Men of Mathematics (63 page)

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 him 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.

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. L
EJEUNE
D
IRICHLET

T
HE 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
x
⁵
−10q
²
x
=
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, Lagrange, 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.

Jacobi seems to have been the first regular mathematical instructor in a university to train students in research by lecturing on his own latest discoveries and letting the students see the creation of a new subject taking place before them. He believed in pitching young men into the icy water to learn to swim or drown by themselves. Many students put off attempting anything on their own account till they have mastered everything relating to their problem that has been done by others. The result is that but few ever acquire the knack of independent work. Jacobi combated this dilatory erudition. To drive home the point to a gifted but diffident young man who was always putting off doing anything until he had learned something more, Jacobi delivered himself of the following parable. “Your father would never have married, and you wouldn't be here now, if he had insisted on knowing
all
the girls in the world before marrying
one.”

Jacobi's entire life was spent in teaching and research except for one ghastly interlude, to be related, and occasional trips to attend scientific meetings in England and on the Continent, or forced vacations to recuperate after too intensive work. The chronology of his life is not very exciting—a professional scientist's seldom is except to himself.

Jacobi's talents as a teacher secured him the position of lecturer at the University of Königsberg in 1826 after only half a year in a similar
position at Berlin. A year later some results which Jacobi had published in the theory of numbers (relating to cubic reciprocity; see chapter on Gauss) excited Gauss' admiration. As Gauss was not an easy man to stir up, the Ministry of Education took prompt notice and promoted Jacobi over the heads of his colleagues to an assistant professorship—quite a step for a young man of twenty three. Naturally the men he had stepped over resented the promotion; but two years later
(1829)
when Jacobi published his first masterpiece,
Fundamenta Nova Theoriae Functionum Ellipticarum
(New Foundations of the Theory of Elliptic Functions) they were the first to say that no more than justice had been done and to congratulate their brilliant young colleague.

*  *  *

In 1832 Jacobi's father died. Up till this he need not have worked for a living. His prosperity continued about eight years longer, when the family fortune went to smash in 1840. Jacobi was cleaned out himself at the age of thirty six and in addition had to provide for his mother, also ruined.