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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.
Ascher, Marcia. Before the conquest. Math. Mag. 65 (1992), no. 4, 211--218. SC: 01A12, MR: 93g:01006.
Discusses the Inca and the Maya. With the Inca, focuses on the quipu. Most quipus were destroyed by the Spanish, who thought them to be the work of the Devil, but some 550 remain. Discusses their basic structure. A fascinating puzzle in the article is a pair of quipus which seem to represent data in a similar yet inexplicable way. With the Maya, focuses on their calendar. Again, much has been destroyed. For example, there only four codices remain, whereas thousands were burned by the Spanish. Fortunately, many stelae still exist. These show a calendar system with a variety of cycles. These cycles to us suggest Chinese Remainder problems. Examples of cycles are the 260 day ritual almanac composed of a cycle of 13 numbers and 20 named dieties, the vague year of 365 days composed of a cycle of 20 numbers within a cycle of 18 named dieties plus 5 unnamed days, their least common multiple (the calendar round of 18,980 days), the long count of days (in effect, multiples of 360 days plus a remainder), a 9 day cycle of Lords of the night associated with gods of the underworld, a lunar cycle of 29 and 30 day months, 13 levels in the heaven, a cycle of 4 cardinal directions (associated with different colors), sometimes used in conjunction with an 819 day cycle of the rain god. The Mayans appear to have had keen astronomical knowledge. The author notes that the error between real and tabulated times of the position of Venus would be off by just two hours in 500 years. Closely related topics: The Inca, The Quipu, The Maya, The Calendar, Astronomy, and Chinese Remainder Problems.
Jones, Phillip S. From Ancient China 'til Today!. Mathematics Teacher 49 (1956), 607--10.
Discusses Chinese remainder problems and their connection with topics such as the Euclidean algorithm and continued fractions. The history is not examined in depth. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topic: Chinese Remainder Problems.
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 topic: Leonardo of Pisa (Fibonacci).
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, Geometry, Analytic Geometry, and Calculus.
Rees, Charles S. Egyptian fractions. Math. Chronicle 10 (1981), no. 1-2, 13--30. (Reviewer: Bruno Poizat.) SC: 10A30 (01A15), MR: 82m:10016.
This article uses the Egyptian preference for dealing with unit fractions (except in the case of 2/3) as a starting point for some interesting problems in number theory. There are several proofs that every fraction can be represented as a sum of unit fractions, and these vary in the number of fractions produced and the maximum size of the denominators (these proofs are given as Fibonacci-Sylvester, Erdös (1950), Golomb (1962), Bleicher (1968, using Farey series), and Bleicher (1972, using continued fractions)). He also discusses various conjectures about unit fractions. For example, Erdös and Strauss conjectured that 4/n can always be written as the sum of three or less Egyptian fractions, and Sierpinski made the same conjecture for numbers of the form 5/n. The author also discusses some interesting results by R. L. Graham (1963). As an example, Graham proves some interesting theorems where the denominators of the unit fractions are required to be squares, or to be cubes, or to be square free. Closely related topics: Ancient Egypt and Arithmetic.
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, and Nichomachus of Gerasa.
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, Gerbert, Pope Sylvester II, Religion, Medieval Europe, and The Islamic World.
Schroeder, Manfred R. Number theory and the real world. Math. Intelligencer 7 (1985), no. 4, 18--26. (Reviewer: M. Mendès France.) SC: 11-02 (00A69 01A99), MR: 87b:11001.
We learn in this interesting article that number theory has applications to, or at least connections, with the real world. The author begins with a discussion of the division of the scale into twelve equal semitones, and how this appears natural from the continued fraction representation of log23. Next, he discusses the acoustics of concert halls, and how ceilings designed with a knowledge of quadratic residues can better convert sound waves traveling longitudinally into lateral waves, and thereby produce a more accurate stereophonic effect. Another suggestion of the author on wave diffraction involves primitive roots. (If the reader wants to really understand this part of the article, some knowledge of physics will be necessary.) The author then discusses of applications of finite fields to error correcting codes and even a verification of Einstein's General Theory of Relativity (the slowing of electromagnetic radiation in a gravitational field, observed with radar echos of the planets Venus and Mercury). The applications of modular arithmetic to cryptography and fast methods of multiplication are more widely known, but will come as a pleasant surprise to the uninitiated. Many other applications are also briefly mentioned. The author has written a book Number Theory in Science and Communication: With Applications in Cryptography, Physics, Biology, Digital Information and Computing (Springer-Verlag, Berlin, 1984) that discusses these and other applications in more detail. Closely related topics: Music, Acoustics, Astronomy, Information Theory, and Arithmetic.
Swift, J. D. Diophantus of Alexandria. American Mathematical Monthly 63 (1956), 163--70.
Discusses the notation, the techniques, and also several problems in Diophantus' Arithmetic. The author finds that Diophantus' methods are similar to those of the Babylonians, and observes that "the work may be viewed as an episode in the decline of Greek mathematics or as the finest flowering of Babylonian algebra." One interesting problem seems to involve an approximation to a square root. Swift also discusses the transmission of Diophantus' work and the resurgence of interest in it in the 1500s and 1600s. There doesn't seem to have been much interest in it in the Hindu or Islamic world. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Diophantus, Indeterminate Equations, Diophantine Equations, and Sumerians and Babylonians.
Taussky, Olga. From Pythagoras' theorem via sums of squares to celestial mechanics. Math. Intelligencer 10 (1988), no. 1, 52--55. (Reviewer: \v Stefan Porubsk\'y.) SC: 01-01 (01A99), MR: 89e:01002.
The author discusses parameterization of Pythagorean triangles, the law of quadratic reciprocity, representation of numbers in a fixed finite number of sums of squares numbers, quadratic forms, and connections with the complex numbers, quaternions, and Cayley numbers. The author tells that H. Levy and E. Isaacson observed the law of quadratic reciprocity in the study of water waves on a sloping beach (if sound waves behaved in an analogous way, would there be an applications in acoustics?). We see a surprising application of the parameterization of Pythagorean triangles in astronomy: E. Stiefel found observed that a straight line u1=c in the parameter plane (u1,u2) gives us triples (x,y,r) corresponding to a parabola, and if one moves along this line at a constant rate, one moves in a parabolic path according to Kepler's second law. Closely related topics: Pythagorean Triangles and Triples, Imaginary and Complex Numbers, Algebra, Acoustics, and Astronomy.