510 alexei volkov
textbook can become the starting point of a professional mathematical
inquiry. Similarly, it well may be possible that in a given mathematical tra-
dition there was no wall separating texts of the two types from each other,
and a special investigation of the social circumstances of the use of given
mathematical texts has to be provided each time in order to avoid histo-
riographic distortions. Unfortunately, even the most outstanding modern
historians have oft en presented Chinese mathematical treatises as if they
were research monographs; this approach to Chinese mathematical texts
is found already in Mikami ( 1913 ) and certainly in Yushkevich ( 1955 ) and
Needham ( 1959 ), not to mention their numerous Chinese counterparts. An
attempt to classify the mathematical problems found in Chinese treatises
was recently made by Martzloff , 2 yet his classifi cation apparently refl ected
the seeming heterogeneity of Chinese mathematical treatises as perceived
by modern historians solely on the basis of the contents of individual
problems rather than the way in which mathematical treatises containing
them were actually read and used in traditional China. Presumably, there
may have existed social settings in which one and the same problem was
treated as belonging to diff erent categories. It can be demonstrated that the
majority of the extant treatises of the late fi rst millennium bce to the fi rst
millennium ce were used as mathematical textbooks in state educational
institutions for several centuries, 3 unlike the mathematical treatises of
the Song (960–1279), Yuan (1279–1368) and Ming (1368–1644) dynas-
ties of which the circumstances of use are oft en unknown. Unfortunately,
all the attempts to off er a plausible reconstruction of the functioning of
these texts in educational context have been thwarted by the lack of data
concerning mathematics instruction in traditional China in the late fi rst to
early second millennium ce , and, in particular, by the lack of the original
examination papers. To circumvent this diffi culty, in what follows I will use(^2) Martzloff 1997 : 54 suggests that the mathematical problems in Chinese treatises belonged
to the four following categories: (1) ‘real problems’ (applicable in real-life situations); (2)
‘pseudo-real problems’ (‘neither plausible nor directly usable’); (3) ‘recreational problems’;
(4) ‘speculative or purely mathematical problems’. Only problems of category (2) thus may
have been used in mathematical instruction, while problems of type (4) represented ‘pure
mathematics’. Martzloff himself ( 1997 : 58) played down the applicability of his classifi cation
when stating that the problems of category (1) also belonged to category (4).
(^3) Th e circumstances of the use of the recently unearthed mathematical treatise Suan shu
shu (Writing on computations with counting rods) as well as the mathematical
treatises and fragments found in Dunhuang caves remain unknown. Here and below I use
the pinyin transliteration of the Chinese characters which nowadays has become a de facto
standard in continental European sinology. I use my own translations of the titles of Chinese
mathematical treatises; for the reader who may be confused by these translations I provide
a list of them in Appendix II together with the translations of the titles as found in Martzloff
1997.