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SCIENTIFIC THOUGHT. 



way the idea of the dimensions of space was extended, 

 and four and more dimensions freely spoken of when 

 really only a limited number is geometrically pres- 

 entable. In the hands of mathematicians these terms 

 are useful, and we may discard the criticism of philo- 

 sophers and laymen as based on misunderstanding. 1 

 The introduction, however, into geometrical work of con- 

 ceptions such as the infinite, the imaginary, and the 

 relations of hyperspace, none of which can be directly 

 imaged, has a psychological significance well worthy 

 of examination. 2 It gives a deep insight into the 

 resources and working of the mind. We arrive at 

 the borderland of mathematics and philosophy. 



1 The most important philosophi- 

 cal criticism of the non-Euclidean 

 geometry is that of Lotze, con- 

 tained in the second book, chap. 

 ii., of the 'Metaphysik' (1879, p. 

 249, &c. ) It must not be forgotten 

 that Lotze wrote at a time when 

 the novel and startling conceptions 

 put forward by popular writers on 

 the subject had been employed in 

 the interest of a spiritualistic philo- 

 sophy, to the delusions of which 

 some even of Lotze's friends had 

 fallen a prey. This explains the 

 severity of Lotze's criticisms, which 

 are of the very same nature as those 

 he pronounced many years earlier 

 on similar aberrations (see ' Kleine 

 Schriften,' vol. iii. p. 329). Those 

 who are interested in following up 

 the subject should refer to] the 

 writings of Friedr. Zb'llner as col- 

 lected in the four vols. of his 

 ' Wissenschaftliche Abhandlungeu ' 

 (Leipzig, 1878-81). They belong 

 to the curiosities of the philosophi- 

 cal and scientific literature of that 

 age, but can hardly claim a place in 

 the history of thought. 



2 See the remark of Cay ley in his 

 Presidential Address (' Coll. Works,' 



vol. xi. p. 434) : " The notion, 

 which is really the fundamental 

 one (and I cannot too strongly 

 emphasise the assertion), under- 

 lying and pervading the whole of 

 modern analysis and geometry, is 

 that of imaginary magnitude in 

 analysis and of imaginary space (or 

 space as a locus in quo of imaginary 

 points and figures) in geometry. I 

 use in each case the word imaginary 

 as including real. This has not 

 been, so far as I am aware, a subject 

 of philosophical discussion or in- 

 quiry. As regards the older meta- 

 physical writers, this would be quite 

 accounted for by saying that they 

 knew nothing, and were not bound 

 to know anything, about it ; but at 

 present, and considering the prom- 

 inent position which the notion 

 occupies say even that the conclu- 

 sion were that the notion belongs 

 to mere technical mathematics or 

 has reference to nonentities, in 

 regard to which no science is pos- 

 sible still it seems to me that (as 

 a subject of philosophical discussion) 

 the notion ought not to be thus 

 ignored ; it should at least be shown 

 that there is a right to ignore it." 



