Methods of Solving Explicit Problems, by Seki Takakazu, collated by Hikosaburo Komatsu

Abstract

Abstract This is a collated text of the first of Seki’s Trilogy. The book is composed of four chapters: Chapter 1 Addition and Subtraction [加減kagen], Chapter 2 Partition and Joining [分合bungō], Chapter 3 Whole Products [全乘zenjō] and Chapter 4 Partial Products [折乘setsujō]. In Chapter 1 easiest examples are given to show how a mathematical problem is interpreted as an algorithm on a counting board. In Chapter 2 the byscript method [傍書法bōsho hō] of Seki is introduced by which we can express a polynomial as a sum of monomials represented by a numerical coefficient of counting rods and literal factors written on its right side. The heading means the distributive law of sums and products. Chapters 3 and 4 deal with the area and the volume of geometric figures. Formulas are not given by the algebraic byscript method but as algorithms on a counting board. The Pythagorean theorem and the Eudoxos formula of the volume of a pyramid are proved by figures. The volume formula of sphere segments is correct but its proof by an illustration is hard to understand.