Men of Mathematics

by E. T. Bell

  All of these successes had a disastrous effect on Karl’s future. Old Weierstrass, like many parents, drew the wrong conclusion from his son’s triumphs. He “reasoned” as follows. Because the boy has won a cartload of prizes, therefore he must have a good mind—this much may be admitted; and because he has kept himself in pocket money by posting the honored female butter and ham merchant’s books efficiently, therefore he will be a brilliant bookkeeper. Now what is the acme of all bookkeeping? Obviously a government nest—in the higher branches of course—in the Prussian civil service. But to prepare for this exalted position, a knowledge of the law is desirable in order to pluck effectively and to avoid being plucked.

  As the grand conclusion of all this logic, paterfamilias Weierstrass shoved his gifted son, at the age of nineteen, headfirst into the University of Bonn to master the chicaneries of commerce and the quibblings of the law.

  Karl had more sense than to attempt either. He devoted his great bodily strength, his lightning dexterity and his keen mind almost exclusively to fencing and the mellow sociability that is induced by nightly and liberal indulgence in honest German beer. What a shocking example for ant-eyed Ph.D.’s who shrink from a spell of school-teaching lest their dim lights be dimmed forever! But to do what Weierstrass did, and get away with it, one must have at least a tenth of his constitution and not less than one tenth of one percent of his brains.

  Bonn found Weierstrass unbeatable. His quick eye, his long reach, his devilish accuracy, and his lightning speed in fencing made him an opponent to admire but not to touch. As a matter of historical fact he never was touched: no jagger scar adorned his cheeks, and in all his bouts he never lost a drop of blood. Whether or not he was ever put under the table in the subsequent celebrations of his numerous victories is not known. His discreet biographers are somewhat reticent on this important point, but to anyone who has ever contemplated one of Weierstrass’ mathematical masterpieces it is inconceivable that so strong a head as his could ever have nodded over a half-gallon stein. His four misspent years in the university were perhaps after all well spent.

  His experiences at Bonn did three things of the greatest moment for Weierstrass: they cured him of his father fixation without in any way damaging his affection for his deluded parent; they made him a human being capable of entering fully into the pathetic hopes and aspirations of human beings less gifted than himself—his pupils—and thus contributed directly to his success as probably the greatest mathematical teacher of all time; and last, the humorous geniality of his boyhood became a fixed life-habit. So the “student years” were not the loss his disappointed father and his fluttering sisters—to say nothing of the panicky Peter—thought they were when Karl returned, after four “empty” years at Bonn, without a degree, to the bosom of his wailing family.

  There was a terrific row. They lectured him—“sick of body and soul” as he was, possibly the result of not enough law, too little mathematics, and too much beer; they sat around and glowered at him and, worst of all, they began to discuss him as if he were dead: what was to be done with the corpse? Touching the law, Weierstrass had only one brief encounter with it at Bonn, but it sufficed: he astonished the Dean and his friends by his acute “opposition” of a candidate for the doctor degree in law. As for the mathematics at Bonn—it was inconsiderable. The one gifted man, Julius Plücker, who might have done Weierstrass some good was so busy with his manifold duties that he had no time to spare on individuals and Weierstrass got nothing out of him.

  But like Abel and so many other mathematicians of the first rank, Weierstrass had gone to the masters in the interludes between his fencing and drinking: he had been absorbing the Celestial Mechanics of Laplace, thereby laying the foundations for his lifelong interest in dynamics and systems of simultaneous differential equations. Of course he could get none of this through the head of his cultured, petty-official father, and his obedient brother and his dismayed sisters knew not what the devil he was talking about. The fact alone was sufficient: brother Karl, the genius of the timorous little family, on whom such high hopes of bourgeois respectability had been placed, had come home, after four years of rigid economy on father’s part, without a degree.

  At last—after weeks—a sensible friend of the family who had sympathized with Karl as a boy, and who had an intelligent amateur’s interest in mathematics, suggested a way out: let Karl prepare himself at the neighboring Academy of Münster for the state teachers’ examination. Young Weierstrass would not get a Ph.D. out of it, but his job as a teacher would provide a certain amount of evening leisure in which he could keep alive mathematically provided he had the right stuff in him. Freely confessing his “sins” to the authorities, Weierstrass begged the opportunity of making a fresh start. His plea was granted, and Weierstrass matriculated on May 22, 1839 at Münster to prepare himself for a secondary school-teaching career. This was a most important stepping stone to his later mathematical eminence, although at the time it looked like a total rout.

  What made all the difference to Weierstrass was the presence at Münster of Christof Gudermann (1798-1852) as Professor of Mathe matics. Gudermann at the time (1839) was an enthusiast for elliptic functions. We recall that Jacobi had published his Fundamenta nova in 1829. Although few are now familiar with Gudermann’s elaborate investigations (published at the instigation of Crelle in a series of articles in his Journal), he is not to be dismissed as contemptuously as it is sometimes fashionable to do merely because he is outmoded. For his time Gudermann had what appears to have been an original idea. The theory of elliptic functions can be developed in many different ways—too many for comfort. At one time some particular way seems the best; at another, a slightly different approach is highly advertised for a season and is generally regarded as being more chic.

  Gudermann’s idea was to base everything on the power series expansion of the functions. (This statement will have to do for the moment; its meaning will become clear when we describe one of the leading motivations of the work of Weierstrass.) This really was a good new idea, and Gudermann slaved over it with overwhelming German thoroughness for years without, perhaps, realizing what lay behind his inspiration, and himself never carried it through. The important thing to note here is that Weierstrass made the theory of power series—Gudermann’s inspiration—the nerve of all his work in analysis. He got the idea from Gudermann, whose lectures he attended. In later life, contemplating the scope of the methods he had developed in analysis, Weierstrass was wont to exclaim, “There is nothing but power series!”

  At the opening lecture of Gudermann’s course on elliptic functions (he called them by a different name, but that is of no importance) there were thirteen auditors. Being in love with his subject the lecturer quickly left the earth and was presently soaring practically alone in the aether of pure thought. At the second lecture only one auditor appeared and Gudermann was happy. The solitary student was Weierstrass. Thereafter no incautious third party ventured to profane the holy communion between the lecturer and his unique disciple. Gudermann and Weierstrass were fellow Catholics; they got along splendidly together.

  Weierstrass was duly grateful for the pains Gudermann lavished on him, and after he had become famous he seized every opportunity—the more public the better—to proclaim his gratitude for what Gudermann had done for him. The debt was not inconsiderable: it is not every professor who can drop a hint like the one—power series representation of functions as a point of attack—which inspired Weierstrass. In addition to the lectures on elliptic functions, Gudermann also gave Weierstrass private lessons on “analytical spherics”—whatever that may have been.

  In 1841, at the age of twenty six, Weierstrass took his examinations for his teacher’s certificate. The examination was in two sections, written and oral. For the first he was allowed six months in which to write out essays on three topics acceptable to the examiners. The third question inspired a fine dissertation on the Socratic method in secondary teaching, a method which Weierstrass
followed with brilliant success when he became the foremost mathematical teacher of advanced students in the world.

  A teacher—at least in higher mathematics—is judged by his students. If his students are enthusiastic about his “beautifully clear lectures,” of which they take copious notes, but never do any original mathematics themselves after getting their advanced degrees, the teacher is a flat failure as a university instructor and his proper sphere—if anywhere—is in a secondary school or a small college where the aim is to produce tame gentlemen but not independent thinkers. Weierstrass’ lectures were models of perfection. But if they had been nothing more than finished expositions they would have been pedagogically worthless. To perfection of form Weierstrass added that intangible something which is called inspiration. He did not rant about the sublimity of mathematics and he never orated; but somehow or another he made creative mathematicians out of a disproportionately large fraction of his students.

  The examination which admitted Weierstrass after a year of probationary teaching to the profession of secondary school work is one of the most extraordinary of its kind on record. One of the essays which he submitted must be the most abstruse production ever accepted in a teachers’ examination. At the candidate’s request Gudermann had set Weierstrass a real mathematical problem: to find the power series developments of the elliptic functions. There was more than this, but the part mentioned was probably the most interesting.

  Gudermann’s report on the work might have changed the course of Weierstrass’ life had it been listened to, but it made no practical impression where it might have done good. In a postscript to the official report Gudermann states that “This problem, which in general would be far too difficult for a young analyst, was set at the candidate’s express request with the consent of the commission.” After the acceptance of his written work and the successful conclusion of his oral examination, Weierstrass got a special certificate on his original contribution to mathematics. Having stated what the candidate had done, and having pointed out the originality of the attack and the novelty of some of the results attained, Gudermann declares that the work evinces a fine mathematical talent “which, provided it is not frittered away, will inevitably contribute to the advancement of science. For the author’s sake and that of science it is to be desired that he shall not become a secondary teacher, but that favorable conditions will make it possible for him to function in academic instruction. . . . The candidate hereby enters by birthright into the ranks of the famous discoverers.”

  These remarks, in part underlined by Gudermann, were very properly stricken from the official report. Weierstrass got his certificate and that was all. At the age of twenty six he entered his trade of secondary teaching which was to absorb nearly fifteen years of his life, including the decade from thirty to forty which is usually rated as the most fertile in a scientific man’s career.

  His work was excessive. Only a man with iron determination and a rugged physique could have done what Weierstrass did. The nights were his own and he lived a double life. Not that he became a dull drudge; far from it. Nor did he pose as the village scholar absorbed in mysterious meditations beyond the comprehension of ordinary mortals. With quiet satisfaction in his later years he loved to dwell on the way he had fooled them all; the gay government officials and the young officers found the amiable school teacher a thoroughly good fellow and a lively tavern companion.

  But in addition to these boon companions of an occasional night out, Weierstrass had another, unknown to his happy-go-lucky fellows •—Abel, with whom he kept many a long vigil. He himself said that Abel’s works were never very far from his elbow. When he became the leading analyst in the world and the greatest mathematical teacher in Europe his first and last advice to his numerous students was “Read Abel!” For the great Norwegian he had an unbounded admiration undimmed by any shadow of envy. “Abel, the lucky fellow!” he would exclaim: “He has done something everlasting! His ideas will always exercise a fertilizing influence on our science.”

  The same might be said for Weierstrass, and the creative ideas with which he fertilized mathematics were for the most part thought out while he was an obscure schoolteacher in dismal villages where advanced books were unobtainable, and at a time of economic stress when the postage on a letter absorbed a prohibitive part of the teacher’s meagre weekly wage. Being unable to afford postage, Weierstrass was barred from scientific correspondence. Perhaps it is as well that he was: his originality developed unhampered by the fashionable ideas of the time. The independence of outlook thus acquired characterized his work in later years. In his lectures he aimed to develop everything from the ground up in his own way and made almost no reference to the work of others. This occasionally mystified his auditors as to what was the master’s and what another’s.

  It will be of interest to mathematical readers to note one or two stages in Weierstrass’ scientific career. After his probationary year as a teacher at the Gymnasium at Münster, Weierstrass wrote a memoir on analytic functions in which, among other things, he arrived independently at Cauchy’s integral theorem—the so-called fundamental theorem of analysis. In 1842 he heard of Cauchy’s work but claimed no priority (as a matter of fact Gauss had anticipated them both away back in 1811, but as usual had laid his work aside to ripen). In 1842, at the age of twenty seven, Weierstrass applied the methods he had developed to systems of differential equations—such as those occurring in the Newtonian problem of three bodies, for example; the treatment was mature and rigorous. These works were undertaken without thought of publication merely to prepare the ground on which Weierstrass’ life work (on Abelian functions) was to be built.

  In 1842 Weierstrass was assistant teacher of mathematics and physics at the Pro-Gymnasium in Deutsch-Krone, West Prussia. Presently he was promoted to the dignity of ordinary teacher. In addition to the subjects mentioned the leading analyst in Europe also taught German, geography, and writing to the little boys under his charge; gymnastics was added in 1845.

  In 1848, at the age of thirty three, Weierstrass was transferred as ordinary teacher to the Gymnasium at Braunsberg. This was something of a promotion, but not much. The head of the school was an excellent man who did what he could to make things agreeable for Weierstrass although he had only a remote conception of the intellectual eminence of his colleague. The school boasted a very small library of carefully selected books on mathematics and science.

  It was in this year that Weierstrass turned aside for a few weeks from his absorbing mathematics to indulge in a little delicious mischief. The times were somewhat troubled politically; the virus of liberty had infected the patient German people and at least a few of the bolder souls were out on the warpath for democracy. The royalist party in power clamped a strict censorship on all spoken or printed sentiments not sufficiently laudatory to their regime. Fugitive hymns to liberty began appearing in the papers. The authorities of course could tolerate nothing so subversive of law and order as this, and when Braunsberg suddenly blossomed out with a lush crop of democratic poets all singing the praises of liberty in the local paper, as yet uncensored, the flustered government hastily appointed a local civil servant as censor and went to sleep, believing that all would be well.

  Unfortunately the newly appointed censor had a violent aversion to all forms of literature, poetry especially. He simply could not bring himself to read the stuff. Confining his supervision to blue-pencilling the dull political prose, he turned over all the literary effusions to schoolteacher Weierstrass for censoring. Weierstrass was delighted. Knowing that the official censor would never glance at any poem, Weierstrass saw to it that the most inflammatory ones were printed in full right under the censor’s nose. This went merrily on to the great delight of the populace till a higher official stepped in and put an end to the farce. As the censor was the officially responsible offender, Weierstrass escaped scot-free.

  The obscure hamlet of Deutsch-Krone has the honor of being the place where Weierstrass (in 1842-4
3) first broke into print. German schools publish occasional “programs” containing papers by members of the staff. Weierstrass contributed Remarks on Analytical Factorials. It is not necessary to explain what these are; the point of interest here is that the subject of factorials was one which had caused the older analysts many a profitless headache. Until Weierstrass attacked the problems connected with factorials the nub of the matter had been missed.

  Crelle, we recall, wrote extensively on factorials, and we have seen how interested he was when Abel somewhat rashly informed him that his work contained serious oversights. Crelle now enters once more, and again in the same fine spirit he showed Abel.

  Weierstrass’ work was not published till 1856, fourteen years after it had been written, when Crelle printed it in his Journal. Weierstrass was then famous. Admitting that the rigorous treatment by Weierstrass clearly exposes the errors of his own work, Crelle continues as follows: “I have never taken the personal point of view in my work, nor have I striven for fame and praise, but only for the advancement of truth to the best of my ability; and it is all one to me whoever it may be that comes nearer to the truth—whether it is I or someone else, provided only a closer approximation to the truth is attained.” There was nothing neurotic about Crelle. Nor was there about Weierstrass.

  Whether or not the tiny village of Deutsch-Krone is conspicuous on the map of politics and commerce it stands out like the capital of an empire in the history of mathematics, for it was there that Weierstrass, without even an apology for a library and with no scientific connections whatever, laid the foundations of his life work—“to complete the life work of Abel and Jacobi growing out of Abel’s Theorem and Jacobi’s discovery of multiply periodic functions of several variables.”

  Abel, he observes, cut down in the flower of his youth, had no opportunity to follow out the consequences of his tremendous discovery, and Jacobi had failed to see clearly that the true meaning of his own work was to be sought in Abel’s Theorem. “The consolidation and extension of these gains—the task of actually exhibiting the functions and working out their properties—is one of the major problems of mathematics.” Weierstrass thus declares his intention of devoting his energies to this problem as soon as he shall have understood it deeply and have developed the necessary tools. Later he tells how slowly he progressed: “The fabrication of methods and other difficult problems occupied my time. Thus years slipped away before I could get at the main problem itself, hampered as I was by an unfavorable environment.”