338 PROCEEDINGS OF THE AMERICAN ACADEMY 



Mr. Pierce remarked that his original views were unchanged 

 in regard to the importance to be attached to the vast dis- 

 crepancies between the predicted and observed orbits of the 

 planet which disturbs the motions of Uranus. 



" Neptune is not the planet designated by geometry, although it is a 

 perfect solution of the problem which analysis had undertaken to in- 

 vestigate, and had really solved, but in a form radically different from 

 the actual solution of nature. This is not a personal question ; it is 

 certainly not one in which the reputations of Adams and Leverrier are 

 concerned. The accuracy of their investigations is not assailed ; but 

 it is expressly admitted that they announced the correct results of most 

 profound analytical researches. 



" The fair consideration of this question cannot be made without re- 

 calling the true office and position of geometry in science, which alone 

 entitles it to the appellation of the key to the physical world. Mathe- 

 matics is the science of exact measurement ; accuracy is its sole aim 

 and object, and it is this which places it in harmony with a creation, 

 which is subject to perfect law and undeviating order. An inaccurate 

 result cannot be a geometrical one ; a result, inaccurate beyond ceiiain 

 well-defined limits, does not belong to the exact science ; an inconsis- 

 tency, which exceeds a certain amount, may not be neglected by him 

 who deals with nothing but more or less, without disturbing the very 

 foundations of his faith. 



"The geometrical statement was distinctly made, that the planet 

 which disturbed Uranus could not be at a less mean distance from the 

 sun than thirty-five times the earth's mean distance from the sun ; that 

 is, that no planet which was within this distance could cause the ob- 

 served irregularities in the motions of Uranus. Neptune's mean dis- 

 tance from the sun is only thirty times the earth's mean distance, and yet 

 Neptune does account for the perturbations of Uranus. It is five hun- 

 dred millions of miles nearer the sun than it was distinctly stated by 

 geometry that it possibly could be, in order to be capable of producing 

 the effect which it actually does produce. The spirit of mathematical 

 accuracy cannot be supposed to be sufficiently elastic to embrace so 

 great an inconsistency, amounting to one sixth part of Neptune's dis- 

 tance from the sun, and to one half of the distance of his orbit from that 

 of Uranus. 



" Whence comes this enormous difference between the theoretical 



