A History of Japanese Mathematics/Chapter 8

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Having considered the contributions of Seki concerning which there can be no reasonable doubt, and having touched upon the question of Western influence,^[1] we now propose to examine the yenri with which his name is less positively connected. The word may be translated "circle principle" or "circle theory", the name being derived from the fact that the mensuration of the circle is the first subject that it treats. It may have been suggested by the title of the Chinese work of Li Yeh (1248), the Tsê-yüan Hai-ching, in which, as we have seen (page 49). Tsî-yüan means "to measure the circle." Seki himself never wrote upon it so far as is positively known, although tradition has assigned its discovery to him, nor is it treated by Ōtaka Yūshō in his Kwatsuyō Sampō of 1712 in connection with the analytic measurement of the circle. After Seki's time there were numerous works treating of the mere numerical measurement of the circle, such as the Taisei Sankyō^[2], commonly supposed to have been written by Takebe Kenkō,^[3] and of which twenty books have come down to us out of a possible forty-three.^[4] There is a story, generally considered as fabulous, told of three other books besides the twenty that are known, that were in possession of Mogami Tokunai^[5] a century ago. ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/156 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/157 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/158 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/159 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/160 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/161 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/162 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/163 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/164 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/165 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/166 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/167 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/168 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/169 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/170 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/171 ​analogy to the ancient Pythagorean brotherhood is seen in the mysticism of the founder. Matsunaga writes^[6] as Pythagoras might have done: "Reason is determinate, but Spirit wanders in the realm of change. Where Reason dwelleth, there is Number found; and wheresoever Spirit wanders, there Number journeys also. Spirit liveth, but Reason and Number are inanimate, and act not of their own accord. The way whereby we attain to Number is called The Art. Heaven is independent, but wherever there are things there is Number. Things, Number,—these are found in nature. What oppresses the high and exalts the humble; what takes from the strong and gives to the weak; what causes plenty here and a void there; what shortens that which is long and lengthens that which is short; what averages up the excess with the defect,—this is the eternal law of Nature. All arts come from Nature, and by the Will alone they cannot exist."

Matsunaga's Hōyen Sankyō is composed of five books, and is devoted entirely to formulas for the circumference and arcs of a circle, no analyses appearing.^[7] His first series is as follows:

This is followed by

a series which is then employed for the evaluation of π to fifty figures. The result is the following:

​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/173 ​He also gives^[8] formulas for the side of the inscribed polygon in terms of the diameter of the circle, for the various diagonals, for the lines joining the mid-points of the diagonals and the various vertices or the mid-points of the sides,^[9] and so on, none of which it is worth while to consider in a work of this nature.

It will be seen that the youri as laid down by Takebe was extended to include solid figures treated somewhat after the manner of Cavalieri, but that it was little more than a rather primitive method of using infinite series in the measurement of the simplest curvilinear figures and the sphere. We shall see, however, that it gradually unfolds into something more claborate, but that it never becomes a great method, remaining always a set of ingenious devices.