By the mid 19^{th} century, China, the “Celestial Empire,” found itself rocked by a series of violent events: western armies had penetrated the Empire seeking imperial concessions, internal revolts and natural disasters ravished the countryside, and the ruling Qing dynasty was in a state of decline. No longer did the dictates of Confucian tradition insure social harmony. Court reformers urged a modernization, a “Self-Strengthening Movement,” of the Chinese world outlook. In particular, it was felt that a new educational system was required, one that stressed the study of western sciences and political theories. Most prominent among these reformers was Prince Kung (I-hsin) of the Imperial Court. Kung pleaded his case in a memorial to the throne, in which he noted (Swetz 1974):

I have learned that when Western scholars make weapons,

They use mathematics for reference and exert their energy

in deep thinking to make daily increases and alterations.

New schools were begun and the study of sciences, particularly mathematics, was stressed. To satisfy the learning needs of this new era of students, textbooks had to be obtained, including some Chinese mathematical classics restored and western texts translated into the Chinese language. An existing British-established London Missionary Society Press Bureau responded to the latter need. The leading missionary translator in the effort to produce appropriate mathematics books was Alexander Wylie (1815-1887). Wylie was proficient in several oriental languages and self-taught in mathematics. He was the first modern western observer to comment on the status and depth of traditional Chinese mathematics. Wylie judiciously chose texts for translation that he believed would best satisfy the needs of the Chinese; among these were contributions by Augustus De Morgan (algebra), Elias Loomis (calculus), and John Herschel (astronomy and mechanics); he was joined in his task by the Chinese scholar and mathematician Li Shanlan. Together, these two men produced several texts that would henceforth change the Chinese conception and appreciation of mathematics.

In 1853, Alexander Wylie published *Shu xue qi meng*, a basic arithmetic book. His opening comments and the “Table of Contents” are provided for general information. The entire book can be viewed in digital form provided by the National Library of Australia.

The cover of Wylie’s *Arithmetic:*

Wylie's introduction to his *Arithmetic:*

Remainder of the introduction and a table of contents:

On the pages below, Wylie demonstrated several kinds of Chinese numerical notation. On the left hand page, each column contains a different set of symbols: the first uses commercial numerals; the second column is traditional rod notation; the third uses the complex characters employed by accountants; and, finally, the fourth column gives the popular standard form beginning with “0” and ending with the character that represents 10^{40}.

The following pages contain a numerical configuration whose meaning remains unclear.

For an algebra text, Wylie chose Augustus De Morgan’s, *The Elements of Algebra*, a popular British book at this time. The translation was entitled *Dai shu xue* and was published in 1859. In his "Prologue," Wylie explained why he believed this text appropriate for the Chinese. In the translation of this work, he was assisted by Li Shanlan. Here is an image of the cover:

In his notes to the western reader at the beginning of the text (see pages i-iv, below), Wylie provided a justification for his decision to use De Morgan’s book. He also gave valuable insights into the traditional algebraic abilities of the Chinese. In particular, he called attention to the “Celestial Element” method for solving equations as explained by Li Zhi in *Ceyuan haijing* *(Sea Mirror of Circle Measurement,* 1248). This is perhaps the first discussion of this Chinese mathematical technique available to a western reader. Note the reversal of page numbers.

Pages within the text. Numbering proceeds from right to left for the columns, 1-13:

### References

Frank Swetz, “The Introduction of Mathematics in Higher Education in China, 1865-1887,” *Historia Mathematica *1 (1974), pp. 167-179.

J. J. O'Connor and E. F. Robertson, "Li Shanlan," *MacTutor History of Mathematics Archive,* December 2003

Index to Mathematical Treasures