Men of Mathematics

by E. T. Bell

  * * *

  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.

  CHAPTER NINETEEN

  An Irish Tragedy

  HAMILTON

  In mathematics he was greater

  Than Tycho Brahe or Erra Pater;

  For he by geometric scale

  Could take the size of pots of ale.

  —SAMUEL BUTLER

  WILLIAM 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?

  By thirteen William was able to brag that he had mastered one language for each year he had lived. At fourteen he composed a flowery welcome in Persian to the Persian Ambassador, then visiting Dublin, and had it transmitted to the astonished potentate. Wishing to follow up his advantage and slay the already slain, young Hamilton called on the Ambassador, but that wily oriental, forewarned by his faithful secretary, “much regretted that on account of a bad headache he was unable to receive me [Hamilton] personally.” Perhaps the Ambassador had not yet recovered from the official banquet, or he may have read the letter. In translation at least it is pretty awful—just the sort of thing a boy of fourteen, taking himself with devastating seriousness and acquainted with all the stickiest and most bombastic passages of the Persian poets, might imagine a sophisticated oriental out on a wild Irish spree would relish as a pick-me-up the morning after. Had young Hamilton really wished to view the Ambassador he should have sent in a salt herring, not a Persian poem.

  Except for his amazing ability, the maturity of his conversation and his poetical love of nature in all her moods, Hamilton was like any other healthy boy. He delighted in swimming and had none of the grind’s interesting if somewhat repulsive pallor. His disposition was genial and his temper—rather unusually so for a sturdy Irish boy—invariably even. In later life however Hamilton showed his Irish by challenging a detractor—who had called him a liar—to mortal combat. But the affair was amicably arranged by Hamilton’s second, and Sir William cannot be legitimately counted as one of the great mathematical duellists. In other respects young Hamilton was not a normal boy. The infliction of pain or suffering on beast or man he would not tolerate. All his life Hamilton loved animals and, what is regrettably rarer, respected them as equals.

  Hamilton’s redemption from senseless devotion to useless languages began when he was twelve and was completed before he was fourteen. The humble instrument selected by Providence to turn Hamilton from the path of error was the American calculating boy, Zerah Colburn (1804-1839), who at the time had been attending Westminster School in London. Colburn and Hamilton were brought together in the expectation that the young Irish genius would be able to penetrate the secret of the American’s methods, whic
h Colburn himself did not fully understand (as was seen in the chapter on Fermat). Colburn was entirely frank in exposing his tricks to Hamilton, who in his turn improved upon what he had been shown. There was but little abstruse or remarkable about Colburn’s methods. His feats were largely a matter of memory. Hamilton’s acknowledgment of Colburn’s influence occurs in a letter written when he was seventeen (August, 1822) to his cousin Arthur.

  By the age of seventeen Hamilton had mastered mathematics through the integral calculus and had acquired enough mathematical astronomy to be able to calculate eclipses. He read Newton and Lagrange. All this was his recreation; the classics were still his serious study, although only a second love. What is more important, he had already made “some curious discoveries,” as he wrote to his sister Eliza.

  The discoveries to which Hamilton refers are probably the germs of his first great work, that on systems of rays in optics. Thus in his seventeenth year Hamilton had already begun his career of fundamental discovery. Before this he had brought himself to the attention of Dr. Brinkley, Professor of Astronomy at Dublin, by the detection of an error in Laplace’s attempted proof of the parallelogram of forces.

  * * *

  Hamilton never attended any school before going to the University but received all his preliminary training from his uncle and by private study. His forced devotion to the classics in preparation for the entrance examinations to Trinity College, Dublin, did not absorb all of his time, for on May 31, 1823, he writes to his cousin Arthur, “In Optics I have made a very curious discovery—at least it seems so to me . . . .”

  If, as has been supposed, this refers to the “characteristic function,” which Hamilton will presently describe for us, the discovery marks its author as the equal of any mathematician in history for genuine precocity. On July 7, 1823, young Hamilton passed, easily first out of one hundred candidates, into Trinity College. His fame had preceded him, and as was only to be expected, he quickly became a celebrity; indeed his classical and mathematical prowess, while he was yet an undergraduate, excited the curiosity of academic circles in England and Scotland as well as in Ireland, and it was even declared by some that a second Newton had arrived. The tale of his undergraduate triumphs can be imagined—he carried off practically all the available prizes and obtained the highest honors in both classics and mathematics. But more important than all these triumphs, he completed the first draft of Part I of his epoch-making memoir on systems of rays. “This young man,” Dr. Brinkley remarked, when Hamilton presented his memoir to the Royal Irish Academy, “I do not say will be, but is, the first mathematician of his age.”

  Even his laborious drudgeries to sustain his brilliant academic record and the hours spent more profitably on research did not absorb all of young Hamilton’s superabundant energies. At nineteen he experienced the first of his three serious love affairs. Being conscious of his own “unworthiness”—especially as concerned his material prospects—William contented himself with writing poems to the young lady, with the usual result: a solider, more prosaic man married the girl. Early in May, 1825, Hamilton learned from his sweetheart’s mother that his love had married his rival. Some idea of the shock he experienced can be inferred from the fact that Hamilton, a deeply religious man to whom suicide was a deadly sin, was tempted to drown himself. Fortunately for science he solaced himself with another poem. All his life Hamilton was a prolific versifier. But his true poetry, as he told his friend and ardent admirer, William Wordsworth, was his mathematics. From this no mathematician will dissent.

  We may dispose here of Hamilton’s lifelong friendships with some of the shining literary lights of his day—the poets Wordsworth, Southey, and Coleridge, of the so-called Lake School, Aubrey de Vere, and the didactic novelist Maria Edgeworth—a litteratrice after Hamilton’s own pious heart. Wordsworth and Hamilton first met on the latter’s trip of September, 1827, to the English Lake District. Having “waited on Wordsworth at tea,” Hamilton oscillated back and forth with the poet all night, each desperately trying to see the other home. The following day Hamilton sent Wordsworth a poem of ninety iron lines which the poet himself might have warbled in one of his heavier flights. Naturally Wordsworth did not relish the eager young mathematician’s unconscious plagiarism, and after damning it with faint praise, proceeded to tell the hopeful author—at great length—that “the workmanship (what else could be expected from so young a writer?) is not what it ought to be.” Two years later, when Hamilton was already installed as astronomer at the Dunsink Observatory, Wordsworth returned the visit. Hamilton’s sister Eliza, on being introduced to the poet, felt herself “involuntarily parodying the first lines of his own poem Yarrow Visited—

  And this is Wordsworth! this the man

  Of whom my fancy cherished

  So faithfully a waking dream,

  An image that hath perished!”

  * * *

  One great benefit accrued from Wordsworth’s visit: Hamilton realized at last that “his path must be the path of Science, and not that of Poetry; that he must renounce the hope of habitually cultivating both, and that, therefore, he must brace himself up to bid a painful farewell to Poetry.” In short, Hamilton grasped the obvious truth that there was not a spark of poetry in him, in the literary sense. Nevertheless he continued to versify all his life. Wordsworth’s opinion of Hamilton’s intellect was high. In fact he graciously said (in effect) that only two men he had ever known gave him a feeling of inferiority, Coleridge and Hamilton.

  Hamilton did not meet Coleridge till 1832, when the poet had practically ceased to be anything but a spurious copy of a mediocre German metaphysician. Nevertheless each formed a high estimate of the other’s capacity, as Hamilton had for long been a devoted student of Kant in the original. Indeed philosophical speculation always fascinated Hamilton, and at one time he declared himself a wholehearted believer—intellectually, but not intestinally—in Berkeley’s devitalized idealism. Another bond between the two was their preoccupation with the theological side of philosophy (if there is such a side), and Coleridge favored Hamilton with his half-digested ruminations on the Holy Trinity, by which the devout mathematician set considerable store.

  * * *

  The close of Hamilton’s undergraduate career at Trinity College was even more spectacular than its beginning; in fact it was unique in university annals. Dr. Brinkley resigned his professorship of astronomy to become Bishop of Cloyne. According to the usual British custom the vacancy was advertized, and several distinguished astronomers, including George Biddell Airy (1801-1892), later Astronomer Royal of England, sent in their credentials. After some discussion the Governing Board passed over all the applicants and unanimously elected Hamilton, then (1827) an undergraduate of twenty two, to the professorship. Hamilton had not applied. “Straight for him was the path of gold” now, and Hamilton resolved not to disappoint the hopes of his enthusiastic electors. Since the age of fourteen he had had a passion for astronomy, and once as a boy he had pointed out the Observatory on its hill at Dunsink, commanding a beautiful view, as the place of all others where he would like to live were he free to choose. He now, at the age of twenty two, had his ambition by the bit; all he had to do was to ride straight ahead.

  He started brilliantly. Although Hamilton was no practical astronomer, and although his assistant observer was incompetent, these drawbacks were not serious. From its situation the Dunsink Observatory could never have cut any important figure in modern astronomy, and Hamilton did wisely in putting his major efforts on his mathematics. At the age of twenty three he published the completion of the “curious discoveries” he had made as a boy of seventeen, Part I of A Theory of Systems of Rays, the great classic which does for optics what Lagrange’s Mécanique analytique does for mechanics and which, in Hamilton’s own hands, was to be extended to dynamics, putting that fundamental science in what is perhaps its ultimate, perfect form.

  The techniques which Hamilton introduced into applied mathematics in this, his first master
piece, are today indispensable in mathematical physics, and it is the aim of many workers in particular branches of theoretical physics to sum up the whole of a theory in a Hamiltonian principle. This magnificent work is that which caused Jacobi, fourteen years later at the British Association meeting at Manchester in 1842, to assert that “Hamilton is the Lagrange of your country”—(meaning of the English-speaking race). As Hamilton himself took great pains to describe the essence of his new methods in terms comprehensible to non-specialists, we shall quote from his own abstract presented to the Royal Irish Academy on April 23, 1827.

  “A Ray, in Optics, is to be considered here as a straight or bent or curved line, along which light is propagated; and a System of Rays as a collection or aggregate of such lines, connected by some common bond, some similarity of origin or production, in short some optical unity. Thus the rays which diverge from a luminous point compose one optical system, and, after they have been reflected at a mirror, they compose another. To investigate the geometrical relations of the rays of a system of which we know (as in these simple cases) the optical origin and history, to inquire how they are disposed among themselves, how they diverge or converge, or are parallel, what surfaces or curves they touch or cut, and at what angles of section, how they can be combined in partial pencils, and how each ray in particular can be determined and distinguished from every other, is to study that System of Rays. And to generalize this study of one system so as to become able to pass, without change of plan, to the study of other systems, to assign general rules and a general method whereby these separate optical arrangements may be connected and harmonised together, is to form a Theory of Systems of Rays. Finally, to do this in such a manner as to make available the powers of the modern mathesis, replacing figures by functions and diagrams by formulas, is to construct an Algebraic Theory of such Systems, or an Application of Algebra to Optics.