Men of Mathematics (80 page)

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 1842-43) 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.”

The whole of Weierstrass' work in analysis can be regarded as a grand attack on his main problem. Isolated results, special developments and even extensive theories—for example that of irrational numbers as developed by him—all originated in some phase or another of the central problem. He early became convinced that for a clear understanding of what he was attempting to do a radical revision of the fundamental concepts of mathematical analysis was necessary, and from this conviction he passed to another, of more significance today perhaps than the central problem itself: analysis must be
founded on the common whole numbers 1, 2, 3, . . .. The irrationals which give us the concepts of limits and continuity, from which analysis springs, must be referred back by irrefrangible reasoning to the integers; shoddy proofs must be discarded or reworked, gaps must be filled up, and obscure “axioms” must be dragged out into the light of critical inquiry till all are understood and all are stated in comprehensible language in terms of the integers. This in a sense is the Pythagorean dream of basing all mathematics on the integers, but Weierstrass gave the program constructive definiteness and made it work.

Thus originated the nineteenth century movement known as
the arithmetization of analysis
—something quite different from Kronecker's arithmetical program, at which we shall glance in a later chapter; indeed the two approaches were mutually antagonistic.

In passing it may be pointed out that Weierstrass' plan for his life work and his magnificent accomplishment of most of what he set himself as a young man to do, is a good illustration of the value of the advice Felix Klein once gave a perplexed student who had asked him the secret of mathematical discovery. “You must have a problem,” Klein replied. “Choose one definite objective and drive ahead toward it. You may never reach your goal, but you will find something of interest on the way.”

From Deutsch-Krone Weierstrass moved to Braunsberg, where he taught in the Royal Catholic Gymnasium for six years, beginning in 1848. The school “program” for 1848-49 contains a paper by Weierstrass which must have astonished the natives:
Contributions to the Theory of Abelian Integrals.
If this work had chanced to fall under the eyes of any of the professional mathematicians of Germany, Weierstrass would have been made. But, as his Swedish biographer, Mittag-Leffler, dryly remarks, one does not look for epochal papers on pure mathematics in secondary-school programs. Weierstrass might as well have used his paper to light his pipe.

His next effort fared better. The summer vacation of 1853 (Weierstrass was then 38) was passed in his father's house at Westernkotten. Weierstrass spent the vacation writing up a memoir on Abelian functions. When it was completed he sent it to Crelle's great
Journal.
It was accepted and appeared in volume 47 (1854).

This may have been the paper whose composition was responsible for an amusing incident in Weierstrass' career as a schoolteacher at
Braunsberg. Early one morning the director of the school was startled by a terrific uproar proceeding from the classroom where Weierstrass was supposed to be holding forth. On investigation he discovered that Weierstrass had not shown up. He hurried over to Weierstrass' dwelling, and on knocking was bidden to enter. There sat Weierstrass pondering by the glimmering light of a lamp, the curtains of the room still drawn. He had worked the whole night through and had not noticed the approach of dawn. The director called his attention to the fact that it was broad daylight and told him of the uproar in his classroom. Weierstrass replied that he was on the trail of an important discovery which would rouse great interest in the scientific world and he could not possibly interrupt his work.

The memoir on Abelian functions published in Crelle's
Journal
in 1854 created a sensation. Here was a masterpiece from the pen of an unknown schoolmaster in an obscure village nobody in Berlin had ever heard of. This in itself was sufficiently astonishing. But what surprised those who could appreciate the magnitude of the work even more was the almost unprecedented fact that the solitary worker had published no preliminary bulletins announcing his progress from time to time, but with admirable restraint had held back everything till the work was completed.

Writing to a friend some ten years later, Weierstrass gives his modest version of his scientific reticence: “. . . the infinite emptiness and boredom of those years [as a schoolteacher] would have been unendurable without the hard work that made me a recluse—even if I was rated rather a good fellow by the circle of my friends among the junkers, lawyers, and young officers of the community. . . . The present offered nothing worth mentioning, and it was not my custom to speak of the future.”

Recognition was immediate. At the University of Königsberg, where Jacobi had made his great discoveries in the field which Weierstrass had now entered with a masterpiece of surpassing excellence, Richelot, himself a worthy successor of Jacobi in the theory of multiply periodic functions, was Professor of Mathematics. His expert eyes saw at once what Weierstrass had done. He forthwith persuaded his university to confer the degree of doctor,
honoris causa,
on Weierstrass and himself journeyed to Braunsberg to present the diploma.

At the dinner organized by the director of the Gymnasium in
Weierstrass' honor Richelot asserted that “we have all found our master in Mr. Weierstrass.” The Ministry of Education immediately promoted him and granted him a year's leave to prosecute his scientific work. Borchardt, the editor of Crelle's
Journal
at the time, hurried to Braunsberg to congratulate the greatest analyst in the world, thus starting a warm friendship which lasted till Borchardt's death a quarter of a century later.

None of this went to Weierstrass' head. Although he was deeply moved and profoundly grateful for all the generous recognition so promptly accorded him, he could not refrain from casting a backward glance over his career. Years later, thinking of the happiness of the occasion and of what that occasion had opened up for him when he was forty years of age, he remarked sadly that “everything in life comes too late.”

*  *  *

Weierstrass did not return to Braunsberg. No really suitable position being open at the time, the leading German mathematicians did what they could to tide over the emergency and got Weierstrass appointed Professor of Mathematics at the Royal Polytechnic School in Berlin. This appointment dated from July
1, 1856;
in the autumn of the same year he was made Assistant Professor (in addition to the other post) at the University of Berlin and was elected to the Berlin Academy.

The excitement of novel working conditions and the strain of too much lecturing presently brought on a nervous breakdown. Weierstrass had also been overworking at his researches. In the summer of
1859
he was forced to abandon his course and take a rest cure. Returning in the fall he continued his work, apparently refreshed, but in the following March was suddenly attacked by spells of vertigo, and he collapsed in the middle of a lecture.

All the rest of his life he was bothered with the same trouble off and on, and after resuming his work—as full professor, with a considerably lightened load—never trusted himself to write his own formulas on the board. His custom was to sit where he could see the class and the blackboard, and dictate to some student delegated from the class what was to be written. One of these “mouthpieces” of the master developed a rash propensity to try to improve on what he had been told to write. Weierstrass would reach up and rub out the amateur's
efforts and make him write what he had been told. Occasionally the battle between the professor and the obstinate student would go to several rounds, but in the end Weierstrass always won. He had seen little boys misbehaving before.

As the fame of his work spread over Europe (and later to America), Weierstrass' classes began to grow rather unwieldy and he would sometimes regret that the quality of his auditors lagged far behind their rapidly mounting quantity. Nevertheless he gathered about him an extremely able band of young mathematicians who were absolutely devoted to him and who did much to propagate his ideas, for Weierstrass was always slow about publication, and without the broadcasting of his lectures which his disciples took upon themselves his influence on the mathematical thought of the nineteenth century would have been considerably retarded.