Euclid index

An introduction to Euclid - Axioms and Postulates

Index --- introduction --- definitions --- axioms and postulates --- propositions --- other

Euclid describes Axioms and Postulates, given below. The Postulates talk about straight lines, circles, right angles and parallel lines (these have been defined already, but here is more information about them). The Axioms are about relationships; what does equal mean, how do you add or subtract things, and so on. Remember that we are talking about lines and angles, not numbers, so adding and subtracting need to be thought about.

In modern mathematics, the first principles of any formal deductive system are 'axioms', so perhaps the Postulates, Axioms and Definitions should all be considered axioms.

Euclid's Axioms (or Common Notions)
Axiom 1 - Things which equal the same thing also equal one another.
Axiom 2 - If equals are added to equals, then the wholes are equal.
Axiom 3 - If equals are subtracted from equals, then the remainders are equal.
Axiom 4 - Things which coincide with one another equal one another.
Axiom 5 - The whole is greater than the part.

Euclid's PostulatesComments
Let the following be postulated: (or assumed)
Postulate 1 - To draw a straight line from any point to any point.You use a straight rule for this. This is a straight edge, like a ruler, but it has no measurements marked on it.
Postulate 2 - To produce a finite straight line continuously in a straight line.If you have a straight line, you can make it longer (and still straight). This is called extending the line.
Postulate 3 - To describe a circle with any centre and radius. You use a pair of compasses for this. Interestingly enough, Euclid has not defined the radius although he has defined the diameter. The radius is half the diameter, of course.
Postulate 4 - That all right angles equal one another.
Postulate 5 - That, 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 are the angles less than the two right angles. This is the tricky one! You can create perfectly valid geometries by keeping all the rest of Euclid's definitions, postulates and axioms, but tweaking this one slightly. These are called non-Euclidean geometries. They even exist in the real world. If you draw 'straight' lines on the surface of a sphere, the two lines can met on both sides. On a hyperbolic surface (saddle-shaped), they may not meet at all. The definitions of 'straight' lines in these cases is that they are the shortest distance between two points, rather than that you can draw them with a ruler!