To expand search, see Archimedes and Calculus. Laterally related topics: Limit and Surface Area.
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Seidenberg, A. On the volume of a sphere. Arch. Hist. Exact Sci. 39 (1988), no. 2, 97--119. (Reviewer: K.-B. Gundlach.) SC: 01A20 (01A15 01A17 01A25 01A32), MR: 89j:01012.
Abraham Seidenberg argues that there is a common source for Pythagorean and Chinese (or Chinese-like) mathematics. He suggests that Old-Babylonian mathematics is a derivative of a more ancient mathematics having a much clearer geometric component (p. 104), and is "in some respects ... is derivative of a Chinese-like mathematics" (p. 109). Van der Waerden holds a similar view on this, and tells us that the mathematics of the Chiu Chang Suan Shu represents the common source more faithfully than the Babylonian does. Seidenberg believes that the common source is most similar to the Sulvasutras. He discusses how questions of the sphere and the circle were treated by the Greeks, Chinese, Egyptians, and to a lesser extent Indians. He discusses the some similarities and differences in the work on the sphere in Greece (Archimedes, with a very brief account of the application of his Method), and in Chinese (first in the Chiu Chang Suan Shu, improved by Liu Hui or perhaps Tsu Ch'ung-Chih, and then further improved by the Tsu Ch'ung-Chih's son Tsu Keng-Chih). He believes that the problem of the volume of a sphere goes back to the common source, to the first part of the second millennium B.C. or earlier. An interesting and related topic is the topic of the equality of the proportionality constants pi that occur in the formulas for the area and circumference of a circle. Seidenberg examines the Moscow Papyrus, Chinese sources, and an Old-Babylonian text and finds that this fact seemed to be recognized in all three groups. He argues that the Egyptian, Babylonian, and Chinese approaches to the volume of a truncated pyramid may have derived from the same common source. He believe that the common source also used infinitesimal, Cavalieri-type, arguments as well. It is interesting as well that Heron, who as Seidenberg notes is sometimes considered to be continuing the Babylonian tradition, gives the formula 1/2(s+p)p+1/14(1/2s)2 for the area of a segment of a circle with chord s and height (sagita, arrow) p (with an Archimedean value of 22/7 for pi), and "that the 'ancients' took [the area as] 1/2(s+p)p and even conjectured that they did so because they took pi = 3." The paper is also interesting in that he discusses the development of some of his ideas from his early papers in the 60s until much later (the paper was received soon before his death). Closely related topics: The Sphere, The Circle, The Pythagoreans, China, The Chiu Chang Suan Shu (Nine Chapters on the Mathematical Art), Sumerians and Babylonians, The Sulvasutras, Archimedes, The Moscow Mathematical Papyrus, Heron, and Abraham Seidenberg.
Swetz, Frank J. The Method of Archimedes. In Swetz, Frank J. From Five Fingers to Infinity. A Journey through the History of Mathematics. Open Court, Chicago, 1994. . 180--181.
Shows how Archimedes used his Method to discover the formula for the volume of a sphere. (Of course Archimedes also gave a rigorous proof using Eudoxus' Method of Exhaustion.) Closely related topics: Archimedes, The Measurement of Area and Volume, and The Sphere.