BENJAMIN PEIRCE. 447 



motions liad the ratios 2 : 5 and 1 : 2, and tliat the reasonings of Le- 

 verrier were thereby vitiated. Not a little controversy has come 

 from these papers of Professor Peirce ; and we cannot say that the 

 last word in regard to the question has even yet been spoken. As is 

 not unusual in like discussions, there is probably a portion of truth 

 and a portion of error with either party. Leverrier and Adams 

 each, as Professor Peirce admits, did point out correctly a place where 

 a planet should be looked for, and assigned paths which that planet 

 could have been travelling for more than one hundred and twenty 

 years previously, and have caused the observed irregularities. Yet 

 the elements of that planet's orbit and its mass and those of Neptune 

 differ widely enough to justify the assertion that they were not cor- 

 rectly given. 



On the other hand, astronomers will not probably agree with Pro- 

 fessor Peirce in regarding the change of character of the perturbations 

 ■when the mean motions of the new planet and of Uranus pass through 

 the exact ratios 3 : 5 and 1:2 as of vital importance. In the usual 

 form of development these fractions do indeed make certain terms 

 infinite. That belongs, however, to the form of the development, not 

 to the perturbations. In solving the question, *' Where is the disturb- 

 ing body ? " the solution need not have involved these forms ; and it 

 has not been shown that they entered into the work of either Le- 

 verrier or Adams in such a way as to vitiate it. 



That the problem was really indeterminate has been steadily held 

 by Professor Peirce. In January, 1878, he read to this Academy a 

 paper, which has not been published, and the conclusions of which, 

 therefore, will not compel the assent of astronomers until some one 

 else shall have gone over the same questions. He showed a chart of 

 the plane of the ecliptic with the orbits of Uranus and Neptune, and 

 having those parts of the plane shaded within any part of which a 

 planet of arbitrary mass might have been situated in Sptember, 184:6, 

 and yet have caused, in the preceding years, the observed irregularities 

 in the motions of Uranus, within reasonable limits of error. With a 

 circular orbit, a large fraction (more than one half) of the ecliptic, as 

 seen from the earth, contained some of the shaded portions. If an 

 eccentricity not greater than one tenth be allowed, the region was 

 greatly enlarged. While, therefore, the solutions of Leverrier and 

 Adams gave a place and a path that explained the disturbances, the 

 problem in its nature was not, he claimed, one having a single answer, 

 or even a finite number of answers. 



In 1852, Professor Bache, then Superintendent of the United States 



