Axiom:Euclid's Postulates

These are the axioms of standard Euclidean Geometry.

They appear at the start of Book I of The Elements by Euclid.

Note that while these are the only axioms that Euclid explicitly uses, he implicitly uses others such as Pasch's Axiom.

Euclid's First Postulate

A straight line segment can be drawn joining any two points.

Euclid's Second Postulate

Any straight line segment can be extended indefinitely to form a straight line.

Euclid's Third Postulate

Given any line segment, a circle can be drawn using the segment as the radius with one endpoint as the center.

Euclid's Fourth Postulate

All right angles are congruent.

Euclid's Fifth Postulate

If a straight line falling on two straight lines make 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 are the angles less than the two right angles.

Euclid's Common Notions

This is a set of axiomatic statements that appear at the start of Book I of The Elements by Euclid.

1. Things which are equal to the same thing are also equal to each other.
2. If equals are added to equals, the wholes are equal.
3. If equals are subtracted from equals, the remainders are equal.
4. Things which coincide with one another are equal to one another.
5. The whole is greater than the part.