The German Genius

by Peter Watson

  The first law may be illustrated by the way it was later taught to Max Planck, the man who, at the turn of the twentieth century, would build on Clausius’s work. Imagine a worker lifting a heavy stone onto the roof of a house. The stone will remain in position long after it has been left there, storing energy until at some point in the future it falls back to earth. Energy, says the first law, can be neither created nor destroyed. Clausius, however, pointed out in his second law that the first law does not give the total picture. In the example given, energy is expended by the worker as he lifts the stone into place and is dissipated in the effort as heat, which among other things causes the worker to sweat. This dissipation, which Clausius was to term “entropy,” was of fundamental importance, he said, because although it did not disappear from the universe, this energy could never be recovered in its original form. Clausius therefore concluded that the world (and the universe) must always tend toward increasing disorder, must always add to its entropy.

  Clausius never stopped refining his theories of heat, becoming in the process interested in the kinetic theory of gases, in particular the notion that the large-scale properties of gases were a function of the small-scale movements of the particles, or molecules, that comprised the gas. Heat, he came to think, was a function of the motion of such particles—hot gases were made up of fast-moving particles, colder gases of slower particles.22 Work was understood as “the alteration in some way or another of the arrangement of the constituent molecules of a body.” This idea that heat was a form of motion was not new. The American Benjamin Thompson had observed that heat was produced when a cannon barrel was bored, and in Britain Humphry Davy had likewise noted that ice could be melted by friction. What attracted Clausius’s interest was the exact form of motion that comprised heat. Was it the vibration of the internal particles, was it their “translational” motion as they moved from one position to another, or was it because they rotated on their own axes?23

  Clausius’s second seminal paper, “On the Kind of Motion That We Call Heat,” was published in the Annalen in 1857, where he argued that the heat of a gas must be made up of all three types of movement and that therefore its total heat ought to be proportional to the sum of these motions. He assumed that the volume occupied by the particles themselves was vanishingly small and that all the particles moved with the same average velocity, which he calculated as being hundreds, if not thousands of meters per second. This brought about the objection from several others that his assumptions and calculations could not be right, since otherwise gases would diffuse far more quickly than they were known to do; he therefore abandoned that approach, introducing instead the concept of the “mean free path,” the average distance that a particle could travel in a straight line before colliding with another one.24

  Others were attracted by Clausius’s efforts, in particular James Clerk Maxwell in Britain, who published “Illustrations of the Dynamical Theory of Gases” in the Philosophical Magazine in 1860, making use of Clausius’s idea of the mean free path. However, where Clausius had assumed that every particle in a gas traveled at the same average velocity, Maxwell relied on the new science of statistics to calculate a random distribution of particle velocities, arguing that the collisions between particles would result in a distribution of velocities about a mean rather than an equalization. (Just what these particles were was never settled, not then, though Maxwell was convinced their very existence “was proof of the existence of a divine manufacturer.”)25

  The statistical—probabilistic—element introduced into physics in this way was a very controversial and yet fundamental advance. In his 1850 paper Clausius had drawn attention in the second law of thermodynamics to the “directionality” of the heat flow—heat tends to pass from a hotter to a colder body. He had not at first bothered with the implications of the irreversibility or otherwise of processes, but in 1854 he argued that the transformation of heat into work and the transformation of heat from a higher temperature into heat of a lower temperature were in effect equivalent and that in some circumstances they could be counteracted—reversed—by the conversion of work into heat, where heat would flow from a colder to a warmer body. This, for Clausius, only emphasized the difference between reversible (man-made) and irreversible (natural) processes: a decayed house never puts itself back together, a broken bottle never spontaneously reassembles.

  It was only later, in 1865, that Clausius proposed the term “entropy” (from the Greek word for “transformation”) for the irreversible processes whereby the tendency for heat to pass from warmer to colder bodies was also described as an instance of the increase in entropy. In doing this Clausius now emphasized the directionality of physical processes, and he described the two laws of thermodynamics as follows: “The energy of the universe is constant” and “The entropy of the universe tends to a maximum.” Time, in some mysterious way, had become a property of matter.26

  For some people, the second law had a much greater significance than even Clausius thought. Another Briton, William Thomson, Lord Kelvin, thought that the irreversibility that was such a feature of the second law—the dissipation of energy—also implied a “progressivist cosmogony,” one that moreover underlined the biblical view about the transitory character of the universe. Thomson drew the implication from the second law that the universe, known by then to be cooling, would “in a finite time” run down and become uninhabitable. Helmholtz had also noticed this implication of the second law, but it was only in 1867 that Clausius himself, who had by then moved back to Germany from Zurich, acknowledged the “heat death” of the universe.27

  THE APPEARANCE OF “STRANGENESS” IN PHYSICS

  The statistical notions aired by Clausius and Maxwell attracted the attention of the Austrian physicist Ludwig Boltzmann.28 Boltzmann (1844–1906) was born in Vienna during the night between Shrove Tuesday and Ash Wednesday, a coincidence which, he half-jokingly complained, helped to explain his frequent and rapid mood swings, which tossed him between unalloyed happiness and deep depression. The son of a tax official, Boltzmann was appointed professor of mathematical physics at the University of Graz in 1869 at the age of only twenty-five. Later he worked with Robert Bunsen at Heidelberg and with Helmholtz in Berlin. In 1873 he joined the University of Vienna as professor of mathematics and remained there until 1902 when he committed suicide during one of his depressions.

  Boltzmann’s main achievement lay in two famous papers, describing in mathematical terms the velocities, spatial distribution, 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.

  THE GOLDEN AGE OF MATHEMATICS

  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 t
he 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’ fascination with prime numbers was introduced, and it was noted how Gauss had uncovered the link between primes and logarithms. His invention of imaginary numbers was also discussed (see Chapter 7). It will be remembered that the link between primes and logarithms allowed Gauss to give a good but still approximate prediction of how many primes there are up to any figure, N. It was Riemann’s achievement to establish and clarify a definitive prediction of the number of primes—and by using Gauss’s other invention, imaginary numbers.

  Riemann worked with something known as the zeta function. This, in one form or another, had been of interest to mathematicians since Pythagoras, in ancient Greece, who had pointed out the link between mathematics and music. Pythagoras found that if he filled an urn with water and banged it with a hammer, it would produce a certain note. If he then removed half the water and banged the urn again, the note had gone up an octave. As he removed water equivalent to a proper integer (1/2, 1/3, 1/4), the notes produced sounded in harmony to his ear, whereas if any intermediate amount were removed, the sound was discordant. Pythagoras came to believe that numbers lay at the root of the order of the universe, giving rise to his famous phrase “the music of the spheres.”

  For other mathematicians, however, this led to an investigation of the behavior of reciprocal numbers (the reciprocal of 2 is 1/2 and the reciprocal of 3 is 1/3). This investigation eventually led to what mathematicians call the zeta function, zeta being represented by the Greek symbol, . The zeta function is represented as follows:

  This function turns up some interesting results, the most celebrated being the discovery in the eighteenth century by the Swiss mathematician Leonhard Euler that when zeta is 2 the sequence becomes:

  and that this may eventually be written as:

  This discovery took the mathematics world by storm, for the number, 1/62, written as a decimal, produces an indefinite progression, like itself. (This is number theory, remember, the sheer behavior of numbers being fascinating for mathematicians, whether that behavior has any use or not.)37

  When Riemann was elected to the Academy in Berlin in November 1859, he wrote a ten-page paper to mark the event, as was (again) customary. This proved every bit as radical as his inaugural lecture. One of the things he did in this paper was to feed Gauss’s other invention—imaginary numbers—into the zeta function, obtaining an entirely unexpected pattern, the most notable feature of which was (when the results of the equations were plotted on a graph) a series of waves and which, he found, could be used to correct Gauss’s calculations regarding primes, to give an exact, error-free prediction of the number of primes in any sequence. And so the apparent randomness of the primes had been shown to have an order. Not a simple order, it is true, but an order nonetheless. Order—however complex—is a form of beauty for mathematicians.

  Felix Christian Klein was born in Düsseldorf on April 25, 1849, and delighted in pointing out that his birth date was a collection of primes squared: 52, 22, 72. He made his most important contribution in group theory, another new field. The son of a government official in the Rhine province, he was appointed to a professorship, in his case at Erlangen, at an even younger age than Boltzmann, when he was twenty-three. He moved to Munich’s Technische Hochschule in 1875, where he taught, among others, Max Planck (he also married Anne Hegel, granddaughter of the philosopher). In 1886 his health deteriorated, and he accepted a quieter life as professor of mathematics at Göttingen. There he consolidated Göttingen as the world’s leading mathematics research center.38

  To explain what Klein was driving at in group theory, imagine two visual experiments. First, imagine a rectangular sheet of paper, its sides measuring A and B inches. Rotate the sheet through forty-five degrees and then photograph it. The photograph will not show a rectangle and the sides will not be A and B inches long, yet the paper will not have changed. What are the mathematics of this foreshortening, the relationship between the original and the photograph? Second, consider an aerial photograph of a particular country—for example, Italy—taken from a satellite fifty miles up in space. Next, view the same country from, say, five miles up. The outline of Italy is the same but many details are now visible that weren’t before—estuaries, small bays, tiny off-shore islands. Again, what transformation has taken place, what has changed, and what has stayed the same, and how can that change/staying-the-same be represented mathematically? This last example was not available in Klein’s day because aerial photography didn’t exist, but the problem, mathematically, is now known as “fractals” and shows how far ahead of his time Klein was. It presaged chaos theory.

  Under Klein’s leadership Göttingen became a mecca to which students from many lands, especially America, flocked.39 The French had led the way at the turn of the nineteenth century, at the École Polytechnique, embracing the work of Joseph-Louis Lagrange, Gaspard Monge, and Jean-Victor Poncelet. The research and inspiration of Gauss, Riemann, and Klein ensured that that leadership passed to Germany. It held that lead—at least in theoretical terms—until the advent of Hitler.40

  18.

  The Rise of the Laboratory: Siemens, Hofmann, Bayer, Zeiss

  No one better illustrates the changes taking place in Germany in the nineteenth century than Werner Siemens. As just one indicator, he became Werner von Siemens in 1888. Born in 1816, the fourth of fourteen children of a tenant farmer in Lenthe, near Hanover, Werner had to leave
his Gymnasium in 1834 without taking his examinations because of the family’s precarious financial situation, so that he could join the Prussian army and gain some engineering training in that way. He had the foresight, while at school, to drop Greek and take extra lessons in mathematics and land surveying.

  He said later that the three years he spent at the Berlin Artillery and Engineering School were the happiest of his life. Among his teachers was Martin Ohm, brother of the physicist Georg Ohm.1 While at the school Werner started to produce the first of the inventions at which he was to prove so adept. The earliest concerned gilding and plating silver, a process he sold to a German silver manufacturer.

  He became interested in the theory of the conservation of energy (he was familiar with the work of both Mayer and Helmholtz) and this fanned his interest in engines (he published some of his early ideas in Poggendorff’s Annalen)—all of which meant he was one of the first to appreciate the great importance of telegraphy.2 His time in the army had taught him, among other things, the need for rapid, reliable communication, and so in 1847 he produced a pointer telegraph, which was notable for its reliability, this dependability laying the foundation for the Siemens & Halske Telegraph Construction Company, which he founded jointly with Johann Georg Halske, himself a mechanic, in Berlin that same year.3

  Once he had a reliable telegraph, Siemens saw its many possibilities. He laid the first long subterranean wire, from Berlin to Grossbeeren, almost twenty miles to the southwest. He recognized that the invention of guttapercha in Britain would enable the lines to be insulated, which meant that telegraph wires could be spread across the world, even in America following the Civil War. An underground line from Berlin to Frankfurt am Main came next, Frankfurt being where the German National Assembly was meeting. The line was buried to keep it safer at times of political trouble.4