







 
   
     
       
         Of the laws of chance, or, A method of calculation of the hazards of game plainly demonstrated and applied to games at present most in use : which may be easily extended to the most intricate cases of chance imaginable.
         Arbuthnot, John, 1667-1735.
      
       
         
           1692
        
      
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         A25748
         Wing A3602
         ESTC R31565
         12165155
         ocm 12165155
         55286
         
           
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             Of the laws of chance, or, A method of calculation of the hazards of game plainly demonstrated and applied to games at present most in use : which may be easily extended to the most intricate cases of chance imaginable.
             Arbuthnot, John, 1667-1735.
          
           [24], 93 p.
           
             Printed by Benj. Motte, and sold by Randall Taylor ...,
             London :
             1692.
          
           
             Attributed to Arbuthnot by Wing and NUC pre-1956 imprints.
             Reproduction of original in the British Library.
          
        
      
    
     
       
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         eng
      
       
         
           Probabilities.
           Games of chance (Mathematics)
           Game theory.
        
      
    
     
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           OF
           THE
           Laws
           of
           Chance
           ,
           OR
           ,
           A
           METHOD
           OF
           
             Calculation
             of
             the
             Hazards
          
           OF
           GAME
           ,
           Plainly
           demonstrated
           ,
           And
           applied
           to
           GAMES
           at
           present
           most
           in
           Use
           ,
           Which
           may
           be
           easily
           extended
           to
           the
           most
           intricate
           Cases
           of
           Chance
           imaginable
           .
        
         
           LONDON
           :
           Printed
           by
           
             Benj.
             Motte
          
           ,
           and
           sold
           by
           
             Randall
             Taylor
          
           near
           Stationers-Hall
           ,
           1692.
           
        
      
       
         
         
         
           PREFACE
           .
        
         
           IT
           is
           thought
           as
           necessary
           to
           write
           a
           Preface
           before
           a
           Book
           ,
           as
           it
           is
           judg'd
           civil
           ,
           when
           you
           invite
           a
           Friend
           to
           Dinner
           ,
           to
           proffer
           him
           a
           Glass
           of
           Hock
           before-hand
           for
           a
           Whet
           :
           And
           this
           being
           maim'd
           enough
           for
           want
           of
           a
           Dedication
           ,
           I
           am
           resolv'd
           it
           shall
           not
           want
           an
           Epistle
           to
           the
           Reader
           too
           .
           I
           shall
           not
           take
           upon
           me
           to
           determine
           ,
           whether
           it
           is
           lawful
           to
           play
           at
           Dice
           or
           not
           ,
           leaving
           
           that
           to
           be
           disputed
           betwixt
           the
           
             Fanatick
             Parsons
          
           and
           the
           Sharpers
           ;
           I
           am
           sure
           it
           is
           lawful
           to
           deal
           with
           playing
           at
           Dice
           as
           with
           other
           Epidemic
           Distempers
           ;
           and
           I
           am
           confident
           that
           the
           writing
           a
           Book
           about
           it
           ,
           will
           contribute
           as
           little
           towards
           its
           Encouragement
           ,
           as
           Fluxing
           and
           Precipitates
           do
           to
           Whoring
           .
        
         
           It
           will
           be
           to
           little
           purpose
           to
           tell
           my
           Reader
           ,
           of
           how
           great
           Antiquity
           the
           playing
           at
           Dice
           is
           ,
           I
           will
           only
           let
           him
           
           
           
           
           
           know
           ,
           that
           by
           the
           
             Aleae
             Ludus
          
           ,
           the
           Antients
           comprehended
           all
           Games
           ,
           which
           were
           subjected
           to
           the
           determination
           of
           mere
           Chance
           ;
           this
           sort
           of
           Gaming
           was
           strictly
           forbid
           by
           the
           Emperor
           
             Justinian
             ,
             Cod.
             Lib.
             3.
             
             Tit.
             43.
          
           under
           very
           severe
           Penalties
           ;
           and
           
             Photius
             Nomocan
             ,
             Tit.
             9.
             
             Cap.
             27.
          
           acquaints
           us
           ,
           that
           the
           Use
           of
           this
           was
           altogether
           denied
           the
           Clergie
           of
           that
           time
           .
           Seneca
           says
           very
           well
           ,
           
             Aleator
             quantò
             in
             arte
             est
             melior
             ,
             tantò
             est
             nequior
          
           ;
           
           That
           by
           how
           much
           the
           one
           is
           more
           skilful
           in
           Games
           ,
           by
           so
           much
           he
           is
           the
           more
           culpable
           ;
           or
           we
           may
           say
           of
           this
           ,
           as
           an
           ingenious
           Man
           says
           of
           Dancing
           ,
           That
           to
           be
           extraordinary
           good
           at
           it
           ,
           is
           to
           be
           excellent
           in
           a
           Fault
           ;
           therefore
           I
           hope
           no
           body
           will
           imagine
           I
           had
           so
           mean
           a
           Design
           in
           this
           ,
           as
           to
           teach
           the
           Art
           of
           Playing
           at
           Dice
           .
        
         
           A
           great
           part
           of
           this
           Discourse
           is
           a
           Translation
           from
           Mons.
           
           Hugen's
           Treatise
           ,
           
             De
             ratiociniis
             in
             ludo
          
           
           Aleae
           ,
           one
           ,
           who
           in
           his
           Improvements
           of
           Philosophy
           ,
           has
           but
           one
           Superior
           ,
           and
           I
           think
           few
           or
           no
           Equals
           .
           The
           whole
           I
           undertook
           for
           my
           own
           Divertisement
           ,
           next
           to
           the
           Satisfaction
           of
           some
           Friends
           ,
           who
           would
           now
           and
           then
           be
           wrangling
           about
           the
           Proportions
           of
           Hazards
           in
           some
           Cases
           that
           are
           here
           decided
           .
           All
           it
           requir'd
           was
           a
           few
           spare
           Hours
           ,
           and
           but
           little
           Work
           for
           the
           Brain
           ;
           my
           Design
           in
           publishing
           it
           ,
           was
           to
           make
           it
           of
           more
           general
           
           Use
           ,
           and
           perhaps
           persuade
           a
           raw
           Squire
           ,
           by
           it
           ,
           to
           keep
           his
           Money
           in
           his
           Pocket
           ;
           and
           if
           ,
           upon
           this
           account
           ,
           I
           should
           ineur
           the
           Clamours
           of
           the
           Sharpers
           ,
           I
           do
           not
           much
           regard
           it
           ,
           since
           they
           are
           a
           sort
           of
           People
           the
           World
           is
           not
           bound
           to
           provide
           for
           .
        
         
           You
           will
           find
           here
           a
           very
           plain
           and
           easie
           Method
           of
           the
           Calculation
           of
           the
           Hazards
           of
           Game
           ,
           which
           a
           man
           may
           understand
           ,
           without
           knowing
           the
           Quadratures
           of
           Curves
           ,
           the
           Doctrin
           of
           
           Series's
           ,
           or
           the
           
           Laws
           of
           Centripetation
           of
           Bodies
           ,
           or
           the
           Periods
           of
           the
           Satellites
           of
           Jupiter
           ;
           yea
           ,
           without
           so
           much
           as
           the
           Elements
           of
           Euclid
           .
           There
           is
           nothing
           required
           for
           the
           comprehending
           the
           whole
           ,
           but
           common
           Sense
           and
           practical
           Arithmetick
           ;
           saving
           a
           few
           Touches
           of
           Algebra
           ,
           as
           in
           the
           first
           Three
           Propositions
           ,
           where
           the
           Reader
           ,
           without
           suspicion
           of
           Popery
           ,
           may
           make
           use
           of
           a
           strong
           implicit
           Faith
           ;
           tho
           I
           must
           confess
           ,
           it
           does
           not
           much
           recommend
           
           it self
           to
           me
           in
           these
           purposes
           ;
           for
           I
           had
           rather
           he
           would
           enquire
           ,
           and
           I
           believe
           he
           will
           find
           the
           speculation
           not
           unpleasant
           .
        
         
           Every
           man's
           Success
           in
           any
           Affair
           is
           proportional
           to
           his
           Conduct
           &
           Fortune
           .
           Fortune
           (
           in
           the
           sense
           of
           most
           People
           )
           signifies
           an
           Event
           which
           depends
           on
           Chance
           ,
           agreeing
           with
           my
           Wish
           ;
           and
           Misfortune
           signifies
           such
           an
           Event
           contrary
           to
           my
           Wish
           :
           an
           Event
           depending
           on
           Chance
           ,
           signifies
           such
           an
           one
           ,
           whose
           immediate
           Causes
           I
           don't
           know
           ,
           and
           consequently
           
           can
           neither
           foretel
           nor
           produce
           it
           (
           for
           it
           is
           no
           Heresie
           to
           believe
           ,
           that
           Providence
           suffers
           ordinary
           matters
           to
           run
           in
           the
           Channel
           of
           second
           Causes
           )
           .
           Now
           I
           suppose
           ,
           that
           all
           a
           wise
           Man
           can
           do
           in
           such
           a
           Case
           is
           ,
           to
           lay
           his
           Business
           on
           such
           Events
           ,
           as
           have
           the
           most
           or
           most
           powerful
           second
           Causes
           ,
           and
           this
           is
           true
           both
           in
           the
           great
           Events
           of
           the
           World
           ,
           and
           in
           ordinary
           Games
           .
           It
           is
           impossible
           for
           a
           Dye
           ,
           with
           such
           a
           determin'd
           force
           and
           direction
           ,
           not
           
           to
           fall
           on
           such
           a
           determin'd
           side
           ,
           only
           I
           don't
           know
           the
           force
           and
           direction
           which
           makes
           it
           fall
           on
           such
           a
           determin'd
           side
           ,
           and
           therefore
           I
           call
           that
           Chance
           ,
           which
           is
           nothing
           but
           want
           of
           Art
           ;
           that
           only
           which
           is
           left
           to
           me
           ,
           is
           to
           wager
           where
           there
           are
           the
           greatest
           number
           of
           Chances
           ,
           and
           consequently
           the
           greatest
           probability
           to
           gain
           ;
           and
           the
           whole
           Art
           of
           Gaming
           ,
           where
           there
           is
           any
           thing
           of
           Hazard
           ,
           will
           be
           reduc'd
           to
           this
           at
           last
           ,
           viz.
           in
           dubious
           Cases
           ,
           to
           
           calculate
           on
           which
           side
           there
           are
           most
           Chances
           ;
           and
           tho
           this
           can't
           be
           done
           in
           the
           midst
           of
           Game
           precisely
           to
           an
           Unite
           ,
           yet
           a
           Man
           who
           knows
           the
           Principles
           ,
           may
           make
           such
           a
           conjecture
           ,
           as
           will
           be
           a
           sufficient
           direction
           to
           him
           ;
           and
           tho
           it
           is
           possible
           ,
           if
           there
           are
           any
           Chances
           against
           him
           at
           all
           ,
           that
           he
           may
           lose
           ,
           yet
           when
           he
           chuseth
           the
           safest
           side
           ,
           he
           may
           partwith
           his
           Money
           with
           more
           content
           (
           if
           there
           can
           be
           any
           at
           all
           )
           in
           such
           a
           Case
           .
        
         
         
           I
           will
           not
           debate
           ,
           whether
           one
           may
           engage
           another
           in
           a
           disadvantageous
           Wager
           .
           Games
           may
           be
           suppos'd
           to
           be
           a
           tryal
           of
           Wit
           as
           well
           as
           Fortune
           ,
           and
           every
           Man
           ,
           when
           he
           enters
           the
           Lists
           with
           another
           ,
           unless
           out
           of
           Complaisance
           ,
           takes
           it
           for
           granted
           ,
           his
           Fortune
           and
           Iudgment
           ,
           are
           ,
           at
           least
           ,
           equal
           to
           those
           of
           his
           Play-Fellow
           ;
           but
           this
           I
           am
           sure
           of
           ,
           that
           false
           Dice
           ,
           Tricks
           of
           
             Leger-de-main
             ,
             &c.
          
           are
           inexcusable
           ,
           for
           the
           question
           in
           Gaming
           
           is
           not
           ,
           Who
           is
           the
           best
           Iugler
           ?
        
         
           The
           Reader
           may
           here
           observe
           the
           Force
           of
           Numbers
           ,
           which
           can
           be
           succesfully
           applied
           ,
           even
           to
           those
           things
           ,
           which
           one
           would
           imagin
           are
           subject
           to
           no
           Rules
           .
           There
           are
           very
           few
           things
           which
           we
           know
           ,
           which
           are
           not
           capable
           of
           being
           reduc'd
           to
           a
           Mathematical
           Reasoning
           ,
           and
           when
           they
           cannot
           ,
           it
           s
           a
           sign
           our
           Knowledg
           of
           them
           is
           very
           small
           and
           confus'd
           ;
           and
           where
           
           a
           mathematical
           reasoning
           can
           be
           had
           ,
           it
           's
           as
           great
           folly
           to
           make
           use
           of
           any
           other
           ,
           as
           to
           grope
           for
           a
           thing
           in
           the
           dark
           when
           you
           have
           a
           Candle
           standing
           by
           you
           .
           I
           believe
           the
           Calculation
           of
           the
           Quantity
           of
           Probability
           might
           be
           improved
           to
           a
           very
           useful
           and
           pleasant
           Speculation
           ,
           and
           applied
           to
           a
           great
           many
           Events
           which
           are
           accidental
           ,
           besides
           those
           of
           Games
           ;
           only
           these
           Cases
           would
           be
           infinitely
           more
           confus'd
           ,
           as
           depending
           on
           Chances
           which
           the
           most
           part
           of
           
           Men
           are
           ignorant
           of
           ;
           and
           as
           I
           have
           hinted
           already
           ,
           all
           the
           Politicks
           in
           the
           World
           are
           nothing
           else
           but
           a
           kind
           of
           Analysis
           of
           the
           Quantity
           of
           Probability
           in
           casual
           Events
           ,
           and
           a
           good
           Politician
           signifies
           no
           more
           ,
           but
           one
           who
           is
           dexterous
           at
           such
           Calculations
           ;
           only
           the
           Principles
           which
           are
           made
           use
           of
           in
           the
           Solution
           of
           such
           Problems
           ,
           can't
           be
           studied
           in
           a
           Closet
           ,
           but
           acquir'd
           by
           the
           Observation
           of
           Mankind
           .
        
         
         
           There
           is
           likewise
           a
           Calculation
           of
           the
           Quantity
           of
           Probability
           founded
           on
           Experience
           ,
           to
           be
           made
           use
           of
           in
           Wagers
           about
           any
           thing
           ;
           for
           Example
           ,
           it
           is
           odds
           ,
           if
           a
           Woman
           is
           
             with
             Child
          
           ,
           but
           it
           shall
           be
           a
           Boy
           ;
           and
           if
           you
           would
           know
           the
           just
           odds
           ,
           you
           must
           consider
           the
           Proportion
           in
           the
           Bills
           that
           the
           Males
           bear
           to
           the
           Females
           :
           The
           Yearly
           Bills
           of
           Mortality
           are
           observ'd
           to
           bear
           such
           Proportion
           to
           the
           live
           People
           as
           1
           to
           30
           ,
           or
           26
           ;
           therefore
           it
           
           is
           an
           even
           Wager
           ,
           that
           one
           out
           of
           thirteen
           ,
           dyes
           within
           a
           Year
           (
           which
           may
           be
           a
           good
           reason
           ,
           tho
           not
           the
           true
           one
           of
           that
           foolish
           piece
           of
           Superstition
           )
           ,
           because
           ,
           at
           this
           rate
           ,
           if
           1
           out
           of
           26
           dyes
           ,
           you
           are
           no
           loser
           .
           It
           is
           but
           1
           to
           18
           if
           you
           meet
           a
           Parson
           in
           the
           Street
           ,
           that
           he
           proves
           to
           be
           a
           Non-Juror
           ,
           because
           there
           is
           but
           1
           of
           36
           that
           are
           such
           .
           It
           is
           hardly
           1
           to
           10
           ,
           that
           a
           Woman
           of
           Twenty
           Years
           old
           has
           her
           Maidenhead
           ,
           and
           almost
           
           the
           same
           Wager
           ,
           that
           a
           Town-Spark
           of
           that
           Age
           has
           not
           been
           clap'd
           .
           I
           think
           a
           Man
           might
           venture
           some
           odds
           ,
           that
           100
           of
           the
           
             Gens
             d'arms
          
           beats
           an
           equal
           Number
           of
           
             Dutch
             Troopers
          
           ;
           and
           that
           an
           
             English
             Regiment
          
           stands
           its
           ground
           as
           long
           as
           another
           ,
           making
           Experience
           our
           Guide
           in
           all
           these
           Cases
           and
           others
           of
           the
           like
           nature
           .
        
         
           But
           there
           are
           no
           casual
           Events
           ,
           which
           are
           so
           easily
           subjected
           to
           Numbers
           ,
           as
           
           those
           of
           Games
           ;
           and
           I
           believe
           ,
           there
           the
           Speculation
           might
           be
           improved
           so
           far
           ,
           as
           to
           bring
           in
           the
           Doctrin
           of
           the
           
           Series's
           and
           Logarithms
           .
           Since
           Gaming
           is
           become
           a
           Trade
           ,
           I
           think
           it
           fit
           the
           Adventurers
           should
           be
           upon
           the
           Square
           ;
           and
           therefore
           in
           the
           Contrivance
           of
           Games
           there
           ought
           to
           be
           a
           strict
           Calculation
           made
           use
           of
           ,
           that
           they
           mayn't
           put
           one
           Party
           in
           more
           probability
           to
           gain
           than
           another
           ;
           and
           likewise
           ,
           if
           a
           Man
           has
           a
           
           considerable
           Venture
           ,
           he
           ought
           to
           be
           allow'd
           to
           withdraw
           his
           Money
           when
           he
           pleases
           ,
           paying
           according
           to
           the
           Circumstances
           he
           is
           then
           in
           :
           and
           it
           were
           easie
           in
           most
           Games
           to
           make
           Tables
           ,
           by
           the
           Inspection
           of
           which
           ,
           a
           Man
           might
           know
           what
           he
           was
           either
           to
           pay
           or
           receive
           ,
           in
           any
           Circumstances
           you
           can
           imagin
           ,
           it
           being
           convenient
           to
           save
           a
           part
           of
           ones
           Money
           ,
           rather
           than
           venture
           the
           loss
           of
           it
           all
           .
        
         
         
           I
           shall
           add
           no
           more
           ,
           but
           that
           a
           Mathematician
           will
           easily
           perceive
           ,
           it
           is
           not
           put
           in
           such
           a
           Dress
           as
           to
           be
           taken
           notice
           of
           by
           him
           ,
           there
           being
           abundance
           of
           Words
           spent
           to
           make
           the
           more
           ordinary
           sort
           of
           People
           understand
           it
           .
        
      
       
         
         
           FOR
           the
           sake
           of
           those
           who
           are
           not
           vers'd
           in
           Mathematicks
           ,
           I
           have
           added
           the
           following
           Explanation
           of
           Signs
           .
        
         
           
             =
             Equal
             .
          
           
             +
             More
             ,
             or
             to
             be
             added
             .
          
           
             −
             Less
             ,
             or
             to
             be
             subtracted
             .
          
           
             ×
             Multiplied
             .
          
        
         
           Example
           .
        
         
           3
           ×
           4
           +
           3
           −
           1
           =
           14
           =
           5
           /
           9
           a
           ,
           is
           to
           be
           read
           thus
           ,
        
         
           3
           multiplied
           in
           4
           more
           by
           3
           less
           by
           1
           is
           equal
           to
           14
           ,
           which
           is
           equal
           to
           five
           ninth
           parts
           of
           a.
           
        
      
    
     
       
         
         
           An
           Exact
           METHOD
           For
           SOLVING
           the
           
             Hazards
             of
             Game
          
           .
        
         
           ALthough
           the
           Events
           of
           Games
           ,
           which
           Fortune
           solely
           governs
           ,
           are
           uncertain
           ,
           yet
           it
           may
           be
           certainly
           determin'd
           ,
           how
           much
           one
           is
           more
           ready
           to
           lose
           than
           gain
           .
           For
           Example
           :
           If
           one
           should
           wager
           ,
           at
           the
           first
           Throw
           with
           one
           Dye
           ,
           to
           throw
           Six
           ,
           it
           's
           an
           
           accident
           if
           he
           gains
           or
           not
           ,
           but
           by
           how
           much
           it
           's
           more
           probable
           he
           will
           lose
           than
           gain
           ,
           is
           really
           determin'd
           by
           the
           Nature
           of
           the
           thing
           ,
           and
           capable
           of
           a
           strict
           Calculation
           .
           So
           likewise
           ,
           if
           I
           should
           play
           with
           another
           on
           this
           condition
           ,
           that
           the
           Victory
           should
           be
           to
           the
           Three
           first
           Games
           ,
           and
           I
           had
           gain'd
           one
           already
           ;
           it
           is
           still
           uncertain
           who
           shall
           first
           gain
           the
           third
           ;
           yet
           by
           a
           demonstrative
           reasoning
           I
           can
           estimate
           both
           the
           Value
           of
           his
           expectation
           and
           mine
           ,
           and
           consequently
           (
           if
           we
           agree
           to
           leave
           the
           Game
           unperfect
           )
           determin
           how
           great
           a
           share
           of
           the
           Stakes
           belong
           to
           me
           ,
           and
           how
           much
           to
           my
           Play-fellow
           ;
           
           or
           if
           any
           were
           desirous
           to
           take
           my
           place
           ,
           at
           what
           rate
           I
           ought
           to
           sell
           it
           .
           Hence
           may
           arise
           innumerable
           Queries
           among
           two
           ,
           three
           ,
           or
           more
           Gamesters
           ;
           and
           since
           the
           Calculation
           of
           these
           things
           is
           a
           little
           out
           of
           the
           common
           road
           ,
           and
           can
           be
           oft-times
           apply'd
           to
           good
           purpose
           ;
           I
           shall
           briefly
           here
           shew
           how
           it
           is
           to
           be
           done
           ,
           and
           afterwards
           explain
           those
           things
           which
           belong
           properly
           to
           the
           Dice
           .
        
         
           In
           both
           cases
           I
           shall
           make
           use
           of
           this
           Principle
           ,
           
             Ones
             Hazard
             or
             Expectation
             to
             gain
             any
             thing
             ,
             is
             worth
             so
             much
             ,
             as
             ,
             if
             he
             had
             it
             ,
             he
             could
             purchase
             the
             like
             Hazard
             or
             Expectation
             again
             in
             a
             just
             and
             equal
             Game
             .
          
        
         
         
           For
           Example
           ,
           If
           one
           ,
           without
           my
           knowledg
           ,
           should
           hide
           in
           one
           Hand
           7
           Shillings
           ,
           and
           in
           his
           other
           3
           Shillings
           ,
           and
           put
           it
           to
           my
           choice
           which
           Hand
           I
           would
           take
           ,
           I
           say
           this
           is
           as
           much
           worth
           to
           me
           ,
           as
           if
           he
           should
           give
           me
           5
           Shillings
           ;
           because
           ,
           if
           I
           have
           5
           Shillings
           ,
           I
           can
           purchase
           as
           good
           a
           Chance
           again
           ,
           and
           that
           in
           a
           fair
           and
           just
           Game
           .
        
         
           
           
             PROPOSITION
             I.
             
          
           
             If
             I
             expect
             a
             or
             b
             ,
             either
             of
             which
             ,
             with
             equal
             probability
             ,
             may
             fall
             to
             me
             ,
             then
             my
             Expectation
             is
             worth
             
             ,
             that
             is
             ,
             the
             half
             Sum
             of
             a
             and
             b.
             
          
           
             THat
             I
             may
             not
             only
             demonstrate
             ,
             but
             likewise
             investigate
             this
             Rule
             ,
             suppose
             the
             Value
             of
             my
             Expectation
             be
             x
             ;
             by
             the
             former
             Principle
             having
             x
             ,
             I
             can
             purchase
             as
             good
             an
             Expectation
             again
             in
             a
             fair
             and
             just
             Game
             .
             Suppose
             then
             I
             play
             with
             another
             on
             these
             
             terms
             ;
             That
             every
             one
             stakes
             x
             ,
             and
             the
             Gainer
             give
             to
             the
             Loser
             a
             ,
             this
             Game
             is
             just
             ,
             and
             it
             appears
             ,
             that
             at
             this
             rate
             ,
             I
             have
             an
             equal
             hazard
             either
             to
             get
             a
             if
             I
             lose
             the
             Game
             ,
             or
             2
             
               x
               −
               a
            
             if
             I
             gain
             ;
             for
             in
             this
             case
             I
             get
             2
             x
             ,
             which
             are
             the
             Stakes
             ,
             out
             of
             which
             I
             must
             pay
             the
             other
             a
             ;
             but
             if
             2
             
               x
               −
               a
            
             were
             worth
             b
             ,
             then
             I
             have
             an
             equal
             hazard
             to
             get
             a
             or
             b
             ;
             therefore
             making
             2
             
               x
               −
               a
               =
               b
            
             ,
             
             ,
             which
             is
             the
             Value
             of
             my
             Expectation
             .
             The
             Demonstration
             is
             easie
             ,
             for
             having
             
             ,
             I
             can
             play
             with
             another
             
             who
             will
             stake
             
             against
             it
             ,
             on
             this
             condition
             ,
             that
             the
             Gainer
             should
             give
             to
             the
             Loser
             a
             ;
             by
             this
             means
             I
             have
             an
             equal
             Expectation
             to
             get
             a
             if
             I
             lose
             ,
             or
             b
             if
             I
             win
             ;
             for
             in
             the
             last
             case
             I
             get
             a+b
             the
             Stakes
             ,
             out
             of
             which
             I
             must
             pay
             a
             to
             my
             Play-fellow
             .
          
           
             In
             Numbers
             :
             If
             I
             had
             an
             equal
             hazard
             to
             get
             3
             or
             7
             ,
             then
             by
             this
             Proposition
             ,
             my
             Expectation
             is
             worth
             5
             ,
             and
             it
             is
             certain
             ,
             having
             5
             ,
             I
             may
             have
             the
             same
             Chance
             ;
             for
             if
             I
             play
             with
             another
             so
             that
             every
             one
             stakes
             5
             ,
             and
             the
             Gainer
             pay
             to
             the
             Loser
             3
             ,
             this
             is
             a
             fair
             way
             of
             Gaming
             ;
             
             and
             it
             is
             evident
             ,
             I
             have
             an
             equal
             hazard
             to
             get
             3
             if
             I
             lose
             ,
             or
             7
             if
             I
             gain
             .
          
        
         
           
             PROP.
             II.
             
          
           
             If
             I
             expect
             
               a
               ,
               b
            
             ,
             or
             c
             ,
             either
             of
             which
             ,
             with
             equal
             facility
             ,
             may
             happen
             ,
             then
             the
             Value
             of
             my
             Expectation
             is
             
             ,
             or
             the
             third
             part
             of
             the
             Sum
             of
             
               a
               b
            
             and
             c.
             
          
           
             FOR
             the
             Investigation
             of
             which
             ,
             suppose
             x
             be
             the
             value
             of
             my
             Expectation
             ;
             then
             x
             must
             be
             such
             ,
             as
             I
             can
             purchase
             with
             it
             the
             same
             Expectation
             in
             a
             just
             Game
             :
             Suppose
             
             the
             Conditions
             of
             the
             Game
             be
             ,
             that
             playing
             with
             two
             others
             ,
             each
             of
             us
             stakes
             x
             ,
             and
             I
             bargain
             with
             one
             of
             the
             Gamesters
             ,
             if
             I
             win
             ,
             to
             give
             him
             b
             ,
             and
             he
             shall
             do
             the
             same
             to
             me
             ;
             but
             with
             the
             other
             ,
             that
             if
             I
             gain
             ,
             I
             shall
             give
             him
             c
             ,
             and
             
               vice
               versâ
            
             ;
             this
             is
             fair
             play
             :
             And
             here
             I
             have
             an
             equal
             hazard
             to
             get
             b
             ,
             if
             the
             first
             win
             ,
             c
             if
             the
             second
             ,
             or
             3
             
               x
               −
               b
               −
               c
            
             if
             I
             gain
             my self
             ;
             for
             then
             I
             get
             3
             
               x
               ,
               viz.
            
             the
             Stakes
             ,
             of
             which
             I
             give
             the
             one
             b
             and
             the
             other
             c
             ;
             but
             if
             3
             
               x
               −
               b
               −
               c
            
             be
             equal
             to
             a
             ,
             I
             have
             an
             equal
             Expectation
             of
             
               a
               ,
               b
            
             ,
             or
             c
             ;
             therefore
             making
             3
             
               x
               −
               b
               −
               c
               =
               a
               ,
            
             
             
             ,
             which
             is
             the
             Value
             of
             my
             Expectation
             .
             After
             the
             same
             method
             you
             will
             find
             ,
             if
             I
             had
             an
             equal
             hazard
             to
             get
             
               a
               b
               c
            
             or
             d
             ,
             the
             Value
             of
             my
             Expectation
             
             that
             is
             the
             fourth
             part
             of
             the
             Sum
             of
             
               a
               b
               c
            
             and
             
               d
               ,
               &c.
            
             
          
        
         
           
             PROP.
             III.
             
          
           
             If
             the
             Number
             of
             Chances
             ,
             by
             which
             a
             falls
             to
             me
             ,
             be
             p
             ,
             and
             the
             Number
             of
             Chances
             ,
             by
             which
             b
             falls
             ,
             be
             q
             ,
             and
             supposing
             all
             the
             Chances
             do
             happen
             with
             equal
             Facility
             ,
             
             then
             the
             Value
             of
             my
             Expectation
             is
             
             ,
             
               i.
               e.
            
             the
             Product
             of
             a
             multiplied
             in
             the
             Number
             of
             its
             Chances
             added
             to
             the
             Product
             of
             b
             ,
             multiplied
             into
             the
             Number
             of
             its
             Chances
             ,
             and
             the
             Summ
             divided
             by
             the
             Number
             of
             Chances
             both
             of
             a
             and
             b.
             
          
           
             SUppose
             ,
             as
             before
             ,
             x
             be
             the
             Value
             of
             my
             Expectation
             ;
             then
             if
             I
             have
             x
             ,
             I
             must
             be
             able
             to
             purchase
             with
             it
             that
             same
             Expectation
             again
             in
             a
             fair
             Game
             :
             For
             this
             I
             shall
             take
             as
             many
             Play-fellows
             as
             ,
             with
             me
             ,
             make
             up
             the
             Number
             
             of
             p+q
             ,
             of
             which
             let
             every
             one
             stake
             x
             ,
             so
             the
             whole
             Stake
             will
             be
             px+qx
             ,
             and
             every
             one
             plays
             with
             equal
             hopes
             of
             winning
             ;
             with
             as
             many
             of
             my
             Fellow-Gamesters
             as
             the
             Number
             9
             stands
             for
             ,
             I
             make
             this
             bargain
             one
             by
             one
             ,
             that
             whoever
             of
             them
             gains
             shall
             give
             me
             b
             ,
             and
             if
             I
             win
             ,
             I
             shall
             do
             so
             to
             them
             ;
             with
             every
             one
             of
             the
             rest
             of
             the
             Gamesters
             ,
             whose
             Number
             is
             
               p
               −
               1
            
             ,
             I
             make
             this
             bargain
             ,
             that
             whoever
             of
             them
             gains
             ,
             shall
             give
             me
             a
             ,
             and
             I
             shall
             give
             every
             one
             of
             them
             as
             much
             ,
             if
             I
             gain
             :
             It
             's
             evident
             this
             is
             fair
             play
             ;
             for
             no
             Man
             here
             is
             injur'd
             ;
             and
             in
             this
             case
             I
             have
             q
             Expectations
             
             to
             gain
             b
             ,
             and
             
               p
               −
               1
            
             Expectations
             to
             gain
             a
             ,
             and
             1
             Expectation
             
               (
               viz.
            
             when
             I
             win
             my
             my self
             )
             to
             get
             
               px+qx
               −
               bqap+a
            
             ;
             for
             then
             I
             am
             to
             deliver
             b
             to
             every
             one
             of
             the
             q
             Players
             ,
             and
             a
             to
             every
             one
             of
             the
             
               p
               −
               1
            
             Gamesters
             ,
             which
             makes
             
               ab+pa
               −
               a
            
             ;
             if
             therefore
             
               qx+bx
               −
               ba
               −
               ap+a
            
             were
             equal
             to
             a
             ,
             I
             would
             have
             p
             Expectations
             of
             a
             (
             since
             just
             now
             I
             had
             
               p
               −
               1
            
             Expectations
             of
             it
             )
             and
             q
             Expectations
             of
             b
             ,
             and
             so
             would
             have
             just
             come
             to
             my
             first
             Expectation
             ;
             therefore
             putting
             
               px+qx
               −
               bq
               −
               ap+a
               =
               a
               ,
            
             then
             is
             
          
           
           
             In
             Numbers
             :
             If
             I
             had
             3
             Chances
             to
             gain
             for
             13
             ,
             and
             2
             for
             8
             ,
             by
             this
             Rule
             ,
             my
             Hazard
             is
             worth
             11
             ;
             for
             13
             multiplied
             by
             3
             gives
             39
             ,
             and
             8
             by
             2
             16
             ,
             these
             two
             added
             ,
             make
             55
             ,
             divided
             by
             5
             is
             11
             ,
             and
             I
             can
             easily
             shew
             ,
             if
             I
             have
             11
             ,
             I
             can
             come
             to
             the
             like
             Expectation
             again
             ;
             for
             playing
             with
             four
             others
             ,
             and
             every
             one
             of
             us
             staking
             11
             ,
             with
             two
             of
             them
             I
             make
             this
             Bargain
             ,
             that
             whoever
             gains
             shall
             give
             me
             8
             ,
             and
             I
             shall
             too
             do
             so
             to
             them
             ;
             with
             the
             other
             two
             I
             make
             this
             Bargain
             ,
             that
             whoever
             gains
             shall
             give
             me
             13
             ,
             and
             I
             them
             as
             much
             if
             I
             gain
             :
             It
             appears
             ,
             by
             this
             means
             I
             have
             two
             Expectations
             
             to
             get
             8
             ,
             viz.
             if
             any
             of
             the
             first
             two
             gain
             ,
             and
             3
             Expectations
             to
             get
             13
             ,
             viz.
             if
             either
             I
             or
             any
             of
             the
             other
             two
             gain
             ;
             for
             in
             this
             case
             I
             gain
             the
             Stakes
             ,
             which
             are
             55
             ,
             out
             of
             which
             I
             am
             oblig'd
             to
             give
             the
             first
             two
             8
             ,
             and
             the
             other
             two
             13
             ,
             and
             so
             there
             remains
             13
             for
             my self
             .
          
        
         
           
             PROP.
             IV.
             
          
           
             That
             I
             may
             come
             to
             the
             Question
             propos'd
             ,
             viz.
             The
             making
             a
             just
             Distribution
             amongst
             Gamesters
             ,
             when
             their
             Hazards
             are
             unequal
             ;
             we
             must
             begin
             with
             the
             most
             easie
             Cases
             .
          
           
           
             SUppose
             then
             I
             play
             with
             another
             ,
             on
             condition
             that
             he
             who
             wins
             the
             three
             first
             Games
             shall
             have
             the
             Stakes
             ,
             and
             that
             I
             have
             already
             gain'd
             two
             ,
             I
             would
             know
             ,
             if
             we
             agree
             to
             break
             off
             the
             Game
             ,
             and
             part
             the
             Stakes
             justly
             ,
             how
             much
             falls
             to
             my
             share
             ?
          
           
             The
             first
             thing
             we
             must
             consider
             in
             such
             Questions
             is
             ,
             the
             Number
             of
             Games
             that
             are
             wanting
             to
             both
             :
             For
             Example
             ,
             If
             it
             had
             been
             agreed
             betwixt
             us
             ,
             that
             he
             should
             have
             the
             Stakes
             who
             gain'd
             the
             first
             20
             Games
             ,
             and
             if
             I
             had
             gain'd
             already
             19
             ,
             and
             my
             Fellow-Gamester
             but
             18
             ,
             my
             Hazard
             
             is
             as
             much
             better
             than
             his
             in
             that
             Case
             ,
             as
             in
             this
             proposed
             ,
             viz
             ?
             When
             of
             3
             Games
             I
             have
             2
             ,
             and
             he
             but
             1
             ,
             because
             in
             both
             cases
             there
             's
             2
             wanting
             to
             him
             ,
             and
             1
             to
             me
             .
          
           
             In
             the
             next
             place
             ,
             to
             find
             the
             portion
             of
             the
             Stakes
             due
             to
             each
             of
             us
             ,
             we
             must
             consider
             what
             would
             happen
             if
             the
             Game
             went
             on
             ;
             it
             is
             certain
             ,
             if
             I
             gain
             the
             first
             Game
             ,
             I
             get
             the
             Stake
             ,
             which
             I
             call
             a
             ;
             but
             if
             he
             gain'd
             ,
             both
             our
             Lots
             would
             be
             equal
             ,
             and
             so
             there
             would
             fall
             to
             each
             of
             us
             ½
             a
             ;
             but
             since
             I
             have
             an
             equal
             Hazard
             to
             gain
             or
             lose
             the
             first
             Game
             ,
             I
             have
             an
             equal
             Expectation
             to
             gain
             a
             ,
             or
             ½
             a
             ,
             which
             ,
             by
             the
             first
             Proposition
             ,
             is
             as
             much
             
             worth
             as
             the
             half
             Sum
             of
             both
             ,
             
               i.
               e.
               ¾a
            
             ,
             so
             there
             is
             left
             to
             my
             Fellow-Gamester
             ¼
             a
             ;
             from
             whence
             it
             follows
             ,
             that
             he
             who
             would
             buy
             my
             Game
             ,
             ought
             to
             pay
             me
             for
             it
             ¾
             a
             ;
             and
             therefore
             ,
             he
             who
             undertakes
             to
             gain
             one
             Game
             before
             another
             gains
             two
             ,
             may
             wager
             3
             to
             1.
             
          
        
         
           
             PROP.
             V.
             
          
           
             Suppose
             I
             want
             but
             one
             Game
             ,
             and
             my
             Fellow-gamester
             three
             ,
             it
             is
             required
             to
             make
             a
             just
             Distribution
             of
             the
             Stake
             :
          
           
             LET
             us
             here
             likewise
             consider
             in
             what
             state
             we
             should
             be
             ,
             if
             I
             or
             he
             gain'd
             the
             
             first
             Game
             ;
             if
             I
             gain
             ,
             I
             have
             the
             Stake
             a
             ,
             if
             he
             ,
             then
             he
             wants
             yet
             2
             Games
             ,
             and
             I
             but
             1
             ,
             and
             therefore
             we
             should
             be
             in
             the
             same
             condition
             which
             is
             supposed
             in
             the
             former
             Proposition
             ;
             and
             so
             there
             would
             fall
             to
             my
             share
             ,
             as
             was
             demonstrated
             there
             ,
             ¾
             a
             ;
             therefore
             with
             equal
             facility
             there
             may
             happen
             to
             me
             a
             ,
             or
             ¾
             a
             ,
             which
             ,
             by
             the
             First
             Proposition
             ,
             is
             worth
             ⅞
             a
             ,
             and
             to
             my
             Fellow-Gamester
             there
             is
             left
             1
             /
             8
             a
             ,
             and
             therefore
             my
             Hazard
             to
             his
             is
             as
             7
             to
             1.
             
          
           
             As
             the
             Calculation
             of
             the
             former
             Proposition
             was
             requisite
             for
             this
             ,
             so
             this
             will
             serve
             for
             the
             following
             .
             If
             I
             should
             suppose
             my self
             to
             want
             but
             
             one
             Game
             ,
             and
             my
             Fellow
             four
             (
             by
             the
             same
             Method
             )
             you
             will
             find
             15
             /
             16
             of
             the
             Stake
             belongs
             to
             me
             ,
             and
             1
             /
             16
             to
             him
             .
          
        
         
           
             PROP.
             VI.
             
          
           
             Suppose
             I
             want
             two
             Games
             ,
             and
             my
             Fellow-Gamester
             three
             .
          
           
             THen
             by
             the
             next
             Game
             it
             will
             happen
             ,
             that
             I
             want
             but
             one
             ,
             and
             he
             three
             ,
             which
             (
             by
             the
             preceding
             Proposition
             )
             is
             worth
             ⅞
             a
             ;
             or
             that
             we
             should
             both
             want
             two
             ,
             whence
             there
             will
             be
             ½
             a
             due
             to
             each
             of
             us
             ;
             now
             I
             being
             in
             an
             equal
             probability
             to
             gain
             or
             lose
             the
             next
             Game
             ,
             I
             have
             an
             equal
             Hazard
             
             to
             gain
             ⅞
             a
             or
             ½
             a
             ,
             which
             ,
             by
             the
             First
             Proposition
             is
             worth
             11
             /
             16
             a
             ,
             and
             so
             there
             are
             eleven
             parts
             of
             the
             Stakes
             due
             to
             me
             ,
             and
             five
             to
             my
             Fellow
             .
          
        
         
           
             PROP.
             VII
             .
          
           
             Let
             us
             suppose
             I
             want
             two
             Games
             ,
             and
             my
             Fellow
             four
             .
          
           
             IF
             I
             gain
             the
             next
             Game
             ,
             then
             I
             shall
             want
             but
             one
             ,
             and
             my
             Fellow
             four
             ;
             but
             if
             I
             lose
             it
             ,
             then
             I
             shall
             want
             two
             ,
             and
             he
             three
             :
             So
             I
             have
             an
             equal
             Hazard
             for
             gaining
             15
             /
             16
             a
             ,
             or
             11
             /
             16
             a
             ,
             which
             ,
             by
             the
             First
             ,
             is
             worth
             13
             /
             16
             a
             :
             So
             it
             appears
             ,
             that
             he
             who
             is
             to
             gain
             two
             Games
             for
             the
             others
             
             four
             ,
             is
             in
             a
             better
             condition
             than
             he
             who
             is
             to
             gain
             one
             for
             the
             others
             two
             ;
             for
             my
             share
             in
             the
             first
             Case
             is
             ¾
             a
             or
             12
             /
             16
             a
             ,
             which
             is
             less
             than
             13
             /
             16
             ,
             my
             share
             in
             the
             last
             .
          
        
         
           
             PROP.
             VIII
             .
          
           
             Let
             us
             suppose
             three
             Gamesters
             ,
             whereof
             the
             first
             and
             second
             want
             1
             Game
             ,
             but
             the
             third
             2.
             
          
           
             TO
             find
             the
             share
             of
             the
             first
             ,
             we
             must
             consider
             what
             would
             happen
             if
             either
             he
             ,
             or
             any
             of
             the
             other
             two
             gain'd
             the
             first
             Game
             ;
             if
             he
             gains
             ,
             then
             he
             has
             the
             Stake
             a
             ;
             if
             the
             second
             
             gain
             ,
             he
             has
             nothing
             ;
             but
             if
             the
             third
             gain
             ,
             then
             each
             of
             them
             would
             want
             a
             Game
             ,
             and
             so
             ⅓
             a
             would
             be
             due
             to
             every
             one
             of
             them
             .
             Thus
             the
             first
             Gamester
             has
             one
             Expectation
             to
             gain
             a
             ,
             one
             to
             gain
             nothing
             ,
             and
             one
             for
             ⅓
             a
             (
             since
             all
             are
             in
             an
             equal
             probability
             to
             gain
             the
             first
             Game
             )
             which
             by
             the
             second
             Proposition
             is
             worth
             4
             /
             9
             a
             :
             Now
             since
             the
             second
             Gamesters
             Condition
             is
             as
             good
             ,
             his
             Share
             is
             likewise
             4
             /
             9
             a
             ,
             and
             so
             there
             remains
             to
             the
             third
             1
             /
             9
             a
             ,
             whose
             Share
             might
             have
             been
             as
             easily
             found
             by
             its
             self
             .
          
        
         
           
           
             PROP.
             IX
             .
          
           
             In
             any
             Number
             of
             Gamesters
             you
             please
             ,
             amongst
             whom
             there
             are
             some
             who
             want
             more
             ,
             some
             fewer
             Games
             :
             To
             find
             what
             is
             any
             ones
             Share
             in
             the
             Stake
             ,
             we
             must
             consider
             ,
             what
             would
             be
             due
             to
             him
             ,
             whose
             Share
             we
             investigate
             ,
             if
             either
             he
             ,
             or
             any
             of
             his
             Fellow-Gamesters
             should
             gain
             the
             next
             following
             Game
             ;
             add
             all
             their
             Shares
             together
             ,
             and
             divide
             the
             Sum
             by
             the
             Number
             of
             the
             Gamesters
             ,
             the
             Quotient
             is
             his
             Share
             you
             were
             seeking
             .
          
           
           
             SUppose
             three
             Gamesters
             ,
             A
             B
             and
             C
             ,
             A
             wants
             1
             Game
             ,
             B
             2
             ,
             and
             C
             likewise
             2
             ,
             I
             would
             find
             what
             is
             the
             Share
             of
             the
             Stake
             due
             to
             B
             ,
             which
             I
             shall
             call
             q.
             
          
           
             First
             we
             must
             consider
             what
             would
             fall
             to
             B's
             share
             ,
             if
             either
             he
             ,
             A
             ,
             or
             C
             ,
             wins
             the
             next
             Game
             ;
             if
             A
             wins
             ,
             the
             Game
             is
             ended
             ,
             so
             he
             gets
             nothing
             ;
             if
             B
             himself
             gain
             ,
             then
             he
             wants
             1
             Game
             ,
             A
             1
             ,
             and
             C
             2
             ;
             therefore
             ,
             by
             the
             former
             Proposition
             ,
             there
             is
             due
             to
             him
             in
             that
             Case
             4
             /
             ●q
             ,
             then
             if
             C
             gains
             the
             next
             Play
             ,
             then
             A
             and
             C
             would
             want
             but
             1
             ,
             and
             B
             2
             ;
             and
             therefore
             ,
             by
             the
             Eigth
             Proposition
             ,
             
             his
             Share
             would
             be
             worth
             1
             /
             9q
             ;
             add
             together
             what
             is
             due
             to
             B
             in
             all
             these
             three
             Cases
             ,
             viz.
             o4
             /
             9q
             ,
             1
             /
             9q
             ,
             the
             Sum
             is
             5
             /
             9q
             ,
             which
             being
             divided
             by
             3
             ,
             the
             Number
             of
             Gamesters
             gives
             5
             /
             27q
             ,
             which
             is
             the
             Share
             of
             B
             sought
             for
             :
             The
             Demonstration
             of
             this
             is
             clear
             from
             the
             Second
             Proposition
             ,
             because
             B
             has
             an
             equal
             Hazard
             to
             gain
             o
             4
             /
             9
             q
             or
             1
             /
             9q
             ,
             that
             is
             ,
             
             ,
             
               i.
               e.
               5
               /
               27q
            
             ;
             now
             it
             's
             evident
             the
             Divisor
             3
             is
             the
             Number
             of
             the
             Gamesters
             .
          
           
             To
             find
             what
             is
             due
             to
             one
             in
             any
             Case
             ,
             viz.
             if
             either
             he
             ,
             or
             any
             of
             his
             Fellow-Gamsters
             win
             the
             following
             Game
             ;
             
             we
             must
             consider
             first
             the
             more
             simple
             Cases
             ,
             and
             by
             their
             help
             the
             following
             ;
             for
             as
             this
             Case
             could
             not
             be
             solv'd
             before
             the
             Case
             of
             the
             Eighth
             Proposition
             was
             calculated
             ,
             in
             which
             ,
             the
             Games
             wanting
             were
             1
             ,
             1
             ,
             2
             ;
             so
             the
             Case
             ,
             where
             the
             Games
             wanting
             are
             1
             ,
             2
             ,
             3
             ,
             cannot
             be
             calculated
             ,
             without
             the
             Calculation
             of
             the
             Case
             ,
             where
             the
             Games
             wanting
             are
             1
             ,
             2
             ,
             2
             ,
             (
             which
             we
             have
             just
             now
             perform'd
             )
             and
             likewise
             of
             the
             Case
             ,
             where
             the
             Games
             wanting
             are
             1
             ,
             1
             ,
             3
             ,
             which
             can
             be
             done
             by
             the
             Eighth
             :
             And
             by
             this
             means
             you
             may
             reckon
             all
             the
             Cases
             comprehended
             in
             the
             following
             Tables
             ,
             and
             an
             infinite
             number
             of
             others
             .
             
             
             
             
             
             
          
           
           
             As
             for
             the
             Dice
             ;
             these
             Questions
             may
             be
             proposed
             ,
             at
             how
             many
             Throws
             one
             may
             wager
             to
             throw
             6
             ,
             or
             any
             Number
             below
             that
             ,
             with
             one
             Dye
             ;
             How
             many
             Throws
             are
             required
             for
             12
             upon
             two
             Dice
             ;
             or
             18
             on
             3
             ;
             and
             several
             other
             Questions
             to
             this
             purpose
             .
          
           
             For
             the
             resolving
             of
             which
             ,
             it
             must
             be
             consider'd
             ,
             that
             in
             one
             Dye
             there
             are
             six
             different
             Throws
             ,
             all
             equally
             probable
             to
             come
             up
             ;
             for
             I
             suppose
             the
             Dye
             has
             the
             exact
             figure
             of
             a
             Cube
             :
             On
             Two
             Dice
             there
             are
             36
             different
             Throws
             ;
             for
             in
             respect
             to
             every
             Throw
             of
             One
             Dye
             ,
             any
             One
             Throw
             of
             the
             6
             of
             the
             other
             Dye
             may
             come
             up
             ;
             and
             6
             times
             
             6
             make
             36
             :
             In
             Three
             Dice
             there
             are
             216
             different
             Throws
             ;
             for
             in
             relation
             to
             any
             of
             the
             36
             Throws
             of
             Two
             Dice
             ,
             any
             one
             of
             the
             six
             of
             the
             Third
             may
             come
             up
             ;
             and
             6
             times
             36
             make
             216
             :
             So
             in
             Four
             Dice
             there
             are
             6
             times
             216
             Throws
             ,
             that
             is
             ,
             1296
             :
             And
             so
             forward
             you
             may
             reckon
             the
             Throws
             of
             any
             Number
             of
             Dice
             ,
             taking
             always
             ,
             for
             the
             addition
             of
             a
             new
             Dye
             ,
             6
             times
             the
             Number
             of
             the
             preceeding
             .
          
           
             Besides
             ,
             it
             must
             be
             observ'd
             ,
             that
             in
             Two
             Dice
             there
             is
             only
             one
             way
             2
             or
             12
             can
             come
             up
             ;
             two
             ways
             that
             3
             or
             11
             can
             come
             up
             ;
             for
             if
             I
             shall
             call
             the
             Dice
             A
             and
             B
             to
             make
             3
             ,
             there
             
             may
             be
             1
             in
             A
             and
             2
             in
             B
             ,
             or
             2
             in
             A
             and
             1
             in
             B
             ;
             so
             to
             make
             11
             ,
             there
             may
             be
             5
             in
             A
             or
             6
             in
             B
             ,
             or
             6
             in
             A
             and
             5
             in
             B
             ;
             for
             4
             there
             are
             three
             Chances
             ,
             3
             in
             A
             and
             1
             in
             B
             ,
             3
             in
             B
             and
             1
             in
             A
             ,
             or
             2
             as
             well
             in
             A
             as
             B
             ;
             for
             10
             there
             are
             likewise
             three
             Chances
             ;
             for
             5
             or
             9
             there
             are
             four
             Chances
             ;
             for
             6
             or
             8
             five
             Chances
             ;
             for
             7
             there
             are
             six
             Chances
             .
          
           
             In
             3
             Dice
             there
             are
             found
             for
             
               
                 3
                 or
                 18
                 1
              
               
                 4
                 or
                 17
                 3
              
               
                 5
                 or
                 16
                 6
              
               
                 6
                 or
                 15
                 10
              
               
                 7
                 or
                 14
                 15
              
               
                 8
                 or
                 13
                 21
              
               
                 9
                 or
                 12
                 25
              
               
                 10
                 or
                 11
                 27
              
            
          
        
         
           
           
             PROP.
             X.
             
          
           
             To
             find
             at
             how
             many
             times
             one
             may
             undertake
             to
             throw
             6
             with
             One
             Dye
             .
          
           
             IF
             any
             should
             undertake
             to
             throw
             6
             the
             first
             time
             ,
             it
             's
             evident
             there
             's
             one
             Chance
             gives
             him
             the
             Stake
             ,
             and
             five
             which
             give
             him
             nothing
             ;
             for
             there
             are
             5
             Throws
             against
             him
             ,
             and
             only
             1
             for
             him
             :
             Let
             the
             Stake
             be
             call'd
             a
             ,
             then
             he
             has
             one
             Expectation
             to
             gain
             a
             ,
             and
             five
             to
             gain
             nothing
             ,
             which
             ,
             by
             the
             Second
             Proposition
             ,
             is
             worth
             1
             /
             6
             a
             ,
             and
             there
             remains
             for
             the
             other
             ⅚
             a
             ;
             so
             he
             who
             undertakes
             ,
             
             with
             one
             Dye
             ,
             to
             throw
             6
             the
             first
             time
             ,
             ought
             to
             wager
             only
             1
             to
             5.
             
          
           
             2.
             
             Suppose
             one
             undertake
             ,
             at
             two
             Throws
             of
             1
             Dye
             ,
             to
             throw
             6
             ,
             his
             Hazard
             is
             calculated
             thus
             ;
             if
             he
             throw
             6
             at
             the
             first
             he
             has
             a
             the
             Stake
             ,
             if
             he
             do
             not
             ,
             there
             remains
             to
             him
             one
             Throw
             ,
             which
             ,
             by
             the
             former
             Case
             ,
             is
             worth
             1
             /
             6
             a
             ;
             but
             there
             is
             but
             one
             Chance
             which
             gives
             him
             6
             at
             the
             first
             Throw
             ,
             and
             five
             Chances
             against
             him
             ;
             so
             there
             is
             one
             Chance
             which
             gives
             him
             a
             ,
             and
             five
             which
             give
             him
             1
             /
             ●
             a
             ,
             which
             ,
             by
             the
             Second
             Proposition
             ,
             is
             worth
             11
             /
             36
             a
             ,
             so
             there
             remains
             to
             his
             Fellow-Gamester
             2●
             /
             3●
             a
             ;
             so
             the
             Value
             of
             my
             Expectation
             to
             his
             ,
             is
             
             as
             11
             to
             25
             ,
             
               i.
               e.
            
             less
             than
             1
             to
             2.
             
          
           
             By
             the
             same
             method
             of
             calculation
             ,
             you
             will
             find
             ,
             that
             his
             Hazard
             who
             undertakes
             to
             throw
             6
             at
             three
             times
             with
             one
             Dye
             ,
             is
             91
             /
             216
             a
             ;
             so
             that
             he
             can
             only
             lay
             91
             against
             125
             ,
             which
             is
             something
             less
             than
             3
             to
             4.
             
          
           
             He
             who
             undertakes
             to
             do
             it
             at
             four
             times
             ,
             his
             Hazard
             is
             671
             /
             1296
             a
             ,
             so
             he
             may
             wager
             671
             against
             625
             ,
             that
             is
             ,
             something
             more
             than
             1
             to
             1.
             
          
           
             He
             who
             undertakes
             to
             do
             it
             at
             five
             times
             ,
             his
             Hazard
             is
             4651
             /
             7776
             a
             ,
             so
             he
             can
             wager
             4651
             against
             3125
             ,
             that
             is
             something
             less
             than
             3
             to
             2.
             
          
           
           
             His
             Hazard
             who
             undertakes
             to
             do
             it
             6
             times
             ,
             is
             3●031
             /
             45656
             a
             ,
             and
             he
             can
             wager
             31031
             against
             15625
             ,
             that
             is
             something
             less
             than
             2
             to
             1.
             
          
           
             Thus
             any
             Numb
             .
             of
             Throws
             may
             be
             easily
             found
             ,
             but
             the
             following
             Proposition
             will
             shew
             you
             a
             more
             compendious
             way
             of
             Calculation
             .
          
        
         
           
             PROP.
             XI
             .
          
           
             To
             find
             at
             how
             many
             times
             one
             may
             undertake
             to
             throw
             12
             with
             Two
             Dice
             .
          
           
             IF
             one
             should
             undertake
             it
             at
             One
             Throw
             ,
             it
             's
             clear
             he
             has
             but
             one
             Chance
             to
             get
             the
             
             Stake
             a
             ,
             and
             35
             to
             get
             nothing
             ;
             so
             ,
             by
             the
             Second
             Proposition
             ,
             he
             has
             much
             1
             /
             36
             a.
             
          
           
             He
             who
             undertakes
             to
             do
             it
             at
             Twice
             ,
             if
             he
             throw
             12
             the
             first
             time
             gains
             a
             ,
             if
             otherwise
             ,
             then
             there
             remains
             to
             him
             One
             Throw
             ,
             which
             ,
             by
             the
             former
             Case
             ,
             is
             worth
             1
             /
             36
             a
             ;
             but
             there
             is
             but
             One
             Chance
             which
             gives
             12
             at
             the
             first
             Throw
             ,
             and
             35
             Chances
             against
             him
             ;
             so
             he
             has
             1
             Chance
             for
             a
             ,
             and
             35
             for
             1
             /
             36
             a
             ,
             which
             ,
             by
             the
             Second
             Proposition
             ,
             is
             worth
             71
             /
             1296
             a
             ,
             and
             there
             remains
             to
             his
             Fellow-Gamester
             1●2●
             /
             1296
             a.
             
          
           
             From
             these
             it's
             easie
             to
             find
             the
             Value
             of
             his
             Hazard
             ,
             who
             undertakes
             it
             at
             four
             times
             ;
             passing
             
             by
             his
             Case
             who
             undertakes
             it
             at
             three
             times
             .
          
           
             If
             he
             who
             undertakes
             to
             do
             it
             at
             four
             times
             throws
             12
             the
             first
             or
             second
             Cast
             ,
             then
             he
             has
             a
             ,
             if
             not
             ,
             there
             remains
             two
             other
             Throws
             ,
             which
             ,
             by
             the
             former
             Case
             ,
             are
             worth
             71
             /
             1296
             a
             ;
             but
             for
             the
             same
             reason
             ,
             in
             his
             two
             first
             Throws
             ,
             he
             has
             71
             Chances
             which
             give
             him
             a
             ,
             against
             1225
             Chances
             ,
             in
             which
             it
             may
             happen
             otherwise
             ;
             therefore
             at
             first
             he
             has
             71
             Chances
             which
             give
             him
             a
             ,
             and
             1225
             which
             give
             him
             71
             /
             1296
             a
             ,
             which
             ,
             by
             the
             Second
             Proposition
             ,
             is
             worth
             19006●5
             /
             167961●
             a
             ,
             which
             shews
             that
             their
             Hazards
             to
             one
             another
             are
             as
             178991
             to
             1500625.
             
          
           
           
             From
             which
             Cases
             it
             is
             easie
             to
             find
             the
             Value
             of
             his
             Expectation
             ,
             who
             undertakes
             to
             do
             it
             at
             8
             times
             ,
             and
             from
             that
             ,
             his
             Case
             who
             undertakes
             to
             do
             it
             at
             16
             times
             ;
             and
             from
             his
             Case
             who
             undertakes
             to
             do
             it
             at
             8
             times
             ;
             and
             his
             likewise
             who
             undertakes
             to
             do
             it
             at
             16
             times
             ;
             it
             is
             easie
             to
             determin
             his
             Expectation
             who
             undertakes
             it
             at
             24
             times
             :
             In
             which
             Operation
             ,
             because
             that
             which
             is
             principally
             sought
             ,
             is
             the
             Number
             of
             Throws
             ,
             which
             makes
             the
             Hazard
             equal
             on
             both
             sides
             ,
             viz.
             to
             him
             who
             undertakes
             ,
             and
             he
             who
             offers
             ,
             you
             may
             ,
             without
             any
             sensible
             Error
             ,
             from
             the
             Numbers
             (
             which
             else
             would
             
             grow
             very
             great
             )
             cut
             off
             some
             of
             the
             last
             Figures
             .
             And
             so
             I
             find
             ,
             that
             he
             who
             undertakes
             to
             throw
             12
             with
             Two
             Dice
             ,
             at
             24
             times
             ,
             has
             some
             Loss
             ,
             and
             he
             who
             undertakes
             it
             at
             25
             times
             ,
             has
             some
             Advantage
             .
          
        
         
           
             PROP.
             XII
             .
          
           
             To
             find
             with
             how
             many
             Dice
             ,
             one
             can
             undertake
             to
             throw
             two
             Sixes
             at
             the
             first
             Cast.
             
          
           
             THis
             is
             as
             much
             ,
             as
             if
             one
             would
             know
             ,
             at
             how
             many
             Throws
             of
             one
             Dye
             ,
             he
             may
             undertake
             to
             throw
             twice
             Six
             ;
             now
             if
             any
             should
             undertake
             it
             at
             two
             Throws
             ,
             by
             
             what
             we
             have
             shewn
             before
             ,
             his
             Hazard
             would
             be
             1
             /
             36
             a
             ,
             he
             who
             would
             undertake
             to
             do
             it
             at
             3
             ;
             times
             ,
             if
             his
             first
             Throw
             were
             not
             6
             ,
             then
             there
             would
             remain
             two
             Throws
             ,
             each
             of
             which
             must
             be
             6
             ,
             which
             (
             as
             we
             have
             said
             )
             is
             worth
             1
             /
             36
             a
             ;
             but
             if
             the
             first
             Throw
             be
             6
             ,
             he
             wants
             only
             one
             6
             in
             the
             two
             following
             Throws
             ,
             which
             by
             the
             Tenth
             Proposition
             ,
             is
             worth
             11
             /
             36
             a
             ;
             but
             since
             he
             has
             but
             one
             Chance
             to
             get
             6
             the
             first
             Throw
             ,
             and
             five
             to
             miss
             it
             ;
             he
             has
             therefore
             ,
             at
             first
             ,
             one
             Chance
             for
             11
             /
             36
             a
             ,
             and
             five
             Chances
             for
             1
             /
             36
             a
             ,
             which
             ,
             by
             the
             Second
             Proposition
             ,
             is
             worth
             16
             /
             216
             a
             ,
             or
             2
             /
             27
             a
             ,
             after
             this
             manner
             still
             assuming
             1
             Chance
             
             more
             ,
             you
             will
             find
             that
             you
             may
             undertake
             to
             throw
             two
             Sixes
             at
             10
             Throws
             of
             one
             Dye
             ,
             or
             1
             Throw
             of
             ten
             Dice
             ,
             and
             that
             with
             some
             Advantage
             .
          
        
         
           
             PROP.
             XIII
             .
          
           
             If
             I
             am
             to
             play
             with
             another
             One
             Throw
             ,
             on
             this
             condition
             ,
             that
             if
             7
             comes
             up
             I
             gain
             ,
             if
             10
             he
             gains
             ;
             if
             it
             happens
             that
             we
             must
             divide
             the
             Stake
             ,
             and
             not
             play
             ,
             to
             find
             how
             much
             belongs
             to
             me
             ,
             and
             how
             much
             to
             him
             .
          
           
             BEcause
             of
             the
             36
             different
             Throws
             of
             the
             Two
             Dice
             ,
             there
             are
             six
             which
             give
             
             7
             and
             3
             ,
             which
             give
             10
             and
             27
             ,
             which
             equals
             the
             Game
             ,
             in
             which
             Case
             there
             is
             due
             to
             each
             of
             us
             ½
             a
             :
             But
             if
             none
             of
             the
             27
             should
             happen
             ,
             I
             have
             6
             ,
             by
             which
             I
             may
             gain
             a
             ,
             and
             3
             ,
             by
             which
             I
             may
             get
             nothing
             ,
             which
             ,
             by
             the
             Second
             Proposition
             ,
             is
             worth
             ⅔
             a
             ;
             so
             I
             have
             27
             Chances
             for
             ½
             a
             ,
             and
             9
             for
             ⅔
             a
             ,
             which
             ,
             by
             the
             second
             Proposition
             ,
             is
             worth
             13
             /
             24
             a
             ,
             and
             there
             remains
             to
             my
             Fellow-Gamester
             11
             /
             24
             a.
             
          
        
         
           
           
             PROP.
             XIV
             .
          
           
             If
             I
             were
             playing
             with
             another
             by
             turns
             ,
             with
             two
             Dice
             ,
             on
             this
             condition
             ,
             that
             if
             I
             throw
             7
             I
             gain
             ,
             and
             if
             he
             throw
             6
             he
             gains
             ,
             allowing
             him
             the
             first
             Throw
             :
             To
             find
             the
             proportion
             of
             my
             Hazard
             to
             his
             .
          
           
             SUppose
             I
             call
             the
             Value
             of
             my
             Hazard
             x
             ,
             and
             the
             Stakes
             a
             ,
             then
             his
             Hazard
             will
             be
             
               a
               −
               x
            
             ;
             then
             whenever
             it
             's
             his
             turn
             to
             throw
             ,
             my
             Hazard
             is
             x
             ,
             but
             when
             it
             's
             mine
             ,
             the
             Value
             of
             my
             Hazard
             is
             greater
             .
             Suppose
             I
             then
             call
             it
             y
             ;
             now
             because
             of
             the
             36
             Throws
             of
             
             Two
             Dice
             ,
             there
             are
             five
             which
             give
             my
             Fellow-Gamester
             6
             ,
             thirty
             one
             which
             bring
             it
             again
             to
             my
             turn
             to
             throw
             ,
             I
             have
             five
             Chances
             for
             nothing
             ,
             and
             thirty
             one
             for
             y
             ,
             which
             ,
             by
             the
             Third
             Proposition
             ,
             is
             worth
             31
             /
             36
             y
             ;
             but
             I
             suppos'd
             at
             first
             my
             Hazard
             to
             be
             x
             ;
             therefore
             31
             /
             36
             
               y
               =
               x
            
             ,
             and
             consequently
             
               y
               =
               36
               /
               3●x
            
             .
             I
             suppos'd
             likewise
             ,
             when
             it
             was
             my
             turn
             to
             throw
             ,
             the
             Value
             of
             my
             Hazard
             was
             y
             ,
             but
             then
             I
             have
             six
             Chances
             which
             give
             me
             7
             ,
             and
             consequently
             the
             Stake
             ,
             and
             thirty
             which
             give
             my
             Fellow
             the
             Dice
             ,
             that
             is
             ,
             make
             my
             Hazard
             worth
             x
             ;
             so
             I
             have
             six
             Chances
             for
             a
             ,
             and
             thirty
             for
             x
             ,
             which
             ,
             by
             
             Prop.
             3.
             is
             worth
             
             but
             this
             by
             supposition
             is
             equal
             to
             y
             ,
             which
             is
             equal
             (
             by
             what
             has
             been
             prov'd
             already
             )
             to
             36
             /
             3●
             x
             ;
             therefore
             
             ,
             and
             consequently
             
               x
               =
               31
               /
               6●a
            
             ,
             the
             Value
             of
             my
             Hazard
             ,
             and
             that
             of
             my
             Fellow-Gamester
             is
             30
             /
             61
             a
             ;
             so
             that
             mine
             is
             to
             his
             as
             31
             to
             30.
             
          
           
             
               Here
               follow
               some
               Questions
               which
               serve
               to
               exercise
               the
               former
               Rules
               .
            
          
           
             1.
             
             A
             and
             B
             play
             together
             with
             two
             Dice
             ,
             A
             wins
             if
             he
             throws
             6
             ,
             and
             B
             if
             he
             throws
             7
             ;
             A
             at
             first
             gets
             one
             Throw
             ,
             then
             B
             two
             ,
             then
             A
             two
             ,
             and
             
             so
             on
             by
             turns
             ,
             till
             one
             of
             them
             wins
             .
             I
             require
             the
             proportion
             of
             A's
             Hazard
             to
             B's
             ?
             Answer
             ,
             It
             is
             as
             10355
             to
             12276.
             
          
           
             2.
             
             Three
             Gamesters
             ,
             A
             ,
             B
             ,
             and
             C
             ,
             take
             12
             Counters
             ,
             of
             which
             there
             are
             four
             white
             and
             eight
             black
             ;
             the
             Law
             of
             the
             Game
             is
             this
             ,
             that
             he
             shall
             win
             ,
             who
             ,
             hood-wink'd
             ,
             shall
             first
             chuse
             a
             white
             Counter
             ,
             and
             that
             A
             shall
             have
             the
             first
             choice
             ,
             B
             the
             second
             ,
             and
             C
             the
             third
             ,
             and
             so
             ,
             by
             turns
             ,
             till
             one
             of
             them
             win
             .
             Quaer
             .
             What
             is
             the
             proportion
             of
             their
             Hazards
             ?
          
           
             3.
             
             A
             wagers
             with
             B
             ,
             that
             of
             40
             Cards
             ,
             that
             is
             ,
             10
             of
             
             every
             Suit
             ,
             he
             will
             pick
             out
             four
             ;
             so
             that
             there
             shall
             be
             one
             of
             every
             suit
             ;
             A's
             Hazard
             to
             B's
             ,
             in
             this
             Case
             ,
             is
             as
             1000
             to
             8139.
             
          
           
             4.
             
             Supposing
             ,
             as
             before
             ,
             4
             white
             Counters
             and
             8
             black
             ,
             A
             wagers
             with
             B
             ,
             that
             out
             of
             them
             ,
             he
             shall
             pick
             7
             Counters
             ,
             of
             which
             there
             are
             3
             white
             .
             I
             require
             the
             proportion
             of
             A's
             Hazard
             to
             B's
             ?
          
           
             5.
             
             A
             and
             B
             taking
             12
             Counters
             ,
             play
             with
             three
             Dice
             after
             this
             manner
             ;
             that
             if
             12
             comes
             up
             ,
             A
             shall
             give
             one
             Counter
             to
             B
             ,
             but
             if
             14
             comes
             up
             ,
             B
             shall
             give
             one
             to
             A
             ,
             and
             that
             
             he
             shall
             gain
             who
             first
             has
             all
             the
             Counters
             .
             A's
             Hazard
             to
             B's
             is
             244140625
             to
             282429
             536481.
             
          
           
             The
             Calculus
             of
             the
             preceeding
             Problems
             is
             left
             out
             by
             Mons.
             Hugens
             ,
             on
             purpose
             that
             the
             ingenious
             Reader
             may
             have
             the
             satisfaction
             of
             applying
             the
             former
             Method
             himself
             ;
             it
             is
             in
             most
             of
             them
             more
             laborious
             than
             difficult
             ;
             for
             Example
             ,
             I
             have
             pitch'd
             upon
             the
             Second
             and
             Third
             ,
             because
             the
             rest
             can
             be
             solv'd
             after
             the
             same
             Method
             .
          
           
             
             
               Problem
               1.
               
            
             
               The
               first
               Problem
               is
               solv'd
               by
               the
               Method
               of
               Prop.
               14.
               only
               with
               this
               difference
               ,
               that
               after
               you
               have
               found
               the
               share
               due
               to
               B
               ,
               if
               A
               were
               to
               get
               no
               first
               Throw
               ,
               you
               must
               subtract
               from
               it
               5
               /
               36
               of
               the
               Stake
               which
               is
               due
               to
               A
               for
               his
               Hazard
               of
               throwing
               Six
               at
               the
               first
               Throw
               .
            
          
           
             
               Probl.
               2
            
             
               As
               for
               the
               second
               Problem
               ,
               it
               is
               solved
               thus
               ,
               Suppose
               A's
               Hazard
               ,
               when
               it
               is
               his
               own
               turn
               to
               chuse
               ,
               be
               x
               ,
               when
               it
               is
               B's
               ,
               be
               y
               ,
               and
               when
               it
               is
               C's
               ,
               
               be
               z
               ;
               it
               is
               evident
               ,
               when
               out
               of
               12
               Counters
               ,
               of
               which
               there
               are
               4
               white
               and
               8
               black
               ,
               he
               endeavours
               to
               chuse
               a
               white
               one
               ,
               he
               has
               four
               Chances
               to
               get
               it
               ,
               and
               eight
               to
               miss
               it
               ,
               that
               is
               ,
               he
               has
               four
               Chances
               to
               get
               the
               Stake
               a
               ,
               and
               eight
               to
               make
               his
               Hazard
               worth
               y
               ;
               so
               
               ,
               and
               consequently
               
               .
               When
               it
               is
               B's
               turn
               to
               chuse
               ,
               then
               he
               has
               four
               Chances
               for
               nothing
               ,
               and
               eight
               for
               z
               ,
               (
               that
               is
               to
               bring
               it
               to
               C's
               turn
               )
               consequently
               
               ;
               this
               equation
               reduc'd
               
               gives
               
               ;
               when
               it
               comes
               to
               C's
               turn
               to
               chuse
               then
               A
               has
               four
               Chances
               for
               nothing
               ,
               and
               eight
               for
               x
               ,
               consequently
               
                 z
                 =
                 8
                 /
                 12x
              
               ,
               therefore
               
               ;
               this
               equation
               reduc'd
               gives
               
                 x
                 =
                 9
                 /
                 19a
              
               ,
               and
               consequently
               there
               remains
               to
               the
               B
               and
               C
               10
               /
               19
               a
               ,
               which
               must
               be
               shar'd
               after
               the
               same
               manner
               ,
               that
               is
               ,
               so
               that
               B
               have
               the
               first
               Choice
               ,
               C
               the
               next
               ,
               and
               so
               on
               ,
               till
               one
               of
               them
               gain
               ;
               the
               reason
               is
               ,
               because
               it
               had
               been
               just
               in
               A
               to
               have
               demanded
               9
               /
               19
               of
               the
               Stake
               for
               not
               playing
               ,
               and
               then
               the
               seniority
               fell
               to
               B
               ;
               
               now
               10
               /
               19
               a
               parted
               betwixt
               B
               and
               C
               ,
               by
               the
               former
               method
               ,
               gives
               6
               /
               19
               to
               B
               ,
               and
               4
               /
               19
               to
               C
               ;
               so
               A
               ,
               B
               ,
               and
               C's
               Hazards
               ,
               from
               the
               beginning
               ,
               were
               as
               9
               ,
               6
               ,
               4.
               
            
             
               I
               have
               suppos'd
               here
               the
               sense
               of
               the
               Problem
               to
               be
               ,
               that
               when
               any
               one
               chus'd
               a
               Counter
               ,
               he
               did
               not
               diminish
               their
               Number
               ;
               but
               if
               he
               miss'd
               of
               a
               white
               one
               ,
               put
               it
               in
               again
               ,
               and
               left
               an
               equal
               Hazard
               to
               him
               who
               had
               the
               following
               Choice
               ;
               for
               if
               it
               be
               otherwise
               suppos'd
               ,
               A's
               share
               will
               be
               55
               /
               123
               ,
               which
               is
               less
               than
               9
               /
               19.
               
            
             
               Prob.
               2.
               
               It
               is
               evident
               ,
               that
               wagering
               to
               pick
               out
               4
               Cards
               out
               of
               40
               ,
               so
               that
               there
               be
               one
               of
               every
               Suit
               ,
               is
               no
               more
               ,
               than
               wagering
               ,
               
               out
               of
               39
               Cards
               to
               take
               3
               which
               shall
               be
               of
               three
               proposed
               Suits
               ;
               for
               it
               is
               all
               one
               which
               Card
               you
               draw
               first
               ,
               all
               the
               Hazard
               being
               ,
               whether
               out
               of
               the
               39
               remaining
               you
               take
               3
               ,
               of
               which
               none
               shall
               be
               of
               the
               Suit
               you
               first
               drew
               .
               Suppose
               then
               you
               had
               gone
               right
               for
               three
               times
               ,
               and
               were
               to
               draw
               your
               last
               Card
               ,
               it
               is
               clear
               ,
               that
               there
               are
               ●7
               Cards
               ,
               
                 (
                 viz.
              
               of
               the
               Suits
               you
               have
               drawn
               before
               )
               of
               which
               ,
               if
               you
               draw
               any
               you
               lose
               ,
               and
               10
               of
               which
               ,
               if
               you
               draw
               any
               ,
               you
               have
               the
               Stake
               a
               ;
               so
               you
               have
               10
               Chances
               for
               a
               ,
               and
               27
               for
               nothing
               ,
               which
               ,
               by
               Prop.
               3.
               is
               worth
               10
               /
               37
               a.
               Suppose
               
               again
               you
               had
               gone
               right
               only
               for
               two
               Draughts
               ,
               then
               you
               have
               18
               Cards
               (
               of
               the
               Suits
               you
               have
               drawn
               before
               )
               which
               make
               you
               lose
               ,
               and
               20
               ,
               which
               put
               you
               in
               the
               Case
               suppos'd
               formerly
               ,
               viz.
               where
               you
               have
               but
               one
               Card
               to
               draw
               ,
               which
               ,
               as
               we
               have
               already
               calculated
               ,
               is
               worth
               10
               /
               37
               a
               ;
               so
               you
               have
               18
               Chances
               for
               nothing
               ,
               and
               20
               for
               10
               /
               37
               a
               ,
               which
               ,
               by
               Prop.
               3.
               is
               worth
               100
               /
               703
               a.
               Suppose
               again
               you
               have
               3
               Cards
               to
               draw
               ,
               then
               you
               have
               9
               (
               of
               the
               Suit
               you
               drew
               first
               )
               which
               make
               you
               lose
               ,
               and
               30
               which
               put
               you
               in
               the
               Case
               suppos'd
               last
               ;
               so
               you
               have
               9
               Chances
               for
               nothing
               ,
               and
               30
               for
               100
               /
               703
               a
               ,
               which
               ,
               by
               
               Prop.
               3.
               is
               worth
               3000
               /
               27417
               a
               ,
               or
               1000
               /
               9139
               a
               ,
               and
               you
               leave
               to
               your
               Fellow-Gamester
               8139
               /
               9139
               a
               ;
               so
               your
               Hazard
               is
               to
               his
               as
               1000
               to
               8139.
               
            
             
               It
               is
               easie
               to
               apply
               this
               Method
               to
               the
               Games
               that
               are
               in
               use
               amongst
               us
               :
               For
               Example
               ,
               If
               A
               and
               B
               ,
               playing
               at
               Backgammon
               ,
               B
               had
               already
               gain'd
               one
               end
               of
               three
               ,
               and
               A
               none
               ,
               and
               if
               A
               had
               the
               Dice
               in
               his
               Hand
               for
               the
               last
               Throw
               of
               the
               second
               end
               ,
               all
               his
               Men
               but
               two
               upon
               the
               Ace
               Point
               being
               already
               cast
               of
               :
               Quaer
               .
               What
               is
               the
               proportion
               of
               As
               Hazard
               to
               Bs
               ?
            
             
               Solution
               :
               There
               being
               of
               the
               36
               Throws
               of
               two
               Dice
               ,
               six
               which
               give
               Doublets
               ;
               if
               A
               
               throw
               any
               of
               the
               Six
               ,
               he
               has
               the
               Stake
               a
               ;
               if
               he
               throw
               any
               of
               the
               other
               Thirty
               ,
               then
               he
               wants
               but
               one
               Game
               ,
               and
               his
               Fellow-Gamester
               three
               ,
               which
               ,
               by
               Prop.
               V.
               is
               worth
               ⅞
               a
               ;
               so
               A
               has
               six
               Chances
               for
               a
               ,
               and
               thirty
               for
               ⅞
               a
               ,
               which
               ,
               by
               Prop.
               3.
               is
               worth
               129
               /
               144
               a
               ,
               and
               there
               remains
               to
               his
               Play-Fellow
               15
               /
               144
               a
               ;
               so
               A's
               Hazard
               to
               B's
               ,
               is
               as
               129
               to
               15
               ,
               that
               is
               ,
               less
               than
               9
               to
               1.
               
            
             
               Supposing
               the
               same
               Case
               ,
               and
               if
               their
               Bargain
               had
               been
               ,
               that
               he
               who
               gain'd
               three
               ends
               before
               the
               other
               gain'd
               one
               ,
               should
               have
               double
               of
               what
               each
               stak'd
               ,
               that
               is
               ,
               the
               Stake
               and
               a
               half
               more
               ,
               then
               there
               had
               been
               due
               to
               A
               282
               /
               285
               of
               the
               
               Stake
               ,
               that
               is
               ,
               B
               ought
               only
               to
               take
               1
               /
               ●●
               ,
               and
               leave
               the
               rest
               to
               A.
               
            
             
               Thus
               likewise
               ,
               if
               you
               apply
               the
               former
               Rule
               to
               the
               
                 Royal
                 Oak-Lottery
              
               ,
               you
               will
               find
               ,
               that
               he
               who
               wagers
               that
               any
               Figure
               shall
               come
               up
               at
               the
               first
               throw
               ,
               ought
               to
               wagers
               1
               against
               31
               ;
               that
               he
               who
               wagers
               it
               shall
               come
               up
               at
               one
               of
               two
               throws
               ,
               ought
               to
               wager
               63
               against
               961
               ;
               that
               he
               who
               wagers
               that
               a
               Figure
               shall
               come
               up
               at
               once
               in
               three
               times
               ,
               ought
               to
               lay
               124955
               against
               923621
               ,
               &c.
               it
               being
               only
               somewhat
               tedious
               to
               calculate
               the
               rest
               .
               Where
               you
               will
               find
               ,
               that
               the
               equality
               will
               not
               fall
               as
               some
               imagin
               on
               16
               Throws
               ,
               no
               more
               than
               the
               equality
               
               of
               wagering
               at
               how
               many
               Throws
               of
               one
               Dye
               6
               shall
               come
               up
               ,
               falls
               on
               three
               ;
               the
               contrary
               of
               which
               you
               have
               seen
               already
               demonstrated
               ;
               you
               will
               find
               by
               calculation
               ,
               that
               he
               has
               the
               Disadvantage
               ,
               who
               wagers
               ,
               that
               1
               of
               the
               32
               different
               Throws
               of
               the
               
                 Royal
                 Oak-Lottery
              
               ,
               shall
               come
               at
               once
               of
               20
               times
               ,
               and
               that
               he
               has
               some
               Advantage
               ,
               who
               wagers
               on
               22
               times
               ;
               so
               the
               nearest
               to
               Equality
               is
               on
               21
               times
               :
               But
               it
               must
               be
               remembred
               ,
               that
               I
               have
               suppos'd
               in
               the
               former
               Calculation
               ,
               the
               Ball
               in
               the
               
                 Royal
                 Oak-Lottery
              
               to
               be
               regular
               ,
               tho
               it
               can
               never
               be
               exactly
               so
               ;
               for
               he
               who
               has
               the
               smallest
               Skill
               in
               
               Geometry
               ,
               knows
               ,
               that
               there
               can
               be
               no
               regular
               Body
               of
               32
               sides
               ,
               and
               yet
               this
               can
               be
               of
               no
               advantàge
               to
               him
               who
               keeps
               it
               .
            
          
        
         
           
             To
             find
             the
             Value
             of
             the
             Throws
             of
             Dice
             as
             to
             the
             Quantity
             .
          
           
             NOthing
             is
             more
             easie
             ,
             than
             by
             the
             former
             Method
             to
             determine
             the
             Value
             of
             any
             Number
             of
             Throws
             of
             any
             Number
             of
             Dice
             ;
             for
             in
             one
             Throw
             of
             a
             Dye
             ,
             I
             have
             an
             equal
             chance
             for
             1
             ,
             2
             ,
             3
             ,
             4
             ,
             5
             ,
             6
             ,
             consequently
             my
             Hazard
             is
             
             worth
             their
             Sum
             21
             divided
             by
             their
             Number
             6
             ,
             that
             is
             ,
             3½
             .
             Now
             if
             one
             Throw
             of
             a
             Dye
             be
             worth
             3½
             ,
             then
             two
             Throws
             of
             a
             Dye
             ,
             or
             one
             Throw
             of
             two
             Dice
             is
             worth
             7
             ,
             two
             Throws
             of
             two
             Dice
             ,
             or
             one
             Throw
             of
             four
             Dice
             is
             worth
             14
             ,
             &c.
             
             The
             general
             Rule
             being
             to
             multiply
             the
             Number
             of
             Dice
             ,
             the
             Number
             of
             Throws
             ,
             and
             3½
             continually
             .
          
           
             This
             is
             not
             to
             be
             understood
             as
             if
             it
             were
             an
             equal
             Wager
             to
             throw
             7
             ,
             or
             above
             it
             ,
             with
             two
             Dice
             at
             one
             Throw
             ;
             for
             he
             who
             undertakes
             to
             do
             so
             ,
             has
             the
             advantage
             by
             21
             against
             15.
             
             The
             meaning
             is
             only
             ,
             if
             I
             were
             to
             have
             a
             Guinea
             ,
             a
             
             Shilling
             ,
             or
             any
             thing
             else
             ,
             for
             every
             Point
             that
             I
             threw
             with
             two
             Dice
             at
             one
             Throw
             ,
             my
             Hazard
             is
             worth
             7
             of
             these
             ,
             because
             he
             who
             gave
             me
             7
             for
             it
             ,
             would
             have
             an
             equal
             probability
             of
             gaining
             or
             losing
             by
             it
             ,
             the
             Chances
             of
             the
             Throws
             above
             7
             being
             as
             many
             ,
             as
             of
             these
             below
             it
             :
             So
             it
             is
             more
             than
             an
             equal
             Wager
             to
             throw
             14
             at
             least
             at
             two
             Throws
             of
             two
             Dice
             ,
             because
             it
             is
             more
             probable
             that
             14
             will
             come
             ,
             than
             any
             one
             Number
             besides
             ,
             and
             as
             probable
             that
             it
             will
             be
             above
             it
             as
             below
             it
             ;
             but
             if
             one
             were
             to
             buy
             this
             Hazard
             at
             the
             rate
             above-mention'd
             ,
             he
             ought
             just
             to
             give
             14
             for
             it
             .
             
             The
             equal
             Wager
             in
             one
             Throw
             of
             two
             Dice
             ,
             is
             to
             throw
             7
             at
             least
             one
             time
             ,
             and
             8
             at
             least
             another
             time
             ,
             and
             so
             
               per
               vices
            
             :
             The
             reason
             is
             ,
             because
             in
             the
             first
             Case
             I
             have
             21
             Chances
             against
             15
             ,
             and
             in
             the
             second
             15
             Chances
             against
             21.
             
          
        
         
           
           
             Of
             RAFFLING
             .
          
           
             IN
             Raiffing
             the
             different
             throws
             and
             their
             Chances
             are
             these
             ;
             Where
             it
             is
             to
             be
             observed
             ,
             that
             of
             the
             216
             different
             Throws
             of
             three
             Dice
             ,
             there
             are
             only
             96
             that
             give
             Doublets
             ,
             or
             two
             ,
             at
             least
             ,
             of
             a
             kind
             ;
             so
             it
             is
             4
             to
             5
             that
             with
             three
             Dice
             
               
                 
                   Throws
                   .
                
                 
                   Chan.
                   
                
              
               
                 
                   3
                   18
                
                 
                   1
                
              
               
                 
                   4
                   17
                
                 
                   3
                
              
               
                 
                   5
                   16
                
                 
                   6
                
              
               
                 
                   6
                   15
                
                 
                   4
                
              
               
                 
                   7
                   14
                
                 
                   9
                
              
               
                 
                   8
                   13
                
                 
                   9
                
              
               
                 
                   9
                   12
                
                 
                   7
                
              
               
                 
                   10
                   11
                
                 
                   9
                
              
            
             you
             shall
             throw
             Doublets
             ,
             and
             it
             is
             1
             to
             35
             that
             you
             throw
             a
             Raffle
             ,
             or
             all
             three
             of
             a
             kind
             .
             
             It
             is
             evident
             likewise
             ,
             that
             it
             is
             an
             even
             Wager
             to
             throw
             11
             or
             above
             it
             ,
             because
             there
             are
             as
             many
             Chances
             for
             11
             ,
             and
             the
             Throws
             above
             it
             ,
             as
             for
             the
             Throws
             below
             it
             ;
             but
             tho
             it
             be
             an
             even
             Wager
             to
             throw
             11
             at
             one
             Throw
             ,
             it
             is
             a
             disadvantage
             to
             wager
             to
             throw
             22
             at
             two
             Throws
             ,
             and
             far
             more
             to
             wager
             to
             throw
             33
             at
             three
             Throws
             ;
             and
             yet
             it
             is
             more
             than
             an
             equal
             Wager
             that
             you
             shall
             throw
             21
             at
             two
             Throws
             in
             Raffling
             ,
             because
             it
             is
             as
             probable
             that
             you
             will
             ,
             as
             that
             you
             will
             not
             throw
             11
             ,
             at
             least
             ,
             the
             first
             time
             ,
             and
             more
             than
             probable
             that
             you
             will
             throw
             10
             ,
             at
             least
             ,
             the
             second
             time
             .
          
           
           
             For
             an
             instance
             of
             the
             plainness
             of
             the
             preceeding
             Method
             ,
             I
             will
             shew
             ,
             how
             by
             simple
             Subtraction
             ,
             the
             most
             part
             of
             the
             former
             Problems
             may
             be
             solv'd
             .
          
           
             Suppose
             A
             and
             B
             ,
             playing
             together
             ,
             each
             of
             'em
             stakes
             32
             Shillings
             ,
             and
             that
             A
             wants
             one
             Game
             of
             the
             Number
             agreed
             on
             ,
             and
             B
             wants
             two
             ;
             to
             find
             the
             share
             of
             the
             Stakes
             due
             to
             each
             of
             '
             em
             .
             It
             's
             plain
             ,
             if
             A
             wins
             the
             next
             Game
             he
             has
             the
             whole
             64
             Shillings
             ;
             if
             B
             wins
             it
             ,
             then
             their
             Shares
             are
             equal
             ;
             therefore
             says
             A
             to
             B
             ,
             If
             you
             will
             break
             off
             the
             Game
             ,
             give
             me
             32
             ,
             which
             I
             am
             sure
             of
             ,
             whether
             I
             win
             or
             lose
             the
             
             next
             Game
             ,
             and
             since
             you
             will
             not
             venture
             for
             the
             other
             32
             ,
             let
             us
             part
             them
             equally
             ,
             that
             is
             ,
             give
             me
             16
             ,
             which
             ,
             with
             the
             former
             32
             ,
             make
             48
             ,
             leaving
             16
             to
             you
             .
          
           
             Suppose
             A
             wanted
             one
             Game
             ,
             and
             B
             three
             ;
             if
             A
             wins
             the
             next
             Game
             ,
             he
             has
             the
             64
             Shillings
             ;
             if
             B
             wins
             it
             ,
             then
             they
             are
             in
             the
             condition
             formerly
             suppos'd
             ,
             in
             which
             Case
             there
             is
             48
             due
             to
             A
             ;
             therefore
             says
             A
             to
             B
             ,
             give
             me
             the
             48
             which
             I
             am
             sure
             of
             ,
             whether
             I
             win
             or
             lose
             the
             next
             Game
             ,
             and
             since
             you
             will
             not
             hazard
             for
             the
             other
             16
             ,
             let
             us
             part
             them
             equally
             ,
             that
             is
             ,
             give
             me
             8
             ,
             which
             ,
             with
             the
             former
             48
             ,
             make
             56
             ,
             leaving
             
             8
             to
             you
             ,
             and
             so
             all
             the
             other
             Cases
             may
             be
             solv'd
             after
             the
             same
             manner
             .
          
           
             Suppose
             A
             wagers
             with
             B
             ,
             that
             with
             one
             Dye
             he
             shall
             throw
             6
             at
             one
             of
             three
             Throws
             ,
             and
             that
             each
             of
             them
             stakes
             108
             Guineas
             :
             To
             find
             what
             is
             the
             proportion
             of
             their
             Hazards
             ;
             Now
             there
             being
             in
             one
             Throw
             of
             a
             Dye
             but
             one
             Chance
             for
             6
             ,
             and
             five
             Chances
             against
             it
             ,
             one
             Throw
             for
             6
             is
             worth
             1
             /
             6
             of
             the
             Stake
             ;
             therefore
             says
             B
             to
             A
             ,
             of
             the
             216
             Guineas
             take
             a
             sixth
             part
             for
             your
             first
             Throw
             ,
             that
             is
             ,
             36
             ;
             for
             your
             next
             Throw
             take
             a
             sixth
             part
             of
             the
             remaining
             180
             ,
             that
             is
             ,
             30
             ;
             and
             for
             your
             third
             
             Throw
             ,
             take
             a
             sixth
             part
             of
             the
             remaining
             150
             ,
             that
             is
             ,
             25
             ,
             which
             in
             all
             make
             91
             ,
             leaving
             to
             me
             125
             ;
             so
             his
             Hazard
             who
             undertakes
             to
             throw
             6
             at
             one
             of
             three
             Throws
             ,
             is
             91
             to
             125.
             
          
           
             Suppose
             A
             had
             undertaken
             to
             throw
             6
             with
             one
             Dye
             at
             one
             Throw
             of
             four
             ,
             and
             that
             the
             whole
             Stake
             is
             1296
             ;
             says
             A
             to
             B
             ,
             Every
             Throw
             for
             6
             of
             one
             Dye
             ,
             is
             worth
             the
             sixth
             part
             of
             what
             I
             throw
             for
             ;
             therefore
             for
             my
             first
             Throw
             give
             me
             216
             ,
             which
             is
             the
             sixth
             part
             of
             1296
             ,
             and
             there
             remains
             1080
             ,
             I
             must
             have
             the
             sixth
             part
             of
             that
             ,
             viz.
             180
             ,
             for
             my
             second
             Throw
             ;
             and
             the
             sixth
             
             part
             of
             the
             remaining
             900
             ,
             which
             is
             150
             ,
             for
             my
             third
             Throw
             ;
             and
             the
             sixth
             part
             of
             the
             last
             remainder
             750
             ,
             which
             is
             125
             for
             my
             fourth
             Throw
             ;
             all
             this
             added
             together
             makes
             671
             ,
             and
             there
             remains
             to
             you
             625
             ;
             so
             it
             is
             evident
             ,
             that
             A's
             Hazard
             ,
             in
             this
             Case
             ,
             is
             to
             B's
             671
             to
             625.
             
          
           
             Suppose
             A
             is
             to
             win
             the
             Stakes
             (
             which
             we
             shall
             suppose
             to
             be
             36
             )
             if
             he
             throws
             7
             at
             once
             of
             twice
             with
             two
             Dice
             ,
             and
             B
             is
             to
             have
             them
             if
             he
             does
             not
             ;
             says
             B
             to
             A
             ,
             the
             Chances
             which
             give
             7
             are
             6
             of
             the
             36
             ,
             which
             is
             as
             '
             much
             as
             1
             of
             6
             ;
             therefore
             for
             your
             first
             Throw
             you
             shall
             have
             a
             sixth
             
             part
             of
             the
             36
             ,
             which
             is
             6
             ;
             and
             for
             your
             next
             Throw
             a
             sixth
             part
             of
             the
             remainder
             30
             ,
             which
             is
             5
             ;
             this
             in
             all
             makes
             11
             ;
             so
             you
             leave
             25
             to
             me
             ;
             so
             A's
             Hazard
             is
             to
             B's
             as
             11
             to
             25.
             
          
           
             It
             were
             easie
             ,
             at
             this
             rate
             to
             calculate
             the
             most
             intricate
             Hazards
             ,
             were
             it
             not
             that
             Fractions
             will
             occur
             ,
             which
             ,
             if
             they
             be
             more
             than
             ½
             ,
             may
             be
             suppos'd
             equal
             to
             an
             Unit
             ,
             without
             causing
             any
             remarkable
             Error
             in
             great
             Numbers
             .
          
           
             It
             will
             not
             be
             amiss
             ,
             before
             I
             conclude
             ,
             to
             give
             you
             a
             Rule
             for
             finding
             in
             any
             Number
             of
             Games
             the
             Value
             of
             the
             first
             ,
             because
             
             Hugens's
             Method
             ,
             in
             
             that
             Case
             ,
             is
             something
             tedious
             .
          
           
             Suppose
             A
             and
             B
             had
             agreed
             ,
             that
             he
             should
             have
             the
             Stakes
             who
             did
             win
             the
             first
             9
             Games
             ,
             and
             A
             had
             already
             won
             one
             of
             the
             9
             ;
             I
             would
             know
             what
             share
             of
             B's
             Mony
             is
             due
             to
             A
             for
             the
             Advantage
             of
             this
             Game
             .
             To
             find
             this
             ,
             take
             the
             first
             eight
             even
             Numbers
             2
             ,
             4
             ,
             6
             ,
             8
             ,
             10
             ,
             12
             ,
             14
             ,
             16
             ,
             and
             multiply
             them
             continually
             ;
             that
             is
             ,
             the
             first
             by
             the
             second
             ,
             the
             product
             by
             the
             third
             ,
             &c.
             take
             the
             first
             eight
             odd
             Numbers
             ,
             1
             ,
             3
             ,
             5
             ,
             7
             ,
             9
             ,
             11
             ,
             13
             ,
             15
             ,
             and
             do
             just
             so
             by
             them
             ,
             the
             product
             of
             the
             even
             Number
             is
             the
             Denominator
             ,
             and
             the
             product
             of
             
             the
             odd
             Number
             the
             Numerator
             of
             a
             Fraction
             ,
             which
             expresseth
             the
             quantity
             of
             B's
             Money
             due
             to
             A
             upon
             the
             winning
             of
             the
             first
             Game
             of
             9
             ;
             that
             is
             ,
             if
             each
             stak'd
             a
             number
             of
             Guineas
             ,
             or
             Shillings
             ,
             &c.
             express'd
             by
             the
             product
             of
             the
             even
             Numbers
             ,
             there
             would
             belong
             to
             A
             ,
             of
             B's
             Money
             ,
             the
             Number
             express'd
             by
             the
             product
             of
             the
             odd
             Numbers
             :
             For
             Example
             ,
             Suppose
             A
             had
             gain'd
             one
             Game
             of
             4
             ,
             then
             by
             this
             Rule
             ,
             I
             take
             the
             three
             first
             even
             Numbers
             ,
             2
             ,
             4
             ,
             6
             ,
             and
             multiply
             them
             continually
             ,
             which
             make
             48
             ,
             and
             the
             first
             three
             odd
             Numbers
             ,
             1
             ,
             3
             ,
             5
             ,
             and
             multiply
             them
             continually
             ,
             
             which
             make
             15
             ;
             so
             there
             belongs
             to
             A
             15
             /
             48
             of
             B's
             Money
             ,
             that
             is
             ,
             if
             each
             stak'd
             48
             ,
             there
             would
             belong
             to
             A
             ,
             besides
             his
             own
             15
             of
             A's
             .
             Now
             by
             
             Hugens's
             Method
             ,
             if
             A
             wants
             but
             three
             Games
             while
             B
             wants
             four
             ,
             there
             is
             due
             to
             A
             21
             /
             32
             of
             the
             Stake
             ;
             by
             this
             Rule
             there
             is
             due
             to
             A
             15
             /
             48
             of
             B's
             Money
             ,
             which
             is
             15
             /
             69
             of
             the
             Stake
             ,
             which
             ,
             with
             his
             own
             48
             /
             96
             of
             the
             Stake
             ,
             makes
             63
             /
             96
             or
             21
             /
             32
             of
             the
             Stake
             ,
             and
             so
             in
             every
             Case
             you
             will
             find
             
             Hugens's
             Method
             and
             this
             will
             give
             you
             the
             same
             Number
             ;
             a
             Demonstration
             of
             it
             you
             may
             see
             in
             a
             Letter
             of
             Monsieur
             Pascals
             to
             Monsieur
             Fermat
             ;
             tho
             it
             be
             otherwise
             express'd
             there
             than
             here
             ,
             
             yet
             the
             consequence
             is
             easily
             supply'd
             .
             To
             prevent
             the
             labour
             of
             Calculation
             ,
             I
             have
             subjoyn'd
             the
             following
             Table
             ,
             which
             is
             calculated
             for
             two
             Gamesters
             ,
             as
             Mons.
             Hugens
             is
             for
             three
             .
          
           
             If
             each
             of
             us
             stake
             256
             Guineas
             in
          
           
           
             There
             belongs
             to
             me
             of
             256
             of
             my
             Play-fellow
          
           
           
             The
             Use
             of
             the
             Table
             is
             plain
             ;
             for
             let
             our
             Stakes
             be
             what
             they
             will
             ,
             I
             can
             find
             the
             Portion
             due
             to
             me
             upon
             the
             winning
             the
             first
             ,
             or
             the
             first
             two
             Games
             ,
             &c.
             of
             2
             ,
             3
             ,
             4
             ,
             5
             ,
             6.
             
             For
             Example
             ,
             If
             each
             of
             us
             had
             stak'd
             4
             Guineas
             ,
             and
             the
             Number
             of
             Games
             to
             be
             plaid
             were
             3
             ,
             of
             which
             I
             had
             gain'd
             1
             ,
             say
             ,
             As
             256
             is
             to
             96
             ,
             so
             is
             4
             to
             a
             fourth
             .
          
           
             256
             :
             96
             :
             :
             4
             :
             1½
          
           
             
               To
               find
               what
               is
               the
               Value
               of
               his
               Hazard
               ,
               who
               undertakes
               ,
               at
               the
               first
               Throw
               ,
               to
               cast
               Doublets
               ,
               in
               any
               given
               Number
               of
               Dice
               .
            
          
           
           
             In
             two
             Dice
             it
             is
             plain
             to
             avoid
             Doublets
             ,
             every
             one
             of
             the
             six
             different
             Throws
             of
             the
             first
             ,
             can
             only
             be
             combin'd
             with
             five
             of
             the
             second
             ,
             because
             one
             of
             the
             six
             is
             of
             the
             same
             kind
             ,
             and
             consequently
             makes
             Doublets
             ;
             for
             the
             samo
             reason
             ,
             the
             thirty
             Throws
             of
             two
             Dice
             ,
             which
             are
             not
             Doublets
             ,
             can
             only
             be
             combin'd
             with
             four
             Throws
             of
             a
             third
             Dice
             ,
             and
             three
             Throws
             of
             a
             fourth
             Dice
             ;
             so
             generally
             it
             is
             this
             Series
             ,
          
           
             6
             ×
             5
             ×
             4
             ×
             3
             ×
             2
             ×
             1
             ×
             0
             ,
             &c.
             
          
           
             6
             ×
             6
             ×
             6
             ×
             6
             ×
             6
             ×
             6
             ×
             6
             ,
             &c.
             
          
           
             The
             second
             Series
             is
             the
             Sum
             of
             the
             Chances
             ,
             and
             the
             first
             the
             
             Number
             of
             Chances
             against
             him
             who
             undertakes
             to
             throw
             Doublets
             ,
             each
             Series
             to
             be
             continu'd
             so
             many
             terms
             ,
             as
             are
             the
             Number
             of
             Dice
             .
             For
             Example
             ,
             If
             one
             should
             undertake
             to
             throw
             Doublets
             at
             the
             first
             Throw
             of
             four
             Dice
             ,
             his
             Adversary's
             Hazard
             is
             
             or
             5
             /
             18
             leaving
             to
             him
             13
             /
             18
             ,
             so
             he
             has
             13
             to
             5.
             
             In
             seven
             Dice
             ,
             you
             see
             the
             Chances
             against
             him
             are
             o
             ,
             because
             then
             there
             must
             necessarily
             be
             Doublets
             .
          
        
         
           
           
             Of
             WHIST
             .
          
           
             
               IF
               there
               be
               four
               playing
               at
            
             Whist
             ,
             
               it
               is
               15
               to
               1
               that
               any
               two
               of
               them
               shall
               not
               have
               the
               four
               Honours
               ,
               which
               I
               demonstrate
               thus
               :
            
          
           
             Suppose
             the
             four
             Gamesters
             be
             A
             ,
             B
             ,
             C
             ,
             D
             :
             If
             A
             and
             B
             had
             ,
             while
             the
             Cards
             are
             a
             dealing
             ,
             already
             got
             three
             Honours
             ,
             and
             wanted
             only
             one
             ,
             since
             it
             is
             as
             probable
             that
             C
             and
             D
             will
             have
             the
             next
             Honour
             ,
             as
             A
             and
             B
             ;
             if
             A
             and
             B
             had
             laid
             a
             Wager
             to
             have
             it
             ,
             there
             is
             due
             to
             them
             but
             ½
             of
             the
             Stake
             :
             If
             A
             and
             B
             
             wanted
             two
             of
             the
             four
             ,
             and
             had
             wager'd
             to
             have
             both
             those
             two
             ,
             then
             they
             have
             an
             equal
             Hazard
             to
             get
             nothing
             ;
             if
             they
             miss
             the
             first
             of
             those
             two
             ,
             or
             to
             put
             themselves
             in
             the
             former
             Case
             if
             they
             get
             it
             ;
             so
             they
             have
             an
             equal
             Hazard
             to
             get
             nothing
             or
             ½
             ,
             which
             ,
             by
             Prop.
             1.
             is
             worth
             ¼
             of
             the
             Stake
             ;
             so
             if
             they
             want
             three
             Honours
             ,
             you
             will
             find
             due
             to
             them
             1
             /
             8
             of
             the
             Stake
             ;
             and
             if
             they
             wanted
             four
             ,
             1
             /
             16
             of
             the
             Stake
             ,
             leaving
             to
             C
             and
             D
             15
             /
             16
             ;
             so
             C
             and
             D
             can
             wager
             15
             to
             1
             ,
             that
             A
             and
             B
             shall
             not
             have
             all
             the
             four
             Honours
             .
          
           
             
               It
               is
               11
               to
               5
               that
            
             A
             and
             B
             
               shall
               not
               have
               three
               of
               the
               four
               
               Honours
               ,
               which
               1
               prove
               thus
               :
            
          
           
             It
             is
             an
             even
             Wager
             ,
             if
             there
             were
             but
             three
             Honours
             ,
             that
             A
             and
             B
             shall
             have
             two
             of
             these
             three
             ,
             since
             't
             is
             as
             probable
             that
             they
             will
             have
             two
             of
             the
             three
             ,
             as
             that
             C
             and
             D
             shall
             have
             them
             ;
             consequently
             ,
             if
             A
             and
             B
             had
             laid
             a
             Wager
             to
             have
             two
             of
             three
             ,
             there
             is
             due
             to
             them
             ½
             of
             the
             Stake
             .
             Now
             suppose
             A
             and
             B
             had
             wager'd
             to
             have
             three
             of
             four
             ,
             they
             have
             an
             equal
             Hazard
             to
             get
             the
             first
             of
             the
             four
             ,
             or
             miss
             it
             ;
             if
             they
             get
             it
             ,
             then
             they
             want
             two
             of
             the
             three
             ,
             and
             consequently
             there
             is
             due
             to
             them
             ½
             of
             the
             Stake
             ;
             if
             they
             miss
             it
             ,
             then
             they
             want
             three
             of
             the
             three
             ,
             and
             consequently
             
             there
             is
             due
             to
             them
             1
             /
             8
             of
             the
             Stake
             ;
             therefore
             ,
             by
             Prop.
             1.
             their
             Hazard
             is
             worth
             5
             /
             16
             ,
             leaving
             to
             C
             and
             D
             11
             /
             16.
             
          
           
             A
             and
             B
             
               playing
               at
            
             Whist
             against
             C
             and
             D
             ;
             A
             and
             B
             
               have
               eight
               of
               ten
               ,
               and
            
             C
             and
             D
             
               nine
               ,
               and
               therefore
               can't
               reckon
               Honors
               ;
               to
               find
               the
               proportion
               of
               their
               Hazards
               .
            
          
           
             There
             is
             5
             /
             16
             due
             to
             C
             and
             D
             upon
             their
             hazard
             of
             having
             three
             of
             four
             Honours
             ;
             but
             since
             A
             and
             B
             want
             but
             one
             Game
             ,
             and
             C
             and
             D
             two
             ,
             there
             is
             due
             to
             C
             and
             D
             but
             ¼
             ,
             or
             4
             /
             16
             more
             upon
             that
             account
             ,
             by
             Prop.
             4.
             this
             in
             all
             makes
             9
             /
             16
             ,
             
             leaving
             to
             A
             and
             B
             7
             /
             16
             ;
             so
             the
             hazard
             of
             A
             and
             B
             to
             that
             of
             C
             and
             D
             ,
             is
             as
             9
             to
             7.
             
          
           
             In
             the
             former
             Calculations
             I
             have
             abstracted
             from
             the
             small
             difference
             of
             having
             the
             Deal
             and
             being
             Seniors
             .
          
           
             All
             the
             former
             Cases
             can
             be
             calculated
             by
             the
             Theorems
             laid
             down
             by
             Monsieur
             Hugens
             ;
             but
             Cases
             more
             compos'd
             require
             other
             Principles
             ,
             for
             the
             easie
             and
             ready
             Computation
             of
             which
             ,
             I
             shall
             add
             one
             Theorem
             more
             ,
             demonstrated
             after
             Mons.
             
             Hugens's
             Method
             .
          
        
         
           
           
             Theor.
             
          
           
             If
             I
             have
             p
             Chances
             for
             
               a
               ,
               q
            
             Chances
             for
             b
             ,
             and
             r
             Chances
             for
             c
             ,
             then
             my
             hazard
             is
             worth
             
             ,
             that
             is
             ,
             a
             multiplied
             into
             the
             number
             of
             its
             Chances
             added
             to
             b
             ,
             multiplied
             into
             the
             number
             of
             its
             Chances
             ,
             added
             to
             c
             multiplied
             into
             the
             number
             of
             its
             Chances
             ,
             and
             the
             Sum
             divided
             by
             the
             Sum
             of
             Chances
             of
             
               a
               ,
               b
               ,
               c.
            
             
          
           
             To
             investigate
             as
             well
             as
             demonstrate
             this
             Theorem
             ,
             suppose
             the
             value
             of
             my
             hazard
             be
             x
             ,
             then
             x
             must
             be
             such
             ,
             as
             having
             it
             ,
             I
             am
             able
             to
             purchase
             as
             
             good
             a
             hazard
             again
             in
             a
             just
             and
             equal
             Game
             .
             Suppose
             the
             Law
             of
             it
             be
             this
             ,
             That
             playing
             with
             so
             many
             Gamesters
             as
             ,
             with
             my self
             ,
             make
             up
             the
             number
             p+q+r
             ,
             with
             as
             many
             of
             them
             as
             the
             nnmber
             p
             represents
             ;
             I
             make
             this
             bargain
             ,
             that
             whoever
             of
             them
             wins
             shall
             give
             me
             a
             ,
             and
             that
             I
             shall
             do
             so
             to
             each
             of
             them
             if
             I
             win
             ;
             with
             the
             Gamesters
             represented
             by
             the
             number
             of
             q
             ,
             I
             bargain
             to
             get
             b
             ,
             if
             any
             of
             them
             win
             ,
             ann
             to
             give
             b
             to
             each
             of
             them
             ,
             if
             I
             win
             my self
             ;
             and
             with
             the
             rest
             of
             the
             Gamesters
             ,
             whose
             number
             is
             
               r
               −
            
             1
             ,
             I
             bargain
             to
             give
             ,
             or
             to
             get
             c
             after
             the
             same
             manner
             :
             Now
             all
             being
             in
             an
             
             equal
             probability
             to
             gain
             ,
             I
             have
             p
             Chances
             to
             get
             
               a
               ,
               q
            
             Chances
             to
             get
             b
             ,
             and
             
               r
               −
            
             1
             Chances
             to
             get
             c
             ,
             and
             one
             Chance
             ,
             viz.
             when
             I
             win
             my self
             ,
             to
             get
             
               px+qx+rx
               −
               ap
               −
               bq
               −
               rc+c
               ,
            
             which
             ,
             if
             it
             be
             suppos'd
             equal
             to
             c
             ,
             then
             I
             have
             p
             Chances
             for
             
               a
               ,
               q
            
             Chances
             for
             b
             ,
             and
             r
             Chances
             for
             c
             (
             for
             I
             had
             just
             now
             
               r
               −
               1
            
             Chances
             for
             it
             )
             therefore
             ,
             if
             
               px+qx+rx
               −
               ap
               −
               bq
               −
               rc+c
               =
               c
               ,
            
             then
             is
             
             .
          
           
             By
             the
             same
             way
             of
             reasoning
             you
             will
             find
             ,
             if
             I
             have
             p
             Chances
             for
             
               a
               ,
               q
            
             Chances
             for
             
               b
               ,
               r
            
             Chances
             for
             c
             ,
             and
             s
             Chances
             
             for
             d
             ,
             that
             my
             hazard
             is
             
             ,
             &c.
             
          
        
         
           
             In
             Numbers
             .
          
           
             If
             I
             had
             two
             Chances
             for
             3
             Shillings
             ,
             four
             Chances
             for
             5
             Shillings
             ,
             and
             one
             Chance
             for
             9
             Shillings
             ,
             then
             ,
             by
             this
             Rule
             ,
             my
             hazard
             is
             worth
             5
             Shillings
             ;
             for
             
             ;
             and
             it
             is
             easie
             to
             prove
             ,
             that
             with
             5
             Shillings
             I
             can
             purchase
             the
             like
             hazard
             again
             ;
             for
             suppose
             I
             play
             with
             six
             others
             ,
             each
             of
             us
             staking
             5
             Shillings
             ;
             with
             two
             of
             them
             I
             bargain
             ,
             that
             if
             either
             of
             them
             win
             ,
             he
             must
             
             give
             me
             3
             Shillings
             ,
             and
             that
             I
             shall
             do
             so
             to
             them
             ;
             and
             with
             the
             other
             four
             I
             bargain
             just
             so
             ,
             to
             give
             or
             to
             get
             5
             Shillings
             :
             This
             is
             a
             just
             Game
             ,
             and
             all
             being
             in
             an
             equal
             probability
             to
             win
             ;
             by
             this
             means
             I
             have
             two
             Chances
             to
             get
             3
             Shillings
             ,
             four
             Chances
             to
             get
             5
             Shillings
             ,
             and
             one
             Chance
             to
             get
             9
             Shillings
             ,
             viz.
             when
             I
             win
             my self
             ;
             for
             then
             out
             of
             the
             Stake
             ,
             which
             makes
             35
             Shillings
             ,
             I
             must
             give
             the
             first
             two
             6
             Shillings
             ,
             and
             the
             other
             four
             20
             Shillings
             ,
             so
             there
             remains
             just
             9
             to
             my self
             .
          
           
             It
             it
             easie
             ,
             by
             the
             help
             of
             this
             Theorem
             ,
             to
             calculate
             in
             the
             Game
             of
             Dice
             ,
             commonly
             call'd
             
             Hazard
             ,
             what
             Mains
             are
             best
             to
             sett
             on
             ,
             and
             who
             has
             the
             Advantage
             ,
             the
             Caster
             or
             Setter
             .
             The
             Scheme
             of
             the
             Game
             ,
             as
             I
             take
             it
             ,
             is
             thus
             ,
             
               
                 
                    
                
                 
                   Throws
                   next
                   following
                   for
                
              
               
                 
                   Mains
                   .
                
                 
                   The
                   Caster
                   .
                
                 
                   The
                   Setter
                   .
                
              
               
                 
                   V.
                   
                
                 
                   V.
                   
                
                 
                   II.
                   III.
                   XI
                   .
                   XII
                   .
                
              
               
                 
                   VI.
                   
                
                 
                   VI.
                   XII
                   .
                
                 
                   XI
                   .
                   II.
                   III.
                   
                
              
               
                 
                   VII
                   .
                
                 
                   VII
                   .
                   XI
                   .
                
                 
                   XII
                   .
                   II.
                   III.
                   
                
              
               
                 
                   VIII
                   .
                
                 
                   VIII
                   .
                   XII
                   .
                
                 
                   XI
                   .
                   II.
                   III.
                   
                
              
               
                 
                   IX
                   .
                
                 
                   IX
                   .
                
                 
                   II.
                   III.
                   XI
                   .
                   XII
                   .
                
              
            
          
           
             By
             an
             easie
             Calculation
             you
             will
             find
             ,
             if
             the
             Caster
             has
             VI.
             and
             the
             Setter
             VII
             ,
             there
             is
             due
             to
             the
             Caster
             ⅓
             of
             the
             Stake
             ;
             if
             he
             has
             
               
               
                 V.
                 against
                 VII
                 .
                 2
                 /
                 5
                 of
                 the
                 Stake
                 ,
              
               
                 VI.
                 against
                 VII
                 .
                 5
                 /
                 11
                 of
                 the
                 Stake
                 ,
              
               
                 IV.
                 against
                 VI.
                 3
                 /
                 8
                 of
                 the
                 Stake
                 ,
              
               
                 V.
                 against
                 VI.
                 4
                 /
                 9
                 of
                 the
                 Stake
                 ,
              
               
                 VI.
                 against
                 V.
                 3
                 /
                 7
                 of
                 the
                 Stake
                 ,
              
            
          
           
             I
             need
             not
             tell
             the
             Reader
             ,
             that
             IV.
             is
             the
             same
             with
             X
             ,
             V.
             with
             IX
             ,
             and
             VI.
             with
             VIII
             .
          
           
             Suppose
             then
             VII
             .
             be
             the
             Main
             :
             To
             find
             the
             proportion
             of
             the
             hazard
             of
             the
             Caster
             to
             that
             of
             the
             Setter
             .
          
           
             By
             the
             Law
             of
             the
             Game
             ,
             the
             Caster
             ,
             before
             he
             throws
             next
             ,
             has
             four
             Chances
             for
             nothing
             ,
             viz.
             these
             II
             ,
             III
             ,
             XII
             ;
             eight
             Chances
             for
             the
             whole
             Stake
             ,
             viz.
             those
             of
             VII
             ,
             XI
             ;
             
             six
             Chances
             for
             ⅓
             ,
             viz.
             those
             IV
             ,
             X
             ;
             eight
             Chances
             for
             2
             /
             5
             ,
             viz.
             those
             of
             V
             ,
             IX
             ;
             and
             ten
             Chances
             for
             5
             /
             11
             ,
             viz.
             these
             of
             VI
             ,
             X
             ;
             so
             his
             hazard
             ,
             by
             the
             preceding
             Theorem
             ,
             is
             
          
           
             Now
             to
             save
             the
             trouble
             of
             a
             tedious
             reduction
             ,
             Suppose
             the
             Stake
             which
             they
             play
             for
             be
             36
             ,
             that
             is
             ,
             the
             Setter
             had
             laid
             down
             18
             ;
             in
             that
             case
             ,
             every
             one
             of
             these
             Fractions
             are
             so
             many
             parts
             of
             an
             Unite
             ,
             which
             ,
             being
             gather'd
             into
             one
             Sum
             ,
             give
             1741
             /
             59
             to
             the
             Caster
             ,
             leaving
             1814
             /
             55
             to
             the
             Setter
             ;
             so
             the
             hazard
             of
             the
             Caster
             is
             to
             that
             of
             the
             Setter
             244
             ,
             251.
             
          
           
           
             Suppose
             VI.
             or
             VIII
             .
             be
             the
             Main
             ,
             then
             the
             Share
             of
             the
             Caster
             is
          
           
             II.
             
          
           
             III.
             VI.
             IV.
             V.
             
          
           
             XI
             .
             XII
             .
             X.
             IX
             .
             VIII
             .
             VII
             .
          
           
             5×0+6×1+6×3
             /
             8+8×4
             /
             9+5×½+6×6
             /
             11
             =
             =
             17229
             /
             396
             ,
             leaving
             to
             the
             Setter
             18167
             /
             396
             ,
             so
             the
             hazard
             of
             the
             Caster
             is
             to
             that
             of
             the
             Setter
             as
             6961
             to
             7295.
             
          
           
             Suppose
             V.
             or
             IX
             .
             be
             the
             Main
             ,
             then
             the
             Share
             of
             the
             Caster
             is
          
           
             II.
             
          
           
             III.
             
          
           
             XI
             .
             IV.
             VI.
             
          
           
             XII
             .
             V.
             X.
             IX
             .
             VIII
             .
             VII
             .
          
           
             6×0+4×1+6×●+4×½+10×●+6×●
             =
             =
             17
             229
             /
             315
             ,
             leaving
             to
             the
             Setter
             
             is
             1886
             /
             315
             ,
             so
             the
             hazard
             of
             the
             Caster
             is
             to
             that
             of
             the
             Setter
             as
             1396
             to
             1493.
             
          
           
             It
             is
             plain
             ,
             that
             in
             every
             Case
             the
             Caster
             has
             the
             Disadvantage
             ,
             and
             that
             V.
             or
             IX
             .
             are
             better
             Mains
             to
             set
             on
             than
             VII
             ,
             because
             ,
             in
             this
             last
             Cast
             the
             Setter
             has
             but
             18
             and
             14
             /
             55
             or
             84
             /
             330
             ;
             whereas
             ,
             when
             V.
             or
             IX
             .
             is
             the
             Main
             ,
             he
             has
             1886
             /
             315
             ;
             likewise
             VI.
             or
             VIII
             .
             are
             better
             Mains
             than
             V.
             or
             IX
             .
             because
             167
             /
             396
             is
             a
             greater
             Froction
             than
             86
             /
             315.
             
          
           
             All
             those
             Problems
             suppose
             Chances
             ,
             which
             are
             in
             an
             equal
             probability
             to
             happen
             ,
             if
             it
             
             should
             be
             suppos'd
             otherwise
             ,
             there
             will
             arise
             variety
             of
             Cases
             of
             a
             quite
             different
             nature
             ,
             which
             ,
             perhaps
             ,
             't
             were
             not
             unpleasant
             to
             consider
             ,
             I
             shall
             add
             one
             Problem
             of
             that
             kind
             ,
             leaving
             the
             Solution
             to
             those
             who
             think
             it
             merits
             their
             pains
             .
          
           
             
               In
               Parallelipipedo
               cujus
               latera
               sunt
               ad
               invicem
               in
               ratione
            
             a
             ,
             b
             ,
             c
             :
             
               Invenire
               quotâ
               vice
               quivis
               suscipere
               potest
               ,
               ut
               datum
               quodvis
               planum
               ,
               v.
               g.
            
             ab
             jaciat
             .
          
        
         
           FINIS
           .
        
      
    
     
       
         
           ERRATA
           .
        
         
           PReface
           ,
           page
           3.
           line
           1.
           read
           in
           .
           p.
           6.
           l.
           5.
           r.
           incur
           .
           p.
           10.
           l.
           8.
           for
           
             is
             left
             to
             me
          
           ,
           r.
           
             properly
             deserves
             the
             name
             of
             Conduct
             .
          
           Book
           ,
           p.
           2.
           l.
           7.
           for
           9
           r.
           q.
           p.
           16.
           l.
           5.
           add
           
             and
             he
             one
          
           .
           p.
           71.
           l.
           5.
           r.
           wins
           .
        
      
       
         
         
           Advertisement
           .
        
         
           
             THe
             whole
             Duty
             of
             Man
             according
             to
             the
             Law
             of
             Nature
             .
          
           By
           that
           famous
           Civilian
           SAMUEL
           PUFFENDORF
           ,
           Professor
           of
           
             The
             Law
             of
             Nature
          
           and
           Nations
           ,
           in
           the
           University
           of
           Heidelberg
           ,
           and
           in
           the
           Caroline
           University
           ,
           afterwards
           Counsellour
           and
           Historiographer
           to
           the
           K.
           of
           Sweden
           ,
           and
           to
           his
           Electoral
           Highness
           of
           Brandenburg
           .
           Now
           made
           English.
           Printed
           for
           
             C.
             Harper
          
           ,
           at
           the
           Flower-de-Luce
           over-against
           St.
           
           Dunstan's
           Church
           in
           Fleetstreet
           .
        
         
      
    
  

