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Published online by Cambridge University Press: 20 January 2009
Much has been written in recent years on the foundations of geometry, chiefly in Germany and Italy, and the relations of the various Non-Euclidean geometries to the Euclidean system are now more generally known among mathematicians. But most of these writings involve a knowledge of more advanced mathematics, while it has been found difficult to represent even the simplest Non-Euclidean geometry—that of Bolyai-Lobatschewsky—in an elementary manner.
page 95 note * For information upon the History of the Theory of Parallels, see specially–
Engel, und Stäekel, Theorie der Parallellinien von Euklid bis Gauss (Leipzig, 1895);Google Scholar and Bonola, –La Geometria Non-Euclide–Esposizione storico-critica del suo sviluppo (Bologna, 1906). Also Gauss, Werke Bd. VIII.Google Scholar
Historical references will also be found in–
Withers, –Euclid&s Parallel Postulate (Chicago, 1905);Google ScholarRussell, –An Essay on the Foundations of Geometry (Cambridge, 1897);Google ScholarKlein, – Nichteuklidische Geometrie (Göttingen, 1893), and elsewhere.Google Scholar
page 96 note * C. Bonola, loc. cit., §94.
page 96 note † The idea of representing the Non-Euclidean plane on the Euclidean half plane, so that semicircles cutting the axis of x at right angles take the place of straight lines, is due primarily to Klein and Poincaré. An analytical treatment on these lines will be found in Liebmann's, Nichteuklidische Geometrie (Leipzig, 1905).Google Scholar The system of spheres orthogonal to a fixed sphere, which includes the system of spheres orthogonal to a plane as a special case, is adopted by Wellstein in Weber-Wellstein's, Encyclopädie der Elementar-Mathematik, Vol. II., §§8–11 (Leipzig, 1905).Google Scholar This paper was suggested by Wellstein's discussion, of which it may be regarded as an extension.
page 100 note * cf.Hilbert, , Grundlagen der Geometrie, §3; (Leipzig, 1899).Google ScholarEnglish Translation, The Foundations of Geometry, p. 5; (Chicago, 1902).Google Scholar
page 105 note * Cf. Bonola, loc. cit., pp. 134, 155, 159, etc.
Poincaré, La Science et l'Hypothèse, p. 56.
page 116 note * Cf. Engel und Stäckel, loc. cit., p. 84. Bonola, loc. cit., p. 26.
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