excessive empiricism. This period marks the beginning of lively discussions upon this subject. Some writers–-Bellavitis, for example–-were able to
see in non-Euclidean geometry and -dimensional space nothing but huge caricatures, or diseased outgrowths of mathematics. Helmholtz's article was entitled Thatsachen, welche der Geometrie zu Grunde liegen, 1868, and contained many of the ideas of Riemann. Helmholtz popularised the subject in lectures, and in articles for various magazines.
Eugenio Beltrami, born at Cremona, Italy, in 1835, and now professor at Rome, wrote the classical paper Saggio di interpretazione della geometria non-euclidea (Giorn. di Matem., 6), which is analytical (and, like several other papers, should be mentioned elsewhere were we to adhere to a strict separation between synthesis and analysis). He reached the brilliant
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 hyperspace were brilliantly expounded and popularised in