Euclid’s Postulates are slightly more complicated than his axioms, but serve a similar purpose.

These axioms and postulates, intended to be self-evident, are sufficient to prove the many propositions he made in his *Elements*.

Postulates, like axioms, are statements made without proofs.

The postulates might be considered slightly “weaker” than axioms: if some results derived from the axioms and postulates are inconsistent, mathematicians would sooner change postulates than axioms.

It happens that Euclid oriented his postulates toward geometry, where the axioms are more general in nature.

## Euclid’s Five Postulates

According to Norman Swartz’s quotations from Sir Thomas Heath’s, *The Elements of Euclid:*

*Let the following be postulated, or assumed to be true.*

*One is able to draw a straight line (segment) from any point to any point.**One is able to produce a finite straight line (segment) continuously in a straight line.**One is able to describe a circle with any centre and distance.**All right angles are equal to one another.**If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, then the straight lines, if produced indefinitely, will meet on that side on which the angles are less that two right angles.*

## What Were Euclid’s Tools?

The first three postulates deal with constructing geometric figures. Euclid would have restricted himself to two specific tools: a compass and straight-edge, or a ruler with no markings on it.

This image shows two compasses. These tools allow you to draw circles, and also to copy a line segment’s length by putting the tips at each endpoint.

A straight-edge is simple a ruler, or “rule,” but without inch or centimetre markings. With a straight-edge, you can draw a straight line, but cannot measure anything. Euclid’s implicit tools are “paper and pencil,” since you need some method of marking a flat surface to construct, measure, and calculate geometrical figures.

Anshuman says

It was really helpfull.

5th postulate was hard to understant

Constantine Roussos says

I think we can and should remove the word “infinite” from explanations in plane Geometry.. For example, instead of saying “they will never meet even if extended infinitely far” we should say “they will never meet no matter how far they are extended”. The concept of infinitely is not needed for plane geometry and its definition (if it exists) is beyond the scope of plane geometry.

dfggf says

that is really clever

ravi raj says

i think 5 postulate is not easy to understand

rita patel says

i think that greg is absoulutely correct , no doubt!

Greg Grant says

I don’t think what you’re saying is correct regarding Postulate 3. If Euclid allowed such an operation then he would not have to prove Proposition 2 in such a complicated manner. Prop 2 states “To place a straight line equal to a given straight line with one end at a given point.” If he could just measure the length of the line and draw a circle of that radius around the point then this would be trivial, but he doesn’t do it that way, he in fact invokes a quite complicated argument to prove this Proposition.

Therefore I think the right way to interpret Postulate 3 is that given any two points, one can draw a circle with center equal to one of the points and passing through the other point. Prior to proving a length can be duplicated, it is exactly this way that he draws circles – there are always two points.

rita patel says

yes, greg is right!!!!!