ChineseMathematics 97
p. 91ff for the Islamic one). Still less are we about to discuss the priority, on which the evidence is
slender to non-existent; a recent evaluation is given by Karine Chemla (1994).
It seems almost inconceivable, however little mingling there was (e.g.) between Chinese and
Muslims at the Mongol court that some knowledge was not shared, although it may have been at a
fairly basic level. Given which, one might ask:
Why did Islamic mathematicians not learn anything of the ‘matrix’ method, and the use of
negative numbers as routinely practised by the Chinese? Why did the Chinese not learn the formula
for the volume of a sphere, the construction of the regular solids, and the properties of conics? The
problem is as much to explain what was not transferred as to find what was.
A different question arises in response to the argument that the occurrence of the same idea
in two cultures must imply copying. We could then ask (putting ourselves imaginatively into the
situation of a medieval Chinese mathematician) how difficult we think its discovery might be.
Pascal’s triangle (the one with the binomial coefficients in, Fig. 9) is a case in point. At one level, it
is a pattern of numbers which one could discover if one were playing with them idly. As Martzloff
comments (1995, p. 91) on Needham’s assertion that it was transferred from China to India: what
was the triangle, in a given culture, being used for? Ancient Indians used something like it for
problems involving combinations, Pascal for probabilities. In Samarkand and Beijing it seems to
have been more an aid towards root-extraction, via the relation of thenth row in the triangle
to the coefficients in(a+b)n. But this is not so difficult that one has to suppose its discovery to
have happened only once. Sometimes at an elementary level, the same ideas occur with the same
Fig. 9Pascal’s triangle (from Zhu Shijie (1303)).