56 karine chemla
that the late history of practices of proof bears witness to circulations and
preservations that challenge the standard account sketched above. 63
At the beginning of his chapter, Volkov recalls how the sinologist
Edouard Biot, in his 1847 Essai sur l’histoire de l’instruction publique en
Chine , dismissively belittled the format of problems in Chinese mathemati-
cal texts, their absence of proof and the elementary level of state education.
With respect to what was discussed above, the additional denigration of
everything classed as educational in the historiography of mathematics may
be partly responsible for the lack of discernment regarding sources that
could have modifi ed Biot’s assessment at least to a certain extent.
In China, the approach to mathematics of the past was strikingly diff er-
ent, if we judge it on the basis of Li Rui’s Detailed Outline of Mathematical
Procedures for the Right-Angled Triangle completed in 1806, which Tian
Miao analyses in her chapter. Th is text illustrates a second form of pro-
longed relevance of ancient practices of proof, which reveals several inter-
esting features.
To be more precise, Li Rui’s practice of proof exemplifi es a revival of past
Chinese practices of proof and shows how they were at that time trans-
formed mathematically while simultaneously reshaped under the infl uence
of – or rather as an alternative to – practices of proof identifi ed as ‘Western’.
Th e topic on which Li Rui chose to write his book, the right-angled tri-
angle, was one in which, as he knew, an interest was documented in both
Chinese and Greek antiquity. Th e ninth of Th e Nine Chapters is devoted
to the right-angled triangle and it is the subject of theorems in the part of
Clavius’ edition of Euclid’s Elements that Ricci and Xu Guangqi translated
into Chinese in 1607.
Li Rui approached the right-angled triangle as was done in the tradition
which descends from Th e Nine Chapters. Among the various identifi able
traces of this approach, one notes that his book takes the form of problems
for which solutions are provided in the form of algorithms. In addition,
Li Rui makes use of the traditional terminology developed throughout
Chinese history and completed in the Song dynasty to designate the quanti-
ties attached to a triangle.
On the other hand, Tian argues, the infl uence of the Elements can be
perceived in the fact that Li Rui provided a systematic set of solutions to
all the problems that can be encountered. Moreover, he organized this set
according to the dependencies of its elements. In the system produced, the63 A similar kind of continuity in the practice of proof is described by François Patte, in his work
on sixteenth-century Sanskrit commentaries; see Patte 2004.