338 A SHORT HISTORY OF SCIENCE 



the wreck of all. ... I have been occupied with the problem over 

 thirty years and I doubt if anyone has given it more serious attention, 

 though I have never published anything concerning it. Gauss(1824). 

 I will add that I have recently received from Hungary a little 

 paper on Non-Euclidean geometry, in which I rediscover all my own 

 ideas and results worked out with great elegance. . . . The writer 

 is a very young Austrian officer, the son of one of my early friends, 

 with whom I often discussed the subject in 1798, although my ideas 

 were at that time far removed from the development and maturity 

 which they have received through the original reflections of this 

 young man. I consider the young geometer von Bolyai a genius of 

 the first rank. Gauss (1832). 



The gradual adoption of new and revolutionary ideas on this 

 subject may be further illustrated by the following passages : 



The characteristic features of our space are not necessities of 

 thought, and the truth of Euclid's axioms, in so far as they specially 

 differentiate our space from other conceivable spaces, must be es- 

 tablished by experience and by experience only. R. S. Ball. 



If the Euclidean assumptions are true, the constitution of those 

 parts of space which are at an infinite distance from us, geometry 

 upon the plane at infinity, is just as well known as the geometry of 

 any portion of this room. In this infinite and thoroughly well-known 

 space the Universe is situated during at least some portion of an 

 infinite and thoroughly well-known time. So that there we have 

 real knowledge of something at least that concerns the Cosmos; 

 something that is true throughout the Immensities and the Eternities. 

 That something Lobatchewski and his successors have taken away. 

 The geometer of today knows nothing about the nature of the actually 

 existing space at an infinite distance ; he knows nothing about the 

 properties of this present space in a past or future eternity. He 

 knows, indeed, that the laws assumed by Euclid are true with an 

 accuracy that no direct experiment can approach, not only in this 

 place where we are, but in places at a distance from us that no as- 

 tronomer has conceived ; but he knows this as of Here and Now ; 

 beyond this range is a There and Then of which he knows nothing at 

 present, but may ultimately come to know more. Clifford. 



Everything in physical science, from the law of gravitation to 

 the building of bridges, from the spectroscope to the art of navigation, 



