228 ON THE EELATION OF THE 



cases which have come under our observation or experience, under gen- 

 eral laws of absolute correctness and extensive application, while in the 

 former such generalization usually presents insurmountable difiSculties. 

 Indeed, in mathematics, the general laws called axioms are so few, so 

 comprehensive, and so evident that they require no i)roof. The whole 

 of the pure mathematics is developed out of the following three axioms : 



" Two magnitudes equal to a third are equal to each other. 



'' Equals added to equals produce equals. 



" Unequals added to equals produce unequals." 



The axioms of geometry and of theoretical mechanics are not more 

 numerous. These sciences are developed out of these few axioms by 

 employing every obtained conclusion in working out subsequent cases. 

 Arithmetic is not confined to the addition of a finite number of magni- 

 tudes, but teaches in higher analysis even to add an infinite number of 

 magnitudes, which increase or decrease in value according to the most 

 varying laws ; in other words, to solve i)roblems which could never be 

 done by direct methods. Here we see the conscious logical operation of 

 our mind in its purest and most perfect form ; here we learn the whole 

 labor and great care with which it must proceed, the accuracy necessary 

 to determine the full value of discovered general laws, the difficulty of 

 forming and understanding abstract ideas ; but we also learn at the 

 same time to gain confidence in the certainty, scope, and fruitfiilness of 

 such mental labors. 



"The latter becomes still more obvious in applied mathematical sciences, 

 especially in mathematical physics, to which must also be added phys- 

 ical astronomy. After Newton had once recognized, from the mechan- 

 ical analysis of the motions of planets, that between all ponderable matter 

 there exists an attraction, inversely proportional to the square of the dis- 

 tance, this simple law was sufficient for calculating with the greatest pre- 

 cision all the motions of the iilanets to the remotest periods of past or 

 future time, if we only have the place, velocity, and mass of the various 

 bodies of our system given for some point of time. We even recognize 

 the effects of the same force iu^the motions of double stars, whose dis- 

 tance from us is so great that their light is years in reaching us, and in 

 those whose distances have never been successfully measured. 



This discovery of the law of gravitation and of its consequences is 

 the most wonderful effort of logical power of which the human mind has 

 ever been capable. I do not assert that no men possessing powers of 

 logical abstraction as great or greater than those of Newton or (^ the 

 other astronomers, who led the way to or developed his discovery, have 

 ever lived ; but that there has never been a better opportunity tlian that 

 of solving the confused motions of the planets, which had before served 

 only to foster a belief in astrology among the uneducated, and which 

 were now brought under a law that was able to account for the slight- 

 est details of their motions. 

 Other branches of physics have also been d^eloped according to the 



