68 Transactions. — Miscellaneous, 



The imperceptible divergence of small portions from the ideal 

 standard is cumulative, and when we take very large portions 

 the divergence accumulates to a very perceptible amount. The 

 difference between the geometry of a cubic mile, if Euclid's 

 assumptions are true, and the geometry of a cubic mile if they 

 are false, we know, by experiment, to be quite insensible : yet 

 by the accumulation of excessively small (though not infinitely 

 small) divergences, it comes about that the geometry of a decil- 

 lion cubic miles {i.e., 10 60 cubic miles) may be so different on 

 the two hypotheses, that while, if Euclid's assumptions are true 

 the decillion cubic miles are but an infinitesimal portion of 

 entire space, if his assumptions are false, all space may actually 

 not hold so large a number of cubic miles. 



20. " The Professor, having perchance, after all, some doubts 

 as to the validity of this deduction, or possibly forgetting he has 

 proved it, essays to prove it again ; he says, 'and this (finiteness 

 of the universe) comes about in a very curious way' " (p. 108). 

 I can assure my critic that Professor Clifford had no such 

 doubts. If the universe is such that two shortest lines may 

 enclose a space, and if, nevertheless, all the other assumptions 

 of Euclid are true, then the extent of space is certainly a finite 

 number of cubic miles. The one statement is logically involved 

 in the other, though it may require a long and intricate process 

 of reasoning to prove it so. 



21. "The qualification put upon straight lines, ' straight 

 according to Leibnitz,' put, no doubt, all in good faith, as explau- 

 ative of straight lines, it does still, I feel assured, confer upon 

 them properties which straight lines have not" (p. 108). It 

 undoubtedly confers upon them properties which Euclidian 

 straight lines have not ; but the lines in question, though not 

 Euclidian straight lines — and if you will, not straight lines at 

 all, for the quarrel need not be over a word when the issue is 

 ono of fact — may nevertheless be the straightest lines that can 

 possibly be constructed (even ideally) in the space in which 

 we actually live. In other words, space may be so constituted 

 tint what Euclid calls straight lines cannot possibly be con- 

 structed in it, any more than a straight line can be constructed 

 on the surface of a sphere. Nevertheless the straightest lines 

 cf nstructible may be of the same shape all along and on all sides, 

 which great circles of a sphere are not : for though of the same 

 shape all along, they are concave on the one side and convex on 

 the other, also they may be shortest lines, which the great 

 circles of a sphere are not, relatively to solid space. The quarrel 

 about the definition of a straight line does not affect the issue in 

 the smallest degree. 



22. "I blame making so much, in this way, of the gap 'in 

 the chain of reasoning,' by which the truths of geometry should 

 be logically connected and represented " (p. 109). They cannot 



