We would here express our grateful acknowledgments to all who have aided in the preparation of this book; to Miss Elizabeth H. Richards, whose successful experience in fitting students for college in Plane Geometry has rendered her criticisms and suggestions most valuable; and to our colleagues, Messrs. W. M. Strong and Joseph Bowden, Jr. Mr. Strong has selected, for the most part, the exercises at the end of the book, and Mr. Bowden has examined critically the references and proof-sheets of the book. ANDREW W. PHILLIPS, YALE UNIVERSITY. SPECIAL TERMS An axiom is a truth assumed as self-evident. A theorem is a truth which becomes evident by a train of reasoning called a demonstration. A theorem consists of two parts, the hypothesis, that which is given, and the conclusion, that which is to be proved. A problem is a question proposed which requires a solution. A proposition is a general term for either a theorem or problem. One theorem is the converse of another when the conclusion of the first is made the hypothesis of the second, and the hypothesis of the first is made the conclusion of the second. The converse of a truth is not always true. Thus, "If a man is in New York City he is in New York State," is true; but the converse, “If a man is in New York State he is in New York City," is not necessarily true. When one theorem is easily deduced from another the first is sometimes called a corollary of the second. A theorem used merely to prepare the way for another theorem is sometimes called a lemma. |