Geometry is the branch of mathematics that deals with size, shape, and space. Probably the most significant step that the ancients took towards modern mathematics was taken in the field of geometry. The ancient Greek mathematicians made geometry, and for that matter all of mathematics, what it is today. Thales was the first human known to prove something through deductive logic, an idea that is fundamental to all of mathematics today. This discovery was unique in human history; while Indians, Chinese, Maya, and other cultures developed advanced mathematics, none developed the idea of deductively proving their discoveries. Another important idea first seen in Greek geometry is that of abstraction—looking at the underlying essence of a concept separate from any real-world objects. For example, one may draw a circle, but that drawing is not the same as a real circle, which is equally smooth everywhere and has no thickness.

One of the great accomplishments of the ancient Greeks was Euclidean geometry, as summarized in the Elements. It is the first subject matter in human history that humans were able to master. Other ancient Greek accomplishments have long been superceded—Galen's works are not used in medical school and the works of Aristotle are nowhere to be found in biology and physics classes—but the ideas in the Elements are still valid, and if it is no longer the standard work in geometry as it was for 2,200 years, it is only because books more suited for modern students and mathematicians are now available.

Euclidean geometry has been shown to consist of two self-contained geometries that have different primitive concepts and axioms. The first, *absolute geometry*, provides a base not only for Euclidean geometry but also for other (non-Euclidean) geometries, such as *hyperbolic geometry*, which do not include the parallel postulate. The second, *affine geometry*, is based firmly on the unique parallel but does not include circles or angle measurement.

Both geometries are *ordered*; that is, the order in which points are placed on a line is meaningful. In geometries such as projective geometry, this order is not meaningful. In projective geometry, every pair of lines in a plane meet in a point; given three lines through a point or three points on a line, we cannot say one lies between the other two.

Topology is sometimes called the most general of all geometries and provides a strong contrast to Euclidean geometry.