The postulates (or axioms) Euclid set out in his famous text, the Elements, are:

1. 1. A straight line may be drawn from any point to any other point.

2. 2. A straight line segment can be extended indefinitely at either end.

3. 3. A circle may be described with any centre and any radius.

4. 4. All right angles are equal.

5. 5. If a straight line (the transversal) meets two other straight lines so that the sum of the two interior angles on one side of the transversal is less than two right angles, then the straight lines, extended indefinitely if necessary, will meet on that side of the transversal. Axiom 5 is the famous parallel postulate.

He also stated definitions of geometrical entities like points and lines, and five ‘common notions’, which are:

1. 1. Things which are equal to the same thing are also equal to one another.

2. 2. If equals are added to equals, the sums are also equal.

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

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

5. 5. The whole is greater than the part.