The German Genius

and collision probabilities of molecules in a gas, all of which determined its temperature. The mathematics were statistical, showing that—whatever the initial state of a gas—Maxwell’s velocity distribution law would describe its equilibrium state. Boltzmann also produced a statistical description of entropy. 29
    What is important about the work of Mayer, Helmholtz, and in particular Clausius and Boltzmann is that, whether one can follow the mathematics or not, they brought probability into physics. 30 How can that be? Matter definitely exists, transformations (as when water freezes) obey invariant laws. What can probability have to do with it? This was the first appearance of “strangeness” in physics, heralding the increasingly bizarre twentieth-century quantum world. These early physicists also made “particles” (atoms, molecules, or something else that was not yet clearly understood) integral to the behavior of substances.
    The understanding of thermodynamics was the high point of nineteenth-century physics and of the marriage between physics and mathematics. It signaled an end to the strictly mechanical Newtonian view of nature, and it would prove decisive in leading to a spectacular new form of energy, nuclear power. This all stemmed, ultimately, from the concept of the conservation of energy.
    T HE G OLDEN A GE OF M ATHEMATICS
     
    In his history of mathematics, Carl Boyer says that the nineteenth century, more than any other preceding period, deserves to be known as the golden age of mathematics. “The additions to the subject during those one hundred years far outweigh the total combined productivity of all preceding ages.” The introduction of such concepts as non-Euclidean geometries, n -dimensional spaces, non-commutative algebras, infinite processes, and non-quantitative structures “all contributed to a radical transformation which changed the appearance as well as the definition of mathematics.” 31 While the French remained strong, and several countries supported mathematics linked to practical activities, such as surveying and navigation, research in pure mathematics—mathematics for the sake of it—was the exception rather than the rule, practiced more than anywhere else in Germany. 32
    The strength of mathematics in Germany owed something to the fact that, as in physics, the subject had an important new journal. Until the nineteenth century, the best mathematical periodicals had come from the École Polytechnique in Paris, but in 1826 August Leopold Crelle (1780–1855) launched his Journal für die reine und angewandte Mathematik (Journal for Pure and Applied Mathematics), though it was often known more simply as “Crelle’s Journal.” 33
    Above all, the golden age—initiated by Gauss—was continued by Bernhard Riemann and Felix Klein. Riemann, frail and shy, was yet another son of a pastor. Born in 1826, he took his doctorate at Göttingen, then spent several semesters in Berlin to study under C. G. J. Jacobi and Peter Dirichlet before returning to Göttingen for a training in physics from Wilhelm Weber. (His subsequent career was split between mathematics and physics.) 34
    In 1854 he was called upon to give an inaugural lecture before the faculty at Göttingen. “The result in Riemann’s case was the most celebrated probationary lecture in the history of mathematics.” 35 In his lecture, titled, “On the Hypotheses which Lie at the Foundation of Geometry,” Riemann urged a totally new view of geometry as the study of “any number of dimensions in any kind of space.” This became known as Riemann geometry. In this paper he envisaged what he called manifolds, surfaces (now known as Riemann surfaces), which are forms of space that are non-Euclidean, where the laws of Euclid no longer apply. The idea of curved space is the best known, because the easiest to understand: a “plane” is in fact the surface of a sphere, and a “straight line” is the great circle of a sphere. Riemann’s results in this area of thinking were so significant that Bertrand Russell described him as “logically the immediate predecessor of Einstein.” 36 Without Riemann’s geometry, general relativity could not have been formulated.
    When Peter Dirichlet, another great mid-century German mathematician, died in 1859, Riemann was appointed to the chair that Carl Gauss had once occupied. In that chair he followed up Gauss’s interest in number theory. In Chapter 7, mathematicians’