The foremost historiographic challenge in interpreting pre-modern Indian mathematics is arguably not anachronism so much as anachorism, the blurring of geographical or cultural rather than chronological distinctions. For example, historians struggle constantly with ways to avoid or explain calling Indian analyses of right-triangle relations “Pythagorean”, or using the term “Diophantine equations” for the type of problems designated in Sanskrit as \kuttaka\ or \varga-\prakrti. Nonetheless, the combination of anachronism and anachorism provides the study of Indian mathematics with a powerful lens, which clarifies even as it distorts. This paper will address such trade-offs between popular misconceptions and deeper insights, especially in the application of concepts from the historiography of early modern European calculus to infinitesimal methods used in Sanskrit mathematics of the early to mid-second millennium.

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.

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.