Mathematics

Laplace published his large book in a new professional and economic situation for science. After the Revolution of 1789 in France, higher education and its institutions there were reformed, with a special emphasis upon engineering. In particular, a new school was created, the École Polytechnique (1794), with leading figures as professors (such as Lagrange) and as examiners (Laplace), and with enrollment of students determined by talent, not birth. A new class of scientists and engineers emerged, with mathematics taught, learned, researched, and published on a scale hitherto unknown.

Of this mass of work only a few main cases can be summarized here. Joseph Fourier (1768–1830) is noteworthy for his mathematical analysis of heat diffusion, both the differential equation to represent it (the first important such equation found outside mechanics) and solutions by certain infinite series and by integrals that both now bear his name. From the 1820s they attracted much attention, not only for their use in heat theory but especially for the "pure" task of establishing conditions for their truth. New techniques for rigor had just become available, mainly from Augustin-Louis Cauchy (1789–1857), graduate of the École Polytechnique and now professor there. He taught a fourth approach to the calculus (and also function and series), based like Newton's upon limits but now fortified by a careful theory of them; although rather unintuitive, its mathematical merits gradually led worldwide to its preference over the other three approaches.

Ironically, Cauchy's own analysis of Fourier series failed, but a beautiful treatment following his approach came in 1829 from J. P. G. Dirichlet (1805–1859)—a French-sounding name of a young German who had studied with the masters in Paris. Dirichlet also exemplifies a novelty of that time: other countries producing major mathematicians. Another contemporary example lies in elliptic functions, which Carl Jacobi (1804–1851) and the young Norwegian Niels Henrik Abel (1802–1829) invented independently following much pioneering work on the inverse function by A. M. Legendre (1752–1833).

Jacobi and Abel drew upon a further major contribution to mathematics made by Cauchy when, by analogy with the calculus, he developed a theory of functions of the complex variable x + √ − 1y (x and y real), complete with an integral. His progress was fitful, from the 1810s to the 1840s; after that, however, his theory became recognized as a major branch of mathematics, with later steps taken especially by the Germans.

Between 1810 and 1830 the French initiated other parts of mathematical physics in addition to Fourier on heat: Siméon-Denis Poisson (1781–1840) on magnetism and electrostatics; André-Marie Ampère (1775–1836) on electrodynamics; and Augustin Jean Fresnel (1788–1827) on optics with his wave theory. Mathematics played major roles: many analogies were taken from mechanics, which itself developed massively, with Carnot's energy approach elaborated by engineers such as Gaspard-Gustave Coriolis (1792–1843), and continuum mechanics extended, especially by Cauchy.

Geometry was also taught and studied widely. Gaspard Monge (1746–1818) sought to develop "descriptive geometry" into a geniune branch of mathematics and gave it prominence in the first curriculum of the École Polytechnique; however, this useful theory of engineering drawing could not carry such importance, and Laplace had its teaching reduced. But former student Jean Victor Poncelet (1788–1867) was partly inspired by it to develop "the projective properties of figures" (Traité des propries projectives de figures, 1822), where he studied characteristics independent of measure, such as the order of points on a line.

The main mathematician outside France at this time was C. F. Gauss (1777–1855), director of the Göttingen University Observatory. Arguably he was the greatest of all, with major work published in number theory, celestial mechanics, and aspects of analysis and probability theory. But he was not socially active, and he left many key insights in his manuscripts (for example, on elliptic functions).

Other major contributors outside France include George Green (1793–1841), who, in An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism (1828), produced a wonderful theorem in potential (his word) theory that related the state of affairs inside an extended body to that on its surface. But he published his book very obscurely, and it became well-known only on the reprint during the 1850s initiated by William Thomson (later Lord Kelvin), who was making notable contributions of his own to the theory.