Introduction

While the modern version of tangents is central to the ideas of the differential calculus, I find students can profit from seeing an earlier and different approach. This minor detour also has the amusing aspect of using quite modern technology to help with an old problem. I use this material at the beginning of Calculus 2, when the students are fairly comfortable with the modern definition of derivative. One class period is used to present Descartes' approach, then students receive a take-home assignment.

Historical Background

In La Géometrie (1637) [2] Renée Descartes presents his general method of drawing a straight line to make right angles with a curve at an arbitrarily chosen point upon it. He praises his own approach as solving “not only the most useful and most general problem in geometry that I know, but even that I have ever desired to know” [2, p. 95]. In our terms, he sought the normal line to a curve at a given point, from which the tangent line can easily be found as well.

Descartes' approach is quite different from the modern one, which raises the question: If his method was as important as he thought,why did it not prevail? I will describe his method and suggest some exercises that can help a student to understand what Descartes was doing and to see why other approaches won out. Using calculators or computer algebra systems allows us to remove much of the drudgery of this historical reenactment.

In the Classroom

Descartes' Approach to Tangents

Figure 13.1 is based very loosely on Descartes' own diagram [2, pp. 94 and 98]. The most important anachronismis the vertical axis.