CHAP. II.] FIEST TRUTHS. 39 



whether such perceptions are objectively true and valid, but 

 to point out that, as a fact, they subjectively 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 necessary truth 

 that (a 2 + a I -f x) + (al - x + I 2 ) = (a -f 1} x (a + 6), let 

 a, &, and x, represent whatever whole numbers they may ; yet 

 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 A fallacy of 



1 Professor 



necessary deductions can even be drawn from such Heimhoitz. 

 deductions. Thus Professor Heimhoitz has conceived * 

 &quot; 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 ; &quot; and he adds, &quot; if 

 such beings lived on the surface of a sphere, their space would 

 be without a limit, but it would not be infinitely extended ; 

 and the 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 ; &quot; also there 

 would be many shorter lines between the same two points if 

 there were two poles. Moreover, he tells us, such beings 

 &quot; would not be able to form the notion of parallel geodetical 

 lines, because every pair of their geodetical lines, when suffi 

 ciently prolonged, would intersect in two points,&quot; &c. This 

 passage is not only interesting as demonstrating our power of 



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



