"measurement") is, with arithmetic, one of the oldest branches of mathematics.
It is concerned with properties of space that are related with distance, shape,
size, and relative position of figures. A mathematician who works in the
field of geometry is called a geometer.
Until the 19th century, geometry was almost exclusively devoted to Euclidean
geometry, which includes the notions of point, line, plane, distance, angle,
surface, and curve, as fundamental concepts.
During the 19th century several discoveries enlarged dramatically the scope
of geometry. One of the oldest such discoveries is Gauss' Theorema Egregium
(remarkable theorem) that asserts roughly that the Gaussian curvature of a
surface is independent from any specific embedding in an Euclidean space.
This implies that surfaces can be studied intrinsically, that is as stand alone
spaces, and has been expanded into the theory of manifolds and
Riemannian geometry.
Later in the 19th century, it appeared that geometries without the parallel
postulate (non-Euclidean geometries) can be developed
without introducing any contradiction. The geometry that
underlies general relativity is a famous application of
non-Euclidean geometry.