Differential Geometry

Abstract

We saw in Chapter 9 that the development of calculus in the seventeenth century was greatly stimulated by problems in the geometry of curves. Differentiation grew out of methods for the construction of tangents, and integration grew out of attempts to find areas and arc lengths. Calculus not only unlocked the secrets of the classical curves and of the algebraic curves defined by Descartes; it also extended the concept of curve itself. Once it became possible to handle slopes, lengths, and areas with precision, it also became possible to use these quantities to define new, nonalgebraic curves. These were the curves called “mechanical” by Descartes (Sections 7.3 and 13.3) and “transcendental” by Leibniz. In contrast to algebraic curves, which could be studied in some depth by purely algebraic methods, transcendental curves were inseparable from the methods of calculus. Hence it is not surprising that a new set of geometric ideas, the ideas of “infinitesimal” or differential geometry, first emerged from the investigation of transcendental curves.