Demonstration in Chinese and Vietnamese mathematics 531
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When constructing his solution, the imaginary Vietnamese examinee pro-
duced a text the structure of which to a large extent resembles the solution
already provided in the treatise, namely, in problem 6 of the same chapter.
One can conjecture that the Chinese examinees of the Tang dynasty were
also supposed to base their solutions on those provided in the respective
mathematical textbooks. Here we come to the focal point of the present
chapter, namely, the role the commentaries found in Chinese mathematical
treatises played in mathematical instruction and examinations. Table 15.3
provides the names of the commentators of the extant ten mathematical
treatises used in the Mathematical College of the Tang dynasty.
Table 15.3 shows that the treatises used for instruction all incorporated
commentaries, unlike the extant treatises listed under numbers 1 and 2.
Th e history of transmission of the treatises is so obscure that even if the
names of the commentators in the extant treatises coincide with those
mentioned in the bibliographies listed in Table 15.3 , it remains unknown
whether the extant commentaries are indeed identical with those used in
the Mathematical College of the Tang dynasty. An inspection of the extant
commentaries listed in Table 15.3 shows that they diff er considerably as far
as their style and contents are considered. Th e commentaries are mainly
focused on the computational procedures designed for solution of the
problems, yet the formats adopted by their authors were not the same.
Liu Hui’s commentary on the Jiu zhang suan shu contains parts written
in diff erent styles: the commentator interpreted the operations with frac-
tions exemplifi ed in the treatise using especially coined mathematical
terms; used diagrams of plane fi gures and descriptions of (probably imagi-
nary) three-dimensional models for solution of geometrical and algebraic
problems; provided detailed computations in case of the calculation of the
value of π close in style to Liu Xiaosun’s cao or left only obscure indications
which, however, may have been referring to some specifi c mathematical
contents. 58 Th e commentaries of another enigmatic fi gure, Zhao Shuang
or Zhao Junqing (conventionally these two names are
believed to be the aliases of the commentator Zhao Ying mentioned in
58 For the original text, translation and discussion see CG2004, as well as the works of other
authors quoted by Chemla and Guo; on the geometrical diagrams see Volkov 2007. Th is
variety of styles can make one ponder over the authenticity of the received commentary
conventionally credited to the authorship of the person known as Liu Hui whose biographical
data remain unknown, yet the latter problem, certainly important, is not pertinent in the
context of the present inquiry.