Kohan Sakabe and His Mathematical Idea on Spherical Trigonometry

Abstract

Relaxation of the book prohibition policy in 1720, the fifth year of Shogun Tokugawa Yoshimune's reign, made possible the importation of astronomical, calendrical and mathematical books from China. As a result "Lixiang kaocheng" <暦象考成> which was published by Chinese scientists under cooperation with the Jesuit in 1723, "Lixiang kaocheng houbian" <暦象考成後編> which was compiled by I. Koegler in 1742, "Lishuan chuanshu" <暦算全書> which was completed by Mei Wending's family in 1723 and so forth were introduced into Japan. Kohan Sakabe <坂部廣胖> (1759-1824) was a mathematician who had a great interest in trigonometry in these scientific books. Basic formulae of the right spherical triangle and the oblique spherical triangle in these books with so many astronomical examples were very useful in establishing his mathematical idea. In 1812 K. Sakabe wrote "Kanki kodo shoho" <管窺弧度捷法>,and in 1815 "Sanpo tenzan shinan-roku" <算法點竄指南録>, a mathematical book which had a good reputation as a textbook among Wasan-ka, was published and in the next year a navigation's book, "Kairo anshin-roku" <海路安心録>, was published. We must point out here that "Kanki kodo shoho", "Sanpo tenzan shinan-roku" and "Kairo anshin-roku" were written under the influence of astronomical, calendrical and mathematical books mentioned above. It will be a proof that he had learned part of western astronomy as a Wasan-ka. Therefore, this paper details his mathematical idea and the background of spherical trigonometry from the following viewpoints: ① His mathematical idea on spherical trigonometry is based on the contents of "Lishuan chuanshu" and "Lixiang kaocheng". ② He try to create new formulae of spherical trigonometry in "Sanpo tenzan shinan-roku". ③ He understand the principle of duality and the polar triangle very well