Geometry is the branch of mathematics that deals with measurements and properties of points, lines, and angles. Geometry is one of the oldest branches of mathematics, used by the Egyptians and Babylonians as early as 2000 B.C. The ancient Egyptians and Babylonians had a practical knowledge of many geometric ideas, which they applied to surveying and construction projects; the first meaning of the word "geometry," which is of Greek origin, was "measurement of the earth." Early tablets have shown that the Egyptians and Babylonians formed reasonable estimates of the value of pi (), and that the Babylonians were aware of the Pythagorean theorem, which gives a relationship between the lengths of the sides of a right triangle. The Egyptians and Babylonians discovered geometric ideas largely through experimental means--from examining the measurements of physical objects.
Geometry came into its own in the time of the Greeks, who developed the notion that geometric relationships could be proven. With that idea, geometry turned from an experimental science into an intellectual pursuit, one that used the laws of logic to deduce complex geometric relationships from simpler ones. The crowning achievement of Greek geometry was Euclid's Elements, a textbook written around 300 B.C. that developed a rigorous presentation of elementary geometry. Euclid followed an axiomatic approach, which has been enormously influential to the development of modern mathematics: he started with five basic assumptions (axioms), and used deductive reasoning to create a huge edifice of theorems that are logical consequences of those axioms. Among his theorems are, for example, the laws of congruent and similar triangles, a proof of the Pythagorean theorem, and the relationships between lengths and areas in circles.
For almost 2000 years after Euclid wrote his Elements, the study of geometry consisted chiefly of understanding the Elements, and enlarging on the ideas and techniques found within it. In the early 17th century, the mathematician and philosopher Rene Descartes made a great step forward by connecting algebra and geometry, through the coordinate system. Descartes realized that the location of a point in the plane could be described by an ordered pair of numbers: one number that described its position on a horizontal axis, one on a vertical axis (according to myth, Descartes conceived this idea while watching a fly walk on the ceiling). With the advent of Cartesian geometry, geometers could use the techniques of algebra, which streamlined geometric proofs enormously. Around the same time, two problems from outside of mathematics began to influence the development of geometry: the new science of mapmaking, through which geometers began to understand the very different nature of the curved earth and the flat plane, and the use of perspective in art, which stimulated the development of projective geometry.
An important question that plagued geometers was the problem of how fundamental Euclid's five axioms really were. The first four of Euclid's axioms dealt with such natural concepts that most geometers were willing to accept that they were truly basic assumptions for any geometric space. The fifth axiom, Euclid's parallel postulate, was much more controversial: it stated that, given a line and a point not contained in the line, there was one and only one line through the point that never met the original line. That statement certainly appeared to be true, but it was nowhere near as simple as the other four postulates. Its complexity led mathematicians to ask whether the fifth axiom was truly a basic assumption, or whether it was really a theorem, something that could be proven from the other four axioms. For many years this question was open, but in the 19th century, Johann Carl Friedrich Gauss, János Bolyai, and Nikolay Lobachevsky shocked the mathematical community by showing that there exist perfectly consistent geometries in which Euclid's fifth axiom is not true.
The discovery of non-Euclidean geometries spurred the development of differential geometry, the study of curved spaces. The Euclidean geometry of the Elements is the geometry of the flat plane, and its theorems and measurements only give information about objects in a flat space. Differential geometry, developed by Gauss and Bernhard Riemann, focused on measurements of lines, circles, triangles, and other objects on surfaces or spaces that are curved, like a sphere or a cylinder. In these curved spaces, the idea of a straight line is replaced by the idea of a geodesic, the shortest path between two points. On a surface, the shortest path between two points can be found by attaching a rubber band to the two points and pulling it taut, always making sure it lies on the surface. So for example, on a sphere, a longitudinal line is a geodesic between the north and south poles. This example exposes some key differences between lines in Euclidean space and geodesics in curved spaces: there does not have to be only one geodesic between two points, and when a geodesic is extended, it may run into itself, instead of extending infinitely far. With this new notion of a "line," differential geometers can ask the same questions about triangles, circles, and angles that Euclidean geometers have asked (although the answers can be wildly different!).
One of the key notions of differential geometry is the idea of the curvature of a surface or space. Curvature is a number associated to a space, which measures the degree and the type of curving on the space. A precise definition requires advanced mathematical tools, but among surfaces, the following are the basic distinctions: a surface shaped like a sphere is said to be positively curved; a surface that is saddle-shaped is negatively curved; a surface that is flat, like the Euclidean plane, has zero curvature. A more complicated surface can have different curvature at different points on the surface.
The notions of curvature and geodesics can be extended to three- and higher-dimensional spaces. One of the most important ideas of Einstein's relativity theory is that space has higher curvature near massive objects; it is this curvature that creates gravitational attraction. Recently, the Hubble telescope has located points in space that are connected by more than one geodesic path, raising the question of the precise geometry of the universe. Differential geometry continues to be fundamental in the attempts of modern physics to advance the quest begun by the Egyptians and Babylonians: to measure the world.
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