The German Genius

loneliness. Although he did not have an easy life, his entry in the Dictionary of Scientific Biography makes it clear that his ideas were influential in thirteen separate areas. 12
    Gauss became famous for his method of least squares, which enabled him to predict the moving orbits of planets; for his ideas about the pattern of prime numbers (divisible only by 1 and themselves), which revealed a hidden order that had totally escaped everyone else and uncovered their relationship to logarithms; for his invention of “clock arithmetic,” which would eventually prove important for the security of the Internet; and for his invention of imaginary numbers, which would also transform understanding and link up, much later, with quantum physics. 13 But it is his conception of non-Euclidean geometry, commutative algebra, and the electric telegraph that really shows the extent to which his imagination was ahead of his time.
    According to his mathematical diary, Gauss was still quite young when he began to consider the possibility that the ancient Greeks—Euclid in particular—had got it wrong with some of their fundamental axioms in geometry. In particular, he had begun to have doubts about parallel lines. Euclid had set out the classical paradigm and identified the classical solution: if you draw a straight line and then a point off that line, there can be only one line that is parallel to the first line and runs through the point. 14 When he was only sixteen, Gauss began to consider—daringly—whether there might be other geometries at variance with the Euclidean. He didn’t publish anything for years, fearing ridicule, because—if he were right—other things followed, such as the fact that the angles of a triangle would not always add up to 180 degrees. Gauss couldn’t get these subversive thoughts out of his mind: he even climbed to the summit of three hilltops to shine beams between them, to see if the angles added up to 180 degrees. This suggests that Gauss had some idea that light might bend in space, anticipating Einstein by nearly a century. It had occurred to Gauss that three-dimensional space might be curved in the way that the two-dimensional surface of the earth was. This thinking developed out of his observation that lines of longitude, along which the shortest path between two points on the surface of the earth is measured, all meet at the poles. They appeared parallel but were not. No one had considered that three-dimensional space might also bend.
    Gauss was to be proved right, as Einstein was proved right, with Arthur Eddington’s confirmation of the bending of light in 1919, but once again Gauss never published his ideas and the friends this troubled man shared his thoughts with were pledged to secrecy. 15
    Noncommutative algebra is the mathematical description of noncommutative geometry, which emerged in the nineteenth century in relation to physics and chemistry. At its simplest it refers to the possibility that, in mathematics, xy , strange as it may seem, is not always equal to yx . We shall meet this phenomenon again in the case of isomers in chemistry and with the benzene ring, where “rightness” and “leftness” determine chemical properties. This, plus the second law of thermodynamics, considered in Chapter 17, which says that time is a fundamental aspect of space, shows that a purely mechanical (i.e., Newtonian) understanding of the universe has to be incomplete. Gauss’s noncommutative algebra was an early attempt to come to grips with this problem. Once more he was well ahead of his time.
    Though the bulk of Gauss’s career was spent in the highly abstract world of numbers, it was bracketed by two very practical discoveries. The first—his calculation of the orbits of moving objects—has already been referred to. The second came when he was in his fifties and already had many abstract, imaginative discoveries to his name. Among the nonmathematical phenomena he was interested in (although of course he was interested in the mathematical aspects), was terrestrial magnetism, in particular the way it varied across the earth, and in the existence of magnetic storms. 16 In 1831, stimulated by Michael Faraday’s discovery of induced current, Gauss collaborated (for once) with the brilliant experimental physicist Wilhelm Weber, one of the (liberal) “Göttingen Seven,” to investigate a number of electrical phenomena. They made several discoveries in static, thermal, and frictional