8 - The Mathematics of Measuring Time

Summary

Introduction

In today's world of electronic clocks and universal calendars, it's easy to forget how important mathematics used to be just for the fundamental task of figuring out what time it was. The standard rigorous approach to the problem involved applying trigonometry to observed positions of the sun or the stars, as described below (“In the Classroom”). But several simpler methods were also developed for use when observations were unavailable or calculation was unappealing. One such practical device was the sinking-bowl water-clock, used for many centuries in India. Students (and teachers) will be impressed by how easy such a clock is to construct and adjust, and how much mathematical labor it can save.

This activity and discussion can be used as part of a module on trigonometry. A more advanced class in calculus may be interested in the theoretical modeling of water-clock construction, and especially in comparing the real mathematics of water-clock design with the artificial assumptions made in typical “related rates” problems about filling and draining water tanks. The construction and testing of the sinking-bowl model can take as little as ten or fifteen minutes (depending on the length of its period): exploring the trigonometry of time-telling may involve fifteen or twenty minutes more.

Historical Background

Any water-clock (or “clepsydra”, Greek for “stealing water”) works on more or less the same principle as an hourglass: it measures a fixed period or interval of time by means of a substance flowing through a hole in a container, and at the end of that interval it must be reset manually to measure another period of the same length.