Men of Mathematics

by E. T. Bell

  It is a satisfaction to record that Gauss was too proud to prostitute mathematics to Napoleon the Great by appealing to the Emperor’s vanity and begging him in the name of his notorious respect for all things mathematical to remit the 2,000-franc fine, as some of Gauss’ mistaken friends urged him to do. Napoleon would probably have been flattered to exercise his clemency. But Gauss could not forget Ferdinand’s death, and he felt that both he and the mathematics he worshipped were better off without the condescension of a Napoleon.

  No sharper contrast between the mathematician and the military genius can be found than that afforded by their respective attitudes to a broken enemy. We have seen how Napoleon treated Ferdinand. When Napoleon fell Gauss did not exult. Calmly and with a detached interest he read everything he could find about Napoleon’s life and did his best to understand the workings of a mind like Napoleon’s. The effort even gave him considerable amusement. Gauss had a keen sense of humor, and the blunt realism which he had inherited from his hardworking peasant ancestors also made it easy for him to smile at heroics.

  * * *

  The year 1811 might have been a landmark in mathematics comparable to 1801—the year in which the Disquisitiones Arithmeticae appeared—had Gauss made public a discovery he confided to Bessel. Having thoroughly understood complex numbers and their geometrical representation as points on the plane of analytic geometry, Gauss proposed himself the problem of investigating what are today called analytic functions of such numbers.

  The complex number x + iy, where i denotes represents the point (x, y). For brevity x + iy will be denoted by the single letter z. As x, y independently take on real values in any prescribed continuous manner, the point z wanders about over the plane, obviously not at random but in a manner determined by that in which x, y assume their values. Any expression containing z, such as z2, or 1/z, etc., which takes on a single definite value when a value is assigned to z, is called a uniform function of z. We shall denote such a function by f(z). Thus iff(z) is the particular function z2, so that here f(z) = (x + iy)2 = x2 + 2ixy + i2y2, = x2 – y2 + 2ixy (because i2 = −1), it is clear that when any value is assigned to z, namely to x + iy, for example x = 2, y = 3, so that z = 2 + 3i, precisely one value of this f(z) is thereby determined; here, for z = 2 + 3i we get z2 = −5 + 12i.

  Not all uniform functions f(z) are studied in the theory of functions of a complex variable; the monogenic functions are singled out for exhaustive discussion. The reason for this will be stated after we have described what “monogenic” means.

  Let z move to another position, say to z′. The function f(z) takes on another value, f(z′), obtained by substituting z′ for z. The difference f(z′)—f(z) of the new and old values of the function is now divided by the difference of the new and old values of the variable, thus [f(z′) +(zy]/(z′ – z), and, precisely as is done in calculating the slope of a graph to find the derivative of the function the graph represents, we here let z′ approach z indefinitely, so that f(z′) approaches f(z) simultaneously. But here a remarkable new phenomenon appears.

  There is not here a unique way in which z′ can move into coincidence with z, for z′ may wander about all over the plane of complex numbers by any of an infinity of different paths before coming into coincidence with z. We should not expect the limiting value of [f(z′)—f(z)]/(z′—z) when z′ coincides with z to be the same for all of these paths, and in general it is not. But iff(z) is such that the limiting value just described is the same for all paths by which z′ moves into coincidence with z, then f(z) is said to be monogenic at z (or at the point representing z). Uniformity (previously described) and monogenicity are distinguishing features of analytic functions of a complex variable.

  Some idea of the importance of analytic functions can be inferred from the fact that vast tracts of the theories of fluid motion (also of mathematical electricity and representation by maps which do not distort angles) are naturally handled by the theory of analytic functions of a complex variable. Suppose such a function f(z) is separated into its “real” part (that which does not contain the “imaginary unit” i) and its “imaginary” part, say f(z) = U + iV. For the special analytic function z2 we have U = x2—y2, V = 2xy. Imagine a film of fluid streaming over a plane. If the motion of the fluid is without vortices, a stream line of the motion is obtainable from some analytic function f(z) by plotting the curve U = a, in which a is any real number, and likewise the equipotential lines are obtainable from V = b (b any real number). Letting a, b range, we thus get a complete picture of the motion for as large an area as we wish. For a given situation, say that of a fluid streaming round an obstacle, the hard part of the problem is to find what analytic function to choose, and the whole matter has been gone at largely backwards: the simple analytic functions have been investigated and the physical problems which they fit have been sought. Curiously enough, many of these artificially prepared problems have proven of the greatest service in aerodynamics and other practical applications of the theory of fluid motion.

  The theory of analytic functions of a complex variable was one of the greatest fields of mathematical triumphs in the nineteenth century. Gauss in his letter to Bessel states what amounts to the fundamental theorem in this vast theory, but he hid it away to be rediscovered by Cauchy and later Weierstrass. As this is a landmark in the history of mathematical analysis we shall briefly describe it, omitting all refinements that would be demanded in an exact formulation.

  Imagine the complex variable z tracing out a closed curve of finite length without loops or kinks. We have an intuitive notion of what we mean by the “length” of a piece of this curve.

  Mark n points P1 P2, . . . , Pn on the curve so that each of the pieces P1P2, P2P3, P3P4, . . . , PnP1 is not greater than some preassigned finite length l. On each of these pieces choose a point, not at either end of the piece; form the value off(z) for the value of z corresponding to the point, and multiply this value by the length of the piece in which the point lies. Do the like for all the pieces, and add the results. Finally take the limiting value of this sum as the number of pieces is indefinitely increased. This gives the “line integral” off(z) for the curve.

  When will this line integral be zero? In order that the line integral shall be zero it is sufficient that f(z) be analytic (uniform and monogenic) at every point z on the curve and inside the curve.

  Such is the great theorem which Gauss communicated to Bessel in 1811 and which, with another theorem of a similar kind, in the hands of Cauchy who rediscovered it independently, was to yield many of the important results of analysis as corollaries.

  * * *

  Astronomy did not absorb the whole of Gauss’ prodigious energies in his middle thirties. The year 1812, which saw Napoleon’s Grand Army fighting a desperate rear-guard action across the frozen plains, witnessed the publication of another great work by Gauss, that on the hyfergeometric series

  the dots meaning that the series continues indefinitely according to the law indicated; the next term is

  This memoir is another landmark. As has already been noted Gauss was the first of the modern rigorists. In this work he determined the restrictions that must be imposed on the numbers a, b, c, x in order that the series shall converge (in the sense explained earlier in this chapter). The series itself was no mere textbook exercise that may be investigated to gain skill in analytical manipulations and then be forgotten. It includes as special cases—obtained by assigning specific values to one or more of a, b, c, x—many of the most important series in analysis, for example those by which logarithms, the trigonometric functions, and several of the functions that turn up repeatedly in Newtonian astronomy and mathematical physics are calculated and tabulated; the general binomial theorem also is a special case. By disposing of this series in its general form Gauss slew a multitude at one smash. From this work developed many applications to the differential equations of physics in the nineteenth century.

  The choice of such an investigation for a serious eff
ort is characteristic of Gauss. He never published trivialities. When he put out anything it was not only finished in itself but was also so crammed with ideas that his successors were enabled to apply what Gauss had invented to new problems. Although limitations of space forbid discussion of the many instances of this fundamental character of Gauss’ contributions to pure mathematics, one cannot be passed over in even the briefest sketch: the work on the law of biquadratic reciprocity. The importance of this was that it gave a new and totally unforeseen direction to the higher arithmetic.

  * * *

  Having disposed of quadratic (second degree) reciprocity, it was natural for Gauss to consider the general question of binomial congruences of any degree. If m is a given integer not divisible by the prime p, and if n is a given positive integer, and if further an integer x can be found such that xn ≡ m (mod p), m is called an n-ic residue of p; when n = 4, m is a biquadratic residue of p.

  The case of quadratic binomial congruences (n = 2) suggests but little to do when n exceeds 2. One of the matters Gauss was to have included in the discarded eighth section (or possibly, as he told Sophie Germain, in the projected but unachieved second volume) of the Disquisitiones Arithmeticae was a discussion of these higher congruences and a search for the corresponding laws of reciprocity, namely the interconnections (as to solvability or non-solvability) of the pair xn ≡ p (mod q), xn ≡ q (mod p), where p, q are rational primes. In particular the cases n = 3, n = 4 were to have been investigated.

  The memoir of 1825 breaks new ground with all the boldness of the great pioneers. After many false starts which led to intolerable complexity Gauss discovered the “natural” way to the heart of his problem. The rational integers 1, 2, 3, . . . are not those appropriate to the statement of the law of biquadratic reciprocity, as they are for quadratic; a totally new species of integers must be invented. These are called the Gaussian complex integers and are all those complex numbers of the form a + bi in which a, b are rational integers and i denotes

  To state the law of biquadratic reciprocity an exhaustive preliminary discussion of the laws of arithmetical divisibility for such complex integers is necessary. Gauss gave this, thereby inaugurating the theory of algebraic numbers—that which he probably had in mind when he gave his estimate of Fermat’s Last Theorem. For cubic reciprocity (n = 3) he also found the right way in a similar manner. His work on this was found in his posthumous papers.

  The significance of this great advance will become clearer when we follow the careers of Kummer and Dedekind. For the moment it is sufficient to say that Gauss’ favorite disciple, Eisenstein, disposed of cubic reciprocity. He further discovered an astonishing connection between the law of biquadratic reciprocity and certain parts of the theory of elliptic functions, in which Gauss had travelled far but had refrained disclosing what he found.

  Gaussian complex integers are of course a subclass of all complex numbers, and it might be thought that the algebraic theory of all the numbers would yield the arithmetical theory of the included integers as a trivial detail. Such is by no means the case. Compared to the arithmetical theory the algebraic is childishly easy. Perhaps a reason why this should be so is suggested by the rational numbers (numbers of the form a/b, where a, b are rational integers). We can always divide one rational number by another and get another rational number: a/b divided by c/d yields the rational number ad/bc. But a rational integer divided by another rational integer is not always another rational integer: 7 divided by 8 gives ⅞ Hence if we must restrict ourselves to integers, the case of interest for the theory of numbers, we have tied our hands and hobbled our feet before we start. This is one of the reasons why the higher arithmetic is harder than algebra, higher or elementary.

  * * *

  Equally significant advances in geometry and the applications of mathematics to geodesy, the Newtonian theory of attraction, and electromagnetism were also to be made by Gauss. How was it possible for one man to accomplish this colossal mass of work of the highest order? With characteristic modesty Gauss declared that “If others would but reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries.” Possibly. Gauss’ explanation recalls Newton’s. Asked how he had made discoveries in astronomy surpassing those of all his predecessors, Newton replied, “By always thinking about them.” This may have been plain to Newton; it is not to ordinary mortals.

  Part of the riddle of Gauss is answered by his involuntary preoccupation with mathematical ideas—which itself of course demands explanation. As a young man Gauss would be “seized” by mathematics. Conversing with friends he would suddenly go silent, overwhelmed by thoughts beyond his control, and stand staring rigidly oblivious of his surroundings. Later he controlled his thoughts—or they lost their control over him—and he consciously directed all his energies to the solution of a difficulty till he succeeded. A problem once grasped was never released till he had conquered it, although several might be in the foreground of his attention simultaneously.

  In one such instance (referring to the Disquisitiones, page 636) he relates how for four years scarcely a week passed that he did not spend some time trying to settle whether a certain sign should be plus or minus. The solution finally came of itself in a flash. But to imagine that it would have blazed out of itself like a new star without the “wasted” hours is to miss the point entirely. Often after spending days or weeks fruitlessly over some research Gauss would find on resuming work after a sleepless night that the obscurity had vanished and the whole solution shone clear in his mind. The capacity for intense and prolonged concentration was part of his secret.

  In this ability to forget himself in the world of his own thoughts Gauss resembles both Archimedes and Newton. In two further respects he also measures up to them, his gifts for precise observation and a scientific inventiveness which enabled him to devise the instruments necessary for his scientific researches. To Gauss geodesy owes the invention of the heliotrope, an ingenious device by which signals could be transmitted practically instantaneously by means of reflected light. For its time the heliotrope was a long step forward. The astronomical instruments he used also received notable improvements at Gauss’ hands. For use in his fundamental researches in electromagnetism Gauss invented the bifilar magnetometer. And as a final example of his mechanical ingenuity it may be recalled that Gauss in 1833 invented the electric telegraph and that he and his fellow worker Wilhelm Weber (1804-1891) used it as a matter of course in sending messages. The combination of mathematical genius with first-rate experimental ability is one of the rarest in all science.

  Gauss himself cared but little for the possible practical uses of his inventions. Like Archimedes he preferred mathematics to all the kingdoms of the earth; others might gather the tangible fruits of his labors. But Weber, his collaborator in electromagnetic researches, saw clearly what the puny little telegraph of Göttingen meant for civilization. The railway, we recall, was just coming into its own in the early 1830’s. “When the globe is covered with a net of railroads and telegraph wires,” Weber prophesied in 1835, “this net will render services comparable to those of the nervous system in the human body, partly as a means of transport, partly as a means for the propagation of ideas and sensations with the speed of lightning.”

  The admiration of Gauss for Newton has already been noted. Knowing the tremendous efforts some of his own masterpieces had cost him, Gauss had a true appreciation of the long preparation and incessant meditation that went into Newton’s greatest work. The story of Newton and the falling apple roused Gauss’ indignation. “Silly!” he exclaimed. “Believe the story if you like, but the truth of the matter is this. A stupid, officious man asked Newton how he discovered the law of gravitation. Seeing that he had to deal with a child in intellect, and wanting to get rid of the bore, Newton answered that an apple fell and hit him on the nose. The man went away fully satisfied and completely enlightened.”

  The apple story has its echo in our own times. When teased
as to what led him to his theory of the gravitational field Einstein replied that he asked a workman who had fallen off a building, to land unhurt on a pile of straw, whether he noticed the tug of the “force” of gravity on the way down. On being told that no force had tugged, Einstein immediately saw that “gravitation” in a sufficiently small region of space-time can be replaced by an acceleration of the observer’s (the falling workman’s) reference system. This story, if true, is also probably all rot. What gave Einstein his idea was the hard labor he expended for several years mastering the tensor calculus of two Italian mathematicians, Ricci and Levi-Civita, themselves disciples of Riemann and Christoffel, both of whom in their turn had been inspired by the geometrical work of Gauss.

  Commenting on Archimedes, for whom he also had a boundless admiration, Gauss remarked that he could not understand how Archimedes failed to invent the decimal system of numeration or its equivalent (with some base other than 10). The thoroughly un-Greek work of Archimedes in devising a scheme for writing and dealing with numbers far beyond the capacity of the Greek symbolism had—according to Gauss—put the decimal notation with its all-important principle of place-value (325 = 3 × 102 + 2 × 10 + 5) in Archimedes’ hands. This oversight Gauss regarded as the greatest calamity in the history of science. “To what heights would science now be raised if Archimedes had made that discovery!” he exclaimed, thinking of his own masses of arithmetical and astronomical calculations which would have been impossible, even to him, without the decimal notation. Having a full appreciation of the significance for all science of improved methods of computation, Gauss slaved over his own calculations till pages of figures were reduced to a few lines which could be taken in almost at a glance. He himself did much of his calculating mentally; the improvements were intended for those less gifted than himself.