The mathematics of proportions in the world of straight lines and circles in flat

2D (or nD) space. '

Geometry' is Greek for 'land measurement'.

Named after Euclid the author/originator of the Elements, the first major text on the subject..

Geometry is useful enough in itself, but Euclid's work got its fame for its approach: it is an impressive piece of formal deductive mathematical logic, showing off the power of that method of reasoning in general.

The idea of that method is to 'capture' a subject in terms of a few logical statements that are known to be true.; all reasoning about the subject will then be performed on these logical statements. In this case, five assumptions were supposed to capture the essence of Euclidean geometry (taken from http://www.britannica.com/bcom/eb/article/single_image/0,5716,15972,00.html):

- given two points there is an interval that joins them
- an interval can be prolonged indefinitely
- a circle can be constructed when its centre, and a point on it, are given
- all right angles are equal
- if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles

For more details, see www.britannica.com.
(This is, actually, a post-rationalization; what Euclid actually wrote is explained in Donald Lancon's beautiful paper at http://www.obkb.com/dcljr/euclid.html.)

It is easy to see that all of the five axioms hold.
In the 18th and 19th century, mathematicians started to wonder if they are all necessary; in particular, they wondered if the fifth (the parallel postulate) didn't follow from the others. It was finally realised that this is not the case; what is more, that you can replace arbitrary axioms with arbitrary replacements and often find a useful interpretation! This game is known as non-Euclidean geometry.

Furthermore, this radical form of mathematical logic was applied to all of mathematics, which led to the debate between David Hilbert and L. E. J. Brouwer on the foundations of mathematics and to the theory of computation on which modern computer science is based.