ABSTRACT
The idea that space may itself be curved and that the axioms and assumptions on which our geometry since the time of Euclid have been based, may not be absolutely and exactly and eternally and universally true has been diligently studied during the last fifty years. The Russian Lobatchewsky, the Hungarian Bolyai and the German Riemann have developed systems of geometry by starting from premisses the opposite of those of Euclid and these systems are just as logical and consistent with themselves as the ordin ary or Euclidean geometry. These non-Euclidean geometries were at first commonly regarded as mere freaks of the mathematical imagination, but they have already proved valuable in leading to a recon sideration of the fundamental principles of our think ing and, if Einstein is right, they may be necessary
to explain physical phenomena. It is hard for the mathematician to discover anything useless. A dis tinguished American mathematician in announcing a new theorem exclaimed: “ And thank Heaven, no possible use can ever be found for it.” But, what ever it was, he made a rash boast for nowadays the mechanic treads on the heels of the mathematician and uses imaginary quantities, actual only in the fourth dimension, like V-1, in figuring out the wind ing of his dynamo.
