A Comparative Review of Recent Researches in Geometry — Prefatory Note & Introduction

Programme on Entering the Philosophical Faculty and the Senate of the University of Erlangen in 1872 — By Prof. Felix Klein (Translated by Dr. M. W. Haskell, Bull. New York Math. Soc. 2, July 1893, pp. 215–217) [1]

Prefatory Note by the Author (1893)

My 1872 Programme (Erlangen, A. Deichert), appearing as a separate publication, had but a limited circulation at first. With this I could be satisfied more easily, as the views developed in the Programme could not be expected at first to receive much attention. But now that the general development of mathematics has taken, in the meanwhile, the direction corresponding precisely to these views, and particularly since Lie has begun the publication in extended form of his Theorie der Transformationsgruppen (Leipzig: Teubner, vol. I 1888, vol. II 1890), it seems proper to give a wider circulation to the expositions in my Programme. An Italian translation by M. Gino Fano was recently published in the Annali di Matematica, ser. 2, vol. 17. A kind reception for the English translation, for which I am much indebted to Mr. Haskell, is likewise desired. The translation is an absolutely literal one; in the two or three places where a few words are changed, the new phrases are enclosed in square brackets [ ]. In the same way are indicated a number of additional footnotes which it seemed desirable to append, most of them having already appeared in the Italian translation. — F. KLEIN.

Introduction

Among the advances of the last fifty years in the field of geometry, the development of projective geometry [2] occupies the first place. Although it seemed at first as if the so-called metrical relations were not accessible to this treatment, as they do not remain unchanged by projection, we have nevertheless learned recently to regard them also from the projective point of view, so that the projective method now embraces the whole of geometry. But metrical properties are then to be regarded no longer as characteristics of the geometrical figures per se, but as their relations to a fundamental configuration, the imaginary circle at infinity common to all spheres.

When we compare the conception of geometrical figures gradually obtained in this way with the notions of ordinary (elementary) geometry, we are led to look for a general principle in accordance with which the development of both methods has been possible. This question seems the more important as, beside the elementary and the projective geometry, are arrayed a series of other methods, which, albeit they are less developed, must be allowed the same right to an individual existence. Such are the geometry of reciprocal radii vectores, the geometry of rational transformations, etc., which will be mentioned and described further on.

In undertaking in the following pages to establish such a principle, we shall hardly develop an essentially new idea, but rather formulate clearly what has already been more or less definitely conceived by many others. But it has seemed the more justifiable to publish connective observations of this kind, because geometry, which is after all one in substance, has been only too much broken up in the course of its recent rapid development into a series of almost distinct theories [3], which are advancing in comparative independence of each other. At the same time I was influenced especially by a wish to present certain methods and views that have been developed in recent investigations by Lie and myself. Our respective investigations, different as has been the nature of the subjects treated, have led to the same generalized conception here presented; so that it has become a sort of necessity to thoroughly discuss this view and on this basis to characterize the contents and general scope of those investigations.

Though we have spoken so far only of geometrical investigations, we will include investigations on manifoldnesses of any number of dimensions [4], which have been developed from geometry by making abstraction from the geometric image, which is not essential for purely mathematical investigations [5]. In the investigation of manifoldnesses the same different types occur as in geometry; and, as in geometry, the problem is to bring out what is common and what is distinctive in investigations undertaken independently of each other. Abstractly speaking, it would in what follows be sufficient to speak throughout of manifoldnesses of $n$ dimensions simply; but it will render the exposition simpler and more intelligible to make use of the more familiar space-perceptions. In proceeding from the consideration of geometric objects and developing the general ideas by using these as an example, we follow the path which our science has taken in its development and which it is generally best to pursue in its presentation.

A preliminary exposition of the contents of the following pages is here scarcely possible, as it can hardly be presented in a more concise form [6]; the headings of the sections will indicate the general course of thought.

At the end I have added a series of notes, in which I have either developed further single points, wherever the general exposition of the text would seem to demand it, or have tried to define with reference to related points of view the abstract mathematical one predominant in the observations of the text.

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Footnotes (1893 Edition)

[1]

Translated by Dr. M. W. Haskell, Assistant Professor of Mathematics in the University of California. Published in Bull. New York Math. Soc. 2 (1892–1893), 215–249.

[2]

See Note I of the appendix.

[3]

See Note II.

[4]

See Note IV.

[5]

See Note III.

[6]

This very conciseness is a defect in the following presentation which I fear will render the understanding of it essentially more difficult. But the difficulty could hardly be removed except by a very much fuller exposition, in which the separate theories, here only touched upon, would have been developed at length.