Page:Popular Science Monthly Volume 68.djvu/26

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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

1. Undefined terms relations.
2. Undemonstrated propositions.

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

1. That a straight line may be drawn from any one point to any other point.
2. That a terminated straight line may be produced to any length in a straight line.
3. And that a circle may be described from any center, at any distance from that center.

His axioms state:

1. Things which are equal to the same thing are equal to one another.
2. If equals be added to equals the wholes are equal.
3. If equals be taken from equals the remainders are equal.
4. If equals be added to unequals the wholes are unequal.
5. If equals be taken from unequals the remainders are unequal.
6. Things which are double or the same thing are equal to one another.
7. Things which are halves of the same thing are equal to one another.
8. Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another.
9. The whole is greater than its part.
10. Two straight lines can not enclose a space.
11. All right angles are equal to one another.
12. If a straight line meet two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles, these straight lines, being continually produced, shall at length meet on that side on which are the angles which are less than two right angles.