The Elements of Non-Euclidean Geometry is a book written by Julian Lowell Coolidge and published in 1909. The book is a comprehensive introduction to the field of non-Euclidean geometry, which is a branch of geometry that explores the properties of geometric figures and spaces that are not consistent with Euclid's axioms. The book begins with an overview of the history of non-Euclidean geometry, including the work of mathematicians such as Lobachevsky and Bolyai. The author then delves into the basic concepts of non-Euclidean geometry, including hyperbolic and elliptic geometry, and provides a detailed explanation of the differences between these two types of geometry. The book also covers a variety of topics related to non-Euclidean geometry, including the geometry of the sphere, the geometry of the hyperbolic plane, and the geometry of n-dimensional spaces. The author provides numerous examples and illustrations throughout the book to help readers understand the concepts being discussed.Overall, The Elements of Non-Euclidean Geometry is a valuable resource for anyone interested in the field of geometry, including students, teachers, and mathematicians. The book is written in a clear and concise style, making it accessible to readers with a range of mathematical backgrounds.This scarce antiquarian book is a facsimile reprint of the old original and may contain some imperfections such as library marks and notations. Because we believe this work is culturally important, we have made it available as part of our commitment for protecting, preserving, and promoting the world's literature in affordable, high quality, modern editions, that are true to their original work.