Axiom
2 of the 6 encyclopedias on this shelf carry an entry for Axiom. Both are reproduced below, so you can see where they agree and where they differ.
Collier's New Encyclopedia (1921)
a Greek word meaning a decision 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. Mathematicians used the word axiom 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 axiom in this sense, though, of course, it is desirable that every axiom be self-evident; thus, Euclid rests the whole of geometry on 15 assumptions, but he proves propositions that are at least as self-evident as some that he takes for granted. Euclid's assumptions are divided into three postulates, or demands, and 12 common notions; the term axiom is of later introduction. The distinction between axioms and postulates is usually stated in this way: an axiom is "a theorem granted without demonstration;" a postulate is "a problem granted without construction"-as, to draw a straight line between two given points.
Aiton's Encyclopedia (1910)
a truth requiring no proof. The term is sometimes applied to any important and generally accepted truth, as, in political economy , "cheap money drives good money out of circulation"; in logic, "he who admits a principle admits its consequence"; in natural history, "mountain ranges restrict and direct migrations"; in geography, "an increased altitude lowers the temperature"; in sociology, "character is affected by associations," etc. The term is restricted more properly, however, to self-evident mathematical truths, as, for instance: "The whole is greater than any of its parts"; "Things which are equal to the same thing are equal to each other"; "Two straight lines cannot enclose a space," etc. Euclid recognized fifteen geometrical axioms.
What Has Changed in the Last Century
Aiton's 1910 definition — "a truth requiring no proof" — is exactly the idea that twentieth-century mathematics abandoned. David Hilbert's Grundlagen der Geometrie (1899) had already recast axioms as arbitrary starting assumptions whose only requirement is consistency, not self-evidence, and the non-Euclidean geometries of Lobachevsky and Riemann had shown that Euclid's parallel postulate could be denied and still yield a coherent system — the sort of curved space Einstein used for general relativity in 1915. Then in 1931 Kurt Gödel proved that any consistent axiom system strong enough for ordinary arithmetic must leave some true statements unprovable within it, ending the hope that a well-chosen list of axioms could settle everything; Paul Cohen showed in 1963 that the continuum hypothesis is likewise undecidable from the standard axioms of set theory. One smaller slip: Aiton gives the economic maxim as "cheap money drives good money out of circulation," which reverses the usual statement of Gresham's law — bad money drives out good.
Written for this edition, 2026.