Excerpt from A Treatise on the Circle and the Sphere Every beginner in the science of geometry knows that the circle and the sphere have always played a central rôle, yet few people realize that the reasons for this are many and various. Attention was first called to these figures by their mechanical simplicity and importance, and the fortunate position thus won was further strengthened by the Euclidean tradition of limiting geometry, on the constructive side, to those operations which can be carried out with the aid of naught but ruler and compass. Yet these facts are far from sufficient to account for the commanding position which the circle and the sphere occupy to-day. To begin with, there would seem no a priori reason why those curves which are the simplest from the mechanical point of view should have the greatest wealth of beautiful properties. Had Euclid started, not with the usual parallel postulate, but with the different assumption either of Lobachevski or Riemann, he would have been unable to prove that all angles inscribed in the same circular arc are equal, and a large proportion of our best elementary theorems about the circle would have been lacking. Again, there is no a priori reason why a curve with attractive geometric properties should be blessed with a peculiarly simple cartesian equation; the cycloid is particularly unmanageable in pure cartesian form. The circle and sphere have simple equations and depend respectively on four and five independent homogeneous parameters. Thus, the geometry of circles is closely related to the projective geometry of three-dimensional space, while the totality of spheres gives our best example of a four-dimensional projective continuum. 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.

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This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work is in the "public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.