Men of Mathematics (72 page)

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What happened on May 29th is not definitely known. Extracts from two letters suggest what is usually accepted as the truth: Galois had run foul of political enemies immediately after his release. These “patriots” were always spoiling for a fight, and it fell to the unfortunate Galois' lot to accommodate them in an affair of “honor.” In a “Letter to All Republicans,” dated 29 May, 1832, Galois writes:

“I beg patriots and my friends not to reproach me for dying otherwise than for my country. I die the victim of an infamous coquette. It is in a miserable brawl that my life is extinguished. Oh! why die for so trivial a thing, die for something so despicable! . . . Pardon for those who have killed me, they are of good faith.” In another letter to two unnamed friends: “I have been challenged by two patriots—it was impossible for me to refuse. I beg your pardon for having advised neither of you. But my opponents had put me on my honor not to warn any patriot. Your task is very simple: prove that I fought in spite of myself, that is to say after having exhausted every means of accommodation. . . . Preserve my memory since fate has not given me life enough for my country to know my name. I die your friend

E. G
ALOIS.”

*  *  *

These were the last words he wrote. All night, before writing these letters, he had spent the fleeting hours feverishly dashing off his scientific last will and testament, writing against time to glean a few of the great things in his teeming mind before the death which he foresaw could overtake him. Time after time he broke off to scribble in the margin “I have not time; I have not time,” and passed on to the next frantically scrawled outline. What he wrote in those desperate last hours before the dawn will keep generations of mathematicians busy for hundreds of years. He had found, once and for all, the true solution of a riddle which had tormented mathematicians for centuries: under what conditions can an equation be solved? But this was only one thing of many. In this great work, Galois used the theory of
groups (see chapter on Cauchy) with brilliant success. Galois was indeed one of the great pioneers in this abstract theory, today of fundamental importance in all mathematics.

*  *  *

In addition to this distracted letter Galois entrusted his scientific executor with some of the manuscripts which had been intended for the Academy of Sciences. Fourteen years later, in
1846,
Joseph Liouville edited some of the manuscripts for the
Journal de Mathématiques pures et appliquées.
Liouville, himself a distinguished and original mathematician, and editor of the great
Journal,
writes as follows in his introduction:

“The principal work of Évariste Galois has as its object the conditions of solvability of equations by radicals. The author lays the foundations of a general theory which he applies in detail to equations whose degree is a prime number. At the age of sixteen, and while a student at the college of Louis-le-Grand . . . Galois occupied himself with this difficult subject.” Liouville then states that the referees at the Academy had rejected Galois' memoirs on account of their obscurity. He continues: “An exaggerated desire for conciseness was the cause of this defect which one should strive above all else to avoid when treating the abstract and mysterious matters of pure Algebra. Clarity is, indeed, all the more necessary when one essays to lead the reader farther from the beaten path and into wilder territory. As Descartes said, 'When transcendental questions are under discussion be transcendentally clear.' Too often Galois neglected this precept; and we can understand how illustrious mathematicians may have judged it proper to try, by the harshness of their sage advice, to turn a beginner, full of genius but inexperienced, back on the right road. The author they censured was before them, ardent, active; he could profit by their advice.

“But now everything is changed. Galois is no more! Let us not indulge in useless criticisms; let us leave the defects there and look at the merits.” Continuing, Liouville tells how he studied the manuscripts, and singles out one perfect gem for special mention.

“My zeal was well rewarded, and I experienced an intense pleasure at the moment when, having filled in some slight gaps, I saw the complete correctness of the method by which Galois proves, in particular, this beautiful theorem:
In order that an irreducible equation of
prime degree be solvable by radicals it is necessary and sufficient that all its roots be rational functions of any two of them”
^II

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Galois addressed his will to his faithful friend Auguste Chevalier, to whom the world owes its preservation. “My dear friend,” he began, “I have made some new discoveries in analysis.” He then proceeds to outline such as he has time for. They were epoch-making. He concludes: “Ask Jacobi or Gauss publicly to give their opinion, not as to the truth, but as to the importance of these theorems. Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.
Je t'embrasse avec effusion.
E. Galois.”

Confiding Galois! Jacobi was generous; what would Gauss have said? What did he say of Abel? What did he omit to say of Cauchy, or of Lobatchewsky? For all his bitter experience Galois was still a hopeful boy.

At a very early hour on the thirtieth of May,
1832,
Galois confronted his adversary on the “field of honor.” The duel was with pistols at twenty five paces. Galois fell, shot through the intestines. No surgeon was present. He was left lying where he had fallen. At nine o'clock a passing peasant took him to the Cochin Hospital. Galois knew he was about to die. Before the inevitable peritonitis set in, and while still in the full possession of his faculties, he refused the offices of a priest. Perhaps he remembered his father. His young brother, the only one of his family who had been warned, arrived in tears. Galois tried to comfort him with a show of stoicism. “Don't cry,” he said, “I need all my courage to die at twenty.”

Early in the morning of May
31, 1832,
Galois died, being then in the twenty first year of his age. He was buried in the common ditch of the South Cemetery, so that today there remains no trace of the grave of Évariste Galois. His enduring monument is his collected works. They fill sixty pages.

I
. That is, so far as actually published work is concerned up to the present (1936). Euler undoubtedly will surpass Cayley in bulk when the full edition of his works is finally printed.

II
. The significance of this theorem will be clear if the reader will glance through the extracts from Abel in Chapter 17.

The theory of Invariants sprang into existence under the strong hand of Cayley, but that it emerged finally a complete work of art, for the admiration of future generations of mathematicians, was largely owing to the flashes of inspiration with which Sylvester's intellect illuminated it.
—P. A. M
AC
M
AHON

“I
T IS DIFFICULT
to give an idea of the vast extent of modern mathematics. The word 'extent' is not the right one:
I
mean extent crowded with beautiful detail—not an extent of mere uniformity such as an objectless plain, but of a tract of beautiful country seen at first in the distance, but which will bear to be rambled through and studied in every detail of hillside and valley, stream, rock, wood, and flower. But, as for every thing else, so for a mathematical theory—beauty can be perceived but not explained.”

These words from Cayley's presidential address in
1883
to the British Association for the Advancement of Science might well be applied to his own colossal output. For prolific inventiveness Euler, Cauchy, and Cayley are in a class by themselves, with Poincaré (who died younger than any of the others) a far second. This applies only to the bulk of these men's work; its quality is another matter, to be judged partly by the frequency with which the ideas originated by these giants recur in mathematical research, partly by mere personal opinion, and partly by national prejudice.

Cayley's remarks about the vast extent of modern mathematics suggest that we confine our attention to some of those features of his own work which introduced distinctly new and far-reaching ideas. The work on which his greatest fame rests is in the theory of invariants and what grew naturally out of that vast theory of which he, brilliantly sustained by his friend Sylvester, was the originator and unsurpassed developer. The concept of invariance is of great importance for modern physics, particularly in the theory of relativity, but
this is not its chief claim to attention. Physical theories are notoriously subject to revision and rejection; the theory of invariance as a permanent addition to pure mathematical thought appears to rest on firmer ground.

Another of the ideas originated by Cayley, that of the geometry of “higher space” (space of
n
dimensions) is likewise of present scientific significance but of incomparably greater importance as pure mathematics. Similarly for the theory of matrices, again an invention of Cayley's. In non-Euclidean geometry Cayley prepared the way for Klein's splendid discovery that the geometry of Euclid and the non-Euclidean geometries of Lobatchewsky and Riemann are, all three, merely different aspects of a more general kind of geometry which includes them as special cases. The nature of these contributions of Cayley's will be briefly indicated after we have sketched his life and that of his friend Sylvester.

The lives of Cayley and Sylvester should be written simultaneously, if that were possible. Each is a perfect foil to the other, and the life of each, in large measure, supplies what is lacking in that of the other. Cayley's life was serene; Sylvester, as he himself bitterly remarks, spent much of his spirit and energy “fighting the world.” Sylvester's thought was at times as turbulent as a millrace; Cayley's was always strong, steady, and unruffled. Only rarely did Cayley permit himself the printed expression of anything less severe than a precise mathematical statement—the simile quoted at the beginning of this chapter is one of the rare exceptions; Sylvester could hardly talk about mathematics without at once becoming almost orientally poetic, and his unquenchable enthusiasm frequently caused him to go off half-cocked. Yet these two became close friends and inspired one another to some of the best work that either of them did, for example in the theories of invariants and matrices (described later).

With two such temperaments it is not surprising that the course of friendship did not always run smoothly. Sylvester was frequently on the point of exploding; Cayley sat serenely on the safety valve, confident that his excitable friend would presently cool down, when he would calmly resume whatever they had been discussing as if Sylvester had never blown off, while Sylvester for his part ignored his hotheaded indiscretion—till he got himself all steamed up for another. In many ways this strangely congenial pair were like a honeymoon couple, except that one party to the friendship never lost his temper.
Although Sylvester was Cayley's senior by seven years, we shall begin with Cayley. Sylvester's life breaks naturally into the calm stream of Cayley's like a jagged rock in the middle of a deep river.

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Arthur Cayley was born on August 16, 1821, at Richmond, Surrey, the second son of his parents, then residing temporarily in England. On his father's side Cayley traced his descent back to the days of the Norman Conquest (1066) and even before, to a baronial estate in Normandy. The family was a talented one which, like the Darwin family, should provide much suggestive material for students of heredity. His mother was Maria Antonia Doughty, by some said to have been of Russian origin. Cayley's father was an English merchant engaged in the Russian trade; Arthur was born during one of the periodical visits of his parents to England.

In 1829, when Arthur was eight, the merchant retired, to live thenceforth in England. Arthur was sent to a private school at Black-heath and later, at the age of fourteen, to King's College School in London. His mathematical genius showed itself very early. The first manifestations of superior talent were like those of Gauss; young Cayley developed an amazing skill in long numerical calculations which he undertook for amusement. On beginning the formal study of mathematics he quickly outstripped the rest of the school. Presently he was in a class by himself, as he was later when he went up to the University, and his teachers agreed that the boy was a born mathematician who should make mathematics his career. In grateful contrast to Galois' teachers, Cayley's recognized his ability from the beginning and gave him every encouragement. At first the retired merchant objected strongly to his son's becoming a mathematician but finally, won over by the Principal of the school, gave his consent, his blessing, and his money. He decided to send his son to Cambridge.

Cayley began his university career at the age of seventeen at Trinity College, Cambridge. Among his fellow students he passed as “a mere mathematician” with a queer passion for novel-reading. Cayley was indeed a lifelong devotee of the somewhat stilted fiction, now considered classical, which charmed readers of the 1840's and '50's. Scott appears to have been his favorite, with Jane Austen a close second. Later he read Thackeray and disliked him; Dickens he could never bring himself to read. Byron's tales in verse excited his admiration, although his somewhat puritanical Victorian taste rebelled at
the best of the lot and he never made the acquaintance of that diverting scapegrace Don Juan. Shakespeare's plays, especially the comedies were a perpetual delight to him. On the more solid—or stodgier—side he read and reread Grote's interminable
History of Greece
and Macaulay's rhetorical
History of England.
Classical Greek, acquired at school, remained a reading-language for him all his life; French he read and wrote as easily as English, and his knowledge of German and Italian gave him plenty to read after he had exhausted the Victorian classics (or they had exhausted him). The enjoyment of solid fiction was only one of his diversions; others will be noted as we go.