by E. T. Bell
One of the leaders in the seminar, Johann Benedict Listing (18081882), may be noted in passing, as he probably influenced Riemann’s thought in what was to be (1857) one of his greatest achievements, the introduction of topological methods into the theory of functions of a complex variable.
It will be recalled that Gauss had prophesied that analysis situs would become one of the most important fields of mathematics, and Riemann, by his inventions in the theory of functions, was to give a partial fulfillment of this prophecy. Although topology (now called analysis situs) as first developed bore but little resemblance to the elaborate theory which today absorbs all the energies of a prolific school, it may be of interest to state the trivial puzzle which apparently started the whole vast and intricate theory. In Euler’s time seven bridges crossed the river Pregel in Königsberg, as in the diagram, the shaded bars representing the bridges. Euler proposed the problem of crossing all seven bridges without passing twice over any one. The problem is impossible.
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The nature of Riemann’s use of topological methods in the theory of functions may be disposed of here, although an adequate description is out of the question in untechnical language. For the meaning of “uniformity” with respect to a function of a complex variable we must refer to what was said in the chapter on Gauss. Now, in the theory of Abelian functions, multiform functions present themselves inevitably; an n-valued function of z is a function which, except for certain values of z, takes precisely n distinct values for each value assigned to z. Illustrating multiformity, or many-valuedness, for functions of a real variable, we note that y, considered as a function of x, defined by the equation y2 = x, is two-valued. Thus, if x = 4, we get y2 = 4, and hence y = 2 or −2; if x is any real number except zero or “infinity,” y has the two distinct values of and In this simplest possible example y and x are connected by an algebraic equation, namely y2—x = 0. Passing at once to the general situation of which this is a very special case, we might discuss the n-valued function y which is defined, as a function of x, by the equation
P0(x)yn + P1(x)yn-1 + . . . + Pn-1(x)y + Pn(x) = 0,
in which the P’s are polynomials in x. This equation defines y as an n-valued function of x. As in the case of y2—x = 0, there will be certain values of x for which two or more of these n values of y are equal. These values of x are the so-called branch points of the n-valued function defined by the equation.
All this is now extended to functions of complex variables, and the function w (also its integral) as defined by
P0(z)wn + P1(z)wn–1 + . . . + Pn-1(z)w + Pn(z) = 0,
in which z denotes the complex variable s + it, where s, t are real variables and The n values of w are called the branches of the function w. Here we must refer (chapter on Gauss) to what was said about the representation of uniform functions of z. Let the variable z (= s + it) trace out any path in its plane, and let the uniform functions f(z) be expressed in the form U + iV, where U, V are functions of s, t. Then, to every value of z will correspond one, and only one, value for each of U, V, and, as z traces out its path in the s, t-plane, f(z) will trace out a corresponding path in the U, V-plane: the path off(z) will be uniquely determined by that of z. But if w is a multiform (many-valued) function of z, such that precisely n distinct values of w are determined by each value of z (except at branch points, where several values of w may be equal), then it is obvious that one w-plane no longer suffices (if n is greater than l) to represent the path, the “march” of the function w. In the case of a two-valued function w, such as that determined by w2 = z, two w-planes would be required and, quite generally, for an n-valued function (n finite or infinite), precisely n such w-planes would be required.
The advantages of considering uniform (one-valued) functions instead of n-valued functions (n greater than 1) should be obvious even to a non-mathematician. What Riemann did was this: instead of the n distinct w-planes, he introduced an rc-sheeted surface, of the sort roughly described in what follows, on which the multiform function is uniform, that is, on which, to each “place” on the surface corresponds one, and only one, value of the function represented.
Riemann united, as it were, all the n planes into a single plane, and he did this by what may at first look like an inversion of the representation of the n branches of the n-valued function on n distinct planes; but a moment’s consideration will show that, in effect, he restored uniformity. For he superimposed n z-planes on one another; each of these planes, or sheets, is associated with a particular branch of the function so that, as long as z moves in a particular sheet, the corresponding branch of the function is traversed by w (the n-valued function of z under discussion), and as z passes from one sheet to another, the branches are changed, one into another, until, on the variable z having traversed all the sheets and having returned to its initial position, the original branch is restored. The passage of the variable z from one sheet to another is effected by means of cuts (which may be thought of as straight-line bridges) joining branch points; along a given cut providing passage from one sheet to another, one “lip” of the upper sheet is imagined as pasted or joined to the opposite lip of the under sheet, and similarly for the other lip of the upper sheet. Diagrammatically, in cross-section,
The sheets are not joined along cuts (which may be drawn in many ways for given branch points) at random, but are so joined that, as z traverses its n-sheeted surface, passing from one sheet to another as a bridge or cut is reached, the analytical behavior of the function of z is pictured consistently, particularly as concerns the interchange of branches consequent on the variable z, if represented on a plane, having gone completely round a branch point. To this circuiting of a branch point on the single z-plane corresponds, on the n-sheeted Riemann surface, the passage from one sheet to another and the resultant interchange of the branches of the function.
There are many ways in which the variable may wander about the n-sheeted Riemann surface, passing from one sheet to another. To each of these corresponds a particular interchange of the branches of the function, which may be symbolized by writing, one after another, letters denoting the several branches interchanged. In this way we get the symbols of certain substitutions (as in chapter 15) on n letters; all of these substitutions generate a group which, in some respects, pictures the nature of the function considered.
Riemann surfaces are not easy to represent pictorially, and those who use them content themselves with diagrammatical representations of the connection of the sheets, in much the same way that an organic chemist writes a “graphical” formula for a complicated carbon compound which recalls in a schematic manner the chemical behavior of the compound but which does not, and is not meant to, depict the actual spatial arrangement of the atoms in the compound. Riemann made wonderful advances, particularly in the theory of Abelian functions, by means of his surfaces and their topology—how shall the cuts be made so as to render the n-sheeted surface equivalent to a plane, being one question in this direction. But mathematicians are like other mortals in their ability to visualize complicated spatial relationships, namely, a high degree of spatial “intuition” is excessively rare.
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Early in November, 1851, Riemann submitted his doctoral dissertation, Grundlagen für eine allegemeine Theorie der Functionen einer veränderlichen complexen Grösse (Foundations for a general theory of functions of a complex variable), for Gauss’ consideration. This work by the young master of twenty five was one of the few modern contributions to mathematics that roused the enthusiasm of Gauss, then an almost legendary figure within four years of his death. When Riemann called on Gauss, after the latter had read the dissertation, Gauss told him that he himself had planned for years to write a treatise on the same topic. Gauss’ official report to the Philosophical Faculty of the University of Göttingen is noteworthy as one of the rare formal pronouncements in which Gauss let himself go.
“The dissertation submitted by Herr Riemann offers convincing evidence of the author’s t
horough and penetrating investigations in those parts of the subject treated in the dissertation, of a creative, active, truly mathematical mind, and of a gloriously fertile originality. The presentation is perspicuous and concise and, in places, beautiful. The majority of readers would have preferred a greater clarity of arrangement. The whole is a substantial, valuable work, which not only satisfies the standards demanded for doctoral dissertations, but far exceeds them.”
A month later Riemann passed his final examination, including the formality of a public “defense” of his dissertation. All went off successfully, and Riemann began to hope for a position in keeping with his talents. “I believe I have improved my prospects with my dissertation,” he wrote to his father; “I hope also to learn to write more quickly and more fluently in time, especially if I mingle in society and if I get a chance to give lectures; therefore am I of good courage.” He also apologizes to his father for not having gone after a vacant assistantship at the Göttingen Observatory more energetically, but as he hopes to be “habilitated” as a Privatdozent the outlook is not as dark as it might be.
For his Habilitationsschrift (probationary essay) Riemann had planned to submit a memoir on trigonometric series (Fourier series). But two and a half years were to pass before he might hang out his shingle as an unpaid university instructor picking up what he could in the way of fees from students not bound to attend his lectures. During the autumn of 1852 Riemann profited by Dirichlet’s presence in Göttingen on a vacation and sought his advice on the embryonic memoir. Riemann’s friends saw to it that the young man met the famous mathematician from Berlin—second only to Gauss—socially.
Dirichlet was captivated by Riemann’s modesty and genius. “Next morning [after a dinner party] Dirichlet was with me for two hours,” Riemann wrote his father. “He gave me the notes I needed for my probationary essay; otherwise I should have had to spend many hours in the library in laborious research. He also read over my dissertation with me and was very friendly—which I could hardly have expected, considering the great distance in rank between us. I hope he will remember me later on.” During this visit of Dirichlet’s there were excursions with Weber and others, and Riemann reported to his father that these human escapes from mathematics did him more good scientifically than if he had sat all day over his books.
From 1853 (Riemann was then twenty seven) onward he thought intensively about mathematical physics. By the end of the year he had completed the probationary essay, after many delays due to his growing passion for physical science.
There was still a trial lecture ahead of him before he could be appointed to the coveted—but unpaid—lectureship. For this ordeal he had submitted three titles for the faculty to choose from, hoping and expecting that one of the first two, on which he had prepared himself, would be selected. But he had incautiously included as his third offering a topic on which Gauss had pondered for sixty years or more—the foundations of geometry—and this he had not prepared. Gauss no doubt was curious to see what a Riemann’s “gloriously fertile originality” would make of such a profound subject. To Riemann’s consternation Gauss designated the third topic as the one on which Riemann should prove his mettle as a lecturer before the critical faculty. “So I am again in a quandary,” the rash young man confided to his father, “since I have to work out this one. I have resumed my investigation of the connection between electricity, magnetism, light, and gravitation, and I have progressed so far that I can publish it without a qualm. I have become more and more convinced that Gauss has worked on this subject for years, and has talked to some friends (Weber among others) about it. I tell you this in confidence, lest I be thought arrogant—I hope it is not yet too late for me and that I shall gain recognition as an independent investigator.”
The strain of carrying on two extremely difficult investigations simultaneously, while acting as Weber’s assistant in the seminar in mathematical physics, combined with the usual handicaps of poverty, brought on a temporary breakdown. “I became so absorbed in my investigation of the unity of all physical laws that when the subject of the trial lecture was given me, I could nor tear myself away from my research. Then, partly as a result of brooding on it, partly from staying indoors too much in this vile weather, I fell ill; my old trouble recurred with great pertinacity and I could not get on with my work. Only several weeks later, when the weather improved and I got more social stimulation, I began feeling better. For the summer I have rented a house in a garden, and since doing so my health has not bothered me. Having finished two weeks after Easter a piece of work I could not get out of, I began at once working on my trial lecture and finished it around Pentecost [that is, in about seven weeks]. I had some difficulty in getting a date for my lecture right away and almost had to return to Quickborn without having reached my goal. For Gauss is seriously ill and the physicians fear that his death is imminent. Being too weak to examine me, he asked me to wait till August, hoping that he might improve, especially as I would not lecture anyhow till fall. Then he decided anyway on the Friday after Pentecost to set the lecture for the next day at eleven thirty. On Saturday I was happily through with everything.”
This is Riemann’s own account of the historic lecture which was to revolutionize differential geometry and prepare the way for the geometrized physics of our own generation. In the same letter he tells how the work he had been doing around Easter turned out. Weber and some of his collaborators “had made very exact measurements of a phenomenon which up till then had never been investigated, the residual charge in a Leyden jar [after discharge it is found that the jar is not completely discharged] . . . I sent him [one of Weber’s collaborators, Kohlrausch] my theory of this phenomenon, having worked it out specially for his purposes. I had found the explanation of the phenomenon through my general investigations of the connection between electricity, light, and magnetism. . . . This matter was important to me, because it was the first time I could apply my work to a phenomenon still unknown, and I hope that the publication [of it] will contribute to a favorable reception of my larger work.”
The reception of Riemann’s probationary lecture (June 10, 1854) was as cordial as even he could have wished in the scared secrecy of his modest heart. The lecture had made him sweat blood to prepare because he had determined to make it intelligible even to those members of the faculty who had but little knowledge of mathematics. In addition to being one of the great masterpieces of all mathematics, Riemann’s essay Ueber die Hypothesen, welche der Geometrie zu Grunde liegen (On the hypotheses which lie at the foundations of geometry), is also a classic of presentation. Gauss was enthusiastic. “Against all tradition he had selected the third of the three topics submitted by the candidate, wishing to see how such a difficult subject would be handled by so young a man. He was surprised beyond all his expectations, and on returning from the faculty meeting expressed to Wilhelm Weber his highest appreciation of the ideas presented by Riemann, speaking with an enthusiasm that, for Gauss, was rare.” What little can be said here about this masterpiece will be reserved for the conclusion of the present chapter.
After a rest at home with his family in Quickborn, Riemann returned in September to Göttingen, where he delivered a hastily prepared lecture (sitting up most of the night to get it ready on short notice) to a convention of scientists. His topic was the propagation of electricity in non-conductors. During the year he continued his researches in the mathematical theory of electricity and prepared a paper on Nobili’s color rings because, as he wrote his sister Ida: “This subject is important, for very exact measurements can be made in connection with it, and the laws according to which electricity moves can be tested.”
In the same letter (October 9, 1854) he expresses his unbounded joy at the success of his first academic lecture and his great satisfaction at the unexpectedly large number of auditors. Eight students had come to hear him! He had anticipated at the most two or three. Encouraged by this unhoped-for popularity, Riemann tells his father, “I have been able to
hold my classes regularly. My first diffidence and constraint have subsided more and more, and I get accustomed to think more of the auditors than of myself, and to read in their expressions whether I should go on or explain the matter further.”
When Dirichlet succeeded Gauss in 1855, Riemann’s friends urged the authorities to appoint Riemann to the security of an assistant professorship, but the finances of the University could not be stretched so far. Nevertheless he was granted the equivalent of two hundred dollars a year, which was better than the uncertainty of half a dozen voluntary students’ fees. His future worried him, and when presently he lost both his father and his sister Clara, making it impossible for him to escape for vacations to Quickborn, Riemann felt poor and miserable indeed. His three remaining sisters went to live with the other brother, a postal clerk in Bremen whose salary was princely beside that of the “economically valueless” mathematician.
The following year (1856; Riemann was then thirty) the outlook brightened a little. It was impossible for a creative genius like Riemann to be downed by despondency so long as he had the wherewithal to keep body and soul together in order that he might work. To this period belong part of his characteristically original work on Abelian functions, his classic on the hypergeometric series (see chapter on Gauss) and the differential equations—of great importance in mathematical physics—suggested by this series. In both of these works Riemann struck out on new directions of his own. The generality, the intuitiveness, of his approach was peculiarly his own. His work absorbed all his energies and made him happy in spite of material worries; possibly, too, the fatal optimism of the consumptive was already at work in him.
