40 LESSONS FROM NATURE. [Cu\v. II. 



transcending experience by conception, but even more so as 

 the solemn enunciation of a transparent fallacy by a man of 

 eminence. Professor Helmholtz concludes : &quot; We may resume 

 the results of these investigations by saying that the axioms 

 on which our geometrical system is based are no necessary 

 truths.&quot; And Professor Clifford * cites with approval the 

 article here quoted, and adopts its conclusions. Nevertheless 

 the fallacy is surely transparent. Unless geometrical axioms 

 were necessary truths, it would be impossible for these pro 

 fessors to declare what would or would not be the necessary 

 results attending such imaginary conditions. And -other 

 systems&quot; could not, as Professor Helmholtz admits t they 

 may, &quot; be developed analytically with perfect logical con 

 sistency.&quot; If such beings as are supposed called the lines re 

 ferred to &quot; straight,&quot; they would mean by that word what we 

 should call &quot; arcs of great circles.&quot; Whether such beings 

 could conceive parallel lines or not, there is no evidence to 

 show, but there is no shadow of foundation for asserting that, 

 if they could conceive them, they would not perceive the im 

 possibility of their ever meeting, as we can perceive the 

 necessary relations of their supposed space conditions which, 

 by the hypothesis, are not ours. 



On this subject Mr. Lewes has observed : J &quot; In a space of 

 two or of four dimensions many geometrical propositions 

 which relate to a space of three dimensions would not be true. 

 Who doubts it ? Who expects that the same results can be 

 the product of different factors ? &quot; 



Mr. Spencer, as we have seen, deems it absolutely in- 

 Mr. spencer s conceivable that an unextended object can offer 



example of . . _ T . . . 



absolute resistance or exercise pressure. JN evertneiess, he 



inconceiv- 1 . 1 . 1 



ability. himselt is able to conceive &quot; body, as really apart 



from extension, and in terms of force only since that which 



s described must be conceived; and he tells us it is 



* Macmillan a Magazine, October 1872, p. 504. 

 t The Academy, vol. i. p. 130. 

 Problems of Life and Mind, vol. i. p. 378. 

 First Principles (2nd edition), p. 167. 



