The Project Gutenberg EBook of The Elements of non-Euclidean Geometry, by Julian Lowell Coolidge This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: The Elements of non-Euclidean Geometry Author: Julian Lowell Coolidge Release Date: August 20, 2008 [EBook #26373] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK NON-EUCLIDEAN GEOMETRY *** Produced by Joshua Hutchinson, David Starner, Keith Edkins and the Online Distributed Proofreading Team at http://www.pgdp.net THE ELEMENTS OF NON-EUCLIDEAN GEOMETRY BY JULIAN LOWELL COOLIDGE Ph.D. ASSISTANT PROFESSOR OF MATHEMATICS IN HARVARD UNIVERSITY OXFORD AT THE CLARENDON PRESS 1909 PREFACE The heroic age of non-euclidean geometry is passed. It is long since the days when Lobatchewsky timidly referred to his system as an ‘imaginary geometry’, and the new subject appeared as a dangerous lapse from the orthodox doctrine of Euclid. The attempt to prove the parallel axiom by means of the other usual assumptions is now seldom undertaken, and those who do undertake it, are considered in the class with circle-squarers and searchers for perpetual motion– sad by-products of the creative activity of modern science. In this, as in all other changes, there is subject both for rejoicing and regret. It is a satisfaction to a writer on non-euclidean geometry that he may proceed at once to his subject, without feeling any need to justify himself, or, at least, any more need than any other who adds to our supply of books. On the other hand, he will miss the stimulus that comes to one who feels that he is bringing out something entirely new and strange. The subject of non-euclidean geome- try is, to the mathematician, quite as well established as any other branch of mathematical science; and, in fact, it may lay claim to a decidedly more solid basis than some branches, such as the theory of assemblages, or the analysis situs. Recent books dealing with non-euclidean geometry fall naturally into two classes. In the one we find the works of Killing, Liebmann, and Manning,1 who wish to build up certain clearly conceived geometrical systems, and are careless of the details of the foundations on which all is to rest. In the other category are Hilbert, Vablen, Veronese, and the authors of a goodly number of articles on the foundations of geometry. These writers deal at length with the consistency, significance, and logical independence of their assumptions, but do not go very far towards raising a superstructure on any one of the foundations suggested. The present work is, in a measure, an attempt to unite the two tendencies. The author’s own interest, be it stated at the outset, lies mainly in the fruits, rather than in the roots; but the day is past when the matter of axioms may be dismissed with the remark that we ‘make all of Euclid’s assumptions except the one about parallels’. A subject like ours must be built up from explicitly stated assumptions, and nothing else. The author would have preferred, in the first chapters, to start from some system of axioms already published, had he been familiar with any that seemed to him suitable to establish simultaneously the euclidean and the principal non-euclidean systems in the way that he wished. The system of axioms here used is decidedly more cumbersome than some others, but leads to the desired goal. There are three natural approaches to non-euclidean geometry. (1) The elementary geometry of point, line, and distance. This method is developed in the opening chapters and is the most obvious. (2) Projective geometry, and the theory of transformation groups. This method is not taken up until Chapter XVIII, not because it is one whit less important than the first, but because it seemed better not to interrupt the natural course of the narrative 1Detailed references given later 1 by interpolating an alternative beginning. (3) Differential geometry, with the concepts of distance-element, extremal, and space constant. This method is explained in the last chapter, XIX. The author has imposed upon himself one or two very definite limitations. To begin with, he has not gone beyond three dimensions. This is because of his feeling that, at any rate in a first study of the subject, the gain in gener- ality obtained by studying the geometry of n-dimensions is more than offset by the loss of clearness and naturalness. Secondly, he has confined himself, al- most exclusively, to what may be called the ‘classical’ non-euclidean systems. These are much more closely allied to the euclidean system than are any oth- ers, and have by far the most historical importance. It is also evident that a system which gives a simple and clear interpretation of ternary and quaternary orthogonal substitutions, has a totally different sort of mathematical signifi- cance from, let us say, one whose points are determined by numerical values in a non-archimedian number system. Or again, a non-euclidean plane which may be interpreted as a surface of constant total curvature, has a more lasting geometrical importance than a non-desarguian plane that cannot form part of a three-dimensional space. The majority of material in the present work is, naturally, old. A reader, new to the subject, may find it wiser at the first reading to omit Chapters X, XV, XVI, XVIII, and XIX. On the other hand, a reader already somewhat familiar with non-euclidean geometry, may find his greatest interest in Chap- ters X and XVI, which contain the substance of a number of recent papers on the extraordinary line geometry of non-euclidean space. Mention may also be made of Chapter XIV which contains a number of neat formulae relative to areas and volumes published many years ago by Professor d’Ovidio, which are not, perhaps, very familiar to English-speaking readers, and Chapter XIII, where Staude’s string construction of the ellipsoid is extended to non-euclidean space. It is hoped that the introduction to non-euclidean differential geometry in Chapter XV may prove to be more comprehensive than that of Darboux, and more comprehensible than that of Bianchi. The author takes this opportunity to thank his colleague, Assistant-Professor Whittemore, who has read in manuscript Chapters XV and XIX. He would also offer affectionate thanks to his former teachers, Professor Eduard Study of Bonn and Professor Corrado Segre of Turin, and all others who have aided and encouraged (or shall we say abetted?) him in the present work. 2 TABLE OF CONTENTS CHAPTER I FOUNDATION FOR METRICAL GEOMETRY IN A LIMITED REGION Fundamental assumptions and definitions . 9 Sums and differences of distances . 10 Serial arrangement of points on a line . 11 Simple descriptive properties of plane and space . 14 CHAPTER II CONGRUENT TRANSFORMATIONS Axiom of continuity . 17 Division of distances . 17 Measure of distance . 19 Axiom of congruent transformations . 21 Definition of angles, their properties . 22 Comparison of triangles . 23 Side of a triangle not greater than sum of other two . 26 Comparison and measurement of angles . 28 Nature of the congruent group . 29 Definition of dihedral angles, their properties . 29 CHAPTER III THE THREE HYPOTHESES A variable angle is a continuous function of a variable distance . 31 Saccheri’s theorem for isosceles birectangular quadrilaterals . 33 The existence of one rectangle implies the existence of an infinite number . 34 Three assumptions as to the sum of the angles of a right triangle . 34 Three assumptions as to the sum of the angles of any triangle, their categorical nature . 35 Definition of the euclidean, hyperbolic, and elliptic hypotheses . 35 Geometry in the infinitesimal domain obeys the euclidean hypothesis . 37 CHAPTER IV THE INTRODUCTION OF TRIGONOMETRIC FORMULAE Limit of ratio of opposite sides of diminishing isosceles quadrilateral . 38 Continuity of the resulting function . 40 Its functional equation and solution . 40 Functional equation for the cosine of an angle . 43 3 Non-euclidean form for the pythagorean theorem . 43 Trigonometric formulae for right and oblique triangles . 45 CHAPTER V ANALYTIC FORMULAE Directed distances . 49 Group of translations of a line . 49 Positive and negative directed distances . 50 Coordinates of a point on a line . 50 Coordinates of a point in a plane . 50 Finite and infinitesimal distance formulae, the non-euclidean plane as a sur- face of constant Gaussian curvature . 51 Equation connecting direction cosines of a line . 53 Coordinates of a point in space . 54 Congruent transformations and orthogonal substitutions . 55 Fundamental formulae for distance and angle . 56 CHAPTER VI CONSISTENCY AND SIGNIFICANCE OF THE AXIOMS Examples of geometries satisfying the assumptions made . 58 Relative independence of the axioms . 59 CHAPTER VII THE GEOMETRIC AND ANALYTIC EXTENSION OF SPACE Possibility of extending a segment by a definite amount in the euclidean and hyperbolic cases . 62 Euclidean and hyperbolic space . 62 Contradiction arising under the elliptic hypothesis . 62 New assumptions identical with the old for limited region, but permitting the extension of every segment by a definite amount . 63 Last axiom, free mobility of the whole system . 64 One to one correspondence of point and coordinate set in euclidean and hy- perbolic cases . 65 Ambiguity in the elliptic case giving rise to elliptic and spherical geometry 65 Ideal elements, extension of all spaces to be real continua . 67 Imaginary elements geometrically defined, extension of all spaces to be perfect continua in the complex domain . 68 Cayleyan Absolute, new form for the definition of distance .