Geometry is a branch of mathematics that is concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the geometry field is called a geometer. Geometry arose independently in numerous early cultures as a body of practical knowledge concerning areas, lengths, and volumes, with elements of a formal mathematical science emerging in the West as early as Thales. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment – Euclidean geometry – set a standard for many centuries to follow.

Archimedes developed ingenious methods for calculating areas and volumes, in many ways anticipating modern integral calculus. The field of astronomy, particularly mapping the positions of the stars and the plants of the celestial sphere and explaining the relationship between movements of celestial bodies, served as a significant source of geometric issues during the next one and a half millennia. Both astronomy and geometry were considered, in the classical world, to be a part of the Quadrivium, a subset of the seven liberal arts that are considered essential for a free citizen to master.

The introduction of coordinates by Rene Descartes and the concurrent developments of algebra marked a new stage for geometry, since geometric figures, such as plane curves, could now be represented analytically. This played a huge role in the emergence of infinitesimal calculus within the 17th century. Furthermore, the theory of perspective displayed that there is more to geometry than just the metric properties of figures: perspective is the origin of projective geometry. The subject of geometry was further augmented by the study of intrinsic structure of geometric objects that originated with Euler and Gauss and led to the formation of topology and differential geometry.

While the visual nature of geometry makes it initially far more accessible than other parts of mathematics such as algebra or number theory, geometric language is also utilized in contexts far removed from its traditional Euclidean origin.

Image Caption: An illustration of Desargues’ theorem, an important result in Euclidean and projective geometry. Jtico/Wikipedia