Karl Weierstrass (1790-1869): Father of mathematical analysis

KARL WEIERSTRASS was born on October 31, 1815, at Ostenfelde, Germany. His father was then a customs officer in the pay of the French. Shortly after Karl's birth the family moved to West- phalia, where there was no school. So he was sent to the adjacent town of Muenster whence he entered at the age of the 14, the catholic Gymansium at Paderborn.

Weierstrass thoroughly enjoyed his school and completed the course in less than the prescribed time, leaving a brilliant academic record. When he left the school at nineteen (1834) prizes fell his way with unfailing regularity.

In addition, he secured a paying job at the age of 15, as an accountant for a prosperous business run by a female merchant. The cartload of prizes he had won and his record as a brilliant bookkeeper made his father shove Karl, at nineteen, into the University of Bonn to master ``the cicaneries of Commerce and the quibblings of law'' It was his father's expectation this would prepare him for an exalted position in the Prussian Civil Service.

However, his four years at the university were mis-spent, as he devoted his bodily strength, his lightning dexterity and his keen mind almost exclusively to fencing and liberal indulgence to German beer. Karl returned without a degree to the bosom of his wailing family.

But, in the interludes between his fencing and drinking, he mastered, by his own efforts, Laplace's ``Celestial Mechanics'' one of the masterpieces ever written for a concentious student to assimilate, because the mathematical reasoning in it is full of gaps and enigmatical declarations that ``it is easy to see.'' He was thereby laying the foundations for his life-long interest in dynamics and systems of simultaneous differential equations.

A good friend of the family, himself an amateur mathematician, stepped in to break the impasse; let Karl prepare for the state teachers' certificate which - though no Ph.D would come out of it, would provide evening leisure in which he could keep alive his mathematical interest. Weierstrass matriculated (May 1839) at the Academy of Muenster and obtained his teacher's certificate in 1841, which equipped him for a career as secondary school teacher. This was an important stepping stone to his later mathematical eminence, although it then looked like a total route.

At Muenster Professor Christof Gudermann exerted a decisive influence on Weierstrass. At his opening lecture on a course of elliptic functions, thirteen students enrolled but in course of time, only one auditor remained. This resulted in the holy communion between the lecturer and his solitary student, Weierstrass.

Weierstrass acknowledged his debt and later proclaimed in public what Gudermann had done for him. It is not every professor who can drop a hint like the one - power series representation of functions as a point of investigation - which inspired Weierstrass.

At the age of 26, Weierstrass entered the profession of secondary school teaching, and spent his time in various schools at Muenster, Deutsch-Krone, Braensberg and finally in the Royal Polytechnic school at Berlin. This activity was to absorb fifteen years of his life (1841-56). It must be noted this included the decade from the age of thirty to forty, which is indisputably the most fertile period in a scientist's career.

The work in the schools was excessive. The creative ideas with which he fertilised mathematics were thought out while he worked in obscure villages where advanced books were unobtainable and when he could not afford to pay the postage from his meagre weekly wage.

So Weierstrass was barred from scientific correspondence. This turned out, in a sense, to be a blessing: his originality developed unhampered by the ideas of the time!

In 1842, he wrote a memour on analytic functions in which he arrived independently at the celebrated Cauchy's integral theorem. But he claimed no priority. This work was not published till 1856, when Crelle printed it in his ``Journal''.

Here was a masterpiece from the pen of an unknown schoolmaster in an obscure village nobody in Berlin had ever heard of. Recognition was immediate.

At the University of Koenigsberg, Professor Richlot saw at once the originality of Weierstrass word and persuaded his university to confer the degree of doctor of philosophy, honoris causa on Weierstrass; he himself journeyed to Braunsberg to present the diploma. There were other recognitions, from the Ministry of Education and the editor of Crelle's Journal. Despite, he could not refrain from looking back over his career. He was now 40 and remarked sadly that ``everything in life becomes too late.''

In July 1856, he was simultaneously made Assistant Professor at the University of Berlin and was elected to the Berlin Academy of Sciences.

Too much lecturing brought on a nervous breakdown and spells of vertigo. So he adopted the custom to sit where he could see the class and blackboard, and dictate to some selected student what was to be written on the board. Occasionally the battle between the professor and obstinate student would go to several rounds, but Weierstrass always won in the end.

During the period (1864-97) as Professor of Mathematics, Berlin, the fame of his work spread over Europe and later in America. His classes began to grow rather unwieldy.

To perfection in his work as a pedagogue, he added that intangible something which is called inspiration. He never orated about the sublimity of mathematics; some-how or other he made creative mathematicians out of a disproptionately large fraction of his students.

In 1873, Mitag - Leffler, his Swedish biographer, arrived in Paris all set to study analysis under the French mathematician Charles Hermite (1822-1901): ``you have made a mistake, Sir,' advised Hermite, ``you should take Weierstrass' course at Berlin. He is the master of all of us.''

Taking the sound advice of Hermite, Mittag-Leffler made a capital discovery of his own, which is to be found in all books on the theory of functions.

In his declining years, Weierstrass experienced great joy at the belated recognition given to his favourite pupil Sophie Kowalesski (1850- 91) by the French Academy of Sciences. She received the Bordin Prize for her memoir ''On the rotation of a solid body about a fixed point``.

(E.T. Bell: Men of Mathematics, Vol II, Penguin Books, London, 1953).

R. Parthasarathy

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