Article ID: CBB222899298

Cyclic quadrilaterals: Solutions of two Japanese problems and their proofs (2023)


Late 18th and early 19th century Japanese mathematicians (wasanka) found solutions of two problems concerning the incircles of the quarter-triangles and skewed sectors of cyclic quadrilaterals. There is a modern proof of the first solution, but it makes extensive use of trigonometry and is therefore unlikely to be what a wasanka would have written. As for the second solution, Aida Yasuaki (1747–1817) gave two proofs for it, the second of which has been summarized in Japanese, but not the first. All three proofs are presented here together with commentary on their mathematical and historical significance. 要旨 円に内接する四角形は対角線があり、その四角形を重なり合わない 4 つの三角形に分割 し、外円を 4 つの準セクターに分割する。 和算家は、三角形か準セクターの内接円の 直径から外円の直径を計算する公式を知っていた。 三角形の解には三角関数を多用し た現代の証明があるが、それは和算家が考えたのとは違うはずである。 会田安明は準 セクターの解の証明を 2つ書いたが、これまでのところ (日本語で) 説明されているの は 2つ目だけである。ここで、それら3つの証明すべてについて詳しく説明し、それ らの歴史的重要性について注意を喚起したいと思う。

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Authors & Contributors
Tsukahara, Togo
Xu, Zelin
Zhou, Chang
Dudley, Leonard
Howell, David L
Jami, Catherine
Ziran Kexueshi Yanjiu (Studies in the History of Natural Sciences)
Historia Scientiarum: International Journal of the History of Science Society of Japan
East Asian Science, Technology, and Medicine
French History
Historia Mathematica
History of Science
Princeton University
Columbia University
Cambridge Scholars Publishing
Columbia University Press
Presses Universitaires de Vincennes
University of Washington Press
East Asia, civilization and culture
Transmission of ideas
Cross-cultural interaction; cultural influence
Wasan (Japanese mathematics)
Transmission of texts
Takebe, Katahiro
Hua, Hengfang
Li, Shanlan
Seki, Takakazu
Siebold, Philipp Franz von
Udagawa, Yoan
Time Periods
18th century
19th century
17th century
Edo period (Japan, 1603-1868)
20th century
Early modern
Java (Indonesia)
Dutch East India Company
North Atlantic Treaty Organization (NATO)

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