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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.
Aiyar, S. Balakrishna. The Ganita-S\=ara-Sangraha of Mah\=av\=\i r\=ac\=arya. Mathematics Teacher 47 (1954), 528--33.
An overview of Mahavira's Ganita-Sara-Sangraha. The author makes the interesting observation that in Jainism, Mahavira's religion, mathematics was very popular, and was "accorded the status of one of the four anuyog\=as, which were the auxiliary sciences, the study of which helped the aspirant to the attainment of soul-liberation." Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Mahaviracarya and Arithmetic.Modify notes on this entry Modify bibliography entry Make comment on this entry
Andersen, Kirsti. The mathematical treatment of anamorphoses from Piero della Francesca to Niceron. History of mathematics: states of the art, 3--28, Academic Press, San Diego, CA, 1996.
Discusses the mathematics of anamorphoses and the history of the subject from the mathematical point of view. Begins with a short discussion of problems stemming from the well-known fact that cylindrical columns seem smaller towards the top. Dürer discussed how one can use letters of different size on such a column so that rows of print will all appear the same size. His student Erhard Schön did some work using anamorphoses proper. (This was about the same time as Hans Holbein's Ambassadors.) Piero della Francesca's De Prospectiva Pingendi includes a discussion of how to construct a particular anamorphic drawing, but little further progress was made until the 1600s. The author notes that artists didn't seem to use the same mathematical techniques when using more extreme perspectives as they used with more normal perspectives. In fact, written works from the time suggest that orthogonal projections were used. The author gives examples from the work of of Daniele Barbaro [Italy 1500s], Paolo Giovanni Lomazzo [Italy 1500s], Egnazio Danti [Italy 1500s], Guidobaldo del Monte [France 1600s], Samuel Marolois [France 1600s], and Salomon de Caus [France 1600s]. (The case of Lomazzo is unclear: he suggested using threads for the construction, but didn't state clearly how they were to be used.) After Niceron, more mathematically accurate techniques were used; the author gives an example of a work by Emmanuel Maignan [France 1600s], who was influenced by Niceron. The problems of mirror anamorphoses apparently originated in China by about 1600. Artists apparently either worked intuitively (as in China), or by using approximate constructions. Approximate constructions still appear today in the work of the 20th century Swedish artist Hans Hamngren. A mathematically precise treatment of the problem (and of a problem using a conical mirror) was given by Jean-Louis Vaulezard in the 1600s, but even Niceron gave only an approximate method. The author suggests that Vaulezard's students were perhaps the only ones who constructed curved-mirror anamorphoses using mathematically accurate methods. (Computer analyses might be useful to verify this.) Using a computer algebra system, the author has derived the equations for the curves which will project to a coordinate grid. The curve is not given in the text, but the author tells us that it is not one of the familiar curves, has degree 6, and has rather complicated coefficients. Closely related topics: Anamorphoses, The Column, Albrecht Dürer, Erhard Schön, Piero della Francesca, China, Jean-Louis Vaulezard, and Jean-François Niceron.Modify notes on this entry Modify bibliography entry Make comment on this entry
Biggs, N. L. The roots of combinatorics. Historia Math. 6 (1979), no. 2, 109--136. (Reviewer: J. Dieudonné.) SC: 05-03 (01A15 01A20 01A25 01A30 01A32 01A40 01A45), MR: 80h:05003.
(1) As the author explains, the most ancient problem connected with combinatorics may be the house-cat-mice-wheat problem of the Rhind Papyrus (Problem 79), which occurs in a similar form in a problem of Fibonacci's Liber Abaci and in an English nursery rhyme. All are concerned with successive powers of 7. (2) The first occurrence of combinatorics per se may be in the 64 hexagrams of the I Ching. (However, the more modern binary ordering of these is first seen in China in the 10th century.) A Chinese monk in the 700s may have had a rule for the number of configurations of a board game similar to go. In Greece, one of the very few references to combinatorics is a statement by Plutarch about the number of compound statements from 10 simple propositions; Plutarch quotes Chrysippus, Hipparchus, and Xenocrates on the subject, so all apparently had some interest in the subject. (Plutarch's statement is also discussed in a recent article in the Monthly.) Boethius apparently had a rule for the number of combinations of n things taken two at a time. The author discusses interest in combinatorics in the Hindu world, by the Jainas, Varahamihira, and Bhaskara (the latter in the Lilavati). The work of Brahmagupta should be relevant, but is not currently available in English. The Arabs seem to have adopted their combinatorics from the Hindus. The author also briefly discusses some interest in combinatorics in the Jewish mathematical tradition; two examples are Rabbi ben Ezra and Levi ben Gerson. (3) Magic squares may first occur in the lo shu diagram, which is often linked with the I Ching. The author discusses how the idea of magic squares may have entered the Islamic world, was then improved, appeared in the work of Manuel Moschopoulos, and possibly through him entered the Western world. What happened in China is less clear. As the author suggests, the the work of Yang Hui suggests that there had been a Chinese tradition of work in magic squares, already dead by Yang Hui's time. For example, the squares Yang Hui gives are not of types found elsewhere. In addition, Yang Hui seems unclear on the techniques for construction. It is interesting that De la Loubère learned of a simple method for constructing magic squares in Siam. The author also discusses: the possibility of a Hindu study of magic squares; the presumably Arab source of Western magic square mysticism; and later developments, such as Euler's questions on orthogonal Latin squares. (4) The author discusses how questions in partitions arose in gambling, such as the throwing of astrogali (huckle bones, which can land 4 ways) or dice (which can land in 6 ways). An early systematic study is in the late Medieval Latin poem De Vetula, which gives the number of ways you can obtain any given total from a throw of 3 dice. Cardano and Galileo examined the subject in more depth. (5) Combinatorial thinking in games and puzzles. Discusses the wolf-goat-cabbage, attributed to Alcuin. [Similar puzzles also occur in a variety of other cultures, but are not discussed in this article.] Also discusses the Josephus problem, based on a process similar to the childhood process of "counting-out". The Josephus problem is named for the Jewish historian Josephus of the 1st century AD, who supposedly saved his life with a correct solution. This problem unexpectedly turned up in Japan. (6) The author discusses how "Pascal's" triangle was possibly known to Omar Khayyam in the context of taking roots. The Hindu scholar Pingala may have known a method, but the case is more cryptic. At any rate, it was known by the time of Halayudha, who may have lived in the 900s AD. A more clear-cut reference occurs in the work of Nasir al-Din al-Tusi in 1265. In China, the triangle appears in the work of Chu Shih-Chieh (1303), but may have been very ancient by then. The triangle was used by Pascal and Fermat to resolve the "problem of points". This problem had the goal of determining how to distribute stakes when a game ends early. ... Excellent article. Closely related topics: Combinatorics, The Rhind/Ahmes Papyrus, Leonardo of Pisa (Fibonacci), The I Ching, Logic, Plutarch, Chrysippus, Hipparchus, Xenocrates, Boethius (Ancius Manlius Torquatus Severinus Boetius), Jainism, Varahamihira, Brahmagupta, Bhaskara, The Islamic World, The Jewish Tradition, Rabbi ben Ezra, Levi ben Gerson, Magic Squares, Manuel Moschopoulos, Yang Hui, Siam, Mathematics and Mysticism, Leonhard Euler, Gambling, De Vetula, Girolamo Cardano, Galileo Galilei, Puzzles, Alcuin, The Josephus Problem, Japan, Pascal's Triangle, Omar Khayyam (abu-l-Fath Omar ibn Ibrahim Khayyam), Pingala, Halayudha, Nasir al-Din al-Tusi, Chu Shih-chieh, Blaise Pascal, and Pierre de Fermat.Modify notes on this entry Modify bibliography entry Make comment on this entry
Deshpande, M. N. Archaeological sources for the reconstruction of the history of sciences of India. Indian J. History Sci. 6 (1971), 1--22. (Reviewer: A. I. Volodarskii.) SC: 01A25 (01A10), MR: 58 #15813.
A broad review of the archaeology of ancient India, focusing on the sciences. Perhaps a third of the article is devoted to a discussion of the Harappan civilization, and particularly Harappa and Mohenjo-Daro. Little is directly known about Harappan mathematics, but there are strong suggestions that there would have been some significant knowledge of surveying and possibly astronomy. The author also discusses the Harappan system of weights and measures. A good area for future research, particularly if some progress is made in reading the Harappan script. Closely related topics: The Harappan Civilization, Surveying, Astronomy, The Balance and the Measurement of Weight, The Measurement of Distance, and Archaeology.Modify notes on this entry Modify bibliography entry Make comment on this entry
Fischer, Irene K. At the dawn of geodesy. Bull. Géodésique 55 (1981), no. 2, 132--142. SC: 01A10 (01A17 01A20 01A25), MR: 83g:01002.
The cultures in ancient Egypt and in Greece, China, and Babylonia all did work in surveying, geodesy, and astronomy. However, they all had different approaches to the subjects. The author explains that "The striking difference between the abstract, geometric approach of Greece and the concrete, algebraic approach of Babylonia and China represent not a difference in talents but a difference in culture-bound interests." The reader should probably have some prior knowledge of the subject matter (and of geodesy in particular) to fully appreciate this article. Closely related topics: Surveying, Astronomy, Ancient Egypt, Greece, China, and Sumerians and Babylonians.Modify notes on this entry Modify bibliography entry Make comment on this entry
Gerdes, P. Reconstruction and extension of lost symmetries: examples from the Tamil of South India. Symmetry 2: unifying human understanding, Part 2. Comput. Math. Appl. 17 (1989), no. 4-6, 791--813. (Reviewer: Marjorie Senechal.) SC: 01A99 (01A10 92K99), MR: 91a:01058d.
Gerdes discusses the designs drawn (or formerly drawn) by Tamil women in South India during the harvest month Margali. The author shows that some of the diagrams may be degradations of earlier patterns that display more symmetry and/or are constructed according to the cultural ideal of having only one line. Gerdes also discusses drawing algorithms; many algorithms work by applying a series of simple transformation rules to a simpler motif. The function of these diagrams appears to be religious. As the author explains, "Margali is the month in which all kinds of epidemics were supposed to occur. Their designs serve the purpose of appeasing the god Siva who presides over Margali." Closely related topics: The Tamil of South India, Continuous Tracing Problems, Symmetry, and Religion.Modify notes on this entry Modify bibliography entry Make comment on this entry
Gerdes, Paulus. Fivefold symmetry and (basket) weaving in various cultures. Fivefold symmetry, 245--261, World Sci. Publishing, River Edge, NJ, 1992. SC: 52B99 (01A07), MR: 1 178 750.
Gerdes suggests that five-fold symmetries arose from efforts to solve problems in basketweaving rather than in observations of five-fold symmetry in natural phenomena (such as starfish). One way five-fold symmetries can arise is by modifying the more obvious six-fold symmetries (such as those used by peasants in Mozambique) to fit a curved surface. The author reports that "these pentagonal-hexagonal baskets are, for instance, also woven by the Ticuna and Omagua Indians (northeastern Brazil), by the Huarani Indians, by the Kha-ko in Laos, and by the Menda in India. One sees them also in China, Japan, and Indonesia." The Malaysian sepak tackraw ball is similar to the soccer ball and is woven in the same way. The author reports that the peasants of the island Roti (Indonesia) may have discovered a way to fold a regular pentagon as a kind of a thimble. The author shows how a similar pentagonal weaving pattern is used in weaving brooms in Mozambique. (A near pentagram then appears inside the knot.) The author notes that a similar method is used in Angola to hold together the bars of a cage. The author in addition discusses how hat weaving techniques can lead naturally to three- and five-fold symmetries. The author's main example is with the hats of the Belu of central Timor, but he notes that related techniques are used in northern Mozambique, southern Tanzania, and by the Kuva of Congo. The author also shows a Chinese hat with five-fold symmetry. Two other particularly interesting examples are "a burden basket ... from the Papago Indians (Arizona) which combines beautifully a global sevenfold symmetry with local fivefold symmetry", and the "center of a Japanese basket, which combines global ninefold symmetry with local fivefold symmetry." Closely related topics: Five Fold Symmetry, Basket Making, Mozambique, Malaysia, and The Belu of Central Timor.Modify notes on this entry Modify bibliography entry Make comment on this entry
Gerdes, Paulus P. J. On ethnomathematical research and symmetry. Symmetry in a kaleidoscope, 2. Symmetry Cult. Sci. 1 (1990), no. 2, 154--170. SC: 01A07, MR: 1 188 949.
Gerdes begins with a discussion of why symmetry is such a common phenomenon in human culture. He notes that some symmetries which are rare in nature (e.g., rotational symmetries of order 2) are common amongst us. Gerdes gives the example of rotational symmetry being used in the tattoos of the Makonde of northern Mozambique. Gerdes explains how symmetries such as the rotational symmetry of order 2 can arise naturally in solving problems in such areas as weaving. Gerdes then turns to the geometry of the line drawings made by the Tamil women in South India (during harvest month) and those made by the Tshokwe. These drawings have some strong similarities, and in both cases show an interest in tracing out a figure with a single continuous line. They also show a strong interest in symmetry, and Gerdes gives examples of how designs which fail to follow the one-line cultural norm may also fail to display the expected symmetries, suggesting that such drawings are degradations of more symmetric ones drawn with one line. The author advances a construction principle that can be used to construct both the Tamil and Tshokwe patterns. (Although the author doesn't note this, it is interesting that this principle is very similar to another principle that has been advanced for Celtic knot friezes!) Gerdes then discusses some mathematical properties of curves made using his construction principle. He also discusses some other interesting topics in his ethnomathematical research. For example, the author mentions that he has a found a new hypothesis on the origin of the Egyptian formula for the volume of a truncated pyramid, and has also found an infinite series proof for the Pythagorean theorem. Closely related topics: Symmetry, The Tamil of South India, TheTshokwe, Continuous Tracing Problems, The Celts, Ancient Egypt, and Pythagorean Triangles and Triples. Also possibly relevant: Mozambique, Tattoos, and Weaving.Modify notes on this entry Modify bibliography entry Make comment on this entry
Grünbaum, Branko. The emperor's new clothes: full regalia, G-string, or nothing? With comments by Peter Hilton and Jean Pedersen. Math. Intelligencer 6 (1984), no. 4, 47--56. (Reviewer: H. S. M. Coxeter.) SC: 01A15 (01A60 05B45 20F32 52A45), MR: 86d:01004.
Grünbaum's article: The author discusses the common misconceptions that the Egyptians and the artists of the Alhambra had used all 17 types of plane patterns. In fact, the Egyptians appear to have missed the five symmetry groups which have three-fold rotations. The sources for these misconceptions are discussed as well. The author has done fairly extensive research on the subject, and has concluded that two of the four plane patterns missing from the Alhambra seem not to appear at all in Islamic art (these are pg and pgg; the two missing at the Alhambra but present elsewhere are p2 and p3m1). A final theme of the author's is that the language of symmetry groups may at times be inadequate to discuss patterns, and can also be misleading in connection with the intentions of the artists themselves.The response by Peter Hilton and Jean Pedersen: The author's acknowledge Grünbaum's correction about the Egyptians. The authors note that the Egyptians and Moore's between them only missed one symmetry group, p3m1. They comment briefly on Chinese and Japanese designs, and quote Schattschneider, who notes that Chinese and Japanese artwork features rotations and glide reflections much more strongly than Islamic art does. Schattschneider also cites an illustration from a Japanese book that seems to suggest that underlying lattices of squares, equilateral triangles, rhombuses, and parallelograms were consciously used in developing symmetry patterns. The authors acknowledge the limitations of group theory in discussing symmetry, but also emphasize its usefulness. Closely related topics: Plane Patterns, Ancient Egypt, The Islamic World, Penrose Tilings, Japan, and China.
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Høyrup, Jens. Sub-scientific mathematics: observations on a pre-modern phenomenon. Hist. of Sci. 28 (1990), no. 79, part 1, 63--87. (Reviewer: David Singmaster.) SC: 01A10 (01A05 01A12 01A80), MR: 91j:01007.
Høyrup makes a distinction between scientific and subscientific mathematics. These fields correspond somewhat to pure and applied mathematics. However, by using this new terminology, the author hopes to avoid suggesting that "subscientific" mathematics is always derived from "scientific" mathematics in the way that "applied" mathematics is derived from "pure" mathematics. Høyrup discusses the distinction between scientific and subscientific mathematics and also their various kinds of relationships. His examples are drawn from Greece, Egypt, India, the Islamic World (with references to the Silk route), and from the Carolingian Propositiones ad acuendos jevenes. (The latter is traditionally associated with Alcuin.) Høyrup touches on relevant work by the mathematicians Hero, Diophantus, and al Khwarizmi. Surveying is discussed as a particularly important type of subscientific mathematics. Closely related topics: Applied Mathematics (General), Greece, Ancient Egypt, India, The Islamic World, Alcuin, Heron, Diophantus, Surveying, and Abu Abdullah Muhammed ibn Musa al Khwarizmi.Modify notes on this entry Modify bibliography entry Make comment on this entry
Kapur, J. N. Encounters of a working mathematician with history of mathematics. Ga\d nita Bh\=arat\=\i 11 (1989), no. 1-4, 30--37. SC: 01A99 (01A32), MR: 91i:01150.
In the process of describing his own encounters with the history of mathematics, the author makes a strong argument for its importance, particularly in mathematics education. He notes that mathematicians are too often unaware even of the history of their own research areas. For example, he mentions "a student who had written a Ph.D. thesis on Banach spaces had no idea who Banach was, to which century he belonged and of what country he was a citizen and why this concept was necessary." As the author notes, such ignorance inevitably weakens mathematics, since it separates mathematics from the applied problems that often motivated it. He discusses the quantity of research currently taking place in India in various fields of mathematics, and in the history of mathematics (and Indian mathematics) in particular. He finds room for improvement, and closes with some some recommendations for correction. Closely related topics: Why Study History Of Math, Education, Applied Mathematics (General), and India.Modify notes on this entry Modify bibliography entry Make comment on this entry
Katz, Victor J. Essay reviews of Ethnomathematics [Brooks/Cole, Pacific Grove, CA, 1991; MR: 92c:01006] by M. Ascher and The crest of the peacock [Tauris, London, 1991; MR: 92g:01004] by G. G. Joseph. Historia Math. 19 (1992), no. 3, 310--315. SC: 01A07 (00A30), MR: 1 177 496.
Katz reviews and contrasts Marcia Ascher's book Ethnomathematics: A Multicultural View of Mathematical Ideas and George Gheverghese Joseph's book The Crest of the Peacock: Non-European Roots of Mathematics. He finds that both correct serious omissions in the literature (and in particular, in Morris Kline's Mathematical Thought from Ancient to Modern Times). Joseph focuses on the history of mathematics in the large civilizations of ancient Egypt, Babylonia, China, India, and the Islamic World. He wanted to highlight "(1) the global nature of mathematical pursuits of one kind or another; (2) the possibility of independent mathematical development within each cultural tradition; and (3) the crucial importance of diverse transmissions of mathematics across cultures, culminating in the creation of the unified discipline of modern mathematics." Katz seems disappointed only in the third thesis, "because the documentary evidence for transmission of mathematical ideas is lacking." (For example, he notes that "whether Diophantus was directly influenced by the Babylonian tradition is a subject of scholarly debate." Joseph's treatment of Indian mathematics seems to be particularly good "especially since it is difficult to find this material in other sources." The focus of Ascher's book is completely different. She looks at traditional non-literate peoples. As Katz notes, "She has no intention of claiming that the mathematics developed in the cultures she discusses had any influence on developments elsewhere. Her main goal is simply to show that mathematical ideas, even if not developed by those called mathematicians, can be found in many societies if one only knows where to look." Katz reports examples as coming from the Inuit, Navajo, Iroquois, and Incas of the Americas, the Malekula, Warlpiri, Maori and Caroline Islanders of Oceania, and the Tshokwe, Bushoong, and Kpelle of Africa. This very useful review concludes by highly recommending both books. Closely related topics: Ancient Egypt, Sumerians and Babylonians, China, India, The Islamic World, The Inuit, The Navajo, The Iroquois, The Inca, The Malekula of Vanuatu, The Warlpiri, The Maori, The Caroline Islands, TheTshokwe, The Bushoong, and The Kpelle of Guinea.Modify notes on this entry Modify bibliography entry Make comment on this entry
Knorr, W. R. The geometer and the archaeoastronomers: on the prehistoric origins of mathematics. Review of: Geometry and algebra in ancient civilizations [Springer, Berlin, 1983; MR: 85b:01001] by B. L. van der Waerden. British J. Hist. Sci. 18 (1985), no. 59, part 2, 197--212. SC: 01A10, MR: 87k:01003.
The reviewer discusses van der Waerden's book Geometry and Algebra in Ancient Civilizations. Although the reviewer clearly admires van der Waerden for his work in algebra and in the history of mathematics in general, he is highly critical of the conclusions reached in van der Waerden's book. A basic theme of the book is that there is a pre-Babylonian ancestor to mathematics in Babylonia, ancient Egypt, Greece, China and India; thus the book can therefore be thought of in part as a further development of Abraham Seidenberg's theories on the ritual origins of ancient mathematics. The reviewer takes issue with several facts cited in the book, and in addition with three assumptions that he sees van der Waerden using explicitly or implicitly in the book: "(1) independent discovery is so rare that it may effectively be discounted as a working hypothesis for relating technical traditions; (2) derivative traditions are inferior to their source traditions; (3) borrowing from one tradition to another is not selective, but entails the adoption of whole bodies of technique." (The phrase "inferior to" in (2) could just as well be replaced by "degraded in".) The reviewer suggests in addition that van der Waerden has not been sufficiently critical in accepting claims by Alexander Thom and others about advanced mathematics in megalithic monuments, and sees these claims as forming "the veritable linchpin of van der Waerden's thesis". The author briefly discusses some of Thom's work in megalithic mathematics, and concludes that he finds no real evidence of the Pythagorean theorem, the ellipse, or a standard unit of distance in neolithic times. The review concludes with the statement "I fear even more the regrettable impact on credulous nonspecialists who may not know to distinguish between the general enterprise of scientific research and the reckless notions of some scientists." Closely related topics: Sumerians and Babylonians, Ancient Egypt, Greece, China, India, The Stone Builders, Alexander Thom, and Pythagorean Triangles and Triples.Modify notes on this entry Modify bibliography entry Make comment on this entry
Kokomoor, F. W. The Status of Mathmatics in India and Arabia during the "Dark Ages" of Europe. Mathematics Teacher 29 (1936), 224--31.
A survey of some of the work in mathematics during the middle ages. The focus is on the Islamic world. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Islamic World, India, China, Medieval Europe, and The Middle Ages.Modify notes on this entry Modify bibliography entry Make comment on this entry
Kudlek, Manfred. Calendar systems. Mathematische Wissenschaften gestern und heute. 300 Jahre Mathematische Gesellschaft in Hamburg, Teil 2. Mitt. Math. Ges. Hamburg 12 (1991), no. 2, 395--428. (Reviewer: J. S. Joel.) SC: 01A99 (00A69), MR: 92j:01079.
A rare and unusually wide ranging look at calendar systems in a variety of cultures. Explains some of the astronomical issues involved. The author discusses calendars of Egypt, Babylonia, the Roman Empire, Greece (Athens), the Islamic World (especially Persia), India, China (only gives a taste, since more than 50 official calendars were used), Japan and Vietnam (their calendars were connected with China), Java, Bali, Guatamala (by the Cakchiquel Indians), revolutionary France, the Mayas, and in the Jewish tradition. Discusses the computation of the date of Easter. (The computation of Easter was of course one of the primary goals of mathematics instruction in the middle ages.) There is information on how to correlate these calendars as well (in terms of Julian dates). Closely related topics: The Calendar, Ancient Egypt, Sumerians and Babylonians, The Roman Empire, Greece, The Islamic World, India, China, Japan, Vietnam, Java, Bali, The Maya, Guatemala (and Cakchiquel Indians), France in the 1700s, The Jewish Tradition, and Religion.Modify notes on this entry Modify bibliography entry Make comment on this entry
Martzloff, Jean-Claude. Pi in the Sky. Unesco Courier (Nov., 1989), 22--28.
Very brief. Includes a bit on the influence of divination, astronomy/astrology, Confucianism, and Taoism on the development of Chinese mathematics. The emphasis on the answer rather than the proof shows a Taoist influence, "on the grounds that the fallacious arguments of the sophists showed its limits". Also a bit on how mathematics and mathematicians fit into Chinese society. Appears in edited form in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: China, Divination, Astronomy, Astrology, Confucianism, and Taoism.Modify notes on this entry Modify bibliography entry Make comment on this entry
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, The Chiu Chang Suan Shu (Nine Chapters on the Mathematical Art), and Abraham Seidenberg.Modify notes on this entry Modify bibliography entry Make comment on this entry
Nagy, Dénes. Symmet-origami (symmetry and origami) in art, science, and technology. Symmetry Cult. Sci. 5 (1994), no. 1, 3--12. SC: 00A69 (01A99), MR: 1 309 239.
Discusses the history and philosophy of origami and then (in a little more depth) discusses some of its applications. The author discusses applications in math and science education, and also in art, design, and technology. A particularly interesting application of paper-folding and the theory of polyhedra is in music education, where one researcher devised "a 'tower' of five octahedra, to illustrate some basic concepts in musicology. His inspiration was from a work by Möbius written in 1861. Ganter's compound polyhedron illustrates geometrically the following concepts and their connections: the vertices correspond to the notes of the chromatic scale, the edges corresponds to the thirds and fifths, and the triangular faces correspond to the triads." He mentions that M. C. Escher was interesting in construction paper models (though it is not really clear how deep that interest lay). It is interesting that the well-known book by T. Sundara Row entitled Geometric Exercises in Paper Folding seems to be independent from the Japanese traditions. Closely related topics: Origami, Symmetry, Japan, Education, Music, M. C. Escher, and August Ferdinand Möbius (1790-1868).Modify notes on this entry Modify bibliography entry Make comment on this entry
Patel, D. M. Symbols for 1, 2, 3, 4, 5, 6, 7, 8, 9 & 0 in Sanskrit and English languages. Math. Ed. (Siwan) 15 (1981), no. 1, B1--B3. (Reviewer: Brij Mohan.) SC: 01A99 (01A32), MR: 82h:01080.
There have been many theories on the origins of the numerals 1 through 9. The numerals for 1, 2, and 3 are frequently thought to based on one two or three tally marks or fingers, drawn in the case of 2 and 3 so that the number is written in one stroke. There have been many theories for the origins of the other numerals. Patel suggests that the Hindu-Arabic numerals 4, 6, 7, 8, and 9 were derived from shapes made with the fingers (perhaps some kind of finger numerals?). It's likely that the last word has not yet been said. He also notes similarities between the Sanskrit and English words for the numbers one through nine; these similarities are however already very well known. Closely related topics: The Hindu-Arabic Numerals, India, and Finger Numerals.Modify notes on this entry Modify bibliography entry Make comment on this entry
Pingree, David. The Mesopotamian origin of early Indian mathematical astronomy. J. Hist. Astronom. 4 (1973), no. 1, 1--12. (Reviewer: A. I. Volodarskii.) SC: 01A15 (01A25), MR: 58 #25.
Some of the most important questions in the history of mathematics are on the interactions between various cultures. Here, the author makes a strong case that early Indian mathematical astronomy (in the Jyotisavedanga) was influenced by Mesopotamia science. His discussion is somewhat technical, and may be hard to follow for those not knowledgeable about the subjects involved. Near the end of the article, the author writes "But there is one further question that we must raise before accepting this hypothesis of transmission. Was this an isolated phenomenon, or part of a general Iranian influence on Indian culture in the fifth and fourth centuries B.C.?" Although, as he notes, "our answer to that question is rather clouded by the scarcity of literary or archaeological data from the period in question", he finds that he is able to conclude with the statement "It is reasonable then, or at least so I believe, to see the origins of mathematical astronomy in India as just one element in a general transmission of Mesopotamian-Iranian cultural forms to northern India during the two centuries that antedated Alexander's conquest of the Achaemenid empire." Closely related topics: India, Sumerians and Babylonians, and Astronomy.Modify notes on this entry Modify bibliography entry Make comment on this entry
Rav, Yehuda. On the interplay between logic and philosophy: a historical perspective. Theoria (San Sebastián) (2) 8 (1993), no. 19, 1--21. (Reviewer: Pierre Kerszberg.) SC: 03A05 (01A99 03-03), MR: 95c:03014.
The author discusses some of the connections between philosophy, logic, mathematics, and language. He focuses mainly on the West but also touches slightly on China. The reader should probably have a relatively strong background in philosophy before attempting this article. There is a long bibliography that should be useful for students making further investigations in these areas. Closely related topics: Philosophy, Logic, Language and Linguistics, and China.Modify notes on this entry Modify bibliography entry Make comment on this entry
Schaaf, William L. Mathematics and World History. Mathematics Teacher 23 (1930), 496--503.
Concerned with the idea the different cultures have different ways of thinking about mathematical concepts. Schaaf takes the number concept as an example. The idea of number and magnitude was concrete and geometric to the Greeks, and was closely tied with the idea of measurement. This notion was changed by Diophantus, who may have been influenced by the mathematics of India and the Middle East. Similar ideas in the Islamic world may have reached Europe in the middle ages. A new concept of number was introduced with Descartes in Analytic Geometry. Since then, mathematics has become still more abstract. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Number Concept, Geometry, Greece, Measurement, Diophantus, India, The Middle East, The Islamic World, Analytic Geometry, and Arithmetic.Modify notes on this entry Modify bibliography entry Make comment on this entry
Schattschneider, Doris. The plane symmetry groups: their recognition and notation. American Mathematical Monthly 86 (1978), 439--450.
Discusses in detail the classification of plane patterns. Although the author avoids group-theoretic notation, she manages to bring out the group theoretic nature of the plane pattern groups more clearly than most other authors discussing these patterns. There is a very useful chart on the seventeen plane patterns that clearly labels the locations of the centers of rotation (with labels that distinguish the 2, 3, 4, and 6-fold centers), the axes of reflection, and the axes of glide-reflection. The chart may give a better understanding of the differences between the different symmetry groups than the flowcharts that appear in some other sources. The author discusses the generating regions for each of the plane patterns, and gives examples for each symmetry group of two set of generators of the group (except in the case of the pattern p1, where there is only one natural set of generators. She illustrates the plane patterns with lattices, most of which are from China. There are a couple of examples from the artwork of M. C. Escher as well. There is also a table cross-referencing notations used by different sources. There are six different notations in all; as the author notes, one the differences results from the common confusion between the groups p3m1 and p31m. Closely related topics: Plane Patterns, Group Theory, Art, M. C. Escher, and China.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, The Chiu Chang Suan Shu (Nine Chapters on the Mathematical Art), 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: The Chiu Chang Suan Shu (Nine Chapters on the Mathematical Art), 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. Closely related topic: The Chiu Chang Suan Shu (Nine Chapters on the Mathematical Art).Modify notes on this entry Modify bibliography entry Make comment on this entry
Swetz, Frank. The Evolution of Mathematics in Ancient China. Mathematics Teacher 52 (1979), 10--19.
An overview of Chinese mathematics, including the discovery of the lo shu magic square (thought to have a plan of universal harmony), square roots, the Chinese remainder theorem, and polynomials of high degree (including a quintic in x2). Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: China, Algebra, and Magic Squares.Modify notes on this entry Modify bibliography entry Make comment on this entry
Swetz, Frank J. Seeking Relevance? Try the History of Mathematics. Mathematics Teacher 77 (1984), 54--62.
Focuses on how the history of mathematics can be used to improve mathematics education. It can not only breath new life into the subject, but also allow students to better understand mathematics as a mode of inquiry. If students see mathematical ideas in other times [and in other cultures], they can appreciate the ideas better in our own. Swetz gives examples from the development of algorithms for arithmetic (including square roots). Ancient demonstrations of mathematical ideas, such as the "husan-thu" proof of the Pythagorean theorem from China can be conceptually more suitable for students than more synthetic modern ones. Ancient "homework problems" from Babylonia, China, and Medieval Italy can be more interesting than the more dry and formulaic modern equivalents. (See Swetz, Was Pythagoras Chinese? for many interesting examples from China.) Although the author doesn't discuss this, the Chinese problems in surveying led to interesting questions in algebra, with fourth and higher degree equations. Swetz discusses how Descartes' idea of a coordinate grid was earlier used by Renaissance artists, ancient Egyptian tomb painters, and various cartographers. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Education, Arithmetic, Computation, China, Algebra, Analytic Geometry, Renaissance Art, Ancient Egypt, and Cartography.Modify notes on this entry Modify bibliography entry Make comment on this entry
Woodruff, Charles E. The Evolution of Modern Numerals from Ancient Tally Marks. American Mathematical Monthly 16 (1909), 125--33.
A theory that the Hindu-Arabic numerals actually started out in China. Gives a possible evolution of each of the digits 1--9. There are many other theories as well, so it would be valuable to find evidence of some of these "missing links". Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Hindu-Arabic Numerals and China.Modify notes on this entry Modify bibliography entry Make comment on this entry
Zimmermann, Francis. Lilavati, Gracious Lady of Arithmetic. Unesco Courier 51 (Nov., 1989), 18--21.
Discusses Bhaskara's Lilavati, but the main interest really seems to be on the character and context of Indian mathematics. Excellent brief introduction. Appears in edited form in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: India and Bhaskara.Modify notes on this entry Modify bibliography entry Make comment on this entry
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