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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.
Archibald, Raymond Clare. Babylonian Mathematics. With Special Reference to Recent Discoveries. Mathematics Teacher 29 (1936), 209--19. (Originally delivered at a joint meeting of the National Council of Teachers of Mathematics, the American Mathematical Society, and The Mathematical Assocation of America, at St. Louis, Mo., on January 1, 1936.)
Surveys some of Neugebauer's remarkable discoveries on Babylonian mathematics, at a time when many of these discoveries were just made. Discusses notation, tables of squares, cubes, and n3+n2. Also exponentials, approximations to compound interest problems where we would use logarithms, a sum of a finite geometric series and a finite sum of squares. Geometric results, including the Pythagorean theorem, proportionality of sides in similar right triangles, a perpendicular bisecting the base in an isosceles triangle, the angle in a semicircle being a right angle, formulas for the circumference and area of a circle (using pi = 3), formulas for the frustum of a square pyramid (at least one incorrect). The relation between chords and sagitas in a circle. Approximations to the square root of a2+b2; both the well known a+b2/2a and the still hypothetical a+(2ab2)/(2a2+b2). An approximation to a square root by comparing with other solutions to an equation x2+D=y2. (The value isn't especially accurate, but the method is interesting.) Equations in five or more unknowns. Problems requiring solutions to apparently general cubic and biquadratic equations. Were the solutions just guessed, or, as Neugebauer suggests, did the Babylonians have some general methods? If so, the most likely theory is that the cubics were solved by effectively reducing them to the form x3+x2, and then using the n3+n2 table. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Sumerians and Babylonians, The Quadratic Formula, Cubics, Quartics, Solutions of Linear Equations, Logarithms, Exponentials, Square Roots, Interpolation, Geometric Theorems, The Circle, and The Pyramid.
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.
Jones, Phillip S. Recent Discoveries in Babylonian Mathematics. III. Trapezoids and Quadratics. Mathematics Teacher 50 (1957), 570--71.
Continues Jones, Phillip S., Recent Discoveries in Babylonian Mathematics. II.. The author discusses a single Babylonian problem. The problem is interesting more as a representative of a "typical" Babylonian problem than as a discovery that gives new insights into Babylonian mathematics. The problem involves the solution to a quadratic. The scribe uses an incorrect "formula" for the area of a trapezoid. The author discusses the solution both using modern notation and in a translation of the scribes actual language. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Sumerians and Babylonians, The Quadratic Formula, and The Measurement of Area and Volume.
Llyod, Daniel B. Further Evidences of Primeval Mathematics. Mathematics Teacher 57 (1966), 668--70.
A tablet from a dig at Tel Dhibayi near Baghdad shows how to find the dimensions of a triangle from its diagonal and area. The solution requires a knowledge of the Pythagorean theorem, and artfully sidesteps the difficulty of solving a quadratic equation by solving a pair of simple linear equations. Many other articles discuss similar tablets and solutions, but few so concisely as this. However, note that in the context of other Babylonian sources, the method of solution may be less obscure than the author seems to suggest. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Sumerians and Babylonians, The Quadratic Formula, and Pythagorean Triangles and Triples.
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).
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, Number Theory, Geometry, Analytic Geometry, and Calculus.
Schattschneider, Doris. The plane symmetry groups: their recognition and notation. American Mathematical Monthly 86 (1978), 439--450.
Discusses in detail the classification of plane patterns. Although the author avoids group-theoretic notation, she manages to bring out the group theoretic nature of the plane pattern groups more clearly than most other authors discussing these patterns. There is a very useful chart on the seventeen plane patterns that clearly labels the locations of the centers of rotation (with labels that distinguish the 2, 3, 4, and 6-fold centers), the axes of reflection, and the axes of glide-reflection. The chart may give a better understanding of the differences between the different symmetry groups than the flowcharts that appear in some other sources. The author discusses the generating regions for each of the plane patterns, and gives examples for each symmetry group of two set of generators of the group (except in the case of the pattern p1, where there is only one natural set of generators. She illustrates the plane patterns with lattices, most of which are from China. There are a couple of examples from the artwork of M. C. Escher as well. There is also a table cross-referencing notations used by different sources. There are six different notations in all; as the author notes, one the differences results from the common confusion between the groups p3m1 and p31m. Closely related topics: Plane Patterns, Group Theory, Art, M. C. Escher, and China.
Swetz, Frank. The Evolution of Mathematics in Ancient China. Mathematics Teacher 52 (1979), 10--19.
An overview of Chinese mathematics, including the discovery of the lo shu magic square (thought to have a plan of universal harmony), square roots, the Chinese remainder theorem, and polynomials of high degree (including a quintic in x2). Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: China and Magic Squares.
Swetz, Frank J. Seeking Relevance? Try the History of Mathematics. Mathematics Teacher 77 (1984), 54--62.
Focuses on how the history of mathematics can be used to improve mathematics education. It can not only breath new life into the subject, but also allow students to better understand mathematics as a mode of inquiry. If students see mathematical ideas in other times [and in other cultures], they can appreciate the ideas better in our own. Swetz gives examples from the development of algorithms for arithmetic (including square roots). Ancient demonstrations of mathematical ideas, such as the "husan-thu" proof of the Pythagorean theorem from China can be conceptually more suitable for students than more synthetic modern ones. Ancient "homework problems" from Babylonia, China, and Medieval Italy can be more interesting than the more dry and formulaic modern equivalents. (See Swetz, Was Pythagoras Chinese? for many interesting examples from China.) Although the author doesn't discuss this, the Chinese problems in surveying led to interesting questions in algebra, with fourth and higher degree equations. Swetz discusses how Descartes' idea of a coordinate grid was earlier used by Renaissance artists, ancient Egyptian tomb painters, and various cartographers. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Education, Arithmetic, Computation, China, Analytic Geometry, Renaissance Art, Ancient Egypt, and Cartography.
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, Number Theory, Acoustics, and Astronomy.