To refine search, see subtopic Graphing. To expand search, see Geometry. Laterally related topics: Symmetry, Trigonometry, Pattern, Geometric Theorems, The Pyramid, Similarity, The Triangle, The Method of Exhaustion, Projective Geometry, Algebraic Geometry, Non-Euclidean Geometry, The Parallel Postulate, The Regular Solids, Irrationals, The Pentagram, The Sphere, The Conic Sections, Polygons, Topology, Spirals, Line-Point Duality, Geometric Fixed Point Principles, The Cycloid, Tilings, and The Square.
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.
Court, Nathan Altshiller. Mathematics in the History of Civilization. The Mathematics Teacher 41 (1948), 104--11.
How different concerns of society influenced mathematics. How the development of the concept of number is reflected in language. How the concept of how many led to arithmetic. How the concept of how much led to geometry. (Taxation and agriculture also contributed to both.) Efforts to keep time led to trigonometry. Navigation and associated astronomical problems led to logarithms [and more trigonometry]. Problems in artillery led to graphs. Both required an understanding of motion. Analytic geometry and calculus were invented in part to better understand motion. Statistics developed to understand problems in the social sciences. Also discusses the nature of mathematics: mathematics for its own sake and the axiomatic method. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Why Study History Of Math, Mathematics in Language, Number Systems, Arithmetic, Geometry, Taxation, Agriculture, Astronomy, The Reckoning of Time, Trigonometry, Artillery, Graphing, Navigation, Dynamics, Force, and Motion, Calculus, Statistics, Social Science, and Proof.
Engels, Hermann. Quadrature of the circle in ancient Egypt. Historia Math. 4 (1977), 137--140. (Reviewer: L. Guggenbuhl.) SC: 01A15, MR: 56 #5124.
Explains the Egyptian formula for the area of a circle in terms of the practices of Egyptian stone masons. In order to form a relief, the stone masons covered their designs with a grid. The hypothesized construction involves an error which would confirm the now commonly held view that the ancient Egyptians did not properly understand the Pythagorean theorem. Closely related topics: Ancient Egypt, The Circle, Coordinates, and Pythagorean Triangles and Triples.
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, Projective Geometry, Algebraic Geometry, Greece, The Islamic World, The Parallel Postulate, and Perspective.
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, and Calculus.
Robins, Gay and Shute, Charles C. D. Mathematical bases of ancient Egyptian architecture and graphic art. Historia Math. 12 (1985), no. 2, 107--122. (Reviewer: Jens Høyrup.) SC: 01A15, MR: 87c:01002.
The authors discuss the slopes that occur in Egyptian pyramids and artwork. The discussion of Egyptian artwork is particularly interesting because of the Egyptian's conscious use of squared grids. The authors find no evidence of circles or the value of pi being used in to determine the overall dimensions of the pyramids, and similarly with the golden ratio. Similarly, the authors find no evidence of pi or the golden ratio being found in slopes of lines in Egyptian artwork. Nevertheless, the authors carefully discuss such claims rather than simply dismissing them out of hand. The authors do, however, find that certain "slopes" seem to have been preferred to others (as the authors note, the Egyptians seem to have preferred to measure slopes as run per unit rise rather than our rise per unit run). The authors buttress their arguments about the artwork through their use of new photographs; these carefully avoid distortion by means of a shift lens. The article is only moderately technical. Closely related topics: Ancient Egypt, The Egyptian Pyramids, The Circle, Proportion and the Golden Ratio, and Coordinates.
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, and Arithmetic.
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.
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, Algebra, Renaissance Art, Ancient Egypt, and Cartography.