NATURAL SCIENCE TO GENERAL SCIENCE. 21 



results of observation and experiment are com- 

 bined under general laws of unexceptionable validity and 

 of an extraordinarily comprehensive character. In the 

 moral sciences, on the other hand, chis is just the point 

 where insuperable difficulties are encountered. In mathe- 

 matics the general propositions 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 arithmetic is 

 developed out of the three axioms : 



6 Things which are equal to the same things are equal 

 to one another.' 



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

 6 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 

 of these few axioms by a continual process of deduction 

 from them in more and more complicated cases. Algebra, 

 however, does not confine itself to finding the sum of the 

 most heterogeneous combinations of a finite number of 

 magnitudes, but in the higher analysis it teaches us to 

 sum even infinite series, the terms of which increase or 

 dimmish 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 con- 

 ceptions. On the other hand, we gain confidence in the 

 certainty, the range, and the fertility of this kind of 

 intellectual work. 



