Xu Yue

Born: about 160 in Donglai, Shandong province, China
Died: about 227 in China

We know a little of Xu Yue but the main text which bears his name, the Shushu jiyi (Notes on Traditions of Arithmetic Methods), is probably the work of a later author trying to claim a certain respectability for his writings.

Xu Yue was a pupil of Liu Hong and he studied mathematics under the famous calendar expert. Liu Hong worked at the Imperial Observatory and it was there that Xu Yue held discussions with him and also with the head of the Astronomical Bureau. Mathematics was used by Liu Hong and others at the Observatory in their studies of astronomy and the related work on the calendar which, of course, was based on the apparent motion of the sun and the moon. It was natural, therefore that Xu Yue would gain expertise from these men in astronomy and calendar science.

It is reported that Xu Yue wrote a commentary on the Nine Chapters on the Mathematical Art. This preceded the major commentary written by Liu Hui in the second half of the third century, and it would appear that Liu Hui commented on a version of the text which did not include Xu Yue's comments. As a result Xu Yue's commentary has not come down to us.

Whether Xu Yue wrote the Shushu jiyi (Notes on Traditions of Arithmetic Methods) is uncertain. It is a rather strange work filled with ideas from Buddhism and from Taoism and describes arithmetical systems intermingled with religious nuances. The version which has come down to us today has a commentary by Zhen Luan written around 566, but that does little to help understand the difficulties in the text.

Fourteen old methods of calculation are mentioned in the text. One of these uses a device resembling the abacus called ball-arithmetic. Three others also uses balls, one involving balls in columns, one involving two balls of different colours which move at right angles to each other suggesting almost the idea of Cartesian coordinates. Needham writes in [4] that Xu Yue:-

... shows an interesting appreciation of coordinate relationships.

Unfortunately there is no great clarity in the descriptions.

There is no doubt that one of the main aims of the text is to introduce a notation which will allow the representation of large numbers. Most historians believe that the aim of the author was to suggest that it was possible to represent any number, no matter how large. Three systems of powers of 10 are given. The lower system is based on the sequence of powers of 10

10, 102, 103, 104, 105, ...

the middle system on powers of 104

104, 108, 1012, 1016, ...

the upper system being based on powers of 10, each being the square of its predecessor

104, 108, 1016, 1032, ...

Finally, Xu Yue talks about calculations of the nine balls. Again what is intended here is unclear, but the commentator Zhen Luan is in no doubt that this refers to a 3 by 3 magic square.

Despite being an obscure text, after editing by Li Chunfeng, the Shushu jiyi (Notes on Traditions of Arithmetic Methods) was selected as a text for the Imperial examinations in 656 and became one of The Ten Classics in 1084.

Article by: J J O'Connor and E F Robertson

December 2003

MacTutor History of Mathematics