These inspired a new approach to the subject by presenting theorems in terms of axioms applied to properties assigned to undefined elements called points and lines. This led to many new geometries, including elliptical, hyperbolic, and parabolic geometries. Modern abstract geometry deals with very general questions of space, shape, size, and other properties of figures. Projective geometry, for example, is an abstract geometry concerned with the geometric properties that remain invariant under the projection of figures onto other figures, as in the case of mathematical perspective. A very useful approach to geometry is found in topology, the study of the properties of a geometric figure that remain the same when a figure is subjected to continuous transformation without loss of identity of any of its parts.
Differential geometry is the study of geometry in terms of infinitesimals.
Analytic Geometry and Trigonometry Analytic geometry combines the generality of algebra with the precision of geometry. It is sometimes called Cartesian geometry, after Descartes, who was the first to exploit the methods of algebra in geometry. Trigonometry is the study of triangles, angles, and their relations. In the 19th century, in response to questions about its rigorous foundations, the calculus was developed in terms of a theory of limits. This required an exact definition of the continuity of the real numbers.