Mathematical proof: a research programme 57
solution to any problem depended only on those before it. Th e proofs of the
correctness of the algorithms were thus a key element for deciding over the
structure of the system.
Tian highlights several mathematical innovations in the book. To
begin with, Li Rui invoked combinatorial methods to state and solve any
problem that could be asked about a right-angled triangle. Moreover, Li
Rui innovatively employed the ancient ‘heavenly unknown ( tianyuan )’ 64
method to establish the correctness of the algorithms which solve each
of the problems in the most uniform way possible. Th e earliest surviving
evidence for this method, which is equivalent to the modern practice of
using polynomial algebra to set up an equation which solves a problem,
dates to 1248, the year that Li Ye completed his Sea-Mirror of the Circle
Measurements ( Ceyuan haijing ). Aft er having been forgotten in China,
the method had been recovered by Mei Juecheng in the fi rst half of the
eighteenth century, thanks to the understanding Mei gained through his
acquaintance with European books of algebra. 65 In particular, Mei deci-
phered the meaning of the algebraic symbolisms for writing down poly-
nomials and equations that had been developed in China a few centuries
earlier and had since been lost.
Li Rui could thus rely on the method and its related symbolisms that had
been rescued from oblivion only a few decades before he wrote his book.
When using the symbolism to establish algebraically the correctness of the
algorithms he stated, Li Rui was using symbols that diff ered in form from
those of Diophantus, but which had played a similar part in the past. Like
Li Ye, Li Rui used these symbolic notations to account for the correctness
of the equation – the ‘procedure’ – yielded to solve a general problem.
However, the way in which Li Rui was now using them modifi ed the status
of the proofs carried out with them. Th e main point that Tian highlights in
this respect is that, when considering given quantities attached to a triangle
as data, Li Rui discriminated among the diff erent categories of triangles
according to the relative size of the data in them. More precisely, in contrast
to Li Ye before him, Li Rui formulated as many problems as there were
distinguishable cases so that he could prove the correctness of the general
equation in a way that would be valid for each case and that would establish
64 Th e literal interpretation of the expression tianyuan is ‘celestial origin’. Th is interpretation
permits the identifi cation of occurrences of the concept before the thirteenth century in the set
of mathematical classics gathered in the seventh century; see above. I shall come back to this
point in a future publication.
65 On this episode, compare Needham and Wang Ling 1959 : 53, Horng Wann-sheng 1993 : 175–6,
Yabuuti Kiyosi 2000 : 141–3.