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
        
      
       Approx. 67 KB of XML-encoded text transcribed from 61 1-bit group-IV TIFF page images.
       
         Text Creation Partnership,
         Ann Arbor, MI ; Oxford (UK) :
         2004-03 (EEBO-TCP Phase 1).
         A25748
         Wing A3602
         ESTC R31565
         12165155
         ocm 12165155
         55286
         
           
            This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of
             Creative Commons 0 1.0 Universal
            . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission.
          
        
      
       
         Early English books online.
      
       
         (EEBO-TCP ; phase 1, no. A25748)
         Transcribed from: (Early English Books Online ; image set 55286)
         Images scanned from microfilm: (Early English books, 1641-1700 ; 1495:1)
      
       
         
           
             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.
          
        
      
    
     
       
         Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford.
         Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. Gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors.
      
       
         EEBO-TCP is a partnership between the Universities of Michigan and Oxford and the publisher ProQuest to create accurately transcribed and encoded texts based on the image sets published by ProQuest via their Early English Books Online (EEBO) database (http://eebo.chadwyck.com). The general aim of EEBO-TCP is to encode one copy (usually the first edition) of every monographic English-language title published between 1473 and 1700 available in EEBO.
         EEBO-TCP aimed to produce large quantities of textual data within the usual project restraints of time and funding, and therefore chose to create diplomatic transcriptions (as opposed to critical editions) with light-touch, mainly structural encoding based on the Text Encoding Initiative (http://www.tei-c.org).
         The EEBO-TCP project was divided into two phases. The 25,363 texts created during Phase 1 of the project have been released into the public domain as of 1 January 2015. Anyone can now take and use these texts for their own purposes, but we respectfully request that due credit and attribution is given to their original source.
         Users should be aware of the process of creating the TCP texts, and therefore of any assumptions that can be made about the data.
         Text selection was based on the New Cambridge Bibliography of English Literature (NCBEL). If an author (or for an anonymous work, the title) appears in NCBEL, then their works are eligible for inclusion. Selection was intended to range over a wide variety of subject areas, to reflect the true nature of the print record of the period. In general, first editions of a works in English were prioritized, although there are a number of works in other languages, notably Latin and Welsh, included and sometimes a second or later edition of a work was chosen if there was a compelling reason to do so.
         Image sets were sent to external keying companies for transcription and basic encoding. Quality assurance was then carried out by editorial teams in Oxford and Michigan. 5% (or 5 pages, whichever is the greater) of each text was proofread for accuracy and those which did not meet QA standards were returned to the keyers to be redone. After proofreading, the encoding was enhanced and/or corrected and characters marked as illegible were corrected where possible up to a limit of 100 instances per text. Any remaining illegibles were encoded as <gap>s. Understanding these processes should make clear that, while the overall quality of TCP data is very good, some errors will remain and some readable characters will be marked as illegible. Users should bear in mind that in all likelihood such instances will never have been looked at by a TCP editor.
         The texts were encoded and linked to page images in accordance with level 4 of the TEI in Libraries guidelines.
         Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements).
         
          Keying and markup guidelines are available at the
           Text Creation Partnership web site
          .
        
      
       
         
         
      
    
     
       
         eng
      
       
         
           Probabilities.
           Games of chance (Mathematics)
           Game theory.
        
      
    
     
        2003-06 TCP
        Assigned for keying and markup
      
        2003-07 Apex CoVantage
        Keyed and coded from ProQuest page images
      
        2004-01 Emma (Leeson) Huber
        Sampled and proofread
      
        2004-01 Emma (Leeson) Huber
        Text and markup reviewed and edited
      
        2004-02 pfs
        Batch review (QC) and XML conversion
      
    
  
   
     
       
         
         
         
           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
           .