Curvature in the Calculus Curriculum

problem was a routine calculation of curvature, requiring the memorization of a particular formula and the mechanics of differentiation. Being conscientious in my duties, I devised a grading scheme whereby a mistake in algebra would cost a few points and a mistakenly computed derivative a few more points, with the largest point deduction reserved for an incorrect formula for curvature. Not only had the material been reduced to rote calculation, but that method of instruction, at least for the calculus sequence, was being promulgated to the next generation of instructors. Possibly the difficulty lies in the subject itself-how can a critical understanding of curvature be tested on a one-hour exam, especially when curvature is merely one of several topics to be tested? Should the students be asked to explain the definition of curvature as the magnitude of the rate of change of the unit tangent with respect to arclength? In subsequent courses I have spent more time motivating that particular definition than actually computing curvature, but find the definition rather far removed from an intuitive idea of curvature. Should the students be asked to sketch the osculating circle to a given curve at a given point, and then notice that as the point of contact changes, the radius of the circle is inversely proportional to the measure of curvature? This seems more intuitive, and remains quite a viable introduction to curvature, although the concept of an osculating circle requires some explanation. Why would anyone have bothered to construct such a circle, our students and teaching assistants may wonder. Perhaps the most complete explanation of curvature lies in its history and offers the best understanding of the subject. Modem calculus textbooks, however, do not present the subject via its history, but opt for opaque definitions, slick formulas, or routine calculations. Although many mathematicians are familiar with the historical development of curvature, it remains unclear how often this is used to present curvature in a calculus course. This may, in part, be due to a lack of appropriate curricular material for calculus students. In this article we suggest a remedy for the problem. 2. TWO PROJECTS. What follows are descriptions and the actual text of two written projects, based on key developments in the history of curvature, that I used recently in a multivariable calculus course. For each project, the students were given about two weeks to write a detailed solution to a sequence of guided questions that begins with an event of historical mathematical significance and culminates with a key observation about curvature. Each project took the place of an in-class examination. The first, "The Radius of Curvature According to Huygens," opens with the Dutch scholar's discovery of the isochronous pendulum [5], [7]. In the project students learn firsthand from an English translation of Huygens's Horologium oscillatorium (The Pendulum Clock) how the radius of curvature, used in the construction of the pendulum, can be described in terms of ratios of segments. The goal of the project is then to reconcile Huygens's very geometric description, typical of mathematics in the seventeenth century, in terms of the modem equation of curvature. An engaging problem that shows how the radius