Thus the only metrical geometry for the whole of projective space is of the elliptic type. But the actual measure-relations (though not their general properties) differ according to the elliptic congruence-group selected for study. In a descriptive space a congruence-group should possess the four characteristics of such a group throughout the whole of the space. Then form the associated ideal projective space. The associated congruence-group for this ideal space must satisfy the four conditions throughout the region of the proper ideal points. Thus the boundary of this region is the absolute. Accordingly there can be no metrical geometry for the whole of a descriptive space unless its boundary (in the associated ideal space) is a closed quadric or a plane. If the boundary is a closed quadric, there is one possible congruence-group of the hyperbolic type. If the boundary is a plane (the plane at infinity), the possible congruence-groups are parabolic; and there is a congruence-group corresponding to each imaginary conic in this plane, together with a Euclidean metrical geometry corresponding to each such group. Owing to these alternative possibilities, it would appear to be more accurate to say that systems of quantities can be found in a space, rather than that space is a quantity.
Lie has also deduced[48] the same results with respect to congruence-groups from another set of defining properties, which explicitly assume the existence of a quantitative relation (the distance) between any two points, which is invariant for any transformation of the congruence-group.[49]
The above results, in respect to congruence and metrical geometry, considered in relation to existent space, have led to the doctrine[50] that it is intrinsically unmeaning to ask which system of metrical geometry is true of the physical world. Any one of these systems can be applied, and in an indefinite number of ways. The only question before us is one of convenience in respect to simplicity of statement of the physical laws. This point of view seems to neglect the consideration that science is to be relevant to the definite perceiving minds of men; and that (neglecting the ambiguity introduced by the invariable slight inexactness of observation which is not relevant to this special doctrine) we have, in fact, presented to our senses a definite set of transformations forming a congruence-group, resulting in a set of measure relations which are in no respect arbitrary. Accordingly our scientific laws are to be stated relevantly to that particular congruence-group. Thus the investigation of the type (elliptic, hyperbolic or parabolic) of this special congruence-group is a perfectly definite problem, to be decided by experiment. The consideration of experiments adapted to this object requires some development of non-Euclidean geometry (see section VI., Non-Euclidean Geometry). But if the doctrine means that, assuming some sort of objective reality for the material universe, beings can be imagined, to whom either all congruence-groups are equally important, or some other congruence-group is specially important, the doctrine appears to be an immediate deduction from the mathematical facts. Assuming a definite congruence-group, the investigation of surfaces (or three-dimensional loci in space of four dimensions) with geodesic geometries of the form of metrical geometries of other types of congruence-groups forms an important chapter of non-Euclidean geometry. Arising from this investigation there is a widely-spread fallacy, which has found its way into many philosophic writings, namely, that the possibility of the geometry of existent three-dimensional space being other than Euclidean depends on the physical existence of Euclidean space of four or more dimensions. The foregoing exposition shows the baselessness of this idea.
Bibliography.—For an account of the investigations on the axioms of geometry during the Greek period, see M. Cantor, Vorlesungen über die Geschichte der Mathematik, Bd. i. and iii.; T.L. Heath, The Thirteen Books of Euclid’s Elements, a New Translation from the Greek, with Introductory Essays and Commentary, Historical, Critical, and Explanatory (Cambridge, 1908)—this work is the standard source of information; W.B. Frankland, Euclid, Book I., with a Commentary (Cambridge, 1905)—the commentary contains copious extracts from the ancient commentators. The next period of really substantive importance is that of the 18th century. The leading authors are: G. Saccheri, S.J., Euclides ab omni naevo vindicatus (Milan, 1733). Saccheri was an Italian Jesuit who unconsciously discovered non-Euclidean geometry in the course of his efforts to prove its impossibility. J.H. Lambert, Theorie der Parallellinien (1766); A.M. Legendre, Éléments de géométrie (1794). An adequate account of the above authors is given by P. Stäckel and F. Engel, Die Theorie der Parallellinien von Euklid bis auf Gauss (Leipzig, 1895). The next period of time (roughly from 1800 to 1870) contains two streams of thought, both of which are essential to the modern analysis of the subject. The first stream is that which produced the discovery and investigation of non-Euclidean geometries, the second stream is that which has produced the geometry of position, comprising both projective and descriptive geometry not very accurately discriminated. The leading authors on non-Euclidean geometry are K.F. Gauss, in private letters to Schumacher, cf. Stäckel and Engel, loc. cit.; N. Lobatchewsky, rector of the university of Kazan, to whom the honour of the effective discovery of non-Euclidean geometry must be assigned. His first publication was at Kazan in 1826. His various memoirs have been re-edited by Engel; cf. Urkunden zur Geschichte der nichteuklidischen Geometrie by Stäckel and Engel, vol. i. “Lobatchewsky.” J. Bolyai discovered non-Euclidean geometry apparently in independence of Lobatchewsky. His memoir was published in 1831 as an appendix to a work by his father W. Bolyai, Tentamen juventutem.... This memoir has been separately edited by J. Frischauf, Absolute Geometrie nach J. Bolyai (Leipzig, 1872); B. Riemann, Über die Hypothesen, welche der Geometrie zu Grunde liegen (1854); cf. Gesamte Werke, a translation in The Collected Papers of W.K. Clifford. This is a fundamental memoir on the subject and must rank with the work of Lobatchewsky. Riemann discovered elliptic metrical geometry, and Lobatchewsky hyperbolic geometry. A full account of Riemann’s ideas, with the subsequent developments due to Clifford, F. Klein and W. Killing, will be found in The Boston Colloquium for 1903 (New York, 1905), article “Forms of Non-Euclidean Space,” by F.S. Woods. A. Cayley, loc. cit. (1859), and F. Klein, “Über die sogenannte nichteuklidische Geometrie,” Math. Annal. vols. iv. and vi. (1871 and 1872), between them elaborated the projective theory of distance; H. Helmholtz, “Über die thatsächlichen Grundlagen der Geometrie” (1866), and “Über die Thatsachen, die der Geometrie zu Grunde liegen” (1868), both in his Wissenschaftliche Abhandlungen, vol. ii., and S. Lie, loc. cit. (1890 and 1893), between them elaborated the group theory of congruence.
The numberless works which have been written to suggest equivalent alternatives to Euclid’s parallel axioms may be neglected as being of trivial importance, though many of them are marvels of geometric ingenuity.
The second stream of thought confined itself within the circle of ideas of Euclidean geometry. Its origin was mainly due to a succession of great French mathematicians, for example, G. Monge, Géométrie descriptive (1800); J.V. Poncelet, Traité des proprietés projectives des figures (1822); M. Chasles, Aperçu historique sur l’origine et le développement des méthodes en géométrie (Bruxelles, 1837), and Traité de géométrie supérieure (Paris, 1852); and many others. But the works which have been, and are still, of decisive influence on thought as a store-house of ideas relevant to the foundations of geometry are K.G.C. von Staudt’s two works, Geometrie der Lage (Nürnberg, 1847); and Beiträge zur Geometrie der Lage (Nürnberg, 1856, 3rd ed. 1860).
The final period is characterized by the successful production of exact systems of axioms, and by the final solution of problems which have occupied mathematicians for two thousand years. The successful analysis of the ideas involved in serial continuity is due to R. Dedekind, Stetigkeit und irrationale Zahlen (1872), and to G. Cantor, Grundlagen einer allgemeinen Mannigfaltigkeitslehre (Leipzig, 1883), and Acta math. vol. 2.
Complete systems of axioms have been stated by M. Pasch, loc. cit.; G. Peano, loc. cit.; M. Pieri, loc. cit.; B. Russell, Principles of Mathematics; O. Veblen, loc. cit.; and by G. Veronese in his treatise, Fondamenti di geometria (Padua, 1891; German transl. by A. Schepp, Grundzüge der Geometrie, Leipzig, 1894). Most of the leading memoirs on special questions involved have been cited in the text; in addition there may be mentioned M. Pieri, “Nuovi principii di geometria projettiva complessa,” Trans. Accad. R. d. Sci. (Turin, 1905); E.H. Moore, “On the Projective Axioms of Geometry,” Trans. Amer. Math. Soc., 1902; O. Veblen and W.H. Bussey, “Finite Projective Geometries,” Trans. Amer. Math. Soc., 1905; A.B. Kempe, “On the Relation between the Logical Theory of Classes and the Geometrical Theory of Points,” Proc. Lond. Math. Soc., 1890; J. Royce, “The Relation of the Principles of Logic to the Foundations of Geometry,” Trans. of Amer. Math. Soc., 1905; A. Schoenflies, “Über die Möglichkeit einer projectiven Geometrie bei transfiniter (nichtarchimedischer) Massbestimmung,” Deutsch. M.-V. Jahresb., 1906.
For general expositions of the bearings of the above investigations, cf. Hon. Bertrand Russell, loc. cit.; L. Couturat, Les Principes des mathématiques (Paris, 1905); H. Poincaré, loc. cit.; Russell and Whitehead, Principia mathematica (Cambridge, Univ. Press). The philosophers whose views on space and geometric truth deserve especial study are Descartes, Leibnitz, Hume, Kant and J.S. Mill.
(A. N. W.)