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Mathematical Treasure: Collected Mathematical Works of Li Shanlan

Frank J. Swetz (The Pennsylvania State University)

Li Shanlan (1811–1882), Chinese mathematician and educational reformer, was a major force in the introduction of western mathematics into 19th-century China. While he assisted in the translation of English-language mathematical materials into the Chinese language, he was also a talented mathematician in his own right. In 1867, Li published Ze gu xi suan xue (Collected Mathematical Works) in six volumes. This work discussed a wide range of mathematical ideas, including the properties of the ellipse, power series, and the use of logarithms and trigonometric functions. The title page and introduction for this work are shown here:


The properties of various solids are considered in the work. The solids are “unfolded” in different patterns to study their structure.

Li presented his reader with a generalization of the “Pascal Triangle”. On the opposite page, he demonstrated sequences of figurative numbers by using the traditional Chinese method of “piling up cubes.” Can you derive the number sequences depicted?

The images above were obtained through the courtesy of the Asian Library, Columbia University.

Li taught mathematics at T'ung-wen-kuan (later part of Imperial University), the college established in Beijing to train Chinese to interact with foreigners. This photograph from the Wellcome Institute shows him (center) with a modern mathematics class around 1868–1870. Scottish traveller John Thomson (1837–1921) took the photograph and published it in his 1873 Illustrations of China and Its People. It is available via a Creative Commons License (CC BY 4.0).

Li Shanlan and pupils at Beijing college, 1868–1870.


Frank Swetz, “The ‘Piling Up of Squares’ in Ancient China,” The Mathematics Teacher (January 1977), 70:72–79.    

Index to Mathematical Treasures

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: Collected Mathematical Works of Li Shanlan," Convergence (September 2020)