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 and surprising conclusion that the theorems of non-Euclidean geometry find their realisation upon surfaces of constant negative curvature. He studied, also, surfaces of constant positive curvature, and ended with the interesting theorem that the space of constant positive curvature is contained in the space of constant negative curvature. These researches of Beltrami, Helmholtz, and Riemann culminated in the conclusion that on surfaces of constant curvature we may have three geometries,—the non-Euclidean on a surface of constant negative curvature, the spherical on a surface of constant positive curvature, and the Euclidean geometry on a surface of zero curvature. The three geometries do not contradict each other, but are members of a system,—a geometrical trinity. The ideas of hyper-space were brilliantly expounded and popularised in 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 $\scriptstyle{n}$-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,