Laplace, preoccupied with this new "science of chance" that he is seeking to master, has an almost experimental model for probability: an urn containing white and black balls. The experiment consists of drawing a ball. In 1814, in his Essai philosophique sur les probabilités, his reflections led him to state a law in simple terms: "The probability that the ratio of the number of white balls drawn to the total number of balls drawn will not depart by more than a given interval from the ratio of the number of white balls to the total number of balls contained in the urn approaches certainty indefinitely as the number of events increases indefinitely, however small that interval may be supposed."
In fact, Laplace published a far more mathematically precise formulation in 1810. The result came as a genuine shock: when Laplace took up a seat in the Senate, he had thought his active involvement in science was effectively over. He even explained as much in the first volume of the Mémoires de physique et de chimie de la Société d’Arcueil.
The proof uses a function that Laplace calls "characteristic", which he feels resembles something Fourier had introduced shortly beforehand in the final part of his Théorie de la propagation de la chaleur. The proof brings together all the earlier research, dating back to 1774, within a single theory. It therefore appears in the Théorie analytique des probabilités and involves the definite integral of an exponential function with exponent −x2/2.
Philosophy enters the picture -----------------------------
Laplace does not want to leave this result as a statement without philosophical consequences and is eager to make his own thinking clear. He returns to ideas from his youth, formed while reading Leibniz, particularly his Essais de théodicée (written in French and published in 1710). He now expresses them in the language of a mathematician, speaking of axioms, without abandoning Aristotle’s old practical language of causes, which saves him from having to develop the formalism too extensively.