Men of Mathematics

by E. T. Bell

  The theorem which Abel thus briefly describes is today known as Abel’s Theorem. His proof of it has been described as nothing more than “a marvellous exercise in the integral calculus.” As in his algebra, so in his analysis, Abel attained his proof with a superb parsimony. The proof, it may be said without exaggeration, is well within the purview of any seventeen-year-old who has been through a good first course in the calculus. There is nothing high-falutin’ about the classic simplicity of Abel’s own proof. The like cannot be said for some of the nineteenth century expansions and geometrical reworkings of the original proof. Abel’s proof is like a statue by Phidias; some of the others resemble a Gothic cathedral smothered in Irish lace, Italian confetti, and French pastry.

  There is ground for a possible misunderstanding in Abel’s opening paragraph. Abel no doubt was merely being kindly courteous to an old man who had patronized him—in the bad sense—on first acquaintance, but who, nevertheless, had spent most of his long working life on an important problem without seeing what it was all about. It is not true that Legendre had discussed the elliptic functions, as Abel’s words might imply; what Legendre spent most of his life over was elliptic integrals, which are as different from elliptic functions as a horse is from the cart it pulls, and therein precisely is the crux and the germ of one of Abel’s greatest contributions to mathematics. The matter is quite simple to anyone who has had a school course in trigonometry; to obviate tedious explanations of elementary matters this much will be assumed in what follows presently.

  For those who have forgotten all about trigonometry, however, the essence, the methodology, of Abel’s epochal advance can be analogized thus. We alluded to the cart and the horse. The frowsy proverb about putting the cart before the horse describes what Legendre did; Abel saw that if the cart was to move forward the horse should precede it. To take another instance: Francis Galton, in his statistical studies of the relation between poverty and chronic drunkenness, was led, by his impartial mind, to a reconsideration of all the self-righteous platitudes by which indignant moralists and economic crusaders with an axe to grind evaluate such social phenomena. Instead of assuming that people are depraved because they drink to excess, Galton inverted this hypothesis and assumed temporarily that people drink to excess because they have inherited no moral guts from their ancestors, in short, because they are depraved. Brushing aside all the vaporous moralizings of the reformers, Galton took a firm grip on a scientific, unemotional, workable hypothesis to which he could apply the impartial machinery of mathematics. His work has not yet registered socially. For the moment we need note only that Galton, like Abel, inverted his problem—turned it upside-down and inside-out, back-end-to and foremost-end-backward. Like Hiawatha and his fabulous mittens, Galton put the skinside inside and the inside outside.

  All this is far from being obvious or a triviality. It is one of the most powerful methods of mathematical discovery (or invention) ever devised, and Abel was the first human being to use it consciously as an engine of research. “You must always invert,” as Jacobi said when asked the secret of his mathematical discoveries. He was recalling what Abel and he had done. If the solution of a problem becomes hopelessly involved, try turning the problem backwards, put the quaesita for the data and vice versa. Thus if we find Cardan’s character incomprehensible when we think of him as a son of his father, shift the emphasis, invert it, and see what we get when we analyse Cardan’s father as the begetter and endower of his son. Instead of studying “inheritance” concentrate on “endowing.” To return to those who remember some trigonometry.

  Suppose mathematicians had been so blind as not to see that sin x, cos x and the other direct trigonometric functions are simpler to use, in the addition formulas and elsewhere, than the inverse functions sin-1 x, cos-1 x. Recall the formula sin (x + y) in terms of sines and cosines of x and y, and contrast it with the formula for sin-1 (x + y) in terms of x and y. Is not the former incomparably simpler, more elegant, more “natural” than the latter? Now, in the integral calculus, the inverse trigonometric functions present themselves naturally as definite integrals of simple algebraic irrationalities (second degree); such integrals appear when we seek to find the length of an arc of a circle by means of the integral calculus. Suppose the inverse trigonometric functions had first presented themselves this way. Would it not have been “more natural” to consider the inverses of these functions, that is, the familiar trigonometric functions themselves as the given functions to be studied and analyzed? Undoubtedly; but in shoals of more advanced problems, the simplest of which is that of finding the length of the arc of an ellipse by the integral calculus, the awkward inverse “elliptic” (not “circular,” as for the arc of a circle) functions presented themselves first. It took Abel to see that these functions should be “inverted” and studied, precisely as in the case of sin x, cos x instead of sin-1 x, cos-1 x. Simple, was it not? Yet Legendre, a great mathematician, spent more than forty years over his “elliptic integrals” (the awkward “inverse functions” of his problem) without ever once suspecting that he should invert.III This extremely simple, uncommonsensical way of looking at an apparently simple but profoundly recondite problem was one of the greatest mathematical advances of the nineteenth century.

  All this however was but the beginning, although a sufficiently tremendous beginning—like Kipling’s dawn coming up like thunder—of what Abel did in his magnificent theorem and in his work on elliptic functions. The trigonometric or circular functions have a single real period, thus sin (x + 2π) = sin x, etc. Abel discovered that his new functions provided by the inversion of an elliptic integral have precisely two periods, whose ratio is imaginary. After that, Abel’s followers in this direction—Jacobi, Rosenhain, Weierstrass, Riemann, and many more—mined deeply into Abel’s great theorem and by carrying on and extending his ideas discovered functions of n variables having 2n periods. Abel himself carried the exploitation of his discoveries far. His successors have applied all this work to geometry, mechanics, parts of mathematical physics, and other tracts of mathematics, solving important problems which, without this work initiated by Abel, would have been unsolvable.

  * * *

  While in Paris Abel consulted good physicians for what he thought was merely a persistent cold. He was told that he had tuberculosis of the lungs. He refused to believe it, wiped the mud of Paris off his boots, and returned to Berlin for a short visit. His funds were running low; about seven dollars was the extent of his fortune. An urgent letter brought a loan from Holmboë after some delay. It must not be supposed that Abel was a chronic borrower on no prospects. He had good reason for believing that he should have a paying job when he got home. Moreover, money was still owed to him. On Holmboë’s loan of about sixty dollars Abel existed and researched from March till May, 1827. Then, all his resources exhausted, he turned homeward and arrived in Kristiania completely destitute.

  But all was soon to be rosy, he hoped. Surely the University job would be forthcoming now. His genius had begun to be recognized. There was a vacancy. Abel did not get it. Holmboë reluctantly took the vacant chair which he had intended Abel to fill only after the governing board threatened to import a foreigner if Holmboë did not take it. Holmboë was in no way to blame. It was assumed that Holmboë would be a better teacher than Abel, although Abel had amply demonstrated his ability to teach. Anyone familiar with the current American pedagogical theory, fostered by professional Schools of Education, that the less a man knows about what he is to teach the better he will teach it, will understand the situation perfectly.

  Nevertheless things did brighten up. The University paid Abel the balance of what it owed on his travel money and Holmboë sent pupils his way. The professor of astronomy took a leave of absence and suggested that Abel be employed to carry part of his work. A well-to-do couple, the Schjeldrups, took him in and treated him as if he were their own son. But with all this he could not free himself of the burden of his dependents. To the last they clung to him, leaving h
im practically nothing for himself, and to the last he never uttered an impatient word.

  By the middle of January, 1829, Abel knew that he had not long to live. The evidence of a hemorrhage is not to be denied. “I will fight for my life!” he shouted in his delirium. But in more tranquil moments, exhausted and trying to work, he drooped “like a sick eagle looking at the sun,” knowing that his weeks were numbered.

  Abel spent his last days at Froland, in the home of an English family where his fiancée (Crelly Kemp) was governess. His last thoughts were for her future, and he wrote to his friend Kielhau, “She is not beautiful; she has red hair and freckles, but she is an admirable woman.” It was Abel’s wish that Crelly and Kielhau should marry after his death; and although the two had never met, they did as Abel had half-jokingly proposed. Toward the last Crelly insisted on taking care of Abel without help, “to possess these last moments alone.” Early in the morning of the sixth of April, 1829, he died, aged twenty six years, eight months.

  Two days after Abel’s death Crelle wrote to say that his negotiations had at last proved successful and that Abel would be appointed to the professorship of mathematics in the University of Berlin.

  * * *

  I. “ . . . ce qu’on peut toujours faire d’un problème quelconque” is what Abel says. This seems a trifle too optimistic; at least for ordinary mortals. How would the method be applied to Fermat’s Last Theorem?

  II. Libri, a soi-disant mathematician, who saw the work through the press adds, “by permission of the Academy,” a smug footnote acknowledging the genius of the lamented Abel. This is the last straw; the Academy might have come out with all the facts or have held its official tongue. But at all costs the honor and dignity of a stuffed shirt must be upheld. Finally it may be recalled that valuable manuscripts and books had an unaccountable trick of vanishing when Libri was round.

  III. In ascribing priority to Abel, rather than “joint discovery” to Abel and Jacobi, in this matter, I have followed Mittag-Leffler. From a thorough acquaintance with all the published evidence, I am convinced that Abel’s claim is indisputable, although Jacobi’s compatriots argue otherwise.

  CHAPTER EIGHTEEN

  The Great Algorist

  JACOBI

  It is the increasingly pronounced tendency of modern analysis to substitute ideas for calculation; nevertheless there are certain branches of mathematics where calculation conserves its rights.—P. G. LEJEUNE DIRICHLET

  THE NAME JACOBI appears frequently in the sciences, not always meaning the same man. In the 1840’s one very notorious Jacobi—M. H.—had a comparatively obscure brother, C. G. J., whose reputation then was but a tithe of M. H.’s. Today the situation is reversed: C. G. J. is immortal—or seemingly so, while M. H. is rapidly receding into the obscurity of limbo. M. H. achieved fame as the founder of the fashionable quackery of galvanoplastics; C. G. J.’s much narrower but also much higher reputation is based on mathematics. During his lifetime the mathematician was always being confused with his more famous brother, or worse, being congratulated for his involuntary kinship to the sincerely deluded quack. At last C. G. J. could stand it no longer. “Pardon me, beautiful lady,’ he retorted to an enthusiastic admirer of M. H. who had complimented him on having so distinguished a brother, “but / am my brother.” On other occasions C. G. J. would blurt out, “I am not his brother, he is mine.’ 1 There is where fame has left the relationship today.

  Carl Gustav Jacob Jacobi, born at Potsdam, Prussia, Germany, on December 10, 1804, was the second son of a prosperous banker, Simon Jacobi, and his wife (family name Lehmann). There were in all four children, three boys, Moritz, Carl, and Eduard, and a girl, Therese. Carl’s first teacher was one of his maternal uncles, who taught the boy classics and mathematics, preparing him to enter the Potsdam Gymnasium in 1816 in his twelfth year. From the first Jacobi gave evidence of the “universal mind” which the rector of the Gymnasium declared him to be on his leaving the school in 1821 to enter the University of Berlin. Like Gauss, Jacobi could easily have made a high reputation in philology had not mathematics attracted him more strongly. Having seen that the boy had mathematical genius, the teacher (Heinrich Bauer) let Jacobi work by himself—after a prolonged tussle in which Jacobi rebelled at learning mathematics by rote and by rule.

  Young Jacobi’s mathematical development was in some respects curiously parallel to that of his greater rival Abel. Jacobi also went to the masters; the works of Euler and Lagrange taught him algebra and the calculus, and introduced him to the theory of numbers. This earliest self-instruction was to give Jacobi’s first outstanding work—in elliptic functions—its definite direction, for Euler, the master of ingenious devices, found in Jacobi his brilliant successor. For sheer manipulative ability in tangled algebra Euler and Jacobi have had no rival, unless it be the Indian mathematical genius, Srinivasa Ramanujan, in our own century. Abel also could handle formulas like a master when he wished, but his genius was more philosophical, less formal than Jacobi’s. Abel is closer to Gauss in his insistence upon rigor than Jacobi was by nature—not that Jacobi’s work lacked rigor, for it did not, but its inspiration appears to have been formalistic rather than rigoristic.

  Abel was two years older than Jacobi. Unaware that Abel had attacked the general quintic in 1820, Jacobi in the same year attempted a solution, reducing the general quintic to the form x5 −10q2x = p and showing that the solution of this equation would follow from that of a certain equation of the tenth degree. Although the attempt was abortive it taught Jacobi a great deal of algebra and he ascribed considerable importance to it as a step in his mathematical education. But it does not seem to have occurred to him, as it did to Abel, that the general quintic might be unsolvable algebraically. This oversight, or lack of imagination, or whatever we wish to call it, on Jacobi’s part is typical of the difference between him and Abel. Jacobi, who had a magnificently objective mind and not a particle of envy or jealousy in his generous nature, himself said of one of Abel’s masterpieces, “It is above my praises as it is above my own works.”

  Jacobi’s student days at Berlin lasted from April, 1821, to May, 1825. During the first two years he spent his time about equally between philosophy, philology, and mathematics. In the philological seminar Jacobi attracted the favorable attention of P. A. Boeckh, a renowned classical scholar who brought out (among other works) a fine edition of Pindar. Boeckh, luckily for mathematics, failed to convert his most promising pupil to classical studies as a life interest. In mathematics not much was offered for an ambitious student and Jacobi continued his private study of the masters. The university lectures in mathematics he characterized briefly and sufficiently as twaddle. Jacobi was usually blunt and to the point, although he knew how to be as subservient as any courtier when trying to insinuate some deserving mathematical friend into a worthy position.

  While Jacobi was diligently making a mathematician of himself Abel was already well started on the very road which was to lead Jacobi to fame. Abel had written to Holmboë on August 4, 1823, that he was busy with elliptic functions: “This little work, you will recall, deals with the inverses of the elliptic transcendents, and I proved something [that seemed] impossible; I begged Degen to read it as soon as he could from one end to the other, but he could find no false conclusion, nor understand where the mistake was; God knows how I shall get myself out of it.” By a curious coincidence Jacobi at last made up his mind to put his all on mathematics almost exactly when Abel wrote this. Two years’ difference in the ages of young men around twenty (Abel was twenty one, Jacobi nineteen) count for more than two decades of maturity. Abel got a tremendous start but Jacobi, unaware that he had a competitor in the race, soon caught up. Jacobi’s first great work was in Abel’s field of elliptic functions. Before considering this we shall outline his busy life.

  Having decided to go into mathematics for all he was worth, Jacobi wrote to his uncle Lehmann his estimate of the labor he had undertaken. “The huge colossus which the works of Euler, Lagran
ge, and Laplace have raised demands the most prodigious force and exertion of thought if one is to penetrate into its inner nature and not merely rummage about on its surface. To dominate this colossus and not to fear being crushed by it demands a strain which permits neither rest nor peace till one stands on top of it and surveys the work in its entirety. Then only, when one has comprehended its spirit, is it possible to work justly and in peace at the completion of its details.”

  With this declaration of willing servitude Jacobi forthwith became one of the most terrific workers in the history of mathematics. To a timid friend who complained that scientific research is exacting and likely to impair bodily health, Jacobi retorted:

  “Of course! Certainly I have sometimes endangered my health by overwork, but what of it? Only cabbages have no nerves, no worries. And what do they get out of their perfect wellbeing?”

  In August, 1825, Jacobi received his Ph.D. degree for a dissertation on partial fractions and allied topics. There is no need to explain the nature of this—it is not of any great interest and is now a detail in the second course of algebra or the integral calculus. Although Jacobi handled the general case of his problem and showed considerable ingenuity in manipulating formulas, it cannot be said that the dissertation exhibited any marked originality or gave any definite hint of the author’s superb talent. Concurrently with his examination for the Ph.D. degree, Jacobi rounded off his training for the teaching profession.

  After his degree Jacobi lectured at the University of Berlin on the applications of the calculus to curved surfaces and twisted curves (roughly, curves determined by the intersections of surfaces). From the very first lectures it was evident that Jacobi was a born teacher. Later, when he began developing his own ideas at an amazing speed, he became the most inspiring mathematical teacher of his time.