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that he also solved the fourteenth problem (see Section 4 of Chapter 9), which
involved an equation of degree 1458. Although the procedure was a mechanical one
using counting boards, prodigious concentration must have been required to execute
it. What a chess player Seki Kowa could have been! As Mikami (1913, p. 160)
remarks, "Perseverance and hard study were a part of the spirit that characterized
Japanese mathematics of the old times."
Seki Kowa was primarily an algebraist who converted the celestial element
method into two sophisticated techniques for handling equations, known as the
method of explanation and the method of clarifying things of obscure origin. He
kept the latter method a secret. According to some scholars, his pupil Takebe Kenko
(1664-1739) refused to divulge the secret, saying, "I fear that one whose knowledge
is so limited as mine would tend to misrepresent its significance." However, other
scholars claim that Takebe Kenko did write an exposition of the latter method, and
that it amounts to the principles of cancellation and transposition. (See Section 2
of Chapter 3.)Determinants. Seki Kowa is given the credit for inventing one of the central ideas
of modern mathematics: determinants. He introduced this subject in 1683 in Kai
Fukudai no Ho (Method of Solving Fukudai Problems).^10 Nowadays determinants
are usually introduced in connection with linear equations, but Seki Kowa developed
them in relation to equations of higher degree as well. The method is explained as
follows. Suppose that we are trying to solve two simultaneous quadratic equations
ax^2 + bx + c = 0
a'x^2 + b'x + c = 0.
When we eliminate x^2 , we find the linear equation
(a'b - ab')x + (a'c - ad) = 0.
Similarly, if we eliminate the constant term from the original equations and then
divide by x, we find
(ac' - a'cjx + (be' - b'c) = 0.
Thus from two quadratic equations we have derived two linear equations. Seki
Kowa called this process tatamu (folding).
We have written out expressions for the simple 2x2 determinants here. For
example,but, as everyone knows, the full expanded expressions for determinants are very
cumbersome even for the 3x3 case. It is therefore important to know ways of
simplifying such determinants, using the structural properties we now call the mul-
tilinear property and the alternating property. Seki Kowa knew how to make use of
the multilinear property to take out a common factor from a given row. He not only
formulated the concept of a determinant but also knew many of their properties,
including how to determine which terms are positive and which are negative in the
expansion of a determinant. It is interesting that determinants were introduced in(^10) The word fukudai seems to be related to fukugen suru, meaning reconstruct or restore. Ac-
cording to Smith and Mikami (1914, P- 124), Seki Kowa's school offered five levels of diploma, the
third of which was called the fukudai menkyo (fukudai license) because it involved knowledge of
determinants.