THE INDESTRUCTIBILITY OF MATTER. 179 



The savage who cannot count the fingers on one hand, 

 can frame no definite thought answering to the statement 

 .that 7 and 5 are 12; still less can he frame the conscious- 

 ness that no other total is possible. 



The boy adding up figures inattentively, says to himself 

 that 7 and 5 are 11; and may repeatedly bring out a wrong 

 result by repeatedly making this error. 



Neither the non-recognition of the truth that 7 and 5 

 are 12, which in the savage results from undeveloped mental 

 structure, nor the assertion, due to the boy's careless mental 

 action, that they make 11, leads us to doubt the necessity of 

 the relation between these two separately-existing numbers 

 and the sum they make when existing together. Nor does 

 failure from either cause to apprehend the necessity of this 

 relation, make us hesitate to say that when its terms are dis- 

 tinctly represented in thought, its necessity will be seen; 

 and that, apart from any multiplied experiences, this neces- 

 sity becomes cognizable when structures and functions are 

 so far developed that groups of 7 and 5 and 12 can be in- 

 tellectually grasped. 



Manifestly, then, there is a recognition of necessary 

 truths, as such, which accompanies mental evolution. Along 

 with acquirement of more complex faculty and more vivid 

 imagination, there comes a power of perceiving to be neces- 

 sary truths, what were before not recognized as truths at all. 

 And there are ascending gradations in these recognitions. 

 A boy who has intelligence enough to see that things 

 which are equal to the same thing are equal to one another, 

 may be unable to see that ratios which are severally equal 

 to certain other ratios that are unequal to each other, are 

 themselves unequal ; though to a more-developed mind this 

 last axiom is no less obviously necessary than the first. 



All this which holds of logical and mathematical truths, 

 holds, with change of terms, of physical truths. There are 

 necessary truths in Physics for the apprehension of which, 

 also, a developed and disciplined intelligence is required; 



