Читать книгу An essay on the foundations of geometry - Bertrand Russell - Страница 6

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12. The originator of the whole system, Gauss, does not appear, as regards strictly non-Euclidean Geometry, in any of his hitherto published papers, to have given more than results; his proofs remain unknown to us. Nevertheless he was the first to investigate the consequences of denying the axiom of parallels[8], and in his letters he communicated these consequences to some of his friends, among whom was Wolfgang Bolyai. The first mention of the subject in his letters occurs when he was only 18; four years later, in 1799, writing to W. Bolyai, he enunciates the important theorem that, in hyperbolic Geometry, there is a maximum to the area of a triangle. From later writings it appears that he had worked out a system nearly, if not quite, as complete as those of Lobatchewsky and Bolyai[9].

It is important to remember, however, that Gauss's work on curvature, which was published, laid the foundation for the whole method of the second period, and was undertaken, according to Riemann and Helmholtz[10], with a view to an (unpublished) investigation of the foundations of Geometry. His work in this direction will, owing to its method, be better treated of under the second period, but it is interesting to observe that he stood, like many pioneers, at the head of two tendencies which afterwards diverged.

13. Lobatchewsky, a professor in the University of Kasan, first published his results, in their native Russian, in the proceedings of that learned body for the years 1829–1830. Owing to this double obscurity of language and place, they attracted little attention, until he translated them into French[11] and German[12]: even then, they do not appear to have obtained the notice they deserved, until, in 1868, Beltrami unearthed the article in Crelle, and made it the theme of a brilliant interpretation.

In the introduction to his little German book, Lobatchewsky laments the slight interest shown in his writings by his compatriots, and the inattention of mathematicians, since Legendre's abortive attempt, to the difficulties in the theory of parallels. The body of the work begins with the enunciation of several important propositions which hold good in the system proposed as well as in Euclid: of these, some are in any case independent of the axiom of parallels, while others are rendered so by substituting, for the word "parallel," the phrase "not intersecting, however far produced." Then follows a definition, intentionally framed so as to contradict Euclid's: With respect to a given straight line, all others in the same plane may be divided into two classes, those which cut the given straight line, and those which do not cut it; a line which is the limit between the two classes is called parallel to the given straight line. It follows that, from any external point, two parallels can be drawn, one in each direction. From this starting-point, by the Euclidean synthetic method, a series of propositions are deduced; the most important of these is, that in a triangle the sum of the angles is always less than, or always equal to two right angles, while in the latter case the whole system becomes orthodox. A certain analogy with spherical Geometry—whose meaning and extent will appear later—is also proved, consisting roughly in the substitution of hyperbolic for circular functions.

14. Very similar is the system of Johann Bolyai, so similar, indeed, as to make the independence of the two works, though a well-authenticated fact, seem all but incredible. Johann Bolyai first published his results in 1832, in an appendix to a work by his father Wolfgang, entitled; "Appendix, scientiam spatii absolute veram exhibens: a veritate aut falsitate Axiomatis XI. Euclidei (a priori haud unquam decidenda) independentem; adjecta ad casum falsitatis, quadratura circuli geometrica." Gauss, whose bosom friend he became at college and remained through life, was, as we have seen, the inspirer of Wolfgang Bolyai, and used to say that the latter was the only man who appreciated his philosophical speculations on the axioms of Geometry; nevertheless, Wolfgang appears to have left to his son Johann the detailed working out of the hyperbolic system. The works of both the Bolyai are very rare, and their method and results are known to me only through the renderings of Frischauf and Halsted[13]. Both as to method and as to results, the system is very similar to Lobatchewsky's, so that neither need detain us here. Only the initial postulates, which are more explicit than Lobatchewsky's, demand a brief attention. Frischauf's introduction, which has a philosophical and Newtonian air, begins by setting forth that Geometry deals with absolute (empty) space, obtained by abstracting from the bodies in it, that two figures are called congruent when they differ only in position, and that the axiom of Congruence is indispensable in all determination of spatial magnitudes. Congruence was to refer to geometrical bodies, with none of the properties of ordinary bodies except impenetrability (Erdmann, Axiome der Geometrie, p. 26). A straight line is defined as determined by two of its points[14], and a plane as determined by three. These premisses, with a slight exception as to the straight line, we shall hereafter find essential to every Geometry. I have drawn attention to them, as it is often supposed that non-Euclideans deny the axiom of Congruence, which, here and elsewhere, is never the case. The stress laid on this axiom by Bolyai is probably due to the influence of Gauss, whose work on the curvature of surfaces laid the foundation for the use made of congruence by Helmholtz.

15. It is important to remember that, throughout the period we have just reviewed, the purpose of hyperbolic Geometry is indirect: not the truth of the latter, but the logical independence of the axiom of parallels from the rest, is the guiding motive of the work. If, by denying the axiom of parallels while retaining the rest, we can obtain a system free from logical contradictions, it follows that the axiom of parallels cannot be implicitly contained in the others. If this be so, attempts to dispense with the axiom, like Legendre's, cannot be successful; Euclid must stand or fall with the suspected axiom. Of course, it remained possible that, by further development, latent contradictions might have been revealed in these systems. This possibility, however, was removed by the more direct and constructive work of the second period, to which we must now turn our attention.