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Excerpt from The Elements of Non-Euclidean Geometry The present work is an extension and elaboration of a course of lectures on non-euclidean Geometry which I delivered at the Colloquium held under the auspices of the Edinburgh Mathematical Society in August, 1913. 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.
Excerpt from The Elements of Non-Euclidean Plane Geometry and Trigonometry IN this little book I have attempted to treat the Elements of non-euclidean Plane Geometry and Trigonometry in such a way as to prove useful to teachers of Elementary Geometry in schools and colleges. Recent changes in the teaching of Geometry in England and America have made it more than ever necessary that the teachers should have some knowledge of the hypotheses on which Euclidean Geometry is built, and especially of that hypothesis on which Euclid's Theory of Parallels rests. The historical treatment of the Theory of Parallels leads naturally to a discussion of the non-euclidean Geometries and it is only when the logical possibility of these non-euclidean Geometries is properly understood that a teacher is entitled to form an independent Opinion upon the teaching of Elementary Geometry. 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.
The Elements of non-Euclidean Geometry: Large Print By Julian Lowell Coolidge Chapters Include: Foundation For Metrical Geometry In A Limited Region - Congruent Transformations - The Three Hypotheses - The Introduction Of Trigonometric Formulae - Analytic Formulae - Consistency And Significance Of The Axioms - The Geometric And Analytic Extension Of Space - The Groups Of Congruent Transformations - Point, Line, And Plane Treated Analytically - The Higher Line Geometry - The Circle And The Sphere - Conic Sections - Quadric Surfaces - Areas And Volumes - Introduction To Differential Geometry - Differential Line-Geometry - Multiply Connected Spaces - The Projective Basis Of Non-Euclidean Geometry - The Differential Basis For Euclidean And Non-Euclidean Geometry - Comprehensive Index We are delighted to publish this classic book as part of our extensive Classic Library collection. Many of the books in our collection have been out of print for decades, and therefore have not been accessible to the general public. The aim of our publishing program is to facilitate rapid access to this vast reservoir of literature, and our view is that this is a significant literary work, which deserves to be brought back into print after many decades. The contents of the vast majority of titles in the Classic Library have been scanned from the original works. To ensure a high quality product, each title has been meticulously hand curated by our staff. Our philosophy has been guided by a desire to provide the reader with a book that is as close as possible to ownership of the original work. We hope that you will enjoy this wonderful classic work, and that for you it becomes an enriching experience.
This book provides an introduction to non-Euclidean geometry, a branch of mathematics that deviates from classical Euclidean geometry in significant ways. Coolidge's approach is accessible and engaging, and provides a valuable entry point to this fascinating field. 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.
In this book Dr. Coolidge explains non-Euclidean geometry which consists of two geometries based on axioms closely related to those specifying Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is replaced with an alternative one. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non-Euclidean geometry. The essential difference between the metric geometries is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line l and a point A, which is not on l, there is exactly one line through A that does not intersect l. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting l, while in elliptic geometry, any line through A intersects l. Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line: In Euclidean geometry, the lines remain at a constant distance from each other (meaning that a line drawn perpendicular to one line at any point will intersect the other line and the length of the line segment joining the points of intersection remains constant) and are known as parallels. In hyperbolic geometry, they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels. In elliptic geometry, the lines "curve toward" each other and intersect.