Page:A History of Mathematics (1893).djvu/295

 opinion that the special intervention, from time to time, of a powerful hand was necessary to preserve order. Now Laplace was able to prove by the law of gravitation that the solar system is stable, and in that sense may be said to have felt no necessity for reference to the Almighty.

We now proceed to researches which belong more properly to pure mathematics. Of these the most conspicuous are on the theory of probability. Laplace has done more towards advancing this subject than any one other investigator. He published a series of papers, the main results of which were collected in his Théorie analytique des probabilités, 1812. The third edition (1820) consists of an introduction and two books. The introduction was published separately under the title, Essai philosophique sur les probabilités, and is an admirable and masterly exposition without the aid of analytical formulæ of the principles and applications of the science. The first book contains the theory of generating functions, which are applied, in the second book, to the theory of probability. Laplace gives in his work on probability his method of approximation to the values of definite integrals. The solution of linear differential equations was reduced by him to definite integrals. One of the most important parts of the work is the application of probability to the method of least squares, which is shown to give the most probable as well as the most convenient results.

The first printed statement of the principle of least squares was made in 1806 by Legendre, without demonstration. Gauss had used it still earlier, but did not publish it until 1809. The first deduction of the law of probability of error that appeared in print was given in 1808 by Robert Adrain in the Analyst, a journal published by himself in Philadelphia.[2] Proofs of this law have since been given by Gauss, Ivory, Herschel, Hagen, and others; but all proofs contain some