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Theory of Parallels

THEORY OF PARALLELS. The fundamental princi-ples of mathematics have not in general received from mathematicians the attention which they deserve. Mathe-matical science might in fact be compared to a building far advanced in construction. As to the firmness of its foundations there can be no doubt, to judge by the weighty superstructure which they carry; but the aspect of the building is not a little marred by the quantity of irrelevant rubbish which lies around those foundations, concealing their real strength and security. The question of the parallel axiom in Euclid's geometry is to some extent an exception. There have been endless discussions concerning it. The difficulty is well known, and will be found succinctly stated in the article GEOMETRY (vol. x. p. 378). Those who have treated the subject have devoted themselves either to criticizing the form of Euclid's axiom, suggesting modifications or substitutes (sometimes with undoubted advantage, e.g., Playfair), or to questioning its necessity, offering either to demonstrate the axiom or to dispense with it altogether. It would serve no useful purpose to attempt a complete account of the literature of the subject; we may refer the reader who is curious in such matters to the various editions of Perronet Thomson's Geometry without Axioms. It will be sufficient to mention Legendre's views, which, although by no means reaching to the root of the matter, may be held as indicating the dawn of the true theory.7 The delicacy of the question may be illustrated by the story which is told of Lagrange, j It is said that towards the end of his life he wrote and \ actually took to the Institute a paper dealing with the ' theory of parallels. He had begun to read it; but, before j he had proceeded very far, something struck him. He stopped reading, muttered "II faut que j'y songe encore," and put the paper in his pocket (De Morgan, Budget of Paradoxes, p. 173). There appears to be no doubt that the true theory first presented itself to the mind of Gauss. The history of the matter is interesting, and deserves to be more generally known than it appears to be. In his earlier days, before his career in life was determined, when he had to consider the possibility of his becoming a teacher of mathematics, he drew up a paper in which he gave a philosophical development of the elements of mathematics. It was probably in the course of this discussion (about 1792) that he first came across the difficulty of the parallel axiom. He arrived at the conclusion that geometry became a logically consistent structure only after the ! parallel axiom was given as part of its foundation; and he convinced himself that this axiom could not be proved, although from experience (for example, from the sum of the angles of the geodesic triangle Brocken, Hohenhagen, Inselberg) we know that it is at least very approximately true. If, on the other hand, this axiom be not granted, there follows another kind of geometry, which he developed to a considerable extent and called the antieuclidian geometry. Writing to Bessel on the 27th January 1829, he says—

" In leisure hours now and then I have again been reflecting on a subject which with me is now nearly forty years old; I mean the first principles of geometry; I do not know if I have ever told you my views on that matter. Here too I have carried many things to farther consolidation, and my conviction that we cannot lay the foundation of geometry completely a priori has become if possible firmer than before. Meantime it will be long before I bring myself to work out my very extensive researches on this subject for publication, perhaps I shall never do so during my lifetime ; for I fear the outcry of the Boeotians, were I to speak out my views on the question.

Bessel entered heartily into the ideas of Gauss, and urged him to publish them regardless of the Boeotians. Concerning the generality of mathematicians in his day, Gauss probably judged rightly, however, for his intimate correspondent Schumacher was, as we learn from their correspondence in 1831, unable to follow the new idea. One of the letters (Gauss to Schumacher, 12th July 1831) is of great interest because it shows us that Gauss was then in full possession of the most important propositions of what is now called hyperbolic geometry. In particular he states that in hyperbolic space the circumference of a circle of radius r is trh^H _ e~T^j, where h is a constant, which we know from experience to be infinitely great compared with any length that we can measure (supposing, he means, the space of our experience to be hyperbolic), and which in Euclid's geometry is infinite.
1856, p. 81.

Gauss never published these researches; and no traces of them seem to have been found among his papers after his death. Our first knowledge of the hyperbolic geometry dates from the publication of the works of N. Lobatscliewsky and W. Bolyai. Lobatsehewsky's views were fiist published in a lecture before the Faculty of Mathematics and Physics in Kasan, 12th February 1826. See Frischauf, Elemente der Absoluten Geometrie, Leipsie, 1876, page 33. Speaking of a German edition of Lobatsehewsky's work, which he had seen published at Berlin in 1840, Gauss says that he finds nothing in it which is materially new to him, but that Lobatsehewsky's method of development is different from his own, and is a masterly performance carried out in the true geometric spirit. The theory received its complement in the famous Habilitalionsschrift of Riemann, in which the elliptic geometry for the first time appears. Beltrami, Helmholtz, Cayley, Klein, and others have greatly developed the subject; but it is unnecessary to pursue its later history here, since all essential details will be found in the article MEASUREMENT, vol. xv. p. 659. All that we need do is to call the attention of those who busy themselves with mental philosophy to this generalization of geometry, as one of the results of modern mathematical research which they cannot afford to overlook. (G. CH.)


For some interesting controversy on this subject see Leslie's Geometry, 3d edition, p. 292; and Legendre, Elements de Géométrie, 12th edition, p. 277.

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