ChineseMathematics 95
8. On ‘transfers’—when, and how?
Seek knowledge, even as far as China. (The Hadith)
Aside from the unusual nature of algebra in Song times—why was it being done, and what were
its rules of operation?—we have what is sometimes an open question, sometimes an argument
about origins. Several apparently similar techniques, most particularly the use of the binomial
triangle (‘Pascal’s triangle’) and of techniques for solving equations by approximation, seem to
have appeared both in China and in the Near East around the same time, say the eleventh to twelfth
centuries. We know of numerous contacts between China and the Islamic world, the earliest
relevant one being the compiling of a calendar—a perennial headache for the government—by a
Muslim named Ma Yize for the Emperor in the late tenth century. From then on, Muslim scholars
seem to have been frequent visitors, until under the Mongols of the Yuan dynasty (thirteenth
century) a large number of Muslim ‘artisans’ were settled in north China; and one Zhamaluding
presented the Emperor Qubilai (‘Kubla Khan’, the son of Chinggis) with yet another calendar,
and astronomical instruments. Arabic or Persian astronomical and mathematical books were also
imported. Given all this, it seems to make for historical economy to suppose that some transmission
of knowledge was taking place one way or the other. However, as with the question of ‘counting
rods’ and the decimal system, it is more complicated than we might wish. For example:
- There is no record of any mathematician of either culture referring to work from another. As
Saidan notes:
Al-Mas‘ ̄ud ̄i...writes much about Hindu wisdom and learning, and refers to Chinese technology and social life,
but never to Chinese science. The learned al-B ̄ir ̄un ̄i does not refer to Chinese science in hisChronologynor does
S.a‘id in hisT.abaq ̄at. (Saidan in al-Uql ̄idis ̄i 1978, p. 455) - The essential works in which the Chinese methods were introduced, such as those of Jia Xian
(eleventh century), are lost and as Martzloff says:
Our knowledge of this subject is very imprecise, since it is based on extracts from 13th century works such as
those of Yang Hui or Zhu Shijie, accessible to us through the medium of 19th century editions. (Martzloff 1995,
p. 17) - So far as wecandraw a parallel between Chinese and Islamic algebra, they seem to have been
very different pursuits.
In non-Chinese algebraic manuals, this character [the equation] is treated with the utmost care and respect; it
is studied in minute detail, in the search for the secrets of the algebraic formulae which unveil the results. This
initially involves a study of equations of degree two, since these are the most docile...But in medieval China, the
degree of equations is of little importance...Chinese equations are not exactly equations, but algebraic forms or
schemes for extracting roots, which...consist of sequences of numbers to be operated on, as though one were
extracting square, cube ornth roots. (Martzloff 1995, p. 261–2)
Given this, and given the specialized technical language of Chinese algebra, it is not surprising
that the question of what could have been transmitted is a hard one. Let us consider a ‘practical’
problem which leads to an equation, fromShushu jiuzhang, III.1. The problem (Libbrecht 1973,
pp. 97–9) is to find the area of a ‘pointed field’ (Fig. 8) when one knows the measurements
shown. Again, the answer could be found trivially (h 1 =36,h 2 =20 by Pythagoras; area=
1
2 (^30 ×(^36 +^20 ))=^840 ). Again, one assumes that Qin knew the answer in setting up the
problem; and, as in the preceding question, that he knew it could be solved easily. Instead, he