Men of Mathematics

by E. T. Bell

  Late in 1842, at the age of twenty, Hermite sat for the entrance examinations to the École Polytechnique. He passed, but only as sixty eighth in order of merit. Already he was a vastly better mathematician than some of the men who examined him were, or were ever to become. The humiliating outcome of this test made an impression on the young master which all the triumphs of his manhood never effaced.

  Hermite stayed only one year at the Polytechnique. It was not his head that disqualified him but his lame foot which, according to a ruling of the authorities, unfitted him for any of the positions open to successful students of the school. Perhaps it is as well that Hermite was thrown out; he was an ardent patriot and might easily have been embroiled in one or other of the political or military rows so precious to the effervescent French temperament. However, the year was by no means wasted. Instead of slaving over descriptive geometry, which he hated, Hermite spent his time on Abelian functions, then (1842) perhaps the topic of outstanding interest and importance to the great mathematicians of Europe. He had also made the acquaintance of Joseph Liouville (1809-1882), a first-rate mathematician and editor of the Journal des Mathématiques.

  Liouville recognized genius when he saw it. In passing it may be amusing to recall that Liouville inspired William Thomson, Lord Kelvin, the famous Scotch physicist, to one of the most satisfying definitions of a mathematician that has ever been given. “Do you know what a mathematician is?” Kelvin once asked a class. He stepped to the board and wrote

  Putting his finger on what he had written, he turned to the class. “A mathematician is one to whom that is as obvious as that twice two makes four is to you. Liouville was a mathematician.” Young Hermite’s pioneering work in Abelian functions, well begun before he was twenty one, was as far beyond Kelvin’s example in unobviousness as the example is beyond “twice two makes four.” Remembering the cordial welcome the aged Legendre had accorded the revolutionary work of the young and unknown Jacobi, Liouville guessed that Jacobi would show a similar generosity to the beginning Hermite. He was not mistaken.

  * * *

  The first of Hermite’s astonishing letters to Jacobi is dated from Paris, January, 1843. “The study of your [Jacobi’s] memoir on quad-ruply periodic functions arising in the theory of Abelian functions has led me to a theorem, for the division of the arguments [Variables] of these functions, analogous to that which you gave . . . to obtain the simplest expression for the roots of the equations treated by Abel. M. Liouville induced me to write to you, to submit this work to you; dare I hope, Sir, that you will be pleased to welcome it with all the indulgence it needs?” With that he plunges at once into the mathematics.

  To recall briefly the bare nature of the problem in question: the trigonometric functions are functions of one variable with one period, thus sin (x + 27r) = sin x, where x is the variable and 2 π is the period; Abel and Jacobi, by “inverting” the elliptic integrals, had discovered functions of one variable and two periods, say f(x + p + q) = f(x), where p, q are the periods (see Chapters 12, 18); Jacobi had discovered functions of two variables and four periods, say

  F(x+ a + b, y + c + d) = F(x, y),

  where a, b, c, d are the periods. A problem early encountered in trigonometry is to express sin or sin or generally sin where n is any given integer, in terms of sin x (and possibly other trigonometric functions of x). The corresponding problem for the functions of two variables and four periods was that which Hermite attacked. In the trigonometric problem we are finally led to quite simple equations; in Hermite’s incomparably more difficult problem the upshot is again an equation (of degree n4), and the unexpected thing about this equation is that it can be solved algebraically, that is, by radicals.

  Barred from the Polytechnique by his lameness, Hermite now cast longing eyes on the teaching profession as a haven where he might earn his living while advancing his beloved mathematics. The career should have been flung wide open to him, degree or no degree, but the inexorable rules and regulations made no exceptions. Red tape always hangs the wrong man, and it nearly strangled Hermite.

  Unable to break himself of his “pernicious originality,” Hermite continued his researches to the last possible moment when, at the age of twenty four, he abandoned the fundamental discoveries he was making to master the trivialities required for his first degrees (bachelor of letters and science). Two harder ordeals would normally have followed the first before the young mathematical genius could be certified as fit to teach, but fortunately Hermite escaped the last and worst when influential friends got him appointed to a position where he could mock the examiners. He passed his examinations (in 1847 −48) very badly. But for the friendliness of two of the inquisitors—Sturm and Bertrand, both fine mathematicians who recognized a fellow craftsman when they saw one—Hermite would probably not have passed at all. (Hermite married Bertrand’s sister Louise in 1848.)

  By an ironic twist of fate Hermite’s first academic success was his appointment in 1848 as an examiner for admissions to the very Polytechnique which had almost failed to admit him. A few months later he was appointed quiz master (répétiteur) at the same institution. He was now securely established in a niche where no examiner could get at him. But to reach this “bad eminence” he had sacrificed nearly five years of what almost certainly was his most inventive period to propitiate the stupidities of the official system.

  * * *

  Having finally satisfied or evaded his rapacious examiners, Hermite settled down to become a great mathematician. His life was peaceful and uneventful. In 1848 to 1850 he substituted for Libri at the Collège de France. Six years later, at the early age of thirty four, he was elected to the Institut (as a member of the Academy of Sciences). In spite of his world-wide reputation as a creative mathematician Hermite was forty seven before he obtained a suitable position: he was appointed professor only in 1869 at the École Normale and finally, in 1870, he became professor at the Sorbonne, a position which he held till his retirement twenty seven years later. During his tenure of this influential position he trained a whole generation of distinguished French mathematicians, among whom Émile Picard, Gaston Darboux, Paul Appell, Émile Borel, Paul Painlevé and Henri Poincaré, may be mentioned. But his influence extended far beyond France, and his classic works helped to educate his contemporaries in all lands.

  A distinguishing feature of Hermite’s beautiful work is closely allied to his repugnance to take advantage of his authoritative position to re-create all his pupils in his own image: this is the unstinted generosity which he invariably displays to his fellow mathematicians. Probably no other mathematician of modern times has carried on such a voluminous scientific correspondence with workers all over Europe as Hermite, and the tone of his letters is always kindly, encouraging, and appreciative. Many a mathematician of the second half of the nineteenth century owed his recognition to the publicity which Hermite gave his first efforts. In this, as in other respects, there is no finer character than Hermite in the whole history of mathematics. Jacobi was as generous—with the one exception of his early treatment of Eisenstein—but he had a tendency to sarcasm (often highly amusing, except possibly to the unhappy victim) which was wholly absent from Hermite’s genial wit. Such a man deserved the generous reply of Jacobi when the unknown young mathematician ventured to approach him with his first great work on Abelian functions. “Do not be put out, Sir,” Jacobi wrote, “if some of your discoveries coincide with old work of my own. As you must begin where I end, there is necessarily a small sphere of contact. In future, if you honor me with your communications, I shall have only to learn.”

  Encouraged by Jacobi, Hermite shared with him not only the discoveries in Abelian functions, but also sent him four tremendous letters on the theory of numbers, the first early in 1847. These letters, the first of which was composed when Hermite was only twenty four, break new ground (in what respect we shall indicate presently) and are sufficient alone to establish Hermite as a creative mathematician of the first rank. The generality
of the problems he attacked and the bold originality of the methods he devised for their solution assure Hermite’s remembrance as one of the born arithmeticians of history.

  The first letter opens with an apology. “Nearly two years have elapsed without my answering the letter full of goodwill which you did me the honor to write to me. Today I shall beg you to pardon my long negligence and express to you all the joy I felt in seeing myself given a place in the repertory of your works. [Jacobi has published parts of Hermite’s letter, with all due acknowledgment, in some work of his own.] Having been for long away from the work, I was greatly touched by such an attestation of your kindness; allow me, Sir, to believe that it will not desert me.” Hermite then says that another research of Jacobi’s has inspired him to his present efforts.

  If the reader will glance at what was said about uniform functions of a single variable in the chapter on Gauss (a uniform function takes only one value for each value of the variable), the following statement of what Jacobi had proved should be intelligible: a uniform function of only one variable with three distinct periods is impossible. That uniform functions of one variable exist having either one period or two periods is proved by exhibiting the trigonometric functions and the elliptic functions. This theorem of Jacobi’s, Hermite declares, gave him his own idea for the novel methods which he introduced into the higher arithmetic. Although these methods are too technical for description here, the spirit of one of them can be briefly indicated.

  Arithmetic in the sense of Gauss deals with properties of the rational integers 1, 2, 3, . . .; irrationals (like the square root of 2) are excluded. In particular Gauss investigated the integer solutions of large classes of indeterminate equations in two or three unknowns, for example as in ax2 + 2bxy + cy2 = m, where a, b, c, m are any given integers and it is required to discuss all integer solutions x, y of the equation. The point to be noted here is that the problem is stated and is to be solved entirely in the domain of the rational integers, that is, in the realm of discrete number. To fit analysis, which is adapted to the investigation of continuous number, to such a discrete problem would seem to be an impossibility, yet this is what Hermite did. Starting with a discrete formulation, he applied analysis to the problem, and in the end came out with results in the discrete domain from which he had started. As analysis is far more highly developed than any of the discrete techniques invented for algebra and arithmetic, Hermite’s advance was comparable to the introduction of modern machinery into a medieval handicraft.

  Hermite had at his disposal much more powerful machinery, both algebraic and analytic, than any available to Gauss when he wrote the Disquisitiones Arithmeticae. With Hermite’s own great invention these more modern tools enabled him to attack problems which would have baffled Gauss in 1800. At one stride Hermite caught up with general problems of the type which Gauss and Eisenstein had discussed, and he at least began the arithmetical study of quadratic forms in any number of unknowns. The general nature of the arithmetical “theory of forms” can be seen from the statement of a special problem. Instead of the Gaussian equation ax2 + 2bxy + cy2 = m of degree two in two unknowns (x, y), it is required to discuss the integer solutions of similar equations of degree n in s unknowns, where n, s are any integers, and the degree of each term on the left of the equation is n (not 2 as in Gauss’ equation). After stating how he had seen after much thought that Jacobi’s researches on the periodicity of uniform functions depend upon deeper questions in the theory of quadratic forms, Hermite outlines his programs.

  “But, having once arrived at this point of view, the problems—vast enough—which I had thought to propose to myself, seemed inconsiderable beside the great questions of the general theory of forms. In this boundless expanse of researches which Monsieur Gauss [Gauss was still living when Hermite wrote this, hence the polite “Monsieur”] has opened up to us, Algebra and the Theory of Numbers seem necessarily to be merged in the same order of analytical concepts, of which our present knowledge does not yet permit us to form an accurate idea.”

  He then makes a remark which, although not very clear, can be interpreted as meaning that the key to the subtle connections between algebra, the higher arithmetic, and certain parts of the theory of functions will be found in a thorough understanding of what sort of “numbers” are both necessary and sufficient for the explicit solution of all types of algebraic equations. Thus, for x3 −1 = 0, it is necessary and sufficient to understand for x5 + ax + b = 0, where a, b are any given numbers, what sort of a “number” x must be invented in order that x may be expressed explicitly in terms of a, b? Gauss of course gave one kind of answer: any root x is a complex number. But this is only a beginning. Abel proved that if only a finite number of rational operations and extractions of roots are permitted, then there is no explicit formula giving x in terms of a, b. We shall return to this question later; Hermite even at this early date (1848; he was then twenty six) seems to have had one of his greatest discoveries somewhere at the back of his head.

  In his attitude toward numbers Hermite was somewhat of a mystic in the tradition of Pythagoras and Descartes—the latter’s mathematical creed, as will appear in a moment, was essentially Pythagorean. In other matters, too, the gentle Hermite exhibited a marked leaning toward mysticism. Up to the age of forty three he was a tolerant agnostic, like so many French men of science of his time. Then, in 1856, he fell suddenly and dangerously ill. In this debilitated condition he was no match for even the least persistent evangelist, and the ardent Cauchy, who had always deplored his brilliant young friend’s open-mindedness on religious matters, pounced on the prostrate Hermite and converted him to Roman Catholicism. Thenceforth Hermite was a devout Catholic, and the practice of his religion gave him much satisfaction.

  Hermite’s number-mysticism is harmless enough and it is one of those personal things on which argument is futile. Briefly, Hermite believed that numbers have an existence of their own above all control by human beings. Mathematicians, he thought, are permitted now and then to catch glimpses of the superhuman harmonies regulating this ethereal realm of numerical existence, just as the great geniuses of ethics and morals have sometimes claimed to have visioned the celestial perfections of the Kingdom of Heaven.

  It is probably right to say that no reputable mathematician today who has paid any attention to what has been done in the past fifty years (especially the last twenty five) in attempting to understand the nature of mathematics and the processes of mathematical reasoning would agree with the mystical Hermite. Whether this modern skepticism regarding the other-worldliness of mathematics is a gain or a loss over Hermite’s creed must be left to the taste of the reader. What is now almost universally held by competent judges to be the wrong view of “mathematical existence” was so admirably expressed by Descartes in his theory of the eternal triangle that it may be quoted here as an epitome of Hermite’s mystical beliefs.

  “I imagine a triangle, although perhaps such a figure does not exist and never has existed anywhere in the world outside my thought. Nevertheless this figure has a certain nature, or form, or determinate essence which is immutable or eternal, which I have not invented and which in no way depends on my mind. This is evident from the fact that I can demonstrate various properties of this triangle, for example that the sum of its three interior angles is equal to two right angles, that the greatest angle is opposite the greatest side, and so forth. Whether I desire to or not, I recognize very clearly and convincingly that these properties are in the triangle although I have never thought about them before, and even if this is the first time I have imagined a triangle. Nevertheless no one can say that I have invented or imagined them.” Transposed to such simple “eternal verities” as 1 + 2 = 3, 2 + 2 = 4, Descartes’ everlasting geometry becomes Hermite’s superhuman arithmetic.

  One arithmetical investigation of Hermite’s, although rather technical, may be mentioned here as an example of the prophetic aspect of pure mathematics. Gauss, we recall, introduced complex integers (nu
mbers of the form a + bi, where a, b are rational integers and i denotes into the higher arithmetic in order to give the law of biquadratic reciprocity its simplest expression. Dirichlet and other followers of Gauss then discussed quadratic forms in which the rational integers appearing as variables and coefficients are replaced by Gaussian complex integers. Hermite passed to the general case of this situation and investigated the representation of integers in what are today called Hermitian forms. An example of such a form (for the special case of two complex variables x1, x2 and their “conjugates” instead of n variables) is

  in which the bar over a letter denoting a complex number indicates the conjugate of that number; namely, if x + iy is the complex number, its “conjugate” is x—iy; and the coefficients a11, a12, a21 a22 are such that aij = āji, for (i, j) = (1, 1), (1, 2), (2, 1), (2, 2), so that a12 and a21 are conjugates, and each of a11, a22 is its own conjugate (so that a11, a22 are real numbers). It is easily seen that the entire form is real (free of i) if all products are multiplied out, but it is most “naturally” discussed in the shape given.