Men of Mathematics (65 page)

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
^-1
x 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).

*  *  *

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 Jacobi. 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.

*  *  *

Jacobi did not suffer an early death from overwork, as his lazier friends predicted that he should, but from smallpox (February 18, 1851) in his forty seventh year. In taking leave of this large-minded man we may quote his retort to the great French mathematical physicist Fourier, who had reproached both Abel and Jacobi for “wasting” their time on elliptic functions while there were still problems in heat-conduction to be solved.

“It is true,” Jacobi says, “that M. Fourier had the opinion that the principal aim of mathematics was public utility and the explanation of natural phenomena; but a philosopher like him should have known that the sole end of science is the honor of the human mind, and that under this title a question about numbers is worth as much as a question about the system of the world.”

If Fourier could revisit the glimpses of the moon he might be disgusted at what has happened to the analysis he invented for “public utility and the explanation of natural phenomena.” So far as mathematical
physics is concerned Fourier analysis today is but a detail in the infinitely vaster theory of boundary-value problems, and it is in the purest of pure mathematics that the analysis which Fourier invented finds its interest and its justification. Whether “the human mind” is honored by these modern researches may be put up to the experts—provided the behaviorists have left anything of the human mind to be honored.

I
. If
n
is odd, the number of ways is 8 times the sum of all the divisors of
n
(l and
n
included); if
n
is even, the number of ways is 24 times the sum of all the odd divisors of
n.

In mathematics he was greater

Than Tycho Brahe or Erra Pater;

For he by geometric scale

Could take the size of pots of ale.

—S
AMUEL
B
UTLER

W
ILLIAM ROWAN HAMILTON
is by long odds the greatest man of science that Ireland has produced. His nationality is emphasized because one of the driving impulses behind Hamilton's incessant activity was his avowed desire to put his superb genius to such uses as would bring glory to his native land. Some have claimed that he was of Scotch descent. Hamilton himself insisted that he was Irish, and it is certainly difficult for a Scot to see anything Scotch in Ireland's greatest and most eloquent mathematician.

Hamilton's father was a solicitor in Dublin, Ireland, where William, the youngest of three brothers and one sister, was born on August
3, 1805.
^I
The father was a first-rate business man with an “exuberant eloquence,” a religious zealot, and last, but unfortunately not least, a very convivial man, all of which traits he passed on to his gifted son. Hamilton's extraordinary intellectual brilliance was probably inherited from his mother, Sarah Hutton, who came of a family well known for its brains.

However, on the father's side, the swirling clouds of eloquence, “both of lips and pen,” which made the jolly toper the life of every party he graced with his reeling presence, condensed into something less gaseous in William's uncle, the Reverend James Hamilton, curate of the village of Trim (about twenty miles from Dublin). Uncle
James was in fact an inhumanly accomplished linguist—Greek, Latin, Hebrew, Sanskrit, Chaldee, Pali, and heaven knows what other heathen dialects, came to the tip of his tongue as readily as the more civilized languages of Continental Europe and Ireland. This polyglot fluency played no inconsiderable part in the early and extremely extensive miseducation of the hapless but eager William, for at the age of three, having already given signs of genius, he was relieved of his doting mother's affection and packed off by his somewhat stupid father to glut himself with languages under the expert tutelage of the supervoluble Uncle James.

Hamilton's parents had very little to do with his upbringing; his mother died when he was twelve, his father two years later. To James Hamilton belongs whatever credit there may be for having wasted young William's abilities in the acquisition of utterly useless languages and turning him out, at the age of thirteen, as one of the most shocking examples of a linguistic monstrosity in history. That Hamilton did not become an insufferable prig under his misguided parson-uncle's instruction testifies to the essential soundness of his Irish common sense. The education he suffered might well have made a permanent ass of even a humorous boy, and Hamilton had no humor.

The tale of Hamilton's infantile accomplishments reads like a bad romance, but it is true: at three he was a superior reader of English and was considerably advanced in arithmetic; at four he was a good geographer; at five he read and translated Latin, Greek, and Hebrew, and loved to recite yards of Dryden, Collins, Milton, and Homer—the last in Greek; at eight he added a mastery of Italian and French to his collection and extemporized fluently in Latin, expressing his unaffected delight at the beauty of the Irish scene in Latin hexameters when plain English prose offered too plebeian a vent for his nobly exalted sentiments; and finally, before he was ten he had laid a firm foundation for his extraordinary scholarship in oriental languages by beginning Arabic and Sanskrit.

The tally of Hamilton's languages is not yet complete. When William was three months under ten years old his uncle reports that “His thirst for the Oriental languages is unabated. He is now master of most, indeed of all except the minor and comparatively provincial ones. The Hebrew, Persian, and Arabic are about to be confirmed by the superior and intimate acquaintance with the Sanskrit, in which he is already a proficient. The Chaldee and Syriac he is grounded in,
also the Hindoostanee, Malay, Mahratta, Bengali, and others. He is about to commence the Chinese, but the difficulty of procuring books is very great. It cost me a large sum to supply him from London, but I hope the money was well expended.” To which we can only throw up our hands and ejaculate Good God! What was the sense of it all?