Article ID: CBB222899298

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

Unger, J. Marshall (Author)

Historia Mathematica
Volume: 65
Pages: 1-13
Publication date: 2023
Language: English

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。

...More
Citation URI
data.isiscb.org/p/isis/citation/CBB222899298

This citation is part of the Isis database.

#### Similar Citations

Article Zhou, Chang; (2008)
The Trailblazer of Analysis in Wasan: Takebe Katahiro (/p/isis/citation/CBB000760557/)

Article Zhou, Chang; Zhang, Jianke; (2008)
Takebe's Mathematical Thought and Methodology (/p/isis/citation/CBB000933525/)

Article Xu, Zelin; (2008)
Wu's Method and Japanese Geometry (/p/isis/citation/CBB000933544/)

Article Xu, Zelin; Wei, Xia; (2011)
Exploration on Yanduan---And on the Evolution of Algebraic Algorithm Mode in East Asia (/p/isis/citation/CBB001210061/)

Thesis William Evan Young; (2015)
Family Matters: Managing Illness in Late Tokugawa Japan, 1750-1868 (/p/isis/citation/CBB563831507/)

Article J. Marshall Unger; (2020)
On the Acceptance of Trigonometry in Wasan: Evidence from a Text of Aida Yasuaki (/p/isis/citation/CBB847455647/)

Article Yulia Frumer; (2018)
Japanese Reverse Compasses: Grounding Cognition in History and Society (/p/isis/citation/CBB035408812/)

Book Catherine Jami; (2014)
Mobilité humaine et circulation des savoirs techniques (xviie-xixe siècles) (/p/isis/citation/CBB903217129/)

Book Robert Hellyer; (2021)
Green with Milk and Sugar: When Japan Filled America’s Tea Cups (/p/isis/citation/CBB751902936/)

Article Takeda, Junko Thérèse; (2014)
Global Insects: Silkworms, Sericulture, and Statecraft in Napoleonic France and Tokugawa Japan (/p/isis/citation/CBB001201419/)

Thesis Marcon, Federico; (2007)
The Names of Nature: The Development of Natural History in Japan, 1600--1900 (/p/isis/citation/CBB001560689/)

Article Luo, Jianjin; (2003)
On the Development of the Formulae for Sums of Integers (/p/isis/citation/CBB000774899/)

Article Pepijn Brandon; Marten Dondorp; (2023)
Nodes of knowledge, managing transfer: Shipbuilding and repair during the transformation from sail to steam (/p/isis/citation/CBB029547827/)

Article Wang, Wenzhi; Mei, Jianjun; (2009)
A Preliminary Study of “Kodo Zuroku” (/p/isis/citation/CBB000933556/)

Book Jakobina K. Arch; Paul S. Sutter; (2018)
Bringing Whales Ashore: Oceans and the Environment of Early Modern Japan (/p/isis/citation/CBB119390015/)

Chapter Osamu Nakamura; (2016)
Patient Work and Family Care at Iwakura, Japan, C. 1799-1970 (/p/isis/citation/CBB402213161/)

Authors & Contributors
Tsukahara, Togo
Xu, Zelin
Zhou, Chang
Dudley, Leonard
Howell, David L
Jami, Catherine
Journals
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
Publishers
Princeton University
Columbia University
Cambridge Scholars Publishing
Columbia University Press
Presses Universitaires de Vincennes
University of Washington Press
Concepts
East Asia, civilization and culture
Transmission of ideas
Mathematics
Cross-cultural interaction; cultural influence
Wasan (Japanese mathematics)
Transmission of texts
People
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
Places
Japan
China
Korea
Java (Indonesia)
France
Netherlands
Institutions
Dutch East India Company
North Atlantic Treaty Organization (NATO)