FOUNDATION OF EUCLIDEAN AND NON-EUCLIDEAN GEOMETRIES ACCORDING TO F. KLEIN BY L. RÉDEI MATHEMATICAL INSTITUTE, JOZSEF ATTILA UNIVERSITY SZEGED, HUNGARY PERGAMON PRESS OXFORD · LONDON · EDINBURGH . NEW YORK TORONTO . SYDNEY · PARIS · BRAUNSCHWEIG Pergamon Press Ltd., Headington Hill Hall, Oxford 4 & 5 Fitzroy Square, London W. 1 Pergamon Press (Scotland) Ltd., 2 & 3 Teviot Place, Edinburgh 1 Pergamon Press Inc., 44-01 21st Street, Long Island City, New York 11101 Pergamon of Canada, Ltd., 6 Adelaide Street East, Toronto, Ontario Pergamon Press (Aust.) Pty. Ltd., Rushcutters Bay, Sydney, N.S.W. e Pergamon Press S.A.R.L., 24 rue des Écoles, Paris 5 Vieweg & Sohn GmbH, Burgplatz 1, Braunschweig Copyright © 1968 AKADÉMIAI KIADO, BUDAPEST First English edition 1968 Library of Congress Catalog Card No. 67-18486 3928/68 PREFACE WHAT is our space like? How is it constructed? These are the questions of crucial importance that geometers have been striving to answer throughout the whole period of our civilization, and efforts made in this direction have been almost continuous. It was a great achievement and to the benefit of mankind when in the last century this question was settled, although there were alternative answers, i.e. we have the choice of three possible geometri- cal systems. The first of these was set up by the Greek Euclid, while of the other two, one was constructed by the Hungarian J. Bolyai and the Russian Ν. I. Lobachevski, and the other by the German B. Riemann: the names of all these heroes of mathematics are written in flaming charac- ters in the sky for ever. Many other outstanding mathematicians have rend- ered great services by contributing to the logical clarification of these three geometries, and nowadays we use the terminology of F. Klein and call them the parabolic, hyperbolic and elliptic geometries, respectively. We shall mention only a few of these men, such as D. Hilbert, M. Dehn, M. Pasch, F. Schur, F. Klein, while disregarding many other great names. There are various methods of laying the foundation of the three geome- tries (new ones have even been appearing recently) but we do not want to sum up all of them; rather, we confine ourselves in this respect to the few references at the end of the book. Among these we especially draw attention to the recently published book of F. Bachmann in which the development of (plane) geometry is based on reflections. The method followed by F. Klein in his lectures, which leads to the goal through a projective extension of space, has not yet found a satisfactory treatment in the literature. The first step in this direction was made in the two volumes of Fr. Schilling's book — written with great enthusiasm — in which, however, only plane geometry is dealt with, and even that in a somewhat sketchy manner. It is the aim of our present book to remedy this deficiency so that the ideas of F. Klein obtain the place they merit in the literature of mathematics. The filling-in of gaps and the extension of the considerations to space has required an unexpectedly great amount of work. So in order not to make our book too long, we have confined ourselves mainly to the foundation of geometry by developing the group of mo- tions and the proof of consistency; this we look upon as complete, ac- cording to the ideas laid down in the "Erlangen Programme" of F. Klein. Nevertheless, at the end we have added three sections dealing with an introduction to measurements of segments and angles (according to the ix χ PREFACE principles of Cayley-Klein) as well as some notions of trigonometry. For further study the reader may avail himself of the references at the end of the book. The development of projective geometry is covered in Chapters I to V, in the course of which we have aimed at a restriction in the number of ideas used; thus some simple concepts of projective geometry have not been introduced, since we did not in fact need them. Throughout the book, however, we do use set-theoretical ideas, as well as the current notation of this theory. Acquaintance with some well-known concepts of algebra, and analysis and complex numbers is presumed. Fami- liarity with the methods of analytical geometry is also assumed. To some extent we have deviated from the common terminology. For example, we deal with axioms of "incidence" (containedness) instead of those of "connection", since we define lines and planes as sets of points; further, we speak of axioms of "betweenness" instead of those of "order", because they are based on the statement "lies between", while the notion "ordering" appears only later as a derived concept. It has always been our aim to use the simplest possible methods for the proofs of statements; we are not, however, convinced that we have succeeded in all cases. Among other methods, we mention the notion of the "associated" Desargues configuration, which was new to us and the use of which has enabled us to obtain a significant shortening of many proofs. Another innovation worth mentioning is that when dealing with space we have introduced plane-coordinates first, and point-coordinates only in the further development of the subject. The fundamental theorem of projective geo- metry has first been proved for the plane, and then — very easily, of course — for space, while the proof for lines (and more generally, for all basic projec- tive configurations of the first degree) comes last. L. Rédei CHAPTER I AXIOMS § 1. Axioms of Incidence Let us agree that if we mention a number of elements, one, two, etc. (of a set), these elements shall be looked upon as distinct, except when we explicitly indicate the contrary or it is clear from the context. The statement "JC is contained by y", where χ is an element or a subset of a set y (i.e. x £ y οτ χ £j y), may also be expressed by saying: "x belongs to y9 or (x( lies in y" etc,, or even — using a typically geometrical mode of expression — that "y passes through x". The empty set is denoted by 0. Unit-sets (i.e. sets consisting of one ele- ment only) are identified by their element: this can be done without running the risk of a misunderstanding. The following axioms will be called "axioms of incidence": AXIOM I Space—denoted by ΐϋ—is the set composed of all points. V AXIOM 1. A straight line is a subset of 9? consisting of at least two points. 2 AXIOM I . For any two points there exists exactly one straight line passing 3 through them. AXIOM I . Any plane is a subset of ίϋ and contains three points which do 4 not all lie in any one straight line. AXIOM I . Just one plane passes through three points which do not all lie 5 in any one straight line. FIG. 1 AXIOM I . A plane contains any straight line two points of which belong to it. e 1 2 AXIOMS AXIOM I . Any two planes with a common point contain at least two common 7 points. AXIOM I . There exist four points which do not all lie in any one plane. 8 Let us refer to these axioms as thç System of Axioms I. We shall often represent the points, lines and planes (as well as some notions appearing later) by simple figures which, however, will not form an organic part of the treatment. Thus e.g. Axiom I may be represented by Fig. 1. For a "straight e line" we use the synonyms "line" or "axis", the latter only in certain con- nections. (Instead of "points on a line" we use "points in aline" since lines are sets (of points).) § 2. Axioms of Betweenness The following axioms will be called "axioms of betweenness" \ AXIOM The basic domain—denoted by W—is a subset of dt not con- tained in any one plane. Β FIG. 2 FIG. 3 AXIOM II . If one of three points lies between the other two, then they are 2 all contained in 9ΐ' and they lie on a line. AXIOM II . If a point A lies between points Β and C, then it lies also between 3 C and B. AXIOM II . Just one of three points of W—contained by a line—lies be- 4 tween the other two. AXIOM II . For any two points A and Β of 9ΐ', the point Β lies between A and 5 some further point. AXIOM II . Let A, B, C be three points offfi not in a line and g a line not 6 passing through any of these points but lying together with it in a common plane and passing through a point lying between A and B, then g passes through a point lying between A and C, or between Β and C. AXIOMS OF ΜΟΉΟΝ 3 Let us refer to these axioms as the System of Axioms IL We may illus- trate these: e.g. Axioms II and II are represented by Figs. 2 and 3, respec- 2 e tively. (On Fig. 2 we have also indicated W but on the subsequent figures we 9 will omit it.) § 3. Axiom of Continuity The axiom of continuity reads as follows: AXIOM III. Inside 9Î' there exist two points A and Β with the property that if the (denumerably) infinite series of points P P,... lies between A and B l9 2 9 where P lies between A and P, and every P(i ^ 2) lies between P and ± 2 t imml P then there exists a point Η {lying between A and Β or identical with B) i+l9 such that for every point X lying between A and Η at least one P lies between t X and Η (Figs. 4, 4a). A P P P Χ Ρ Η Β Α Ρ, P X P B~H 1 2 3 { 3 { FIG. 4 FIG. 4a Instead of "Axiom III" we sometimes say "System of Axioms III": e.g. when we mention "Systems of Axioms I, II, III" we mean the Systems of Axioms I, II and Axiom III. § 4. Axioms of Motion The axioms of motion are as follows: AXIOM IV^ Any motion is a one-to-one mapping of?R onto itself such that every three points of a line will be transformed into (three) points of a line. AXIOM IV . The identical mapping of SR is a motion. 2 AXIOM IV. The product of two motions is a motion. 3 AXIOM IV. The inverse mapping of a motion is also a motion. 4 AXIOM IV. If we have two planes a, a', two lines g, g' and two points P 5 9 P' such that Ρ Ç g da and P' Ç g' CI a', then there exists a motion mapping a onto a', g onto g' and Ρ onto P'. AXIOM IV. There is a plane a, a line g and a point P, where P f^ Ca 6 such that there exist just four motions mapping cc g and Ρ onto themselves, 9 respectively, and not more than two of these motions may have every point of g as a fixed point, while there is only ope of them (i.e. the identity) for which every point of a is fixed. 4 AXIOMS AXIOM IV . There exist three points Α, Β, Ρ offfi lying in a line g such that 7 Ρ is between A and Β and for every point C(^P) between A and Β there is a point D lying between C and Ρ for which no motion with Ρ as fixed point can be found that will map C onto a point lying between D and P. AXIOM IV. There exists a line g such that g Π 9î'#0 having the following 8 property: If A, B, C are three points ofgCilfl' and Β lies between A and C, and M is a motion such that AM = B> (AMZ =) BM = C, mapping the set γ of points between A and Β onto the set of points lying between Β and C: then every point of g is either one of the AM i or a point of one of the γΜ' (/=» 0, ± 1 ,...)· (Here and also later, when we wish to denote the images of elements or subsets of a set prodoced by mappings, the mappings will be written as exponents.) We shall refer to these axioms as System of Axioms IV. We can see that Axioms Γν can be linked to form the (combined) 2 34 statement that the motions form a group, called the group of motions of 9Î. The following discussion has been arranged so that first of all we take only the System of Axioms I as established, and then we shall add System II, Axiom III and System IV in succession. CHAPTER II CONSEQUENCES OF THE SYSTEM OF AXIOMS I HENCEFORTH the System of Axioms I will be presumed. In this chapter we shall obtain some conclusions from it; at the same time we shall intro- duce some ideas which will be used later on, all these being related to space % even if this is not explicitly stated. § 5. Simple Properties of Straight Lines and Planes A (geometric) configuration (figure) or, more exactly, a point-configura- tion is a subset of SR. Accordingly, a set of lines or planes will be called a configuration of lines or planes, respectively. A set of configurations lying in a line or in a plane is called collinear or coplanar, respectively. A set of lines or planes passing through a point may be called concurrent, while planes having a line in common are called co- axial. We say that two point-configurations 21 and 93 intersect (in 31 Π S3) if neither of them is contained by the other and 21 Π 93^0, which means that 21 and 93 must each contain at least two points and that neither 21 nor S3 is equal to 9Î. If 21 and 93 intersect and their intersection 21 Π 93 happens to be a point or a Une, we call this the point of intersection or line of inter- section of 2Ï and 93. The line passing through A and B, which according to Axiom I is unique- 3 ly determined, will be denoted by AB (= ΒΑ). Similarly the plane deter- mined uniquely by three non-collinear points (according to Axiom I ) has 5 the following notation: ABC (= ACB = BAC =...): FIG. 5 THEOREM 1. Two intersecting lines intersect in one point. This theorem states that two Unes a and b having a common point Ρ (Fig. 5) cannot have any other common points. To prove this let us suppose that we have another point Q (#P), where QÇ.a,b. Since P Q £ a and 9 5 6 CONSEQUENCES OF THE SYSTEM OF AXIOMS I Λ Q £ b we have (according to Axiom I ): a = PQ as well as b = Ρβ, i.e. 3 a = b. This is a contradiction, and Theorem 1 is proved. FIG. 6 FIG. 7 THEOREM 2. ^4 plane and a line intersect, if at all, in one point. Let α be a plane and g a line not lying in it with a common point Ρ (Fig. 6). We have to prove that α Π g = P. Unless this is true, we have a point Q (ΦΡ) such that β Ç a, g. Then since P, β Ç we have ^ = Ρβ. Hence since Ρ, β £ α it follows from Axiom I that # lies in a. This contradiction e proves the theorem. THEOREM 3. Two intersecting planes intersect in one line. To prove this let us consider two planes α, β with a common point P. By Axiom I they have another common point Q (φΡ) (Fig. 7) which fur- 7 ther implies, by Axiom I , that PQCKx and PQ (Ζ β, i.e. α Π β Ξ> Ρβ. We e must now show that in this formula the equality holds. Unless this is so, it follows that there exists a point R outside the Une Ρβ and lying in α Π β, consequently also in α and β. Owing to Axiom I this would mean that both 5 α and β were equal to PQR. This contradiction proves Theorem 3. THEOREM 4. A point and a line are coplanar and so are two intersecting 9 lines. We can restrict ourselves to the proof of the second part of the statement. FIG. 8 Let a and b be two intersecting lines with a common point Ρ (Fig. 8). We take two further points A and Β in a and b, respectively, and consider the