AXIOX, a Greek word meaning a demand or assumption, is commonly used to signify a general proposition, which the understanding recognizes as true, as soon as the import of the words conveying it is apprehended. Such a proposition is therefore known directly, and does not need to be deduced from any other. Of this kind, for example, are all propositions whose predicate is a property essential to our notion of the subject. Every rational science requires such fundamental propositions, from which all the truths composing it arc derived; the whole of geometry, for instance, rests on, compara tively, a very few axioms. Whether there is, for the whole of human knowledge, any single, absolutely first A., from which all else that is known may be deduced, is a ques tion that has given rise to much disputation: but the fact, that human knowledge may have various starting-points, answers it in the negative. Mathematicians use the word A. to denote those propositions which they must assume as known from some other source than deductive reasoning, and employ in proving all the other truths of the science. The rigor of method requires that no more be assumed than are absolutely
necessary. Every self-evident proposition, therefore, is not an A. in this sense, though. of course, it is desirable that every A. be self-evident; thus, Euclid rests the whole of geometry on 15 assumptions, but he proves propositions that are at ]east as self-evident as some that he takes for granted. That "any two sides of a triangle are greater than the third," is as self-evident as that "all right angles are equal to one another," and much more So than his assumption about parallels, which, it has been remarked, is neither self-evident nor even easily made evident. See PARALLELS. Euclid's assump tions are divided into 3 "postulates" or demands, and 12 "common notions "—the term A. Is of later introduction. The distinction between axioms and postulates is usually stated in this way: an A. is "a theorem granted without demonstration ;" a postulate. is " a, problem granted without construction "—as, to draw a straight lino between two given points.