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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.
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.
Artmann, Benno. The cloisters of Hauterive. Math. Intelligencer 13 (1991), no. 2, 44--49. SC: 00A69 (01A99), MR: 1 098 219.
The author discusses geometric principles behind Gothic tracery. The Gothic style developed in France about 1150, but spread widely in the next few centuries. Examples are taken from Reims, Haina, Strasbourg, and Esslingen. The geometric principles are by no means trivial; some make rather challenging exercises. The author discusses the windows of the cloisters of Hauterive in some detail. Hauterive is a Cistercian monastery near Fribourg in Switzerland, and the cloister dates from 1320-1328. The windows there are unusually geometric, and the author advances the theory that the windows amount to a kind of commentary on Book IV of Euclid's Elements. One window, however, can not be constructed with straightedge and compass: it involves the construction of a regular 9-gon. The author notes that a regular 15-gon may have originally been envisioned, but that "esthetic considerations overwhelmed mathematics." Interesting article. A number of illustrations, a few of which appear in Artmann, Benno; Swetz, Frank J., The Geometry of Gothic Church Windows. Closely related topics: Medieval Europe, France in the Middle Ages, Fractals in Art, Similarity, Rotational Symmetry Groups (Rosettes), Polygons, The Circle, Euclid, and Religion.
Artmann, Benno; Swetz, Frank J. The Geometry of Gothic Church Windows. In Swetz, Frank J. From Five Fingers to Infinity. A Journey through the History of Mathematics. Open Court, Chicago, 1994. 228.
Illustrations adapted from Artmann, Benno, The cloisters of Hauterive. The tracery in European Gothic churches uses arcs of a circle, fitted together in ingenious ways. Some of the ingenious ways have mathematical principles underlying them. Although this brief excerpt does not mention it, it is not uncommon for the construction to be repeated in the same tracery in a different scale---a kind of reaching to infinity that is reminiscent of fractals. Closely related topics: Medieval Europe, France in the Middle Ages, Fractals in Art, Similarity, Rotational Symmetry Groups (Rosettes), Polygons, The Circle, and Religion.
Bérczi, Sz. Symmetry and technology in ornamental art of old Hungarians and Avar-Onogurians from the archaeological finds of the Carpathian Basin, seventh to tenth century A.D. Symmetry 2: unifying human understanding, Part 2. Comput. Math. Appl. 17 (1989), no. 4-6, 715--730. (Reviewer: Marjorie Senechal.) SC: 01A99 (01A10 92K99), MR: 91a:01058b.
Analysis of symmetries can be very helpful in better understanding archaeological art and artifacts. The types of symmetries not only show what the author describes as "intuitive mathematical development in ornamental art" but can also help trace relationships between different communities. Such studies are now relatively new, but with time should become "an accepted, standard part of the description of archaeological finds". In this article, the author discusses how all 7 types of strip/frieze patterns occur in Old Hungarian ornamental art, and develops a notion of a double frieze pattern, which is intermediary between frieze patterns and plane patterns. A number of these patterns occur (sometimes individualized) in Avar-Onogurian artifacts. The author's classification of double frieze patterns focuses on how the patterns are generated horizontally and vertically, and may be more useful for archaeological purposes than classification by the related plane patterns. The author gives examples of some plane patterns that came up somewhat naturally, including patterns from weaving, chained ring structures, and the optimal fitting of furs (a pmg plane pattern). The author compares the frequencies of certain symmetry patterns in collections from several cultures. Closely related topics: Hungary in the Middle Ages, Frieze Patterns, Plane Patterns, Double Frieze Patterns, Archaeology, and Metal Work.
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.
Byrne, Catriona. The left-handed Pythagoras. Math. Intelligencer 12 (1990), no. 3, 52--53. SC: 01A99, MR: 1 059 227.
The author notes an relief at Notre Dame de Chartres (dating from the 1100s) where Pythagoras is depicted as being left-handed. The author suggests that left-handedness is distinctly higher among mathematicians than in a random population. It would be interesting to know if any such association were perceived in the middle ages. Closely related topics: Medieval Europe and Biology.
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.
Dahlke, Richard; Fakler, Robert A. and Morash, Ronald P. A sketch of the history of probability theory. Math. Ed. 5 (1989), no. 4, 218--232. (Reviewer: William J. Adams.) SC: 01A99 (60-03), MR: 91i:01148.
Focuses on the history of probability theory, but also touches on the development of statistics. Considers one ancient root of probability theory to be the gambling with astrogali. Mentions the related ancient Egyptian game "Hounds and Hackals", of c. 3500 BC. Discusses the table of frequencies of tosses of 3 die in De Vetula, and Cardano's and Galileo's explanations of the probabilities of such events. Galileo's telescope led him to consider some of the theory of errors, and he arrived, in effect, at some of the features of the normal probability distribution. (It is interesting that later on, Gauss refined some of his own work in statistics to rediscover the planetoid Ceres.) Discusses the "division of stakes" problem and its solution by Pascal and Fermat. The first book actually published on games of chance was written by Huygens. In addition, as the author explains, "Huygens was the first to use probability in studying vital statistics of humans. He used John Graunt's (London) now famous book displaying vital statistics to construct a mortality curve and to define the notions of mean and probable duration of life. Shortly thereafter, probability theory was being applied to annuities." The article continues through the beginning of the 1900s. Much of this later material is of course beyond the scope of these pages. Closely related topics: Probability, Statistics, Gambling, De Vetula, Girolamo Cardano, Galileo Galilei, Astronomy, Blaise Pascal, Pierre de Fermat, Christiaan Huygens, and Insurance.
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.
Eves, Howard. On the Practicality of the Rule of False Position. Mathematics Teacher 51 (1958), 606--8.
Eves shows how the method of false position can be simpler than our own methods by giving one example from the Ahmes Papyrus, three from the Greek Anthology of c. 500 AD, and two of his own. One of his examples is from surveying, and Eves says that it is the method a surveyor would probably use. In the other example of his own, he likens the rule of false position to the method of similitude in geometric constructions. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Method of False Position, Ancient Egypt, Medieval Europe, Surveying, and Geometry.
Gardner, Arthur O. The History of Mathematics as a Part of the History of Mankind. Mathematics Teacher 61 (1968), 524--26.
Briefly discusses how factors such as religion and warfare have influenced the development of mathematics. Attributes the success of Leonardo of Pisa (Fibonacci) to the unconventional ideas of his sovereign, Emperor Frederick II of the house of Hanover. Martin Luther is an example of an important theologian who supported mathematics: "If I had children, they should not only study language and history, but they should also learn singing and music, together with the whole of mathematics." Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Religion, Warfare, and Leonardo of Pisa (Fibonacci).
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.
Høyrup makes a distinction between scientific and subscientific mathematics. These fields correspond somewhat to pure and applied mathematics. However, by using this new terminology, the author hopes to avoid suggesting that "subscientific" mathematics is always derived from "scientific" mathematics in the way that "applied" mathematics is derived from "pure" mathematics. Høyrup discusses the distinction between scientific and subscientific mathematics and also their various kinds of relationships. His examples are drawn from Greece, Egypt, India, the Islamic World (with references to the Silk route), and from the Carolingian Propositiones ad acuendos jevenes. (The latter is traditionally associated with Alcuin.) Høyrup touches on relevant work by the mathematicians Hero, Diophantus, and al Khwarizmi. Surveying is discussed as a particularly important type of subscientific mathematics. Closely related topics: Applied Mathematics (General), Greece, Ancient Egypt, India, The Islamic World, Alcuin, Heron, Diophantus, Surveying, and Abu Abdullah Muhammed ibn Musa al Khwarizmi.
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.
King, Charles. Leonardo Fibonacci. Fibonacci Quarterly 1 (1963), 15--19.
A brief survey of the work of Fibonacci, Leonardo of Pisa. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topic: Leonardo of Pisa (Fibonacci).
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, and Medieval Europe.
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.
Loeb, A. L. The magic of the pentangle: dynamic symmetry from Merlin to Penrose. Symmetry 2: unifying human understanding, Part 1. Comput. Math. Appl. 17 (1989), no. 1-3, 33--48. (Reviewer: Marjorie Senechal.) SC: 01A99 (01A10 52-03), MR: 91a:01058a.
In this interesting and entertaining article, Merlin the magician assists Arthur and Key in exploring the secrets of dynamic symmetry (in a problem with four beetles in a square always walking towards each other), in the logarithmic spiral (the curve generated by the beetles), the golden rectangle (and its own associated spiral), and the Fibonacci numbers. The article closes with a discussion of the pentangle, which the author says "is central to the late fourteenth-century 'Sir Gawain and the Green Knight', to medieval sign theory as well as to recent research in quasi-periodic alloy crystals. The Socratic discussions here could be turned used as active learning exercises for talented students. Highly recommended. Closely related topics: England in the Middle Ages, Dynamic Symmetry, Spirals, Proportion and the Golden Ratio, Leonardo of Pisa (Fibonacci), The Pentagram, and Education.
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).
McClendon, R. B. Leonardo of Pisa and His Liber quadratorum. American Mathematical Monthly 26 (1919), 1--8.
The author discusses some of the most important work in Fibonacci's Liber quadratorum, and convincingly makes the case that Leonardo was the greatest genius in number theory between Diophantus and Fermat. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Leonardo of Pisa (Fibonacci) and Number Theory.
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.
Miller, G. A. Gerbert's Letter to Adelbold. School Science and Mathematics 21 (1921), 649--53.
Gerbert puts circles and squares inside an equilateral triangle, and attempts to explain why they give different answers for the area. We think of these answers as estimates, but Gerbert's letter contains no hint of a limiting process. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Abacus, Gerbert, Pope Sylvester II, The Measurement of Area and Volume, and Limit.
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.
Petruso, Karl M. Additive progression in prehistoric mathematics: a conjecture. Historia Math. 12 (1985), no. 2, 101--106. (Reviewer: Garry J. Tee.) SC: 01A10 (01A15), MR: 86m:01005.
A collection of stone balance weights was recovered from a Late Bronze Age ship (c. 1200 BC) that sank off the coast of southern Turkey (near Cape Gelidonya, modern Finike). Some of these weights are sphendonoid in shape ("approximately the shape of an olive pit"), and appear to be multiples 1, 3, 5, 7, 12, 31, 50, and 54 of a hypothetical unit weight of 9.3 grams (the error is within about 2 percent). There are five weights of 7, and one weight of each of the others. Initially, these balance weights defied analysis, but the author (Petruso) realized that they nearly form a Fibonacci series; he posits the existence of missing weight of 2 and 19. Two problems with this interpretation are the fact that a weight of 7 occurs instead of a weight of 8, and the fact that the weight of 54 does not fit into his system. He suggests that the weight of 8 is a "purposeful and quite utilitarian shift in the basic Fibonacci series .... [to] allow the generation of a 50-unit (rather than 55-unit) mass further along the series." He also notes that the units of 19+31+50 would conveniently add up to 100. As for the 54 unit weight, "it might well have had a specific, idiosyncratic (industrial) purpose which is now lost to us." The author notes that one particular advantage of the Fibonacci-like system is that the accuracy of the individual weights could be quickly checked: for example, one can weigh the 12 against the 5 and the 7. Altogether a fascinating theory, readily readable. Closely related topics: The Balance and the Measurement of Weight, Leonardo of Pisa (Fibonacci), Archaeology, and The Late Bronze Age.
Pressman, Ian and Singmaster, David. The jealous husbands and the missionaries and cannibals. Math. Gaz. 73 (1989), no. 464, 73--81. (Reviewer: E. Keith Lloyd.) SC: 01A99 (05A99), MR: 92b:01086.
There are three river crossing problems in the Propositiones ad Acuendos, which is generally attributed to Alcuin: the problem of three jealous husbands (each of whom won't let another man be alone with his wife), the problem of the wolf, goat, and cabbage, and the problem of "the two adults and two children where the children weigh half as much as the adults." The authors discusses modifications of these problems and attempted solutions by Luca Pacioli, Tartaglia, and others. Modifications include the addition of more people, an island in the center, and a bigger boat. A more recent version is the problem of the Missionaries and the Cannibals, where the cannibals must never outnumber the missionaries. The authors give some solutions and theorems on minimality, although they leave their discovery of a 16 move solution to the four-couples-with-an-island problem as "a nice exercise for the reader". The authors don't discuss this, but problems similar to the wolf-goat-cabbage problem have appeared in a variety of cultures. Closely related topics: Alcuin, Discrete Mathematics, Luca Pacioli, Niccolò Fontana (Tartaglia) (1499?-1557), and Mathematics in Recreation.
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.
Schaaf, William L. Mathematics and World History. Mathematics Teacher 23 (1930), 496--503.
Concerned with the idea the different cultures have different ways of thinking about mathematical concepts. Schaaf takes the number concept as an example. The idea of number and magnitude was concrete and geometric to the Greeks, and was closely tied with the idea of measurement. This notion was changed by Diophantus, who may have been influenced by the mathematics of India and the Middle East. Similar ideas in the Islamic world may have reached Europe in the middle ages. A new concept of number was introduced with Descartes in Analytic Geometry. Since then, mathematics has become still more abstract. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Number Concept, Geometry, Greece, Measurement, Diophantus, India, The Middle East, The Islamic World, Analytic Geometry, and Arithmetic.
Schaaf, William L. Mathematics as a Cultural Heritage. Arithmetic Teacher 8 (1961), 5--9.
Briefly discusses some of the key characteristics of the mathematics of the Babylonians, Egyptians, Greeks, and of Medieval Europe. Then discusses adoption of the Hindu-Arabic numerals, the development of computation, and more abstract mathematics. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Ancient Egypt, Greece, Medieval Europe, and The Hindu-Arabic Numerals.
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.
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 a diffusion 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 analogy The Lord : His people = the shepherd : his sheep, the analogy The 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 equation The Lord's people = the stars of the heaven to The Lord's people = the sand upon the seashore that one would expect to find a ritual counting of sand. In fact, he finds the notion of Counter of the Sands both in Buddhism and among the Ancient Greeks. The equation The Lord = The Counter seems to be confirmed in two of the ninety-nine beautiful names of Allah, namely The Counter and the Reckoner; 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.
Seidenberg, A. and Casey, J. The ritual origin of the balance. Arch. Hist. Exact Sci. 23 (1980/81), no. 3, 179--226. (Reviewer: M. P. Closs.) SC: 01A10, MR: 82j:01008.
The author's trace the beginnings of the balance back to a rituals where principals contended against each other on a kind of see-saw (somewhat similar sports are of course known from medieval times). The grain-crusher and water-lifter are similar, and perhaps derived from, the see-saw; the fact that one stands on these suggested to the authors that the contestants may have been standing on the see-saw. The authors note that in ancient Egypt, one's heart was believed to be weighed against a feather in order to decide whether one would be able to enter the afterlife. Other parts of the body, such as hair, can be used to represent an individual, and in other instances these may have been weighed instead; the authors give examples of rites where hair is weighed. An interesting use of the balance in Greece is from the Iliad where Zeus weighs Achilles and Hector on pans of a balance. "That of Hector sinks toward Hades and Hector falls, slain by Achilles." An even more interesting weighing ritual was once common in the far east, where a ruler was balanced against a quantity of a precious substance such as gold, and gave that substance (and thereby symbolically himself) to his people. The authors found many other interesting examples in a wide variety of cultures and world religions. The authors believe that only items of ritual significance were weighed at first, and that widespread commercial use came much later. Although the authors don't focus greatly on this, they also briefly discuss the different kinds of balances (and the balance-like instrument used to carry loads on the shoulders) and the weight multiples that were used on balances. Closely related topics: The Balance and the Measurement of Weight, Myth and Ritual, Religion, Sports, Ancient Egypt, Greece, The Islamic World, and Abraham Seidenberg.
Shloming, Robert. Th\^abit ibn Qurra and the Pythagorean Theorem. Mathematics Teacher 63 (1970), 519--28.
Discusses the life and work of Th\^abit ibn Qurra, focusing on his work on the Pythagorean Theorem. Th\^abit gave two proofs of this theorem (both independently rediscovered in the early 1900s), and also a generalization to triangles that are not necessarily right-angled (independently rediscovered about 1665 by John Wallis). The author also discusses the Ishaq-Th\^abit translation of Euclid's Elements, which was the basis for the translation by Gerard of Cremona. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Th\^abit ibn Qurra and Pythagorean Triangles and Triples.
Sleight, E. R. The Art of Nombryng. Mathematics Teacher 35 (1942), 112--16.
The Art of Nombryng is from England in the 1400s, and is a translation of de Arte Numerandi, which was in turn written in the 1200s and is attributed to Sacrobosco. It explains how to do the basic operations of arithmetic, including mediation and duplication, and going as far as the extraction of square and cube roots. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Arithmetic, Sacrobosco (John of Holywood), and England in the 1400s.
Sleight, E. R. The Craft of Nombrynge. Mathematics Teacher 32 (1939), 243--48.
As we are told, The Craft of Nombrynge is based on the Canto de Algorismo by Alexander de Villa Dei (1220). It explains how to add, subtract, double, and divide by two, but does not discuss general division or the extraction of roots. (The method of multiplication is essentially the galley method.) Topics are introduced from the Latin Canto, and the remaining text is given in English. Arithmetic (algorism) is attributed to a supposed King Algor of India. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Arithmetic, The Extraction of Roots, Alexander de Villa Dei, and England in the 1400s.
Smith, Thomas M. Some Uses of Graphing before Descartes. Mathematics Teacher 54 (1961), 565--67.
Briefly discusses how graphing was used before the 1600s. The De Configurationibus qualitatum of Nicole Oresme is particularly important in this regard. Oresme even points out that if the two axes represent time and velocity, then the enclosed area represents distance. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Graphing, Nicole Oresme, Dynamics, Force, and Motion, and Calculus.
Struik, D. J. Omar Khayyam, Mathematician. Mathematics Teacher 51 (1958), 280--84.
An excellent introduction to the work of Omar Khayyam. He discusses Omar's solution of cubic equations (by intersections of cubics), his possible understanding of the binomial theorem (occurring in the the work of al Kashi, and later in the work of Michael Stifel), his work on the parallel postulate (including his reduction to the cases of an acute angle, an obtuse angle, and a right angle; his proof of the parallel postulate rests on other axioms, including the axiom that a straight line can be indefinitely prolonged and on the Axiom of Archimedes), and his implicit recognition that a ratio can be expressed by ratios of integers to any desired degree of accuracy. He closes with his view of the real flesh and blood Omar Khayyam. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topic: Omar Khayyam (abu-l-Fath Omar ibn Ibrahim Khayyam).
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, Magic Squares, Mathematics in Language, Frieze Patterns, Plane Patterns, The Islamic World, and Anthropology, General.