18 OX THE RELATION OF 



see the peculiar characteristics of the natural and physical 

 sciences fully brought out. 



The essential differentia of these sciences seems to me to 

 consist in the comparative ease with which the individual 

 results of observation and experiment are combined under 

 general laws of unexceptionable validity and of an extra- 

 ordinarily comprehensive character. In the moral sciences, on 

 the other hand, this is just the point where insuperable diffi- 

 culties are encountered. In mathematics the general propo- 

 sitions which, under the name of axioms, stand at the head of 

 the reasoning, are so few in number, so comprehensive, and so 

 immediately obvious, that no proof whatever is needed for 

 them. Let me remind you that the whole of algebra and 

 aiithmetic is developed out of the three axioms : 



' Things which are equal to the same things are equal to 

 one another.' 



' If equals be added to equals, the wholes are equal.' 



' If unequals be added to equals, the wholes are unequal.' 

 And the axioms of geometry and mechanics are not more 

 numerous. The sciences we have named are developed out ot 

 these few axioms by a continual process of deduction from 

 them in more and more complicated cases. Algebra, however, 

 does not confine itsslf to finding the sum of the most hetero- 

 geneous combinations of a finite number of magnitudes, but in 

 the higher analysis it teaches us to sum even infinite series, 

 he terms of which increase or diminish according to the most 

 various laws ; to solve, in fact, problems which could never be 

 completed by direct addition. An instance of this kind shows 

 us the conscious logical activity of the mind in its purest and 

 most perfect form. On the one hand we see the laborious nature 

 of the process, the extreme caution with which it is necessary 

 to advance, the accuracy required to determine exactly the 

 scope of such universal principles as have been attained, the 

 difficulty of forming and understanding abstract conceptions. 

 On the other hand, we gain confidence in the certainty, the 

 range, and the fertility of this kind of intellectual work. 



The fertility of the method comes out moie strikingly in 



