Men of Mathematics (96 page)

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.

*  *  *

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 thorough 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.

Riemann's development of the theory of Abelian functions is as unlike that of Weierstrass as moonlight is unlike sunlight. Weierstrass' attack was methodical, exact in all its details, like the advance of a perfectly disciplined army under a generalship that foresees everything and provides for all contingencies. Riemann, for his part, looked over the whole field, seeing everything but the details, which he left to take care of themselves, and was content to have grasped the key positions of the general topography in his imagination. The method of Weierstrass was arithmetical, that of Riemann geometrical and intuitive. To say that one is “better” than the other is meaningless; both cannot be seen from a common point of view.