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                Julian Lowell Coolidge
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                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
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                           THEELEMENTSOF
                   NON-EUCLIDEANGEOMETRY
                                          BY
                        JULIAN LOWELL COOLIDGE Ph.D.
                             ASSISTANT PROFESSOR OF MATHEMATICS
                                  IN HARVARD UNIVERSITY
                                       OXFORD
                                 ATTHECLARENDONPRESS
                                         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.
                              Thesystemofaxiomshereusedisdecidedlymorecumbersomethansomeothers,
                              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.
                Theauthortakesthisopportunitytothankhiscolleague, 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.
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