Frankland — The Non-Euclidian Geometry Vindicated. 67 



are omitted. The original reads thus: — "In this case the 

 universe, as known, is again a valid conception," &c. Professor 

 Clifford very clearly explains what he means by this, in an 

 earlier part of the lecture from which I quoted. Referring 

 to the state of science before Lobatchewsky he says, " 

 the laws of space and motion that we are presently going 

 to examine, implied an infinite space and an infinite duration, 

 about whose properties as space and time everything was 

 accurately known. The very constitution of those parts of it 

 which are at an infinite distance from us, ' geometry upon the 

 plane at infinity,' is just as well known, if the Euclidian 

 assumptions are true, 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 here we have real 

 knowledge of something at least that concerns the cosmos; 

 something that is true throughout the immensities and 

 eternities. That something Lobatchewsky and his successors 

 have taken away. The geometer of to-day knows nothing 

 about the nature of actually existing space at an infinite 

 distance : he knows nothing about the properties of this 

 present space in a past or a 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 at places at a distance from us 

 which no astronomer has conceived : but he knows this as of 

 here, and now ; beyoud his range is a there, and a then, of 

 which he knows nothing at present, but may ultimately come 

 to know more. So, you see, there is a real parallel between 

 the work of Copernicus and his successors on the one hand, and 

 the work of Lobatchewsky and his successors on the other. 

 In both of these the knowledge of immensity and eternity is 

 replaced by knowledge of here and now. And in virtue of these 

 two revolutions the idea of the universe, the macrocosm, the all, 

 as a subject of human knowledge, and therefore of human 

 interest, has fallen to pieces." 



Well, then : If space should turn out to be of finite extent, 

 the idea of the universe (the universe of matter at any rate) would 

 be reinstated, as in a certain measure an object of knowledge 

 throughout its entire extent, as it was supposed to be before 

 Lobatchewsky arose, when Euclidian geometers could tell us the 

 exact constitution of the whole of space. 



19. " To make the conclusion agree with the premises, it 

 should have gone no further than to affirm that the universe 

 may not differ sensibly from an infinite one" (p. 108). By no 

 means: The surface of a sheet of still water does not differ 

 sensibly from a Euclidian plane, but the surface of the Pacific 

 Ocean, even if perfectly calm, differs very sensibly from a plane. 



