England by Clifford.
William Kingdon Clifford (1845–1879) was born at Exeter, educated at Trinity College, Cambridge, and from 1871 until his death professor of applied mathematics in University College, London. His premature death left incomplete several brilliant researches which he had entered upon. Among these are his paper On Classification of Loci and his Theory of Graphs. He wrote articles On the Canonical Form and Dissection of a Riemann's Surface, on Biquaternions, and an incomplete work on the Elements of Dynamic. The theory of polars of curves and surfaces was generalised by him and by Reye. His classification of loci, 1878, being a
general study of curves, was an introduction to the study of -dimensional space in a direction mainly projective. This study has been continued since chiefly by G. Veronese
of Padua, C. Segre of Turin, E. Bertini, F. Aschieri, P. Del Pezzo
of Naples.
Beltrami's researches on non-Euclidean geometry were followed, in 1871, by important investigations of Felix Klein,
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 dimensions
was studied along a line mainly metrical by a host of writers, among whom may be mentioned Simon Newcomb of