Demonstration in Chinese and Vietnamese mathematics 535
as ‘As for those [texts/papers] without commentaries’ can be understood
in at least three diff erent ways: (1) it refers to a commentary expected to
be written by the examinee in his examination paper but omitted for some
reason; (2) it refers to a commentary missing in one of the two treatises of
the ‘advanced programme’ which constituted the topic of the examination,
and (3) the word ‘commentary’ zhu had here the technical meaning ‘to
preappoint a candidate to a position’. 61 Th e third option hardly seems to
be relevant in this particular context. Siu and Volkov ( 1999 ) have argued
for the fi rst option mainly on the basis of the inspection of the only extant
treatise of the ‘advanced programme’, the Qi gu suan jing by
Wang Xiaotong in which almost all the problems are provided with com-
mentaries. However, a large part of the original treatise is lost: according to
the bibliographical sections of the Jiu Tang shu and Xin Tang shu , the book
originally contained four juan ( JTS 47: 6b; XTS 59: 14a) while the Song
shi mentions only one juan (SS 207: 1a). Th e extant version contains only
twenty problems; the texts of problems 17–20 and of the respective com-
mentaries are partly lost ( SJSSb : 434–5). It is therefore impossible to know
whether every single problem of the Tang dynasty version of the treatise
was commented upon by Li Chunfeng, or whether a certain number of
the problems were left without commentaries. 62 Moreover, nothing can
be known about Li Chunfeng’s commentaries on the second book of the
‘advanced programme’, the Zhui shu by Zu Chongzhi, since the book had
already been lost by the time of the Song dynasty; it is equally possible
that only some problems contained commentaries. If this was the case,
the phrase wu zhu zhe , ‘as for those without commentaries’, may have
referred to paradigmatic problems from the treatises used as textbooks in
the ‘advanced programme’ which did not contain commentaries on certain
problems. Th is option leads to the following hypothesis: in the ‘advanced
programme’ examination tasks were compiled on the basis of problems
from the Qi gu suan jing and Zhui shu ; if the original problem contained
a commentary, the examination criteria were the same as in the ‘regular
programme’ examination: the examinee had to ‘elucidate numbers’ and to
‘elucidate in detail the internal structure of the [computational] procedure’,
that is, to compile a text similar to the original commentary. If the problem
taken as the model for the examination task did not contain a commen-
tary, the candidate was not asked to provide ‘elucidations’ but to ‘make the
numerical data coherent’, and ‘not to make mistakes in the meaning and
61 Des Rotours 1934: 43, 49, 217, 244, 266, 268; Hucker 1985 : 182, nos. 1407–8.
62 Th e interested reader will fi nd the annotated translation by Berezkina 1975 highly useful.