Seki’s Trilogy: Methods of Solving Explicit Problems, Methods of Solving Implicit Problems and Methods of Solving Concealed Problems

Abstract

Seki Takakazu (1642?–1708) classified mathematical problems into three categories: explicit problems, which can be solved by arithmetic, implicit problems, which can be solved by an algebraic equation of one unknown, and concealed problems, which needs simultaneous algebraic equations with more than one unknowns. He is believed to have written for each category of problems a book of their solutions. The three books: Methods of Solving Explicit Problems, Methods of Solving Implicit Problems and Methods of Solving Concealed Problems are called Seki’s Trilogy and used as the standard textbooks in the later Seki School of Mathematics. If a student masters a book of them, he is given a license which certificates his degree of understanding in mathematics.

Each part of these books is without doubt Seki’s writing but we don’t think that the books as they are now are the same as what Seki wrote. The Methods of Solving Implicit Problems may be his original but the others seem to be the results of edition by Yamaji Nushizumi (1704–1772), who instituted Seki’s School of Mathematics. The following are our restorations of the Yamaji editions around 1726 or later.

Supported by JSPS Grant-in-Aid Scientific Research (C) 16540119 (2004–2006), MEXT Grant-in-Aid on Priority Areas 17083006 (2005–2010) and JSPS Grant-in-Aid Scientific Research (C) 20540107 (2008–2011).