144 Herbert Spencer 



those which are positively and actively inconceivable, 

 because they are clearly known by the mind to be 

 absolutely and universally impossible. At present we have 

 not to consider whether such perceptions are objectively 

 true and valid ; but to point out that, as a fact, they sub- 

 jectively exist. 



Let us, then, first note certain propositions which the 

 mind seems impotent to imagine, but which the intellect can 

 both understand and believe. The intellect clearly conceives 

 a force varying inversely as the square of the distance 

 between two bodies it reciprocally affects ; yet this variation 

 cannot be adequately represented by any image to the 

 imagination. We can, again, conceive an infinite addition 

 of fractions, which shall yet never attain to unity ; but such 

 a conception is utterly beyond the power of the imagination. 

 Again, we can not only conceive but it is evidently a neces- 

 sary truth that (a^ -f a h-\-x) + (a 6 -cc + 6^)= (a + &) x 

 {a + 6), let a, 6, and x represent whatever whole numbers 

 they may ; but this can by no means be directly represented 

 by the imagination. 



But conceptions may be formed as to modes of existence 

 of which we have had no experience whatever, and necessary 

 deductions can even be dra^vn from such conceptions. Thus 

 Professor Helmholtz has conceived ^ 'beings living and 

 moving along the surface of a solid body, who are able to 

 perceive nothing but what exists on this surface, and 

 insensible to all beyond it ' ; and he adds, * If such beings 

 lived on the surface of a sphere, their space would be with- 

 out a limit, but it would not be infinitely extended ; and their 

 axioms of geometry would turn out very different from 

 ours, and from those of the inhabitants of a plane. The 

 shortest lines which the inhabitants of a spherical surface 

 could draw would be arcs of greater circles'; also there 



1 The Academy, vol. i. p. 128. 



