Cyclic quadrilaterals: Solutions of two Japanese problems and their proofs

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...

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Bibliographic Details
Published in:Historia mathematica 2023-11, Vol.65, p.1-13
Main Author: Unger, J. Marshall
Format: Article
Language:English
Subjects:
Cyclic quadrilaterals
Harmonic mean
Incircles
Resultant
Trigonometry
Wasan
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Summary: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. 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。
ISSN:0315-0860
1090-249X
DOI:10.1016/j.hm.2023.08.001