvius
           ,
           he
           will
           find
           there
           a
           ●ir
           Cut
           of
           it
           .
        
         
           
             Among
             divers
             seats
             which
             this
             Ma●hine
             performed
             (
             as
             sounding
             Trumpets
             ,
             ●hrowing
             Stones
             ,
             &c.
             )
             one
             use
             of
             it
             was
             ,
             ●o
             shew
             the
             Hours
             (
             which
             were
             unequal
             ●n
             that
             age
             )
             through
             every
             month
             of
             the
             ●ear
             .
             The
             words
             of
          
           Vitruvius
           are
           ,
           Aequa●iter
           influens
           aqua
           sublevat
           Scaphum
           inversum
           (
           quod
           ●b
           artificibus
           Phellos
           sive
           Tympanam
           ●icitur
           )
           in
           quo
           collocata
           regula
           ▪
           versatilia
           ●●mpana
           denticuli●
           a
           ▪
           qualibus
           s●nt
           perfecta
           .
           ●ui
           denticuli
           alius
           alium
           impelientes
           ,
           ver●●●iones
           modicas
           faciant
           ,
           ac
           motiones
           .
           Item
           〈◊〉
           Reg●loe
           ,
           aliaque
           Tympana
           ad
           eundem
           
           modum
           dentata
           ,
           quoe
           una
           motione
           coacta
           ,
           versando
           faciunt
           effectus
           ,
           varietatesque
           motionum
           :
           in
           quibus
           moventur
           Sigilla
           ,
           vertuntur
           Metoe
           ,
           calculi
           aut
           Tona
           projiciuntur
           ,
           Buccinoe
           canunt
           ,
           &c.
           
           In
           his
           etiam
           ,
           aut
           in
           colu●na
           ,
           an
           t
           parastatica
           Horoe
           describuntur
           ;
           quas
           Sigillum
           egrediens
           ab
           imo
           virguloe
           ,
           significat
           ,
           in
           diem
           totum
           :
           quarum
           brevitates
           ant
           crescentias
           ,
           cuneorum
           adjectus
           aut
           exemptus
           ,
           in
           singulis
           diebus
           &
           mensibus
           ,
           perficere
           cogit
           .
        
         
           The
           Inventer
           of
           this
           famous
           Machine
           ,
           Vitruvius
           says
           ,
           was
           one
           Ctesibius
           ,
           a
           Barbers
           Son
           of
           Alexandria
           .
           Which
           Ctesibius
           flourished
           under
           
             Ptolomy
             Euergetes
          
           ,
           says
           
           Athenoeus
           ,
           l.
           4.
           
           And
           if
           so
           ,
           he
           lived
           about
           240
           years
           before
           our
           Saviours
           days
           ;
           and
           might
           be
           cotemporary
           with
           Archimedes
           .
        
         
           §
           3.
           
           Thus
           having
           given
           a
           small
           account
           of
           the
           ancient
           ways
           of
           Measuring
           Time
           ,
           it
           is
           time
           to
           come
           closer
           to
           our
           business
           ,
           and
           say
           something
           more
           particularly
           of
           Clock-work
           .
        
         
           Which
           is
           thought
           to
           be
           a
           much
           younger
           invention
           ,
           than
           the
           fore-mentioned
           Pieces
           ;
           and
           to
           have
           had
           its
           beginning
           in
           Germany
           ,
           within
           less
           than
           these
           200
           years
           .
        
         
           It
           is
           very
           probable
           ,
           that
           our
           Ballance
           
           Clocks
           ,
           and
           some
           other
           Automata
           ,
           might
           have
           their
           beginning
           there
           ;
           or
           that
           Clock-work
           (
           which
           had
           long
           been
           buried
           in
           oblivion
           )
           might
           be
           revived
           there
           .
           But
           that
           Clock
           work
           was
           the
           Invention
           of
           that
           age
           purely
           ,
           I
           utterly
           deny
           ;
           having
           (
           besides
           what
           goes
           before
           )
           two
           instances
           to
           the
           contrary
           ,
           of
           much
           earlier
           date
           .
        
         
           §
           4.
           
           The
           first
           example
           is
           the
           Sphere
           of
           Archimedes
           ;
           who
           lived
           about
           200
           years
           before
           our
           Saviours
           days
           .
           There
           is
           no
           mention
           of
           this
           Sphere
           in
           Archimedes
           his
           extant
           works
           :
           but
           we
           have
           an
           account
           of
           it
           in
           others
           .
           Cicero
           speaks
           of
           it
           more
           than
           once
           .
           In
           his
           2d
           Book
           
             De
             Nature
             Deorum
          
           ,
           are
           these
           words
           ;
           
             
               Archimedem
               arbitrantur
               plus
               valuisse
               in
               imitandis
               Sphoeroe
               conversionibus
               ,
               quam
               Naturam
               in
               efficiendis
               ,
            
             &c.
             
          
           And
           in
           his
           Tusculane
           Questions
           ,
           
           the
           Collocutor
           ,
           proving
           the
           Soul
           to
           be
           of
           a
           Divine
           Nature
           ,
           argues
           from
           this
           Contrivance
           of
           Archimedes
           ,
           and
           says
           ,
           
             Nam
             cum
             Archimedes
             Lunoe
             ,
             Solis
             ,
             quinque
             Errantium
             motus
             in
             Sphoeram
             illigavit
             ,
             effecit
             ,
          
           &c.
           
           The
           Sense
           is
           ,
           That
           Archimedes
           contrived
           a
           Sphere
           ,
           which
           shewed
           the
           motion
           of
           the
           Moon
           ,
           Sun
           ,
           and
           five
           Planets
           .
        
         
         
           
           But
           the
           most
           accurate
           description
           is
           that
           of
           Claudian
           ,
           in
           these
           words
           .
        
         
           
             Jupiter
             in
             parvo
             cum
             cerneret
             oethera
             vitro
             ,
          
           
             Risit
             ,
             &
             ad
             Superos
             talia
             dicta
             dedit
             :
          
           
             Huccine
             mortalis
             progressa
             potentia
             curoe
             ?
          
           
             Jam
             meus
             in
             fragili
             luditur
             or
             be
             labor
             .
          
           
             Jura
             poli
             ,
             rerumque
             fidem
             ,
             legesque
             Deorum
             ,
          
           
             Ecce
             Syracusius
             transtulit
             arte
             Senex
             .
          
           
             Inclusus
             variis
             famulatur
             Spiritus
             astris
             ,
          
           
             Et
             vivum
             certis
             motibus
             urget
             opus
             .
          
           
             Percurrit
             proprium
             mentitus
             Signifer
             annum
             .
          
           
             Et
             simulata
             novo
             Cynthia
             mense
             redit
             .
          
           
             Jamque
             suum
             volvens
             audax
             industria
             mundum
          
           
             Gaudet
             ,
             &
             humana
             Sidera
             mente
             regit
             .
          
           
             Quid
             falso
             insontem
             tonitru
             Salmonea
             miror
             ?
          
           
             Aemula
             Naturoe
             parva
             reperta
             manus
             .
          
        
         
           
             In
             English
             thus
             :
          
           
             When
             Jove
             espy'd
             in
             Glass
             his
             Heavens
             made
             ,
          
           
             He
             smil'd
             ,
             and
             to
             the
             other
             Gods
             thus
             said
             :
          
           
             Strange
             feats
             !
             when
             human
             art
             so
             far
             proceeds
             ,
          
           
             To
             ape
             in
             brittle
             Orbs
             my
             greatest
             deeds
             .
          
           
             The
             heav'nly
             motions
             ,
             Natures
             constant
             course
             ,
          
           
             Lo
             !
             here
             old
             Archimede
             to
             art
             transfers
             .
          
           
             Th'
             inclosed
             Spirit
             here
             each
             Star
             doth
             drive
             ;
          
           
             And
             to
             the
             living
             work
             sure
             motions
             give
             .
          
           
           
             The
             Sun
             in
             counterfeit
             his
             year
             doth
             run
             ,
          
           
             And
             Cynthia
             too
             her
             monthly
             circle
             turn
             .
          
           
             Since
             now
             bold
             man
             has
             Worlds
             of
             's
             own
             descry'd
          
           
             He
             joys
             ,
             and
             th'
             Stars
             by
             human
             art
             can
             guide
             .
          
           
             Why
             should
             we
             so
             admire
             proud
             Salmons
             cheats
          
           
             When
             one
             poor
             hand
             Natures
             chief
             work
             repeats
             .
          
        
         
           From
           this
           description
           it
           appeareth
           ,
           that
           in
           this
           Sphere
           ,
           the
           Sun
           ,
           Moon
           ,
           and
           other
           heavenly
           bodies
           ,
           had
           their
           proper
           motion
           :
           and
           that
           this
           motion
           was
           effected
           by
           some
           enclosed
           Spirit
           .
           What
           this
           
             enclosed
             Spirit
          
           was
           ,
           I
           cannot
           tell
           ,
           but
           suppose
           it
           to
           be
           Springs
           ,
           Wheels
           or
           Pullies
           ,
           or
           some
           such
           means
           of
           Clock-work
           :
           Which
           be●ng
           hidden
           from
           vulgar
           eyes
           ,
           might
           be
           ●●ken
           for
           some
           Angel
           ,
           Spirit
           ,
           or
           Divine
           power
           ;
           unless
           by
           Spirit
           here
           ,
           you
           un●erstand
           some
           aerious
           ,
           subtiliz'd
           liquor
           ,
           〈◊〉
           vapours
           .
           But
           how
           this
           ,
           or
           indeed
           any
           ●hing
           ,
           but
           Clock-work
           ,
           could
           give
           such
           ●ue
           ,
           and
           regular
           motions
           ,
           I
           am
           not
           able
           ●o
           guess
           .
        
         
           §
           5.
           
           The
           next
           instance
           I
           have
           met
           with
           〈◊〉
           ancient
           Clock-work
           ,
           is
           that
           famous
           ●ne
           in
           Cicero
           ,
           which
           ,
           among
           other
           irre●agable
           
           arguments
           ▪
           is
           brought
           in
           to
           ●ove
           ,
           
             That
             there
             is
             some
             intelligent
             ,
             
             divine
             ,
             and
             wise
             Being
             ,
             that
             inhabiteth
             ,
             ruleth
             in
             ,
             and
             is
             as
             an
             Architect
             of
             so
             great
             a
             work
             ,
             as
             the
             World
             is
             ,
          
           as
           the
           Collocutor
           expresseth
           himself
           ▪
           His
           words
           (
           so
           far
           as
           they
           relate
           to
           my
           present
           purpose
           )
           are
           these
           :
           
             Cum
             Solarium
             vel
             descriptum
             ,
             aut
             ex
             Aqua
             contemplere
             ,
             intelligere
             declarari
             horas
             arte
             ,
             non
             casu
             ,
             &c.
             
          
           And
           a
           little
           after
           ,
           
             Quod
             si
             in
             Scythiam
             ,
             aut
             in
             Britanniam
             ,
             Sphaeram
             aliquis
             tulerit
             hanc
             ,
             quam
             nuper
             familiaris
             noster
             effecit
             Posidonius
             ,
             cujus
             singulae
             Conversiones
             idem
             efficiunt
             in
             Sole
             ,
             &
             in
             Luna
             ,
             &
             in
             quinque
             Stellis
             errantibus
             ,
             quod
             efficitur
             in
             coelo
             singulis
             diebus
             ,
             &
             noctibus
             ;
             quis
             in
             illa
             barbarie
             dubitet
             ,
             quin
             ea
             Sphaera
             sit
             perfecta
             ratione
             ?
          
           The
           sum
           of
           the
           Authors
           meaning
           is
           ,
           
             That
             there
             were
             Sun-dials
             described
             ,
             or
             drawn
          
           [
           with
           Lines
           ,
           after
           the
           manner
           as
           our
           Sun-dials
           are
           :
           ]
           
             and
             some
             made
             with
             Water
          
           (
           which
           were
           the
           Clepsydrae
           ,
           or
           Hour-glasses
           ,
           before-mentioned
           .
           )
           
             That
             Posidonius
             had
             lately
             contrived
             a
             Sphere
             ,
             whose
             Motions
             were
             the
             same
             in
             the
             Sun
             ,
             Moon
             ,
             and
             5
             Planets
             ,
             as
             were
             performed
             in
             the
             heavens
             each
             day
             and
             night
             .
          
        
         
         
           The
           age
           wherein
           this
           Sphere
           was
           in●ented
           ,
           was
           
           Cicero's
           time
           ,
           which
           was
           a●out
           80
           years
           before
           our
           Saviours
           birth
           .
        
         
           And
           that
           it
           was
           a
           piece
           of
           Clock-work
           ,
           ●s
           not
           (
           I
           think
           )
           to
           be
           doubted
           ,
           if
           it
           be
           ●onsidered
           ,
           that
           it
           kept
           time
           with
           those
           ●elestial
           bodies
           ,
           imitating
           both
           their
           an●ual
           ,
           and
           diurnal
           motions
           ,
           as
           from
           the
           ●escription
           we
           may
           gather
           it
           did
           .
        
         
           It
           may
           be
           questioned
           ,
           whether
           those
           Machines
           were
           common
           or
           not
           :
           I
           believe
           ●hey
           were
           rarities
           then
           ,
           as
           well
           as
           Mr
           
           Wat●n's
           and
           others
           are
           accounted
           now
           .
           But
           ●nethinks
           it
           is
           hard
           to
           imagine
           ,
           that
           so
           ●seful
           an
           Invention
           should
           not
           be
           reduced
           ●to
           common
           use
           ;
           it
           being
           natural
           ,
           and
           ●sie
           to
           apply
           it
           to
           the
           measuring
           of
           ●ours
           (
           tho
           unequal
           )
           especially
           in
           two
           ●●ch
           Ages
           ,
           as
           those
           of
           Archimedes
           and
           ●ully
           were
           ,
           in
           which
           the
           Liberal
           Arts
           so
           greatly
           flourished
           .
        
         
           §
           6.
           
           After
           the
           times
           last
           mentioned
           ,
           I
           ●nd
           little
           worth
           remark
           ,
           till
           the
           last
           Age
           ;
           〈◊〉
           which
           Clock-work
           was
           revived
           ,
           or
           ●holly
           invented
           anew
           in
           
             Germany
             ▪
          
           as
           is
           ●enerally
           thought
           ,
           because
           the
           ancient
           ●ieces
           are
           German
           work
           .
           But
           who
           was
           ●e
           Inventer
           ,
           or
           in
           what
           time
           ,
           I
           cannot
           
           
           discover
           .
           Some
           think
           
             Sever.
             Boethius
          
           invented
           it
           about
           the
           year
           510.
           
           Perhaps
           it
           was
           in
           Regiomontanus
           his
           time
           (
           if
           not
           so
           early
           as
           Boethius
           )
           which
           was
           above
           200
           years
           ago
           .
           It
           is
           very
           manifest
           ,
           it
           was
           before
           
           Cardan's
           time
           ,
           because
           he
           speaketh
           of
           it
           ,
           as
           a
           thing
           common
           then
           .
           And
           he
           lived
           about
           150
           years
           since
           .
        
         
           §
           7.
           
           As
           to
           those
           curious
           contrivances
           in
           Clock-work
           ,
           which
           perform
           strange
           ,
           surprizing
           feats
           ,
           I
           shall
           say
           little
           .
           Dr.
           Heylin
           
           tells
           us
           of
           a
           famous
           Clock
           and
           Dial
           in
           the
           Cathedral
           Church
           of
           Lunden
           in
           Denmark
           .
           
             In
             the
             Dial
             (
             saith
             he
             )
             are
             to
             be
             seen
             distinctly
             the
             Year
             ,
             Month
             ,
             Week-day
             ,
             and
             Hour
             of
             every
             day
             throughout
             the
             Year
             ;
             with
             the
             Feasts
             ,
             both
             moveable
             and
             fixed
             ;
             together
             with
             the
             Motion
             of
             the
             Sun
             and
             Moon
             ,
             and
             their
             passage
             thro
             each
             degree
             of
             the
             Zodiack
             .
             Then
             for
             the
             Clock
             ,
             it
             is
             so
             framed
             by
             artifical
             Engines
             ,
             that
             whensoever
             it
             is
             to
             strike
             ,
             two
             Horse-men
             encounter
             one
             another
             ,
             giving
             as
             many
             blows
             apiece
             ,
             as
             the
             Bell
             sounds
             hours
             :
             And
             on
             the
             opening
             of
             a
             door
             ,
             there
             appeareth
             a
             Theatre
             ,
             the
             Virgin
             Mary
             on
             a
             Throne
             ,
             with
             Christ
             in
             her
             arms
             ,
             and
             the
             three
             
             Kings
             or
             Magi
             (
             with
             their
             several
             trains
             )
             marching
             in
             order
             ,
             doing
             humble
             reverence
             ,
             and
             presenting
             severally
             their
             Gifts
             ;
             two
             Trumpeters
             sounding
             all
             the
             while
             ,
             to
             adorn
             the
             Pomp
             of
             that
             Procession
             .
          
        
         
           
           To
           this
           I
           might
           add
           many
           more
           such
           curious
           performances
           ;
           but
           I
           rather
           chuse
           to
           refer
           the
           Reader
           to
           Schottus
           ,
           where
           he
           may
           find
           a
           great
           variety
           ,
           to
           please
           him
           .
        
      
       
         
           CHAP.
           VII
           .
           Of
           the
           Invention
           of
           Pendulum
           Watches
           .
        
         
           §
           1.
           
           THe
           first
           that
           invented
           the
           way
           of
           applying
           Pendulums
           to
           Watch-work
           ,
           was
           Mr
           
             Christian
             Hugens
          
           of
           Zulichem
           ;
           as
           he
           affirmeth
           of
           himself
           ,
           with
           very
           cogent
           reasons
           .
        
         
           This
           excellent
           invention
           ,
           he
           says
           ,
           he
           put
           first
           in
           practice
           in
           the
           year
           1657
           ▪
           and
           
           in
           the
           following
           year
           1658
           ,
           he
           printed
           a
           delineation
           and
           description
           of
           it
           .
        
         
         
           Others
           have
           claimed
           the
           honour
           of
           this
           Invention
           ;
           among
           which
           ,
           the
           great
           Galileo
           hath
           the
           most
           to
           be
           said
           on
           his
           side
           .
           Dr.
           
             John
             Joachim
             Becher
          
           (
           who
           printed
           a
           Book
           when
           he
           was
           in
           England
           ,
           entituled
           ;
           
             De
             Nova
             Temporis
             dimetiendi
             ratione
             Theoria
             ,
          
           &c.
           which
           he
           dedicated
           to
           the
           English
           Royal
           Society
           ,
           Anno
           1680.
           )
           he
           ,
           I
           say
           ,
           tells
           
           us
           ,
           
             That
             the
             
               Count
               Magalotti
            
             (
             the
             Great
             Duke
             of
             
             Tuscany's
             Resident
             at
             the
             Emperors
             Court
             )
             told
             him
             the
             whole
             History
             of
             these
             Pendulum
             Clocks
             ,
             and
             denied
             Mr
             Zulichem
             to
             be
             the
             Author
             of
             them
             .
          
           Also
           ,
           
             That
             one
             Treffler
             (
             Clock-maker
             to
             the
             Father
             of
             the
             then
             G.
             Duke
             of
             Tuscany
             )
             related
             to
             him
             the
             like
             History
             :
             and
             said
             moreover
             ,
             That
             he
             had
             made
             the
             first
             Pend.
             Clock
             ,
             at
             Florence
             ,
             by
             the
             command
             of
             the
             Great
             Duke
             ,
             and
             by
             the
             direction
             of
             his
             Mathematician
             
               Galilaeus
               a
               Galilaeo
            
             ;
             a
             pattern
             of
             which
             was
             brought
             into
             Holland
             .
             And
             further
             he
             saith
             ,
             That
             one
             
               Caspar
               Doms
            
             ,
             a
             Fleming
             ,
             and
             Mathematician
             to
             
               John
               Philip
               a
               Schonborn
            
             ,
             the
             late
             Elector
             of
             Mentz
             )
             told
             him
             ,
             that
             he
             had
             seen
             at
             Prague
             ,
             in
             the
             time
             of
             Rudolphus
             the
             Emperor
             ,
             a
             Pend.
             Clock
             ,
             made
             by
             the
             famous
             
               Justus
               Borgen
            
             ,
             Mechanick
             
             and
             Clock-maker
             to
             the
             Emperor
             :
             which
             Clock
             the
             great
             Tycho-Brahe
             used
             in
             his
             Astronomical
             observations
             .
          
        
         
           
           Thus
           far
           Becher
           .
           To
           which
           I
           may
           add
           ,
           what
           is
           said
           by
           the
           
             Academie
             del
             Cimento
             ,
             viz.
          
           
           
             It
             was
             thought
             good
             to
             apply
             the
             Pendulum
             to
             the
             Movement
             of
             the
             
               Clock
               ▪
            
             a
             thing
             which
             Galilaeo
             first
             invented
             ,
             and
             his
             Son
             
               Vincenzio
               Galilei
            
             put
             in
             practice
             in
             the
             year
             1649.
             
          
        
         
           As
           to
           these
           matters
           thus
           related
           by
           hear-say
           by
           Becher
           ,
           and
           so
           expressly
           affirmed
           by
           the
           Academy
           ,
           I
           have
           little
           to
           reply
           ,
           but
           that
           Mr
           Hugens
           does
           expressly
           say
           ,
           He
           was
           the
           Inventer
           ,
           and
           that
           if
           Galilaeo
           ever
           
           thought
           of
           any
           such
           thing
           ,
           he
           never
           brought
           it
           to
           any
           perfection
           .
           It
           is
           certain
           ,
           that
           this
           Invention
           never
           flourished
           till
           Mr
           Hugens
           set
           it
           abroad
           .
        
         
           §
           2.
           
           After
           Mr
           Hugens
           had
           thus
           invented
           these
           Pendulum
           Watches
           ,
           and
           caused
           several
           to
           be
           made
           in
           Holland
           ,
           Mr
           Fromantil
           ,
           a
           Dutch
           Clock-maker
           ,
           came
           over
           into
           England
           ,
           and
           made
           the
           first
           that
           ever
           were
           made
           here
           :
           which
           was
           about
           the
           year
           1662.
           
           One
           of
           the
           first
           Pieces
           that
           was
           made
           in
           England
           ,
           is
           now
           in
           Gresham-Colledg
           ,
           given
           to
           that
           Honorable
           Society
           by
           
           the
           late
           eminent
           Seth
           ,
           Lord
           Bishop
           of
           Salisbury
           :
           which
           is
           made
           exactly
           according
           to
           Mr
           
           Zulichem's
           directions
           .
        
         
           §
           3.
           
           For
           several
           years
           this
           way
           of
           Mr
           Zulichem
           was
           the
           only
           method
           ,
           viz.
           Crown-wheel
           Pendulums
           ,
           to
           play
           between
           two
           cycloidal
           cheeks
           ,
           &c.
           But
           afterwards
           Mr
           
             W.
             Clement
          
           ,
           a
           London
           Clock-maker
           ,
           
           contrived
           them
           (
           as
           Mr
           Smith
           saith
           )
           to
           go
           with
           less
           weight
           ,
           an
           heavier
           Ball
           (
           if
           you
           please
           )
           and
           to
           vibrate
           but
           a
           small
           compass
           .
           Which
           is
           now
           the
           universal
           method
           of
           the
           Royal
           Pendulums
           .
           But
           Dr
           Hook
           denies
           Mr
           Clement
           to
           have
           invented
           this
           ;
           and
           says
           that
           it
           was
           his
           Invention
           ,
           and
           that
           he
           caused
           a
           piece
           of
           this
           nature
           to
           be
           made
           ,
           which
           he
           shewed
           before
           the
           
             R.
             Society
          
           ,
           soon
           after
           the
           Fire
           of
           London
           .
        
         
           §
           4.
           
           The
           Use
           of
           these
           Pendulum
           Watches
           Mr.
           Hugens
           setteth
           forth
           in
           several
           instances
           .
           Particnlarly
           ,
           he
           giveth
           two
           examples
           of
           their
           great
           use
           at
           Sea
           ,
           in
           discovering
           the
           Difference
           of
           Meridians
           ,
           more
           exactly
           than
           any
           other
           way
           :
           which
           he
           deduceth
           from
           the
           Observations
           of
           an
           English
           ,
           and
           French
           Ship.
           
        
         
           On
           Land
           ,
           they
           were
           found
           very
           serviceable
           ,
           among
           other
           uses
           ,
           particularly
           
           to
           these
           two
           .
           1.
           
           To
           measure
           the
           time
           more
           exactly
           ,
           and
           equally
           than
           the
           Sun.
           ●
           .
           To
           be
           (
           as
           Sir
           
             Christoph
             .
             Wren
          
           first
           proposed
           )
           a
           perpetual
           ,
           and
           universal
           Measure
           ,
           or
           Standard
           ,
           to
           which
           all
           Lengths
           may
           be
           reduced
           ,
           and
           by
           which
           they
           may
           be
           udged
           ,
           in
           all
           ages
           ,
           and
           countries
           .
           For
           as
           our
           
             Royal
             Society
          
           ,
           Mr
           Hugens
           ,
           and
           Mountonus
           have
           proposed
           after
           Sir
           
             Christopher
             Wren
          
           )
           this
           
             Horary
             foot
          
           ,
           or
           Tripedal
           length
           ,
           which
           vib●teth
           Seconds
           ,
           will
           fit
           ill
           ages
           and
           places
           ▪
           But
           then
           respect
           must
           be
           had
           to
           the
           Center
           of
           Oscillation
           ,
           which
           you
           have
           an
           account
           of
           in
           Mr
           Hugens
           his
           ●soresaid
           book
           ,
           
             de
             Horologio
             Oscillatorio
          
           ,
           ●s
           hath
           before
           been
           said
           .
        
         
           §
           5.
           
           There
           is
           one
           contrivance
           more
           of
           Pendulums
           ,
           still
           behind
           ,
           viz.
           the
           
             Circular
             Pendulum
          
           :
           which
           is
           mentioned
           by
           Mr
           Hugens
           as
           his
           own
           ,
           but
           is
           claimed
           by
           the
           ngenious
           Dr.
           Hook
           as
           really
           his
           .
           This
           ●end
           ▪
           doth
           not
           vibrate
           backward
           and
           forward
           ,
           as
           ●hose
           we
           have
           been
           speaking
           of
           ●o
           ;
           but
           always
           round
           ,
           round
           ;
           the
           String
           being
           suspended
           above
           ,
           at
           the
           tripedal
           length
           ,
           and
           the
           Ball
           fi●ed
           below
           ,
           as
           suppose
           at
           the
           end
           of
           the
           fly
           of
           a
           common
           ●ack
           .
        
         
         
           The
           motion
           of
           this
           Circular
           Pend
           ▪
           is
           as
           regular
           ,
           and
           much
           the
           same
           with
           what
           goeth
           before
           :
           and
           was
           thus
           far
           made
           very
           useful
           in
           Astronomical
           observations
           ,
           by
           the
           said
           Dr
           
             Hook
             ,
             viz.
          
           To
           give
           warning
           at
           any
           moment
           of
           its
           circumgyration
           ,
           either
           when
           it
           had
           turned
           but
           a
           quarter
           ,
           half
           ,
           or
           any
           lesser
           ,
           or
           greater
           part
           of
           its
           circle
           .
           So
           that
           here
           you
           had
           notice
           not
           only
           of
           a
           Second
           ,
           but
           of
           the
           most
           minute
           part
           of
           a
           Second
           of
           Time.
           You
           may
           find
           a
           description
           of
           this
           Pendulum
           ,
           and
           other
           matters
           belonging
           to
           it
           ,
           in
           Dr
           
           Hook's
           
             Lectiones
             Cutlerianoe
             :
             Animad
             .
             in
             Hevelius
             Mach
             ▪
             Caelest
             .
             p.
          
           60.
           
        
      
       
         
           CHAP.
           VIII
           .
           Of
           the
           Invention
           of
           those
           Pocket-Watches
           ,
           commonly
           called
           Pendulum
           Watches
           .
        
         
           §
           1.
           
           THe
           reason
           they
           are
           called
           Pendulum-Watches
           ,
           is
           from
           the
           regularity
           of
           their
           Strokes
           ,
           and
           Motion
           .
           
           Which
           exactness
           is
           effected
           by
           the
           government
           of
           a
           small
           Spiral
           Spring
           ,
           running
           round
           the
           upper
           part
           of
           the
           Verge
           of
           the
           Ballance
           :
           which
           Spring
           is
           called
           the
           Regulator
           .
        
         
           §
           2.
           
           The
           first
           Inventer
           hereof
           ,
           was
           ●hat
           ingenious
           and
           learned
           member
           of
           our
           Royal-Society
           ,
           Dr
           Hook
           :
           who
           contrived
           va●ious
           ways
           of
           Regulation
           .
           One
           way
           was
           ●ith
           a
           Load-stone
           :
           another
           was
           with
           a
           ●nder
           strait
           Spring
           ▪
           one
           end
           whereof
           ●layed
           backward
           ,
           and
           forward
           ,
           with
           the
           ●allance
           :
           So
           that
           the
           ballance
           was
           to
           this
           ●pring
           as
           the
           bob
           of
           a
           Pendulum
           ,
           and
           the
           ●ttle
           Spring
           ,
           as
           the
           Rod
           thereof
           .
           And
           se●eral
           other
           contrivances
           he
           had
           besides
           of
           ●is
           nature
           .
        
         
           §
           3.
           
           But
           the
           Invention
           which
           best
           an●ered
           expectation
           ,
           was
           at
           first
           ,
           with
           two
           ●llances
           :
           of
           which
           I
           have
           seen
           two
           sorts
           ,
           ●ho
           there
           were
           several
           others
           .
           One
           ●ay
           was
           without
           Spiral
           Springs
           ,
           the
           ●her
           with
           .
           They
           both
           agreed
           in
           this
           ,
           ●hat
           the
           outward
           Rims
           of
           both
           the
           Bal●nces
           ,
           had
           alike
           number
           of
           Teeth
           ;
           which
           ●nning
           in
           each
           other
           ,
           caused
           each
           Bal●nce
           to
           vibrate
           alike
           .
        
         
         
           But
           as
           to
           the
           former
           of
           these
           ,
           which
           had
           no
           Spiral
           Spring
           ▪
           the
           Verges
           of
           its
           Ballances
           ,
           had
           each
           but
           one
           Pallet
           apiece
           ,
           about
           the
           middle
           of
           the
           Verge
           .
           The
           Crown-wheel
           lay
           (
           contrary
           to
           others
           )
           reversed
           ,
           in
           the
           middle
           of
           the
           Watch
           ,
           in
           the
           place
           ,
           and
           after
           the
           manner
           of
           the
           Contrate-wheel
           .
           The
           teeth
           of
           this
           Crown-wheel
           ,
           were
           cut
           after
           the
           manner
           of
           Contrate-wheel
           teeth
           ,
           viz.
           lying
           upwards
           ,
           but
           very
           wide
           apart
           ,
           so
           as
           that
           the
           Pallets
           (
           which
           were
           about
           one
           tenth
           of
           an
           inch
           long
           ,
           and
           n●rrow
           )
           might
           play
           in
           and
           out
           between
           each
           tooth
           .
           The
           Verges
           of
           the
           two
           Ballances
           ,
           were
           set
           one
           on
           one
           side
           ,
           the
           other
           on
           the
           other
           side
           of
           the
           Crown-wheel
           ,
           so
           that
           the
           Pallets
           might
           play
           freely
           in
           its
           teeth
           .
           And
           when
           the
           Crown-wheel
           in
           moving
           round
           ,
           had
           delivered
           its
           self
           of
           one
           Pallet
           ,
           the
           other
           Pallet
           on
           the
           opposite
           side
           ,
           was
           drawn
           on
           to
           make
           its
           Beat
           ,
           by
           means
           of
           the
           motion
           which
           the
           other
           Ballance
           had
           given
           its
           Ballance
           ,
           (
           the
           two
           Ballances
           moving
           one
           another
           ,
           as
           hath
           been
           said
           in
           the
           beginning
           of
           this
           Paragraph
           .
           )
           And
           so
           the
           same
           back
           again
           .
        
         
         
           It
           may
           be
           here
           noted
           ,
           that
           for
           the
           more
           clear
           understanding
           of
           the
           last
           contrivance
           ,
           I
           have
           described
           the
           two
           Ballances
           ,
           as
           having
           Teeth
           on
           the
           edges
           of
           their
           Rims
           ,
           running
           in
           one
           another
           .
           But
           the
           contrivance
           was
           really
           thus
           ▪
           There
           was
           a
           small
           Wheel
           under
           each
           Ballance
           ,
           proportioned
           to
           the
           width
           of
           the
           Crown-wheel
           .
           But
           the
           Ballances
           were
           much
           larger
           .
           And
           so
           the
           Teeth
           of
           these
           two
           little
           foresaid
           Wheels
           or
           Ballances
           ,
           running
           in
           one
           another
           ,
           moved
           the
           larger
           Ballances
           above
           them
           ,
           all
           one
           ,
           as
           if
           these
           two
           great
           Ballances
           had
           been
           toothed
           and
           played
           in
           each
           other
           .
        
         
           §
           4.
           
           The
           other
           way
           ,
           with
           two
           Ballances
           also
           ,
           moving
           each
           other
           (
           as
           was
           said
           in
           the
           beginning
           of
           the
           last
           §
           )
           had
           a
           Spiral
           Spring
           to
           each
           Ballance
           ,
           for
           its
           Regulator
           .
           In
           this
           Invention
           ,
           only
           one
           Ballance
           had
           the
           Pallets
           ,
           as
           the
           common
           Ballances
           have
           :
           and
           the
           Crown-wheel
           operated
           upon
           it
           ,
           according
           to
           the
           usual
           way
           .
           But
           then
           when
           this
           Ballance
           vibrateth
           ,
           it
           giveth
           the
           same
           motion
           backward
           and
           forward
           ,
           to
           the
           other
           ballance
           ;
           as
           hath
           been
           said
           .
        
         
         
           The
           first
           of
           these
           two
           ways
           ,
           was
           never
           prosecuted
           so
           far
           ,
           as
           perhaps
           it
           deserved
           .
           And
           the
           excellency
           of
           the
           latter
           is
           ,
           that
           no
           jirk
           ,
           or
           the
           most
           confused
           shake
           ,
           can
           in
           the
           least
           alter
           its
           Vibrations
           .
           Which
           it
           will
           do
           in
           the
           best
           Pendulum
           Watch
           with
           one
           ballance
           now
           commonly
           used
           .
           For
           if
           you
           lay
           one
           of
           these
           Watches
           upon
           a
           Table
           ,
           and
           by
           the
           Pendant
           jirk
           it
           backward
           and
           forward
           ,
           you
           will
           put
           it
           into
           the
           greatest
           hurry
           ;
           whereas
           the
           last
           mentioned
           Watch
           ,
           with
           two
           ballances
           ,
           will
           be
           nothing
           affected
           with
           it
           .
           But
           notwithstanding
           this
           inconvenience
           ,
           yet
           the
           Watch
           with
           one
           ballance
           and
           one
           Spring
           (
           which
           was
           also
           Dr.
           Hooks
           Invention
           )
           prevailed
           ,
           and
           grew
           common
           ,
           being
           now
           the
           universal
           Mode
           :
           but
           of
           the
           other
           very
           few
           were
           ever
           made
           .
           The
           reason
           hereof
           ,
           I
           judge
           ,
           was
           the
           great
           trouble
           and
           vast
           niceness
           required
           in
           it
           ,
           and
           perhaps
           a
           little
           foulness
           in
           the
           ballance-teeth
           may
           retard
           the
           motion
           of
           the
           ballances
           .
           But
           the
           other
           is
           easier
           made
           ,
           and
           performeth
           well
           enough
           ,
           and
           in
           a
           pocket
           is
           scarce
           subject
           to
           the
           aforesaid
           disorder
           ,
           which
           is
           caused
           rather
           by
           a
           turn
           ,
           than
           a
           shake
           .
        
         
         
           §
           5.
           
           The
           time
           of
           these
           Inventions
           was
           about
           the
           year
           1658
           ,
           as
           appears
           (
           among
           other
           evidence
           )
           from
           this
           inscription
           ,
           upon
           one
           of
           the
           aforesaid
           double
           Ballance-Watches
           ,
           presented
           to
           K.
           Charles
           II
           ,
           
             viz.
             Robert
             Hook
             inven
          
           .
           1658.
           
           
             T.
             Tompion
             fecit
          
           1675.
           
        
         
           This
           Watch
           was
           wonderfully
           approved
           of
           by
           the
           King
           ;
           and
           so
           the
           Invention
           grew
           into
           reputation
           ,
           and
           was
           much
           talked
           of
           at
           home
           ,
           and
           abroad
           .
           Particularly
           its
           same
           flew
           into
           France
           ,
           from
           whence
           the
           Dauphine
           sent
           for
           two
           ;
           which
           that
           eminent
           Artist
           Mr.
           Tompion
           made
           for
           him
           .
        
         
           §
           6.
           
           Dr.
           Hook
           had
           long
           before
           this
           ,
           caused
           several
           pieces
           of
           this
           nature
           ,
           to
           be
           made
           ,
           altho
           they
           did
           not
           take
           till
           after
           1675.
           
           However
           he
           had
           before
           so
           far
           proceeded
           herein
           ,
           as
           to
           have
           a
           Patent
           (
           drawn
           ,
           tho
           not
           sealed
           )
           for
           these
           ,
           and
           some
           other
           Contrivances
           ,
           about
           Watches
           ,
           in
           the
           year
           1660.
           
           But
           the
           reason
           why
           that
           Patent
           did
           no
           further
           proceed
           ,
           was
           some
           disagreement
           about
           some
           Articles
           in
           it
           ,
           with
           some
           Noble
           Persons
           who
           were
           concerned
           for
           the
           procuring
           it
           .
           The
           same
           ingenious
           Dr.
           had
           also
           a
           Grant
           for
           a
           Patent
           for
           this
           last
           way
           of
           Spring
           Watches
           
           in
           the
           year
           1675
           :
           but
           he
           omitted
           the
           taking
           it
           out
           ,
           as
           thinking
           it
           not
           worth
           the
           while
           .
        
         
           §
           7.
           
           After
           these
           Inventions
           of
           Dr.
           Hook
           ,
           and
           (
           no
           doubt
           )
           after
           the
           Publication
           of
           Mr.
           
           Hugens's
           book
           
             de
             Horolog
             .
             Oscil
          
           .
           at
           Paris
           1673
           (
           for
           there
           is
           not
           a
           word
           of
           this
           ,
           tho
           of
           several
           other
           Contrivances
           )
           after
           this
           ,
           I
           say
           ,
           Mr.
           
           Hugen's
           Watch
           with
           a
           Spiral
           Spring
           came
           abroad
           ,
           and
           made
           a
           great
           noise
           in
           England
           ,
           as
           if
           the
           Longitude
           could
           be
           now
           found
           .
           One
           of
           these
           the
           Lord
           Bruncker
           sent
           for
           out
           of
           France
           ,
           (
           where
           Mr
           
           Hugen
           ●
           had
           a
           Patent
           for
           them
           )
           which
           I
           have
           seen
           .
        
         
           This
           Watch
           of
           Mr.
           
           Zulichem's
           agreed
           with
           Dr.
           
           Hook's
           ,
           in
           the
           application
           of
           the
           Spring
           to
           the
           ballance
           :
           only
           Mr.
           
           Zulichem's
           had
           a
           longer
           Spiral
           Spring
           ,
           and
           the
           Pulses
           and
           Beats
           were
           much
           slower
           .
           That
           wherein
           it
           differs
           ,
           is
           1.
           
           The
           Verge
           hath
           a
           Pinion
           instead
           of
           Pallets
           ;
           and
           a
           Contrate-wheel
           runs
           therein
           ,
           and
           drives
           it
           round
           ,
           more
           than
           one
           turn
           .
           2.
           
           The
           Pallets
           are
           on
           the
           Arbor
           of
           this
           Contrate-wheel
           .
           3.
           
           Then
           followeth
           the
           Crown
           wheel
           ,
           &c.
           4.
           
           The
           ballance
           ,
           instead
           of
           turning
           scarce
           quite
           round
           (
           as
           Dr.
           
           Hook's
           )
           doth
           turn
           several
           rounds
           every
           vibration
           .
        
         
         
           §
           8.
           
           As
           to
           the
           great
           abilities
           of
           Mr.
           Hugens
           ,
           no
           man
           can
           doubt
           ,
           that
           is
           acquainted
           with
           his
           Books
           ,
           and
           his
           share
           in
           the
           Philosophical
           Transactions
           ,
           &c.
           
           But
           I
           have
           some
           reason
           to
           doubt
           ,
           whether
           his
           fancy
           was
           not
           first
           set
           on
           work
           ,
           by
           some
           Intelligence
           he
           might
           have
           of
           Dr
           
           Hook's
           Invention
           ,
           from
           Mr
           Oldenburgh
           ,
           or
           others
           his
           correspondents
           here
           in
           England
           .
        
         
           But
           whether
           or
           no
           that
           ingenious
           person
           doth
           owe
           any
           thing
           herein
           to
           our
           ingenious
           Dr
           Hook
           ,
           it
           is
           however
           a
           very
           pretty
           ,
           and
           ingenious
           contrivance
           ;
           but
           subject
           to
           some
           defects
           :
           viz.
           When
           it
           standeth
           still
           ,
           it
           will
           not
           vibrate
           ,
           until
           it
           is
           set
           on
           vibrating
           :
           which
           ,
           tho
           it
           be
           no
           defect
           in
           a
           Pendulum
           Clock
           ,
           may
           be
           one
           in
           a
           Pocket-Watch
           ,
           which
           is
           exposed
           to
           continual
           jogs
           .
           Also
           ,
           it
           doth
           somewhat
           vary
           in
           its
           Vibrations
           ,
           making
           sometimes
           longer
           ,
           sometimes
           shorter
           turns
           ,
           and
           so
           some
           slower
           some
           quicker
           vibrations
           .
        
         
           I
           have
           seen
           some
           other
           contrivances
           of
           this
           sort
           ,
           which
           I
           mention
           not
           ,
           because
           they
           are
           of
           younger
           standing
           .
           But
           these
           two
           (
           of
           Dr
           Hook
           and
           Mr
           Hugens
           )
           I
           have
           taken
           notice
           of
           ,
           because
           they
           were
           the
           first
           that
           ever
           appeared
           in
           the
           world
           .
        
      
       
         
         
           CHAP.
           IX
           .
           The
           Invention
           of
           Repeating
           Clocks
           .
        
         
           §
           1.
           
           THe
           Clocks
           I
           now
           shall
           speak
           of
           ,
           are
           such
           as
           by
           pulling
           of
           a
           String
           ,
           &c.
           do
           strike
           the
           Hour
           ,
           Quarter
           ,
           or
           Minute
           ,
           at
           any
           time
           of
           the
           day
           and
           night
           .
        
         
           §
           2.
           
           These
           Clocks
           are
           a
           late
           Invention
           of
           one
           Mr
           Barlow
           ,
           of
           no
           longer
           standing
           than
           the
           latter
           end
           of
           K.
           Charles
           II.
           about
           the
           year
           1676.
           
        
         
           This
           ingenious
           Contrivance
           (
           scarce
           so
           much
           as
           thought
           of
           before
           )
           soon
           took
           air
           ,
           and
           being
           talked
           of
           among
           the
           London
           Artists
           ,
           set
           their
           heads
           to
           work
           ;
           who
           presently
           contrived
           several
           ways
           to
           effect
           such
           a
           performance
           .
           And
           hence
           arose
           the
           divers
           ways
           of
           
             Repeating
             work
          
           ,
           which
           so
           early
           might
           be
           observed
           to
           be
           about
           the
           Town
           ,
           every
           man
           almost
           practising
           ,
           according
           to
           his
           own
           Invention
           .
        
         
         
           §
           3.
           
           This
           Invention
           was
           practised
           chief●
           ,
           if
           not
           only
           ,
           in
           larger
           Movements
           ,
           〈◊〉
           K.
           James
           II.'s
           Reign
           :
           at
           which
           time
           it
           as
           transferred
           into
           Pocket-Clocks
           .
           But
           ●ere
           being
           some
           little
           contest
           concern●g
           the
           Author
           hereof
           ,
           I
           shall
           relate
           the
           ●●e
           matter
           of
           fact
           ,
           leaving
           the
           Reader
           to
           ●own
           judgment
           .
        
         
           About
           the
           latter
           end
           of
           K.
           James
           II.'s
           ●gn
           ,
           Mr
           Barlow
           (
           the
           ingenious
           Inventer
           ●ore-mentioned
           )
           contrived
           to
           put
           his
           ●ention
           into
           Pocket
           ▪
           watches
           ;
           and
           en●voured
           (
           with
           the
           Lord
           Chief
           Justice
           ●bone
           ,
           and
           some
           others
           )
           to
           get
           a
           Patent
           ●it
           .
           And
           in
           order
           to
           it
           ,
           he
           set
           Mr
           Tom●
           the
           famous
           Artist
           ,
           to
           work
           upon
           it
           :
           ●o
           accordingly
           made
           a
           Piece
           according
           ●is
           directions
           .
        
         
           ●r
           Quare
           (
           a
           very
           ingenious
           Watch
           ▪
           ●er
           in
           London
           )
           had
           some
           years
           before
           〈◊〉
           thinking
           of
           the
           like
           Invention
           :
           but
           bringing
           it
           to
           perfection
           ,
           he
           laid
           by
           thoughts
           of
           it
           ,
           until
           the
           talk
           of
           Mr
           Bar●
           Patent
           revived
           his
           former
           thoughts
           ;
           ●ch
           he
           then
           brought
           to
           effect
           .
           This
           ●g
           known
           among
           the
           Watch-makers
           ,
           〈◊〉
           all
           pressed
           him
           to
           endeavour
           to
           hin●
           Mr
           
           Barlow's
           Patent
           .
           And
           accordingly
           
           applications
           were
           made
           at
           Court
           ,
           and
           a
           Watch
           of
           each
           Invention
           ,
           produced
           before
           the
           King
           and
           Council
           .
           The
           King
           ,
           upon
           tryal
           of
           each
           of
           them
           ,
           was
           pleased
           to
           give
           the
           preference
           to
           Mr
           
           Quare's
           :
           of
           which
           ,
           notice
           was
           given
           soon
           after
           in
           the
           Gazette
           .
        
         
           The
           difference
           between
           these
           two
           Inventions
           was
           ,
           Mr
           
           Barlow's
           was
           made
           to
           Repeat
           by
           pushing
           in
           two
           pieces
           on
           each
           side
           the
           Watch-box
           :
           one
           of
           which
           Repeated
           the
           Hour
           ,
           the
           other
           the
           Quarter
           ▪
           Mr
           
           Quare's
           was
           made
           to
           Repeat
           ,
           by
           a●
           Pin
           that
           stuck
           out
           near
           the
           Pendant
           which
           being
           thrust
           in
           (
           as
           now
           't
           is
           done
           by
           thrusting
           in
           the
           Pendant
           )
           did
           Repeat
           both
           the
           Hour
           ,
           and
           Quarter
           ,
           with
           the
           sam●
           thrust
           .
        
         
           It
           would
           (
           I
           think
           )
           be
           very
           frivolous
           ,
           to●
           speak
           of
           the
           various
           contrivances
           ,
           and
           methods
           of
           Repeating
           work
           ,
           and
           the
           Inventers
           of
           them
           ;
           and
           therefore
           I
           shall
           sa●
           nothing
           of
           them
           .
        
      
       
         
         
           CHAP.
           X.
           
        
         
           
             Numbers
             for
             several
             sorts
             of
             Movements
             .
          
           
             I
             Think
             it
             may
             be
             very
             convenient
             to
             set
             down
             some
             Numbers
             ,
             fit
             for
             several
             Movements
             ;
             partly
             ,
             to
             be
             as
             Examples
             to
             exercise
             the
             young
             Reader
             ,
             in
             the
             foregoing
             Art
             of
             Calculation
             :
             and
             partly
             ,
             to
             serve
             such
             ,
             who
             want
             leisure
             or
             understanding
             to
             attain
             to
             this
             Art.
             
          
           
             §
             1.
             
             But
             first
             it
             may
             be
             requisite
             ,
             to
             shew
             the
             usual
             way
             of
             Watch-makers
             writing
             down
             their
             Numbers
             ,
             which
             is
             somewhat
             different
             from
             that
             in
             the
             preceding
             Book
             .
          
           
             Their
             way
             representeth
             the
             Wheel
             and
             Pinion
             ,
             on
             the
             same
             Spindle
             ;
             not
             as
             they
             play
             in
             one
             another
             .
             Thus
             the
             numbers
             of
             an
             old
             House-Watch
             ,
             of
             12
             hours
             ,
             is
             written
             thus
             ;
          
           
           
             
               
                 
                   My
                   way
                   :
                
                 
                   The
                   Watch-makers
                   way
                   .
                
              
               
                 
                   4
                   )
                   48
                
                 
                   ▪
                   48
                
              
               
                 
                   7
                   )
                   56
                
                 
                   56
                   —
                   4
                
              
               
                 
                   6
                   )
                   54
                
                 
                   54
                   —
                   7
                
              
               
                 
                   19
                
                 
                   19
                   —
                   6
                
              
            
          
           
             According
             to
             my
             way
             ,
             the
             Pin.
             of
             Report
             [
             4
             ]
             drives
             the
             Dial-wheel
             [
             48
             :
             ]
             the
             Pinion
             [
             7
             ]
             plays
             in
             the
             Great-wheel
             [
             56
             ]
             &c.
             
             But
             according
             to
             the
             other
             way
             ,
             the
             Dial-wheel
             stands
             alone
             ;
             the
             Great-wheel
             hath
             the
             Pinion
             of
             Report
             on
             the
             same
             arbor
             :
             the
             Wheel
             [
             54
             ]
             hath
             the
             Pin
             :
             [
             7
             ]
             and
             the
             Crown-wheel
             [
             19
             ]
             the
             Pin
             :
             [
             6
             ]
             on
             the
             same
             Spindles
             .
          
           
             This
             latter
             way
             (
             tho
             very
             inconvenient
             in
             Calculation
             )
             representeth
             a
             piece
             of
             work
             handsomely
             enough
             ,
             and
             somewhat
             naturally
             .
          
           
             §
             2.
             
             Numbers
             of
             an
             8
             
               day
               Piece
            
             ,
             with
             16
             turns
             the
             Barrel
             ,
             the
             Pend.
             vibrates
             Seconds
             ,
             the
             shews
             Minutes
             ,
             Seconds
             ,
             &c.
             
          
           
             
               
                 
                   The
                   Watch-part
                   .
                
                 
                   The
                   Clock
                   part
                   .
                
              
               
                 
                   8
                   )
                   96
                
                 
                   8
                   )
                   78
                
              
               
                 
                   8
                   )
                   60
                   —
                   48
                   )
                   48
                   —
                   6
                   )
                   72
                
                 
                   6
                   )
                   48
                   8
                   pins
                   .
                
              
               
                 
                   7
                   )
                   56
                
                 
                   6
                   )
                   48
                
              
               
                 
                   30
                
                 
                   6
                   )
                   48
                
              
            
          
           
           
             In
             the
             Watch-part
             ,
             the
             Wheel
             60
             is
             the
             Minute-wheel
             ,
             which
             is
             set
             in
             the
             middle
             of
             the
             Clock
             ,
             that
             its
             Spindle
             may
             go
             thro
             the
             middle
             of
             the
             Dial-plate
             to
             carry
             the
             Minute-hand
             .
          
           
             Also
             on
             this
             Spindle
             is
             a
             Wheel
             48
             ,
             which
             driveth
             another
             Wheel
             of
             48
             ,
             which
             last
             hath
             a
             Pinion
             6
             ,
             which
             driveth
             round
             the
             Wheel
             72
             in
             12
             hours
             .
             Note
             here
             two
             things
             :
             1.
             
             That
             the
             two
             Wheels
             48
             ,
             are
             of
             no
             other
             use
             ,
             but
             to
             set
             the
             Pinion
             6
             at
             a
             convenient
             distance
             from
             the
             Minute-wheel
             ,
             to
             drive
             the
             Wheel
             72
             ,
             which
             is
             concentrical
             with
             the
             Minute-wheel
             .
             For
             a
             Pinion
             6
             driving
             a
             Wheel
             72
             ,
             would
             be
             sufficient
             ,
             if
             the
             Minute-hand
             and
             Hour-hand
             had
             two
             different
             centers
             .
             ▪
             2.
             
             These
             numbers
             ,
             60
             ▪
             48
             )
             48-6
             )
             72
             ,
             set
             thus
             ,
             ought
             (
             according
             to
             the
             last
             §
             )
             be
             thus
             read
             ,
             viz.
             The
             Wheel
             60
             ,
             hath
             another
             Wheel
             48
             on
             the
             same
             Spindle
             ;
             which
             Wheel
             48
             divideth
             ,
             playeth
             in
             ,
             or
             turns
             round
             another
             Wheel
             48
             ;
             which
             hath
             a
             Pinion
             6
             concentrical
             with
             it
             :
             which
             Pinion
             driveth
             ,
             or
             divideth
             a
             Wheel
             of
             72.
             
             For
             a
             Line
             parting
             two
             numbers
             (
             as
             60-48
             )
             denoteth
             those
             two
             numbers
             to
             be
             concentrical
             ,
             or
             to
             be
             placed
             upon
             
             the
             same
             Spindle
             .
             And
             when
             two
             numbers
             have
             a
             hook
             between
             them
             (
             as
             48
             )
             48
             )
             it
             signifies
             one
             to
             run
             in
             the
             other
             ,
             as
             hath
             before
             been
             hinted
             .
          
           
             In
             the
             Striking-part
             ,
             there
             are
             8
             Pins
             on
             the
             Second
             wheel
             48.
             
             The
             Count-wheel
             may
             be
             fixed
             unto
             the
             Great-wheel
             ,
             which
             goeth
             round
             once
             in
             12
             hours
             .
          
           
             §
             3.
             
             A
             Piece
             of
             32
             days
             ,
             with
             16
             ,
             or
             12
             turns
             both
             parts
             :
             the
             Watch
             sheweth
             Hours
             ,
             Minutes
             ,
             and
             Seconds
             ;
             and
             the
             Pend.
             vibrateth
             Seconds
             ▪
          
           
             
               
                 The
                 Watch-part
                 ,
              
               
                 
                   With
                   16
                   turns
                   .
                
                 
                   With
                   12
                   turns
                   .
                
              
               
                 
                   16
                   )
                   96
                
                 
                   12
                   )
                   96
                
              
               
                 
                   9
                   )
                   72
                
                 
                   9
                   )
                   72
                
              
               
                 
                   8
                   )
                   60
                   —
                   48
                   )
                   48
                   —
                   6
                   )
                   72
                
                 
                   8
                   )
                   60
                   —
                   48
                   )
                   48
                   —
                   6
                   )
                   72
                
              
               
                 
                   7
                   )
                   56
                
                 
                   7
                   )
                   56
                
              
               
                 
                   30
                
                 
                   30
                
              
            
          
           
             
               
                 The
                 Striking
                 part
                 .
              
               
                 
                   With
                   16
                   turns
                   .
                
                 
                   With
                   12
                   turns
                   .
                
              
               
                 
                   10
                   )
                   130
                
                 
                   8
                   )
                   128
                
              
               
                 
                   8
                   )
                   96
                
                 
                   24
                   pins
                
                 
                   8
                   )
                   104
                
                 
                   26
                   pins
                
              
               
                 
                   12
                   )
                   39
                
                 
                   8
                   )
                   24
                
              
               
                 
                   6
                   )
                   72
                   Double
                   hoop
                   .
                
                 
                   8
                   )
                   96
                   Double
                   hoop
                   .
                
              
               
                 
                   6
                   )
                   60
                
                 
                   8
                   )
                   80
                
              
            
          
           
           
             The
             Pinion
             of
             Report
             is
             fixed
             on
             the
             ●nd
             of
             the
             arbor
             of
             the
             Pin
             ▪
             wheel
             .
             This
             Pinion
             in
             the
             first
             is
             12
             ,
             the
             Count-wheel
             39
             ;
             thus
             ,
             12
             )
             39.
             
             Or
             it
             may
             be
             8
             )
             26.
             ●n
             the
             latter
             (
             with
             12
             turns
             )
             it
             may
             be
             6
             )
             18
             ,
             or
             8
             )
             24.
             
          
           
             §
             4.
             
             
               A
               two
               month
               Piece
            
             ,
             of
             64
             days
             ;
             with
             16
             turns
             ;
             Pend.
             vibrateth
             Seconds
             ,
             and
             sheweth
             Minutes
             ,
             Seconds
             ,
             &c.
             
          
           
             
               
                 
                   Watch-part
                   .
                
                 
                   Clock-part
                   .
                
              
               
                 
                   9
                   )
                   90
                
                 
                   10
                   )
                   80
                
              
               
                 
                   8
                   )
                   76
                
                 
                   10
                   )
                   65
                
              
               
                 
                   8
                   )
                   60
                   —
                   48
                   )
                   48
                   —
                   6
                   )
                   72
                
                 
                   9
                   )
                   54
                
                 
                   12
                   pins
                
              
               
                 
                   7
                   )
                   56
                
                 
                   —
                   8
                   )
                   52
                
              
               
                 
                    
                
                 
                   5
                   )
                   60-Double
                   Hoop
                
              
               
                 
                   30
                
                 
                   5
                   )
                   50
                
              
            
          
           
             Here
             the
             third
             Wheel
             is
             the
             Pin-wheel
             ;
             which
             also
             carrieth
             the
             Pinion
             of
             Report
             8
             ,
             driving
             the
             Count-wheel
             52.
             
          
           
             
               
                 Or
                 thus
                 .
              
               
                 
                   Watch-part
                   .
                
                 
                   Clock-part
                   .
                
              
               
                 
                   8
                   )
                   80
                
                 
                   6
                   )
                   144
                
              
               
                 
                   8
                   )
                   76
                
                 
                   6
                   )
                   78
                
                 
                   26
                   pins
                
              
               
                 
                   8
                   )
                   60
                   —
                   48
                   )
                   48
                   —
                   6
                   )
                   72
                
                 
                   —
                   8
                   )
                   24
                
              
               
                 
                   7
                   )
                   56
                
                 
                   6
                   )
                   72-Double
                   Hoop
                
              
               
                 
                   30
                
                 
                   6
                   )
                   60
                
              
            
          
           
           
             §
             5.
             
             A
             piece
             of
             13
             weeks
             ,
             with
             Pendulum
             ,
             Turns
             ,
             and
             Motions
             ,
             as
             before
             .
          
           
             
               
                 The
                 Watch
                 part
                 .
              
               
                 
                   8
                   )
                   96
                   Or
                   thus
                   .
                
                 
                   6
                   )
                   72
                
              
               
                 
                   8
                   )
                   88
                
                 
                   6
                   )
                   66
                
              
               
                 
                   8
                   )
                   60
                   —
                   48
                   )
                   48
                   —
                   6
                   )
                   72
                
                 
                   6
                   )
                   48
                   —
                   48
                   )
                   48
                   —
                   6
                   )
                   72
                
              
               
                 
                   7
                   )
                   56
                
                 
                   6
                   )
                   45
                
              
               
                 
                   30
                
                 
                   30
                
              
            
          
           
             
               
                 The
                 Clock
                 part
                 .
              
               
                 
                   8
                   )
                   72
                   Or
                   thus
                   .
                
                 
                   5
                   )
                   145
                
              
               
                 
                   8
                   )
                   64
                   —
                   37
                   )
                   30
                
                 
                   6
                   )
                   90
                
                 
                   —
                   30
                   pins
                
              
               
                 
                   8
                   )
                   48
                   —
                   12
                   pins
                
                 
                   —
                   24
                   )
                   62
                
              
               
                 
                   6
                   )
                   48
                   Double
                   Hoop
                
                 
                   6
                   )
                   72
                
              
               
                 
                   5
                   )
                   40
                
                 
                   6
                   )
                   60
                
              
            
          
           
             §
             6.
             
             A
             
               Seven
               Month
            
             Piece
             ,
             with
             Turns
             ,
             Pendulum
             ,
             and
             Motions
             ,
             as
             before
             .
          
           
             
               
                 
                   The
                   Watch.
                   
                
                 
                   The
                   Clock
                   .
                
              
               
                 
                   8
                   )
                   60
                
                 
                   8
                   )
                   96
                
              
               
                 
                   8
                   )
                   56
                
                 
                   8
                   )
                   88
                   —
                   27
                   )
                   12
                
              
               
                 
                   8
                   )
                   48
                
                 
                   8
                   )
                   64
                   —
                   16
                   pins
                
              
               
                 
                   6
                   )
                   45
                   —
                   48
                   )
                   48
                   —
                   6
                   )
                   72
                
                 
                   6
                   )
                   48
                   Double
                   Hoop
                
              
               
                 
                   5
                   )
                   40
                
                 
                   6
                   )
                   48
                
              
               
                 
                   30
                
                 
                    
                
              
            
          
           
           
             §
             7.
             
             A
             
               Year
               Piece
            
             ,
             of
             384
             days
             ,
             with
             Turns
             ,
             Pendulum
             ,
             and
             Motions
             ,
             as
             before
             .
          
           
             
               
                 
                   The
                   Watch.
                   
                
                 
                   The
                   Clock
                   .
                
              
               
                 
                   12
                   )
                   108
                
                 
                   10
                   )
                   120
                
              
               
                 
                   9
                   )
                   72
                
                 
                   8
                   )
                   96
                   —
                   36
                   )
                   9
                
              
               
                 
                   8
                   )
                   64
                
                 
                   6
                   )
                   78
                   26
                   pins
                
              
               
                 
                   8
                   )
                   60
                   —
                   48
                   )
                   48
                   ▪
                   6
                   )
                   72
                
                 
                   6
                   )
                   72
                   Double
                   Hoop
                
              
               
                 
                   7
                   )
                   56
                
                 
                   6
                   )
                   60
                
              
               
                 
                   30
                
                 
                    
                
              
            
          
           
             If
             you
             had
             rather
             have
             the
             Pinion
             of
             Report
             ,
             on
             the
             Spindle
             of
             the
             Pin-wheel
             ,
             it
             must
             be
             13
             )
             39.
             
          
           
             §
             8.
             
             A
             Piece
             of
             30
             Hours
             ,
             Pendulum
             about
             6
             inches
             .
          
           
             
               
                 
                   The
                   Watch.
                   
                
                 
                   The
                   Clock
                   .
                
              
               
                 
                   12
                   )
                   48
                
                 
                   8
                   )
                   48
                
              
               
                 
                   6
                   )
                   78
                
                 
                   6
                   )
                   78
                   13
                   pins
                
              
               
                 
                   6
                   )
                   60
                
                 
                   6
                   )
                   60
                
              
               
                 
                   6
                   )
                   42
                
                 
                   6
                   )
                   48
                
              
               
                 
                   15
                
                 
                    
                
              
            
          
           
             §
             9.
             
             A
             Piece
             of
             8
             days
             ,
             with
             16
             turns
             ,
             Pendulum
             about
             6
             inches
             ,
             to
             shew
             Minutes
             ,
             Seconds
             ,
             &c.
             
          
           
           
             
               
                 
                   The
                   Watch.
                   
                
                 
                   The
                   Clock
                   may
                   be
                   the
                   same
                   with
                   the
                   8
                   day
                   piece
                   before
                   ,
                   §
                   2.
                   
                
              
               
                 
                   8
                   )
                   96
                
              
               
                 
                   8
                   )
                   64
                   —
                   48
                   )
                   48
                   —
                   6
                   )
                   72
                
              
               
                 
                   8
                   )
                   60
                
              
               
                 
                   8
                   )
                   40
                   The
                   Seconds
                   Wheel
                   .
                
              
               
                 
                   15
                
              
            
          
           
             §
             10.
             
             A
             
               Month
               Piece
            
             of
             32
             days
             ,
             with
             Pendulum
             ,
             Turns
             ,
             and
             Motions
             ,
             as
             the
             last
             .
          
           
             
               
                 
                   The
                   Watch.
                   
                
                 
                   The
                   Clock
                   may
                   have
                   the
                   same
                   numbers
                   ,
                   as
                   the
                   Clock
                   §
                   3.
                   
                
              
               
                 
                   8
                   )
                   64
                
              
               
                 
                   8
                   )
                   48
                
              
               
                 
                   6
                   )
                   48
                   —
                   48
                   )
                   48
                   —
                   6
                   )
                   72
                
              
               
                 
                   6
                   )
                   45
                
              
               
                 
                   6
                   )
                   30
                   Seconds
                   Wheel
                   .
                
              
               
                 
                   15
                
              
            
          
           
             §
             11.
             
             A
             
               Year
               Piece
            
             of
             384
             days
             ,
             with
             Pendulum
             ,
             Turns
             ,
             &c.
             as
             the
             last
             .
          
           
           
             
               
                 The
                 Watch
                 part
                 .
              
               
                 
                   10
                   )
                   90
                
                 
                   Or
                   thus
                   ▪
                   with
                   a
                   Wheel
                   less
                   ,
                   and
                   not
                   to
                   shew
                   Minutes
                   and
                   Seconds
                   .
                
              
               
                 
                   8
                   )
                   64
                
                 
                    
                
              
               
                 
                   7
                   )
                   56
                
                 
                   8
                   )
                   96
                
              
               
                 
                   6
                   )
                   48
                
                 
                   6
                   )
                   72
                   —
                   36
                   )
                   9
                
              
               
                 
                   6
                   )
                   45
                   —
                   48
                   )
                   48
                   —
                   6
                   )
                   72
                
                 
                   6
                   )
                   66
                
              
               
                 
                   6
                   )
                   30
                
                 
                   6
                   )
                   60
                
              
               
                 
                   Seconds
                   Wheel
                   .
                
                 
                   6
                   )
                   54
                
              
               
                 
                   15
                
                 
                   19
                
              
            
          
           
             In
             the
             latter
             of
             these
             two
             Numbers
             ,
             the
             Pinion
             of
             Report
             is
             36
             ,
             on
             the
             Second
             Wheel
             .
             The
             Dial
             Wheel
             is
             9.
             
          
           
             The
             Clock-part
             may
             have
             the
             same
             Numbers
             ,
             as
             the
             Year-piece
             before
             §
             7.
             
          
           
             §
             12.
             
             An
             8
             Day
             Piece
             ,
             to
             shew
             the
             Hour
             and
             Minute
             ,
             Pend.
             about
             3
             inches
             long
             .
          
           
             
               
                 
                   6
                   )
                   96
                
                 
                   The
                   Clock
                   may
                   have
                   the
                   same
                   numbers
                   ,
                   as
                   the
                   8
                   day
                   piece
                   before
                   ,
                   §
                   2.
                   
                
              
               
                 
                   8
                   )
                   64
                   —
                   6
                   )
                   72
                
              
               
                 
                   7
                   )
                   49
                
              
               
                 
                   6
                   )
                   36
                
              
               
                 
                   19
                
              
            
          
        
         
           
           
             Automata
             shewing
             the
             Motion
             of
             the
             Celestial
             Bodies
             .
          
           
             §
             1.
             
             Numbers
             for
             the
             Motion
             of
             the
             Sun
             and
             Moon
             .
             See
             before
             in
             Chap.
             2.
             
             Sect.
             5.
             
             §
             3
             ,
             4.
             
          
           
             §
             2.
             
             Numbers
             to
             shew
             the
             Revolution
             of
             the
             Planet
             Saturn
             ,
             which
             consists
             of
             10759
             days
             .
          
           
             
               
                 
                   On
                   the
                   Dial-wheel
                   .
                
                 
                   If
                   you
                   would
                   make
                   it
                   depend
                   upon
                   a
                   Wheel
                   going
                   round
                   in
                   a
                   year
                   ,
                   thus
                   .
                
              
               
                 
                   5
                   )
                   69
                
              
               
                 
                   4
                   )
                   52
                
              
               
                 
                   4
                   )
                   48
                
                 
                   10
                   )
                   59
                   or
                   thus
                   ,
                   4
                   )
                   118
                
              
               
                 
                   4
                   )
                   40
                
                 
                   6
                   )
                   30
                
              
            
          
           
             Note
             ,
             The
             lowermost
             Pinion
             in
             these
             ,
             and
             the
             following
             numbers
             ,
             is
             to
             be
             fixed
             concentrical
             to
             the
             Wheel
             ,
             which
             is
             to
             drive
             the
             Motion
             ,
             viz.
             the
             Dial-wheel
             ,
             Year-wheel
             ,
             or
             &c.
             
          
           
             §
             3.
             
             Numbers
             for
             the
             Planet
             Jupiter
             ,
             whose
             Revolution
             is
             4332
             ½
             days
             .
             On
             the
             Dial-wheel
             .
          
           
             
               
                 
                   4
                   )
                   48
                
                 
                   Or
                   thus
                   ,
                   on
                   the
                   Year-wheel
                   .
                
              
               
                 
                   4
                   )
                   40
                
                 
                   6
                   )
                   71
                
              
               
                 
                   4
                   )
                   36
                
                 
                    
                
              
               
                 
                   4
                   )
                   32
                
                 
                    
                
              
            
          
           
             Note
             here
             ,
             That
             the
             two
             last
             numbers
             
             of
             Saturn
             ,
             may
             be
             the
             two
             first
             of
             Jupiter
             also
             .
          
           
             By
             the
             permission
             of
             my
             ingenious
             friend
             Mr
             Flamsteed
             ,
             I
             here
             insert
             a
             description
             of
             Mr
             
               Olaus
               Romer
            
             ,
             the
             French
             King
             's
             Mathematician's
             Instrument
             ,
             to
             represent
             the
             Motion
             of
             
             Jupiter's
             Satellites
             ;
             a
             copy
             of
             which
             he
             sent
             to
             Mr
             Flamsteed
             in
             1679.
             
          
           
             Upon
             an
             axis
             (
             which
             turns
             round
             once
             in
             7
             days
             )
             are
             four
             Wheels
             fixed
             :
             one
             of
             87
             teeth
             ;
             a
             second
             of
             63
             ;
             the
             third
             42
             ;
             and
             the
             last
             of
             28
             teeth
             .
             On
             another
             axis
             run
             4
             other
             Wheels
             (
             or
             Pinions
             you
             may
             call
             them
             )
             which
             are
             driven
             by
             the
             asoresaid
             Wheels
             .
             The
             first
             is
             a
             Wheel
             ,
             or
             Pinion
             of
             22
             leaves
             ,
             driven
             by
             the
             Wheel
             87
             ,
             which
             carrieth
             round
             the
             first
             Satellite
             .
             The
             second
             is
             32
             ,
             driven
             by
             the
             Wheel
             63
             ,
             which
             carrieth
             round
             the
             second
             Satellite
             .
             The
             third
             hath
             43
             leaves
             ,
             driven
             by
             the
             Wheel
             42
             ,
             which
             carrieth
             the
             third
             Satellite
             .
             And
             lastly
             ,
             is
             the
             Pinion
             67
             ,
             driven
             by
             the
             Wheel
             28
             ,
             which
             carrieth
             round
             the
             fourth
             Satellite
             .
          
           
             On
             the
             first
             axis
             is
             an
             Index
             ,
             that
             pointeth
             to
             a
             circle
             divided
             into
             168
             parts
             ,
             which
             are
             the
             hours
             in
             7
             days
             .
          
           
           
             On
             the
             other
             axis
             all
             the
             Pinions
             run
             concentrically
             ,
             by
             means
             of
             their
             being
             hollow
             in
             the
             middle
             .
             In
             the
             midst
             of
             them
             all
             ,
             the
             axis
             of
             Jupiter
             himself
             is
             fixed
             ,
             with
             a
             little
             Ball
             at
             the
             top
             ,
             representing
             
             Jupiter's
             body
             .
             On
             the
             ends
             of
             4
             small
             Wires
             ,
             fixed
             in
             the
             four
             several
             Sockets
             of
             the
             aforesaid
             Pinions
             ,
             may
             4
             lesser
             Globules
             be
             placed
             (
             at
             their
             due
             distance
             from
             
             Jupiter's
             Globule
             )
             to
             represent
             the
             4
             Satellites
             going
             round
             that
             Planet
             .
          
           
             §
             4.
             
             Numbers
             for
             Mars
             ,
             whose
             Revolution
             is
             1
             year
             322
             days
             .
          
           
             
               
                 
                   On
                   the
                   Dial-wheel
                   ▪
                
                 
                    
                
              
               
                 
                   4
                   )
                   48
                
                 
                   The
                   two
                   last
                   Numbers
                   of
                   Saturn
                   may
                   be
                   the
                   two
                   first
                   of
                   Mars
                   also
                   .
                
              
               
                 
                   4
                   )
                   40
                
              
               
                 
                   4
                   )
                   45
                
              
            
          
           
             §
             5.
             
             Numbers
             for
             Venus
             ,
             whose
             Revolution
             is
             in
             224
             days
             .
          
           
             
               
                 
                   On
                   the
                   Dial-wheel
                   .
                
                 
                    
                
              
               
                 
                   4
                   )
                   32
                
                 
                   Note
                   ,
                   The
                   last
                   number
                   of
                   Jupiter
                   may
                   be
                   the
                   first
                   of
                   Venus
                   .
                
              
               
                 
                   4
                   )
                   32
                
              
               
                 
                   4
                   )
                   28
                
              
            
          
           
             §
             6.
             
             Numbers
             for
             Mercury
             ,
             whose
             Revolution
             is
             near
             88
             days
             .
          
           
           
             
               
                 
                   On
                   the
                   Dial-wheel
                   .
                
              
               
                 
                   4
                   )
                   56
                
              
               
                 
                   4
                   )
                   52
                
              
            
          
           
             §
             7.
             
             Numbers
             to
             represent
             the
             Motion
             of
             the
             
             Dragon's
             Head
             and
             Tail
             ,
             (
             near
             19
             years
             )
             to
             shew
             the
             Eclipses
             of
             the
             Sun
             and
             Moon
             .
          
           
             
               
                 
                   On
                   the
                   Dial-wheel
                   .
                
                 
                   On
                   the
                   Year-wheel
                   .
                
              
               
                 
                   4
                   )
                   48
                
                 
                   4
                   )
                   76
                
              
               
                 
                   4
                   )
                   40
                
                 
                   Note
                   ,
                   The
                   two
                   last
                   numbers
                   of
                   Saturn
                   may
                   be
                   the
                   two
                   first
                   of
                   this
                   on
                   the
                   Dial-wheel
                   .
                
              
               
                 
                   4
                   )
                   44
                
              
               
                 
                   4
                   )
                   42
                
              
            
          
           
             As
             to
             the
             placing
             these
             several
             Motions
             on
             the
             Dial-plate
             ,
             I
             shall
             leave
             it
             wholly
             to
             the
             Work-mans
             contrivance
             .
             He
             may
             perhaps
             make
             them
             to
             represent
             the
             Copernican
             ,
             or
             some
             other
             Sys●em
             .
          
        
         
           
             Numbers
             for
             Pocket
             ▪
             Watches
             .
          
           
             §
             1.
             
             A
             Watch
             to
             go
             8
             Days
             ,
             with
             1●
             turns
             ,
             to
             shew
             Minutes
             and
             Seconds
             ,
             the
             Train
             16000.
             
          
           
             
               
                 
                   6
                   )
                   96
                
                 
                    
                
              
               
                 
                   6
                   )
                   48
                   —
                   12
                   )
                   48
                   —
                   12
                   )
                   36.
                   
                
              
               
                 
                   6
                   )
                   45
                
                 
                   On
                   the
                   Wheel
                   [
                   42
                   ]
                   is
                   the
                   Second's
                   hand
                   placed
                   ,
                   and
                   on
                   the
                   Wheel
                   [
                   48
                   ]
                   the
                   Minute
                   hand
                   .
                
              
               
                 
                   6
                   )
                   42
                
              
               
                 
                   19
                
              
            
          
           
           
             §
             2.
             
             Another
             of
             the
             same
             ,
             without
             Minutes
             and
             Seconds
             ,
             to
             go
             with
             only
             8
             turns
             .
          
           
             
               
                 
                   20
                   )
                   10
                
              
               
                 
                   6
                   )
                   66
                
              
               
                 
                   6
                   )
                   60
                
              
               
                 
                   5
                   )
                   50
                
              
               
                 
                   5
                   )
                   45
                
              
               
                 
                   19
                
              
            
          
           
             §
             3.
             
             A
             Pocket-Watch
             of
             32
             Hours
             ,
             with
             8
             turns
             ,
             to
             shew
             Minutes
             and
             Seconds
             ,
             Train
             as
             the
             last
             .
          
           
             
               
                 
                   12
                   )
                   48
                
              
               
                 
                   6
                   )
                   48
                   —
                   12
                   )
                   48
                   —
                   12
                   )
                   36
                
              
               
                 
                   6
                   )
                   45
                
              
               
                 
                   6
                   )
                   42
                   —
                   Seconds
                   Hand
                   .
                
              
               
                 
                   19
                
              
            
          
           
             §
             4.
             
             The
             usual
             Numbers
             of
             30
             hours
             Pendulum
             Watches
             ,
             with
             8
             turns
             ,
             to
             shew
             the
             Hour
             and
             Minute
             .
          
           
             
               
                 
                   12
                   )
                   48
                
              
               
                 
                   6
                   )
                   54
                   —
                   12
                   )
                   48
                   —
                   12
                   )
                   36
                
              
               
                 
                   6
                   )
                   48
                
              
               
                 
                   6
                   )
                   45
                
              
               
                 
                   15
                
              
            
          
           
           
             §
             5.
             
             The
             usual
             Numbers
             of
             the
             old
             30
             hours
             Pocket-watches
             .
          
           
             
               
                 
                   With
                   5
                   Wheels
                   .
                
                 
                   With
                   4
                   Wheels
                   .
                
              
               
                 
                   10
                   )
                   30
                
                 
                   6
                   )
                   32
                
              
               
                 
                   7
                   )
                   63
                
                 
                   6
                   )
                   66
                
              
               
                 
                   6
                   )
                   42
                
                 
                   5
                   )
                   50
                
              
               
                 
                   6
                   )
                   36
                
                 
                   5
                   )
                   45
                
              
               
                 
                   6
                   )
                   32
                
                 
                   13
                
              
               
                 
                   15
                
              
            
          
           
             If
             any
             of
             the
             Numbers
             of
             the
             preceding
             Wheels
             and
             Pinions
             should
             not
             please
             the
             Reader
             ,
             he
             may
             easily
             correct
             them
             to
             his
             mind
             ,
             by
             the
             Instructions
             in
             the
             foregoing
             Book
             .
             The
             way
             in
             short
             is
             this
             :
             Divide
             the
             Wheel
             by
             the
             Pinion
             ,
             and
             so
             find
             the
             number
             of
             turns
             ,
             according
             to
             Chap.
             2.
             
             Sect.
             1.
             
             §
             2.
             
             Multiply
             the
             Pinion
             you
             like
             better
             ,
             by
             this
             number
             of
             turns
             ,
             and
             the
             Product
             is
             the
             Wheel
             .
             Thus
             in
             the
             8
             day
             Pocket-watch
             §
             1
             ,
             if
             you
             think
             the
             Great-wheel
             too
             large
             ,
             you
             make
             it
             instead
             of
             6
             )
             96
             (
             16
             thus
             ,
             viz.
             5
             )
             80
             (
             16
             :
             
               i.
               e.
            
             chusing
             the
             Pinion
             only
             5
             ,
             and
             multiplying
             it
             by
             16
             (
             the
             turns
             )
             the
             Wheel
             will
             be
             80.
             
          
        
      
       
         
         
           CHAP.
           XI
        
         
           
             Tables
             of
             Time
             relating
             to
             Watch-work
             .
          
           
             
               
                 A
                 Table
                 of
                 Time.
                 
              
               
                 
                   Seconds
                   .
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   60
                
                 
                   Minutes
                   .
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   3600
                
                 
                   60
                
                 
                   Hours
                   .
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   86400
                
                 
                   1440
                
                 
                   24
                
                 
                   Day
                   .
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   604800
                
                 
                   10080
                
                 
                   168
                
                 
                   7
                
                 
                   Week
                   .
                
                 
                    
                
                 
                    
                
              
               
                 
                   2592000
                
                 
                   43200
                
                 
                   720
                
                 
                   30
                
                 
                   4
                
                 
                   Month.
                   
                
                 
                    
                
              
               
                 
                   31536000
                
                 
                   525600
                
                 
                   8760
                
                 
                   365
                
                 
                   52
                
                 
                   12
                
                 
                   Year
                   .
                
              
            
          
           
             The
             foregoing
             Table
             will
             be
             of
             good
             use
             in
             Calculation
             ,
             for
             the
             ready
             finding
             out
             the
             parts
             of
             Time
             :
             which
             is
             thus
             .
             Find
             the
             parts
             of
             time
             you
             seek
             for
             ,
             the
             number
             in
             the
             concurrence
             of
             Squares
             ,
             is
             the
             answer
             to
             your
             question
             .
             Thus
             ,
             suppose
             you
             seek
             for
             the
             number
             of
             Seconds
             
             ●n
             a
             Year
             :
             in
             the
             Square
             under
             Seconds
             ,
             and
             in
             the
             same
             line
             with
             Year
             (
             which
             is
             the
             ●owermost
             Square
             on
             the
             left
             hand
             )
             is
             the
             number
             sought
             ,
             viz.
             315
             ,
             &c.
             
             So
             Minutes
             in
             a
             Month
             ,
             are
             43200.
             
          
           
             If
             you
             would
             know
             any
             number
             ,
             where
             there
             is
             the
             addition
             of
             an
             odd
             number
             to
             it
             ,
             as
             the
             Seconds
             in
             a
             Month
             and
             one
             day
             ;
             add
             the
             Seconds
             in
             a
             month
             (
             which
             are
             259
             —
             )
             and
             the
             Seconds
             in
             a
             Day
             (
             which
             are
             86
             —
             )
             and
             you
             have
             the
             number
             sought
             ,
             viz.
             2678400.
             
          
           
             
               
                 A
                 Table
                 to
                 set
                 a
                 Watch
                 by
                 the
                 Fixed
                 Stars
                 ▪
              
               
                 
                   Night
                
                 
                   Hour
                   .
                
                 
                   Min.
                   
                
                 
                   Sec.
                   
                
                 
                   Night
                
                 
                   Hour
                   .
                
                 
                   Min.
                   
                
                 
                   Sec.
                   
                
              
               
                 
                   1
                
                 
                   0
                
                 
                   3
                
                 
                   57
                
                 
                   16
                
                 
                   1
                
                 
                   3
                
                 
                   20
                
              
               
                 
                   2
                
                 
                   0
                
                 
                   7
                
                 
                   54
                
                 
                   17
                
                 
                   1
                
                 
                   7
                
                 
                   17
                
              
               
                 
                   3
                
                 
                   0
                
                 
                   11
                
                 
                   51
                
                 
                   18
                
                 
                   1
                
                 
                   11
                
                 
                   14
                
              
               
                 
                   4
                
                 
                   0
                
                 
                   15
                
                 
                   47
                
                 
                   19
                
                 
                   1
                
                 
                   15
                
                 
                   11
                
              
               
                 
                   5
                
                 
                   0
                
                 
                   19
                
                 
                   44
                
                 
                   20
                
                 
                   1
                
                 
                   19
                
                 
                   8
                
              
               
                 
                   6
                
                 
                   0
                
                 
                   23
                
                 
                   41
                
                 
                   21
                
                 
                   1
                
                 
                   23
                
                 
                   5
                
              
               
                 
                   7
                
                 
                   0
                
                 
                   27
                
                 
                   38
                
                 
                   22
                
                 
                   1
                
                 
                   27
                
                 
                   1
                
              
               
                 
                   8
                
                 
                   0
                
                 
                   31
                
                 
                   35
                
                 
                   23
                
                 
                   1
                
                 
                   30
                
                 
                   58
                
              
               
                 
                   9
                
                 
                   0
                
                 
                   35
                
                 
                   32
                
                 
                   24
                
                 
                   1
                
                 
                   34
                
                 
                   55
                
              
               
                 
                   10
                
                 
                   0
                
                 
                   39
                
                 
                   29
                
                 
                   25
                
                 
                   1
                
                 
                   38
                
                 
                   52
                
              
               
                 
                   11
                
                 
                   0
                
                 
                   43
                
                 
                   26
                
                 
                   26
                
                 
                   1
                
                 
                   42
                
                 
                   49
                
              
               
                 
                   12
                
                 
                   0
                
                 
                   47
                
                 
                   23
                
                 
                   27
                
                 
                   1
                
                 
                   46
                
                 
                   46
                
              
               
                 
                   13
                
                 
                   0
                
                 
                   51
                
                 
                   29
                
                 
                   28
                
                 
                   1
                
                 
                   50
                
                 
                   43
                
              
               
                 
                   14
                
                 
                   0
                
                 
                   55
                
                 
                   26
                
                 
                   29
                
                 
                   1
                
                 
                   54
                
                 
                   40
                
              
               
                 
                   15
                
                 
                   0
                
                 
                   59
                
                 
                   23
                
                 
                   30
                
                 
                   1
                
                 
                   58
                
                 
                   36
                
              
            
          
        
         
           
           
             Explanation
             of
             the
             Table
             .
          
           
             This
             Table
             shews
             how
             much
             the
             Sidereal
             ,
             goeth
             faster
             than
             the
             Solar
             day
             ,
             in
             any
             number
             of
             nights
             for
             a
             month
             .
             So
             that
             observing
             by
             your
             Watch
             ,
             the
             nice
             time
             when
             any
             fixed
             Star
             cometh
             to
             the
             Meridian
             ,
             or
             any
             other
             point
             of
             the
             Heavens
             :
             if
             after
             one
             Revolution
             of
             that
             same
             Star
             to
             the
             same
             point
             ,
             your
             Watch
             goeth
             3′
             .
             57″
             ●lower
             than
             the
             Star
             ;
             or
             after
             two
             nights
             7′
             .
             54″
             ;
             or
             16
             nights
             ,
             1
             h.
             3′
             .
             20″
             ,
             &c.
             then
             doth
             your
             Watch
             keep
             time
             rightly
             with
             the
             Mean
             motion
             of
             the
             Sun.
             If
             it
             vary
             from
             the
             Table
             ,
             you
             must
             alter
             the
             length
             of
             your
             Pend
             ▪
             to
             make
             it
             so
             keep
             time
             .
          
           
             To
             observe
             the
             time
             nicely
             ,
             when
             the
             Star
             cometh
             again
             to
             the
             same
             point
             of
             the
             heavens
             ,
             't
             is
             necessary
             to
             make
             the
             observation
             with
             a
             Telescope
             ,
             that
             hath
             cross
             threads
             in
             the
             focus
             of
             the
             object-glass
             ;
             and
             so
             leaving
             the
             Telescope
             fixed
             in
             the
             same
             posture
             ,
             till
             a
             second
             Observation
             .
             You
             may
             do
             this
             with
             the
             telescopular
             sights
             of
             a
             Quadrant
             ,
             or
             Sextans
             ,
             and
             so
             leaving
             it
             standing
             until
             anoher
             night
             of
             Observation
             .
             Or
             for
             want
             of
             this
             more
             nice
             way
             ,
             you
             may
             do
             it
             by
             looking
             along
             by
             the
             edge
             of
             two
             Strings
             ,
             suspended
             with
             Plumbets
             ,
             
             in
             a
             room
             ,
             at
             some
             distance
             from
             one
             another
             .
             Or
             by
             looking
             at
             the
             edge
             of
             a
             Chimney
             ,
             &c.
             as
             Mr
             Watson
             hath
             directed
             ,
             at
             the
             end
             of
             Mr
             
             Smith's
             
               Horo●
               .
               Disquis
            
             .
             But
             to
             make
             a
             tolerable
             observation
             any
             of
             these
             last
             ways
             ,
             't
             is
             necessary
             to
             have
             a
             Candle
             shine
             upon
             the
             edge
             of
             the
             furthermost
             String
             ,
             or
             Chimney
             ;
             without
             which
             you
             cannot
             see
             exactly
             when
             the
             Star
             cometh
             thereto
             .
          
           
             
               
                 A
                 Table
                 shewing
                 the
                 Variations
                 made
                 in
                 the
                 true
                 Hour
                 of
                 the
                 Day
                 ,
                 by
                 the
                 Refraction
                 of
                 ▪
                 the
                 Sun
                 in
                 the
                 Equator
                 ,
                 and
                 both
                 the
                 Solstices
                 .
              
               
                 
                   Sun's
                   altitude
                   .
                   Deg.
                   
                
                 
                   Sun's
                   Refraction
                   .
                
                 
                   Variation
                   at
                   the
                   N.
                   Solstice
                   .
                
                 
                   Variation
                   at
                   the
                   Equator
                   .
                
                 
                   Variation
                   at
                   the
                   S.
                   Solstice
                   .
                
              
               
                 
                   ′
                
                 
                   ″
                
                 
                   ′
                
                 
                   ″
                
                 
                   ′
                
                 
                   ″
                
                 
                   ′
                
                 
                   ″
                
              
               
                 
                   00
                
                 
                   33
                
                 
                   00
                
                 
                   4
                
                 
                   34
                
                 
                   3
                
                 
                   32
                
                 
                   4
                
                 
                   38
                
              
               
                 
                   1
                
                 
                   23
                
                 
                   00
                
                 
                   2
                
                 
                   34
                
                 
                   2
                
                 
                   28
                
                 
                   3
                
                 
                   19
                
              
               
                 
                   2
                
                 
                   17
                
                 
                   00
                
                 
                   2
                
                 
                   24
                
                 
                   1
                
                 
                   49
                
                 
                   2
                
                 
                   31
                
              
               
                 
                   3
                
                 
                   13
                
                 
                   30
                
                 
                   1
                
                 
                   46
                
                 
                   1
                
                 
                   27
                
                 
                   2
                
                 
                   3
                
              
               
                 
                   4
                
                 
                   11
                
                 
                   30
                
                 
                   1
                
                 
                   29
                
                 
                   1
                
                 
                   12
                
                 
                   1
                
                 
                   40
                
              
               
                 
                   5
                
                 
                   9
                
                 
                   30
                
                 
                   1
                
                 
                   12
                
                 
                   1
                
                 
                   1
                
                 
                   1
                
                 
                   33
                
              
               
                 
                   6
                
                 
                   7
                
                 
                   30
                
                 
                   0
                
                 
                   56
                
                 
                   0
                
                 
                   49
                
                 
                   1
                
                 
                   17
                
              
               
                 
                   7
                
                 
                   7
                
                 
                   00
                
                 
                   0
                
                 
                   52
                
                 
                   0
                
                 
                   44
                
                 
                   1
                
                 
                   16
                
              
               
                 
                   8
                
                 
                   6
                
                 
                   00
                
                 
                   0
                
                 
                   43
                
                 
                   0
                
                 
                   39
                
                 
                   1
                
                 
                   8
                
              
               
                 
                   9
                
                 
                   5
                
                 
                   00
                
                 
                   0
                
                 
                   36
                
                 
                   0
                
                 
                   34
                
                 
                   1
                
                 
                   2
                
              
               
                 
                   10
                
                 
                   4
                
                 
                   40
                
                 
                   0
                
                 
                   25
                
                 
                   0
                
                 
                   29
                
                 
                   1
                
                 
                   2
                
              
            
          
        
         
           
           
             Remarks
             upon
             the
             Table
             .
          
           
             The
             Column
             of
             the
             Sun's
             Refractions
             ,
             I
             owe
             to
             that
             accurate
             observer
             of
             the
             celestial
             motions
             ,
             Mr
             ▪
             Flamsteed
             .
             Which
             Refractions
             ,
             altho
             in
             the
             Table
             the
             same
             ,
             yet
             do
             differ
             at
             different
             seasons
             of
             the
             year
             ,
             nay
             perhaps
             ,
             according
             to
             the
             different
             temperature
             of
             the
             air
             sometimes
             ,
             in
             the
             same
             day
             .
             Thus
             Mr
             Flamsteed
             found
             the
             Refractions
             in
             February
             ,
             very
             different
             from
             those
             in
             April
             :
             and
             it
             is
             observed
             ,
             that
             the
             Refractions
             are
             commonly
             greater
             ,
             when
             the
             Mercury
             is
             higher
             in
             the
             Barometer
             .
          
           
             The
             Table
             therefore
             doth
             not
             shew
             what
             the
             Refractions
             always
             are
             ,
             but
             only
             about
             the
             middle
             quantity
             of
             them
             ,
             at
             every
             degree
             ,
             of
             the
             10
             first
             of
             the
             Sun's
             altitude
             .
             And
             accordingly
             I
             have
             calculated
             the
             Variations
             thereby
             made
             in
             the
             hour
             of
             the
             day
             .
          
           
             These
             Variations
             of
             the
             hour
             are
             greater
             or
             lesser
             ,
             according
             as
             the
             angle
             of
             the
             Sun
             's
             diurnal
             motion
             is
             acuter
             with
             the
             horizon
             .
             The
             reason
             is
             plain
             ;
             because
             as
             the
             Sun
             appears
             by
             refraction
             higher
             than
             
             really
             he
             is
             ;
             so
             this
             false
             height
             doth
             affect
             the
             hours
             in
             Winter
             ,
             more
             than
             the
             Summer
             half
             year
             .
          
           
             There
             is
             no
             ray
             indeed
             of
             the
             Sun
             ,
             but
             what
             cometh
             refracted
             to
             a
             Sun
             ▪
             dial
             ▪
             and
             consequently
             ,
             there
             is
             no
             Dial
             but
             what
             goeth
             more
             or
             less
             false
             (
             except
             at
             Noon
             in
             Dials
             that
             cast
             a
             Shade
             ,
             where
             the
             refraction
             makes
             no
             variation
             ▪
             )
             But
             the
             Refraction
             decreaseth
             apace
             ,
             as
             the
             Sun
             gets
             higher
             ,
             and
             causeth
             a
             variation
             of
             not
             above
             half
             a
             minute
             ,
             at
             10
             degrees
             of
             the
             Sun's
             altitude
             ;
             except
             when
             the
             Sun
             is
             in
             ,
             or
             near
             the
             Southern
             Tropick
             .
             Nearer
             than
             half
             a
             minute
             ,
             few
             common
             Sun-dials
             shew
             the
             time
             .
             And
             therefore
             ,
             partly
             for
             this
             reason
             ,
             and
             partly
             ,
             because
             Mr.
             
             Flamsteed's
             observations
             reach
             not
             much
             farther
             ,
             I
             have
             calculated
             my
             Table
             to
             only
             10
             degrees
             .
          
           
             The
             Table
             needs
             little
             explication
             .
             For
             having
             the
             Sun's
             height
             ,
             you
             have
             against
             it
             ,
             in
             the
             next
             Column
             ,
             the
             Refraction
             ▪
             and
             in
             the
             3
             next
             the
             alterations
             of
             the
             hour
             ,
             at
             3
             times
             of
             the
             year
             .
             Taking
             therefore
             by
             a
             Quadrant
             the
             Sun's
             altitude
             ,
             and
             observe
             at
             the
             same
             time
             ,
             the
             hour
             of
             the
             day
             by
             a
             Sun-dial
             ,
             by
             the
             Table
             ,
             
             you
             see
             how
             many
             minutes
             ,
             and
             seconds
             ,
             the
             Dial
             is
             too
             fast
             .
             As
             at
             the
             Sun-rising
             a
             Sun-dial
             is
             too
             fast
             4′
             .
             34″
             ,
             about
             June
             11
             ,
             and
             3′
             .
             32″
             ,
             about
             Mar.
             10.
             and
             Sept.
             12
             ,
             and
             4′
             38″
             about
             Dec.
             11.
             
          
        
      
    
     
       
         
         
           Addenda
           .
        
         
           TO
           the
           Fifth
           part
           of
           the
           Rule
           in
           §
           6.
           p.
           21.
           
           If
           you
           have
           occasion
           to
           lay
           the
           Pinion
           of
           Report
           upon
           any
           other
           Wheel
           ,
           and
           not
           the
           great-Wheel
           ,
           you
           may
           do
           it
           by
           this
           Rule
           ,
           
             As
             the
             Beats
             in
             one
             turn
             of
             any
             Wheel
             ;
             To
             the
             Beats
             in
             an
             hour
             :
             :
             So
             are
             the
             hours
             of
             the
             Dial
             ;
             To
             the
             Quotient
             of
             the
             Hour-wheel
             divided
             by
             the
             Pinion
             of
             Report
             .
          
        
         
           To
           page
           66.
           
           Suppose
           in
           altering
           an
           old
           Watch
           ,
           you
           would
           have
           it
           shew
           minutes
           ,
           as
           well
           as
           hours
           ;
           you
           may
           do
           it
           thus
           :
           Divide
           the
           Beats
           in
           one
           turn
           of
           the
           Great-wheel
           ,
           by
           the
           Beats
           in
           an
           hour
           ;
           the
           Quotient
           will
           shew
           in
           how
           many
           hours
           the
           Great-wheel
           goeth
           round
           once
           .
           If
           the
           Beats
           in
           the
           Great
           wheel
           exceed
           the
           Train
           ,
           you
           must
           chuse
           your
           Minute-wheel
           first
           ,
           and
           multiply
           it
           by
           the
           Quotient
           ;
           this
           will
           give
           the
           Pin.
           of
           Report
           .
           But
           if
           the
           Train
           exceeds
           the
           Beats
           of
           the
           Great-wheel
           ,
           you
           must
           chuse
           the
           Pin.
           of
           Rep.
           and
           multiply
           the
           Quotient
           by
           it
           :
           the
           product
           is
           the
           Minute-wheel
           .
        
         
         
           But
           it
           often
           falls
           out
           ,
           that
           the
           Train
           and
           Beats
           of
           the
           Great-wheel
           will
           not
           exactly
           measure
           one
           another
           :
           if
           so
           ,
           the
           best
           way
           is
           to
           half
           the
           two
           numbers
           ,
           as
           far
           as
           they
           will
           equally
           admit
           of
           halfing
           ;
           or
           divide
           them
           by
           some
           common
           divisor
           ,
           and
           so
           having
           brought
           them
           to
           as
           small
           numbers
           as
           you
           can
           ,
           you
           may
           suppose
           them
           to
           be
           a
           Wheel
           and
           Pinion
           ,
           and
           reduce
           them
           to
           lesser
           numbers
           ▪
           by
           Chap.
           2.
           
           Sect.
           2.
           
           §
           5.
           
           Thus
           suppose
           you
           would
           make
           the
           old
           dull
           Movement
           there
           mentioned
           ,
           a
           Minute-watch
           ;
           you
           may
           reduce
           the
           numbers
           of
           the
           Great-wheel
           2188●
           ▪
           and
           the
           Train
           9368
           ,
           to
           a
           Pinion
           and
           Wheel
           28
           )
           12.
           
           Which
           Pin.
           28
           being
           set
           upon
           the
           Spingle
           of
           the
           Gr.
           Wh.
           will
           drive
           a
           Wheel
           12
           round
           once
           in
           an
           hour
           ,
           to
           shew
           Minutes
           .
           If
           you
           make
           this
           ▪
           Wh.
           12
           drive
           another
           of
           48
           ;
           (
           concentrical
           to
           which
           ,
           is
           a
           Pin.
           12
           driving
           a
           Wheel
           36
           (
           which
           Wheel
           is
           concentrical
           with
           the
           Minute-wheel
           )
           this
           will
           carry
           a
           Hand
           round
           in
           12
           hours
           .
           But
           in
           this
           case
           ,
           you
           must
           place
           the
           Pin.
           28
           on
           the
           Spindle
           of
           the
           Gr.
           Wh.
           so
           as
           to
           slide
           round
           stiffly
           ,
           when
           you
           turn
           the
           Minute-hand
           to
           rectifie
           the
           Watch.
           
        
         
           FINIS
           .
        
      
       
         Notes, typically marginal, from the original text
         
           Notes for div A35722-e4360
           
             Oughtred
             of
             Autom
             .
             sect
             .
             4.
             
          
           
             
               
                 
                   4
                   )
                
                 
                   36
                
                 
                   (
                   9
                
              
               
                 
                   5
                   )
                
                 
                   55
                
                 
                   (
                   11
                
              
               
                 
                   5
                   )
                
                 
                   45
                
                 
                   (
                   9
                
              
               
                 
                   5
                   )
                
                 
                   40
                
                 
                   (
                   8
                
              
               
                 
                    
                
                 
                    
                
                 
                   17
                
              
            
          
           
             By
             the
             Quotients
             I
             commonly
             mean
             the
             ●umber
             of
             Turns
             ;
             which
             number
             is
             set
             on
             the
             right
             hand
             ,
             without
             a
             hook
             ,
             as
             is
             shewn
             in
             the
             last
             Paragraph
             .
             Which
             I
             no●e
             ●ere
             now
             once
             for
             all
             .
          
           
             
               
                 
                   5
                   )
                
                 
                   5●
                
                 
                   (
                   11
                
              
               
                 
                   5
                   )
                
                 
                   45
                
                 
                   (
                   9
                
              
               
                 
                   5
                   )
                
                 
                   40
                
                 
                   (
                   8
                
              
            
          
           
             
               
                 
                   8
                   )
                
                 
                   80
                
                 
                   (
                   10
                
              
               
                 
                   6
                   )
                
                 
                   54
                
                 
                   (
                   9
                
              
               
                 
                   5
                   )
                
                 
                   40
                
                 
                   (
                   8
                
              
               
                 
                    
                
                 
                    
                
                 
                   15
                
              
            
          
           
             
               
                 
                   
                     4
                     )
                  
                   
                     32
                  
                   
                     (
                     8
                  
                
                 
                   
                     5
                     )
                  
                   
                     55
                  
                   
                     (
                     11
                  
                
                 
                   
                     5
                     )
                  
                   
                     45
                  
                   
                     (
                     9
                  
                
                 
                   
                     5
                     )
                  
                   
                     40
                  
                   
                     (
                     8
                  
                
                 
                   
                      
                  
                   
                      
                  
                   
                     17
                  
                
              
            
          
           
             Sir
             
               J.
               Moor
               ▪
            
             Mat.
             Com.
             p.
             109.
             
          
           
             Ibid.
             p.
             116.
             
          
           
             Oughtred
             Autom
             .
             Sect.
             14.
             
          
           
             
               
                 
                   
                     28
                     )
                  
                   
                     1440
                  
                
              
            
          
           
             Ought
             .
             ib.
             
          
           
             Id.
             ib.
             
          
           
             
               
                 
                   
                     9
                  
                   
                      
                  
                   
                     8
                  
                
                 
                   
                     36
                  
                   
                     X
                  
                   
                     8
                  
                
                 
                   
                     4
                  
                   
                      
                  
                   
                     1
                  
                
                 
                   
                     32
                  
                   
                     X
                  
                   
                     9
                  
                
              
            
          
           
             Id.
             ib.
             
          
           
             Oughtred
             Sect.
             12.
             
             Sir
             
               J.
               Moor
            
             Ibid.
             p.
             109.
             
          
           
             
               
                 
                   
                     4
                     )
                  
                   
                     36
                  
                   
                     (
                     9
                  
                
                 
                   
                     5
                     )
                  
                   
                     55
                  
                   
                     (
                     11
                  
                
                 
                   
                     5
                     )
                  
                   
                     45
                  
                   
                     (
                     9
                  
                
                 
                   
                     5
                     )
                  
                   
                     40
                  
                   
                     (
                     8
                  
                
                 
                   
                      
                  
                   
                      
                  
                   
                     17
                  
                
              
            
          
           
             Oughtred
             Sect.
             21.
             
          
           
             §
             8.
             
          
           
             
               
                 
                   
                     4
                     )
                  
                   
                     28
                  
                   
                     (
                     7
                  
                
                 
                   
                     5
                     )
                  
                   
                     55
                  
                   
                     (
                     11
                  
                
                 
                   
                     5
                     )
                  
                   
                     45
                  
                   
                     (
                     9
                  
                
                 
                   
                     5
                     )
                  
                   
                     40
                  
                   
                     (
                     8
                  
                
                 
                   
                      
                  
                   
                      
                  
                   
                     17
                  
                
              
            
          
           
             Id.
             ib.
             §
             22.
             
          
           
             
               
                 
                   
                     24
                     )
                  
                   
                     20
                  
                   
                     20
                     /
                     24
                  
                
                 
                   
                     6
                     )
                  
                   
                     60
                  
                   
                     (
                     10
                  
                
                 
                   
                     6
                     )
                  
                   
                     48
                  
                   
                     (
                     8
                  
                
                 
                   
                     5
                     )
                  
                   
                     40
                  
                   
                     (
                     8
                  
                
                 
                   
                     5
                     )
                  
                   
                     33
                  
                   
                     (
                     6⅗
                  
                
                 
                   
                      
                  
                   
                      
                  
                   
                     17
                  
                
              
            
          
           
             Horol
             .
             Disq
             .
          
           
             Sect.
             1.
             
             §
             6.
             
          
           
             
               
                 
                   
                     8
                     )
                  
                   
                     40
                  
                   
                     (
                     5
                  
                
                 
                   
                      
                  
                   
                      
                  
                   
                     15
                  
                
              
            
          
           
             
               
                 
                   
                     8
                     )
                  
                   
                     64
                  
                   
                     (
                     8
                  
                
                 
                   
                     8
                     )
                  
                   
                     60
                  
                   
                     (
                     7½
                  
                
                 
                   
                     8
                     )
                  
                   
                     40
                  
                   
                     (
                     5
                  
                
                 
                   
                      
                  
                   
                      
                  
                   
                     15
                  
                
              
            
          
           
             §
             6.
             
             Par.
             3.
             and
             §
             7.
             
          
           
             
               
                 
                   
                     ●
                     )
                  
                   
                     108
                  
                   
                     (
                     12
                  
                
                 
                   
                     ●
                     )
                  
                   
                     64
                  
                   
                     (
                     8
                  
                
                 
                   
                     ●
                     )
                  
                   
                     60
                  
                   
                     (
                     7½
                  
                
                 
                   
                     ●
                     )
                  
                   
                     40
                  
                   
                     (
                     5
                  
                
                 
                   
                      
                  
                   
                      
                  
                   
                     15
                  
                
              
            
          
           
             Sir
             
               J.
               Moo●
            
             Ibid.
             p.
             116.
             
          
           
             V.
             Sect.
             1.
             
             §
             ▪
             6.
             
          
           
             
               
                 
                   
                     8
                     )
                  
                   
                     96
                  
                   
                     (
                     12
                  
                
                 
                   
                     8
                     )
                  
                   
                     64
                  
                   
                     (
                     8
                  
                
                 
                   
                     8
                     )
                  
                   
                     60
                  
                   
                     (
                     7½
                  
                
                 
                   
                      
                  
                   
                      
                  
                   
                     30
                  
                
              
            
          
           
             
               
                 
                   
                     8
                     )
                  
                   
                     48
                  
                   
                     (
                     6
                  
                
                 
                   
                     6
                     )
                  
                   
                     78
                  
                   
                     (
                     13
                     pins
                  
                
                 
                   
                     6
                     )
                  
                   
                     60
                  
                   
                     (
                     10
                  
                
                 
                   
                     6
                     )
                  
                   
                     48
                  
                   
                     (
                     8
                  
                
              
            
          
           
             §
             1.
             
             Infer
             2●
             
          
           
             
               
                 
                   
                     15
                     )
                  
                   
                     195
                  
                   
                     (
                     13
                  
                
                 
                   
                     13
                     )
                  
                   
                     195
                  
                   
                     (
                     15
                  
                
              
            
          
           
             
               
                 
                   
                     10
                     )
                  
                   
                     65
                  
                   
                     (
                     6●
                  
                
                 
                   
                     8
                     )
                  
                   
                     48
                  
                   
                     (
                     6
                  
                
                 
                   
                     6
                     )
                  
                   
                     48
                  
                   
                     (
                     8
                     pins
                  
                
              
            
          
           
             V.
             Sect.
             1
             ▪
             §.
             3.
             
          
           
             
               
                 
                   
                     8
                     )
                  
                   
                     104
                  
                   
                     (
                     13
                  
                
                 
                   
                     6
                     )
                  
                   
                     72
                  
                   
                     (
                     12.
                     24
                     pins
                  
                
              
            
          
           
             
               
                 
                   
                     12
                     )
                  
                   
                     39
                  
                   
                     (
                     3¼
                  
                
              
            
          
           
             
               
                 
                   
                     10
                     )
                  
                   
                     120
                  
                   
                     (
                     12
                  
                
                 
                   
                     8
                     )
                  
                   
                     96
                  
                   
                     (
                     12
                  
                
                 
                   
                      
                  
                   
                     78
                  
                   
                     (
                     26
                     pins
                  
                
              
            
          
           
             
               
                 
                   
                     13
                     )
                  
                   
                     39
                  
                   
                     (
                     3
                  
                
              
            
          
           
             Ch.
             2.
             
             Sect.
             2.
             
             §
             7.
             
          
           
             Oughtred
             .
             §
             26.
             
          
           
             
               
                 
                   
                     4
                     )
                  
                   
                     62
                  
                   
                     (
                     15½
                  
                
                 
                   
                     5
                     )
                  
                   
                     20
                  
                   
                     (
                     4
                  
                
              
            
          
           
             
               
                 
                   
                     4
                     )
                  
                   
                     62
                  
                   
                     (
                     15½
                  
                
                 
                   
                     10
                  
                   
                     (
                     40
                  
                   
                     (
                     4
                  
                
              
            
          
           
             Id.
             ib.
             
          
           
             
               
                 
                   
                     10
                     )
                  
                   
                     5
                     ,
                     9
                  
                   
                     (
                     ●9
                  
                
                 
                   
                     4
                     )
                  
                   
                     40
                  
                   
                     (
                     10
                  
                
              
            
             
               
                 
                   
                     4
                     )
                  
                   
                     59
                  
                   
                     (
                     14¾
                  
                
                 
                   
                     10
                     )
                  
                   
                     40
                  
                   
                     (
                     4
                  
                
              
            
          
           
             Id.
             ib.
             
          
           
             
               
                 
                   
                     4
                     )
                  
                   
                     73
                  
                   
                     (
                     18¼
                  
                
                 
                   
                     4
                     )
                  
                   
                     40
                  
                   
                     (
                     10
                  
                
                 
                   
                     5
                     )
                  
                   
                     20
                  
                   
                     (
                     4
                  
                
              
            
             
               
                 
                   
                     4
                     )
                  
                   
                     73
                  
                   
                     (
                     18¼
                  
                
                 
                   
                     4
                     )
                  
                   
                     32
                  
                   
                     (
                     8
                  
                
                 
                   
                     4
                     )
                  
                   
                     20
                  
                   
                     (
                     5
                  
                
              
            
          
           
             Autom
             .
             §
             35.
             
             Id.
             ib.
             
          
           
             Mat.
             Com.
             p.
             117.
             
          
           
             V.
             Sect.
             1
             §
             4
             ,
             5.
             
          
           
             
               De
               Subtil
            
             .
             l.
             17.
             
          
        
         
           Notes for div A35722-e11990
           
             
               
                 
                   
                     4
                     )
                  
                   
                     48
                  
                   
                     (
                     12
                  
                
                 
                   
                     7
                     )
                  
                   
                     56
                  
                   
                     (
                     8
                  
                
                 
                   
                     6
                     )
                  
                   
                     54
                  
                   
                     (
                     9
                  
                
                 
                   
                      
                  
                   
                      
                  
                   
                     19
                  
                
              
            
          
           
             
               Horol
               .
               Dis
            
             .
             p.
             54.
             
          
           
             
               
                 
                   
                     2736
                     )
                  
                   
                     9368
                  
                   
                     (
                     3½
                  
                
              
            
          
           
             
               
                 
                   
                     4
                     )
                  
                   
                     48
                  
                   
                     (
                     12
                  
                
                 
                   
                     7
                     )
                  
                   
                     56
                  
                   
                     (
                     8
                  
                
                 
                   
                     6
                     )
                  
                   
                     54
                  
                   
                     (
                     9
                  
                
                 
                   
                     6
                  
                   
                     (
                     2●
                  
                   
                     (
                     3½
                  
                
                 
                   
                      
                  
                   
                      
                  
                   
                     19
                  
                
              
            
          
           
             V.
             Sect.
             1
             ▪
             §
             6.
             
          
           
             
               
                 
                   
                     4
                     )
                  
                   
                     48
                  
                   
                     (
                     12
                  
                
                 
                   
                     7
                     )
                  
                   
                     56
                  
                   
                     (
                     8
                  
                
                 
                   
                     6
                     )
                  
                   
                     54
                  
                   
                     (
                     9
                  
                
                 
                   
                     6
                     )
                  
                   
                     36
                  
                   
                     (
                     6
                  
                
                 
                   
                      
                  
                   
                      
                  
                   
                     11
                  
                
              
            
          
           
             
               
                 
                   
                     6
                     )
                  
                   
                     30
                  
                   
                     (
                     5
                  
                
                 
                   
                     7
                     )
                  
                   
                     56
                  
                   
                     (
                     8
                  
                
                 
                   
                     6
                     )
                  
                   
                     54
                  
                   
                     (
                     9
                  
                
                 
                   
                     6
                     )
                  
                   
                     48
                  
                   
                     (
                     8
                  
                
                 
                   
                      
                  
                   
                      
                  
                   
                     19
                  
                
              
            
          
           
             
               
                 
                   
                     4
                     )
                  
                   
                     39
                  
                   
                     ●¾
                  
                
                 
                   
                     7
                     )
                  
                   
                     56
                  
                   
                     (
                     8
                     pins
                  
                
                 
                   
                     6
                     )
                  
                   
                     54
                  
                   
                     (
                     9
                  
                
                 
                   
                     6
                     )
                  
                   
                     48
                  
                   
                     (
                     8
                  
                
              
            
          
           
             
               
                 
                   
                     5
                     )
                  
                   
                     24
                  
                   
                     (
                     7
                     /
                     18
                     /
                     6
                  
                
                 
                   
                     7
                     )
                  
                   
                     56
                  
                   
                     8.
                     16
                     pins
                  
                
                 
                   
                     6
                     )
                  
                   
                     54
                  
                   
                     (
                     9
                  
                
                 
                   
                     6
                     )
                  
                   
                     48
                  
                   
                     (
                     8
                  
                
              
            
          
        
         
           Notes for div A35722-e13570
           
             Sir
             
               J.
               Moor
            
             Mat.
             Com.
             R.
             5.
             
          
        
         
           Notes for div A35722-e13690
           
             Id.
             ib.
             Rule
             3.
             
          
           
             
               De
               Horol
               .
               Oscil
            
             .
             p.
             10
             ,
             11
             ,
             12.
             
          
           
             Machina
             Pneumat
             .
             Exp.
             26.
             
          
           
             Ibid.
             
          
           
             Ibid.
             
          
           
             Fiugenius
             ●●●i
             supra
             ,
             
               p.
               141.
               
               Sir
            
             J.
             Moor
             ibid.
             
          
           
             Hugen
             ▪
             Moor
             ,
             
               ib●
               ▪
            
          
           
             Horolog
             .
             Disquis
             .
          
           
             Ibid.
             
          
           
             Ibid.
             de
             
               Centro
               Oscil
            
             .
             Prop.
             23.
             
          
        
         
           Notes for div A35722-e15650
           
             2
             
             Kings
             ●0
             ▪
             11.
             
             Isai
             .
             38.
             8.
             
          
           
             Lexic
             .
             in
             verbo
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
          
           
             De
             die
             Natali
             
               c.
               23.
            
             
          
           
             Ibid.
             
          
           
             
               Nat.
               Hist
            
             .
             l.
             2.
             c.
             76.
             
          
           
             
               De
               Archit
            
             .
             l.
             6.
             c.
             48.
             
          
           
             In
             the
             Life
             of
             Dions
          
           
             Euseb
             .
             Vit.
             Const
             ▪
             
               l.
               3.
            
             
          
           
             
               De
               Subtil
               ▪
            
             l.
             17.
             
          
           
             
               De
               Architect
            
             .
             l.
             9.
             c.
             9●
             
          
           
             V●d
             .
             Phi●●nd
             .
             not
             .
             in
             Vitruv.
             
          
           
             Lib.
             1.
             
             §
             25.
             
             
               Edit
               .
               El●ivir
            
             .
          
           
             Epigr.
             in
             Sphoer
             .
             Archimed
             .
             Vid.
             Card.
             de
             Subtil
             .
             
               l.
               17.
            
             
          
           
             
               De
               Nat.
               Deor.
            
             Lib.
             2.
             
             §
             34.
             
          
           
             M●lyneaux
             ,
             Scioth
             .
             Telescop
             .
             Ep.
             Dedic
             .
          
           
             Cosmog
             .
             l.
             2.
             
          
           
             Magia
             Univers
             .
             P.
             1.
             
             Proleg
             :
             &
             Magia
             Thaumaturg
             .
          
        
         
           Notes for div A35722-e17680
           
             
               Hor.
               Oscil
            
             .
             p.
             3
             ▪
             Edit
             .
             Paris
             .
          
           
             p.
             8.
             
          
           
             Exper
             ▪
             made
             in
             the
             Acad
             ▪
             
               del
               Cimento
            
             by
             Mr.
             Wal●er
             ,
             p.
             12.
             
          
           
             Hugen
             .
             ib.
             
          
           
             
               Horolog
               .
               Disquis
            
             .
             p.
             3.