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NON-EUCLIDEAN] 4/3R2; hence, returning to general axes, the same is the quotient when the terms of the fourth order in (1) are divided by the square of the triangle whose vertices are (0, 0, . . . 0), (z1, z2, z3, . . . zn), (dz1, dz2, dz3 . . . dzn). But − of this quotient is defined by Riemann as the measure of curvature. Hence the measure of curvature is −1/R2, i.e. is constant and negative. The properties of parallels, triangles, &c., are as in the Saggio. It is also shown that the analogues of limit surfaces have zero curvature; and that spheres of radius have constant positive curvature 1/R2 sinh2 (/R), so that spherical geometry may be regarded as contained in the pseudo-spherical (as Beltrami calls Lobatchewsky’s system).

The Saggio, as we saw, gives a Euclidean interpretation confined to two dimensions. But a consideration of the auxiliary plane suggests a different interpretation, which may be extended to any number of dimensions. If, instead of referring to the pseudosphere, we merely define

distance and angle, in the Euclidean plane, as those functions of the coordinates which gave us distance and angle on the pseudosphere, we find that the geometry of our plane has become Lobatchewsky’s. All the points of the limiting circle are now at infinity, and points beyond it are imaginary. If we give our circle an imaginary radius the geometry on the plane becomes elliptic. Replacing the circle by a sphere, we obtain an analogous representation for three dimensions. Instead of a circle or sphere we may take any conic or quadric. With this definition, if the fundamental quadric be xx = 0, and if xx′ be the polar form of xx, the distance between x and x′ is given by the projective formula

cos(/k) = xx′ / {xx·x′x′}$1⁄2$.

That this formula is projective is rendered evident by observing that e−2i/k is the anharmonic ratio of the range consisting of the two points and the intersections of the line joining them with the fundamental quadric. With this we are brought to the third or projective period. The method of this period is due to Cayley; its application to previous non-Euclidean geometry is due to Klein. The projective method contains a generalization of discoveries already made by Laguerre in 1853 as regards Euclidean geometry. The arbitrariness of this procedure of deriving metrical geometry from the properties of conics is removed by Lie’s theory of congruence. We then arrive at the stage of thought which finds its expression in the modern treatment of the axioms of geometry.

The projective method leads to a discrimination, first made by Klein, of two varieties of Riemann’s space; Klein calls these elliptic and spherical. They are also called the polar and antipodal forms of elliptic space. The latter names will here be used. The difference is strictly

analogous to that between the diameters and the points of a sphere. In the polar form two straight lines in a plane always intersect in one and only one point; in the antipodal form they intersect always in two points, which are antipodes. According to the definition of geometry adopted in section VII. (Axioms of Geometry), the antipodal form is not to be termed “geometry,” since any pair of coplanar straight lines intersect each other in two points. It may be called a “quasi-geometry.” Similarly in the antipodal form two diameters always determine a plane, but two points on a sphere do not determine a great circle when they are antipodes, and two great circles always intersect in two points. Again, a plane does not form a boundary among lines through a point: we can pass from any one such line to any other without passing through the plane. But a great circle does divide the surface of a sphere. So, in the polar form, a complete straight line does not divide a plane, and a plane does not divide space, and does not, like a Euclidean plane, have two sides. But, in the antipodal form, a plane is, in these respects, like a Euclidean plane.

It is explained in section VII. in what sense the metrical geometry of the material world can be considered to be determinate and not a matter of arbitrary choice. The scientific question as to the best available evidence concerning the nature of this geometry is one beset with difficulties of a peculiar kind. We are obstructed by the fact that all existing physical science assumes the Euclidean hypothesis. This hypothesis has been involved in all actual measurements of large distances, and in all the laws of astronomy and physics. The principle of simplicity would therefore lead us, in general, where an observation conflicted with one or more of those laws, to ascribe this anomaly, not to the falsity of Euclidean geometry, but to the falsity of the laws in question. This applies especially to astronomy. On the earth our means of measurement are many and direct, and so long as no great accuracy is sought they involve few scientific laws. Thus we acquire, from such direct measurements, a very high degree of probability that the space-constant, if not infinite, is yet large as compared with terrestrial distances. But astronomical distances and triangles can only be measured by means of the received laws of astronomy and optics, all of which have been established by assuming the truth of the Euclidean hypothesis. It therefore remains possible (until a detailed proof of the contrary is forthcoming) that a large but finite space-constant, with different laws of astronomy and optics, would have equally explained the phenomena. We cannot, therefore, accept the measurements of stellar parallaxes, &c., as conclusive evidence that the space-constant is large as compared with stellar distances. For the present, on grounds of simplicity, we may rightly adopt this view; but it must remain possible that, in view of some hitherto undiscovered discrepancy, a slight correction of the sort suggested might prove the simplest alternative. But conversely, a finite parallax for very distant stars, or a negative parallax for any star, could not be accepted as conclusive evidence that our geometry is non-Euclidean, unless it were shown—and this seems scarcely possible—that no modification of astronomy or optics could account for the phenomenon. Thus although we may admit a probability that the space-constant is large in comparison with stellar distances, a conclusive proof or disproof seems scarcely possible.

Finally, it is of interest to note that, though it is theoretically possible to prove, by scientific methods, that our geometry is non-Euclidean, it is wholly impossible to prove by such methods that it is accurately Euclidean. For the unavoidable errors of observation must always leave a slight margin in our measurements. A triangle might be found whose angles were certainly greater, or certainly less, than two right angles; but to prove them exactly equal to two right angles must always be beyond our powers. If, therefore, any man cherishes a hope of proving the exact truth of Euclid, such a hope must be based, not upon scientific, but upon philosophical considerations.

—The bibliography appended to section VII. should be consulted in this connexion. Also, in addition to the citations already made, the following works may be mentioned.

For Lobatchewsky’s writings, cf. Urkunden zur Geschichte der nichteuklidischen Geometrie, i., Nikolaj Iwanowitsch Lobatschefsky, by F. Engel and P. Stäckel (Leipzig, 1898). For John Bolyai’s Appendix, cf. Absolute Geometrie nach Johann Bolyai, by J. Frischauf (Leipzig, 1872), and also the new edition of his father’s large work, Tentamen. . ., published by the Mathematical Society of Budapest; the second volume contains the appendix. Cf. also J. Frischauf, Elemente der absoluten Geometrie (Leipzig, 1876); M. L. Gérard, Sur la géométrie non-Euclidienne (thesis for doctorate) (Paris, 1892); de Tilly, Essai sur les principes fondamentales de la géométrie et de la mécanique (Bordeaux, 1879); Sir R. S. Ball, “On the Theory of Content,” ''Trans. Roy. Irish Acad.'' vol. xxix. (1889); F. Lindemann, “Mechanik bei projectiver Maasbestimmung,” ''Math. Annal. vol. vii.; W. K. Clifford, “Preliminary Sketch of Biquaternions,” Proc.'' ''of Lond. Math. Soc. (1873), and Coll. Works''; A. Buchheim, “On the Theory of Screws in Elliptic Space,” ''Proc. Lond. Math. Soc.'' vols. xv., xvi., xvii.; H. Cox, “On the Application of Quaternions and Grassmann’s Algebra to different Kinds of Uniform Space,” Trans. ''Camb. Phil. Soc.'' (1882); M. Dehn, “Die Legendarischen Sätze über die Winkelsumme im Dreieck,” Math. Ann. vol. 53 (1900), and “Über den Rauminhalt,” ''Math. Annal.'' vol. 55 (1902).

For expositions of the whole subject, cf. F. Klein, Nicht-Euklidische Geometrie (Göttingen, 1893); R. Bonola, La Geometria non-Euclidea (Bologna, 1906); P. Barbarin, La Géométrie non-Euclidienne (Paris, 1902); W. Killing, Die nicht-Euklidischen Raumformen in analytischer Behandlung (Leipzig, 1885). The last-named work also deals with geometry of more than three dimensions; in this connexion cf. also G. Veronese, Fondamenti di geometria a più dimensioni ed a più specie