What is the Earliest Chinese Origin of Indeterminate Analysis -- Find out the Truth of the Mystery behind the Ta Yen Shu

by Deng Honghai

As Dr. Joseph Needham, the author of Science and Civilisation in China, pointed out, indeterminate analysis was always a marked mathematical interest of Chinese. Why had Chinese been so interested in this mathematical field? Although giving an excellent introduction to the post-4th century Chinese mathematicians’ contributions to this field, he did not examine this question because his research was limited within the available Chinese mathematical literatures.

Fortunately, he connected time indeterminate analysis with Ta yen shu (大衍術 Great Extension Mathematics) and introduced two important works in its study and application: I Ching’s(一行) Ta Yen Li Shu (大衍曆書 Book of the Ta Yen Calendar) and Chhin Chiu Shao’s(秦九韶) Shu Shu Chiu Chang (數書九章 Nine Chapters of Mathematics). The former only contains the results from the calendrical calculation by applying the Ta yen shu; the latter focused on how to apply the Ta yen shu to calendrical calculation and other public works.

He realized that the name Ta yen shu was derived from an obscure statement in the I Ching (Book of Changes) that the ‘Great Extension Number (大衍之數) ‘ is 50. Having known the mathematical procedure of the classical method of consulting the oracle of the I-Ching, he concluded: “ indeterminate analysis was connected with, if not actually derived from, an ancient method of divination using yarrow stalks.” (Joseph Needham: Science and Civilisation in China, Cambridge at the University Press, 1959, P. 119-122).His conferment of the connection between indeterminate analysis and the classical method of divination using yarrow stalks is correct. However, since he could not understand the statement, he could not further find the truth of the development of indeterminate analysis in China.

In fact, Chhin Chiu Shao already indicated in his book that the classical method of divination using yarrow stalks was one of the applied disciplines of the Ta yen shu, just like the calendrical calculation as one of its applied disciplines. Because the classical method of divination had been deified since the Xia dynasty, the truth of Ta yen shu was lost. He approached the historical development and extensive applications of the Ta yen shu since remote antiquity. Although having shown his mathematical proof, he did not have archaeological evidences to prove his logical conclusion.

This paper will have both mathematical and archaeological evidences to prove the natural reasonability of the statement in the I Ching (Book of Changes) that the ‘Great Extension Number ‘ is 50. The statement means that as a key number in Fu Xi numbering system, 50 is so important that a set basic parameters of the ancient Sifen (古四分曆 four quadrantal) Calendar, the algebraic formulation of the Pythagoras theorem (勾股定理) and it applications in terrestrial and celestial measurements, and the mathematics of circle and square (方圓術) and its applications in article makings and engineering and so on were derived from it. This was the reason for Chinese to be so interested in this mathematical field. Based on these evidences, the process of evolution of the Ta yen shu from an ancient method of calendrical calculation to the classical method of divination using yarrow stalks will logically and historically be recovered and hence the truth of the mystery behind the Ta Yen Shu will be found out.