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 resting upon Cayley's Sixth Memoir on Quantics, 1859. The question whether it is not possible to so express the metrical properties of figures that they will not vary by projection (or linear transformation) had been solved for special projections by Chasles, Poncelet, and E. Laguerre (1834–1886) of Paris, but it remained for Cayley to give a general solution by defining the distance between two points as an arbitrary constant multiplied by the logarithm of the anharmonic ratio in which the line joining the two points is divided by the fundamental quadric. Enlarging upon this notion, Klein showed the independence of projective geometry from the parallel-axiom, and by properly choosing the law of the measurement of distance deduced from projective geometry the spherical, Euclidean, and pseudospherical geometries, named by him respectively the elliptic, parabolic, and hyperbolic geometries. This suggestive investigation was followed up by numerous writers, particularly by G. Battaglini of Naples, E. d' Ovidio of Turin, R. de Paolis of Pisa, F. Aschieri, A. Cayley, F. Lindemann of Munich, E. Schering of Göttingen, W. Story of Clark University, H. Stahl of Tübingen, A. Voss of Würzburg, Homersham Cox, A. Buchheim.[55] The geometry of $$\scriptstyle{n}$$ dimensions was studied along a line mainly metrical by a host of writers, among whom may be mentioned Simon Newcomb of the Johns Hopkins University, L. Schläfli of Bern, W. I. Stringham of the University of California, W. Killing of Münster, T. Craig of the Johns Hopkins, R. Lipschitz of Bonn. R. S. Heath and Killing investigated the kinematics and mechanics of such a space. Regular solids in $\scriptstyle{n}$-dimensional space were studied by Stringham, Ellery W. Davis of the University of Nebraska, R. Hoppe of Berlin, and others. Stringham gave pictures of projections upon our space of regular solids in four dimensions, and Schlegel at Hagen constructed models of such projections. These are