**Li Shanlan** was the greatest Chinese mathematician of the 19th century who translated Western works and gave many summation formulas.

- There are two slightly different versions of how Li first came upon mathematics.
- One suggests that when he was eight years old he found a copy of the Nine Chapters on the Mathematical Art in the library of a private school.
- In 1825, when he was fourteen years old, Li studied the first six books of Euclid's Elements which had been translated into Chinese in 1607 by Xu Guangqi and Matteo Ricci.
- One was the Ce yuan hai jing Ⓣ(Sea mirror of circle measurements), originally written by Li Zhi in 1248, which was of fundamental importance in the development of Chinese algebra.
- The other was Gougu geyuan ji Ⓣ(Record of measuring segments of a circle) which was a trigonometry text written by Dai Zhen, who is famous as an editor of the Nine Chapters on the Mathematical Art, and had been appointed as an editor for compiling an encyclopaedia of knowledge by the Emperor Qianlong in 1773.
- The "tian yuan" or "coefficient array method" or "method of the celestial unknown" of setting up equations, which Li learnt from Li Zhi's famous text, had a huge influence on him and he began to push these algebraic techniques forward solving a whole variety of new problems.
- Although he had been largely self educated in mathematics, Li now made contact with others having the same interests.
- He wanted to make a career for himself in mathematics but at this time in China the subject was not considered of sufficient importance that one could make a living as a professional mathematician.
- The best way for a scholar to earn his living, yet have sufficient time to push forward his research, was to be employed as a private tutor.
- This is because up until about 1850 it had relatively little impact on him.
- The British claimed that they were supporting the Chinese people against their leaders (always a good policy!).
- Certainly the Chinese government had not been popular and after this its prestige declined further.
- The Taiping Rebellion began in 1850 when a religious group based on Christianity began to gain huge support.
- Well organised militarily, they made gains defeating the Imperial army in April 1852.
- Li looked to avoid getting caught up in the Taiping Rebellion and, in 1852, he went to Shanghai which looked safe.
- From a scientific prospective the move to Shanghai was very significant for Li since there he met with Protestant missionaries from the London Missionary Society, and in particular he met Alexander Wylie.
- With Alexander Wylie, Li translated Elements of Analytical Geometry and of the Differential and Integral Calculus which had been written by Elias Loomis and published in New York in 1851.
- Their Chinese translation was published in 1859 and became the first book to introduce Newton's calculus into China.
- Li also worked with the protestant missionary Joseph Edkins on a translation of W Whewell's An elementary treatise on mechanics.
- It was not published until several years after the translation was made, finally being published under the title Zhongxue Ⓣ(Middle school) in 1867.
- Another major translation which Li undertook with Alexander Wylie was the translation of the final nine books of Euclid's Elements.
- Other mathematics books translated by Li include De Morgan's Elements of algebra but Li did not just translate mathematics book, however, for with Alexander Williamson and Joseph Edkins he translated John Lindley's Botany.
- He also translated John Herschel's Outlines of astronomy.
- For eight years Li worked with the missionaries of the London Missionary Society in Shanghai.
- During this period the Taiping Rebellion had at first had great success with the capture of Nanking in 1853.
- However, internal conflicts between those who led the rebellion weaken it but in 1860 they tried to re-establish their progress and take Shanghai but the attempt failed.
- Li left Shanghai, probably before the attack on the city, and moved to join the staff of Xu Youren who was Governor of Jiangsu province.
- They financially supported the publication of Li's complete mathematical works which were published in Nanking in 1867.
- Li was recommended to the newly established T'ung-wen-kuan (College of combined learning) in 1864.
- However Li did not take up the appointment until 1866 for he did not wish to be part of a translating school.
- However the government realised the importance of mathematics in the development of the country and in 1868 the T'ung wen-kuan was upgraded to a college and a department of mathematics and astronomy was added.
- After this, in July 1869, Li was happy to accept the position of Professor of Mathematics.
- There, Li worked with William A P Martin (1827-1916), who served as president of the college from the time of its upgrading in 1869 until 1882, teaching mathematics and preparing translations of scientific works.
- Although Li Shanlan is important as a translator of Western science texts, it is not in this capacity that he is most famous.
- These were remarkable in that although he became familiar with Western mathematics he actually based his research on ancient Chinese mathematics.
- One has to praise him very highly for this approach for his neither threw away the heritage of Chinese mathematics nor did he live in the past by ignoring the progress which had been made in the West.
- Li wrote Duoji bilei Ⓣ(Summing finite series) (published in 1867 as part of his collected works) where, in Chapter 4, he gave fascinating formulae relating binomial coefficients, Stirling numbers, Eulerian numbers and many others.
- The works of the great astronomer Guo Shoujing concerning the inequalities of the solar and lunar motion, Wang Lai's iterated sums, Dong Fangli's cyclotomical computations, and lastly the summation of series which appear in the algebra and the differential calculus of the Westerners constitute the major part of this chapter.
- His use of a generalised version of Pascal's triangle is also explained.
- Li worked out his own form of integration to compute volumes.
- Li also wrote on prime numbers.

Born 2 January 1811, Haining, Zhejiang Province, China. Died 9 December 1882, Peking (now Beijing), China.

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Ancient Chinese, Astronomy, Chinese, Origin China

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive