The Elements of non-Euclidean Geometry

Julian Lowell Coolidge

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• Publié le: 08 décembre 2010
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• Langue: English
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Title: The Elements of

nonEuclidean Geometry

Author: Julian Lowell Coolidge

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Language: English

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NONEUCLIDEAN GEOMETRY ***

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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 1 classes. In the one we find the works of Killing, Liebmann, and Manning, 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 1 Detailed 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 ofn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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Division of distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measure of distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axiom of congruent transformations . . . . . . . . . . . . . . . . . . . . . . Definition of angles, their properties . . . . . . . . . . . . . . . . . . . . . . Comparison of triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Side of a triangle not greater than sum of other two . . . . . . . . . . . . . Comparison and measurement of angles . . . . . . . . . . . . . . . . . . . . Nature of the congruent group . . . . . . . . . . . . . . . . . . . . . . . . . Definition of dihedral angles, their properties . . . . . . . . . . . . . . . . .

CHAPTER III THE THREE HYPOTHESES A variable angle is a continuous function of a variable distance . . . . . . . Saccheri’s theorem for isosceles birectangular quadrilaterals . . . . . . . . . The existence of one rectangle implies the existence of an infinite number . Three assumptions as to the sum of the angles of a right triangle . . . . . . Three assumptions as to the sum of the angles of any triangle, their categorical nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition of the euclidean, hyperbolic, and elliptic hypotheses . . . . . . . Geometry in the infinitesimal domain obeys the euclidean hypothesis . . . .

CHAPTER IV THE INTRODUCTION OF TRIGONOMETRIC FORMULAE Limit of ratio of opposite sides of diminishing isosceles quadrilateral . . . . Continuity of the resulting function . . . . . . . . . . . . . . . . . . . . . . Its functional equation and solution . . . . . . . . . . . . . . . . . . . . . . Functional equation for the cosine of an angle . . . . . . . . . . . . . . . . .

3

17 17 19 21 22 23 26 28 29 29

31 33 34 34

35 35 37

38 40 40 43

Noneuclidean form for the pythagorean theorem . . . . . . . . . . . . . . . Trigonometric formulae for right and oblique triangles . . . . . . . . . . . .

CHAPTER V ANALYTIC FORMULAE Directed distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Group of translations of a line . . . . . . . . . . . . . . . . . . . . . . . . . Positive and negative directed distances . . . . . . . . . . . . . . . . . . . . Coordinates of a point on a line . . . . . . . . . . . . . . . . . . . . . . . . Coordinates of a point in a plane . . . . . . . . . . . . . . . . . . . . . . . . Finite and infinitesimal distance formulae, the noneuclidean plane as a sur face of constant Gaussian curvature . . . . . . . . . . . . . . . . . Equation connecting direction cosines of a line . . . . . . . . . . . . . . . . Coordinates of a point in space . . . . . . . . . . . . . . . . . . . . . . . . . Congruent transformations and orthogonal substitutions . . . . . . . . . . . Fundamental formulae for distance and angle . . . . . . . . . . . . . . . . .

CHAPTER VI CONSISTENCY AND SIGNIFICANCE OF THE AXIOMS Examples of geometries satisfying the assumptions made . . . . . . . . . . Relative independence of the axioms . . . . . . . . . . . . . . . . . . . . . .

CHAPTER VII THE GEOMETRIC AND ANALYTIC EXTENSION OF SPACE Possibility of extending a segment by a definite amount in the euclidean and hyperbolic cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclidean and hyperbolic space . . . . . . . . . . . . . . . . . . . . . . . . . Contradiction arising under the elliptic hypothesis . . . . . . . . . . . . . . New assumptions identical with the old for limited region, but permitting the extension of every segment by a definite amount . . . . . . . . . . Last axiom, free mobility of the whole system . . . . . . . . . . . . . . . . . One to one correspondence of point and coordinate set in euclidean and hy perbolic cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ambiguity in the elliptic case giving rise to elliptic and spherical geometry Ideal elements, extension of all spaces to be real continua . . . . . . . . . . Imaginary elements geometrically defined, extension of all spaces to be perfect continua in the complex domain . . . . . . . . . . . . . . . . . . . Cayleyan Absolute, new form for the definition of distance . . . . . . . . . Extension of the distance concept to the complex domain . . . . . . . . . . Case where a straight line gives a maximum distance . . . . . . . . . . . . .

4

43 45

49 49 50 50 50

51 53 54 55 56

58 59

62 62 62

63 64

65 65 67

68 70 71 73

CHAPTER VIII THE GROUPS OF CONGRUENT TRANSFORMATIONS Congruent transformations of the straight line . . . . . . . . . . . . . . . . ,, ,, ,, hyperbolic plane . . . . . . . . . . . . . . ,, ,, ,, elliptic plane . . . . . . . . . . . . . . . . ,, ,, ,, euclidean plane . . . . . . . . . . . . . . ,, ,, ,, hyperbolic space . . . . . . . . . . . . . . ,, ,, ,, elliptic and spherical space . . . . . . . . Clifford parallels, or paratactic lines . . . . . . . . . . . . . . . . . . . . . . The groups of right and left translations . . . . . . . . . . . . . . . . . . . . Congruent transformations of euclidean space . . . . . . . . . . . . . . . . .

CHAPTER IX POINT, LINE, AND PLANE TREATED ANALYTICALLY Notable points of a triangle in the noneuclidean plane . . . . . . . . . . . . Analoga of the theorems of Menelaus and Ceva . . . . . . . . . . . . . . . . Formulae of the parallel angle . . . . . . . . . . . . . . . . . . . . . . . . . . Equations of parallels to a given line . . . . . . . . . . . . . . . . . . . . . . Notable points of a tetrahedron, and resulting desmic configurations . . . . Invariant formulae for distance and angle of skew lines in line coordinates . Criteria for parallelism and parataxy in line coordinates . . . . . . . . . . . Relative moment of two directed lines . . . . . . . . . . . . . . . . . . . . .

CHAPTER X THE HIGHER LINE GEOMETRY Linear complex in hyperbolic space . . . . . . . . . . . . . . . . . . . . . . . The cross, its coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . The use of the cross manifold to interpret the geometry of the complex plane Chain, and chain surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hamilton’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chain congruence, synectic and nonsynectic congruences . . . . . . . . . . Dual coordinates of a cross in elliptic case . . . . . . . . . . . . . . . . . . . Condition for parataxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Clifford angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chain and strip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chain congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER XI THE CIRCLE AND THE SPHERE Simplest form for the equation of a circle . . . . . . . . . . . . . . . . . . . Dual nature of the curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Curvature of a circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radical axes, and centres of similitude . . . . . . . . . . . . . . . . . . . . . Circles through two points, or tangent to two lines . . . . . . . . . . . . . .

5

76 76 77 78 78 80 80 80 81

83 85 87 88 89 91 93 95

96 96 98 98 99 100 102 103 104 106 107

109 109 111 112 112

Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Poincaré’s sphere to sphere transformation from euclidean to noneuclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER XII CONIC SECTIONS Classification of conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equations of central conic and Absolute . . . . . . . . . . . . . . . . . . . . Centres, axes, foci, focal lines, directrices, and director points . . . . . . . . Relations connecting distances of a point from foci, directrices, &c., and their duals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conjugate and mutually perpendicular lines through a centre . . . . . . . . Auxiliary circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Normals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Confocal and homothetic conics . . . . . . . . . . . . . . . . . . . . . . . . Elliptic coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER XIII QUADRIC SURFACES Classification of quadrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . Central quadrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Planes of circular section and parabolic section . . . . . . . . . . . . . . . . Conjugate and mutually perpendicular lines through a centre . . . . . . . . Confocal and homothetic quadrics . . . . . . . . . . . . . . . . . . . . . . . Elliptic coordinates, various forms of the distance element . . . . . . . . . . String construction for the ellipsoid . . . . . . . . . . . . . . . . . . . . . .

CHAPTER XIV AREAS AND VOLUMES Amplitude of a triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relation to other parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Limiting form when the triangle is infinitesimal . . . . . . . . . . . . . . . . Deficiency and area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Area found by integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . Area of circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Area of whole elliptic or spherical plane . . . . . . . . . . . . . . . . . . . . Amplitude of a tetrahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . Relation to other parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simple form for the differential of volume of a tetrahedron . . . . . . . . . . Reduction to a single quadrature of the problem of finding the volume of a tetrahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Volume of a cone of revolution . . . . . . . . . . . . . . . . . . . . . . . . . Volume of a sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Volume of the whole of elliptic or of spherical space . . . . . . . . . . . . .

6

115

116

119 119 120

120 124 127 127 128 128

130 132 133 134 135 135 140

143 144 146 147 148 150 150 150 150 152

155 155 156 156

CHAPTER XV INTRODUCTION TO DIFFERENTIAL GEOMETRY Curvature of a space or plane curve . . . . . . . . . . . . . . . . . . . . . Analoga of direction cosines of tangent, principal normal, and binormal . Frenet’s formulae for the noneuclidean case . . . . . . . . . . . . . . . . . Sign of the torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolutes of a space curve . . . . . . . . . . . . . . . . . . . . . . . . . . . Two fundamental quadratic differential forms for a surface . . . . . . . . Conditions for mutually conjugate or perpendicular tangents . . . . . . . Lines of curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dupin’s theorem for triply orthogonal systems . . . . . . . . . . . . . . . Curvature of a curve on a surface . . . . . . . . . . . . . . . . . . . . . . . Dupin’s indicatrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Torsion of asymptotic lines . . . . . . . . . . . . . . . . . . . . . . . . . . Total relative curvature, its relation to Gaussian curvature . . . . . . . . Surfaces of zero relative curvature . . . . . . . . . . . . . . . . . . . . . . Surfaces of zero Gaussian curvature . . . . . . . . . . . . . . . . . . . . . Ruled surfaces of zero Gaussian curvature in elliptic or spherical space . . Geodesic curvature and geodesic lines . . . . . . . . . . . . . . . . . . . . Necessary conditions for a minimal surface . . . . . . . . . . . . . . . . . Integration of the resulting differential equations . . . . . . . . . . . . . .

CHAPTER XVI DIFFERENTIAL LINEGEOMETRY

. . . . . . . . . . . . . . . . . . .

Analoga of Kummer’s coefficients . . . . . . . . . . . . . . . . . . . . . . . . Their fundamental relations . . . . . . . . . . . . . . . . . . . . . . . . . . . Limiting points and focal points . . . . . . . . . . . . . . . . . . . . . . . . Necessary and sufficient conditions for a normal congruence . . . . . . . . . MalusDupin theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isotropic congruences, and congruences of normals to surfaces of zero curva ture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spherical representation of rays in elliptic space . . . . . . . . . . . . . . . . Representation of normal congruence . . . . . . . . . . . . . . . . . . . . . . Isotropic congruence represented by an arbitrary function of the complex variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Special examples of this representation . . . . . . . . . . . . . . . . . . . . . Study’s ray to ray transformation which interchanges parallelism and para taxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resulting interchange among the three special types of congruence . . . . .

7

157 158 159 161 161 163 164 165 166 168 170 170 171 172 173 174 175 178 179

182 183 185 188 191

191 193 194

194 197

198 199

CHAPTER XVII MULTIPLY CONNECTED SPACES Repudiation of the axiom of free mobility of space as a whole . . . . . . . . Resulting possibility of one to many correspondence of points and coordinate sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiply connected euclidean planes . . . . . . . . . . . . . . . . . . . . . . Multiply connected euclidean spaces, various types of line in them . . . . . Hyperbolic case little known; relation to automorphic functions . . . . . . . Nonexistence of multiply connected elliptic planes . . . . . . . . . . . . . . Multiply connected elliptic spaces . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER XVIII THE PROJECTIVE BASIS OF NONEUCLIDEAN GEOMETRY Fundamental notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axioms of connexion and separation . . . . . . . . . . . . . . . . . . . . . . Projective geometry of the plane . . . . . . . . . . . . . . . . . . . . . . . . Projective geometry of space . . . . . . . . . . . . . . . . . . . . . . . . . . Projective scale and cross ratios . . . . . . . . . . . . . . . . . . . . . . . . Projective coordinates of points in a line . . . . . . . . . . . . . . . . . . . . Linear transformations of the line . . . . . . . . . . . . . . . . . . . . . . . Projective coordinates of points in a plane . . . . . . . . . . . . . . . . . . . Equation of a line, its coordinates . . . . . . . . . . . . . . . . . . . . . . . Projective coordinates of points in space . . . . . . . . . . . . . . . . . . . . Equation of a plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Collineations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Imaginary elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axioms of the congruent collineation group . . . . . . . . . . . . . . . . . . Reappearance of the Absolute and previous metrical formulae . . . . . . . .

200

200 202 203 205 207 208

210 210 211 212 216 220 221 221 222 222 223 224 224 226 229

CHAPTER XIX THE DIFFERENTIAL BASIS FOR EUCLIDEAN AND NONEUCLIDEAN GEOMETRY Fundamental assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 Coordinate system and distance elements . . . . . . . . . . . . . . . . . . . 232 Geodesic curves, their differential equations . . . . . . . . . . . . . . . . . . 233 Determination of a geodesic by two near points . . . . . . . . . . . . . . . . 234 Determination of a geodesic by a point and direction cosines of tangent thereat 234 Definition of angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Axiom of congruent transformations . . . . . . . . . . . . . . . . . . . . . . 235 Simplified expression for distance element . . . . . . . . . . . . . . . . . . . 236 Constant curvature of geodesic surfaces . . . . . . . . . . . . . . . . . . . . 237 Introduction of new coordinates; integration of equations of geodesic . . . . 239 Reappearance of familiar distance formulae . . . . . . . . . . . . . . . . . . 240 Recapitulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

8

CHAPTER I

FOUNDATION FOR METRICAL GEOMETRY IN A LIMITED REGION

In any system of geometry we must begin by assuming the existence of certain fundamental objects, the raw material with which we are to work. What names we choose to attach to these objects is obviously a question quite apart from the nature of the logical connexions which arise from the various relations assumed to exist among them, and in choosing these names we are guided principally by tradition, and by a desire to make our mathematical edifice as well adapted as possible to the needs of practical life. In the present work we shall assume the existence of two sorts of objects, called respectivelypointsand 2 distancesexplicit assumptions shall be as follows:—. Our AxiomI.There exists a class of objects, containing at least two members, called points. It will be convenient to indicate points by large Roman letters asA,B,C.

AxiomII.The existence of any two points implies the existence of a unique object called their distance. If the points beAandBit will be convenient to indicate their distance by ABorBA. We shall speak of this also as the distancebetweenthe two points, or from one to the other. We next assume that between two distances there may exist a relation ex pressed by saying that the one iscongruentIn place of the wordsto the other.

2 There is no logical or mathematical reason why the point should be taken as undefined rather than the line or plane. This is, however, the invariable custom in works on the founda tions of geometry, and, considering the weight of historical and psychological tradition in its favour, the point will probably continue to stand among the fundamental indefinables. With regard to the others, there is no such unanimity. Veronese,Fondamenti di geometria, Padua, 1891, takes the line, segment, and congruence of segments. Schur, ‘Ueber die Grundlagen der Geometrie,’Mathematische AnnalenHilbert,, vol. lv, 1902, uses segment and motion. Die Grundlagen der Geometrie, Leipzig, 1899, uses practically the same indefinables as Veronese. Moore, ‘The projective Axioms of Geometry,’Transactions of the American Mathematical Society, vol. iii, 1902, and Veblen, ‘A System of Axioms for Geometry,’ same Journal, vol. v, 1904, use segment and order. Pieri, ‘Della geometria elementare come sistema ipotetico dedut tivo,’Memorie della R. Accademia delle Scienze di Torino, Serie 2, vol. xlix, 1899, introduces motion alone, as does Padoa, ‘Un nuovo sistema di definizioni per la geometria euclidea,’ Periodico di matematica, Serie 3, vol. i, 1903. Vahlen,Abstrakte Geometrie, Leipzig, 1905, uses line and separation. Peano, ‘La geometria basata sulle idee di punto e di distanza,’Atti della R. Accademia di Torino, vol. xxxviii, 19023, and Levy, ‘I fondamenti della geometria metricaproiettiva,’Memorie Accad. TorinoI have made, Serie 2, vol. liv, 1904, use distance. the same choice as the lastnamed authors, as it seemed to me to give the best approach to the problem in hand. I cannot but feel that the choice of segment or order would be a mistake for our present purpose, in spite of the very condensed system of axioms which Veblen has set up therefor. For to reach congruence and measurement by this means, one is obliged to introduce the sixparameter group of motions (as in Ch. XVIII of this work), i.e. base metrical geometry on projective. It is, on the other hand, an inelegance to base projective geometry on a nonprojective conception such as ‘betweenness’, whereas writers like Vahlen require both projective and ‘affine’ geometry, before reaching metrical geometry, a very roundabout way to reach what is, after all, the fundamental part of the subject.

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