Complete Book of Mathematics Vol. 10: Geometry, by Seki Takakazu, Takebe Kataakira and Takebe Katahiro, collated by Hikosaburo Komatsu

Volume 10 is named Algorithms for Appearances [形法 keihō] as the first of five volumes on geometry. It consists of four chapters. Chapter 1 Algorithms for Squares [方法 hōhō] deals with squares. Chapter 2 is entitled Algorithms for Rectangles [直法 chokuhō]. The longer side of a rectangle [直 choku] is called [縦 tate] or length [長 chō] and the shorter side [横 yoko] or [平 hei] or width [闊 katsu]. Their product is the area [積 seki]. In the following Chapter 3 Algorithms for Right Triangles [勾股法 kōko hō] this is used to prove the Pythagorean theorem. The shorter leg hook [勾 kō] and the longer leg leg [股 ko] of a right triangle stand for the right triangle itself. The hypothenuse is called chord [弦 gen]. There are no formal treatments of proportions as in Euclid’s Elements but it is remarked that rectangles or right triangle with a fixed ratio of heights to widths make a straight line. There are two appendices to Chapter 3. The first one deals with Pythagorean triplets, i. e. the integral solutions of x 2 +y 2 =z 2. The second appendix is a brief introduction to the methods of survey described in Sea Islands Mathematics [海島算經 Haidao Suanjing] by Liu Hui. In the last Chapter 4 Algorothms for Polygons [斜法 shahō] the authors develop a general theory of triangles [三斜 sansha], quadrilaterals [四 斜 shisha], pentagons [五斜] and hexagons [六斜] with the use of their elimination theory