94 A History ofMathematics
Exercise 5.Find the equation which solves Li’s problem, and check that x= 240 is a root.A note on ‘equations’
At various points in earlier history it has been necessary to exercise caution about using the word
‘equation’—for the activities of the Babylonian scribes, for Euclid’s book II, and so on. Still more
have we tried to emphasize that, if we write a Babylonian problem in terms ofxs, it is to help
us read it and not to indicate how the authors thought of it. If this caution is noticeably absent
in those who write about the Song mathematicians, this is because what they wrote down does
lookremarkably like an equation, even if it has noxand no ‘=0’. To see this, look at Fig. 7,
which reproduces the ‘equation’ for Qin’s problem 6.2 (Martzloff 1995, p. 233ff). We translate
this as−x^4 +15,245x^2 −6,262,506.25= 0The coefficients are written in a column, from the constant term downwards
(1, 0, 15,245, 0, 6,262,506.25); below each one is its ‘rank’, a verbal/symbolic description of
the power of the unknown to which it belongs. Positive (cong) and negative (yi) coefficients are
distinguished by writing the relevant word by them. It is not an equation as we know it—but it can
be seen as a convenient translation of one to the language of arrays which had been so success-
ful in theNine Chapters; and Qin’s subsequent manipulations seem to be related to thefangcheng
method. No more an equation than an array is a matrix, it is a clearly defined tool of equivalent
sophistication. Perhaps we should still be cautious about translating it into our own terms; but we
hardly need a dictionary to do so.
Fig. 7The equation for Qin’s problem 6.2.