Laterally related topics: Religion, Time and Space, Mathematics in Recreation, Art, Language and Literature, Music, Measurement, Arithmetic, Mathematics and Mysticism, Geometry, Discrete Mathematics, Optimization, Philosophy, Calculus, Statistics, Social Science, Logic, Computation, Probability, Applied Mathematics (General), Education, Algebra, Number Theory, Archaeology, Medicine, Creativity, Business, Fractals, and Science.
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.
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, Leveling, The Measurement of Distance, Frotinus, Heron, Surveying, and The Sundial.
Hildebrandt, Stefan and Tromba, Anthony. The parsimonious universe. Shape and form in the natural world. Copernicus, New York, 1996. xiv+330 pp. ISBN: 0-387-97991-3. SC: 00A05 (01A99 49Q15), MR: 97c:00001.
This book has many interesting examples of how problems in optimization have been important both historically and in the world around us. For our purposes, we focus on Chapter 2, The Heritage of Ancient Science. The authors start here with a survey the history of some of the mathematics and applied mathematics of the Babylonians, Egyptians, and Greeks. They consider aspects such as astronomy, burning mirrors, and the discovery of the irrationals (they include a modulo 10 proof that the square root of two is irrational). Of course, this part of the book is not intended to be authoritative; the reader should beware of comments about the Egyptians and the Pythagorean theorem. The book continues with discussions of the Ptolemaic system (which they said was once thought to have been handed down from above) and of the heliocentric system. One of the more appealing parts of Chapter 2 is a discussion of the problem where Queen Dido of Carthage obtained the largest possible area that can be enclosed by the hide of an ox. She supposedly cut the hide into strips and formed it into a semicircle bounded by the sea. Elsewhere in the book there is quite a bit of discussion on optical shortest path problems. There are many fine illustrations both here and elsewhere. Example from Chapter 2 include the music of the spheres as imagined by Kepler, an illustration of Dido's minimization problem from the 1630s, pictures of medieval towns built with an optimization principle à la Dido, and a fronticepiece of a treatise on optics from the 1200s where refraction and burning mirrors are clearly illustrated. This book can be a fine educational resource for teachers trying to motivate ideas such as minimization problems in Calculus. Closely related topics: Optimization, Astronomy, Irrationals, The Circle, Carthage, and Education.