Excerpt from An Elementary Treatise on Cross-Ratio Geometry: With Historical Notes

The development of the theory of cross-ratio is due, quite independently of each other, to Mobius, Der barycentrische Calcul, 1827, and to Chasles, Apercu Historique, 1829 - 1837, followed by the Geometric Supérieure, 1852, where the subject is treated very fully as regards the point and straight line, its application to the conic being given in the Traite des Sections Coniques, 1865. Some employment of its principles is met with in the various treatises on what is sometimes called Modern Geometry which have subsequently appeared, but as far as I am aware there is no English text-book exclusively devoted to it.

The power of the method of cross-ratio, as an instrument of analysis, it is not easy to over-rate. In the facility with which it deals alike with the range and pencil, with the points and line at infinity, with questions relating to concurrency and collinearity, loci and envelopes, it can compare not unfavourably with the methods of analytical geometry, and in those questions to which it is specially applicable, the steps necessary to establish any result are few in number, and are mostly of the same character, dealing as a rule with the homography of certain ranges or pencils, with the additional advantage that the geometrical meaning of each step is in general obvious.

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.