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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.
Anagnostakis, Christopher and Goldstein, Bernard R. On an error in the Babylonian table of Pythagorean triples. Centaurus 18 (1973/74), 64--66. (Reviewer: E. M. Bruins.) SC: 01A15, MR: 58 #20994.
The authors explain a well-known mistake in the Babylonian tablet Plimpton 322 (column I, entry 10) as a consequence of a certain method of computation and of the neglect of a medial zero. It is a very appealing theory, and could give us some insight the way Babylonians did their mathematics. Other solutions have also been proposed. A good example of how we can learn from mistakes! Closely related topics: Sumerians and Babylonians, Pythagorean Triangles and Triples, and Algorithms.
Archibald, Raymond Clare. Babylonian Mathematics. With Special Reference to Recent Discoveries. Mathematics Teacher 29 (1936), 209--19. (Originally delivered at a joint meeting of the National Council of Teachers of Mathematics, the American Mathematical Society, and The Mathematical Assocation of America, at St. Louis, Mo., on January 1, 1936.)
Surveys some of Neugebauer's remarkable discoveries on Babylonian mathematics, at a time when many of these discoveries were just made. Discusses notation, tables of squares, cubes, and n3+n2. Also exponentials, approximations to compound interest problems where we would use logarithms, a sum of a finite geometric series and a finite sum of squares. Geometric results, including the Pythagorean theorem, proportionality of sides in similar right triangles, a perpendicular bisecting the base in an isosceles triangle, the angle in a semicircle being a right angle, formulas for the circumference and area of a circle (using pi = 3), formulas for the frustum of a square pyramid (at least one incorrect). The relation between chords and sagitas in a circle. Approximations to the square root of a2+b2; both the well known a+b2/2a and the still hypothetical a+(2ab2)/(2a2+b2). An approximation to a square root by comparing with other solutions to an equation x2+D=y2. (The value isn't especially accurate, but the method is interesting.) Equations in five or more unknowns. Problems requiring solutions to apparently general cubic and biquadratic equations. Were the solutions just guessed, or, as Neugebauer suggests, did the Babylonians have some general methods? If so, the most likely theory is that the cubics were solved by effectively reducing them to the form x3+x2, and then using the n3+n2 table. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Sumerians and Babylonians, The Quadratic Formula, Cubics, Quartics, Solutions of Linear Equations, Logarithms, Exponentials, Square Roots, Interpolation, Geometric Theorems, The Circle, and The Pyramid.
Arndt, A. B. Al-Khwarizmi. Mathematics Teacher 76 (1983), 668--70.
An introduction to the work of al Khwarizmi. Focuses on his algebra, the Al-Kitab Al-jabr wa'l muqabalah and its influence on the West. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topic: Abu Abdullah Muhammed ibn Musa al Khwarizmi.
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.
Bruins, Evert M. The division of the circle and ancient arts and sciences. Janus 63 (1976), no. 1--3, 61--84. (Reviewer: J. L. Berggren.) SC: 01A15 (01A20), MR: 57 #12015.
One Etruscan cup, made in Caere about 500 BC, and now in the Museum of Fine Arts in Budapest, has both an 11-gon and a 14-gon inscribed on it. As the author notes, one possible reason why both were given together could be that the sum of the sides of an 11-gon and of a 14-gon imperceptibly deviates from the radius of a circle inscribing them. Moreover, methods known in the old Babylonian period could be used to provide excellent approximations to the lengths of the sides. All this raises questions about the level of Etruscan mathematical development, about which little is still known (their language still being poorly understood). The author also discusses Heron's rather accurate method for approximating the area of a circle. The article is very interesting, but the reader should be forewarned that it is a bit technical. Closely related topics: The Etruscans, Sumerians and Babylonians, The Circle, Polygons, and Heron.
Chorbachi, W. K. In the tower of Babel: beyond symmetry in Islamic design. Symmetry 2: unifying human understanding, Part 2. Comput. Math. Appl. 17 (1989), no. 4-6, 751--789. (Reviewer: Marjorie Senechal.) SC: 01A99 (01A30 92K99), MR: 91a:01058c.
An interesting and personal account of how the author discovered geometric manuscripts written for Islamic artisans. With this discover, the author gives a new historical and scientific basis to the study of certain kinds of Islamic art. Much work preceding the author's had focused on religious, mystical, or perceptual interpretations of the work. Many ideas were primarily hypothetical, such as the (incorrect) idea that all Islamic art derives from the circle. The author suggests that many religious and mystical interpretations of Islamic geometric art should not be regarded as being historically based. Instead, the author shows how some Islamic art is highly mathematical, showing concerns with such topics as Pythagorean triangles and the notion of similarity (he gives an example where a shape appears in three different scales, each similar shape being derived from the last by a clever process). Much of the article discusses these in the context of a cyclic quadrilateral appearing in Islamic art with sides 1, 2, 2, 71/2. The author even noted an Islamic anticipation of a shape used to produce Penrose tilings. The author suggests that symmetry groups, while useful, can not alone give a full understanding of Islamic art. Closely related topics: The Islamic World, Art, Plane Patterns, Pythagorean Triangles and Triples, Penrose Tilings, Religion, and Mathematics and Mysticism.
Crowe, D. W. and Washburn, D. K. Groups and geometry in the ceramic art of San Ildefonso. Proceedings of the conference on groups and geometry, Part A (Madison, Wis., 1985). Algebras Groups Geom. 2 (1985), no. 3, 263--277. (Reviewer: H. S. M. Coxeter.) SC: 05B45 (00A05 01A12 20F32 52A45), MR: 87k:05055.
Discusses the types of frieze patterns and bichromatic strip patterns occurring in the pottery of the pueblo of San Ildefonso in New Mexico. The people of San Ildefonso are Tewa speaking and are thought to be of Anasazi descent. However, it should be noted that the pottery has apparently been influenced by the Spanish and by attempts to make it more readily salable. All 7 of the strip patterns and 14 of the 17 possible bichromatic strip patterns are exhibited. (The authors supply the missing 3 bichromatic strip patterns in a similar style. The authors supplement their discussion with an explanation of the appealing Coxeter notation for classifying the bichromatic patterns (the standard classification system is cumbersome) and give a table of the correspondences between various systems. A historical aside briefly discusses the study of plane patterns in the context of the Alhambra, where there is still some disagreement on which patterns are represented. Closely related topics: The Pueblo of San Ildefonso, Frieze Patterns, Bichromatic Strip Patterns, Plane Patterns, Pottery, Archaeology, The Islamic World, and Spain in the Middle Ages.
Dilke, O. A. W. Mathematics and measurement. Reading the Past, 2. University of California Press, Berkeley, CA; British Museum Publications, Ltd., London, 1987. 64 pp. ISBN: 0-520-06072-5. (Reviewer: Richard L. Francis.) SC: 01A05 (01A15 01A20), MR: 89f:01003.
This very interesting book discusses many aspects of mathematics in the Roman empire, Egypt, Babylonia, Greece, and sometimes other cultures. The book discusses systems of measurement of length, area, volume, and weight, mathematical or para-mathematical subjects such as surveying, cartography, interest rates, taxes, time keeping, games, and numerology. Also discusses number systems. Much of the discussion on number systems may be familiar, but here there is also a little that may be a little less familiar, such as the use of Etruscan letters in the early Roman numerals. In a work of this scope, the author of the book is not to be faulted that there may be some disagreement with occasional facts. The discussions on the mathematics of the Romans are particularly interesting; there are few other studies touching on Roman mathematical practices at all. Closely related topics: The Roman Empire, Ancient Egypt, Sumerians and Babylonians, Greece, The Measurement of Distance, The Measurement of Area and Volume, The Balance and the Measurement of Weight, Surveying, Cartography, Banking, Taxation, The Reckoning of Time, Games, Numerology, and Number Systems.
Eves, Howard. Omar Khayyam's Solution of Cubic Equations. Mathematics Teacher 51 (1958), 285--86.
Shows how Omar Khayyam solved the equation x3+b2x+a3=cx2 using the intersection of a circle and a rectangular hyperbola. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Omar Khayyam (abu-l-Fath Omar ibn Ibrahim Khayyam), Cubics, and The Conic Sections.
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.
Gerdes, Paulus. On mathematics in the history of sub-Saharan Africa. Historia Math. 21 (1994), no. 3, 345--376. SC: 01A13, MR: 95f:01003.
This paper broadly surveys the recent research in sub-Saharan mathematics (and some related areas as well). Areas discussed include prehistoric mathematics (e.g., the Ishango and Border Cave bones), number systems and symbolism (including algorithms and education), games and puzzles (for example, a leopard-goat-cassava leaf river crossing problem and a "topological" puzzle), symmetry in African art, graphs or networks (e.g. Tschokwe sand drawings), architecture (one case involving magic squares; also a brief reference to fractals). Gerdes mentions string figures as a possibly productive future research area; he gives some starting points. He also discusses related areas, such as technology, and studies on language and mathematical concepts. A goal of the studies mentioned is apparently to better understand mathematics learning in Africa. Some studies focus on logic. Questions on interaction with ancient Egypt are still largely open. A better understanding of Islamic mathematics in sub-Saharan Africa is desirable as well. The author also touches on factors connected with the slave trade; e.g., the remarkable but not perhaps entirely atypical abilities of Thomas Fuller. Includes an extensive bibliography. Closely related topics: Sub-Saharan Africa, TallySystems, Games, Puzzles, Topology, Symmetry, Continuous Tracing Problems, Architecture, Magic Squares, Fractals in Art, String Figures, Ancient Egypt, The Reckoning of Time, Education, Mathematics in Language, Logic, The Islamic World, and Thomas Fuller (1710-1790).
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.
Grünbaum, Branko, Grünbaum, Zdenka; Shephard, G. C. Symmetry in Moorish and other ornaments. Computers \& Mathematics with Applications. Part B 12 (1986), no. 3--4, 641--653.
It is observed that 13 of the 17 plane patterns are represented at the Alhambra. Two of the four missing groups have been found in Toledo, Spain, and dating from about the same period (one, p3, was found in a church, and the other, p3m1, was found in a synagogue). The authors note that the remaining two patterns (pg and pgg) seem not to appear in Islamic art at all. The authors note that features of Islamic art are not always fully described by the symmetry groups alone; such features can include color changes and interlace patterns. The color-symmetry groups are only a partial solution to the former, since colors are often in ratios "2:1:1, 4:2:1:1, 6:2:1, 6:3:1:1:1 or some similar ratio... The mathematical theory of such colorings still awaits development." The authors also attack the commonly held view that the artists of the Alhambra exhausted the possibilities of symmetry in art, and illustrate their points with pictures. Moreover, the authors suggest that ideas of local structure are as important as ideas of global structure. "The various kinds of symmetry groups are useful in the description of many of the artifacts, but more general approaches (based on 'adjacency relations' or other 'local' criteria) are necessary for a better understanding of the ornaments and artwork, and of the ways their creator thought about them." Closely related topics: The Islamic World and Plane Patterns.
Grünbaum, Branko; Shephard G. C. Interlace patterns in Islamic and Moorish art. The Visual Mind, 147--155, Leonardo Book Series, MIT Press, Cambridge, Mass., 1993.
Many Islamic and Moorish patterns exhibit what the authors call interlace patterns, where the patterns seem to be made of strands that alternately go over and other strands. This is a phenomenon that makes these Islamic artworks appear something like a 2-D extension of the Celtic knot friezes; the over/under rule is of course also common in weaving. The authors focus on the seemingly curious phenomenon that many of the Moorish and Islamic interlace patterns can be viewed as being made of a small number of basic shapes, often one or two. The authors analyze this phenomenon for the symmetry groups p4m and p6m, and find that it arises in a mathematically natural way, especially if artists used stencils, as is sometimes now thought. The article gives give propositions without proof; proofs of these should be within reach of a good undergraduate with the requisite knowledge of group theory. Closely related topics: The Islamic World, Plane Patterns, Knots and Knotwork, and Weaving.
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.
Hĝyrup 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. Hĝyrup 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.) Hĝyrup 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.
Jones, Phillip S. Recent Discoveries in Babylonian Mathematics. I. Zero, Pi, and Polygons. Mathematics Teacher 50 (1957), 162--65.
Supplements Archibald, Raymond Clare, Babylonian Mathematics, discussing some work by Neugebauer and others 1936 and 1957. Discusses the invention of the zero in (later) Babylonia and its appearance in Greece. (Zero was apparently first regarded as a true number by Aristotle.) Also discusses a value of 3 1/8 for pi (reported by M.E.M. Bruins, anticipated by Neugebauer), a problem to determine the radius of a circle circumscribing an isosceles triangle with two sides of 50 and one of 60 (an often discussed example, originally discovered by Bruins, that is still a good algebra problem, using only the Pythagorean theorem), and a table giving areas of pentagons, hexagons, and heptagons from the square of a side. Not all are accurate, but agree with analogous values given later by Heron (c. 75 AD). Heron's table included the regular nonagon as well. The article is continued in Jones, Phillip S., Recent Discoveries in Babylonian Mathematics. II., which however, has a somewhat smaller scope. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Sumerians and Babylonians, The Circle, Zero, Aristotle, The Measurement of Area and Volume, and Heron.
Jones, Phillip S. Recent Discoveries in Babylonian Mathematics. II. The Earliest Known Problem Text. Mathematics Teacher 50 (1957), 442--44.
Continues Jones, Phillip S., Recent Discoveries in Babylonian Mathematics. I.. Discusses a very old Babylonian problem text (c. 2000 BC), that seems to show an understanding of the proportionality of sides in similar right triangles. Continued in Jones, Phillip S., Recent Discoveries in Babylonian Mathematics. III., which has a different character from both of its predecessors. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Sumerians and Babylonians, Similarity, and The Triangle.
Jones, Phillip S. Recent Discoveries in Babylonian Mathematics. III. Trapezoids and Quadratics. Mathematics Teacher 50 (1957), 570--71.
Continues Jones, Phillip S., Recent Discoveries in Babylonian Mathematics. II.. The author discusses a single Babylonian problem. The problem is interesting more as a representative of a "typical" Babylonian problem than as a discovery that gives new insights into Babylonian mathematics. The problem involves the solution to a quadratic. The scribe uses an incorrect "formula" for the area of a trapezoid. The author discusses the solution both using modern notation and in a translation of the scribes actual language. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Sumerians and Babylonians, The Quadratic Formula, and The Measurement of Area and Volume.
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.
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.
Knuth, Donald E. Ancient Babylonian algorithms. Twenty-fifth anniversary of the Association for Computing Machinery. Comm. ACM 15 (1972), no. 7, 671--677; errata, ibid. 19 (1976), no. 2, 108; MR: 52#13133. SC: 01A15, MR: 52 #13132.
Were there computer scientists among the ancient Babylonians? Probably not. However, some of the ideas in computer science occurred to the ancient Babylonians as well. The author here discusses Babylonian algorithms in particular. Most algorithms are of course given as examples, but Knuth notes one text that is an exception: "Length and width is to be equal to the area. You should proceed as follows. Make two copies of one parameter. Subtract 1. Form the reciprocal. Multiply by the parameter you copied. This gives the width." Knuth explains, "In other words, if x+y=xy, it is possible to compute y by the procedure y=(x-1)-1x. The fact that no numbers are given made this passage particularly hard to decipher, and it was not properly understood for many years; hence we can see the advantages of numerical examples. The above procedure reads surprisingly like a program for a 'stack' machine like the Burroughs B5500!". Knuth finds a table involving compound interest where he finds evidence of a "DO I = 1 TO N" loop and something like a "WHILE" clause. He also discusses how one tablet may have been obtained by sorting a large set of numbers. "Thus, Inakibit seems to have the distinction of being the first man in history to solve a computational problem that takes longer than one second of time on a modern electronic computer!" [However, note that this statement was made in 1972.] Some tablets cited are available here in English for the first time (Knuth translated them using German and French translations, and at times Akkadian and Sumerian vocabularies as well). See errata in Knuth, Donald E., Errata: "Ancient Babylonian algorithms" (Comm. ACM 15 (1972), no. 7, 671--677). Closely related topics: Sumerians and Babylonians, Computation, Algorithms, and Logarithms.
Knuth, Donald E. Errata: "Ancient Babylonian algorithms" (Comm. ACM 15 (1972), no. 7, 671--677). Comm. ACM 19 (1976), no. 2, 108. SC: 01A15, MR: 52 #13133.
An errata to Knuth, Donald E., Ancient Babylonian algorithms. The table that was sorted was not as extensive as Knuth previously believed, and involved a "file" of about 500 instead of about 800. As Knuth notes "My italicized statement on p. 676 that 'this table contains every one' of the 231 regular sexagesimal numbers of six digits or less, is false; the table contains only 136 of those 231." The misunderstanding was due to a failure "to read the accompanying German commentary carefully enough, since [Neugebauer] departed from his usual custom in this particular case. Many of the lines in his rendition of the table were not on the original clay tablet at all, they were interpolated to show what the tablet would have looked like if it had been complete." Closely related topics: Sumerians and Babylonians, Computation, Algorithms, and Logarithms.
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.
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.
Llyod, Daniel B. Further Evidences of Primeval Mathematics. Mathematics Teacher 57 (1966), 668--70.
A tablet from a dig at Tel Dhibayi near Baghdad shows how to find the dimensions of a triangle from its diagonal and area. The solution requires a knowledge of the Pythagorean theorem, and artfully sidesteps the difficulty of solving a quadratic equation by solving a pair of simple linear equations. Many other articles discuss similar tablets and solutions, but few so concisely as this. However, note that in the context of other Babylonian sources, the method of solution may be less obscure than the author seems to suggest. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Sumerians and Babylonians, The Quadratic Formula, and Pythagorean Triangles and Triples.
Lorch, Richard. The sphera solida and related instruments. Special issue dedicated to Olaf Pedersen on his sixtieth birthday. Centaurus 24 (1980), 153--161. (Reviewer: K.-B. Gundlach.) SC: 01A99 (85-03), MR: 82a:01057.
The sphera solida or "solid sphere" is "essentially a globe, on which the stars and principal celestial circles are depicted, and a frame of horizon and meridian circles." Related instruments include the astrolabe, and particularly the spherical astrolabe. On the other hand, the sphera solida should not be confused with the armillary sphere. As an example how the sphera solida was used, the author explains that "To align the sphere with the Heavens in the daytime, and so obtain the configuratio celi, a pin is stuck into the degree of the sun in the ecliptic and the sphere is turned until the pin has no shadow. At night the same can be achieved by the less spectacular method of taking the altitude of a known star and shifting the sphere till the representation of the star has the same altitude--just as in a plane astrolabe." (p. 157) Much of the article focuses on the literary sources on the sphera solida, which are "at least as old as the fourteenth century." The author concludes that the ultimate source may be Arabic, and mentions a related Islamic globe made in 1279. "But unfortunately there is no clear Arabic exemplar for the text of the Sphera solida." This article has a rather scholarly tone, was doubtless difficult to research; it ends with the unusual note "Finit tractatus. Deo gratias." Closely related topics: Medieval Europe, The Islamic World, and The Astrolabe and Related Instruments.
Lumpkin, Beatrice. From Egypt to Benjamin Banneker: African origins of false position solutions. Vita mathematica (Toronto, ON, 1992; Quebec City, PQ, 1992), 279--289, MAA Notes, 40, Math. Assoc. America, Washington, DC, 1996. SC: 01A05 (01A13), MR: 1 391 748.
Discusses the work of the Benjamin Banneker, who is perhaps the most interesting early American mathematician. The author gives a fine introduction to Banneker's life; this is necessarily brief, because as the author observes, his house burned down on the day of his funeral, destroying almost all his papers. She notes that there were hints of his genius starting with his building of a wood clock at the age of 22 (he used a borrowed pocket watch as a model; unfortunately, the clock was destroyed in the fire); he thereafter became famous for his ability to solve and create mathematical puzzles. "People sent him puzzles from all over the colonies and later from the new republic." His work became more serious when he was 57 and borrowed some books and astronomy instruments from a neighbor. He taught himself the mathematics he needed to become an astronomer, and published local almanacs including things such as the planetary positions and the times of sunrise, sunset, moonrise, moonset, eclipses, and tides. "Based on Banneker's work on his almanac, he was appointed an astronomer on the team of surveyors that drew up the outline for the new nation's capital, Washington, DC. Banneker was appointed because he was one of the few in the country capable of doing such work. Charles Leadbetter, author of an astronomy book that Banneker studied, wrote that knowledge of astronomy in London was 'so rare, ... not one of 20,000 hath attained to it.' Knowledge of astronomer", Lumpkin continues, "was even rarer in the new United States. Banneker's work so impressed Thomas Jefferson, then Secretary of State, that he wrote Banneker that he was sending a copy of the almanac to the Paris Academy of Sciences." Most amazing of all is that Banneker accomplished all this as an African American who had spent most of his life thus far hard physical labor. After this introduction, the author focuses on how Banneker and other mathematicians used the rule of false position. She notes, the rule of false position was used by the Egyptians in the time of the Rhind Papyrus and in a variety of other Egyptian sources (e.g., the Kahun and Berlin papyri), in the work of Alexandrian Greeks like Diophantus (c. 250 AD), in the work of Islamic mathematicians such as Abu Kamil (b. 850 AD), and in the work of the mathematician Leonardo of Pisa (Fibonacci) (who was also influenced by the work in Northern Africa). The author then discusses some interesting false position problems from Banneker's own work. Closely related topics: Benjamin Banneker, The Method of False Position, The Rhind/Ahmes Papyrus, Ancient Egypt, Diophantus, Abu Kamil (b. 850), and Leonardo of Pisa (Fibonacci).
Meserve, Bruce E. The Evolution of Geometry. Mathematics Teacher 49 (1956), 372--82.
Discusses the history of geometry starting with the Egyptians and Babylonians and continuing into modern times. The rise and decline of Greek geometry, the logical structure of Greek proofs. Contributions by the Islamic world on the parallel postulate. Contributions of Renaissance artists studying perspective. Analytic geometry. More on the parallel postulate. Non-Euclidean geometry. The development of projective geometry and algebraic geometry. The article concludes with a discussion of how computational technology might change the nature of mathematics. Reprinted in edited form in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Geometry, Analytic Geometry, Projective Geometry, Algebraic Geometry, Greece, The Islamic World, The Parallel Postulate, and Perspective.
Pazwah, Hormoz; Mavrigian, Gus. The Contributions of Karaji---Successor to al-Khwarizmi. Mathematics Teacher 79 (1986), 538--41.
An introduction to the work of al Karaji (often known as al Karkhi). Includes a little on arithmetic, algebra, geometry, and surveying. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topic: Abu Abdullah Muhammed ibn Musa al Khwarizmi.
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.
Powell, Marvin A., Jr. The antecedents of old Babylonian place notation and the early history of Babylonian mathematics. Historia Math. 3 (1976), 417--439. (Reviewer: Richard L. Francis.) SC: 01A15, MR: 58 #9990.
The Mesopotamian positional notation is generally thought to have originated in the Old Babylonian period (c. 2000--1600 BC), but the author argues that it actually dates back even further, before the end of the Third Dynasty of Ur (c. 2112--2004 BC) or even to the middle of the third millennium BC. The author looks at several texts, and finds evidence of a positional way of thinking in the way units of measurement were used and in the kinds of errors made by students. As is often the case, errors can be very useful in understanding the procedures that were used to do mathematics. In one example, the author compares the errors made by two different students: One tablet is "rather a text ... written by a bungler who did not know the front from the back of his tablet, did not know the difference between standard numerical notation and area notation, and succeeded in making half a dozen writing errors in as many lines, but nevertheless was not without a modicum of ability and probably finished school with a low passing grade, took a post with the government and became a bureaucrat. The writer of no. 50 [the other tablet] no doubt became a scholar and died penniless. However probable these postulated eventualities may be, the modern scholar may well be more grateful to our third millennium bungler than to his competent classmate." (p. 432) Closely related topics: Sumerians and Babylonians, Number Systems, and Measurement.
Rashed, Roshdi. Where Geometry and Algebra Intersect. UNESCO Courier (Nov., 1989), 37--41.
The interaction of Islamic algebra with algebra and geometry. Ways in which Islamic methods anticipated discoveries in Europe that were centuries later. Examples include the solution of cubics with intersecting curves (al-Khayyam, often attributed to Descartes) and the notion of maxima and minima of an algebraic expression (al-Tusi). Appears in edited form in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Islamic World, Algebra, Number Theory, Geometry, Analytic Geometry, and Calculus.
Ritter, James. Prime Numbers. Unesco Courier (November 1989), 12--17.
The title is a bit misleading. Discusses the work of Babylonian and Egyptian scribes and how they fit into society. Although neither society had a word for a mathematician, the ability to do mathematics was highly valued. One Mesopotamian king boasted of his academic achievements by stating proudly "I am perfectly able to subtract and add, [clever in] counting and accounting", and another says "I can find the difficult reciprocals and products which are not in the tables." In Babylonia and Egypt, mathematics was taught by creating a "network of typical examples in which a new problem can be related---by a form of interpolation---to those already known." An edited version appears in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Education, Sumerians and Babylonians, and Ancient Egypt.
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 Islamic World, Analytic Geometry, and Arithmetic.
Schmandt-Besserat, Denise. Oneness, Twoness, Threeness. The Sciences 27 (1987), 44--48.
Writing developed in Sumeria from attempts to represent numbers. Objects such as animals and bushels of grain were represented in a one-to-one correspondence with small clay tokens--animals with cylinders and bushels of grain with spheres. When Sumerian society became more complex, new complex tokens were invented. These represented finished items such as garments, metalworks, jars of oil, and loaves of bread. The complex tokens could have elaborate markings and a wide variety of shapes. What made things change was the habit of putting plain tokens in solid clay envelopes to record quantities in legal documents. Since breaking the envelopes symbolically "broke the deal", accountants began impressing the tokens on the surface. Later, they realized that the envelopes themselves were unnecessary. Soon, the Sumerians also copied the markings on complex tokens onto a two-dimensional surface. Writing had been invented. The symbols for small and large quantities of grain (a wedge and a circle) came to be used to represent the numbers 1 and 10 when used in conjunction with two-dimensional representations of complex tokens. Abstract numbers had been invented as well. Not long after, the pictographs came to represent sounds. This worked fairly well until the first fully phonetic alphabet was invented by the Phoenicians, perhaps 1400 years later. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Development of Writing, Sumerians and Babylonians, and Number Systems.
Schrader, Dorothy V. The Arithmetic of the Medieval Universities. Mathematics Teacher 60 (1967), 264--75.
The history of the notion of the liberal arts, particularly in the middle ages. The role of arithmetic (computational and theoretical). The abacus of Gerbert. The computation of Easter. The influence of the Arabic texts. Different attitudes towards arithmetic at different times and in different places. An excellent introduction to the mathematics of the middle ages, though of course it omits much on topics such as geometry and astronomy. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Liberal Arts, Arithmetic, Number Theory, Gerbert, Pope Sylvester II, Religion, Medieval Europe, and The Islamic World.
Scriba, Christoph J. Mathematics and music. (Danish) Normat 38 (1990), no. 1, 3--17, 52. SC: 01A99 (00A69), MR: 91i:01154.
The author discusses the relationship between mathematics and music from Pythagorean through modern times. His story begins in in Pythagorean times, and as he explains, the notes of the musical scale were then determined by the ratio of a perfect fifth, i.e. 3:2. Twelve intervals of a fifth are roughly equal to seven octaves, but are in reality slightly more than seven octaves, the discrepancy being the "Pythagorean comma" of 312:219, or roughly 74:73. Whole steps in the scale were in the ratio 9:8, and half steps were in the ratio 256:243. Thus two half steps were slightly less than one whole step. In fact, Philolaus noted that one whole note is equal to two half notes plus a Pythagorean comma. Archytas showed that intervals like the octave 2:1, fifth 3:2, fourth 4:3, and whole tone 9:8, or any other interval in the ratio (n+1):n cannot in fact be divided with rational numbers into two equal intervals. However, he noted that the product of the arithmetic mean and the harmonic mean is equal to the square of the geometric mean, so this gave a way of dividing the fifth of 3:2 into the product of 5:4 and 6:5. 5:4 can be thought of as a major third, and 6:5 can be though of as a minor third. So the ratio 3:2 is divided as 6:5:4. Similarly, the fourth of 4:3 can be divided into the product of 7:6 and 8:7, so the ratio 4:3 is divided as 8:7:6. The interval 7:6 can be though of as a shrunken minor third and 8:7 can be though of as an enlarged whole tone. Scriba suggests that the germs of the idea of making this division lie with the Babylonians.In the Renaissance, the musical scale was modified to take some of these ideas into account through the work of theoreticians like Ludovico Fogliano and Giusseppe Zarlino. For example, the ratio for the notes E:C and A:F were changed from the Pythagorean 81:64 (two whole tones) to the ratio 5:4. B moved to stay a whole tone of 9:8 above A. Thus the half tones F:E and c:B were now in the ratio 16:15 rather than the Pythagorean 256:243. The whole tones C:D, F:G, and A:B remained in the ratio 9:8, but the whole tones D:E and G:A were now in the ratio 10:9. (It was roughly in the same time interval that intervals of a third began to be considered consonant.) Sharps and flats did not coincide: C sharp and D flat were for example different notes. However, it wasn't long before there were efforts to make a scale of 12 uniform steps. The first to attempt to do so was Galileo Galilei's father, Vincenso Galilei. He tried to make each step of size 18:17, though that of course led to problems. It was Simon Stevin who first had the idea of making uniform steps of size 21/12.
Later on, some mathematicians even began to question the division of the scale into 12 tones, with the idea that a division into a different number of notes might lead to a more perfect representation of the intervals. For example Christiaan Huygens defined a 31-tone system of temperament in his Lettre touchant le cycle harmonique. One source even suggests that this has "led indirectly to a tradition of 31-tone music in the Netherlands in this century". Leonhard Euler's efforts involved an attempt to reconcile the ideal "octave" 2:1 with the ideal "fifth" 3:2. He analyzed the problem by using a continued fraction representation of the ratio log 2:log 3/2. The convergent 12/7 corresponds to the popular division of 7 octaves into a circle of 12 fifths. Other convergents include 17/12, 29/17, 41/24, and 53/31. In the last case, for example, 31 octaves would be divided into 53 fifths. These didn't answer the question of what kind of equally tempered scale best reconciles the intervals of an octave, fifth, and third (2:1, 3:2, and 5:4) simultaneously. This may or may not influence the course of music, but Scriba shows how an algorithm by the Norwegian mathematician Viggo Brun (1885-1978) gives an answer. If the best answers are written in terms of the number of steps in the three intervals, the best approximations are (2,1,1), (3,2,1), (5,3,2), (7,4,2), (12,7,4), (19,11,6), (31,18,10), (34,20,11), (53,31,17), (87,51,28), .... The triple (12,7,4) is the common case with 12 semitones in an octave, 7 in a "major fifth", and 4 in a "major third". As Scriba explains, the case of the 31 tone scale has been especially important historically. In fact, Scriba tells us that it was back in the middle of the 1600s that Nicolas Vicentino described a "archicembalo" with six manuals with the octave divided into 31 parts; as mentioned above, Huygens clarified this. Moreover, Scriba tells us that Zarlino and Salinas shortly thereafter discussed the division of the octave into 19 equal parts. There is apparently an organ built according to the principles of the Dutch physicist D. Fokker (1887-1972) that also divides the octave into 31 parts (it is now in the Teylers Museum in Haarlem). Along a different line, Euler tried to design a mathematical system to quantify the dissonance of chords, but it apparently did not work very well.
The next part of the article discusses some of the work of Wolfgang Graesers (1906-1928), who tried to do a mathematical study of Bach's Art of the Fugue (this was published under the name Bachs "Kunst der Fuge" (German) in the Bach-Jahrbuch 1924, pages 1-104). Here, group theoretic notions reflect the kinds of transformations, such as inversion, that can be used in a fugue. A background in music theory may be useful in understanding Graesers's work.