306
Chapter 9
As already remarked, Takebe evidently discovered his series by trial and error, not to mention hard soroban handling. The formal concept of the Taylor series had yet to be invented. A more theoretical approach was introduced by Wada Yasushi, 3 who was born samurai in 1787 into the Bansyu clan of Hyogo province. Later he moved to Edo to become a Master of Mathematics in an astronomy institute. As we shall see momentarily, Wada’s main work was on definite integrals. He wrote sixty- six manuscripts but they were all destroyed by fire. Anxious to help his friend, and angered that other mathematicians were plagiarizing Wada’s results, Koide Kanemasa (1797–1865) published the Enri Sankei (1842), or Mathematics of the Enri, making clear that it was Wada’s work. Wada’s sole passion was mathematics and he disdained honors, with the consequence that he spent his entire life in poverty, dying in 1840. In the Enri Sankei, or the Mathematics of Enri, Wada deduces the expansion of 1 − x by assuming that 1 − x = a 0 + a1x + a 2x 2 + a 3x 3 + . . . . Squaring both he gets 1 − x = a 02 + 2a 0a1x + (a12 + 2a 0a 2 )x 2 + . . . , which implies a 0 = 1, a1 = –1/2, a 2 = –1/8, a 3 = –3/48, . . . , or 1 1 3 3 15 4 105 5 . . . x − x2 − x − x − x − 2 8 48 384 3840
1−x = 1−
(3)
in agreement with the binomial expansion. Similarly, 1 1−x
= 1+
1 3 15 3 105 4 945 5 . . . x + x2 + x + x + x + . 2 8 48 384 3840
(4)
Definite Integration With such expansions, Wada and other Japa nese mathematicians were in a position of being able to calculate various definite integrals. Their basic procedure was essentially what every calculus student learns when first calculating the areas under curves y = f(x): Divide the domain of integration from 0 to xf into small subintervals Δx = (x f )/N; approximate the area under the curve as A = Σ nN=1 f (x n )Δx , and let N → ∞. The difficult part of this procedure is evaluating Σ f(xn). Taking the function f(x) = x 2 as a standard example, we let xn = (x f ) · n/N. Then N
A=
∑ n =1
3
Sometimes Wada Nei.
N
x n2 Δx =
∑ n =1
x f3 .
x f2 n2 = N3 N3
N
∑n n =1
2.