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They are not intended as reviews.
However, some items have been reviewed in *Mathematical Reviews*,
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Angell, Ian O. Megalithic mathematics, ancient almanacs or neolithic
nonsense. *Bull. Inst. Math. Appl.* ** 14 ** (1978),
no. 10, 253--258. (Reviewer: C. R. Fletcher.) SC: 01A10, MR: 80f:01002.

Discusses different explanations for the shapes of megalithic stone rings. The author briefly discusses some of the theories of Alexander Thom, which involve an astronomical calendar and an effort to make the circumference equal to 3 times the "diameter" rather than the irrationalpi. He then discusses two new theories of his own. One explains the shapes of the stone rings as extensions of the ellipse, generated with three or four pegs and a string rather than with just the usual two. The other explains the shapes as an effort to store shadow lengths. Neither theory may be given entirely in earnest. A theme of the paper is how theories may start as intellectual games, go out of control, and be changed into pseudo-science. Closely related topics: The Stone Builders, Astronomy, The Calendar, The Ellipse, Pseudoscience, and Alexander Thom.

Aveni, Anthony F.; Morandi, Steven J. and Peterson, Polly A. The Maya
number of time: intervalic time reckoning in the Maya codices. I.
*Archaeoastronomy* No. 20, suppl. J. Hist. Astronom. 26 (1995),
S1--S28. (Reviewer: M. P. Closs.) SC: 01A12, MR: 97a:01004.

Some almanacs in the surviving Mayan codices have surprising irregularities. The authors explain how these almanacs may have been formed from more regular tables by a variety of factors, including astronomical, political, ritual, and numerological. Although parts of the paper may seem a bit dry, there is quite a bit that would merit further investigation arithmetically, astronomically, statistically, or archaeologically. Closely related topics: The Maya, Astronomy, and Numerology.

Barit, Julian. The Lore of Number. *Mathematics Teacher*
**61** (1968), 779--83.

Number symbolism among the Greeks, Hebrews, and in cases also Egyptians, Druids, Hindus. Discusses numbers up through 13. For further reading, suggests a book by W. Wynn Westcott calledNumbers, Their Occult Powers and Mystic Virtuesby the Theosophical Publishing Society, London, 1911. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topic: Numerology.

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'sLiber Abaciand 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 theI 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 theMonthly.) Boethius apparently had a rule for the number of combinations ofnthings taken two at a time. The author discusses interest in combinatorics in the Hindu world, by the Jainas, Varahamihira, and Bhaskara (the latter in theLilavati). 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 thelo shudiagram, which is often linked with theI 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 ofastrogali(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 poemDe 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 theJosephus 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, 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.

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, 7^{1/2}. 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, and Religion.

Diana, Lind Mae. The Peruvian Quipu. *Mathematics Teacher*
**60** (1967), 623--28.

An introduction to the Quipu. The author observes that the quipu was used not only in Peru but also in other areas of South America. These others have not been as well preserved as those found in dry graves in coastal Peru. Discusses Nordenskiöld's theory that the burial quipus contain numerological and astronomical secrets. Briefly discusses the unusual Incan abacus. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Maya, The Quipu, Numerology, Astronomy, and The Abacus.

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.

Emmer, Michele. Art and mathematics: the Platonic solids. *The Visual
Mind*, 215--220, *Leonardo Book Series, MIT Press, Cambridge,
Mass.*, 1993.

The author begins by mentioning some ancient representations of Platonic solids. These include a pair of Egyptian die from the Ptolemaic dynasty, an Etruscan dodecahedron (at least 2500 years old), two Celtic dodecahedra, and a West German dodecahedron from the 2nd century BC. The author continues with a discussion of the regular solids in Plato'sTimaeus. The author notes that Dürer'sMelancholia, which includes a truncated rhombohedron, is sometimes thought to show the influence of Luca Pacioli. The magic square in the painting gives some evidence for this; Dürer's engraving may be one of the earliest depictions of a magic squares in the West, but an earlier manuscript by Pacioli showed an interest in them. On the other hand, Luca Pacioli'sDe Divina Proportionerelied heavily on, and perhaps even appropriated the work of Piero della Francesca. The book is also notable for its pictures of the regular solids, attributed to Leonardo da Vinci. Also discusses work on the regular solids due to Johannes Kepler, including Kepler's recognition of a duality and his idea of a combination of two tetrahedra called astella octangula. The author notes that the notion of thestella octangulaalso appears in Pacioli'sDe Divina Proportione. In addition, Kepler's stellated dodecahedron occurs in mosaics in the San Macro Cathedral in Venice; this work is thought to have been done by Paolo Uccello. Regarding Uccello, the author quotes Donatello as saying to his close friend "Ah Paolo, this perspective of yours makes you neglect what we know for what we don't know. These things are no use except for marquetry." (The source is Vasari'sVita di Paolo Uccello.) The author, Michele Emmer, collaborated on the filmArt and Mathematics. Closely related topics: The Regular Solids, Plato, Art, The Etruscans, Germany in Ancient Times, The Celts, Albrecht Dürer, Luca Pacioli, Magic Squares, Piero della Francesca, Leonardo da Vinci (1452-1519), Paolo Uccello (1397-1475), Johannes Kepler (1571-1630), and Perspective.

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).

Grattan-Guinness, I. Mozart 18, Beethoven 32: hidden shadows of integers
in classical music. * History of mathematics, * 29--47,
*Academic Press, San Diego, CA,* 1996. SC: 01A99 (00A69), MR:
97a:01075.

Discusses number symbolism in the works of Mozart and Beethoven. With Mozart, discusses in particularDie Zauberflöteand the last three symphonies (and particularly the Symphony in g of 1788). There is also some evidence that Mozart used gematria. Literary sources also attest to Mozart's interest in numerology. With Beethoven, focuses primarily on Piano Sonata op. 111 (no. 32), theDiabelli Variations, and theMissa Solemnis. The choice of opus numbers themselves appear to show an interest in numerology. The author suggests that some knowledge of the history and conventions of numerology would be useful before reading this article. The author's own article in theCompanion Encyclopedia of the History and Philosophy of the Mathematical Sciencesmay be useful in this regard. The author also suggests some avenues for future research. Closely related topics: Music, Numerology, Gematria, Wolfgang Amadeus Mozart (1756-1791), and Beethoven.

Grattan-Guinness, I. Some numerological features of Beethoven's output.
*Ann. of Sci.* ** 51 ** (1994), no. 2, 103--135. SC:
01A99 (00A69), MR: 1 278 119.

The author discusses possible occurrences of number symbolism in Beethoven's compositions. A large number of examples are used to buttress his arguments, and some prior familiarity with Beethoven's work might be useful. In some cases, numbers occur as the number of measures or notes of a them or motif, and in other cases in Beethoven's choice of opus numbers. (In contrast with the common practice of the time, Beethoven chose his opus numbers himself, and the numbers chosen could at times be seriously at variance with the order of composition.) The author's conclusions have been controversial, partly because Beethoven has often been regarded as being quite poor at arithmetic. The author discusses this objection and aspects of methodology in some detail. Closely related topics: Numerology, Music, and Beethoven.

Manansala, Paul. Sungka mathematics of the Philippines. *Indian J.
Hist. Sci.* ** 30 ** (1995), no. 1, 13--29. (Reviewer:
J. S. Joel.) SC: 01A29 (01A13), MR: 96g:01009.

The author discusses the Sungka Board, which may once have been used as a kind of abacus. The wordsungkais from the Philippines, but the author tells us that a similar board is "known over a wide area of the Malayo-Polynesian world from Madagascar to Polynesia, and also through Southeast Asia, India, and even mainland Africa." As the author notes, "documentation for this usage is very hard to come by". The arithmetical algorithms that the author advances for the sungka board have few surprises to someone familiar with abacus systems, but the article has some interesting remarks about other uses of the sungka board and about some number systems from India, the Philippines, and elsewhere in Asia that used mixed number bases. The author is particularly interested in eight-based counting systems, and believes that the Sungka board is particularly relevant in this regard: "The board has two large wells at each end, with each large well having a corresponding row of seven smaller wells. These two rows of seven are parallel and thus the board has a total of 16 wells divided into two groups of eight." The wells were apparently once filled with various numbers of things such as cowrie shells. In the examples given, the wells are used for powers of 10. Apparently the sungka board is now used at least as much for divination. As the author explains, "Its main purpose in modern times is to serve as a sedentary game. In the Philippines, and probably elsewhere, the Sungka Board is also still occasionally used for popular divination, especially by elders enquiring on whether travel by youths is auspicious on a certain day, or by girls interested in finding out whether and when they will get married." Closely related topics: The Philippines, The Abacus, Divination, Indo-Malay Archipelago, Polynesia, and Africa.

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.

Seidenberg, A. The ritual origin of counting. *Arch. Hist. Exact
Sci.* **2** (1962b), 1-40.

It is common to argue that counting and other elementary mathematics arose spontaneously throughout the world in response to a practical, or perhaps psychological, need. Abraham Seidenberg argues instead for adiffusion theory, that counting arose only once, and then spread throughout the world. In fact, many common associations with numbers suggest such a common origin. One such association that Seidenberg is the idea that odd numbers are male and even numbers are female; this is certainly well known from the Pythagoreans, but turns out to be nearly universal. Seidenberg proposes that counting in fact originally arose in a ritual context. Seidenberg draws from a wide variety of anthropological sources for rituals and myths that hint at what this common origin might have been. He finds that counting "was frequently the central feature of a rite, and that participants in ritual were numbered." He focuses more specifically on creation rituals. He suggests that in the enaction of creation myths, men and women may have come onto the scene alternately, easily explaining the odd/male even/female association. He finds that his ideas clarify "pure 2-counting, which is the oldest stratum of counting we can detect." In pure-2 counting, there are separate words for one and two and these are used to form all other number words. He illustrates this with number words from diverse languages such as the Gumulgal of Australia, the Bakairi of South America, and the Bushmen of South Africa. He sheds additional light on his hypothesis with discussions of the possible origin of counting taboos (and connections with ritual sacrifice), of ancient one-one-correspondence "tally" systems (e.g., counting people with stones), of taxation systems, of money, and of gematria. Seidenberg also gives us some fascinating examples of counting in world religions. These include the analogyThe Lord:His people=the shepherd:his sheep, the analogyThe shepherd:his sheep=the moon:the stars. These two lead one to expect the moon to count the stars; and Seidenberg in fact finds evidence of this in ancient Babylonia. He argues from the equationThe Lord's people=the stars of the heaventoThe Lord's people=the sand upon the seashorethat one would expect to find a ritual counting of sand. In fact, he finds the notion ofCounter of the Sandsboth in Buddhism and among the Ancient Greeks. The equationThe Lord=The Counterseems to be confirmed in two of the ninety-nine beautiful names of Allah, namelyThe Counterand theReckoner; and there is further confirmation in Chapter's XV and XIX of the Qu'ran. This is a fascinating article, connecting mathematics with a wide variety of disciplines. Closely related topics: Myth and Ritual, Storytelling Traditions, Anthropology, General, Counting, TallySystems, Taxation, Number Words, The Pythagoreans, Gematria, Religion, The Islamic World, and Abraham Seidenberg.

Swetz, Frank. The Evolution of Mathematics in Ancient China.
*Mathematics Teacher* **52** (1979), 10--19.

An overview of Chinese mathematics, including the discovery of thelo shumagic square (thought to have a plan of universal harmony), square roots, the Chinese remainder theorem, and polynomials of high degree (including a quintic inx^{2}). Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: China, Algebra, and Magic Squares.

Wren, R. L. and Rossmann, Ruby. Mathematics Used by American Indians
North of Mexico. *School Science and Mathematics*
**33** (1933), 363--72.

Surveys the use of numbers and geometric shapes in various North American indigenous peoples. Includes sacred numbers, number words, including an unusual instance of subtractive number words in the Bellacoola of British Columbia, number systems, reckoning of time and seasons. Also includes geometric characteristics of dwellings and (briefly) textiles, basketry, pottery, and tattooing. Often pottery designs were borrowed from textile art. A common principle in weaving is that no line, curved or otherwise could intersect itself. (Is this principle partly responsible for the popularity of spirals?) Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Indigenous Mathematics of North America, Numerology, Number Words, The Bellacoola, The Reckoning of Time, Pattern, Weaving, Basket Making, Pottery, and Tattoos.

Zaslavsky, Claudia. Africa counts. Number and pattern in African culture.
*Prindle, Weber & Schmidt, Inc., Boston, Mass.,* 1973. x+328
pp. SC: 01A10, MR: 58 #20993.

This book is an excellent introduction to the mathematics of (primarily sub-Saharan) Africa. The best tribute to its importance may be in Gerdes, Paulus, On mathematics in the history of sub-Saharan Africa. Gerdes writes "In her classical studyAfrica Counts: Number and Pattern in African Culture..., Claudia Zaslavsky presented an overview of the available literature on mathematics in the history of sub-Saharan Africa. She discussed written, spoken, and gesture counting, number symbolism, concepts of time, numbers and money, weights and measures, record-keeping (sticks and strings), mathematical games, magic squares, graphs, and geometric forms, while Donald Crowe contributed a chapter on geometric symmetries in African art." Regarding geometric symmetries, it is primarily the frieze patterns and plane patterns that are discussed; there is surely more work to be done on the bichromatic frieze and plane patterns. Many readers will wish to explore further. Gerdes' paper should be invaluable for this, not least for its extensive bibliography. Another useful resource is the newsletter distributed by the African Mathematical Union's Commission on the History of Mathematics in Africa (AMUCHMA). Closely related topics: Sub-Saharan Africa, TallySystems, Finger Numerals, Counting, Numerology, The Reckoning of Time, Money, Measurement, Games, Continuous Tracing Problems, Architecture, Magic Squares, Mathematics in Language, Frieze Patterns, Plane Patterns, The Islamic World, and Anthropology, General.