Introduction

The cycloid was an important “new curve” attracting mathematicians' attention in the seventeenth and eighteenth centuries. It turned out to be particularly significant in the study of the behavior of objects falling under the force of gravity: the cycloid is not only the brachistochrone (path of descent in shortest time) but also the tautochrone (path of descent in equal time from any point on the path). New mathematical tools such as the calculus made it possible to apply the study of such curves, and of concepts such as their “evolutes” and “involutes”, to mechanical problems.

The significance of these developments is often lost on students who find them unfamiliar and remote. The story of Huygens' cycloid pendulum clock is an intriguing, easy-to-understand application of these mathematical ideas to a very practical problem. And it supplies a hands-on construction project that reinforces students' comprehension of how the cycloid and evolutes of curves actually work.

Huygens and the cycloid

Timekeeping problems and the tautochrone curve

In the middle of the seventeenth century, the scientific revolution and nautical discovery were in full swing. The expansion of trade and colonization meant an increasing need for accuracy in determining longitude at sea. An accurate clock would solve the problem of measuring time differences precisely enough to determine longitude; it would also be useful in many scientific experiments. The trouble was that clockmaking technology at that time wasn't developed enough to produce a sufficiently accurate clock.