(2) Statement without proof of some leading methods and results, so as to give a bird's-eye view of the subject. (3) A small number of references indicating what the reader may profitably take up after he has mastered the contents of the monograph." Both the plan itself, and the invitation to act as author, were most cordially received; work on the monographs was promptly begun, has been carried through substantially as planned, and the results are presented herewith. The manuscripts have, whenever feasible, been read carefully by at least one collaborator other than myself, and in consequence various questions and suggestions have been submitted to the authors and acted upon by them. Each author, however, retains sole responsibility for his monograph as it now appears. No attempt has been made to secure uniformity in style of treatment; each monograph is an independent unit, that can be read without reference to the others. The amount of technical mathematical knowledge that is presupposed on the part of the reader varies with the different subjects. A large part of the book presupposes only knowledge of elementary geometry and algebra, together with a certain measure of mathematical maturity. On the other hand, there is much that will repay careful and detailed study by advanced students. So far as the subject-matter permits, the less difficult topics are taken up first in each monograph. J. W. A. YOUNG. CONTENTS I. THE FOUNDATIONS OF GEOMETRY. Introduction—The Assumption of Order—Order on a Line— The Triangle and the Plane-Regions in a Plane-Congruence of Introduction-Simple Elements in Geometry-The Principle of Duality-Principle of Continuity-Points at Infinity-Funda- mental Theorem-Metric Properties-Anharmonic Ratios- Elementary Geometric Forms-Correlation of Elementary Forms-Curves and Sheaves of Rays of the Second Order- Introduction-Parallel Lines-The Euclidean Assumption— The Lobachevskian Assumption-The Riemannian Assumption -The Sum of the Angles of a Triangle-Areas-Non-Euclidean Trigonometry-Non-Euclidean Analytic Geometry-Representa- tion of the Lobachevskian Geometry on a Euclidean Plane— Introduction-The Addition of Angles and the Multiplication of Distances-The Abstract Theory of these Operations- Geometric Example of the Algebra of Complex Quantities: The System of Points in the Plane-The Abstract Theory of the Algebra of Complex Quantities-Appendix: Other Examples of the Algebra of Complex Quantities-Geometric Proof that General Introduction-Historical Sketch and Definitions- Equations with One Unknown and with Literal Coefficients— Equations with One Unknown and with Numerical Coefficients -Simultaneous Equations-A Few References. Introduction - Factors - Diophantine Equations Con- Introduction Analytic Criterion for Constructibility-Graph- 'gon-Regular Polygon of 17 Sides-Construction of the Regular Polygon of 17 Sides-Gauss's Theory of Regular Polygons- Primitive Roots of Unity-Gauss's Lemma-Irreducibility of The Nature of the Problem-The History of the Problem— The Transcendence of e-The Transcendence of . * A fuller Table of Contents precedes the Monograph itself. |