Men of Mathematics (86 page)

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
n
⁴
),
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
ax
²
+
2bxy
+
cy
²
= 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
ax
²
+
2bxy
+
cy
²
= 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.