by E. T. Bell
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.
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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.
Gauss all this time had been watching Jacobi’s phenomenal activity with more than a mere scientific interest, as many of Jacobi’s discoveries overlapped some of those of his own youth which he had never published. He had also (it is said) met the young man personally: Jacobi called on Gauss (no account of the visit has survived) in September, 1839, on his return trip to Königsberg after a vacation in Marienbad to recuperate from overwork. Gauss appears to have feared that Jacobi’s financial collapse would have a disastrous effect on his mathematics, but Bessel reassured him: “Fortunately such a talent cannot be destroyed, but I should have liked him to have the sense of freedom which money assures.”
The loss of his fortune had no effect whatever on Jacobi’s mathematics. He never alluded to his reverses but kept on working as assiduously as ever. In 1842 Jacobi and Bessel attended the meeting of the British Association at Manchester, where the German Jacobi and the Irish Hamilton met in the flesh. It was to be one of Jacobi’s greatest glories to continue the work of Hamilton in dynamics and, in a sense, to complete what the Irishman had abandoned in favor of a will-o-the-wisp (which will be followed when we come to it).
At this point in his career Jacobi suddenly attempted to blossom out into something showier than a mere mathematician. Not to interrupt the story of his scientific life when we take it up, we shall dispose here of the illustrious mathematician’s singular misadventures in politics.
The year following his return from the trip of 1842, Jacobi had a complete breakdown from overwork. The advancement of science in the 1840’s in Germany was in the hands of the benevolent princes and kings of the petty states which were later to coalesce into the German Empire. Jacobi’s good angel was the King of Prussia, who seems to have appreciated fully the honor which Jacobi’s researches conferred on the Kingdom. Accordingly, when Jacobi fell ill, the benevolent King urged him to take as long a vacation as he liked in the mild climate of Italy. After five months at Rome and Naples with Borchardt (whom we shall meet later in the company of Weierstrass) and Dirichlet, Jacobi returned to Berlin in June, 1844. He was now permitted to stay on in Berlin until his health should be completely restored but, owing to jealousies, was not given a professorship in the University, although as a member of the Academy he was permitted to lecture on anything he chose. Further, out of his own pocket, practically, the King granted Jacobi a substantial allowance.
After all this generosity on the part of the King one might think that Jacobi would have stuck to his mathematics. But on the utterly imbecilic advice of his physician he began meddling in politics “to benefit his nervous system.” If ever a more idiotic prescription was handed out by a doctor to a patient whose complaint he could not diagnose it has yet to be exhumed. Jacobi swallowed the dose. When the democratic upheaval of 1848 began to erupt Jacobi was ripe for office. On the advice of a friend—who, by the way, happened to be one of the men over whose head Jacobi had been promoted some twenty years before—the guileless mathematician stepped into the arena of politics with all the innocence of an enticingly plump missionary setting foot on a cannibal island. They got him.
The mildly liberal club to which his slick friend had introduced him ran Jacobi as their candidate for the May election of 1848. But he never saw the inside of parliament. His eloquence before the club convinced the wiser members that Jacobi was no candidate for them. Quite properly, it would seem, they pointed out that Jacobi, the King’s pensioner, might possibly be the liberal he now professed to be, but that it was more probable he was a trimmer, a turncoat, and a stoolpigeon for the royalists. Jacobi refuted these base insinuations in a magnificent speech packed with irrefutable logic—oblivious of the axiom that logic is the last thing on earth for which a practical politician has any use. They let him hang himself in his own noose. He was not elected. Nor was his nervous system benefited by the uproar over his candidacy which rocked the beer halls of Berlin to their cellars.
Worse was to come. Who can blame the Minister of Education for enquiring the following May whether Jacobi’s health had recovered sufficiently for him to return safely to Königsberg? Or who can wonder that his allowance from the King was stopped a few days later? After all even a King may be permitted a show of petulance when the mouth he tries to feed bites him. Nevertheless Jacobi’s desperate plight was enough to excite anybody’s sympathy. Married and practically penniless he had seven small children to support in addition to his wife. A friend in Gotha took in the wife and children, while Jacobi retired to a dingy hotel room to continue his researches.
He was now (1849) in his forty fifth year and, except for Gauss, the most famous mathematician in Europe. Hearing of his plight, the University of Vienna began angling for him. As an item of interest here, Littrow, Abel’s Viennese friend, took a leading part in the negotiations. At last, when a definite and generous offer was tendered, Alexander von Humboldt talked the sulky King round; the allowance was restored, and Jacobi was not permitted to rob Germany of her second greatest man. He remained in Berlin, once more in favor but definitely out of politics.
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The subject, elliptic functions, in which Jacobi did his first great work, has already been given what may seem like its share of space; for after all it is today more or less of a detail in the vaster theory of functions of a complex variable which, in its turn, is fading from the ever changing scene as a thing of living interest. As the theory of elliptic functions will be mentioned several times in succeeding chapters we shall attempt a brief justification of its apparently unmerited prominence.
No mathematician would dispute the claim of the theory of functions of a complex variable to have been one of the major fields of nineteenth
century mathematics. One of the reasons why this theory was of such importance may be repeated here. Gauss had shown that complex numbers are both necessary and sufficient to provide every algebraic equation with a root. Are any further, more general, kinds of “numbers” possible? How might such “numbers” arise?
Instead of regarding complex numbers as having first presented themselves in the attempt to solve certain simple equations, say x2 + 1 = 0, we may also see their origin in another problem of elementary algebra, that of factorization. To resolve x2—y2 into factors of the first degree we need nothing more mysterious than the positive and negative integers: (x2 —y2) = (x + y)(x—y). But the same problem for x2 + y2 demands “imaginaries”: . Carrying this up a step in one of many possible ways open, we might seek to resolve x2 + y2 + z2 into two factors of the first degree. Are the positives, negatives, and imaginaries sufficient? Or must some new kind of “number” be invented to solve the problem? The latter is the case. It was found that for the new “numbers” necessary the rules of common algebra break down in one important particular: it is no longer true that the order in which “numbers” are multiplied together is indifferent; that is, for the new numbers it is not true that a × b is equal to b × a. More will be said on this when we come to Hamilton. For the moment we note that the elementary algebraic problem of factorization quickly leads us into regions where complex numbers are inadequate.
How far can we go, what are the most general numbers possible, if we insist that for these numbers all the familiar laws of common algebra are to hold? It was proved in the latter part of the nineteenth century that the complex numbers x + iy, where x, y are real numbers and are the most general for which common algebra is true. The real numbers, we recall, correspond to the distances measured along a fixed straight line in either direction (positive, negative) from a fixed point, and the graph of a function f(x), plotted as y = f(x), in Cartesian geometry, gives us a picture of a function y of a real variable x. The mathematicians of the seventeenth and eighteenth centuries imagined their functions as being of this kind. But if the common algebra and its extensions into the calculus which they applied to their functions are equally applicable to complex numbers, which include the real numbers as a very degenerate case, it was but natural that many of the things the early analysts found were less than half the whole story possible. In particular the integral calculus presented many inexplicable anomalies which were cleared up only when the field of operations was enlarged to its fullest possible extent and functions of complex variables were introduced by Gauss and Cauchy.
The importance of elliptic functions in all this vast and fundamental development cannot be overestimated. Gauss, Abel, and Jacobi, by their extensive and detailed elaboration of the theory of elliptic functions, in which complex numbers appear inevitably, provided a testing ground for the discovery and improvement of general theorems in the theory of functions of a complex variable. The two theories seemed to have been designed by fate to complement and supplement one another—there is a reason for this, also for the deep connection of elliptic functions with the Gaussian theory of quadratic forms, which considerations of space force us to forego. Without the innumerable clues for a general theory provided by the special instances of more inclusive theorems occurring in elliptic functions, the theory of functions of a complex variable would have developed much more slowly than it did—Liouville’s theorem, the entire subject of multiple periodicity with its impact on the theory of algebraic functions and their integrals, may be recalled to mathematical readers. If some of these great monuments of nineteenth century mathematics are already receding into the mists of yesterday, we need only remind ourselves that Picard’s theorem on exceptional values in the neighborhood of an essential singularity, one of the most suggestive in current analysis, was first proved by devices originating in the theory of elliptic functions. With this partial summary of the reason why elliptic functions were important in the mathematics of the nineteenth century we may pass on to Jacobi’s cardinal part in the development of the theory.
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The history of elliptic functions is quite involved, and although of considerable interest to specialists, is not likely to appeal to the general reader. Accordingly we shall omit the evidence (letters of Gauss, Abel, Jacobi, Legendre, and others) on which the following bare summary is based.
First, it is established that Gauss anticipated both Abel and Jacobi by as much as twenty seven years in some of their most striking work. Indeed Gauss says that “Abel has followed exactly the same road that I did in 1798.” That this claim is just will be admitted by anyone who will study the evidence published only after Gauss’ death. Second, it seems to be agreed that Abel anticipated Jacobi in certain important details, but-that Jacobi made his great start in entire ignorance of his rival’s work.
A capital property of the elliptic functions is their double periodicity (discovered in 1825 by Abel): if E(x) is an elliptic function, then there are two distinct numbers, say p1, p2, such that
E(x + p1) = E(x), and E(x + p2) = E(x)
for all values of the variable x.
Finally, on the historical side, is the somewhat tragic part played by Legendre. For forty years he had slaved over elliptic integrals (not elliptic functions) without noticing what both Abel and Jacobi saw almost at once, namely that by inverting his point of view the whole subject would become infinitely simpler. Elliptic integrals first present themselves in the problem of finding the length of an arc of an ellipse. To what was said about inversion in connection with Abel the following statement in symbols may be added. This will bring out more clearly the point which Legendre missed.
If R(t) denotes a polynomial in t, an integral of the type
is called an elliptic integral if R(t) is of either the third or the fourth degree; if R(t) is of degree higher than the fourth, the integral is called Abelian (after Abel, some of whose greatest work concerned such integrals). If R(t) is of only the second degree, the integral can be calculated out in terms of elementary functions. In particular
(sin-1x is read, “an angle whose sine is x”). That is, if
we consider the upper limit, x, of the integral, as a function of the integral itself, namely of y. This inversion of the problem removed most of the difficulties which Legendre had grappled with for forty years. The true theory of these important integrals rushed forth almost of itself after this obstruction had been removed—like a log-jam going down the river after the king log has been snaked out.
When Legendre grasped what Abel and Jacobi had done he encouraged them most cordially, although he realized that their simpler approach (that of inversion) nullified what was to have been his own masterpiece of forty years’ labor. For Abel, alas, Legendre’s praise came too late, but for Jacobi it was an inspiration to surpass himself. In one of the finest correspondences in the whole of scientific literature the young man in his early twenties and the veteran in his late seventies strive to outdo one another in sincere praise and gratitude. The only jarring note is Legendre’s outspoken disparagement of Gauss, whom Jacobi vigorously defends. But as Gauss never condescended to publish his researches—he had planned a major work on elliptic functions when Abel and Jacobi anticipated him in publication—Legendre can hardly be blamed for holding a totally mistaken opinion. For lack of space we must omit extracts from this beautiful correspondence (the letters are given in full in vol. 1 of Jacobi’s Werke—in French).
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The joint creation with Abel of the theory of elliptic functions was only a small if highly important part of Jacobi’s huge output. Only to enumerate all the fields he enriched in his brief working life of less than a quarter of a century would take more space than can be devoted to one man in an account like the present, so we shall merely mention a few of the other great things he did.
Jacobi was the first to apply elliptic functions to the theory of numbers. This was to become a favorite diversion with some of the greatest mathematicians who followed J
acobi. It is a curiously recondite subject, where arabesques of ingenious algebra unexpectedly reveal hitherto unsuspected relations between the common whole numbers. It was by this means that Jacobi proved the famous assertion of Fermat that every integer 1, 2, 3, . . . is a sum of four integer squares (zero being counted as an integer) and, moreover, his beautiful analysis told him in how many ways any given integer may be expressed as such a sum.I
For those whose tastes are more practical we may cite Jacobi’s work in dynamics. In this subject, of fundamental importance in both applied science and mathematical physics, Jacobi made the first significant advance beyond Lagrange and Hamilton. Readers acquainted with quantum mechanics will recall the important part played in some presentations of that revolutionary theory by the Hamilton-Jacobi equation. His work in differential equations began a new era.
In algebra, to mention only one thing of many, Jacobi cast the theory of determinants into the simple form now familiar to every student in a second course of school algebra.
To the Newton-Laplace-Lagrange theory of attraction Jacobi made substantial contributions by his beautiful investigations on the functions which recur repeatedly in that theory and by applications of elliptic and Abelian functions to the attraction of ellipsoids.
Of a far higher order of originality is his great discovery in Abelian functions. Such functions arise in the inversion of an Abelian integral, in the same way that the elliptic functions arise from the inversion of an elliptic integral. (The technical terms were noted earlier in this chapter.) Here he had nothing to guide him, and for long he wandered lost in a maze that had no clue. The appropriate inverse functions in the simplest case are functions of two variables having four periods; in the general case the functions have n variables and 2n periods; the elliptic functions correspond to n = 1. This discovery was to nineteenth century analysis what Columbus’ discovery of America was to fifteenth century geography.