Sānjiǎo héjiào shù 三角和較術
Sum-and-Difference Methods in Triangle Geometry by 項名達 (撰)
About the work
項名達 Xiàng Míngdá’s (1789–1850) treatise in 4 juàn on the héjiào 和較 (sum-and-difference) methods in sānjiǎo 三角 (triangle) geometry — the application of identities expressing one side or angle of a triangle in terms of sums and differences of the others to solve trigonometric problems.
Abstract
The héjiào methods in trigonometry are the systematic application of the sum-and-difference identities — sin(A±B), cos(A±B), and their generalisations to sides and angles of a general triangle — to solve oblique-triangle problems. In the European tradition, these identities are central to Renaissance and post-Renaissance plane and spherical trigonometry (Vièta, Napier, etc.). In the Chinese tradition they entered via the Jesuit translation programme and the imperially-commissioned Lǜlì yuānyuán 律曆淵源 (1722–1723).
Xiàng Míngdá’s 4-juàn treatise presents a systematic exposition of the héjiào methods applied across the plane-triangle problem-domain. The treatise extends well beyond the Lǜlì yuānyuán presentation to cover the systematic derivation of higher-order trigonometric identities (including some that arise naturally from the cyclotomic-series approach of KR3fc069 Xiàngshù yīyuán).
The work is closely tied to Xiàng Míngdá’s broader theoretical project: where the Xiàngshù yīyuán develops the cyclotomic infinite-series expansions for trigonometric functions, the Sānjiǎo héjiào shù develops the finite identities (sum-and-difference relations) that govern the same functions. The two works together cover the analytic and the synthetic sides of the trigonometric apparatus.
Dating: Xiàng Míngdá lived 1789–1850. notBefore 1820 (mature productive period); notAfter 1850.
Translations and research
- Wú Wénjùn 吳文俊, ed. 1985. Zhōng-guó shù-xué shǐ dà-xì 中國數學史大系, vol. 7–8.
- Tián Miǎo 田淼. 2003. Zhōng-guó shù-xué de xī-huà lì-chéng 中國數學的西化歷程.
Links
- Companion: KR3fc069 Xiàngshù yīyuán (cyclotomic infinite-series counterpart)
- Other works by 項名達: KR3fc070 Gōugǔ liùshù, KR3fc072
- CBDB (author): https://cbdb.fas.harvard.edu/cbdbapi/person.php?id=84043