Historical Activities for the Calculus Classroom

The history of the calculus is a fascinating story, inspired by the search for solutions to interesting problems. We do our students a disservice when we fail to share with them some of this exciting history. Over the last two years, with support from the National Science Foundation, I have been developing modules for teaching calculus concepts in a way that integrates the historical evolution of these concepts. Below are some examples. Included are several interactive Java applets, produced using Geometer's Sketchpad, which require a Java-enabled Web browser.

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Module 1: Curve Drawing Then and Now

It was René Descartes (1596–1650) who dramatically changed the mathematical landscape with his book Discours de la méthode pour bien conduire sa raison et chercher la vérité dans les sciences (A Discourse on the Method of Rightly Conducting the Reason and Seeking Truth in the Sciences). In La Géométrie, one of three appendices of this book, Descartes showed how to describe geometric objects like curves by means of an algebraic equation. This enabled him to take advantage of the power of algebra to solve geometric problems. This idea of describing points in the plane with coordinates appears also in a work by Pierre de Fermat from 1637. In fact, while Descartes used this idea to derive algebraic equations of certain curves that he was interested in, it was Fermat who first used equations to define new curves. Later, Jacob Bernoulli (1654–1705) and Isaac Newton (1643–1727) generalized this concept to other kinds of coordinate systems such as polar. The curves that Descartes was interested in were ones that could be described by some mechanical motion.

The popularity of Descartes' La Géométrie is due largely to Franz Van Schooten (1615–1660), a Dutch professor of mathematics who translated Descartes' work into Latin and wrote a commentary on it filling in many gaps. Van Schooten was particularly interested in conic section drawers and wrote a treatise on them. He devised several instruments for drawing conic curves, as shown in Figure 2. Like Descartes' instruments, these generally consisted of a collection of straight rods hinged together in some way.

Today's graphing calculators easily generate graphs of curves by using an algebraic equation to generate the coordinates of many points on the curve and then plotting them. Many, many points must be plotted in order to obtain an accurate graph, so this would not have been a very practical method before the age of computers. In contrast, the ancients drew curves by constructing specialized instruments or “curvedrawers”. In the same way that ruler and compass can be used to draw straight lines and circles in a continuous movement, the curve drawers devised by Van Schooten and others were used to draw other curves of interest, particularly the conic sections, namely parabolas, ellipses, and hyperbolas.