Einführung in die Theorie de Curven in der Ebene und im Raume

Abstract

THIS volume is the first of two which will make a complete work under the title “Anwendung der Differential- und Integral-Rechnung auf Geometrie.” The subject-matter of the two volumes may be said to be, roughly, the infinitesimal geometry of curves and surfaces respectively. The first volume is divided into three sections, dealing with plane curves, curves in space, and developable surfaces. The first section does not attempt to be a complete exposition of the subject, and must be regarded as an introduction to what follows, intended to accustom readers who are already well grounded in differential and integral calculus to the style and methods which are employed later. The theory of the curvature of plane curves is based on the definition of contact of an assigned order, which is explained with great exactness. The differential invariants of a curve for the group of movements in the plane are fully investigated, and their properties established in an elementary manner without introducing notions of groups or partial differential equations. Envelopes, evolutes, singular points, and the geometrical significance of differential equations of the first order and degree are discussed shortly. In connection with the trajectories of a family of curves, the problem is completely solved of finding all curves for which the product of the normal and radius of curvature is constant. The remainder of the first section is devoted to an explanation of curvilinear coordinates.

Einführung in die Theorie de Curven in der Ebene und im Raunie.

By Dr. Georg Scheffers. Pp. viii + 360. (Leipzig: Veit and Co., 1901.) M. 10.