Now it is all right purposely to avoid knowing what it is that we are talking about, so that the names of these things shall constitute mere blank forms which may be filled in, when and if we wish, by the names of any things in the universe of which our "axioms" turn out to be true. But what about these axioms themselves? When we lay them down in ignorance of the identity of the elements to which they may eventually apply, they cannot by any possibility be "self-evident." We may, at pleasure, accept as self-evident a statement about points and lines and planes; or one about electrons, centimeters and seconds; or one about integers, fractions, and irrational numbers; or one about any other concrete thing or things whatever. But we cannot accept as self-evident a statement about chings, changs and chungs. So we must base our "axioms" on some other ground than this; and our modern geometer has his ground ready and waiting. He accepts his axioms on the ground that it pleases him to do so. To avoid all suggestion that they are supposed to be self-evident, or even necessarily true, he drops the term "axiom" and substitutes for it the more color-less word "postulate." A postulate is merely something that we agreed to accept, for the time being, as a basis of further argument. If it turns out to be true, or if we can find circumstances under which and elements to which it applies, any conclusions which we deduce from it by trustworthy processes are valid within the same limitations. And the propositions which tell us that, if our postulates are true, such and such conclusions are true--they, too, are valid, but without any reservation at all! Perhaps an illustration of just what this means will not be out of place. Let it be admitted, as a postulate, that $7+20$ is greater, by 1, than $7+19$. Let us then consider the statement: "If $7 + 19 = 65$, then $7 + 20 = 66$." We know--at least we are quite certain--that $7 + 19$ is not equal to 65, if by "7" and "19" and "65" we mean what you think we mean. We are equally sure, on the same grounds, that $7 + 20$ is not equal to 66. But, under the one assumption that we have permitted ourselves, it is unquestionable that if $7 + 19$ were equal to 65, then $7 + 20$ certainly would be equal to 66. So, while the conclusion of the proposition which I have put in quotation marks is altogether false, the proposition itself, under our assumption, is entirely true. I have taken an illustration designed to be striking rather than to possess scientific interest; I could just as easily have shown a true proposition leading to a false conclusion, but of such sort that it would be of decided scientific interest as telling us one of the consequences of a certain assumption. WHAT MAY WE TAKE FOR GRANTED?