From the PREFACE TO THE THIRD EDITION. Researches in the non-Euclidean geometries of Bolyai-Lobachévski and Riemann, while renewing the traditional admiration for Euclid, yet emphasize the advantages of a comparative study of pure spherics and plane geometry before similar figures. Putting pure spherics as Book II is thus a beginning toward comparative geometry. Again, the non-Euclidean geometry has given the key to the artificiality in Euclid's order of propositions. But it is approaching metric geometry from pure projective geometry which decides in favor of symmetry as a guiding principle. George Bruce Halsted.
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