Methods of Solving Implicit Problems, by Seki Takakazu, collated by Hikosaburo Komatsu
This is a collated text of the second of Seki’s Trilogy. This is a very concise monograph of Celestial Element Method [天元術 tengen jutsu] and Root Extraction Method [開方術 kaihō jutsu]. In Chapter 1: Setting Element [立元], Celestial Element [天元 tengen], that is, the unknown is represented by a counting rod of 1 in the modulus [方hō] class of a counting board. Polynomials [式 shiki] in Celestial Element with numerical coefficients and their operations [演段 endan] are discussed in Chapter 2: Addition and Subtraction [加減 kagen], and Chapter 3: Mutual Multiplications [相乗 sōjō] as manipulations of counting rods on a counting board. Two polynomials representing the same quantity define an Equation [開方式 kaihōsiki] by subtracting one from the other as shown in Chapter 4: Mutual Cancellation [相 消]. The last Chapter 5: Root Extraction [開方] is devoted to the Numerical Extraction of Roots [開方術], that is the same as the so called Horner method which is erroneously attributed to W. G. Horner (1819). Japanese learnt these methods from Zhu Shijie [朱世傑] by his Introduction to Mathematics [算学啓蒙] (1299) but the methods described here are more refined. At the end of the book, Newton’s method of approximation is explained very succinctly.
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