Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/25

Rh (4) Shao-kang (Shaou-kwang). This relates to the extraction of square and cube roots, the process being much like that of the present time.

(5) Shang-kung. This has reference to the mensuration of such solids as the prism, cylinder, pyramid, circular cone, frustum of a cone, tetrahedron, and wedge.

(6) Kin-shu (Kiun-shoo, Ghün-shu) treats of allegation.

(7) Ying-pu-tsu (Ying-yu, Yin-nuh). This chapter treats of “Excess and deficiency”, and follows essentially the old rule of false position.

(8) Fang-ch’êng (Fang-chêng, Fang-ching). This chapter relates to linear equations involving two or more unknown quantities, in which both positive (ching) or negative (foo) terms are employed. The following example is a type: “If 5 oxen and 2 sheep cost 10 taels of gold, and 2 oxen and 8 sheep cost 8 taels, what is the price of each?” It is probable that this chapter contains the earliest known mention of a negative quantity, and if the ancient text has not been corrupted, it places this kind of number between 2000 and 3000 B. C.

(9) Kou-ku, a term meaning a right triangle. The essential feature of this chapter is the Pythagorean theorem, which is stated as follows: “The first side and the second side being each squared and added, the square root of the sum is the hypotenuse.” One of the twenty-four problems in this section involves the equation x+(20+14)x–2×20×1775=0, and a rule is laid down that is equivalent to the modern formula for the quadratic. If these problems were in the original text, and that text has the antiquity usually assigned to it, concerning neither of which we are at all certain, then they contain the oldest known quadratic equation. The interrelation of ancient mathematics is seen in two problems in this chapter. One is that of the reed growing 1 foot above the surface in the center of a pond 10 feet square, which just reaches the surface when drawn to the edge of the pond, it being required to find the