韓國數學家南秉吉對「角」的概念與「相似形」的理解

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.