韓國數學家南秉吉對「角」的概念與「相似形」的理解

The Korean Mathematician Nam Pyŏng-Gil's Ideas of "Angle" and His Understanding of "Similar Shapes"

英家銘(Jia-Ming Ying)

《中華科技史學會學刊》 13期 (2009/12) Pp. 1-14

https://doi.org/10.7094/BAFHS.200912.0001

摘要

本文討論韓國朝鮮王朝後期數學家南秉吉(Nam Pyŏng-Gil, 1820-1869)對歐氏幾何中「角」與「相似形」概念的理解。韓國數學受中國數學影響甚鉅，而中國古代可能並無相當於西方幾何學中「角」的概念；關於測量的計算，也僅比較線段長度，而不討論兩圖形的「相似」。「角」與「相似形」的概念，隨著《幾何原本》前六卷(1607)的翻譯進入中國，並在明、清兩代傳入朝鮮王朝的數學、天文學著作中，進入韓國。 本文中主要引用南秉吉所著之《測量圖解》(1858)與《九章術解》(1864)來說明他的理解。從本文舉的例子中，我們可以看出，受到西方數學影響的南秉吉，對於「角」的概念，大致上來說與西方幾何中的「角」意義相同，僅有一例，是他用「角」稱呼頂點，除此之外，他對「角」的理解，就是指兩條線段之間所撐開的空間。關於「相似形」的理解，對三角形而言，各種不同方向的相似三角形，都出現在他的論證中，也至少有一例可說明，他知道兩三角形對應角相等則兩三角形相似。不過，若是觸及四邊以上的多邊形，在南秉吉的論證中，有一例可看出，他認為「對應角相等兩四邊形即相似」。南秉吉可能是將三角形的論證方式類比到其他的多邊形中，以致於產生論證的瑕疵。 由上面也可以映證一個韓國與中國的相同之處，就是在18至19世紀，強調嚴格演繹邏輯的論證方式，沒有成為數學家寫作數學文本的重要依據。南秉吉所關心的，或許是類比性的思考，以及用快速、簡潔或直覺的手法解決問題。

關鍵字

韓國數學 ； 南秉吉 ； 角 ； 相似形 ； 歐氏幾何

並列摘要

This paper discusses the ideas of ”angle” and understanding of ”similar shapes” in Euclidean Geometry by Nam Pyŏng-Gil (1820-1869), a mathematician in the late Chosŏn Dynasty of Korea. Korean mathematics was immensely influenced by Chinese mathematics, but in ancient China there does not seem to exist the concept of ”angle” as that in Euclidean Geometry. As for the case of measurements, the objects being compared are not two ”similar shapes”, but pairs of proportional line segments. The concepts of Euclidean angle and similarity was introduced to China with the translations of first six books of Euclid’s Elements in 1607, and they were later transmitted to Korea by the mathematical and astronomical treatises published in Ming and Qing Dynasties, the most important of which is the Shuli jingyun 數理精蘊.
I mainly quote Nam's Ch'ŭkryang tohae測量圖解(1858) and Kujang sulhae 九章術解(1864) to explain his understanding. Nam, being highly influenced by the Sino-Western mathematics introduced in Qing texts, had his idea about ”angle” which was generally the same as that in Euclidean Geometry, except in one case in which he referred to a vertex also with the word jiao角. For the understanding of ”similarity”, there are pairs of similar triangles of different orientations, and he knew that if corresponding angles are equal, then two triangles are similar. However, for polygons of four or more sides, there is at least one case to show he believed that as long as the corresponding angles are equal, those shapes are similar. He did not consider corresponding sides in that case. Nam might have made an analogy between triangles and polygons of more sides to reach that conclusion and had a flaw in his argument.
This paper also gives an example of a common phenomenon between Korean and Chinese mathematics: rigorous deductive argument was not the way of writing mathematical texts. What Nam Pyŏng-Gil cared about was analogical thinking and solving problems with intuition and quick methods.

並列關鍵字

Korean mathematics ； Nam Pyŏng-Gil ； angle ； similar shapes ； Euclidean Geometry

參考文獻

Cullen, Christopher, Astronomy and Mathematics in Ancient China: The Zhou Bi Suan Jing. Cambridge University Press (Cambridge), 1996.

Jun, Yong Hoon, “Mathematics in context: A case study in early nineteen-century Korea”, Science in Context, 2006, no. 19, pp.475-512.

Oh, Young Sook, "Suri chõngon pohae (數理精蘊補解): An 18th Century Korean Supplement to Shuli jingyun (數理精蘊)". In W. S. Horng, Y. C. Lin, T. C. Ning & T. Y. Tso (Eds.), Proceedings of Asia-Pacific HPM 2004 Conference, National Taichung Teachers College (Taichung), 2004.

Volkov, Alexei, Geometrical diagrams in traditional Chinese mathematics. In F. Bray, V. Dorofeeva-Lichtmann and G. Metailie (Eds.), Graphics and Text in the Production of Technical Knowledge in China. Brill (Leiden), 2007.

王渝生，〈幾何原本提要〉，收入郭書春主編，《中國科學技術典籍通彙》數學卷第五分冊，河南教育出版社（鄭州）1993 年，頁 1145-1150。