32 2. MATHEMATICAL CULTURES Iby the Mongol emperor Kublai Khan and its capital established at what is now
Beijing. Unification of the country enabled Zhu Shijie to travel more widely than
had previously been possible. As a result, his Suan Shu Chimeng (Introduction to
Mathematical Studies), although based on the Jiu Zhang Suanshu, went beyond
it, discussing the latest methods of Chinese algebra. The original of this book
was lost in Chinese, but a Korean version was later exported to Japan, where
it had considerable influence. Eventually, a translation back from Korean into
Chinese was made in the nineteenth century. According to Zharov (2001), who
analyzed four fragments from this work, it shows some influence of Hindu or Arabic
mathematics in its classification of large numbers. Zharov also proposed that the
title be translated as "Explanation of some obscurities in mathematics" but says
that his Chinese colleagues argued that the symbols for chi and meng were written
as one and should be considered a single concept.The Suan Fa Tong Zong of Cheng Dawei. A later work, the Suan Fa Tong Zong
(Treatise on Arithmetic) by Cheng Dawei (1533-1606), was published in 1592. This
book is well described by its title. It contains a systematic treatment of the kinds
of problems handled in traditional Chinese mathematics, and at the end has a
bibliography of some 50 other works on mathematics. The author, according to
one of his descendents, was fascinated by books discussing problems on fields and
grain, and assembled this book of problems over a lifetime of purchasing such
books. Like the book of Zhu Shijie, Cheng Dawei's book had a great influence
on the development of mathematics in Korea and Japan. According to Li and
Du (1987, p. 186), Cheng Dawei left a record of his mathematical studies, saying
that he had been involved in travel and trade when young and had sought teachers
everywhere he went. He retired from this profession while still young and spent 20
years consolidating and organizing his knowledge, so that "finally I rooted out the
false and the nonsensical, put all in order, and made the text lucid."3.2. China's encounter with Western mathematics. Jesuit missionaries who
entered China during the late sixteenth century brought with them some mathe-
matical works, in particular Euclid's Elements, the first six books of which the
missionary Matteo Ricci and the Chinese scholar Xu Guangchi (1562-1633) trans-
lated into Chinese (Li and Du, 1987, p. 193). The version of Euclid that they used,
a Latin translation by the German Jesuit Christopher Clavius (1538-1612) bearing
the title Euclidis elementorum libri XV (The Fifteen Books of Euclid's Elements),
is still extant, preserved in the Beijing Library. This book aroused interest in China
because it was the basis of Western astronomy and therefore offered a new approach
to the calendar and to the prediction of eclipses. According to Mikami (1913, p.
114), the Western methods made a correct prediction of a solar eclipse in 1629,
which traditional Chinese methods got wrong. It was this accurate prediction that
attracted the attention of Chinese mathematicians to Euclid's book, rather than
the elaborate logical structure which is its most prominent distinguishing charac-
teristic. Martzloff (1993) has studied a commented (1700) edition of Euclid by
the mathematician Du Zhigeng and has noted that it was considerably abridged,
omitting many proofs of propositions that are visually or topologically obvious. As
Martzloff says, although Du Zhigeng retained the logical form of Euclid, that is, the
definitions, axioms, postulates, and propositions, he neglected proofs, either omit-
ting them entirely or giving only a fraction of a proof, "a fraction not necessarily
containing the part of the Euclidean argument relative to a given proposition and