It has been said that the tridosha theory in Ayurveda originated from the theory of the three elements of the universe. The names of these three doshas, which are roughly equivalent to humour, are vata (wind), pitta (bile),
and kapha (phlegm), corresponding to the three elements of the universe: air, fire, and water. On the other hand,
Buddhist medicine which has a close relation to Ayurveda is based on the theory of the four elements of the
universe which includes the earth as well as the three elements mentioned above. Greek medicine on the other
hand, is founded on the theory of the four humours, i. e. blood, yellow bile, black bile, and phlegm. Furthermore,
even in Ayurveda, like in "Sushruta Samhita", the theory of the four humours can be found: This includes the abovd-mentioned tridosha plus blood as the forth humour. "Timaios" by Plato also mentions this. We compared these various theories and pointed out that the tridosha theory had its origin in the theory of the four elements of the universe. The process of the formation of the tridosha theory is considered as follows:(1) "Earth" was segregated from the four elements of the universe owing to its solid properties, and was rearranged into the seven elments of the body called "dhatu"; and the other three elements, uwatern, "fire", and "air"，were integrated as the tridosha theory, namely, the theory of the three humours, owing to th&r properties of fluid; (2)"Blood", assigned to the element of "earth", was segregated from the tridosha because "blood" was considered to be comprised of the properties of every humour without having its own peculiar properties. Therefore, the diseases caused by deranged "blood" were regarded as an aggregate disease caused by the other three deranged humours. Then the category of the disease, caused by deranged "earth", did not appear.
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
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