T plus and black were minus. They had no symbols to Takebe Katahiro express power; the position of the rods on the counting board indicated the powers. Therefore, the system could not indicate complex expressions. For example, JOCHI SHIGERU 1/(x − 1), that is (x − 1)−1, was very difficult to indicate by that system. Seki Kowa abandoned the counting Takebe Katahiro was born in 1664 at Edo (now Tokyo). rods system and created algebraic symbols, which His father, Takebe Naotsune, was a Yuhitsu (secretary) used Chinese characters for calculation with figures. of the Shogun. In 1676, when he was 13 years old, he This might be the first example of creating Japanese and his elder brother Takebe Kataaki (1661–1716) mathematics from the Chinese. became pupils of SEKI Kowa (d. 1708) and studied Takebe commented on the most important Chinese mathematics. The Takebe brothers and Seki Kowa were algebraic text in Japan at that time, Suanxue Qimeng colleagues in the Shogun’s government, and their (Introduction to Mathematical Studies, Zhu Shijie, 1299). 1 families were the same rank: 300 koku. He published the Sangaku Keimo Genkai Taisei Takebe’s mathematical works are in three fields. One (Complete Works of Commentaries on Suan Xue Qi concerns completing the tenzan-jutsu or endan-jutsu Meng) in 1690. Seki Kowa and the Takebe brothers (lit. addition and subtraction methods, Japanese algebra started to compile an edition of the mathematical works of system), which was created by Seki Kowa. In the Seki-ryu, Seki Kowa’s school. After Seki died, Takebe second work Takebe created the tetsu-jutsu (inductive Katashiro continued this work and published Taisei methods). Using these methods he obtained the formula Sankyo (Complete Mathematical Manual) about 1710. 2 of (arcsin θ) . In the third, for computing the approximate Takebe’s main work is the tetsu-jutsu method, a sort value of fractions, he solved the Diophanine equations of inductive method. He computed small natural using the reiyaku-jutsu (continual division method). numbers and then predicted infinite numbers. The Takebe also worked in astronomy and geography. computation was helped by the algebraic symbols of In 1683, Takebe wrote his first work, Kenki Sampo the tenzan-jutsu method. Using these methods, Takebe (Studies for Mathematical Methods). The book provided obtained the formula of (arcsin θ)2. He computed the counter-arguments for Saji Ippei’s Sampo Nyumon length of curve AB (hereafter s) using the diameter d (Introduction to Mathematical Methods, 1680). Saji had and the length of straight line AB (h). criticized Seki Kowa’s tenzan-jutsu system in the Hatsubi Takebe set up d = 10 and h =10−5. Then letting the Sampo (Mathematical Methods for Finding Details, half point of straight line AB be C , and the half point ’ 2 1674) and solved Ikeda Masoki sproblemsintheSugaku of curve AB be B2, he computed the length of AB2 (h2). Jojo Orai (Textbook of Mathematical Multiplication and Then he computed the length of AB4 as h4, and Division, 1672). Takebe made good use of the tenzan- continued to compute h , h , h and h . Takebe ’ 8 16 32 64 jutsu method for solving Ikeda s remainder problems. computed h∞ using a sort of infinity series zoyaku-jutsu ’ Takebe commented on Seki Kowa s Hatsubi Sampo (extra division method), and obtained h∞. and published the Sampo Endan Genkai (Commentar- ies for Japanese Algebra System), in 1685. This is one s 2 ¼ 10À4 Â 1:000 000 333 333 411 111 225 396 906 of the best books for studying the tenzan-jutsu method. 2 Chinese mathematicians in the Song and Yuan 666 728 234 776 947 959 587 ... ð Þ¼ À5 dynasties used counting rods to solve higher degree h AC2B 10 , 1 equations of more than the fourth degree. There were dðB2ODÞ¼10 : two color symbols in the counting rods: red rods were 1 His annual salary was 300 pyo (1 pyo was 60 kgs of rice), which is the same as a landlord of 300 person village. 2078 Takebe Katahiro Second, he indicated this value using h and d. The First, divide 10,000,000 by 1,415,926; the quotient q1 −4 power 10 is h by d, and the approximate value of the is 7 and the remainder r1 is 88,518. Next, divide the coefficient is 1. Therefore, former divisor 1,415,926 by the remainder r ; the 1 s 2 quotient of q2 is 15 and the remainder r2 is 88,156. ¼ 1hd þ 10À10 Â 0:333 333 511 111 ...: 2 Next, divide the former divisor r1 by the newer remainder r2, and the quotient q3 is 1 and the remainder Then he used the same method. The power 10−10 is h2, r3 is 362. Continuing this algorithm, the quotients are: and the approximate value of the coefficient is 1/3 using the reiyaku-jutsu method, {q1, q2, q3,…, qn} = {7, 15, 1, 243, 1, 1, 9, 1, 1, 4}, s 2 1 p =3+1/7− 22/7 > π , ¼ 1hd þ þ 10À16 Â 0:177 777 992 ...: 1 n 2 3h2 p2 = 3 + 1/(7 + 1/15) = 333/106 < πn, Takebe continued to compute as above, and he set the p3 = 3 + 1/(7 + 1/(15 + 1/1)) = 355/113 > πn. series as p2k = q0 + 1/(q1 + 1/(q2 +1/q3 + 1/(q4 ++1/q2n s 2 π ¼ A þ A þ A þ A þ A þÁÁÁ: + 1))) < n, 2 0 1 2 3 4 p2k+1 = q0 + 1/(q1 +1/q2 + 1/(q3 + 1/(q4 ++1/q2n He obtained π + 1))) > n. A1 h A0 ¼ hd, ¼ a1 , The value of p and p had already been computed A d 1 2 0 by the Chinese mathematician, Zu Chongzhi (429–500). A2 h A3 h Takebe concluded that Zu Chongzhi also invented the ¼ a2 , ¼ a3 , ... A1 d A2 d same method as his own reiyaku-jutsu and Zu Chongz- 1 8 9 32 hi’s significant figures were seven decimal places, a ¼ , a ¼ , a ¼ , a , ... 1 3 2 15 3 14 4 45 the same as the above computation. He admired Zu Chongzhi and named his book Fukyu Tetsu-jutsu; the title He expanded them to the general series of an, which was was connected with Zu Chongzhi’s Zhui Shu. 2n2 In 1723, Takebe made a map of Japan under the A ¼ : n ðn þ 1Þðn þ 2Þ order of Tokugawa Yoshimune, the eighth Shogun, this map is now lost. That year Takebe became a Yoriai Therefore Takebe obtained the formula (adviser) of the Shogun. ! X1 2 þ Shogun Yoshimune already permitted the import of s 2 ðn! Â 2nÞ hn 1 ¼ 2 Â : foreign scientific books in 1720, if they were not ð þ Þ! nÀ1 d n¼0 2n 2 d related to missionaries. Knowledge of Western astron- omy was imported in some Chinese translations. Takebe had no notion of triangle functions. However, Takebe and his student, Nakane Genkei (1662–1733) if we set rffiffiffi translated Mei Juecheng’s Li Suan Quan Shu (Com- h plete Works of Calendar and Mathematics, 1723) and ¼ , d sent the manuscript (Japanese name Rekisan Zensho)to Yoshimune in 1733. It was published and read by many θ we can obtain arcsin = s/(2d). Therefore, his formula Japanese scholars. Kepler’s newest opinion, the ellipti- has the same value as the formula cal orbit of planets, however, was hidden by the X1 n 2 missionaries who advised Mei Juecheng. It became ðn! Â 2 Þ þ ðarcsin Þ2 ¼ 2 Â 2n 2: known to the Japanese after Asada Goryu (1734–1799) ð2n þ 2Þ! n¼0 translated Kepler’s (Part 2 of ) Li Suan Quan Shu. Takebe was held in honor by Yoshimune, and he This work was described in the Fukyu Tetsu-jutsu became a Hoi (Knight), and held successively better (Inductive Methods) in 1722, and the special manu- positions. In 1733, he retired, and he received a life script was sent to to Shogun in 1730. annuity of 300 pyo. He died on July 20, 1739 (August 24, The key method of tetsu-jutsu was computing the 1739 in the present calendar) at Edo. approximate value of decimal fractions. He named it Takebe was one of the best mathematicians in Japan. the reiyaku-jutsu, which used the Euclidean algorithm The Mathematical Society of Japan has presented the to compute the value of continuing fractions. For Takebe Katahiro Prize since 1996. example, to try to compute the approximative value of π using this method, we set; See also: ▶Computation: The Chinese Rod Numeral 1; 415; 926 System, ▶Asada Goryu, ▶Seki Kowa, ▶Mathematics pn ¼ 3:1415926, or 3 þ : 10; 000; 000 in Japan Tao Hongjing 2079 References years (Shang 1989). His working methodology and epistemology were praised and copied by later scholars Fujiwara, Matsusaburo. Meiji-zen Nihon Sugakushi (Mathe- matics in Japan before the Meiji Era). 5 vols. Tokyo: in the same field. Tang exerted an especially profound Iwanami Shoten, 1954. influence on the work of the great naturalist of the Ming Hirayama, Akira et al. eds. Seki Kowa Zenshi. Osaka: Osaka Dynasty, Li Shizhen. Kyoiku Tosho, 1974. Mikami, Yoshio. The Development of Mathematics in China See also: ▶Medical Ethics in China – Li Shizhen and Japan. 2nd ed. New York: Chelsea, 1913. Shimodaira, Kazuo. Wasan no Rekishi (History of Japanese Mathematics in the Edo Period). 2 vols. Tokyo: Fuji Junior References College Press, 1965–1970. Takebe Kataaki. Rokkaku Sasaki Yamanouchi-ryu Takebe-shi Shang, Zhijun, et al. Essentials of Chinese Materia Medica of Denki (Biographies of Takebe Families). Edo, 1715. Successive Ages.