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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.
Altshiller-Court, Nathan. The Dawn of Demonstrative Geometry. Mathematics Teacher 57 (1964), 163--66.
The author argues that it seems unlikely that the Greeks could have invented their notion of proof so rapidly and in isolation. Instead, he suggests that the notion of geometric proof was a secret that was jealously guarded from all but the "inner sanctum" of the Egyptian priesthood. (Of course, since his argument implies by its very nature that Egyptian proofs were unlikely to have been written down, this will be a hard argument to either prove or disprove.) Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Geometry, Proof, and Ancient Egypt.
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.
Bernal, Martin. Response to a paper by R. Palter: "Black Athena, Afro-centrism, and the history of science" [Hist. Sci. 31 (1993), no. 93, part 3, 227--287; MR: 94i:01001]. With comments by Palter. Hist. Sci. 32 (1994), no. 98, part 4, 445--468. (Reviewer: Donald Cook.) SC: 01A16 (01A07 01A20 01A70 01A80), MR: 96c:01005.
An important question in the history of Greek mathematics is how much Greek mathematics was influenced by the mathematics of the Egyptians. Bernal suggests in Black Athena that the influence may be much greater than previously thought. Palter's review article Black Athena, Afro-centrism, and the history of science disagreed with a number of Bernal's points. Bernal responds here to Palter's review, Palter, Robert, Black Athena, Afro-centrism, and the history of science, and then Palter comments on Bernal's response. The response and comment provide an excellent introduction to some of the issues involved in the question of Egyptian influence and also to some of the issues of modern scholarship. It might be useful to have a class read and comment about this article. It is interesting that questions in the history of medicine play a more important role in this controversy than one might at first expect. If the Greeks borrowed heavily from the Egyptians in medicine, it seems more reasonable that they borrowed in the other sciences as well. Closely related topics: Ancient Egypt and Medicine.
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.
Brendan, Brother T. How Ptolemy Constructed Trigonometry Tables. Mathematics Teacher 58 (1965), 141--49.
Discusses how Ptolemy may have constructed his trigonometry tables, which in effect give a table of sines for every quarter degree between 0o and 90o correct to four decimal places. Ptolemy's first theorem shows how he could have constructed the chords of 36o and 72o. Ptolemy's second theorem can be used to find sum and difference angle formulas, and a half angle formula. Since the chord of 60o is simple, he can thus find chords of 12o, 6o, 3o, 3/2o, and 3/4o. The sticky part is then to find the chord of 1o [one sees this also in the Islamic world, where in one instance an approximate solution was found to a cubic]. Ptolemy uses a clever argument and the values for 3/2o and 3/4o to find an accurate answer for the chord of 1o. The table also includes a method to interpolate values of chords at every minute of arc (in effect, sines of every half minute). The author does not discuss the method of interpolation in detail. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Ptolemy (Claudius Ptolemaeus), Trigonometry, and Interpolation.
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.
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, 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'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, Magic Squares, Piero della Francesca, Leonardo da Vinci (1452-1519), Paolo Uccello (1397-1475), Johannes Kepler (1571-1630), and Perspective.
Fields, Margaret. Practical Mathematics of Roman Times. Mathematics Teacher 26 (1933), 77--84.
Surveys Roman mathematics. Some of the most interesting examples come from the De Architectura of Vitruvius, which discusses principles of symmetry and proportion and how to use them in architecture. Vitruvius goes as far as how to correct for an optical illusion on the capitals of columns. He also discusses geometric procedures to be used in laying out a town (to shut out winds), and various Roman instruments, including leveling instruments and an instrument for measuring distance called a hodometer. The hodometer is used for "telling the number of miles while sitting on a carriage or sailing by sea", and is particularly ingenious. Second to Vitruvius, the most important source on Roman engineering may be the Urbis Romae of Frotinus, which includes mathematical rules (not entirely successful) to determine the flow of an aqueduct. Surviving Roman bridges show a high level of skill; there were surely mathematical principles behind their design, but no detailed study has survived. Roman tunnels are equally impressive. Heron discusses how to use an instrument called the "dioptra" to survey for tunnels, measure the width of a river, and so on. Roman sundials were relatively unsophisticated. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Vitruvius, Architecture, Symmetry, Proportion and the Golden Ratio, Optics, Leveling, The Measurement of Distance, Frotinus, Heron, Surveying, and The Sundial.
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, China, and Sumerians and Babylonians.
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), Ancient Egypt, India, The Islamic World, Alcuin, Heron, Diophantus, Surveying, and Abu Abdullah Muhammed ibn Musa al Khwarizmi.
Hansen, David W. The Dependence of Mathematics on Reality. Mathematics Teacher 64 (1971), 715--19.
Discusses how the greatest mathematicians have been vitally concerned with the real world. Uses Archimedes, Newton, and Gauss as examples. Archimedes did so much applied work that it is hard to see how he fits Plutarch's description of considering mechanical work ignoble and inferior. The case of Newton is of course well known. An interesting example is Gauss, who used the motto "Thou, nature art my goddess;to thy laws/My services are bound" from Shakespeare's King Lear. Newton and Gauss were also very interested in religion. Philosophy was very important to Gauss. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Applied Mathematics (General), Archimedes, Isaac Newton (1642-1727), Karl Friedrich Gauss (1777-1855), Religion, and Philosophy. Also possibly relevant: Literature.
Jones, Phillip S. Irrationals or Incommensurables. I. Their discovery, and a "Logical Scandal". Mathematics Teacher 49 (1956), 123--27.
The discovery of irrationals. Discusses an appealing theory, due to Kurt von Fritz, that the discovery of irrationals grew out of a study of the pentagram. Von Fritz is in support of the traditional theory that discovery or irrationals was due to Hippasus of Metapontum. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Irrationals, The Pentagram, and Hippasus of Metapontum.
Jones, Phillip S. Irrationals or Incommensurables. III. The Greek solution. Mathematics Teacher 49 (1956), 282--85.
Shows how Eudoxus' Method of Exhaustion is used to prove that circles are to one another as the squares on their diameters. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Method of Exhaustion, Eudoxus, The Measurement of Area and Volume, and The Circle.
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.
Kilmister, C. W. Zeno, Aristotle, Weyl and Shuard: two-and-a-half millenia of worries over number. Math. Gaz. 64 (1980), no. 429, 149--158. (Reviewer: K. E. Hirst.) SC: 01A99 (00A05 03A05), MR: 82i:01075.
Ever since Zeno's paradoxes, mathematicians, philosophers, and logicians have been discussing the nature of the infinite. The author starts by discussing one of Zeno's four paradoxes, the Dichotomy. This leads to a discussion of Aristotle's views of the infinite. Needless to say, philosophical problems remained, and Hermann Weyl made one attempt to rectify them. Weyl advised caution in dealing with impredicative definitions, which he believed could lead to a vicious circle. Unfortunately, as Weyl notes "This vicious circle, which has crept into analysis through the foggy nature of the usual set and function concepts, is not a minor, easily avoided form of error in analysis." And in fact, if impredicative definitions are abandoned entirely, we must also abandon the notion that a bounded infinite set has a least upper bound and of course the related theorem (Bolzano-Wierestrass) that a bounded infinite set has a limit point. As the author notes, "On 9 February 1918, Polya and Weyl made a bet in Zürich, with twelve witnesses (all mathematicians). About [the least upper bound property], Weyl prophesied 'A. Within twenty years, Polya, or a majority of leading mathematicians, will admit that the concepts of number, set and countability involved are completely vague; and that there is no more point in asking about the truth of [the least upper bound property] than of the main assertions of Hegel's physics. B. It will be recognized by Polya, or a majority of leading mathematicians, that in any wording [the least upper bound property] is false...'" When the bet was called, everyone agreed that Polya had won with the single exception of Kurt Gödel. The author notes "if the construction of the real numbers contains subtleties that troubled such an acute intellect as Weyl's as recently as 1917, and still worried Gödel in 1940, it is not to be wondered at that some of our first-year undergraduates find it hard to stomach. Perhaps they are wiser than we are." Closely related topics: Zeno, Aristotle, Hermann Weyl, Infinity, Paradox, and Philosophy.
Knight, Gordon. The geometry of Maori art---spirals. New Zealand Math. Mag. 22 (1985), no. 1, 4--7. (Reviewer: H. S. M. Coxeter.) SC: 51N20 (01A10), MR: 87m:51060.
The Maoris frequently use spirals in their tattoos and wood carvings. These appear very much like the spirals of Archimedes, but often interlace two or more such spirals. Although the easiest way to construct a spiral similar to the spiral of Archimedes may be to use sets of concentric semicircles (or other segments of circles) offset with respect to one another, the author believes that the Maoris didn't use this technique. "In Spirals of Archimedes, and, it seems, in Maori spirals, there is a gradual, rather than an abrupt, change in curvature." The author gives several examples from Maori artwork; there are examples with 2, 3, and 4 interlaced spirals. The author notes that the 3 spiral form is more common in tattooing patterns than in carving. Apparently there was once a 6 spiral pattern on one of the figures guarding the gateway of Papawai Pa. The center of the spiral can be varied somewhat; for example, two spirals can come together in an S-curve. In one case, "the plain ridges, which form an S-curve, are made to cross over the notched spirals, giving a woven effect. According to Phillips this was chiefly an Arawa modification." The author concludes with a note that the spiral of Archimedes should perhaps have a Maori name instead. He suggests that an investigation of these spirals might be useful in mathematics education (when polar coordinates are studied). Closely related topics: Spirals, The Maori, Tattoos, Wood Carving, Archimedes, and Education.
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, China, India, The Stone Builders, Alexander Thom, and Pythagorean Triangles and Triples.
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, The Islamic World, India, China, Japan, Vietnam, Java, Bali, The Maya, Guatemala (and Cakchiquel Indians), France in the 1700s, The Jewish Tradition, and Religion.
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).
Mainzer, Klaus. Symmetry and beauty in arts and mathematical sciences. Physis Riv. Internaz. Storia Sci. (N.S.) 32 (1995), no. 1, 91--103. SC: 01A99 (00A69), MR: 96h:01043.
As this article explains, symmetry appears in a variety of disciplines over a variety of ages. The author begins by briefly discussing the natural and philosophical reasons for studying symmetry (starting in ancient Greek times). He then discusses the appearance of the 7 frieze groups and 17 ornamental groups of the plane and related groups in mathematics and crystallography. Next, he discusses appearances of symmetry and symmetry breaking in modern physics, in the theory of relativity, and in quantum mechanics and superstring theory. He finds that symmetry considerations are important in chemistry and biology as well: "In biochemistry macromolecules (for example L-amino acids or D-sugars) possess a characteristic homochirality ('dissymetry') which is assumed to be caused by parity violations of weak atomic forces." He also explains that "The emergence of pattern structure can be described by symmetry breaking not only in chemistry, but in biology. Since the pioneering work of the famous English logician and mathematician A. Turing on the chemical basis of morphogenesis in biology (1952), there has been an increasing interest in this topic." He then proceeds to discuss "Symmetry and Symmetry Breaking in the Computer World", focusing on dynamical systems. For example, he write, "Nevertheless the Feigenbaum diagram is self-similar. Every part of the tree contains the Feigenbaum diagram infinitely often like Russian dolls. It follows that mathematical chaos can be highly symmetric." He closes with a discussion of modern architecture, where he finds that symmetry concerns are important as well: "But the variety of historical reminiscences and asymmetrical elements in architecture does not mean a movement back to historicism or eclecticism. It is the expression of a sceptic and ironic view of the world which no longer believes in an omnipotent technical rationality and its claim to solve all human problems. It underlines individuality and the importance of accidental details, and has doubts about universal harmony and rationality. So it prefers symmetry breaking as a chance of variety, pluralism, and individual freedom." And this is a theme that nicely rounds of his article: "But variety and pluralism need not be in conflict with unity. It was Leibniz who suggested that the unity of the world can only be experienced by man under special aspects. So his motto was 'unity in variety.' It dates back to the old philosophical idea of Heraclitus that even symmetry breaking is related to a sometimes hidden symmetry." Interesting and thought-provoking article. Closely related topics: Symmetry, Philosophy, Physics, Chemistry, Biology, Alan Turing, Computation, Fractals, and Architecture.
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, The Islamic World, The Parallel Postulate, and Perspective.
Nagy, Dénes. The 2,500-year old term symmetry in science and art and its "missing link" between the antiquity and the modern age. Symmetry: natural and artificial, 1 (Washington, DC, 1995). Symmetry Cult. Sci. 6 (1995), no. 1, 18--28. SC: 01A99, MR: 1 371 622.
Documents the evolution of the word symmetry from its beginnings in ancient Greece. As the author explains, the word originally had a somewhat different meaning: symmetry = syn together + metron measure, suggesting the notion of commensurability. The word was adopted into Latin but was apparently rare in the middle ages. It's reappearance can probably be credited to the importance to the Renaissance of the De architectura libri decem of Vitruvius (1st century BC). The author discusses the Hebrew, Indian, and Chinese words for symmetry as well. At the end of the article the author enumerates some modern generalizations and uses of symmetry. For example, the author mentions "Noether's theorems connecting symmetry transformations (invariances) and conservation laws", Gell-Mann and Ne'eman's classification of elementary particles, and "Graeser's reconstruction of Bach's Kunst der Fuge". Closely related topics: Symmetry, Language and Literature, Vitruvius, Physics, and Music.
Palter, Robert. Black Athena, Afro-centrism, and the history of science. Hist. Sci. 31 (1993), no. 93, part 3, 227--287. (Reviewer: Donald Cook.) SC: 01A16 (01A07 01A20 01A70), MR: 94i:01001.
Martin Bernal's Black Athena created a bit of a sensation when it first came out. Robert Palter discusses aspects of Bernal's article and also other arguments of afro-centrists. Palter particularly focuses on the question of whether Egyptian mathematics and science influenced the Greeks. Bernal suggests that the influence may be quite large, and Palter argues that all existing evidence points to the influence being quite small. An important area in Palter's discussions is ancient astronomy, where Palter discusses the general character of Egyptian astronomy, and argues that some claims about it have been vastly exaggerated; much of this discussion focuses on discrediting claims made by John Pappademos. Palter then notes that Peter Tompkins, author of Secrets of the Great Pyramid, seems to suggest that Newton was led by Egyptian science to discover his law of gravitation. About Tompkins, Bernal writes that "it it a tragedy that Tompkins's brilliant and scholarly book has been stripped of its scholarly apparatus". Palter writes "It seems never to have occurred to Bernal that the absence of scholarly apparatus in Tompkins's account of Newton has a very simple explanation: no scholarly evidence exists to support that account." When discussing Egyptian mathematics proper, Palter focuses discusses the general character, and then square roots (or a relative lack of them), the value of pi, the controversial problem in the Moscow papyrus on the surface area of a basket, the Pythagorean theorem (or the relative lack of it, arguments on the special case of involving the diagonal of the square), and the notion (or absence of notion) of an irrational number. Palter attacks claims by Cheikh Anta Diop (see Civilization or barbarism: An authentic anthropology) that Archimedes stole some of his most famous mathematics from the Egyptians. Palter then discusses pyramidology, and some of the claims cited by Bernal that "one can find such relations as pi, phi, the 'golden number' and Pythagoras' triangle from them." The final section, discusses the similarities and differences between Egyptian and Greek medicine. Although Mathematics is not so directly involved here, strong Egyptian influence in Greek medicine could argue for the plausibility of influence of other Egyptian science on Greek science as well. A very interesting paper. Apart from the fact that Palter's article serves as a kind of review of Bernal's book, it is worth reading for its discussions on the nature of Egyptian mathematics and science. Bernal responds to Palter's article in Bernal, Martin, Response to a paper by R. Palter: "Black Athena, Afro-centrism, and the history of science" [Hist. Sci. 31 (1993), no. 93, part 3, 227--287; MR: 94i:01001]. Closely related topics: Ancient Egypt, Astronomy, Archimedes, The Egyptian Pyramids, Pythagorean Triangles and Triples, and Medicine.
Phillips, Anthony. The topology of Roman mazes. The Visual Mind, 65--73, Leonardo Book Series, MIT Press, Cambridge, Mass., 1993.
In this fascinating article, the author analyzes Roman mosaic mazes topologically and uses his conclusions to suggest some reconstructions for a number of damaged Roman mazes. His research allows him to conclude that all mazes occurring in antiquity are meander mazes; the exceptions appear to be because of faulty restoration or recording (p. 66). Roman mazes generally appear to be made of copies (usually four) of identical submazes (he calls this the "standard scheme"). The last of the copies is occasionally varied so that the side holding the entrance gate would only have one path towards the center. Otherwise the standard scheme would dictate that there are two paths running on the side with the entrance gate but only one for sides between the other components. The author calls this variation "the Pompeian Variation", and it seems to be well standardized. The last submaze apparently varies in a fairly standardized way. The submazes themselves are commonly made up of stacks of elementary submazes gamma4, although other cases also occur; the author includes a table listing the submazes and the number of examples from among the Roman mazes that are sufficiently well preserved to be intelligible. The author's systematic treatment makes his proposed restorations seem very plausible. He notes that the basic ideals for the Roman maze seems to have originated in Crete, where there is a famous association between Crete and the legend of the Minotaur. Most significantly, Phillips suggests that the Cretans had an understanding of the topological structure of their mazes: "The cons of Knossos bear at least two other designs relevant to this study. One [K:50] is the four-level maze with level sequence 0 3 2 1 4. It appears on a coin dated circa 431-350 B.C. and is evidence that the Cretans had gone beyond the labyrinth game to analyze the structure of the Cretan maze, because in fact the Cretan maze can be realized as two copies of 0 3 2 1 4, one nested inside the other." A fine article, highly recommended. Closely related topics: The Roman Empire, Mazes, and Topology.
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, 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, Medieval Europe, and The Hindu-Arabic Numerals.
Schrader, Dorothy V. De arithmetica, Book I, of Boethius. Mathematics Teacher 61 (1968), 615--28.
Paraphrases Book I of Boethius' De arithmetica, which is in turn based on the Arithmetica of Nichomachus. This book is somewhere between simple arithmetic and elementary number theory, but develops the subjects quite differently than we do today. Boethius begins what we might think of as modular arithmetic (even and odd, and later evenly-even, evenly-odd, oddly-even), but the classification of numbers and parts of numbers soon acquires an unexpected complexity. The article gives an excellent introduction to the character of Medieval arithmetic/number theory. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Boethius (Ancius Manlius Torquatus Severinus Boetius), Arithmetic, Number Theory, and Nichomachus of Gerasa.
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.