Lessons for Classics from the History of Mathematics

Abstract

Presented here are examples of two problems set by ancient Greek mathe maticians that engaged scholars for centuries and could be included in a course on Greek civilization. Also presented are samples from mathematical works published in Latin and some Latin anagrams produced by mathematicians and scientists, all of which could be introduced into the Latin classroom. The Teaching Innovation Program at the University of Mary Washington provided the opportunity for a Teaching Partner Exchange that granted two faculty members in different disci plines a course release to attend one another's classes. The faculty members were to engage themselves fully, doing all the work, side by side with the regular students. The anticipated results included a greater awareness of another discipline and a healthy reminder of what it means to be a student. Participating in this program, the mathematician among us took Elementary Latin and the classicist The History of Mathematics.1 With a continued interest in the intersection of our disciplines, we have sought ways to build on those interdisci plinary connections. While the importance of ancient Greece to the history and development of mathematics is common knowledge, and while most classicists are aware that mathematical treatises were published in scholarly Latin into the Renaissance and beyond,2 it is rare to see the history of mathematics incorporated into either courses in classical civilization or Latin classes. We therefore offer some examples from the history of mathematics and science that could reasonably be inserted into Classics courses at the high school or college level. Plato and the Three Classical Construction Problems Plato's impact on the development of mathematics cannot be overstated. While he developed little original mathematics himself, he used the subject to train the intellect, and his insistence on its impor tance produced an environment in which the discipline flourished. 1 The textbook used in the latter course, Burton (2007), is an excellent reference for the history of mathematics, especially in antiquity. 2 Mathematical works published in Latin, in addition to Barrow (1655), Cardano [1663] (1967) and Heiberg (1883-6) discussed below, include Newton's Philosophiae Natu ralis Principia Mathematica (1687) and Fibonacci's Liber Abaci (1202). No copy of Fibonac ci's work from 1202 is extant. See Boncompagni (1857-62) for Fibonacci's 1228 edition. THE CLASSICAL JOURNAL 104.4 (2009) 351-62 This content downloaded from 207.46.13.156 on Sat, 10 Sep 2016 06:10:11 UTC All use subject to http://about.jstor.org/terms 352 LIANE HOUGHTALIN AND SUZANNE SUMNER Legend has it that Plato even affixed a sign over the doors of his Academy with the warning, "Let no man ignorant of geometry enter here."3 His personal inclination, however, was to value theoretical mathematics and to display contempt for applying the subject to any practical use. Instead, Plato believed that all mathematics should be created from the ideal forms of circles and lines, and he accordingly restricted the tools allowed to a straightedge (a ruler with no grid for measuring) to draw lines, and a compass to construct circles with any center and radius.4 Over the course of the centuries, mathematicians realized that three mathematical problems, called the Three Classical Construction Problems, were unsolvable under Plato's limitations, but were solv able with looser restrictions. These problems are Squaring a Circle (constructing a square with the same area as a given circle), Trisect ing a General Angle (dividing an arbitrary angle into thirds) and Duplicating a Cube (constructing a cube with double the volume of a given cube). Figure 1 illustrates the essence of the cube duplication problem. The cube on the left with edge a will have a volume of a3, whereas the cube on the right with edge x will have a volume of x3, which will be double the volume of the first cube if x3 equals 2a3.