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‎Théorie des fonctions doublement périodiques et, en particulier, des fonctions elliptiques.‎

‎Paris, Mallet-Bachelier, 1859. 8vo. Contemporary half calf. XXIV,342 pp.‎

‎First edition of this important textbook on function theory. Charles Auguste Briot (1817-1882) was one of the most eminent teachers of his time. Already during his time in elementary school Briot had decided that he wanted to be a mathematics teacher. He entered the École Normale Supériure in 1838 were he was ranked second. Just three years later he received agrégation in mathematics with the highest rank. In March 1842 he received his doctorate of science, having presented his thesis on the movement of a solid body round a fixed point. Despite his marked academic success in research, Briot still wanted to follow his chosen career as a teacher. He first obtained a professorship at the Orléans Lycée and afterward at University of Lyons, where he re-encountered his school friend Jean-Claude Bouquet, with whom he collaborated throughout his career.During the years 1814-1831 Agustin Cauchy created the basic framework of complex function theory. The results obtained by Cauchy were, however, still only to be found throughout his numerous journal papers, and although Cauchy is known for having stimulated the development of mathematical rigor throughout the field, his own research papers often used intuitive, not rigorous, methods. The joint scientific work of Briot and Bouquet was a profound study and clarification of the analytic work of Cauchy. In a memoire that has remained famous since its publication in 1853, they proposed to establish precisely the conditions that a function must fulfill in order to be developable into an entire series. They also perfected the analysis by which Cauchy had, for the first time, established the existence of the integral of a differential equation. They opened the way to research on singular points and showed their importance for knowledge of the integral. All this work culminated in their extremely influential text book 'Théorie des fonctions doublement périodiques et, en particulier, des fonctions elliptiques'. This work played a decisive role in spreading Cauchy's methods and results to European mathematicians, and for a long time it remained the standard text of the French School. It was translated into German in 1862, and went through several editions. For his outstanding contributions to mathematics the Académie des Sciences in Paris awarded Briot their Poncelet Prize in 1882 shortly before he died.‎