Sources page biographical material

6.AN. VOLUME OF THE INTERSECTION OF TWO CYLINDERS
Archimedes. The Method: Preface, 2. In: T. L. Heath; The Works of Archimedes, with a supplement "The Method of Archimedes"; (originally two works, CUP, 1897 & 1912) = Dover, 1953. Supplement, p. 12, states the result. The proof is lost, but pp. 48 51 gives a reconstruction of the proof by Zeuthen.

Liu Hui. Jiu Zhang Suan Chu Zhu (Commentary on the Nine Chapters of the Mathematical Art). 263. ??NYS -- described in Li & Du, pp. 73 74 & 85. He shows that the ratio of the volume of the sphere to the volume of Archimedes' solid, called mou he fang gai (two square umbrellas), is π/4, but he cannot determine either volume.

Zu Geng. c500. Lost, but described in: Li Chunfeng; annotation to Jiu Zhang (= Chiu Chang Suan Ching) made c656. ??NYS. Described on pp. 86 87 of: Wu Wenchun; The out in complementary principle; IN:  Ancient China's Technology and Science; compiled by the Institute of the History of Natural Sciences, Chinese Academy of Sciences; Foreign Languages Press, Beijing, 1983, pp. 66 89. [This is a revision and translation of parts of: Achievements in Science and Technology in Ancient China [in Chinese]; China Youth Publishing House, Beijing(?), 1978.]

He considers the shape, called fanggai, within the natural circumscribed cube and shows that, in each octant, the part of the cube outside the fanggai has cross section of area h² at distance h from the centre. This is equivalent to a tetrahedron, whose volume had been determined by Liu, so the excluded volume is ⅓ of the cube.

Li & Du, pp. 85 87, and say the result may have been found c480 by Zu Geng's father, Zu Chongzhi.

Lam Lay-Yong & Shen Kangsheng. The Chinese concept of Cavalieri's Principle and its applications. HM 12 (1985) 219-228. Discusses the work of Liu and Zu.

Shiraishi Chōchū. Shamei Sampu. 1826. ??NYS -- described in Smith & Mikami, pp. 233-236. "Find the volume cut from a cylinder by another cylinder that intersects is orthogonally and touches a point on the surface". I'm not quite sure what the last phrase indicates. The book gives a number of similar problems of finding volumes of intersections.

P. R. Rider, proposer; N. B. Moore, solver. Problem 3587. AMM 40 (1933) 52 (??NX) & 612. Gives the standard proof by cross sections, then considers the case of unequal cylinders where the solution involves complete elliptic integrals of the first and second kinds. References to solution and similar problem in textbooks.

Leo Moser, solver; J. M. Butchart, extender. MM 25 (May 1952) 290 & 26 (Sep 1952) 54. ??NX. Reproduced in Trigg, op. cit. in 5.Q: Quickie 15, pp. 6 & 82 83. Moser gives the classic proof that V = 16r³/3. Butchart points out that this also shows that the shape has surface area 16r².