Euclid, Elements
Thomas L. Heath, Sir Thomas Little Heath, Ed.

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“and readily grasped, I mean both the postulate and the axiom; but the postulate bids us contrive and find some subject-matter (ὕλη) to exhibit a property simple and easily grasped, while the axiom bids us assert some essential attribute which is self-evident to the learner, just as is the fact that fire is hot, or any of the most obvious things¹.”

Again, says Proclus, “some claim that all these things are alike postulates, in the same way as some maintain that all things that are sought are problems. For Archimedes begins his first book on Inequilibrium² with the remark ’I postulate that equal weights at equal distances are in equilibrium,’ though one would rather call this an axiom. Others call them all axioms in the same way as some regard as theorems everything that requires demonstration³.”

“Others again will say that postulates are peculiar to geometrical subject-matter, while axioms are common to all investigation which is concerned with quantity and magnitude. Thus it is the geometer who knows that all right angles are equal and how to produce in a straight line any limited straight line, whereas it is a common notion that things which are equal to the same thing are also equal to one another, and it is employed by the arithmetician and any scientific person who adapts the general statement to his own subject⁴.”

The third view of the distinction between a postulate and an axiom is that of Aristotle above described⁵.

The difficulties in the way of reconciling Euclid's classification of postulates and axioms with any one of the three alternative views are next dwelt upon. If we accept the first view according to which an axiom has reference to something known, and a postulate to something done, then the 4th postulate (that all right angles are equal) is not a postulate; neither is the 5th which states that, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the straight lines, if produced indefinitely, will meet on that side on which are the angles less than two right angles. On the second view, the assumption that two straight lines cannot enclose a space, “which even now,” says Proclus, “some add as an axiom,” and which is peculiar to the subject-matter of geometry, like the fact that all right angles are equal, is not an axiom. According to the third (Aristotelian) view, “everything which is confirmed (πιστοῦται) by a sort of demonstration ”