309
The Mysterious Enri
In modern notation we would write this as 1 1⋅3 1⋅3 ⋅5 ⎡ ⎤ l = 2r arcsin(h /r ) = 2h ⎢1 + (h /r )2 + (h /r )4 + (h /r )6 + . . .⎥, 2⋅3 2⋅4⋅5 2⋅4⋅6⋅7 ⎣ ⎦
or, with x = h/r, l = arcsin x = x +
1 3 1⋅ 3 5 1⋅ 3 ⋅ 5 7 . . . . x + x + x + 2⋅3 2⋅4 ⋅5 2⋅4 ⋅6 ⋅7
Two decades earlier, in 1822, Sakabe Ko¯ han (1759–1824) had employed the same procedure to to find the length of an ellipse, a result he wrote down in his manuscript Sokuen Syukai, or “Circumference of Ellipse.” Sakabe was the first Japanese mathematician who succeeded in doing this.
Problem 3 Here are three of Wada’s expansions from plate 9.3. Two are for the area of a circle and one for the circumference. Confirm their correctness. 1 1 1 1 ⎛ ⎞ (1) A = d 2 ⎜ 1 − + − + − . . .⎟ , ⎝ ⎠ 3 5 7 9 1 1 3 15 105 ⎛ . . .⎞⎟ , (2) A = d 2 ⎜ 1 − − − − − ⎝ ⎠ 2 ⋅ 3 5 ⋅ 8 7 ⋅ 48 9 ⋅ 384 11 ⋅ 3840 1 3 15 105 945 . . .⎞ ⎛ (3) C = 2d ⎜ 1 + + + + + ⎟. ⎝ ⎠ 2 ⋅ 3 5 ⋅ 8 7 ⋅ 48 9 ⋅ 384 11 ⋅ 3840 Solutions are on page 311. As a final example of the Enri, also from Uchida’s Sanpo¯ Kyu ¯ seki Tsu-ko, we return to problem 19 from chapter 6, one of the most difficult problems in the wasan. We were to find the surface area cut out of an elliptic cylinder by the sectors of two right circular cylinders. The problem resulted in the integral S = 4d Dd
∫
1
0
t 1/2 1 − (d /D )t
1 − (1 − b 2/a 2 )(d 2/a 2 )t 2 dt , 1 − (d 2/a 2 )t 2
which we evaluated numerically. Although the author of the problem, Matsuoka Makota, did not write down his detailed calculations on the sangaku, he evidently did much the same but in a fashion more suitable for soroban calculations. The integrand contains three square roots, two of which are of the same form. We use the binomial expansions above for 1 − x and 1/ 1 − x , applying the fi rst expansion to the two square roots in the