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Excerpt from Essays on Mathematical Education It is first necessary to lay down definitions as precise as may be possible, of the terms axiom, postulate, proof. It is not implied that these definitions should be insisted on, or the terms used, in elementary teaching; nothing could be more likely to lead to failure. But a full comprehension of each is essential to every teacher of mathematics, and is too often lacking in current usage. An axiom, or common notion in Euclid's language, is a statement which is true of all processes of thought, whatever be the subject matter under discussion. Thus the following are axioms: If A is identical with B, and C is different from B, then C is different from A If E is a necessary consequence of A, and also C of B, then C is a necessary consequence of A. But Two and two make four and The straight line is the shortest distance between two points are not axioms, although they may be considered no less obvious. A statement is not an axiom because it is obvious, but because it concerns uni versal forms of thought, and not a particular subject matter such as arithmetic, geometry, and the like. A postulate is a statement which is assumed concerning a particular subject matter; for example, The whole is greater than a part (subject matter, finite aggregates); All right angles are equal (subject matter, Euclidean space). It is essential to Observe that, whereas an axiom is an axiom once for all, a postulate in one treatment of a science may not be a postulate in another. In Euclid's development of geometry, the statement that any two sides of a triangle are together greater than the third Side is not a postulate, because it is deduced from other statements (postulates) which are avowedly assumed; but in many current developments it is adopted at once, without refer ence to other statements, and is therefore a postulate in such cases. To use an unconventional but expressive term, postulates are jumping-off places for the logical explo ration of a subject. Their number and nature are immaterial; they may be readily acceptable, or difficult of credence. Their one function is to supply a basis for reasoning. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.

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