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References
Zhang Yong, Duoji bilei shu zheng [Proofs and Commentaries of (Finite Summation Formulas) in theDuoji bilei],Kexue 23 (1939), 647–663. Paul Turân’s proof appears on pp. 661-663 of Zhang’s article. Here,Duoji bilei is the exact title of a book by Li Shanlan, the literal meaning of which is “Piles of Heaps Summed Analogically.” Perhaps a more idiomatic translation of this would be “Finite Summation Formulas Derived by Analogical Reasoning.”
Paul Turân, A kinai matematika torténetének egy problémâjâroi [A Problem from Chinese Mathematics],Matematikai Lapok (1954), 1-6. See, also, the papers by Lajos Takacs, Jânos Surânyi, Géza Huszâr, and Jânos Maté inMatematikai Lapok (1955), 27-29; (1955), 30-35; (1955), 36-38; and (1956), 112-113, respectively. In Combinatorial Identities (New York: Wiley, 1968), John Riordan treats these ideas in a more modern setting.
On Li Shanlan’s life, see, for example, Arthur W. Rummel,Eminent Chinese of the Ch’ing Period, Washington: n.p. (1943); reprint ed., Taipei: Ch’eng Wen Publishing Co. (1970), pp. 479-480; Wang Ping,Xifang lisuanxue zhi shurn [The Introduction of Western Astronomical and Mathematical Sciences into China], Taipei & Nankang Monograph Series No. 17, Taipei: Institute of Modern History, Academia Sinica, Republic of China (1966), pp. 144-182; and Wang Yusheng, Li Shanlan yanjiu [Researches on Li Shanlan], inMing-Qing shuxue shi lunivrn ji [Collected Papers in the History of Chinese Mathematics in the Ming and Qing Periods], Nanking: Jiaoyu chubanshe (1990), pp. 334-406.
Wang Ping, p. 144.
Alexander Wylie,Jottings on the Science of the Chinese, Arithmetic, North China Herald, Aug.-Nov. 1852, Nos. 108, 111, 112, 113, 116, 117, 119, 120, 121. Wylie’s “jottings” have been reprinted often. For extensive references to these reprintings, see Joseph Needham,Science and Civilization in China, vol. 3, Cambridge: University Press (1959).
Augustus DeMorgan,The Elements of Algebra Preliminary to the Differential Calculus, and Fit for the Higher Classes of Schools …, London: J. Taylor (1835).
Elias Loomis,Elements of Analytical Geometry and of Differential and Integral Calculus, New York: Harper & Brothers (1851). In fact, according to theDictionary of American Biography, Loomis’s books were also translated into Arabic. See Allen Johnson, Dumas Malone, et al. (ed.),Dictionary of American Biography, 10 vols, and 8 suppls., New York: Charles Scribner’s Sons (1927–1990), s. v. “Loomis, Elias,” by David Eugene Smith.
William Whewell,An Elementary Treatise on Mechanics, Cambridge: J. Deighton & Sons (1819).
John F. W. Herschel,Outlines of Astronomy, London: Longman, Brown, Green, and Longmans (1849).
Probably from John Lindley,An Introduction to Botany, London: Longman, Rees, Orme, Brown, Green, and Longmans (1832).
Wylie, Jottings in Alexander Wylie,Chinese Researches, Taipei: Ch’eng-wen Publishing Co. (1966), 193. The exact quotation is: “Li Shen-lan [sic] … who has recently published a small work called Tuy-soo-tan-yuan ‘ Discovery of the source of logarithms,’ in which he details an entirely new method for their computation, based on geometrical formulas, which he says in his introduction is ‘ ten thousand times easier than the methods used by Europeans,’ and that ‘ although they can just calculate the numbers, yet they [i.e., the Europeans] are ignorant of the principle.’ “
Wann-Sheng Horng, “Li Shanlan, the impact of Western mathematics in China during the late 19th century,” Ph.D. dissertation, The City University of New York, March, 1991.
J. Worpitzki, “Studien über die Bernoullischen und Eulerschen Zahlen,”J. Reine Ange. Math. 94 (1883), 202–232.
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Parshall, K.V.H., Martzloff, JC. Years Ago. The Mathematical Intelligencer 14, 32–37 (1992). https://doi.org/10.1007/BF03024470
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DOI: https://doi.org/10.1007/BF03024470