by E. T. Bell
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.
Overwork and lack of reasonable comforts brought on a nervous breakdown early in his thirty first year, and Riemann was forced to spend a few weeks with a friend in the Hartz mountain country, where he was joined by Dedekind. The three took long tramps together into the mountains and Riemann soon recovered. Relieved of the strain of having to keep up academic appearances, Riemann indulged his sense of humor and kept his companions amused with his spontaneous wit. They also talked shop together—most mathematicians do when they get together, just as lawyers or doctors or business men do, provided they do not have to talk drivel to maintain the social conventions. One evening after a strenuous hike Riemann dipped into Brewster’s life of Newton and discovered the letter to Bentley in which Newton himself asserts the impossibility of action at a distance without intervening media. This delighted Riemann and inspired him to an impromptu lecture. Today the “medium” which Riemann extolled is not the luminiferous ether, but his own “curved space,” or its reflection in the space-time of relativity.
At last, in 1857, at the age of thirty one, Riemann got his assistant professorship. His salary was the equivalent of about three hundred dollars a year, but as he had had little all his life he missed less. However, a real disaster presently descended on him: his brother died and the care of three sisters fell to his lot. It figured out at exactly seventy five dollars a year for each of them. Love on nothing a year in a cottage may be paradise; existence on next to nothing in a university community is just plain hell. It was but little different in Riemann’s day. No wonder he contracted consumption. However, the Lord, who had so generously given, shortly relieved Riemann of his youngest sister, Marie, so the individual budgets skyrocketed to one hundred dollars a year. If rations had to be watched, affection was free, and Riemann was more than repaid for his sacrifices by the self-confidence inspired in him by his sisters’ devotion and encouragement. The Lord may have known that if ever a struggling mortal needed encouragement, poor Riemann did; still, it seems rather an odd way of providing what was required.
In 1858 Riemann produced his paper on electrodynamics, of which he told his sister Ida, “My discovery concerning the close connection between electricity and light I have dedicated to the Royal Society [of Göttingen]. From what I have heard, Gauss had devised another theory regarding this close connection, different from mine, and communicated it to his intimate friends. However, I am fully convinced that my theory is the correct one, and that in a few years it will be recognized as such. As is known, Gauss soon withdrew his memoir and did not publish it; probably he himself was not satisfied with it.” Riemann would seem here to have been overoptimistic; Clerk Maxwell’s electromagnetic theory is the one which today holds the field—in macroscopic phenomena. The present status of theories of light and the electromagnetic field is too complicated to be described here; it is sufficient to note that Riemann’s theory has not survived.
Dirichlet died on May 5, 1859. He had always appreciated Riemann and had done his best to help the struggling young man along. This interest of Dirichlet’s and Riemann’s rapidly mounting reputation caused the government to promote Riemann to succeed Dirichlet. At thirty three Riemann thus became the second successor of Gauss. To ease his domestic difficulties the authorities let him reside at the Observatory, as Gauss had done. Recognition of the sincerest kind—praise from mathematicians who, although older than himself, were in some degree his rivals—now came in abundance. On a visit to Berlin he was feted by Borchardt, Kummer, Kronecker, and Weierstrass. Learned societies, including the Royal Society of London and the French Academy of Sciences, honored him with membership, and in short he got the usual highest distinctions that can come to a man of science. A visit to Paris in 1860 acquainted him with the leading French mathematicians, particularly Hermite, whose admiration for Riemann was unbounded. This year, 1860, is memorable in the history of mathematical physics as that in which Riemann began intensive work on his memoir Über eine Frage der Wärmeleitung (On a Question in the Conduction of Heat), in which he develops the whole apparatus of quadratic differential forms (to be noticed in connection with Riemann’s work in the foundations of geometry), which is today basic in the theory of relativity.
His material affairs having improved considerably with his appointment to the full professorship, Riemann was in a position to marry at the age of thirty six. His wife, Elise Koch, was a friend of his sisters. Barely a month after his marriage, Riemann fell ill in July 1862 with pleurisy. An incomplete recovery ended in consumption. Influential friends induced the Government to grant Riemann the funds for convalescence in the mild climate of Italy, where he spent the winter. The following spring on his return trip to Germany he took great delight in the art treasures of the many Italian cities he visited. This was the brief summer of his life.
Full of hope he left his beloved Italy, only to fall more seriously ill on reaching Göttingen. On the return journey he had grown careless, and while walking through deep snow in the Splügen Pass, had taken a severe chill. The following August (1863) he returned to Italy, stopping first at Pisa, where his daughter Ida (named after his older sister) was born. The winter was exceptionally harsh, the river Arno being frozen over. In May he moved to a small villa in the suburbs of Pisa. There his younger sister Helene died. His own illness, complicated by jaundice, grew steadily graver. To his great regret he was obliged to refuse a professorship offered to him at the University of Pisa. Göttingen generously extended his leave of absence to enable him to spend the following winter in Pisa, surrounded by his Italian mathematical friends. But further complications made him long for home, and after vainly seeking health in Leghorn and Genoa, he returned in October to Göttingen, where he spent a tolerable winter.
All this time he worked when he had the strength. At Göttingen he often expressed the desire to speak with Dedekind of the works he had not completed, but never felt quite strong enough to stand a visit. One of his last projects was a work on the mechanics of the ear, which he left incomplete. He had hoped to finish this, also some other things which he considered of great importance, and in a final attempt to regain his strength returned to Italy. His last days were spent in a villa at Selasca, Lago Maggiore.
Dedekind tells how his friend died. “But his strength declined rapidly; he felt himself that his end was near. The day before his death he worked under a fig tree, his soul filled with joy at the glorious landscape around him. . . . His life ebbed gently away, without strife or death agony; it seemed as though he followed with interest the separation of the soul from the body; his wife had to give him bread and wine . . . he said to her, ‘Kiss our child.’ She repeated the Lord’s prayer with him; he could no longer speak; at the words ’Forgive us our trespasses’ he looked up devoutly; she felt his hand grow colder in hers, and with a few last sighs his pure, noble heart had ceased to beat. The gentle mind which had been implanted in him in his father’s house remained with him all his life, and he served his God faithfully, as his father had, but in a different way.”
Thus Riemann died, in the full glory of his matured genius, on July 20, 1866, aged thirty nine. The inscription on his tombstone, erected by his Italian friends, closes with the words “Denen die Gott lieben müssen alle Dinge zum Besten dienen”, or as it is usually put in English, “All things work together for good to them t
hat love the Lord.”
* * *
Riemann’s greatness as a mathematician resides in the powerful generality and unbounded scope of the methods and new points of view which he revealed to both pure and applied mathematics. Details never oppressed him; he saw the whole of a vast problem as a coherent unity. Even the fragmentary notes on uncompleted projects usually hint at some haunting novelty and sharpen our regret that Riemann died so long before his time. Only one of his great masterpieces can be described here, the memoir of 1854 on the foundations of geometry, and although it may not be quite fair to Clifford to use him merely to introduce another, we shall quote in its entirety his daring paper of 1870, On the space-theory of matter, as a singularly prophetic introduction to the body and spirit of Riemann’s geometry. Clifford was no servile copyist but a man with a brilliantly original mind of his own, of whom it may be said, as Newton said of Cotes, “If he had lived we might have known something.” The reader who is acquainted with any of the better available popular accounts of relativistic physics and the wave theory of electrons will recognize several curious adumbrations of current theories in Clifford’s brief prophecy.
“Riemann has shown that as there are different kinds of lines and surfaces, so there are different kinds of space of three dimensions; and that we can only find out by experience to which of these kinds the space in which we live belongs. In particular, the axioms of plane geometry are true within the limits of experiment on the surface of a sheet of paper, and yet we know that the sheet is really covered with a number of small ridges and furrows, upon which (the total curvature being not zero) these axioms are not true. Similarly, he says, although the axioms of solid geometry are true within the limits of experiment for finite portions of our space, yet we have no reason to conclude that they are true for very small portions; and if any help can be got thereby for the explanation of physical phenomena, we may have reason to conclude that they are not true for very small portions of space.
“I wish here to indicate a manner in which these speculations may be applied to the investigation of physical phenomena. I hold in fact
(1) That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them.
(2) That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave.
(3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or ethereal.
(4) That in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity.
“I am endeavoring in a general way to explain the laws of double refraction on this hypothesis, but have not yet arrived at any results sufficiently decisive to be communicated.”
Riemann also believed that his new geometry would prove of scientific importance, as is shown by the conclusion of his memoir (Clifford’s translation):
“Either therefore the reality which underlies space must form a discrete manifold, or we must seek the ground of its metric relations outside it, in binding forces which act upon it.
“The answer to these questions can only be got by starting from the conception of phenomena which has hitherto been justified by experience, and which Newton assumed as a foundation, and by making in this conception the successive changes required by facts which it cannot explain.” And he goes on to say that researches like his own, starting from general notions, “can be useful in preventing this work from becoming hampered by too narrow views, and progress of knowledge of the interdependence of things from being checked by traditional prejudices.
“This leads us into the domain of another science, that of physics, into which the object of this work does not allow us to go today.”
Riemann’s work of 1854 put geometry in a new light. The geometry he visions is non-Euclidean, not in the sense of Lobatchewsky and Johann Bolyai, nor in that of Riemann’s own elaboration of the hypothesis of the obtuse angle (as explained in chapter 16), but in a more comprehensive sense depending on the conception of measurement. To isolate measure-relations as the nerve of Riemann’s theory is to do it an injustice; the theory contains much more than a workable philosophy of metrics, but this is one of its main features. No paraphrase of Riemann’s concise memoir can bring out all that is in it; nevertheless, we shall attempt to describe some of his basic ideas, and we shall select three: the concept of a manifold, the definition of distance, and the notion of curvature of a manifold.
A manifold is a class of objects (at least in common mathematics) which is such that any member of the class can be completely specified by assigning to it certain numbers, in a definite order, corresponding to “numberable” properties of the elements, the assignment in the given order corresponding to a preassigned ordering of the “number-able” properties. Granted that this may be even less comprehensible than Riemann’s definition, it is nevertheless a working basis from which to start, and all that it amounts to in plain mathematics is this: a manifold is a set of ordered “n-tuples” of numbers (x1x2, . . . , xn), where the parentheses, (), indicate that the numbers x1x2, . . . , xn are to be written in the order given. Two such n-tuples, (x1, x2, . . . , xn) and (y1, y2, . . . , yn) are equal when, and only when, corresponding numbers in them are respectively equal, namely, when, and only when, x1 = y1 x2 = y2, . . . , xn = yn.
If precisely n numbers occur in each of these ordered n-tuples in the manifold, the manifold is said to be of n dimensions. Thus we are back again talking coordinates with Descartes. If each of the numbers in (x1, x2, . . . , xn) is a positive, zero, or negative integer, or if it is an element of any countable set (a set whose elements may be counted off 1, 2, 3, . . .), and if the like holds for every n-tuple in the set, the manifold is said to be discrete. If the numbers x1, x2, . . . , xn, may take on values continuously (as in the motion of a point along a line), the manifold is continuous.
This working definition has ignored—deliberately—the question of whether the set of ordered n-tuples is “the manifold” or whether something “represented by” these is “the manifold.” Thus, when we say (x, y) are the coordinates of a point in a plane, we do not ask what “a point in a plane” is, but proceed to work with these ordered couples of numbers (x, y) where x, y run through all real numbers independently. On the other hand it may sometimes be advantageous to fix our attention on what such a symbol as (x, y) represents. Thus if x is the age in seconds of a man and y his height in centimeters, we may be interested in the man (or the class of all men) rather than in his coordinates, with which alone the mathematics of our enquiry is concerned. In this same order of ideas, geometry is no longer concerned with what “space” “is”—whether “is” means anything or not in relation to “space.” Space, for a modern mathematician, is merely a number-manifold of the kind described above, and this conception of space grew out of Riemann’s “manifolds.”
Passing on to measurement, Riemann states that “Measurement consists in a superposition of the magnitudes to be compared. If this is lacking, magnitudes can be compared only when one is part of another, and then only the more or less, but not the how much, can be decided.” It may be said in passing that a consistent and useful theory of measurement is at present an urgent desideratum in theoretical physics, particularly in all questions where quanta and relativity are of importance.
Descending once more from philosophical generalities to less mystical mathematics, Riemann proceeded to lay down a definition of distance, extracted from his concept of measurement, which has proved to be extremely fruitful in both physics and mathematics. The Pythagorean proposition that where a is the length of the longest side of a right-angled triangle and b, c are the lengths of the other two sides, is the fundamental formula for the measurement of distances in a plane. How shall this be extended to a curved surface?. To straight lines
on the plane correspond geodesics (see chapter 14) on the surface; but on a sphere, for example, the Pythagorean proposition is not true for a right-angled triangle formed by geodesies. Riemann generalized the Pythagorean formula to any manifold as follows:
Let (x1x2, . . . , xn), (x1 + x1’, x2 + x2’, . . . , xn + xn’) be the coordinates of two “points” in the manifold which are “infinitesimally near” one another. For our present purpose the meaning of “infinitesimally near” is that powers higher than the second of X1’, x2’, . . . , xn’, which measure the “separation” of the two points in the manifold, can be neglected. For simplicity we shall state the definition when n = 4—giving the distance between two neighboring points in a space of four dimensions: the distance is the square root of
g11x1′2 + g22x2′2 + g33x3′2 + g44x4′2 + g12x1′x2′ + g13x1′x3′ + g14x1′x4′ + g23x2′x3′ + g24x2′x′4 + g34x3′x4′
in which the ten coefficients gn, . . ., gu are functions of x1, x2, x3, x4. For a particular choice of the g’s, one “space” is defined. Thus we might have = 1, g11 = 1, £33 = 1, gu = −1, and all the other g’s zero; or we might consider a space in which all the g’s exceptg’s and ¿•34 were zero, and so on. A space considered in relativity is of this general kind in which all the g’s except g11, g11, g22, g33, g44 are zero, and these are certain simple expressions involving x1, x2, x3, x4.
In the case of an n-dimensional space the distance between neighboring points is defined in a similar manner; the general expression contains ½n (n + 1) terms. The generalized Pythagorean formula for the distance between neighboring points being given, it is a solvable problem in the integral calculus to find the distance between any two points of the space. A space whose metric (system of measurement) is defined by a formula of the type described is called Riemannian.
Curvature, as conceived by Riemann (and before him by Gauss; see chapter on the latter) is another generalization from common experience. A straight line has zero curvature; the “measure” of the amount by which a curved line departs from straightness may be the same for every point of the curve (as it is for a circle), or it may vary from point to point of the curve, when it becomes necessary again to express the “amount of curvature” through the use of infinitesimals. For curved surfaces, the curvature is measured similarly by the amount of departure from a plane, which has zero curvature. This may be generalized and made a little more precise as follows. For simplicity we state first the situation for a two-dimensional space, namely for a surface as we ordinarily imagine surfaces. It is possible from the formula
