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 researches in finite differences and in determinants, the establishment of the expansion theorem in determinants which had been previously given by Vandermonde for a special case, the determination of the complete integral of the linear differential equation of the second order. In the Mécanique Céleste he made a generalisation of Lagrange's theorem on the development of functions in series known as Laplace's theorem.

Laplace's investigations in physics were quite extensive. We mention here his correction of Newton's formula on the velocity of sound in gases by taking into account the changes of elasticity due to the heat of compression and cold of rarefaction; his researches on the theory of tides; his mathematical theory of capillarity; his explanation of astronomical refraction; his formulæ for measuring heights by the barometer.

Laplace's writings stand out in bold contrast to those of Lagrange in their lack of elegance and symmetry. Laplace looked upon mathematics as the tool for the solution of physical problems. The true result being once reached, he spent little time in explaining the various steps of his analysis, or in polishing his work. The last years of his life were spent mostly at Arcueil in peaceful retirement on a country-place, where he pursued his studies with his usual vigour until his death. He was a great admirer of Euler, and would often say, "Lisez Euler, lisez Euler, c'est notre maître à tous."

Abnit-Théophile Vandermonde (1735–1796) studied music during his youth in Paris and advocated the theory that all art rested upon one general law, through which any one could become a composer with the aid of mathematics. He was the first to give a connected and logical exposition of the theory of determinants, and may, therefore, almost be regarded as the founder of that theory. He and Lagrange originated the method of combinations in solving equations.[20]

Adrien Marie Legendre (1752–1833) was educated at the