Shǎoguǎng bǔyí 少廣補遺
Supplement to [the] Less-Width [Tradition] (heap-piling and discrete-summation methods) by 陳世仁 (Chén Shìrén, 1676–1722, 清, zhuàn 撰)
About the work
Chén Shìrén’s 1-juan focused treatise on duīduǒ 堆垜 (heap-piling) computation — the calculation of summed series for various pyramidal-or-prismatic heap-shapes. The work develops 12 distinct procedures (shíèr fǎ 十二法) covering: the one-sided pyramidal heap (yīmiàn jiān duī 一面尖堆); the square-base pyramidal heap (fāngdǐ 方底); the triangular-base heap (sānjiǎo dǐ 三角底); the hexagonal-base heap (liùjiǎo dǐ 六角底); and the various half-heaps (gè bànduī 各半堆). Supplementary material covers the chōujī (drawing odd) and chōuǒu (drawing even) series-summation procedures.
The work’s contribution is to address an underdeveloped problem-class in the Jiǔzhāng Shǎoguǎng tradition. The Sìkù 提要 articulates the methodological-and-historical situation:
The Shǎoguǎng-class problem of duīduǒ (heap-piling) is similar to but methodologically distinct from the Shānggōng-class problem of solid-volume-of-prism-or-pyramid (jiānzhuī tǐ 尖錐體). The latter treats continuous-and-filled solids whose external surface is plane-and-the-interior-is-solid; the former treats discrete unit-counts arranged in heap-shapes whose external surface is pointed-or-stepped and whose interior contains intervening gaps. The two computational methods differ, and the Shǎoguǎng bǔyí is the principal Chinese systematic treatment of the heap-piling case.
The ancient Shǎoguǎng tradition had treated only one direction of the duīduǒ problem: computing the summation given the base-edge-count and the layer-count. Chén Shìrén systematically addresses the inverse problem: computing edge-count and layer-count given the summation. The 12 procedures are the principal cases. The work applies these procedures with auxiliary chōujī / chōuǒu summations for arithmetic-sequence problems.
The Sìkù 提要 commends the work as a substantive contribution despite its narrow scope. The principal limitation: “although the diagrams-and-explanations are not complete (so as to make the student peep into the meaning of the law-establishment)“. I.e., the work presents results without full geometric proof. But the procedures themselves are correct and the contribution is genuine.
For Chén Shìrén’s biography, see 陳世仁. For the broader Shǎoguǎng tradition this work supplements, see KR3f0032 Jiǔzhāng suànshù’s Shǎoguǎng chapter.
Tiyao
[Full text in source file. Dated Qiánlóng 46 (1781), fifth month.]
Translations and research
- Limited substantial secondary literature.
- Han Qi 韓琦, Tōng-tiān zhī xué 通天之學, Beijing: Sānlián, 2018.