% LaTeX source for Huygens' book on chance

% LaTeX source for Huygens' book on chance
% Text of the main text entered by Peter M. Lee
% found at http://www1.york.ac.uk/depts/maths/histstat/huygens.htm
% Lightly edited 15 Jan 2003 by Peter G. Doyle
% More severely edited and expanded with prefaces 
% May 2003 - May 2005 by Kees C. Verduin





                          C {\scriptsize H R I S T I A N I} \ H {\scriptsize U G E N I I} \\
                               L \ I \ B \ E \ L \ L \ U \ S \\
                                  D E \\
                             {\LARGE R A T I O C I N I I S} \\
                                  I N \\
                              {\LARGE\it LUDO  ALE\AE.} \\
\centerline{\vbox{\hrule width 7cm}}
                                  O R, \\
                           The V A L U E of all \\
                               {\Huge\bf C H A N C E S} \\
                                  I N \\
                          {\LARGE\bf Games of Fortune;} \\
                        C A R D S,  D I C E,  W A G E R S, \\
                            L O T T E R I E S,  \&c. \\
                     Mathematically Demonstrated. \\
\centerline{\vbox{\hrule width 7cm}}
                               \textit{L O N D O N:}  \\
            Printed  by  S. K{\scriptsize EIMER}  for  T. W{\scriptsize OODWARD},  near \\
            the  \textit{Inner Temple-Gate}  in  \textit{Fleetstreet}.  1714.

[This document consists of a reprint of an English translation of the first 
book on probability theory ever published, viz.\ Christiaan Huygens' 
\textit{De Ratiociniis in Ludo Aleae} which was published in Latin in 1657.  
The reprint of the translation is complete.  The original pagination 
is indicated in square brackets.]
T O\\


Physician to St. \textit{Thomas}'s\\
\indent Hospital and  Fellow of\\
\indent the R{\scriptsize OYAL} S{\scriptsize OCIETY}.\\

\indent\textit{HONOURED SIR},\\

\dropping{4}{W}\textit{HEN I consider the Subject
of the following papers, I can
no more forbear dedicating them
to Your Name, than I can refuse
giving my assent to any }Proposition\textit{ in
these }Sciences\textit{, which I have already seen
clearly demonstrated. The Reason is plain,
for as You have contributed to the greatest
Lustre and Glory to an very considerable part
of the }Mathematicks\textit{, by introducing them
into their noblest Province, the }Theory of
Physick\textit{ ; the Publisher of any Truths of
that Nature, who is desirous of seeing them
come to their utmost Perfection, nust of
course beg Your Patronage and Application
of them. By so prudent a Course as this, he 
may perhaps see those }Propositions\textit{ which}
[d2]\textit{ it was his utmost Ambition to make capable
only of directing Men in the Management
of their Purses, and instructing
them to what Chances and Hazards they
might safely commit their Money ; turn'd
some time or other to a much more glorious
End, and made instrumental likewise
towards the securing their Bodies
from the Tricks of that too successful Sharper,
}Death\textit{, and countermining the underhand 
Dealings of secret and over-reaching
\indent THE most celebrated Endeavours of the 
greatest }GENIUS\textit{ that ever appear'd before
in }Mathematical Learning\textit{, have been able
to carry it no farther, than to calculate by
its means the Motions of the }Heavenly Bodies\textit{
more exactly, explain the Mystery of 
}Tides\textit{, The Doctrine of }Sounds\textit{ and }Light\textit{,
and other curious }Ph\oe nomena\textit{ in the
Works of Nature, which are without us:
But You, }Great Sir\textit{, have made this wonderful
Clew conduct us in Paths much more
intricate and obscure, and guide us thro'
the far more more curious and puzzling Labyrinths
of }Human Body\textit{ ; and have as far 
out-done those noblest improvements, as the
glorious Frame of Man exceeds all the rest
of Nature's Works, and as the Knowledge
of ourselves is preferable to all other parts of
}Science\textit{ whatever. All our Hopes that this
Sort of Learning will meet with a warmer
and more welcome Reception from the present}
[d3]\textit{ and future Ages, than it has from those
past, are entirely owing to Your instructing
us in the true Value of }Mathematical
Disquisitions\textit{, and discovering to us their most
excellent Uses which our Ancestors were perfectly
ignorant of. The }Mathematicks\textit{ being
formerly believ'd of no further Service, than
as they assisted us in the Business of our Hands,
by directing our Mechanical Operations, or
diverted our Heads with pleasant and amusing
}Problems\textit{, where then but particular
Studies, and cultivated only by some few
as a necessary help to their employments,
and by fewer as a Diversion ; but since You, to
our extreme Surprize and Satisfaction, have
shown their scarce dream'd of Excellencies
in promoting the }Profession of Physick\textit{ almost
to a pitch of Certainty, and demonstrated
them capable, by a due Application of adding
Days and Years to the Length of our
Lives, they must now necessarily become the 
general Concern of all Mankind and the 
greatest Subject of their Admiration and Esteem.\\
\indent PERHAPS the few following }Propositons\textit{,
may to the World appear so foreign
to this Purpose, as to be thought perfectly
incapable of being directed to such a 
mighty End. For what can there be in
common between the Value of a }Chance\textit{ in
a }Game\textit{, and the }Knowledge\textit{ and }Cure\textit{ of
a }Distemper\textit{? And how can the nicest Determination
of the former, any way influence}
[d4] \textit{or illustrate the latter? But these
apparent Inconsistencies, Sir, are not such
as can impose upon Your clearer Judgement,
at the Light of which they vanish like
Phantoms at the break of Day. You have
reconciled Truths that have seem'd to us
much more inconsistent than these, and even
demonstrated what the wiser part of Mankind
before, had always look'd upon as absurd
and impossible, That the State of our
Bodies on Earth, are subject to surprizing 
Alterations and Changes from the various
Positions of those two heavenly ones, the
}SUN\textit{ and }MOON\textit{. So that we can
now talk of their Conjunctions, Oppositions,
and Quadratures, in relation to Human
Diseases, whithout being any longer
liable to the usual Reproaches of Ignorance
and Infatuation, justly chargeable
upon the old judicial }Astrology\textit{. Your late
learned Friend, Dr. }PITCAIRNE\textit{,
made no difficulty of lifting this Subject into
the Service of }Physick\textit{, when he gave
our }Third Proposition\textit{ a very honourable
Post in his }Dissertat. de Circulatione sanguinis\textit{,
and found it of Force enough to
overthrow and defeat for ever, that erroneous
and long standing Doctrine of }Secretion\textit{,
which supposes the Orifice of each Secretory
Duct to be of a different Figure, and as
admitting only those Particles of the Blood
that are of a similar Figure and Magnitude.
Yes, there is but too much Chance and} 
[d5] \textit{Uncertainty in Human Constitutions, and
the Duration of this mortal Life ; and if
a proper Application of this Doctrine
might assist us to get a true and exact Value
upon that Uncertainty, perhaps the }Physicians\textit{
will not be the only Persons that are
like to benefit by it ; but the }Divines\textit{ likewise 
may be furnish'd from thence with the
most convincing and, I think, only Arguments
they have left to make use of ; for 
that eminent Body  have so far improv'd in 
their Admonitions and Reasonings with
Mankind, that nothing but Mathematical 
Demonstration seems able to do them any
further Service. Such an Application as
this, together with all the Advantages the 
other Parts of the }Mathematicks\textit{ can confer
upon the Science of }Physick\textit{, are what
we must only expect from Your Pen ; whenever
Your Hurry or Business will allow You
a few leisure Intervals, and a convenient
Opportunity to resume it.\\
\indent YOU have prophesy'd, Sir, in the Preface to
Your excellent Piece upon }Poisons\textit{, that the 
true }Physician\textit{ shall hereafter be distinguish'd
from the }QUACK\textit{, not only by a perfect acquaintance
with the learned Languages, 
but a considerable Proficiency likewise in the 
}Mathematicks\textit{: and the Prognostick is 
pretty surely grounded; for that happy Distinction
is not only foretold, but establish'd
by your Writings, the Pleasure of reading
which, will most certainly compleat that}
[d6] Prediction. \textit{ For the Veneration the World
pays to Your Works, is so equal to their
Value, that we find none desirous of being
accounted }Physicians\textit{, who have not first
made themselves thoroughly acquainted with
them ; and consequently none are like to be
so, without the Qualifications by You precribed.\\
\indent I have made no mention, Sir, of the
many Personal Obligations, I lie under to
You, and the singular Favours and Encouragement
I have met with at Your Hands, because
they cannot be imagin'd to have been
any Motives to this Address ; which is so
far from being able to discharge the smallest
of that great Debt, that on the part
of Mr. }HUYGENS\textit{, it is only my Duty,
and but Justice done You, and on my own,
a very considerable Addition to what I
already owe Your unlimited Goodness and
Generosity, and too great an Honour done
\indent SIR,\\
        Your most Obedient,\\
            Obliged, and humble Servant,\\
                            W. Browne.\\
[a1] \\
\dropping{3}{A}S alle Mathematical Studies in general
are unaccountably bewitching and 
delightful to those that are once happily
engaged in them ; so that part which
considers and estimates our \textit{Expectations} of
\textit{Events} that are themselves \textit{uncertain}, and
depend entirely upon \textit{Chance} and \textit{Hazard},
cannot fail of giving a particular Pleasure
and Satisfaction. To reduce the unconstant
and irregular Proceedings of blind \textit{Fortune}
to certain Rules and Limits, and to set a definite
\textit{Value} upon her capricious \textit{Favours} and
\textit{Smiles}, seem to be Undertakings of so chim{\ae}rical
a Nature, that there is no Body but
must be delightfully surprized with that Art
which discovers them both really possible,
and with a little Application easily practicable\\
\indent But I think there is little need of saying in
Commendation either of the Subject, any thing
or of the Manner in which M. H{\scriptsize UYGENS} has 
consider'd it ; the great scarcity of his little
[a2] Treatise upon it, and the general want 
of a new Edition making that altogether 
superfluous. The just value every one has
for his Performance this Way, as well as for
his other admirable Pieces, abundantly show
that they are sufficiently sensible of his Excellencies;
and is the greatest Commendation it 
can possibly receive. Besides the Latin Editions
it has passed thro', the learned Dr. A{\scriptsize RBUTHNOTT}
publish'd an \textit{English} one, together 
with an Application of the General Doctrine
to some particular \textit{Games} then most in use;
which is so entirely dispers'd Abroad, that
an Account of it is all we can now meet with.
A late \textit{French Author} has indeed illustrated this 
Work, with a plentifull Number of Instances
where it may be servicable ; but all that he 
has done serves only to demonstrate what an
inexhaustible Treasure this little Book of
M. H{\scriptsize UYGENS} contains, and how easily it will
satisfy every ones Occasions, that will but be
at the Pains to make use of it. Our excellent 
Analyst M. D{\scriptsize E} M{\scriptsize OIVRE} likewise had wonderfully
improv'd the Subject, and besides
taking the different Dexterities of the Gamesters
into the Account, has much shorten'd the
Calculation, and made the whole more
general and compleat. But nothing can reflect
more Honour upon M. H{\scriptsize UYGENS} than this
last admirable Performance, which must needs
show us what an excellent Foundation that
learned Man laid, that cou'd support and
give Strenght to such a glorious Structure as
M. D{\scriptsize E} M{\scriptsize OIVRE} has rais'd upon it. M. D{\scriptsize E}
M{\scriptsize OIVRE'S} Piece therefore will be far 
from lessening the Worth of M. H{\scriptsize UYGENS'S};
[a3] and the superficial Mathematicians will still
be glad to satisfy their enquiries by this last
Author's easy tho' more tedious Method, as
not being able to understand the other's more
comprehensive and general one; while those
of a greater depth, will with no less delight
first read M. H{\scriptsize UYGENS'S} Treatise, in order to
proceed with so much the greater Pleasure
afterwards to peruse M. D{\scriptsize E} M{\scriptsize OIVRE'S} Additions,
and to perceive by what just Steps and
Degrees the Subject has from these two Great
Men receiv'd both its first Consideration and
last Perfection.\\
\indent M{\scriptsize Y} Design in publishing this Edition, was
to have made it as Useful as possible, by an
Addition of a very large \textit{Appendix} to it, containing
a Solution of some of the most servicable
and intricate Problems I cou'd think
of, and such as have not as yet, that I know of,
met with a particular Consideration : But an
Information I have within these few Days 
receiv'd, that M. M{\scriptsize ONMORT'S} \textit{French} Piece is
just newly reprinted at \textit{Paris} with very considerable 
Additions, has made me put a Stop 
to the \textit{Appendix}, till I can procure a Sight of
what has been added anew, for fear some
part of it may possibly have been honour'd
with the Notice and Consideration of that ingenious
Author. If the  Reader is of Opinion 
that I shou'd likewise till then, have deferr'd
the Publication of these few Pages, tho'
printed off ; and thinks much that he has not
either all that was design'd him or none at all;
or if he has any Exceptions to their appearing 
in the \textit{English} Language, the Bookseller is the
only Person that is to answer for his not receiving
Satisfaction in both those Points.


\noindent\textit{Just Published the following BOOKS, Printed for
and Sold by} T{\scriptsize HO.} W{\scriptsize OODWARD}, \textit{next the} Inner 
Temple-Gate \textit{in} Fleetstreet.\\

\dropping{2}{A}Short and easy Method to understand
 G{\scriptsize EOGRAPHY}, wherein are described the 
Form of Government of each Country, its
Qualities, the Manners of its Inhabitants, and 
whatsoever is most Remarkable in it. To
which are added, Observations upon those
Things of Importance that have happen'd in
each State: With an A{\scriptsize BRIDGEMENT} of the
\textit{Sphere}, and the Description and Use of the
\textit{Globe}, \textit{Geographical Maps} and \textit{Sea-Charts},
\textit{Englished} by a Gentleman of \textit{Cambridge}, from
the \textit{French} of Monsieur \textit{A. D. Fer}, Geographer
to the \textit{French} King.\\
\indent The Seventh Edition of an Help and Exhortation
to \textit{Worthy Communicating} ; or, a Treatise
describing the Meaning, worthy Reception,
Duty and Benefits of the \textit{Holy Sacrament};
And answering the Doubts of Conscience, and 
other Reasons, which most generally detain
Men from it. Together with suitable Devotions
added. By J{\scriptsize OHN} K{\scriptsize ETTLEWELL}, late Vicar
of \textit{Coles-hill} in \textit{Warwickshire}.\\
\indent The Fifth Edition of a S{\scriptsize ERMON} preach'd to
the \textit{Protestants of Ireland}, now in London, at 
the Parish-Church of St. \textit{Mary-le Bow, Oct.}23.
1714. Being the Day appointed by Act of 
Parliament in \textit{Ireland}, for an \textit{Anniversary
Thanksgiving} for the Deliverance of the \textit{Protestants}
of that Kingdom from the B{\scriptsize LOODY}
M{\scriptsize ASSACRE} begun by the \textit{Irish Papists}, on the 
23\textit{d} of \textit{October}, 1641. By J{\scriptsize OHN} R{\scriptsize AMSEY}, Rector
of \textit{Langdon} in \textit{Kent}.\\

\textit{And shortly will be published,}

\indent A large \textit{Appendix} to this Piece of Mr. H{\scriptsize UYGENS}
on the \textit{Value of Chances}; to be printed
and sold single, in order to be bound up with it.\\


                           THE \\
                      {\Huge\bf V \ A \ L \ U \ E} \\
                            OF \\ 
                     {\Huge\it C \ H \ A \ N \ C \ E \ S.} \\

\dropping{5}{A}L{\scriptsize THOUGH} in Games depending entirely upon Fortune, the Success is always 
uncertain; yet it may be exactly determin'd at the same time, how much more 
likely one is to win than lose.  As, if any one shou'd lay that he wou'd 
throw the Number \textit{Six} with a single Die the first throw, it is indeed 
uncertain whether he will win or lose; but how much more probability there 
is that he shou'd lose than win, is presently determin'd, and easily calculated.  
So likewise, if I agree with another to play the first Three Games for a 
certain Stake, and I have won one of my Three, it is yet uncertain which of 
us shall first get his third Game; but the Value of my Expectation and his 
likewise may be exactly discover'd; and consequently it may be determin'd, 
if we shou'd both agree to give over play, and leave the remaining Games 
unfinish'd, how much more of the Stake comes to my Share than his; 
[2] or, if another desired to purchase my Place and Chance, how much I might 
just sell it for.  And from hence an infinite Number of Questions may arise 
between two, three, four, or more Gamesters: The satisfying of which being 
a thing neither vulgar nor useless, I shall here demonstrate in few Words, 
the Method of doing it; and then likewise explain particularly the Chances 
that belong more properly to Dice.

                               P{\scriptsize O S T U L A T}

A{\scriptsize S} a Foundation to the following \textit{Proposition}, I shall take Leave to 
lay down this Self-evident Truth: \textit{That my Chance or Expectation 
to win any thing is worth just such a Sum, as wou'd procure me in the same 
Chance and Expectation at a fair Lay}.  As for Example, if any one shou'd put 
3 Shillings in one Hand, without letting me know which, and 7 in the other, 
and give me Choice of either of them; I say, it is the same thing as if he 
shou'd give me 5 Shillings; because with 5 Shillings I can, at a fair Lay, 
procure the same even Chance or Expectation to win 3 or 7 Shillings.

                               P R O P.\enspace I.

\textit{If I expect $a$ or $b$, and have an equal Chance of gaining either of 
them, my Expectation is worth $\frac{a + b}{2}$.}

[3]\quad T{\scriptsize O} trace this Rule from its first Foundation, as well as demonstrate 
it, having put $x$ for the value of my Expectation, I must with $x$ be able to 
procure the same Expectation at a fair Lay.  Suppose then that I play with 
another upon this Condition, That each shall stake $x$, and he that wins give 
the Loser $a$.  'Tis plain, the Play is fair, and that I have upon this 
Agreement an even Chance to gain $a$, if I lose the Game; or $2x - a$, if I 
win it: for I then have the whole stake $2x$, out of which I am to pay my 
Adversary $a$.  And if $2x - a$ be supposed equal to $b$, then I have an even 
Chance to gain either $a$ or $b$.  Therefore putting $2x - a = b$, we have 
$x = \frac{a + b}{2}$, for the Value of my Expectation.  \textit{Q.E.I.}

T{\scriptsize HE} Demonstration of which is very easy:  For having $\frac{a + b}{2}$, I can 
play with another, who shall likewise stake $\frac{a + b}{2}$, upon Condition 
that the Winner shall pay the Loser $a$.  By which means I must necessarily 
have an equal Expectation to gain $a$, if I am Loser, or $b$, if I am Winner; 
for then I win $\frac{a + b}{2}$, the Whole Stake, out of which I am to pay 
the Loser $a$.  \textit{Q.E.D.}

I{\scriptsize N} Numbers.  If I have an equal Chance to 3 or 7, then my Expectation is, by 
this \textit{Proposition}, worth 5, and it is certain I can with 5, again 
procure the same Expectation: For if Two of us stake 5 a piece upon this 
Condition, That he that wins pay the other 3, 'tis plain the Lay is just 
[4] and that I have an even Chance to come off with 3, if I lose, or 7 if 
I win; for then I gain 10, and pay my Adversary 3 out of it.  \textit{Q.E.D.}

                               P R O P.\enspace II.

\textit{If I expect $a$, $b$, or $c$, and each of them be equally likely 
to fall to my Share, my Expectation is worth $\frac{a + b + c}{3}$.}

T{\scriptsize O} calculate which, I again put $x$ for the value of my Expectation: Therefore 
having $x$, I must be able, by fair Gaming, to procure the same Expectation.  
Supposing then I play with two others upon this Condition, That every one of 
us stake $x$; and I agree with one of them, that which soever of us Two wins, 
shall give the loser $b$; and with the other, that which soever of us Two 
wins, shall give the Loser $c$.  It appears evidently, that the Lay is very 
fair, and that I have by this means an equal Chance to gain $b$, if the first 
wins; or $c$ if the second wins; or $3x - b - c$, if I win my self; for then 
I have the whole Stake $3x$, but of which I give $b$ to one, and $c$ to the 
other.  But if $3x - b - c$ be supposed equal to $a$, then I have equal
Expectation of $a$, $b$, or $c$, Therefore putting $3x - b - c = a$, we 
shall find $x = \frac{a + b + c}{3}$, for the value of my Expectation.  \textit{Q.E.I.}

A{\scriptsize FTER} the same Manner, an even Chance to $a$, $b$, $c$ or $d$, will be found 
worth $\frac{a + b + c + d}{4}$.  And so on.

                              P R O P.\enspace III.

\textit{If the number of Chances I have to gain $a$, be $p$, and the number of 
Chances I have to gain $b$, be $q$,  supposing the Chances equal; my 
Expectation will then be worth $\frac{ap + bq}{p + q}$.}

T{\scriptsize O} investigate this Rule, I again put $x$ for the value of my Expectation 
which must consequently procure me the same Expectation in fair Gaming.  I 
take therefore such a Number of Gamesters as may, including my self, be equal 
to $p + q$, every one of which stakes $x$; so that the whole stake is 
$px + qx$, and all play with an equal Expectation of winning.  With so many 
Gamesters as are expressed by the Number $q$, I agree singly, that whoever of 
them wins, shall give me $b$; and if I win, he shall have $b$ of me: And with 
the rest, express'd by $p - 1$, I singly make this Agreement, That whoever of 
them wins, shall give me $a$; and if I win, he shall receive $a$ of me.  It 
is evident, our playing upon this Condition is fair, no Body having any injury 
done him; and that my Expectation of $b$ is $q$; my Expectation of $a$ is 
$p - 1$; and my Expectation of $px + qx - bq - ap + a$ (\textit{i.e.}\ of 
winning) is 1: for then I gain the whole Stake $px + qx$, out of which I must 
pay $b$, to every one of the Gamesters $q$, and $a$ to every one of the 
Gamesters $p - 1$, which together makes $bq + ap - a$.  If therefore 
$px + qx - bq - ap + a$ be equal to $a$, I shou'd have $p$ Expectations of 
$a$ (for I had $p - 1$ Expectations of $a$, and 
[6] 1 Expectation of $px + qx - bq - ap \leftarrow a$, which is now supposed 
equal to $a$,) and $q$ Expectations of $b$; and consequently am again come to 
my first Expectation.  Therefore $px + qx - bq - ap + a = a$, and consequently 
$x = \frac{ap + bq}{p + q}$, is the value of my Expectation.  \textit{Q.E.I.}

I{\scriptsize N} Numbers.  If I have 3 Expectations of 13 and 2 Expectations of 8, the value 
of my Expectations wou'd by this Rule be 11.  And it is easy to show, that 
having 11, I cou'd again come to the same Expectations.  For playing against 
Four more, and every one of us staking 11; with Two of them I agree singly, 
that he that wins shall give me 8; or to give him 8, if I win: And with the 
other Two in like manner, that which soever wins, shall give me 13; or give 
him so much if I win.  The Play is manifestly fair, and I have just 2 
Expectations of 8, if either of the Two that promis'd me 8 shou'd win; and 3 
Expectations of 13, if either of the Two that are to pay me 13 shou'd win, 
or if I win my self, for then I gain the whole Stake, which is 55; from which, 
deducting 13, a piece for the last Two I bargain'd with, and 8 a piece for the 
other Two, there remain 13 for my self.  \textit{Q.E.D.}

                               P R O P.\enspace IV.

\textit{To come to the Question first propos'd, How to make a fair 
Distribution of the Stake among the several Gamesters, whose Chances 
are unequal?  The best way will be to begin with the most easy Cases of 
that Kind.}

S{\scriptsize UPPOSING} therefore that I play with another upon this Condition, That he who 
gets the first three Games shall have the Stake; and that I have won two of 
the three, and he only one.  I desire to know, if we agree to leave off and 
divide the Stake, how much falls to my Share?

I{\scriptsize N} the first place we must consider the number of Games still wanting to 
either Party: For it is plain, that supposing we had agreed the Stake shou'd 
be deliver'd to him that shou'd win the first twenty Games, and I had won 
nineteen of them, and the other only eighteen; my Chances wou'd have then 
been just so much better than his, as it is in the present Case, where I am 
supposed to have won two out of the three, and he only one: because in both 
Cases there remains but one Game for me to win and two for him.

M{\scriptsize OREOVER} to find how to share the Stake, we must have regard to what would 
happen, if both play'd on: For it is manifest, that if I win the next Game, 
my Number is compleated, and the Stake, which call $a$, is mine.  But if the 
other gets the next Game, then both our Chances will be even, because we want 
but one Game apiece, and each of them 
[8] worth $\frac{1}{2}a$.  But it is plain, I have an equal Chance to win or 
lose the next Game; and consequently an equal Chance to gain $a$, or 
$\frac{1}{2}a$; which by \textit{Prop.}\ 1.\ is worth half the Sum of them 
both, \textit{i.e.}\ $\frac{3}{4}a$.  \textit{Q.E.I.}

M{\scriptsize Y} Playfellow's Share, which of course must be the remaining $\frac{1}{4}a$, 
might be first found after  the same manner.  From whence it 
appears, That he who would play in my room, ought to give me $\frac{3}{4}a$ 
for my Chance; and consequently that whoever undertakes to win one Game, 
before another shall win two, may lay 3 to 1 Odds.

                               P R O P.\enspace V.

\textit{Suppose I want one Game of being up, and my Adversary wants three; 
How must the Stakes be divided?}

L{\scriptsize ET} us again consider what wou'd be the Consequence, if I shou'd get the next 
Game; 'tis plain I should win the Stake, suppose a; but if the other shou'd 
get it, he wou'd still want two Games and I but one; and consequently our Case 
would be the same with that mention'd in the foregoing Proposition and my 
Share, $\frac{3}{4}a$, as is there demonstrated.   
[9] Therefore, since I have an equal Chance to gain $a$, or $\frac{3}{4}a$, 
my Expectation must by \textit{Prop.}\ 1.\ be worth $\frac{7}{8}a$; and my 
Adversary's Share, the remaining $\frac{1}{8}a$.  So that my Chance is to his, 
as 7 to 1.  \textit{Q.E.I.}

A{\scriptsize ND} as the \textit{Solution} of the foregoing Case is necessary to solving 
this last, so is the \textit{Solution} of this last necessary to solving 
the following one, where I am supposed to want but one Game, and my Adversary 
four; for then my Share will be found, after the same manner, to be 
$\frac{15}{16}$ of the Stake, and his, $\frac{1}{16}$.

                               P R O P.\enspace VI.

\textit{Suppose I have two Games to get, and my Adversary three.}

T{\scriptsize HEREFORE} after the next Game, I shall want but one more, and he three (in 
which Case my Share, by the foregoing \textit{Prop.}\ is $\frac{7}{8}a$) or 
we shall want two a piece, and then my Share is $\frac{1}{2}a$, both our 
Chances being equal.  But I have an even Chance to win or lose the next Game, 
and consequently have an equal Expectation of obtaining $\frac{7}{8}a$, or 
$\frac{1}{2}a$, wch 
[10] by \textit{Prop.}\ 1 is worth $\frac{11}{16}a$.  So that eleven Parts of 
the Stake fall to my Share, and five to his.  \textit{Q.E.I.}

                              P R O P.\enspace VII.

\textit{Suppose I want two Games, and my Adversary four.}

T{\scriptsize HEREFORE} it will either fall out, that by winning the next Game, I shall 
want but one more, and he four, or by losing it I shall want two, and he 
shall want three.  So that by \textit{Schol.\ Prop.}\ 5.\ and \textit{Prop.}\ 
6., I shall have an equal Chance for $\frac{15}{16}a$ or $\frac{11}{16}a$, 
which, by \textit{Prop.}\ 1 is just worth $\frac{13}{16}a$.  \textit{Q.E.I.}

                                C O R O L L.

F{\scriptsize ROM} whence it appears, that he that is to get two Games, before another shall 
get four, has a better Chance than he is to get one, before another gets two 
Games.  For in this last Case, namely of 1 to 2 his Share by Prop.\ 4 is but 
$\frac{3}{4}a$, which is less than $\frac{13}{16}a$

                              P R O P.\enspace VIII.

\textit{Suppose now there are three Gamesters and that the first and second 
want a game a piece, and the third wants two Games.}

[11]\quad T{\scriptsize O} find therefore the Share of the first Gamester, we must again 
examine what he wou'd gain, if either he himself, or one of the other two 
gets the next Game: If he gets it, he wins the whole Stake $a$; if the 
second gets it, because he likewise wanted but one Game, he has the Stake, 
and the first gets 0; and if the third gets it, then they will all three want one 
Game a piece, and the Share of each of them will consequently be 
$\frac{1}{3}a$.  The first Gamester therefore has an equal Expectation of 
gaining $a$, or 0, or $\frac{1}{3}a$, (since each has the same likelihood 
of winning the next Game,) which, by \textit{Prop.}\ 2.\ is worth 
$\frac{4}{9}a$.  The Share of the second will be likewise $\frac{4}{9}a$, and 
there will be $\frac{1}{9}a$ remaining for the third; whose Share might, after 
the same manner, be found separately from the others, and theirs determin'd 
by that.  \textit{Q.E.I.}

                               P R O P.\enspace IX.

\textit{To find the several Shares of as many Gamesters, as we please, some of 
which shall want more Games, others fewer; we must consider what he, whose 
Share we want to find, wou'd gain, if he, or any one of the others wins the 
next Game:  Then adding together what he wou'd gain in all those particular 
Cases, and dividing the sum by the Number of Gamesters, the Quotient gives 
the particular Share required.}

[12]\quad S{\scriptsize UPPOSE}, for Example, there were three Gamesters $A$, $B$, and $C$, 
and $A$ wanted one Game of being up, and $B$ and $C$ wanted two apiece, and I 
desire to find the Share that $B$ has in the Stake $q$.

F{\scriptsize IRST} of all we must see what will happen to $B$, if either he, or $A$, or 
$C$ wins the following Game.

I{\scriptsize F} $A$ wins it, he is up and consequently $B$ gets 0.  If $B$ himself wins it, 
he and $A$ will still want a Game apiece, and $C$ will want two; and 
consequently $B$, by \textit{Prop.}\ 8. gets $\frac{4}{9}q$.  But lastly, if 
$C$ wins it, then $A$ and $C$ will still want one Game apiece, and $B$ will want
two; and consequently $B$, by 
\textit{Prop.}\ 8.\ will in this Case get $\frac{1}{9}q$.

N{\scriptsize OW} the three several Gains of $B$ in all these particular Cases, which are 
0, $\frac{4}{9}q$, and $\frac{1}{9}q$, added together make $\frac{5}{9}q$; 
which Sum being divided by 3, the number of Gamesters, gives $\frac{5}{27}q$
for the Share of $B$.  \textit{Q.E.I.}

T{\scriptsize HE} Demonstration of this is plain from \textit{Prop.}\ 2.  For since $B$ has 
an equal Chance to 0, $\frac{4}{9}q$ or $\frac{1}{9}q$, his Expectation is, 
by that \textit{Prop.}\ worth $\frac{0 + \frac{4}{9}q + \frac{1}{9}q}{3}$ 
\textit{i.e.}\ $\frac{5}{27}q$.  And 'tis evident, this Divisor 3, is the 
Number of Gamesters.  \textit{Q.E.D.}

[13]\quad B{\scriptsize UT} in order to find what any one will gain in every particular Case, 
supposing himself or any of the rest should win the next Game; the more simple 
and intermediate Cases must first be investigated.  For as this last Case cou'd
not have been resolv'd without that 
calculated in \textit{Prop.}\ 8.\ where the Games wanting were 1,1,2; so 
likewise every single Person's Share, in case the Games wanting were 1,2,3, 
cannot be found, without first determining the Case where the Games wanting 
are 1,2,2 (which is already done in \textit{Prop.}\ 9.\ and that likewise where 
the Games wanting are 1,1,3, which may, by \textit{Prop.}\ 8.\ be easily 
calculated.  And after this manner may be resolv'd all the Cases comprehended 
in the following Table, and others, \textit{ad Infinitum}.


[This page set out landscape in the original]

                        TABLE \textit{for Three Gamesters}

  \textit{Games wanting} \\
  \textit{Shares.} \\
  \ \\
  \textit{Games wanting} \\
  \textit{Shares.} \\
  \ \\
  \textit{Games wanting} \\
  \textit{Shares.} \\
  \ \\
  \textit{Games wanting} \\
  \textit{Shares.} \\
  \ \\
  \text{[Continuation of same} \\
  \text{line in original]} \\
  1.\ 1.\ 2. & 1.\ 2.\ 2. & 1.\ 1.\ 3. & 1.\ 2. \ 3. \\
  \mbox{$\frac{ 4.\  4.\ 1.}{ 9.}$} & \mbox{$\frac{17.\ 5.\ 5.}{27.}$} &
  \mbox{$\frac{13.\ 13.\ 1.}{27.}$} & \mbox{$\frac{19.\ 6.\ 2.}{27.}$} \\ 
  \ & \ & \ & \ \\ \hline
  1.\ 1.\ 4. & 1.\ 1.\ 5. & 1.\ 2.\ 4. & 1.\ 2. \ 5. \\
  \mbox{$\frac{40.\ 40.\ 1.}{81.}$} & \mbox{$\frac{121.121.1.}{243.}$} &
  \mbox{$\frac{178.58.7.}{243.}$} & \mbox{$\frac{542.179.8.}{729.}$} \\ 
  \ & \ & \ & \ \\ \hline
  1.\ 3.\ 3. & 1.\ 3.\ 4. & 1.\ 3.\ 5. & \\
  \mbox{$\frac{65.\  8.\ 8.}{81.}$} & \mbox{$\frac{616.82.31.}{729.}$} &
  \mbox{$\frac{629.87.13.}{729.}$} & \\ 
  \ & \ & \ & \ \\
  2.\ 2.\ 3. & 2.\ 2.\ 4. & 2.\ 2.\ 5. & 2.\ 3. \ 3. \\
  \mbox{$\frac{34.\  34.\ 13.}{81.}$} & \mbox{$\frac{338.338.53.}{729.}$} &
  \mbox{$\frac{353.353.23.}{729.}$} & \mbox{$\frac{133.55.55.}{243.}$} \\ 
  \ & \ & \ & \ \\
  & & 2.\ 3.\ 4. & 2.\ 3. \ 5. \\
  & &
  \mbox{$\frac{451.195.83.}{729.}$} & \mbox{$\frac{1433.635.119.}{2187.}$} \\ 
  \ & \ & \ & \ 

[15]\quad A{\scriptsize S} to what belongs to D{\scriptsize ICE}, the Questions propos'd concerning
them are, In how many Times we may venture to throw \textit{Six}, or any
other Number, with a single Die, or two \textit{Sixes} with two Dice, or
three \textit{Sixes} with three Dice; and such-like.

T{\scriptsize O} resolve which, we must observe, First, That there are six several Throws 
upon one Die, which all have an equal probability of coming up.  That upon 
two Dice there are 36 several Throws, equally liable to be thrown, for any 
one of the six Throws of one Die may come up with every one of the six Throws 
of the other; and so 6 times 6 will make 36 Throws.  So likewise, that three 
Dice have 216 several Throws; for the 36 Throws on the two Dice may happen 
together with any one of the 6 Throws of the third Die; and so 6 times 36 
will make 216 Throws.  After the same manner, 'tis plain, four dice will 
have 6 times 216, \textit{i.e.}\ 1296 Throws; and so on, may we calculate 
the Throws upon any Number of Dice, taking always at the addition of every 
Die 6 times the preceding Number of Throws.

I{\scriptsize T} is farther to be observ'd, that upon two Dice there is only one Throw that 
can produce 2 or 12, and two Throws that can produce 3 or 11.  For if we call 
the Dice $A$ and $B$, 'tis plain, the throw 3 may be made up of the 1 of $A$ 
and 2 of $B$; or of the 1 of $B$ and 2 of $A$.  So likewise will 11 be 
produced by the 5 of $A$, and 6 of $B$; or by the 5 of $B$, and 6 of $A$.  
The Number 4 may be thrown three Ways, by 1 of $A$ and 3 of $B$; or 3 of $A$ 
and 1 of $B$; or 2 of $A$ and 2 of $B$.

[16]\quad X may likewise be thrown three several Ways.

   V or IX has 4 several Throws.

   VI or VIII has 5 Throws.

   VII has 6 Throws.

  \text{Upon 3 Dice,} \\
  \text{the Numbers}
    3 \text{\ or\ } 18 \\
    4 \text{\ or\ } 17 \\
    5 \text{\ or\ } 16 \\
    6 \text{\ or\ } 15 \\
    7 \text{\ or\ } 14 \\
    8 \text{\ or\ } 13 \\
    9 \text{\ or\ } 12 \\
   10 \text{\ or\ } 11
  \rotatebox{90}{may come up by}
    1  \\
    3  \\
    6  \\
   10  \\
   15  \\
   21  \\
   25  \\
  \rotatebox{90}{several Throws}
\] \\

                               PROP.\ X.

\textit{To find how many Throws one may undertake to throw the Number 6 
with a single Die.}

I{\scriptsize F} any one wou'd venture to throw \textit{Six} the first Throw, 'tis plain 
there is but 1 Chance by which he might win the Stake; and 5 Chances by which 
he might lose it: For there are 5 Throws against him and only 1 for him.  Let 
the stake be called $a$.  Since therefore he has one Expectation of $a$, and 5 
Expectations of 0, his Chance is, by \textit{Prop.}\ 2.\ worth $\frac{1}{6}a$; 
and consequently there remains to his Adversary $\frac{5}{6}a$.  So that he 
that wou'd undertake to throw \textit{Six} the first Throws, must lay only 1 
to 5.

[17]\quad H{\scriptsize E} that wou'd venture to throw \textit{Six} once in two Throws, may 
calculate his Chance after the following manner: If he throws \textit{Six} 
the first Throw, he gains $a$; if the contrary happens, he has still another 
Throw remaining, which by the foregoing Case, is worth $\frac{1}{6}a$.
But he has only 1 Chance to throw \textit{Six} the first Throw and 5 Chances to 
for the contrary: Therefore before he throws, he has one Chance for $a$ and five
Chances for $\frac{1}{6}a$,
the Value of which, by \textit{Prop.}\ 3.\ is $\frac{11}{36}a$.  And consequently 
there remains $\frac{25}{36}a$, to the other that lays with him.  So that 
their several Chances, or the Values of their several Expectations, bear the 
Proportion of 11 to 25, \textit{i.e.}\ less than 1 to 2.

H{\scriptsize ENCE}, after the same Method, the Chance of him who wou'd venture to throw 
\textit{Six} one in three Throws, may be investigated and found worth 
$\frac{91}{216}a$, so that he may lay 91 against 125; which is a little 
less than 3 to 4.

H{\scriptsize E} who undertakes to throw it once in four Times, has a Chance worth 
$\frac{671}{1296}a$, and may lay 671 to 625, \textit{i.e.}\ something more than 
1 to 1.

H{\scriptsize E} who undertakes to throw it once in five Times, has a Chance worth 
$\frac{4651}{7776}a$, and may lay 4651 against 3125, \textit{i.e.}\ something 
less than 3 to 2.

[18]\quad H{\scriptsize E} who undertakes the same in 6 Throws has a Chance worth 
$\frac{31031}{46656}a$, and may lay 31031 to 15625, \textit{i.e.}\ a little 
less than 2 to 1.

A{\scriptsize ND} thus the \textit{Problem} be resolv'd in what Number of Throws we please.  

B{\scriptsize UT} it is possible for us to proceed after a more compendious Method, as 
shall be shown in the following \textit{Proposition}, without which the Calculation 
wou'd otherwise be much more prolix.

                               PROP.\ XI.

\textit{To find in how many Throws one may venture to throw the Number 12 
with two Dice.}

I{\scriptsize F} any one shou'd pretend to throw 12 the first throw, 'tis plain he has but 
one Chance of winning, \textit{i.e.}\ of gaining $a$; and 35 Chances of 
losing, or gaining 0, because there are in all, 36 several Throws.  And 
consequently his Expectation, by \textit{Prop.}\ 3.\ is worth $\frac{1}{36}a$.

H{\scriptsize E} that undertakes to do it in two Throws, if it comes up in the first Throw, 
will obtain $a$; and if it does not he has yet one Throw to come, which, by 
what has been said before, is worth $\frac{1}{36}a$.  But there is only one 
Chance for throwing 12 the first Throw and 35 Chances against it:  
Therefore since he has 1 Expectation of $a$ and 35 of $\frac{1}{36}a$, his 
[19] Chance by \textit{Prop.}\ 3.\ is worth $\frac{71}{1296}a$; and his 
Adversary's the remaining $\frac{1225}{1296}a$

F{\scriptsize ROM} these Two Cases we can determine the value of his Chance who ventures to 
do the same in Four Throws, without considering the Chance of him that 
undertakes to do it in Three.

F{\scriptsize OR} he that ventures to throw 12 once in four times throwing, if he does not 
throw it the first or second time, gains $a$; and if the contrary happens, he 
has still two Throws more, which, by what has been said before, are worth 
$\frac{71}{1296}a$: For which Reason likewise, he must have 71 Chances for 
throwing 12 in one of the two first, and 1225 Chances against it.  Therefore 
at his beginning to throw he has 71 Expectations of $a$, and 1225 Expectations of 
$\frac{71}{1296}a$, which, by 
\textit{Prop.}\ 2., are worth $\frac{178991}{1679616}a$: And the value of his 
Chance that plays against him will be the remaining 
$\frac{1500625}{1679616}a$.  Which shows that their Chances are to one 
another, as 178991 to 1500625.

F{\scriptsize ROM} which likewise, without calculating any other Cases for that Purpose, 
may be found by the same Way of Reasoning, the worth of his Expectation who 
undertakes to throw two \textit{Sixes} once in 8 Throws.  And from thence the 
worth of his Expectation [20] who ventures to do the same in 16 Throws.  And 
from the Expectation of this last being found, together with his also who 
ventures it in 8 Throws, may be determined the value of his Expectation who 
undertakes it in 24 Throws.  In which Operation, because the principal 
Question is, In what Number of Throws one may lay an even Wager to throw two 
\textit{Sixes}; we may cut off some of the hindermost Figures from the long 
Numbers that arise in the midst of the Calculations, and which wou'd otherwise 
encrease prodigiously.  And by this Means, I find he that undertakes it in 24 
Throws wants something of an even Chance to win; and he that lays to do it in 
25 Throws, has something the better side of the Wager.

                              PROP.\ XII.

\textit{To find with how many Dice one may undertake to throw two Sixes the 
first Throw.}

T{\scriptsize HIS} is the same thing as if we wou'd know, in how many Throws one may 
undertake to throw two \textit{Sixes} with a single Die. He that ventures
to do this in two Throws, has, by what has been demonstrated in  
\textit{Prop.} 11. a Chance worth $\frac{1}{36}a$. 
He that undertakes 
it in three Throws, if he does not happen to throw one \textit{Six} the first 
Throw, has yet two more to come, which as before, are worth $\frac{1}{36}a$.  
But if the first Throw chance to be a \textit{Six}, he has two Throws more to 
throw one \textit{Six}, which by \textit{Prop.}\ 10.\ are worth 
[21] Now 'tis plain that there is one Chance for throwing a \textit{Six} the 
first Throw, and five Chances to the contrary: So that before he throws, he 
has one Chance for $\frac{11}{36}a$, 5 Chances of $\frac{1}{36}a$, which by 
\textit{Prop.}\ 3., are worth $\frac{16}{216}a$, or $\frac{2}{27}a$.  After 
this manner, by continually taking one more throw, we find that we may in 
10 Throws with one Die, or in one Throw with 10 Dice, undertake to throw two 
\textit{Sixes}, and that with Advantage.
                              PROP.\ XIII.

\textit{Supposing I lay with another to take one Throw with a pair of Dice 
upon these Terms, That if the Number 7 comes up, I shall win, and if 10 comes 
up, he shall win; and after this Bargain made, we consent to draw Stakes by a 
fair Division, according to the Value of our Chances in the present Contract: 
To find what shall be our several Shares.}

B{\scriptsize ECAUSE}, of the 36 Throws upon two Dice, there are but 6 which consist of the 
Number 7 a piece, and 3 which consist of the Number 10 a piece; there remain 
27 Throws, which, if any one of them chance to come up, will make us neither 
win nor lose, and consequently entitle each of us to $\frac{1}{2}a$.  But 
if none of those Throws shou'd happen, I have 6 Chances for winning $a$, or and 
Chances of losing, or having 0, which by \textit{Prop.}\ 3.\ is as good as if 
[22] I had $\frac{2}{3}a$.  Therefore I have from the beginning 27 Chances for 
$\frac{1}{2}a$ and 9 Chances for $\frac{2}{3}a$, which by \textit{Prop.}\ 3.\ 
is worth $\frac{13}{24}a$; and there remains to him that plays against me 

                              PROP.\ XIV.

\textit{If my self and another play by turns with a pair of Dice upon these 
Terms, That I shall win if I throw the Number 7, or he if he throws 6 soonest, 
and he to have the Advantage of first Throw: To find the Proportion of our 

S{\scriptsize UPPOSE} my Chance worth $x$, and call the Stake $a$; therefore his Chance will 
be $= a - x$.  'Tis plain then, that whenever it comes to his Turn to throw, my 
Chance ought to be $= x$.  But when it is my Turn to throw, my Chance must be 
worth something more.  Let its Value, then, be express'd by $y$.  Now because 
of the 36 Throws upon a pair of Dice, 5 are made up of the Number 6 apiece, 
and may make my Adversary win, and 31 of them are against him, \textit{i.e.}\ 
promote my Turn of throwing; I have, before he begins to throw, 5 Chances of 
obtaining 0, and 31 Chances of obtaining $y$, which, by \textit{Prop.}\ 3.\ 
are worth $\frac{31}{36}y$.  But my Chance from the beginning was supposed 
worth $x$, and therefore $\frac{31}{36}y = x$, and consequently 
[23] $y = \frac{36x}{31}$.  It was further supposed that in my Turns of 
throwing, my Chance was worth $y$.  But when I'm to throw, I have 6 Chances 
of gaining $a$, because there are 6 Throws of the Number 7 apiece, which 
wou'd give me the Game; and 30 Chances, which will bring it to my Adversary's 
Turn to throw, \textit{i.e.}\ make me gain $x$; which, by \textit{Prop.}\ 
3.\ are worth $\frac{6a + 30x}{36}$.  And because this $= y$, which was 
before found $= 36x$; therefore $\frac{6a + 30x}{36} = \frac{36x}{31}$.   
From which Equation will be had $x = \frac{31}{61}a$, the Value of my 
Chance.  And by consequence my Adversary's will be worth $\frac{30}{61}a$, 
so that the Proportion of my Chance to his is, as 31 to 30.

\textit{For a concluding Ornament to this Work, we shall subjoin the following} 

                              PROBLEM I.

$A$ and $B$ play together with a pair of Dice upon this Condition, That $A$ 
shall win if he throws 6, and $B$ if he throws 7; and $A$ is to take one 
Throw first, and then $B$ two Throws together, then $A$ to take two Throws 
together, and so on both of them the same, till one wins.  The Question is, 
\textit{What Proportion their Chances bear to one another?}
Answ.  As 10355 to 12276.

                              PROBLEM II.

T{\scriptsize HREE} Gamesters, $A$, $B$, and $C$, taking 12 Counters, 4 of which are white, 
and 8 black, play upon these Terms: That the first of them that shall 
blindfold choose a white Counter shall win; and $A$ shall have the first 
Choice, $B$ the second, and $C$ the third; and then $A$ to begin again, and 
so on in their Turns.  \textit{What is the Proportion of their Chances?}

                             PROBLEM III.

$A$ lays with $B$, that out of 40 Cards, \textit{i.e.}\ 10 of each different 
Sort, he will draw 4, so as to have one of every Sort.  And the Proportion of 
his Chance to that of $B$, is found to be as 1000 to 8139.

                              PROBLEM IV.

H{\scriptsize AVING} chosen 12 Counters as before, 8 black and 4 white, $A$ lays with $B$ 
that he will blindfold take 7 out of them, among which there shall be 3 black 
ones.  \textit{Quaere, What is the Proportion of their Chances?}

                              PROBLEM V.

$A$ and $B$ taking 12 Pieces of Money each, play with 3 Dice on this 
Condition, That if the Number 11 is thrown, $A$ shall give $B$ one Piece, 
but if 14 be thrown, then $B$ shall give one to $A$; and he shall win 
the Game that first gets all the Pieces of Money.  And the Proportion of 
$A$'s Chance to $B$'s is found to be, as 244,140,625 to 282,429,536,481.


                              \textit{F I N I S.}