Thomas Singleton Dr. Eugene Boman MATH October 2014 Takakazu Seki Kowa: The Japanese Newton

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1 Singleton 1 Thomas Singleton Dr. Eugene Boman MATH October 2014 Takakazu Seki Kowa: The Japanese Newton Takakazu Seki Kowa was a Japanese mathematician, probably born in March 1642 in the town of Fujioka (Selin 1992). This, of course, is the same year that Sir Isaac Newton was born. Seki's birth father, a samurai named Nagaakira Utiyama, sent him to live with an accountant by the name of Seki Gorozayemon at a young age. Seki Kowa was young enough that he took his adoptive father's name. (Knight 298) According to Knight, it was not long before Seki Kowa demonstrated an incredibly gifted ability as a mathematician, even earning the title of 'divine child' by his adoptive father and his associates (298). Seki eventually studied mathematics under Takahara Yoshitane, a disciple of another great Japanese mathematician by the name of Mori Shigeyoshi (Selin 1992). Seki was a better mathematician than Takahara and taught himself using the minimal Japanese works and the available collection of Chinese works (Selin ). It is no surprise that Japanese mathematics, like much of their culture and written language, have a strong Chinese basis. Chinese mathematics at the time could solve only singlevariable equations, to which Seki made considerable advancements (Knight 299). Seki Kowa married, but had no children (Knight 298). It seems quite possible that this was due to a heavy focus on his work. Tokugawa Tsunashige, the Lord of Koshu that eventually became Shogun, hired Seki as an auditor (Selin 1992). Once Tokugawa became Shogun, Seki was made a samurai of the Shogun due to both his connections

2 Singleton 2 with the shogun, and the fact that he was born into the samurai caste (Knight 298). While it seems that a lot of people like to focus on the warlike aspects of samurai, there are plenty that are better known as scholars, poets, artists and apparently mathematicians. Even after rising to such a level in the state, Seki Kowa focused on mathematics and began to teach and write on the subject (Knight 298). Seki died in October of 1708, within the town of Edo, after being granted the title of 'Master of Ceremonies' by the Shogun (Knight 298). Seki became famous among his peers and students for solving 15 supposedly unsolvable problems published in He published his most notable paper, Hatubi sanpo, in 1674, solving each of these problems (Knight 298). However, according to Knight, it was not the Japanese custom to show how one arrived at one's solutions, and it appears that even Seki Kowa's students remained unaware of his methodologies (298). We would discover his methods from a posthumous publication (Knobloch 187). Algebra was unknown to Japanese mathematicians during Seki's lifetime, which makes his work and abilities particularly remarkable (Knight 299). Seki Kowa made major contributions to Japanese mathematics with barely any input from other scholars (Knight 299). Though thousands of miles away from Europe, Seki Kowa made similar discoveries to Sir Isaac Newton. Like Newton, Seki discovered a method to approximate the root of a numerical equation. He also created his own table of determinants. (Knight 299). As one of his greatest achievements, it is not surprising that an example of Seki

3 Singleton 3 Kowa's methodology for calculating determinants is on his stamp. In particular, the stamp displays Seki Kowa's expansion of a fourth-order determinant (Knobloch). Seki's other great publication, from which we know most about his methods, came after his death. Katsuyo Sanpo is a comprehensive collection of his work on mathematics (Knobloch, 187). Within Katsuyo Sanpo, we find many of Seki Kowa's greatest discoveries. The first book within the publication contains his research on the sum of powers of natural numbers, and his formula to count said sum of powers (Knobloch, 187). The methodology here is ultimately similar to Bernoulli's, and Seki Kowa even invented a way to get what we would call the Bernoulli numbers. However, Seki Kowa did this without any previously established notation for such things, nor even much background of advanced mathematics to build upon (Knobloch ). The second book of Katsuyo Sanpo focuses entirely upon Seki Kowa's research into the solution for indeterminate equations of natural numbers (Knobloch 189). The third focuses upon polygons with between three and twenty sides. Even with an approximate translation towards a more Western notation, some of the formulas he derived are extremely difficult to understand. Seki successfully discovered a numerical relation between the lengths of the sides of the polygon, and the radii of both the circum- and inscribed circles. (Knobloch ) The fourth and final book in Katsuyo Sanpo focused upon his prior work on π, and the research and ultimately the methods Seki Kowa used to calculate the length of an arc and the volume of a sphere (Knobloch 191). The classical Japanese methodology to calculate the circumference of a circle was actually just an

4 Singleton 4 approximation using the perimeter of an inscribed regular polygon. Seki used polygons with many sides, up to 131,072 (Knobloch 191). His accuracy in calculating π is quite impressive: Let a,b,c be the following values, the length of the diameter being put equal to 1. a = the perimeter of the regular polygon with 32,768 sides. b = the perimeter of the regular polygon with 65,636 sides. C = the perimeter of the regular polygon with 131,072 sides. (Knobloch 191). Seki then used the formula b + [(b-a) (c-b)]/[(b-a)-(c-b)] to estimate π, and arrived at (Knobloch 191). He then declared that this is the exact number of the ratio of the length of the circumference of a circle, against its diameter (Knobloch 191). Rather than always having to always write such a long constant, Seki Kowa determined an approximate fraction to be used in its place, arriving at 335/113. He made the obvious decision to begin with 3/1, and then added the fractions 7/2, 10/3, (n 3+1)/n until he arrived at a value he determined was close enough (Knobloch 192). His work to determine the volume of a sphere was similarly impressive, considering that Japan did not have the same mathematical background that the more Western civilizations did. He used the same formula used to calculate π as above, though applied differently, to assist in calculating the volume of a sphere. He instead sliced a sphere with a diameter of 10 units 50, 100 and 200 times. (Knobloch 192) He added the square of the diameters for each of these disks, keeping each summation separate. Consider these sums to be a, b and c respectively for the formula above. He multiplied this result with his approximation of π and divided by four, and then further divided this result by (Knobloch 192) Ultimately, he found the volume rate of a

5 Singleton 5 sphere to be 355/678, and stated that this should be multiplied with diameter of the sphere cubed (Knobloch 192). Knobloch's summarized description of the problem, paraphrased above, while it does simplify Seki Kowa's work, still leaves the result far too complex. It can be simplified as V = (355/678) d 3 where d is the diameter, and further to V = (355/678) 8 r 3. As it turns out, multiplying 8 (355/678) results in 2840/678, which differs from 4/3 π by just percent. That is extremely accurate, and very likely as accurate as his European contemporaries! Seki Kowa is considered to be the founder of Japanese mathematics by some; not just of modern Japanese mathematics, but of all Japanese mathematics (Selin 1993). I disagree, since there was at least one other great Japanese mathematician before him. However, Seki may be the most important. In addition to the brief summaries of his work above, it should be noted that Seki Kowa more or less invented Japanese algebra (Selin 1993). His work also covered the following: solutions and properties of high degree equations, infinite series, approximations of fractions, the method of interpolation, Newton's formula (with the Kyusho method), computing the area of rings, conics, 'magic' squares and circles, the equivalent of the Josephus question, and probably much more. (Selin 1993). While some of his conclusions and contributions may seem trivial to Western scholars who are familiar with advanced mathematics, the truth is that Seki Kowa's contributions to Japanese mathematics were absolutely phenomenal. They have merit

6 Singleton 6 for the whole world in that they provide further proof for many of the concepts that modern mathematics are built around. It seems to me that Seki Kowa's brilliance was on the level of Sir Isaac Newton and many of the other great Western mathematicians. I imagine that Seki Kowa could have made even more impressive discoveries if he did not have to catch up on hundreds of years of mathematics versus Western society, or at the very least had similarly brilliant contemporaries to work with.

7 Singleton 7 Sources Cited Knight, Judson. "Takakazu Seki Kowa." Science and Its Times. Ed. Neil Schlager and Josh Lauer. Vol. 3: 1450 to Detroit: Gale, Gale Virtual Reference Library. Web. 6 Sept Knobloch, Eberhard, Hikosaburo Komatsu, and Dun Liu, ed. Seki, Founder of Modern Mathematics in Japan. Tokyo: Springer, PDF e-book. Selin, Helaine, ed. Seki Kowa. Encylopaedia of the History of Science, Technology and Medicine in Non-Western Cultures. Netherlands: Springer, PDF e-book. pp

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