Foundation of Euclidean and non-Euclidean geometries according to F. Klein

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 geometrical 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 N. I. Lobachevski, and the other by the German B. Riemann: the names of all these heroes of mathematics are written in flaming characters in the sky for ever. Many other outstanding mathematicians have rendered 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 geometries (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 motions and the proof of consistency; this we look upon as complete, according 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 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. Familiarity 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 geometry 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 projective configurations of the first degree) comes last.