# Elements of Geometry: With Exercises for Students and an Introduction to Modern Geometry

Fb&c Limited, Jun 15, 2015 - Mathematics - 376 pages
Excerpt from Elements of Geometry: With Exercises for Students and an Introduction to Modern Geometry

A new treatise on Geometry, to be of sufficient merit to claim attention, must be both conservative and progressive. It should lay firm hold on the past, embody the present state of the science, and anticipate future developments. A work claiming to be wholly new might, perhaps with justice, be at once discarded as worthless; while one containing no improvements could not justify its own existence.

The geometrical objects, - points, lines, surfaces, solids, and angles, - constitute the subject-matter of the science; the definitions are the tests by which these objects are discriminated and their classification determined; the axioms are the warrants for the steps taken in the course of demonstration; the postulates justify the assumption of magnitudes having any position, form, and extent.

The logical principles which underlie the demonstrations of this volume have been carefully discriminated and illustrated. The discussion of the axioms and postulates is the result of research, and intent and prolonged thought. That fundamental principles have been reached is manifest from their underivability, and the simplicity of the deduction of the ordinary so-called axioms from them as corollaries.

Mr. Bain has observed of the principle, If A be greater than B, and B greater than C, much more is A greater than C: "If it can not be deductively inferred from the proper axioms, it will have to be received as a third axiom." Not only can this principle be inferred (23, 20), but also Mr. Bain's so-called proper axioms (23, 3, 6).