The standard fare of high school geometry consists mostly of material collected two millennia ago when geometry stood at the center of natural philosophy. The ancient Greeks, however, did not limit their investigations to the circles and straight lines of Euclid's Elements. Archimedes, using methods and ideas that were later to underpin the analysis of infinitesimal quantities, could calculate the length of a spiral and the areas and volumes of other complex figures. In elementary algebra we learn to graph simple quadratic functions such as the parabola and hyperbola, and then in analytic geometry we learn to handle space figures such as the ellipsoid. The methods involved are essentially due to Descartes, who connected the ideas of geometry with those of algebra, and their application is largely limited to objects whose describing equations are of the second order. Finally, in elementary calculus we study formulas that permit us to calculate the length of a curve given in Cartesian coordinates, etc. These more powerful methods are now incorporated into a branch of mathematics known as differential geometry.

Differential geometry allows us to characterize curves and figures of a very general nature. The practical importance of this is well illustrated by the problem of optimal pursuit, wherein one object tries to catch another moving object in the shortest possible time. ^[1] Of course, we shall often make use of standard figures such as circles, parabolas, etc., as specific examples since we are fully familiar with their properties; in this way...