The device known as the method of least squares, for reducing numerous equations of condition to the number of unknown quantities to be determined, had been adopted as a practically convenient rule by Gauss and Legendre; but Laplace first treated it as a problem in probabilities, and proved by an intricate and difficult course of reasoning that it was also the most advantageous, the mean of the probabilities of error in the determination of the elements being thereby reduced to a minimum.
Laplace published in 1779 the method of generating functions, the foundation of his theory of probabilities, and the first part of his Théorie analytique is devoted to the exposition of its principles, which in their simplest form consist in treating the successive values of any function as the coefficients in the expansion of another function with reference to a different variable. The latter is therefore called the generating function of the former. A direct and an inverse calculus is thus created, the object of the former being to determine the coefficients from the generating function, of the latter to discover the generating function from the coefficients. The one is a problem of interpolation, the other a step towards the solution of an equation in finite differences. The method, however, is now obsolete owing to the more extended facilities afforded by the calculus of operations.
The first formal proof of Lagrange’s theorem for the development in a series of an implicit function was furnished by Laplace, who gave to it an extended generality. He also showed that every equation of an even degree must have at least one real quadratic factor, reduced the solution of linear differential equations to definite integrals, and furnished an elegant method by which the linear partial differential equation of the second order might be solved. He was also the first to consider the difficult problems involved in equations of mixed differences, and to prove that an equation in finite differences of the first degree and the second order might always be converted into a continued fraction.
In 1842, the works of Laplace being nearly out of print, his widow was about to sell a farm to procure funds for a new impression, when the government of Louis Philippe took the matter in hand. A grant of 40,000 francs having been obtained from the chamber, a national edition was issued in seven 4to vols., bearing the title Œuvres de Laplace (1843-1847). The Mécanique céleste with its four supplements occupies the first 5 vols., the 6th contains the Système du monde, and the 7th the Th. des probabilités, to which the more popular Essai philosophique forms an introduction. Of the four supplements added by the author (1816-1825) he tells us that the problems in the last were contributed by his son. An enumeration of Laplace’s memoirs and papers (about one hundred in number) is rendered superfluous by their embodiment in his principal works. The Th. des prob. was first published in 1812, the Essai in 1814; and both works as well as the Système du monde went through repeated editions. An English version of the Essai appeared in New York in 1902. Laplace’s first separate work, Théorie du mouvement et de la figure elliptique des planètes (1784), was published at the expense of President Bochard de Saron. The Précis de l’histoire de l’astronomie (1821), formed the fifth book of the 5th edition of the Système du monde. An English translation, with copious elucidatory notes, of the first 4 vols. of the Mécanique céleste, by N. Bowditch, was published at Boston, U.S. (1829-1839), in 4 vols. 4to.; a compendium of certain portions of the same work by Mrs Somerville appeared in 1831, and a German version of the first 2 vols. by Burckhardt at Berlin in 1801. English translations of the Système du monde by J. Pond and H. H. Harte were published, the first in 1809, the second in 1830. An edition entitled Les Œuvres complètes de Laplace (1878), &c., which is to include all his memoirs as well as his separate works, is in course of publication under the auspices of the Academy of Sciences. The thirteenth 4to volume was issued in 1904. Some of Laplace’s results in the theory of probabilities are simplified in S. F. Lacroix’s Traité élémentaire du calcul des probabilités and De Morgan’s Essay, published in Lardner’s Cabinet Cyclopaedia. For the history of the subject see A History of the Mathematical Theory of Probability, by Isaac Todhunter (1865). Laplace’s treatise on specific heat was published in German in 1892 as No. 40 of W. Ostwald’s Klassiker der exacten Wissenschaften.
Authorities.—Baron Fourier’s Éloge, Mémoires de l’institut, x. lxxxi. (1831); Revue encyclopédique, xliii. (1829); S. D. Poisson’s Funeral Oration (Conn. des Temps, 1830, p. 19); F. X. von Zach, Allg. geographische Ephemeriden, iv. 70 (1799); F. Arago, Annuaire du Bureau des Long. 1844, p. 271, translated among Arago’s Biographies of Distinguished Men (1857); J. S. Bailly, Hist. de l’astr. moderne, t. iii.; R. Grant, Hist. of Phys. Astr. p. 50, &c.; A. Berry, Short Hist. of Astr. p. 306; Max Marie, Hist. des sciences t. x. pp. 69-98; R. Wolf, Geschichte der Astronomie; J. Mädler, Gesch. der Himmelskunde, i. 17; W. Whewell, Hist. of the Inductive Sciences, ii. passim; J. C. Poggendorff, Biog-lit. Handwörterbuch.
(A. M. C.)
[1] “Recherches sur le calcul intégral,” Mélanges de la Soc. Roy. de Turin (1766-1769).
[2] “Plan de l’Ouvrage,” Œuvres, tom. i. p 1.
[3] Journal des savants (1850).