Addendum: The following survey on geometries is taken mainly from Jan Gullberg, Mathematics from the Birth of Numbers. New York, WW Norton & Company, 1997.

Euclidean Geometries

• Euclidean geometry: also referred to as classical or elementary geometry. Mainly concerned with points, lines, circles, polygons, polyhedra and conic sections. It is based on definitions and axioms described by Euclid (c.330 - c.275 B.C.) in the treatise Elements, a compendium of all the geometric knowledge of his time. Solid geometry is primarily concerned with spheres, cylinders and cones, and was developed by Archimedes (287 - 221 B.C.) some years later. Conic sections were the subject of Apollonius studies by the same time (c.260 - after 200 B.C.).
  
  
• Trigonometry: the geometry of triangles. Hipparchus of Nicea (? - after 127 B.C.) is credited for the invention of this branch of geometry as a tool for resolving astronomical distances. It can be subdivided into plane trigonometry, for triangles on a plane, and spherical trigonometry, for triangles on the surface of a sphere.
  
  
• Projective geometry: concerned with the properties of plane figures that are unchanged when a given set of points is projected onto a second plane. It became into use in the 15th and 16th centuries through its application in architecture by the Italian master Leone Alberti (1404 - 1472) and the French mathematician Girard Desargues (1591 - 1661), although it is sometimes associated (together with descriptive geometry) to Ptolemy of Alexandria (c. A.D. 100 - c.170).
  
  
• Analytical geometry: Invented by René Descartes (1596 - 1650), it works with geometric problems by means of coordinate systems and their transformation into algebraic problems. It is also subdivided into plane analytical geometry, for equations with two variables, and solid analytical geometry, for equations with three variables.
  
  
• Differential geometry: This one came into being when mathematicians in the 18th century, following Descartes' discoveries, added differential and integral calculus to curves, surfaces, and other geometrical entities.
  
  
• Vector analysis: studies quantities that have magnitude and direction. Known since Aristotle's times and even more by Simon Stevin in the 1580s, the modern theory dates from the early 19th century.

Non-Euclidean Geometries. In the nineteenth century, mathematicians started to develop other kinds of geometry for which at least one of the Euclidean axioms does not hold. Hence non-Euclidean geometries flourished.

• Hyperbolic geometry: Credited independently to Nicolai Lobachevski (1792 - 1856) and János Bólyai (1802 - 1860), it rejects the parallel postulate of Euclidean geometry, and states that "Through a given point outside a given straight line pass more than one line not intersecting the given line."
  
  
• Elliptic geometry: also rejects the Euclidean parallel postulate, and states that "there are no parallel lines and, if extended far enough, any two straight lines in a plane will meet." Bernhard Riemann (1820 - 1866) is credited with its invention.
  
  
• Topology: Also from the 19th century, it began with the Dutch astronomer Augustus Möbius (1790 - 1868) and a wealth of other mathematicians that later included David Hilbert (1862 - 1943), Oswald Veblen (1880 - 1966) and Henry Whitehead (1904 - 1960). It deals with properties that are not altered by continuous deformations, such as bending, stretching and twisting.
  
  
• Fractal geometry: A very recent addition to the realm of geometry, it studies shapes and figures that have self-similarity and fractional dimension. The leading soul in fractal geometry is Dr. Benoît Mandelbrot.