# 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 nth 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 sth power of the first letter in the ​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 combinations. 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 combinations. From this it is easy to conclude that the number of arrangements of which $s$ letters are susceptible 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 $s$ by $s$ 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 possible or of the combinations of $n$ letters taken $r$ by $r$ 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 $n$ 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 $r$ ​numbers taken $s$ by $s$ times, and whose denominator is the number of combinations of $n$ numbers taken similarly $s$ by $s$ 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 $\scriptstyle {\frac {5}{90}}$, or $\scriptstyle {\frac {1}{18}}$; the 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 $\scriptstyle {\frac {1}{4005}}$, and the lottery ought to give four hundred and a half times the stake; it ought to give 11748 times for a given tray, 511038 times for a quaternary, and 43949268 times for a quint. The lottery is far from giving the player these advantages.
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$a$ by $b$ 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 $n$th power of the binomial $a+b$ ​expresses the number of all the cases possible in n 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 p and q, the probability of drawing m white balls in n draws will be the term multiplied by the mth power of p in the development of the nth power of the binomial p + q; we may see that the sum p + q is 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 probability when we increase the variables by their respective 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 consider 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 ​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 — 1 and win it thus at the following play. But if in place of winning the play n — 1 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 — 1 and will win it is equal to the probability that the game will end precisely with this play; and as this player ought to win the following play in order that the game may be finished at the nth 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 when 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 1/2. Hence by virtue of the preceding equation we conclude that the successive probabilities of the end of the game are 1/4 for the third play, 1/8 for the fourth play, and so ​on; and in general 1/2 raised to the power n — 1 for the nth play. The sum of all these powers of 1/2 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 proportional to the respective probabilities of winning the game. The question is reduced then to the determination of these probabilities. They depend evidently upon the number of points which each player lacks of having attained the given number. Hence the probability 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 1/2; the function sought is then equal to one half of this function in which we diminish by unity the first ​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 observing 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. Supposing thus that the player A lacks only one point, we find that his probability is 1/2, 3/4, 7/8, etc., according as B lacks one point, two, three, etc. Generally it is then unity less the power of 1/2, 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 1/4, 1/2, 11/16, 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 according 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 ​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 connected by a general equation among several consecutive terms, taken as much in the horizontal as in the vertical 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 differences 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 ​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 consequently 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 coefficient. 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 equation of this new progression. In this condition he multiplies it by the ratio of the terms of the first progression, 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 ​similar to that which he has found for the first progression. 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 progressions an equation, of the second degree, whose roots are these ratios. Thus Moivre decomposes the recurrent series into two geometrical progressions, each multiplied 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 equations of infinitely small differences with constant coefficients, 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 V 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 V by a function of the same variable, such, for example, ​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 function of the index x in V plus twice the same function in which the index is diminished by unity. This function 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 δ 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$^2$, we shall form a third discriminant function, in which the coefficient of the xth power of t will be a derivative similar to the corresponding coefficient of the preceding product; it may be expressed by the same character δ 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 nth power of T, we shall have the coefficient of the xth power of t in the product of V by the nth power of T by placing before the primitive function the character δ with n for an exponent.
Let us suppose, for example, that T be unity divided ​by t; then in the product of V by T the coefficient of the xth 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 Δ placed before the primitive function the coefficient of the xth 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 character Δ 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 V by Z will be the coefficient of the x + 1 power of t in V less the coefficient of the xth power. It will be then the finite difference of the primitive function of the index x. Then the character Δ 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 indicate the finite nth difference of this function. If we suppose that T be unity divided by t, we shall have T equal to the binomial Z + 1. The product of V by the nth power of T will then be equal to the product of V by the nth power of the binomial Z + 1. Developing 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 ​corresponding finite differences of the primitive function of the index.
Now the product of V by the nth 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 + 1, 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 binomial T — 1; 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 nth power of the binomial T — 1, in which we substitute 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 δ 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 Δ expressing the same derivative in the product of V by Z, we are led ​by that which precedes to this general result: whatever may be the function of the variable t 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 δ and Δ 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 transform 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 t — 1 , and that Z be always unity divided by t — 1; then the coefficient of the x power of t in the product of V by T will be the coefficient of the x + i power of t in V less the coefficient of the x power of t; 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 + 1 and unity. The nth power of T is equal to the nth power of this difference. If in this equality we substitute in place of T and Z the characters δ and Δ, 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 ​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 δ and Δ indicate the integrals. Indeed the nth 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 primitive function which is the nth integral of this difference has for a discriminant function that of the same difference 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 Δ corresponds; this power indicates then an integral of the same order, the index x varying by unity; and the negative powers of δ 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 δ placed before the primitive function is zero, we shall have an equation of finite differences, and V will 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 differences; 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 ​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 xth power of t in the development of the preceding 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 V be a function of the two variables t and t′ developed according to the powers and products of these variables; the coefficient of any product of the powers x and x′ of t and t′ will be a function of the exponents or indices x and x′ of these powers; this function I shall call the primitive function of which V is the discriminant function.
Let us multiply V by 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 ​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 δ placed before the primitive function, this derivative, the product of V by the nth power of T, will be the discriminant function of the derivative of the primitive function before which one places the nth power of the character δ. Hence result the theorems analogous to those which are relative to functions of a single variable.
Suppose the function indicated by the character δ 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 V 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 t by t′; but V may include separately the powers of t and those of t′, that is to say, an arbitrary function of t and an arbitrary function of 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 xth power of t by the x′ power of t′ 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 ​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 light upon the obscure points of their calculus, which have been the subject of great discussions among geometricians. It is thus that I have demonstrated the possibility 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 understanding, general signs whose true meaning may be ascertained 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 nth 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. ​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 nth 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 indeterminant. 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 subsecant; 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 difference 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 subsecants, a limit which is evidently the subtangent.
This singularly happy method of obtaining the subtangent is due to Fermat, who has extended it to ​transcendent curves. This great geometrician expresses 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 differential calculus the subtangents of the curves, their points of inflection, the 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 notation which, by indicating the passage from the finite to the infinitely small, adds to the advantage of expressing 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 formulæ in order to know with what probability the results are indicated, which the events develop by multiplication. It is necessary especially to have the ​law according to which this probability 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 differentials multiplied by the factors raised to great powers would give by integration the formulæ 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 remarkable 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 circumference to the diameter; sometimes it depends upon other transcendents whose number is infinite.
An important remark which pertains to great generality 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 ​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 negative 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 accordingly 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 compared 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 ​germs of new calculi. This is rendered appreciable by this example.
Wallis, who in his work entitled Arithmetica Infinitorum, one of those which have most contributed to the progress of analysis, has interested himself especially in following the thread of induction and analogy, considered 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 fraction—namely, of the quantity raised to a power indicated 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 ​number. The observation then of the law of the numbers which express these integrals, a series of interpolations 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 fractional 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 coefficients of the terms of its development in the case where this exponent is integral and positive, he determined 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 constitutes exponential calculus, one of the most fruitful ​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 differentiations 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 formulæ which are as curious as useful on the transformations of differences and of integrals the ones into the others when the variables have divers finite increments and when these increments are infinitely small. But he has not given the demonstrations of it which appear to him difficult. 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.