### Topic Page: Euclidean geometry

**Euclidean geometry**from

*The Penguin Dictionary of Mathematics*

Geometry based on the definitions and axioms set out in Euclid's Elements. Book I starts out with 23 ‘definitions’ of the type ‘a point is that which has no part’ and ‘a line is a length without breadth’. Then follow ten axioms, which Euclid divided into five ‘common notions’ and five propositions. His common notions were:

Things that are equal to the same thing are equal to one another.

If equals are added to equals, the wholes are equal.

If equals are subtracted from equals, the remainders are equal.

Things that coincide with one another are equal to one another.

The whole is greater than the part.

Euclid's postulates were:

A straight line can be drawn from any point to any other point.

A straight line can be extended indefinitely in any direction.

It is possible to describe a circle with any centre and radius.

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, then the two straight lines, if produced indefinitely, will meet on that side on which the angles are less than two right angles.

With these basic assumptions, Euclid went on to prove propositions (theorems) about geometrical figures. Euclid's system of geometry was regarded as logically sound for 2000 years, although in fact it contained many concealed assumptions. In 1899, Hilbert, in Grundlagen der Geometrie (Foundations of Geometry), recast Euclidean geometry using three undefined entities (point, line, and plane). He introduced 28 assumptions, known as Hilbert's axioms. See also non-Euclidean geometry; parallel postulate.

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