22 - An “Impossible” Problem, Courtesy of Leonhard Euler

Summary

Introduction

The second semester of calculus may well be the busiest course in the standard undergraduate mathematics curriculum. Between applications of integration, integration techniques, polar coordinates, parametric representations of curves, sequences and infinite series, one usually has no time to give conic sections their due. For quite some time, therefore, I have been looking for interesting things to say about conics that tie in well with students' recently acquired calculus tools.

Recently I got lucky. I happened upon an article published in 1755 by the great Swiss mathematician Leonhard Euler, which considers a problem that fits the bill perfectly. Euler's treatment of the problem synthesizes a number of ideas from elementary calculus: trigonometric identities, techniques of integration including partial fractions, representation of curves by polar equations, and separable differential equations, with a particular conic section—the parabola-leading off the action.

Historical Setting

Suppose that you are given a parabola, and that you draw an arbitrary line through its focus F, meeting the parabola at points M and N. The tangent lines to these points will always meet at a right angle! One possible approach to a proof is to work from the reflection property of parabolas, as follows:

In Figure 22.1, the points P and Q are chosen so that PN and QM are parallel to the central axis of the parabola. By the reflection property, a ray of light traveling from P to N will bounce off the parabola and head toward F, with PN and NF making equal angles, of measure α let us say, to the tangent line YZ at point N.