Mathematics

By the 1840s Britain and the Italian and German states were producing quality mathematicians to complement and even rival the French, and new posts were available in universities and engineering colleges everywhere. Among the Germans, two figures stand out.

From around 1860 Karl Weierstrass (1815–1897) gave lecture courses on many aspects of real-and complex-variable analysis and parts of mechanics at Berlin University, attended by students from many countries who then went home and taught likewise. Meanwhile at Göttingen, Bernhard Riemann (1826–1866) rethought complex-variable analysis and revolutionized the understanding of both Fourier series and the foundations of geometry. Much of this work was published only after his early death in 1866, but it soon made a great impact. The work on Fourier series led Georg Cantor (1845–1918) to develop set theory from the 1870s. On geometry Riemann showed that the Euclidean was only one of many possible geometries, and that each of them could be defined independently of any embedding space. The possibility of non-Euclidian geometries, using alternatives to the parallel axiom, had been exhibited around 1830 in little-recognized work by Janos Bolyai (1802–1860) and Nicolai Lobachevsky (1793–1856) (and, in manuscript, Gauss); Riemann, however, went much further and brought us proper understanding of the plurality of geometries.

Weierstrass emulated and indeed enhanced Cauchy-style rigor, carefully formulating definitions and distinctions and presenting proofs in great detail. By contrast, Riemann worked intuitively, offering wonderful but often proof-free insights grounded upon some "geometric fantasy," as Weierstrass described it. A good example is their revisions of Cauchy's complex-variable analysis: Weierstrass relied solely on power series expansions of the functions, whereas Riemann invented surfaces now named after him that were slit in many remarkable ways. Among many consequences of the latter, the German Felix Klein (1849–1925) and the Frenchman Henri Poincaré (1854–1912) in the early 1880s found beautiful properties of functions defined on these surfaces, which they related to group theory as part of the rise of abstract algebras.

Another example of the gap between Riemann and Weierstrass is provided by potential theory. Riemann used a principle employed by his mentor Dirichlet (and also envisaged by Green) to solve problems in potential theory, but in 1870 Weierstrass exposed its fallibility by a counterexample, and so methods became far more complicated.

Better news for potential theory had come at midcentury with the "energetics" physics of Thomson, Hermann von Helmholtz (1821–1894), and others. The work expression of engineering mechanics was extended into the admission of potentials, which now covered all physical factors (such as heat) and not just the mechanical ones that had split Carnot from Lagrange. The latter's algebraic tradition in mechanics had been elaborated by Jacobi and by the Irishman William Rowan Hamilton (1805–1865), who also introduced his algebra of quaternions.

Among further related developments, the Scot James Clerk Maxwell (1831–1879) set out theories of electricity and magnetism (including, for him, optics) in his Treatise on Electricity and Magnetism (1873). Starting out from the electric and magnetic potentials as disturbances of the ether rather than Newton-like forces acting at a distance through it, he presented relationships between his basic notions as differential equations (expressed in quaternion form). A critical follower was the Englishman Oliver Heaviside, who also analyzed electrical networks by means of a remarkable but mysterious operator algebra. Other "Maxwellians" preferred to replace dependence upon fields with talk about "things," such as electrons and ions; the relationship between ether and matter (J. J. Larmor, Aether and Matter, 1900) was a major issue in mathematical physics at century's end.