307
The Mysterious Enri
y
Δx
xn
x x
Figure 9.2. In this standard calculus example, the function is y = x 2. The interval along the x-axis between 0 and xf, the final value of x, is divided into subintervals each with width Δx = xf /N, where N is some large integer. An intermediate point xn is given the value xn = nΔx = nxf /N, where n < N is another integer. As explained in the text, taking the limit N→∞ leads to the definite integral of x 2.
f
But from chapter 2, problem 4-7, we know how this series sums, and so A=
x 3f ⎛ N 3 N 2 N ⎞ + + ⎟. ⎜ 3 6⎠ N3 ⎝ 3
Taking the limit N → ∞, yields the well-known result A = x 3f /3. Calculus students, though, quickly learn more efficient methods of evaluating integrals. Employing the universal method of substitution, we usually bring complicated functions into polynomial form, which is then easy to integrate. The traditional Japanese geometers did not go that far. Regardless of the function, they expanded it in a series and calculated the definite integral as we have just done. Wada’s Enri Sankei, for example, includes many tables, the Enri Hy¯o 4 of definite integrals of irrational functions. Wada’s near contemporary, Uchida Kyu ¯ mei (?–1868), made a detailed study of such integrals. It is worth saying a few words about Uchida. Like most of the other mathematicians we met earlier, Uchida was a samurai, from the Hikone clan in Shiga province, and he eventually become mathematics teacher to the lord of the clan himself, Ii Naosuke. Uchida’s main work is the Sanpo¯ Kyu ¯ seki Tsu-ko of 1844, the five–volume Theory of Integrals, from which we have already taken solutions to problems 22 and 23 in chapter 5 and problem 19 chapter 6. Here are further examples of definite integration from the Enri Sankei and the Sanpo¯ Kyu ¯ seki Tsu-ko that employ the expansion of 4
1 − x 2 and 1/ 1 − x .
Literally “folding tables,” from the Japa nese term for integration.