A Philosophical Essay on Probabilities

by marquis de Pierre Simon Laplace

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 i 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.