To expand search, see China. Laterally related topics: Yang Hui, The I Ching, and Chu Shih-chieh.
The Mathematics and the Liberal Arts pages are intended to be a resource for student research projects and for teachers interested in using the history of mathematics in their courses. Many pages focus on ethnomathematics and in the connections between mathematics and other disciplines. The notes in these pages are intended as much to evoke ideas as to indicate what the books and articles are about. They are not intended as reviews. However, some items have been reviewed in Mathematical Reviews, published by The American Mathematical Society. When the mathematical review (MR) number and reviewer are known to the author of these pages, they are given as part of the bibliographic citation. Subscribing institutions can access the more recent MR reviews online through MathSciNet.
Mathews, Jerold. A Neolithic oral tradition for the van der Waerden/Seidenberg origin of mathematics. Arch. Hist. Exact Sci. 34 (1985), no. 3, 193--220. (Reviewer: M. Folkerts.) SC: 01A10 (01A25), MR: 88b:01005.
Abraham Seidenberg advanced a theory that mathematics arose from a common origin, and that some the mathematics was preserved by an oral tradition, and very likely a religious tradition, perhaps one like the one seen in the Indian Sulvasutras. Van der Waerden's book Geometry and Algebra in Ancient Civilizations takes a similar views, and in fact van der Waerden credits Seidenberg for making him look at the history of mathematics a new way. As Mathews notes, the Chinese Chiu Chang Suan Shu is very important in van der Waerden's work. Mathews relies heavily on this work as well to "give a small, coherent, and basic core of geometry concerning rectangles and their parts, ..., which may serve as what van der Waerden has called an 'oral tradition current in the Neolithic age.'" He states the he hoped "to give this hypothesized ancient core some credence through its relation to the Chiu Chang and its explanatory power. After giving a thorough discussion of this geometry, he then carefully analyzes the ninth chapter of the Chiu Chang in terms of this core. He is able to find a strong match, though his conclusions on one of the problems (Problem 20) are not consistent with those of some other researchers, who find in problem 20 instead suggestions of something like Horner's method. A very interesting article. Hopefully future papers will discuss how well the author's geometry agrees with the ancient geometry of other cultures. As he notes, "Until I can thoroughly test his conjecture on, say, the Babylonian corpus, I can argue for the merits of my conjecture only on such grounds as the simplicity of explanation it allows, or its congruence with received results or figures." Closely related topics: The Neolithic Era, Religion, Geometry, The Sulvasutras, and Abraham Seidenberg.Modify notes on this entry Modify bibliography entry Make comment on this entry
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, Sumerians and Babylonians, The Sulvasutras, Archimedes, Archimedes' Method, The Moscow Mathematical Papyrus, Heron, and Abraham Seidenberg.Modify notes on this entry Modify bibliography entry Make comment on this entry
Swetz, Frank. The "Piling Up of Squares" in Ancient China. Mathematics Teacher 70 (1977), 72--79.
Chapter IX of the Chiu Chang Suan Shu has a series of interesting problems on the Pythagorean Theorem, many requiring a little resourcefulness to solve, even today. Two methods are used in Chapter IX. This article discusses one of these, the Chi-Chü, or "piling up of squares". This is a dissection method; thus areas are disassembled and reassembled in a different way. The author gives several examples. The last two are among the most interesting. They find the largest square and circle that can be drawn in a right triangle; only the case where the square includes the right angle seems to be considered. The methods are ingenious, and would make appealing classroom demonstrations. The Chi-Chü method is also used in problems that at first seem to have little to do with areas. Problem 14 is an example:Two men starting from the same point begin walking in different directions. Their rates of travel are in the ratio 7:3. The slower walks towards the east. His faster companion walks to the south 10 pu and then turns towards the northeast and proceeds until both men meet. How many pu did each man walk?The author also discusses problem 6, the famous problem of a reed in a square pond:In the center of a square pond whose side measures 10 ch'ih grows a cattail whose top reaches 1 ch'ih above the water level. If we pull the reed toward the bank, its top becomes even with the waters surface. What is the depth of the pond and the length of the plant?As the author observes, this problem is very similar to a much later problem of Bh\=askara, where even the ratios involved are the same:In a certain lake, swarming with red geese, the tip of a bud of a lotus was seen a span (9 inches) above the surface of the water. Forced by the wind, it gradually advanced and was submerged at a distance of two cubits (approximately 40 inches). Compute quickly, mathematician, the depth of the pond.The question of Chinese influence on Indian mathematicians is still unsettled. One can't but wonder how the Chinese became so amazingly successful with the Chi-Chü method. The author mentions the possibility that familiarity with the tangram exercises may have contributed to their skill. Excellent article. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Pythagorean Triangles and Triples and The Tangrams.Modify notes on this entry Modify bibliography entry Make comment on this entry
Swetz, Frank. The Amazing Chiu Chang Suan Shu. Mathematics Teacher 65 (1972), 423--30.
A chapter by chapter survey of the Chiu Chang Suan Shu. One interesting example (of many) is a pursuit problem, which anticipated Alcuin's hound-pursuing-a-hare problem by nearly a thousand years (Chapter VI). Swetz' article leads one to understand how the president of the Bureau of Foreign Affairs in China (Prince Kung) might have felt justified in his claim (in the 1860s) that "Western sciences borrowed their roots from ancient Chinese mathematics. Westerners still regard their mathematics as coming from the Orient. It is only because of the careful, inquiring minds of the Westerners that they are good at developing something new out of the old... China invented the method, Westerners adopted it..." Reprinted in Swetz, Frank J., From Five Fingers to Infinity.Modify notes on this entry Modify bibliography entry Make comment on this entry
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