By Bruno Belhoste

A nice hassle dealing with a biographer of Cauchy is that of delineating the curious interaction among the guy, his occasions, and his medical endeavors. Professor Belhoste has succeeded admirably in assembly this problem and has therefore written a bright biography that's either readable and informative. His topic sticks out as some of the most remarkable, flexible, and prolific fig ures within the annals of technology. approximately 200 years have now handed because the younger Cauchy set approximately his activity of clarifying arithmetic, extending it, utilizing it at any place attainable, and putting it on an organization theoretical footing. via Belhoste's paintings we're afforded a close, particularly custom-made photograph of ways a primary expense mathematician labored at his self-discipline - his strivings, his inspirations, his triumphs, his disasters, and especially, his conflicts and his errors.

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T. "4'..... { Title page of the manuscript of the paper on the polygons and polyhedrons. January 20, 1812. Published by permission of the Ecole Poly technique. 28 2. Sojourn at Cherbourg mean the theorem by which it is proved that two triangles are equal if their three sides are equal. If one should establish this latter theorem without using either trigonometry or reductio ad absurdum, I would agree that my proof ought not be admitted. It thus seems impossible to banish the reductio ad absurdum proof from geometry; and this is particularly true in the present case.

In connection with his work in number theory, he made a careful study of Gauss' Disquisitiones Arithmeticae, which had been published in 1801 and translated into French in 1807. He seems to have mastered very quickly the methodology and sense of Gauss; and grasping the importance of the theory offorms that Gauss had used in proving Fermat's theorem on triangular numbers, he managed to simplify it and to generalize some of the results, particularly the ones on discriminants. These investigations on number theory, he declared in the first article, led him to work on the 'theory of combinations' and from there to prove a theorem that was more general than the results obtained by Lagrange and by Ruffini on the number of values of a function of n given quantities: namely, if the number of distinct values assumed by a function of n quantities is less than the largest prime factor p of n, it is less than or equal to 2.

Sojourn at Cherbourg mean the theorem by which it is proved that two triangles are equal if their three sides are equal. If one should establish this latter theorem without using either trigonometry or reductio ad absurdum, I would agree that my proof ought not be admitted. It thus seems impossible to banish the reductio ad absurdum proof from geometry; and this is particularly true in the present case. In fact, in order to prove that under certain conditions only one polyhedron can be constructed, it is necessary to see that after the first figure has been constructed subject to the given conditions then one cannot construct a second figure without encountering a contradiction.