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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.
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, 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.
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's Timaeus. The author notes that Dürer's Melancholia, 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's De Divina Proportione relied 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 a stella octangula. The author notes that the notion of the stella octangula also appears in Pacioli's De 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's Vita di Paolo Uccello.) The author, Michele Emmer, collaborated on the film Art and Mathematics. Closely related topics: The Regular Solids, Plato, Art, The Etruscans, Germany in Ancient Times, The Celts, Albrecht Dürer, Luca Pacioli, 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, Fractals in Art, String Figures, Ancient Egypt, The Reckoning of Time, Education, Mathematics in Language, Logic, The Islamic World, and Thomas Fuller (1710-1790).
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 and Algebra.
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 study Africa 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, Mathematics in Language, Frieze Patterns, Plane Patterns, The Islamic World, and Anthropology, General.