# A Philosophical Essay on Probabilities/Chapter 5

CHAPTER V.

*CONCERNING THE ANALYTICAL METHODS OF *
THE CALCULUS OF PROBABILITIES.

The application of the principle which we have just expounded to the various questions of probability requires methods whose investigation has given birth to several methods of analysis and especially to the theory of combinations and to the calculus of finite differences.

If we form the product of the binomials, unity plus the first letter, unity plus the second letter, unity plus the third letter, and so on up to *n* letters, and subtract unity from this developed product, the result will be the sum of the combination of all these letters, taken one by one, two by two, three by three, etc., each combination having unity for a coefficient. In order to have the number of combinations of these *n* letters taken *s* by *s* times, we shall observe that if we suppose these letters equal among themselves, the preceding product will become the *n*th power of the binomial one plus the first letter; thus the number of combinations of *n* letters taken *s* by *s* times will be the coefficient of the *s*th power of the first letter in the THE CALCULUS OF PROBABILITIES. 27
development in this binomial ; and this number is
obtained by means of the known binomial formula.
Attention must be paid to the respective situations
of the letters in each combination, observing that if a
second letter is joined to the first it may be placed in
the first or second position which gives two combina-
tions. If we join to these combinations a third letter,
we can give it in each combination the first, the second,
and the third rank which forms three combinations
relative to each of the two others, in all six combina-
tions. From this it is easy to conclude that the
number of arrangements of which s letters are suscepti-
ble is the product of the numbers from unity to s. In
order to pay regard to the respective positions of the
letters it is necessary then to multiply by this product
the number of combinations of n letters s hy s times,
which is tantamount to taking away the denominator
of the coefficient of the binomial which expresses this
number.
Let us imagine a lottery composed of n numbers, of
which r are drawn at each draw. The probability is
demanded of the drawing of s given numbers in one
draw. To arrive at this let us form a fraction whose
denominator will be the number of all the cases possi-
ble or of the combinations of n letters taken r hy r
times, and whose numerator will be the number of all
the combinations which contain the given s numbers.
This last number is evidently that of the combinations
of the other numbers taken 71 less s by n less s times.
This fraction will be the required probability, and we
shall easily find that it can be reduced to a fraction
whose numerator is the number of combinations of /■ 2 8 A PHILOSOPHICAL ESSAY ON PROBABILITIES.
numbers taken s hy s times, and whose denominator is
the number of combinations of n numbers taken
similarly s hy s times. Thus in the lottery of France,
formed as is known of 90 numbers of which five are
drawn at each draw, the probability of drawing a given
combination is -^^, or y^! ^^e lottery ought then for the
equality of the play to give eighteen times the stake.
The total number of combinations two by two of the
90 numbers is 4005, and that of the combinations two
by two of 5 numbers is 10. The probability of the
drawing of a given pair is then ^j^V?' ^^^ ^^^ lottery
ought to give four hundred and a half times the stake ;
it ought to give 11748 times for a given tray, 51 1038
times for a quaternary, and 43949268 times for a quint.
The lottery is far from giving the player these advan-
tages.
Suppose in an urn a white balls, b black balls, and
after having drawn a ball it is put back into the urn ;
the probability is asked that in n number of draws m
white balls and n — m black balls will be drawn. It
is clear that the number of cases that may occur at
each drawing is a -{- b. Each case of the second
drawing being able to combine with all the cases of the
first, the number of possible cases in two drawings is
the square of the binomial a -- b. In the development
of this square, the square of a expresses the number of
cases in which a white ball is twice drawn, the double
product of « by 3 expresses the number of cases in
which a white ball and a black ball are drawn. Finally,
the square of b expresses the number of cases in which
two black balls are drawn. Continuing thus, we see
generally that the nth. power of the binomial a -- b THE CALCULUS OF PROBABILITIES. 29
7t
expresses the number of all the cases possible in
draws; and that in the development of this power the
term multiplied by the mth. power of a expresses the
number of cases in which m white balls and n — m
black balls may be drawn. Dividing then this term
by the entire power of the binomial, we shall have the
probability of drawing m white balls and n — m black
balls. The ratio of the numbers a and a -^ b being
the probability of drawing one white ball at one draw;
and the ratio of the numbers b and a -- b being the
probability of drawing one black ball ; if we call these
probabilities/ and q, the probability of drawing m white
balls in n draws will be the term multiplied by the m'Cn.
power of/ in the development of the «th power of the
binomial p -- q; we may see that the sum p -{- q %
unity. This remarkable property of the binomial is
very useful in the theory of probabilities. But the
most general and direct method of resolving questions
of probability consists in making them depend upon
equations of differences. Comparing the successive
conditions of the function which expresses the prob-
ability when we increase the variables by their respect-
ive differences, the proposed question often furnishes a
very simple proportion between the conditions. This
proportion is what is called equation of ordinary or
partial differentials ; ordinary when there is only one
variable, partial when there are several. Let us con-
sider some examples of this.
Three players of supposed equal ability play together
on the following conditions : that one of the first two
players who beats his adversary plays the third, and if
he beats him the game is finished. If he is beaten, the 3° A PHILOSOPHICAL ESSAY ON PROBABILITIES.
victor plays against the second until one of the players
has defeated consecutively the two others, which ends
the game. The probability is demanded that the game
will be finished in a certain number n of plays. Let
us find the probability that it will end precisely at the
nth play. For that the player who wins ought to enter
the game at the play n — i and win it thus at the fol-
lowing play. But if in place of winning the play n — i
he should be beaten by his adversary who had just
beaten the other player, the game would end at this
play. Thus the probability that one of the players will
enter the game at the play n — i and will win it is
equal to the probability that the game will end pre-
cisely with this play; and as this player ought to win
the following play in order that the game may be
finished at the wth play, the probability of this last case
will be only one half of the preceding one. This
probability is evidently a function of the "number n; this
function is then equal to the half of the same function
v/hen n is diminished by unity. This equality forms
one of those equations called ordinary finite differential
equations.
We may easily determine by its use the probability
that the game will end precisely at a certain play. It
is evident that the play cannot end sooner than at the
second play; and for this it is necessary that that one
of the first two players who has beaten his adversary
should beat at the second play the third player ; the
probability that the game will end at this play is .
Hence by virtue of the preceding equation we conclude
that the successive probabilities of the end of the game
are for the third play, for the fourth play, and so THE CALCULUS OF PROBABILITIES. 31
on ; and in general ^ raised to the power « — i for the
nth. play. The sum of all these powers of ^ is unity
less the last of these powers ; it is the probability that
the game will end at the latest in n plays.
Let us consider again the first problem more difficult
which may be solved by probabilities and which Pascal
proposed to Fermat to solve. Two players, A and B,
of equal skill play together on the conditions that the
one who first shall beat the other a given number of
times shall win the game and shall take the sum of the
stakes at the game; after some throws the players
agree to quit without having finished the game : we ask
in what manner the sum ought to be divided between
them. It is evident that the parts ought to be propor-
tional to the respective probabilities of winning the
game. The question is reduced then to the determina-
tion of these probabilities. They depend evidently
upon the number of points which each player lacks of
having attained the given number. Hence the prob-
ability of A is a function of the two numbers which we
will call indices. If the two players should agree to
play one throw more (an agreement which does not
change their condition, provided that after this new
throw the division is always made proportionally to the
new probabilities of winning the game), then either A
would win this throw and in that case the number of
points which he lacks would be diminished by unity,
or the player B would win it and in that case the
number of points lacking to this last player would be
less by unity. But the probability of each of these
cases is J; the function sought is then equal to one half
of this function in which we diminish by unity the first 32 A PHILOSOPHICAL ESSAY ON PROBABILITIES.
index plus the half of the same function in which the
second variable is diminished by unity. This equality
is one of those equations called equations of partial
differentials .
We are able to determine by its use the probabilities
of A by dividing the smallest numbers, and by observ-
ing that the probability or the function which expresses
it is equal to unity when the player A does not lack a
single point, or when the first index is zero, and that
this function becomes zero with the second index. Sup-
posing thus that the player A lacks only one point, we
find that his probability is , |, |, etc., according as B
lacks one point, two, three, etc. Generally it is then
unity less the power of ^, equal to the number of points
which B lacks. We will suppose then that the player
A lacks two points and his probability will be found
equal to J, , , etc., according as B lacks one point,
two points, three points, etc. We will suppose again
that the player A lacks three points, and so on.
This manner of obtaining the successive values of a
quantity by means of its equation of differences is long
and laborious . The geometricians have sought methods
to obtain the general function of indices that satisfies
this equation, so that for any particular case we need
only to substitute in this function the corresponding
values of the indices. Let us consider this subject in
a general way. For this purpose let us conceive a
series of terms arranged along a horizontal line so that
each of them is derived from the preceding one accord-
ing to a given law. Let us suppose this law expressed
by an equation among several consecutive terms and
their index, or the number which indicates the rank that THE C/tLCULUS OF PROBABILITIES. 33
they occupy in the series. This equation I call the
equation of finite differences by a single index. The
order or the degree of this equation is the difference of
rank of its two extreme terms. We are able by its use
to determine successively the terms of the series and to
continue it indefinitely ; but for that it is necessary to
know a number of terms of the series equal to the
degree of the equation . These terms are the arbitrary
constants of the expression of the general term of the
series or of the integral of the equation of differences.
Let us imagine now below the terms of the preceding
series a second series of terms arranged horizontally;
let us imagine again below the terms of the second
series a third horizontal series, and so on to infinity;
and let us suppose the terms of all these series con-
nected by a general equation among several consecutive
terms, taken as much in the horizontal as in the ver-
tical sense, and the numbers which indicate their rank
in the two senses. This equation is called the equation
of partial finite differences by two indices.
Let us imagine in the same way below the plan of
the preceding series a second plan of similar series,
whose terms should be placed respectively below those
of the first plan ; let us imagine again below this second
plan a third plan of similar series, and so on to infinity;
let us suppose all the terms of these series connected
by an equation among several consecutive terms taken
in the sense of length, width, and depth, and the three
numbers which indicate their rank in these three senses.
This equation I call the equation of partial finite differ-
ences by three indices.
Finally, considering the matter in an abstract way and independently of the dimensions of space, let us imagine generally a system of magnitudes, which should be functions of a certain number of indices, and let us suppose among these magnitudes, their relative differences to these indices and the indices themselves, as many equations as there are magnitudes; these equations will be partial finite differences by a certain number of indices.

We are able by their use to determine successively these magnitudes. But in the same manner as the equation by a single index requires for it that we known a certain number of terms of the series, so the equation by two indices requires that we know one or several lines of series whose general terms should be expressed each by an arbitrary function of one of the indices. Similarly the equation by three indices requires that we know one or several plans of series, the general terms of which should be expressed each by an arbitrary function of two indices, and so on. In all these cases we shall be able by successive eliminations to determine a certain term of the series. But all the equations among which we eliminate being comprised in the same system of equations, all the expressions of the successive terms which we obtain by these eliminations ought to be comprised in one general expression, a function of the indices which determine the rank of the term. This expression is the integral of the proposed equation of differences, and the search for it is the object of integral calculus.

Taylor is the first who in his work entitled *Metodus incrementorum* has considered linear equations of finite differences. He gives the manner of integrating those THE CALCULUS OF PROBABILITIES. 35
of the first order with a coefficient and a last term,
functions of the index. In truth the relations of the
terms of the arithmetical and geometrical progressions
which have always been taken into consideration are
the simplest cases of linear equations of differences ; but
they had not been considered from this point of view.
It was one of those which, attaching themselves to
general theories, lead to these theories and are conse-
quently veritable discoveries.
About the same time Moivre was considering under
the name of recurring series the equations of finite
differences of a certain order having a constant coeffi-
cient. He succeeded in integrating them in a very
ingenious manner. As it is always interesting to follow
the progress of inventors, I shall expound the method
of Moivre by applying it to a recurring series whose
relation among three consecutive terms is given. First
he considers the relation among the consecutive terms
of a geometrical progression or the equation of two
terms which expresses it. Referring it to terms less
than unity, he multiplies it in this state by a constant
factor and subtracts the product from the first equation.
Thus he obtains an equation among three consecutive
terms of the geometrical progression. Moivre considers
next a second progression whose ratio of terms is the
same factor which he has just used. He diminishes
similarly by unity the index of the terms of the equa-
tion of this new progression. In this condition he
multiplies it by the ratio of the terms of the first pro-
gression, and he subtracts the product from the equation
of the second progression, which gives him among three
consecutive terms of this progression a relation entirely 3<> A PHILOSOPHICAL ESSAY ON PROBABILITIES.
similar to that which he has found for the first progres-
sion. Then he observes that if one adds term by term
the two progressions, the same ratio exists among any
three of these consecutive terms. He compares the
coefficients of this ratio to those of the relation of the
terms of the proposed recurrent series, and he finds for
determining the ratios of the two geometrical progres-
sions an equation, of the second degree, whose roots are
these ratios. Thus Moivre decomposes the recurrent
series into two geometrical progressions, each multi-
plied by an arbitrary constant which he determines by
means of the first two terms of the recurrent series.
This ingenious process is in fact the one that d'Alembert
has since employed for the integration of linear equa-
tions of infinitely small differences with constant coeffi-
cients, and Lagrange has transformed into similar
equations of finite differences.
Finally, I have considered the linear equations of
partial finite differences, first under the name of recurro-
recurrent series and afterwards under their own name.
The most general and simplest manner of integrating
all these equations appears to me that which I have
based upon the consideration of discriminant functions,
the idea of which is here given.
If we conceive a function F of a variable t developed
according to the powers of this variable, the coefficient
of any one of these powers will be a function of the
exponent or index of this power, which index I shall
call X. V is what I call the discriminant function of
this coefficient or of the function of the index.
Now if we multiply the series of the development of
Vhya function of the same variable, such, for example, THE CALCULUS OF PROBABILITIES. 37
as unity plus two times this variable, the product will
be a new discriminant function in which the coefficient
of the power x of the variable t will be equal to the
coefficient of the same power in V plus twice the
coefficient of the power less unity. Thus the function
of the index x in the product will be equal to the func-
tion of the index x m V plus twice the same function
in which the index is diminished by unity. This func-
tion of the index x is thus a derivative of the function
of the same index in the development of V, a function
which I shall call the primitive function of the index.
Let us designate the derivative function by the letter S
placed before the primitive function. The derivation
indicated by this letter will depend upon the multiplier
of V, which we will call T and which we will suppose
developed- like V by the ratio to the powers of the
variable t. If we multiply anew by T the product of
V by T, which is equivalent to multiplying V by T^,
we shall form a third discriminant function, in which
the coefficient of the ;rth power of t will be a derivative
similar to the corresponding coefficient of the preceding
product ; it may be expressed by the same character d
placed before the preceding derivative, and then this
character will be written twice before the primitive
function of x. But in place of writing it thus twice we
give it 2 for an exponent.
Continuing thus, we see generally that if we multiply
V by the «th power of T, we shall have the coefficient
of the ;irth power of t in the product of V by the «th
power of T by placing before the primitive function the
character S with 7i for an exponent.
Let us suppose, for' example, that T be unity divided 38 A PHILOSOPHICAL ESSAY ON PROBABILITIES.
by t; then in the product of Fby 7" the coefificient of
the ;rth power of t will be the coefficient of the power
greater by unity in V; this coefficient in the product
of V by the nth. power of T will then be the primitive
function in which x is augmented by n units.
Let us consider now a new function Z of t, developed
like V and T according to the powers of t; let us
designate by the character A placed before the primi-
tive function the coefficient of the jrth power of t in the
product of V by Z; this coefficient in the product of V
by the nth. power of Z will be expressed by the char-
acter A affected by the exponent n and placed before
the primitive function of x.
If, for example, Z is equal to unity divided by t less
one, the coefficient of the xth power of t in the product
of Vhy Z will be the coefficient of the x -- i power
of ^ in V less the coefficient of the ;irth power. It will
be then the finite difference of the primitive function of
the index x. Then the character J indicates a finite
difference of the primitive function in the case where
the index varies by unity; and the nth power of this
character placed before the primitive function will indi-
cate the finite nth difference of this function. If we
suppose that T be unity divided by i, we shall have 7
equal to the binomial Z-- i. The product of F by
the nth power of T will then be equal to the product
of V by the nth power of the binomial Z-- i. Develop-
ing this power in the ratio of the powers of Z, the
product of V by the various terms of this development
will be the discriminant functions of these same terms
in which we substitute in place of the powers of Z the THE CALCULUS OF PROBABILITIES. 39
corresponding finite differences of the primitive function
of the index.
Now the product of V by the «th power of T is the
primitive function in which the index x is augmented
by n units ; repassing from the discriminant functions
to their coefficients, we shall have this primitive function
thus augmented equal to the development of the nth.
power of the binomial Z -- i, provided that in this
development we substitute in place of the powers of Z
the corresponding differences of the primitive function
and that we multiply the independent term of these
powers by the primitive function. We shall thus
obtain the primitive function whose index is augmented
by any number n by means of its differences.
Supposing that T and Z always have the preceding
values, we shall have Z equal to the binonrial T — i;
the product of V by the nth power of Z will then be
equal to the product of V by the development of the
nth power of the binomial T — 1. Repassing from the
discriminant functions to their coefficients as has just
been done, we shall have the nth difference of the
primitive function expressed by the development of the
«th power of the binomial T — i, in which we substi-
tute for the powers of T this same function whose index
is augmented by the exponent of the power, and for
the independent term of t, which is unity, the primitive
function, which gives this difference by means of the
consecutive terms of this function.
Placing 8 before the primitive function expressing the
derivative of this function, which multiplies the x power
of t in the product of V by T, and d expressing the
same derivative in the product of V by Z, we are led 40 A PHILOSOPHICAL ESSAY ON PROBABILITIES.
by that which precedes to this general result: whatever
may be the function of the variable i represented by T
and Z, we may, in the development of all the identical
equations susceptible of being formed among these
functions, substitute the characters 3 and ^ in place of
T and Z, provided that we write the primitive function
of the index in series with the powers and with the
products of the powers of the characters, and that we
multiply by this function the independent terms of these
characters.
We are able by means of this general result to trans-
form any certain power of a difference of the primitive
function of the index x, in which x varies by unity, into
a series of differences of the same function in which x
varies by a certain number of units and reciprocally.
Let us suppose that T be the i power of unity divided
by ^ — I , and that Z be always unity divided by ^ — i ;
then the coefficient of the x power of t in the pro-
duct of V by T will be the coefficient of the x -- i
power of i* in V less the coefficient of the x power of ^;
it will then be the finite difference of the primitive
function of the index x in which we vary this index by
the number i. It is easy to see that T is equal to the
difference between the i power of ,the binomial Z -{- i
and unity. The nth. power of T is equal to the nth.
power of this difference. If in this equality we substi-
tute in place of T and Z the characters d and J, and
after the development we place at the end of each term
the primitive function of the index x, we shall have the
nth. difference of this function in which x varies by t
units expressed by a series of differences of the same
function in which x varies by unity. This series is THE CALCULUS OF PROBABILITIES. 41
only a transformation of the difference which it
expresses and which is identical with it; but it is in
similar transformations that the power of analysis
resides.
The generality of analysis permits us to suppose in
this expression that n is negative. Then the negative
powers of d and A indicate the integrals. Indeed the
«th difference of the primitive function having for a
discriminant function the product of V by the nth power
of the binomial one divided by t less unity, the primi-
tive function which is the nth integral of this difference
has for a discriminant function that of the same differ-
ence multiplied by the nth power taken less than the
binomial one divided by t minus one, a power to which
the same power of the character A corresponds; this
power indicates then an integral of the same order, the
index x varying by unity; and the negative powers of
d indicate equally the integrals x varying by i units.
We see, thus, in the clearest and simplest manner the
rationality of the analysis observed among the positive
powers and differences, and among the negative powers
and the integrals.
If the function indicated by d placed before the
primitive function is zero, we shall have an equation of
finite differences, and Fwill be the discriminant function
of its integral. In order to obtain this discriminant
function we shall observe that in the product of V by
T all the powers of t ought to disappear except the
powers inferior to the order of the equation of differ-
ences; V is then equal to a fraction whose denominator
is T and whose numerator is a polynomial in which the
highest power of t is less by unity than the order of the 42 A PHILOSOPHICAL ESSAY ON PROBABILITIES.
equation of differences. The arbitrary coefficients of
the various powers of t in this polynomial, including
the power zero, will be determined by as many values
of the primitive function of the index when we make
successively x equal to zero, to one, to two, etc.
When the equation of differences is given we determine
T by putting all its terms in the first member and zero
in the second; by substituting in the first member unity
in place of the function which has the largest index;
the first power of t in place of the primitive function in
which this index is diminished by unity; the second
power of t for the primitive function where this index
is diminished by two units, and so on. The coefficient
of the jrth power of t in the development of the preced-
ing expression of V will be the primitive function of x
or the integral of the equation of finite differences.
Analysis furnishes for this development various means,
among which we may choose that one which is most
suitable for the question proposed ; this is an advantage
of this method of integration.
Let us conceive now that F be a function of the two
variables / and f developed according to the powers
and products of these variables ; the coefficient of any
product of the powers x and x' of t and /' will be a
function of the exponents or indices x and x' of these
powers; this function I shall call ^& primitive function
of which V is the discriminant function.
Let us multiply V hy a. function T of the two
variables t and t' developed like V in ratio of the
powers and the products of these variables ; the product
will be the discriminant function of a derivative of the
primitive function; if T, for example, is equal to the THE CALCULUS OF PROBABILITIES. 43
variable t plus the variable t' minus two, this derivative
will be the primitive function of which we diminish by-
unity the index x plus this same primitive function of
which we diminish by unity the index x' less two
times the primitive function. Designating whatever T
may be by the character d placed before the primitive
function, this derivative, the product of V by the «th
power of T, will be the discriminant function of the
derivative of the primitive function before which one
places the «th power of the character S. Hence result
the theorems analogous to those which are relative to
functions of a single variable.
Suppose the function indicated by the character S be
zero ; one will have an equation of partial differences.
If, for example, we make as before T equal to the
variable t plus the variable t' — 2, we have zero equal
to the primitive function of which we diminish by unity
the index x plus the same function of which we diminish
by unity the index x' minus two times the primitive
function. The discriminant function F of the primitive
function or of the integral of this equation ought then
to be such that its product by T does not include at
all the products of ^ by ^' ; but Fmay include separately
the powers of t and those of t' , that is to say, an arbi-
trary function of t and an arbitrary function oi t'; V is
then a fraction whose numerator is the sum of these two
arbitrary functions and whose denominator is T. The
coefficient of the product of the j^h power of t by the
x' power of f in the development of this fraction will
then be the integral of the preceding equation of partial
differences. This method of integrating this kind of
equations seems to me the simplest and the easiest by 44 A PHILOSOPHICAL ESSAY ON PROBABILITIES.
the employment of the various analytical processes for
the development of rational fractions.
More ample details in this matter would be scarcely
understood without the aid of calculus.
Considering equations of infinitely small partial
differences as equations of finite partial differences in
which nothing is neglected, we are able to throw hght
upon the obscure points of their calculus, which have
been the subject of great discussions among geometri-
cians. It is thus that I have demonstrated the possi-
bility of introducing discontinued functions in their
integrals, provided that the discontinuity takes place
only for the differentials of the order of these equations
or of a superior order. The transcendent results of
calculus are, like all the abstractions of the understand-
ing, general signs whose true meaning may be ascer-
tained only by repassing by metaphysical analysis to
the elementary ideas which have led to them; this
often presents great difficulties, for the human mind
tries still less to transport itself into the future than to
retire within itself. The comparison of infinitely small
differences with finite differences is able similarly to
shed great light upon the metaphysics of infinitesimal
calculus.
It is easily proven that the finite nth. difference of a
function in which the increase of the variable is E
being divided by the «th power of E, the quotient
reduced in series by ratio to the powers of the increase
E is formed by a first term independent of E. In the
measure that E diminishes, the series approaches more
and more this first term from which it can differ only
by quantities less than any assignable magnitude. THE CALCULUS OF PROBABILITIES. 45
This term is then the limit of the series and expresses
in differential calculus the infinitely small nth difference
of the function divided by the «th power of the infinitely
small increase.
Considering from this point of view the infinitely
small differences, we see that the various operations of
differential calculus amount to comparing separately in
the development of identical expressions the finite
terms or those independent of the increments of the
variables which are regarded as infinitely small ; this
is rigorously exact, these increments being indetermi-
nant. Thus differential calculus has all the exactitude
of other algebraic operations.
The same exactitude is found in the applications of
differential calculus to geometry and mechanics. If
we imagine a curve cut by a secant at two adjacent
points, naming E the interval of the ordinates of these
two points, E will be the increment of the abscissa from
the first to the second ordinate. It is easy to see that
the corresponding increment of the ordinate will be the
product of E by the first ordinate divided by its sub-
secant; augmenting then in this equation of the curve
the first ordinate by this increment, we shall have the
equation relative to the second ordinate. The differ-
ence of these two equations will be a third equation
which, developed by the ratio of the powers of E and
divided by E, will have its first term independent of E,
which will be the limit of this development. This
term, equal to zero, will give then the limit of the sub-
secants, a limit which is evidently the subtangent.
This singularly happy method of obtaining the sub-
tangent is due to Fermat, who has extended it to 46 A PHILOSOPHICAL ESSAY ON PROBABILITIES.
transcendent curves. This great geometrician ex-
presses by the character E the increment of the
abscissa; and considering only the first power of this
increment, he determines exactly as we do by differen-
tial calculus the subtangents of the curves, their points
of inflection, ihe maxima and minima of their ordinates,
and in general those of rational functions. We see
likewise by his beautiful solution of the problem of the
refraction of light inserted in the Collection of the
Letters of Descartes that he knows how to extend his
methods to irrational functions in freeing them from
irrationalities by the elevation of the roots to powers.
Fermat should be regarded, then, as the true discoverer
of Differential Calculus. Newton has since rendered this
calculus more analytical in his Method of Fluxions, and
simplified and generalized the processes by his beautiful
theorem of the binomial. Finally, about the same time
Leibnitz has enriched differential calculus by a nota-
tion which, by indicating the passage from the finite to
the infinitely small, adds to the advantage of express-
ing the general results of calculus that of giving the
first approximate values of the differences and of the
sums of the quantities; this notation is adapted of itself
to the calculus of partial differentials.
We are often led to expressions which contain so
many terms and factors that the numerical substitutions
are impracticable. This takes place in questions of
probability when we consider a great number of events.
Meanwhile it is necessary to have the numerical value
of the formulse in order to know with what probability
the results are indicated, which the events develop by
multiplication. It is necessary especially to have the THE CALCULUS OF PROBABILITIES. 47
law according to which this probabiHty continually
approaches certainty, which it will finally attain if the
number of events were infinite. In order to obtain this
law I considered that the definite integrals of differen-
tials multiplied by the factors raised to great powers
would give by integration the formula; composed of
a great number of terms and factors. This remark
brought me to the idea of transforming into similar
integrals the complicated expressions of analysis and
the integrals of the equation of differences. I fulfilled
this condition by a method which gives at the same
time the function comprised under the integral sign
and the limits of the integration. It offers this remark-
able thing, that the function is the same discriminant
function of the expressions and the proposed equations ;
this attaches this method to the theory of discriminant
functions of which it is thus the complement. Further,
it would only be a question of reducing the definite
integral to a converging series. This I have obtained
by a process which makes the series converge with as
much more rapidity as the formula which it represents
is more complicated, so that it is more exact as it
becomes more necessary. Frequently the series has
for a factor the square root of the ratio of the circum-
ference to the diameter; sometimes it depends upon
other transcendents whose number is infinite.
An important remark which pertains to great gen-
erality of analysis, and which permits us to extend this
method to formulae and to equations of difference which
the theory of probability presents most frequently, is
that the series to which one comes by supposing the
limits of the definite integrals to be real and positive 48 A PHILOSOPHICAL ESSAY ON PROBABILITIES.
take place equally in the case where the equation which
determines these limits has only negative or imaginary
roots. These passages from the positive to the nega-
tive and from the real to the imaginary, of which I first
have made use, have led me further to the values of
many singular definite integrals, which I have accord-
ingly demonstrated directly. We may then consider
these passages as a means of discovery parallel to
induction and analogy long employed by geometricians,
at first with an extreme reserve, afterwards with entire
confidence, since a great number of examples has
justified its use. In the mean time it is always necessary
to confirm by direct demonstrations the results obtained
by these divers means.
I have named the ensemble of the preceding methods
the Calculus of Discriminant Functions; this calculus
serves as a basis for the work which I have published
under the title of the Analytical Theory of Probabilities.
It is connected with the simple idea of indicating the
repeated multiplications of a quantity by itself or its
entire and positive powers by writing toward the top of
the letter which expresses it the numbers which mark
the degrees of these powers.
This notation, employed by Descartes in his Geometry
and generally adopted since the publication of this
important work, is a little thing, especially when com-
pared with the theory of curves and variable functions
by which this great geometrician has established the
foundations of modern calculus. But the language of
analysis, most perfect of all, being in itself a powerful
instrument of discoveries, its notations, especially when
they are necessary and happily conceived, are so many THE CALCULUS OF PROBABILITIES. 49
germs of new calculi. This is rendered appreciable by
this example.
Wallis, who in his work entitled Arithmetica Infini-
torum, one of those which have most contributed to the
progress of analysis, has interested himself especially
in following the thread of induction and analogy, con-
sidered that if one divides the exponent of a letter by
two, three, etc., the quotient will be accordingly the
Cartesian notation, and when division is possible the
exponent of the square, cube, etc., root of the quantity
which represents the letter raised to the dividend
exponent. Extending by analogy this result to the
case where division is impossible, he considered a
quantity raised to a fractional exponent as the root of
the degree indicated by the denominator of this frac-
tion — namely, of the quantity raised to a power indi-
cated by the numerator. He observed then that,
according to the Cartesian notation, the multiplication
of two powers of the same letter amounts to adding
their exponents, and that their division amounts to
subtracting the exponents of the power of the divisor
from that of the power of the dividend, when the second
of these exponents is greater than the first. Wallis
extended this result to the case where the first
exponent is equal to or greater than the second, which
makes the difference zero or negative. He supposed
then that a negative exponent indicates unity divided
by the quantity raised to the same exponent taken
positively. These remarks led him to integrate
generally the monomial differentials, whence he inferred
the definite integrals of a particular kind of binomial
differentials whose exponent is a positive integral so A PHILOSOPHICAL ESSAY ON PROBABILITIES.
number. The observation then of the law of the num-
bers which express these integrals, a series of inter-
polations and happy inductions where one perceives
the germ of the calculus of definite integrals which has
so much exercised geometricians and which is one of
the fundaments of my new Theory of Probabilities,
gave him the ratio of the area of the circle to the square
of its diameter expressed by an infinite product, which,
when one stops it, confines this ratio to limits more and
more converging; this is one of the most singular
results in analysis. But it is remarkable that Wallis,
who had so well considered the fractional exponents
of radical powers, should have continued to note these
powers as had been done before him. Newton in his
Letters to Oldembourg, if I am not mistaken, was the
first to employ the notation of these powers by frac-
tional exponents. Comparing by the way of induction,
of which Wallis had made such a beautiful use, the
exponents of the powers of the binomial with the
coefiScients of the terms of its development in the case
where this exponent is integral and positive, he deter-
mined the law of these coefficients and extended it by
analogy to fractional and negative powers. These
various results, based upon the notation of Descartes,
show his influence on the progress of analysis. It has
still the advantage of giving the simplest and fairest
idea of logarithms, which are indeed only the exponents
of a magnitude whose successive powers, increasing by
infinitely small degrees, can represent all numbers.
But the most important extension that this notation
has received is that of variable exponents, which con-
stitutes exponential calculus, one of the most fruitful THE CALCULUS OF PROBABILITIES. 51
branches of modern analysis. Leibnitz was the first
to indicate the transcendents by variable exponents, and
thereby he has completed the system of elements of
which a finite, function can be composed; for every
finite explicit function of a variable may be reduced in
the last analysis to simple magnitudes, combined by
the method of addition, subtraction, multiplication, and
division and raised to constant or variable powers.
The roots of the equations formed from these elements
are the implicit functions of the variable. It is thus
that a variable has for a logarithm the exponent of the
power which is equal to it in the series of the powers
of the number whose hyperbolic logarithm is unity, and
the logarithm of a variable of it is an implicit function.
Leibnitz thought to give to his differential character
the same exponents as to magnitudes ; but then in place
of indicating the repeated multiplications of the same
magnitude these exponents indicate the repeated differ-
entiations of the same function. This new extension
of the Cartesian notation led Leibnitz to the analogy of
positive powers with the differentials, and the negative
powers with the integrals. Lagrange has followed this
singular analogy in all its developments; and by series
of inductions which may be regarded as one of the
most beautiful applications which have ever been made
of the method of induction he has arrived at general
formulae which are as curious as useful on the trans-
formations of differences and of integrals the ones into
the others when the variables have divers finite incre-
ments and when these increments are infinitely small.
But he has not given the demonstrations of it which
appear to him difificult. The theory of discriminant functions extends the Cartesian notations to some of its characters; it shows with proof the analogy of the powers and operations indicated by these characters; so that it may still be regarded as the exponential calculus of characters. All that concerns the series and the integration of equations of differences springs from it with an extreme facility.