Theorie der Potenzial- oder der cyklisch- hyperbolischen Functionen. - [GUDERMANNIAN FUNCTION]

Berlin, G. Reimer, 1830. 4to. In "Journal für die reine und angewandte Mathematik, 6. Band, 1 Heft, 1830". In the original printed wrappers, without backstrip. Fine and clean. [Gudermann:] Pp. 1-39. [Entire issue: Pp. 106 pp. + 1 folded plate. ]. First printing of Gudermann extensive and important paper on hyperbolic functions. During 1830ies Gudermann focused his work on these functions and published extensive on the subject, the present paper being the first. It was later coined the Gudermannian function.The function was introduced by Johann Heinrich Lambert in the 1760s at the same time as the hyperbolic functions. He called it the "transcendent angle," and it went by various names until 1862 when Cayley suggested it be given its current name as a tribute to Gudermann's work in the 1830ies on the theory of special functions. The present paper together with two later published papers were collected in Theorie der potenzial- oder cyklisch-hyperbolischen functionen (1833), a book which expounded sinh and cosh to a wide audience.The issue also contain a paper by the famous Norwegian mathematician Niels Henrik Abel."Gudermann devoted much more attention to the theory of special functions. After the earlier works of Leonhard Euler, John Landen, and A. M. Legendre (Gauss's results were still in manuscript), Niels Abel's studies on elliptical functions, published mostly in A. L. Crelle's Journal für reine und angewandte Mathematik, represented an important divide in treating this area. In 1829 Carl Jacobi's book Fundamenta nova theoriae functionum ellipticarum was published. At the time Gudermann was one of the first mathematicians to expand on these results. Beginning with volume 6 (1830) of Crelle's Journal, he published a series of papers which he later summarized in two books: Theorie der Potenzialoder cyklisch-hyperbolischen Functionen (1833) and Theorie der Modular-Functionen und der Modular-Integrale (1844), which were to have had a sequel which was never written." (DSB).Gudermann is known today for being the teacher of Karl Weierstrass, who took Gudermann's course in elliptic functions in 1839, the first to be taught in any institute. Weierstrass was greatly influenced by this course, which marked the direction of his own research.…