There is reason to suppose that Saccheri’s work had some influence on subsequent thought, although its full significance was certainly not perceived. The parallel axiom continued to be investigated, and the total effect of all these efforts was to induce a doubt concerning the absolute _necessity_ of the Euclidean geometry. Such a doubt was very daring; for two thousand years the postulates of Euclid had been accepted as absolutely true; the fact of their existence had profoundly influenced philosophy, and, indeed, theology. But the doubt persisted and grew, until finally, early in the nineteenth century, a perfectly logical and consistent non-Euclidean geometry, one explicitly denying the parallel postulate, was published to the world. As so often happens, the great step was taken by two men independently of one another, Lobatschewski, a Russian, and Bolyai, a Hungarian. It appeared, however, that both had been preceded by that great mathematical genius, Gauss, although he had been too timid to publish his conclusions. The new geometry developed the consequences of that one of Saccheri’s alternatives which supposed the interior angles of a triangle to be less than two right angles. The whole outlook on geometry now assumed a new complexion. Riemann tried the effect of denying the infinity of the straight line and of developing Saccheri’s other alternative. He found he was led to no contradictions. But with Riemann’s work we come to a yet further extension of geometry--the extension to space of four, five, or any number of dimensions. And these investigations, which seemed for some time to constitute the most gratuitous, although the most profound and subtle, exercises of the mind, have now received their complete justification by flowering into the Generalised Principle of Relativity. THE NEW SCIENTIFIC HORIZON