Page:Popular Science Monthly Volume 68.djvu/26

 22 verbs, are conjunctive of elements, and correspond to the notion relation. You observe that no formal definition is here made of the words element and relation. I simply try to call up a distinction which I suppose to exist in the reader's mind.

The postulates and axioms of Euclid are so little to be distinguished from each other that in various editions some of the postulates are put among the axioms. The axioms (common notions) were regarded by Euclid's editors and the world at large, if not by Euclid himself, as a list of fundamental truths without granting which no reasoning process is possible. It was nearly as great a heresy in the middle ages to deny Euclid's axioms as to contradict the Bible. Without emphasizing further the historical fact that the axioms were regarded as necessary a priori truth, nor the fact that this belief is now largely outgrown, I wish to call attention to a mathematically more important feature. If the axioms are necessarily true, and if they are to be used in proving all things else, they themselves are not capable of demonstration. For mathematical purposes, the axioms are a set of unproved propositions. Out of Euclid's definitions and axioms we therefore select for emphasis the presence of

The postulates of Euclid are as follows. Let it be granted,

His axioms state: