Page:First six books of the elements of Euclid 1847 Byrne.djvu/30

xxvi the use of which is permitted in Euclid, or plain Geometry. To declare this restriction is the object of the postulates. The Axioms of geometry are certain general propositions, the truth of which is taken to be self-evident and incapable of being established by demonstration.

Propositions are those results which are obtained in geometry by a process of reasoning. There are two species of propositions in geometry, problems and theorems.

A Problem is a proposition in which something is proposed to be done; as a line to be drawn under some given conditions, a circle to be described, some figure to be constructed, &c. The solution of the problem consists in showing how the thing required may be done by the aid of the rule or straight-edge and compasses. The demonstration consists in proving that the process indicated in the solution really attains the required end.

A Theorem is a proposition in which the truth of some principle is asserted. This principle must be deduced from the axioms and definitions, or other truths previously and independently established. To show this is the object of demonstration.

A Problem is analogous to a postulate.

A Theorem resembles an axiom. A Postulate is a problem, the solution of which is assumed.

An Axiom is a theorem, the truth of which is granted without demonstration.

A Corollary is an inference deduced immediately from a proposition.

A Scholium is a note or observation on a proposition not containing an inference of sufficient importance to entitle it to the name of a corollary.

A Lemma is a proposition merely introduced for the purpose of establishing some more important proposition.