Surfaces Telles Que la Somme des Rayons de Courbure Principaux est Proportionnelle a la Distance d'un Point Fixe au Plan Tangent

1888 ◽  
Vol 10 (3) ◽  
pp. 187 ◽  
Author(s):  
E. Goursat
Keyword(s):  
Plan Tangent

Henri Lebesgue, 1875 - 1941

1944 ◽  
Vol 4 (13) ◽  
pp. 483-490 ◽  
Keyword(s):  
Royal Society ◽  
London Mathematical Society ◽  
Continuous Functions ◽  
Honorary Member ◽  
Riemann Integral ◽  
Discontinuous Functions ◽  
Collège De France ◽  
Académie Des Sciences ◽  
Plan Tangent ◽  
Foreign Member

Henri Léon Lebesgue was born in 1875 and died in 1941. He studied at the Ecole Normale Superieure, and his first post appears to have been that of maitre des conferences at Rennes, which he held until 1906. He then moved to Poitiers, where he was described first as ‘charge de cours a la faculte des Sciences’ and later as professor. In or about 1912 he was called to Paris as maitre des conferences and he afterwards became professor at the College de France. He was elected to the Academie des Sciences in 1922. He was made an honorary member of the London Mathematical Society in 1924, and a foreign member of the Royal Society in 1934. Towards the close of the last century, the development of mathematical analysis may be said broadly to have reached the stage at which a piece of work wherein only continuous functions were encountered could be carried through. For example, the Riemann integral solved the problem of finding a function having a given derivative if the derivative was continuous. Again, everything was known about the length of a curve and its expression as an integral if it had a continuously turning tangent. Artificial and unaesthetic restrictions had repeatedly to be made to keep out discontinuous functions. Jordan, in the preface to his Cours d'Analyse (2nd edition, 1893), wrote: ‘Certains points presentent encore quelque obscurite. Pour en citer un exemple, nous n’avons pu arriver a definir d’une maniere satisfaisante l’aire d’une surface gauche, que dans le cas ou la surface a un plan tangent, variant suivant une loi continue’. In 1898 vital steps were taken by Baire and by Borel. The thése of Baire was a systematic and penetrating discussion of discontinuous functions. And Borel in the small compass of four pages of his tract, Leçons sur la théorie des fonctions , propounded his theory of measure, fundamentally more powerful than Jordan’s theory of content in being additive for an enumerable infinity of sets. All was ready for the rapid advances of the next decade which made the theory of real functions into a satisfying whole; in this transformation the leading part was played by Lebesgue.

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