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

This is the last of Seki’s Trilogy. Here he gives a general procedure to eliminate a common unknown from two polynomial equations, which is the same as the elimination theory of Étienne Bézout. The book has five chapters. In Chapter 1: Real and Fictitious [真虚 shinkyo] and Chapter 2: Two Equations [兩式 ryōshiki] we learn how to formulate a system of algebraic equations as a succession of two equations in an unknown to be eliminated with polynomials of the other unknowns as coefficients. Chapter 3: Estimates of Degrees [定乘 teijō] gives an estimate of the degree of eliminated equations. The elimination is carried out in two steps. Chapter 4: Transformed Equations [換式 kanshiki] shows how to construct

n

equations of degree less than

n

out of a system of two equations of degree

≤ n

. Then the eliminated equation is obtained as the determinant of their coefficients equated to 0 as shown in Chapter 5: Creative and Annihative Terms [生剋 seikoku], where Seki gives the expansion of determinants of order up to 4. The disorder of table happens in the case of order 4, for which we append an amendment for disorder and alterations of sheets 14 and 15

1

. If Seki had stopped here, he would have been praised for having written the most concise and complete book on ellimination. His errors occurred in the expansion of determinant of order 5, for which we refer the reader to Goto–Komatsu [1].