xvn.] THE LAW OF ERROR. 391 



sound in air, it would be absurd to go back to the old 

 experiments which made the velocity from 1200 to 1474 

 feet per second ; for we know that the old observers did 

 not guard against errors arising from wind and other 

 causes. Old chemical experiments are valueless as re- 

 gards quantitative results. The old chemists found the 

 atmosphere in different places to differ in composition 

 nearly ten per cent., whereas modern accurate experi- 

 menters find very slight variations. Any method of 

 measurement which we know to avoid a source of error 

 is far to be preferred to others which trust to probabilities 

 for the elimination of the error. As Flamsteed says, 1 " One 

 good instrument is of as much worth as a hundred in- 

 different ones." But an instrument is good or bad only in 

 a comparative sense, and no instrument gives invariable 

 and truthful results. Hence we must always ultimately 

 fall back upon probabilities for the selection of the final 

 mean, when other precautions are exhausted. 



Legendre, the discoverer of the method of Least Squares, 

 recommended that observations differing very much from 

 the results of his method should be rejected. The subject 

 has been carefully investigated by Professor Pierce, who has 

 proposed a criterion for the rejection of doubtful observa- 

 tions based on the following principle : 2 observations 

 should be rejected when the probability of the system of 

 errors obtained by retaining them is less than that of the 

 system of errors obtained by their rejection multiplied by 

 the probability of making so many and no more abnormal 

 observations." Professor Pierce's investigation is given 

 nearly in his own words in Professor W. Chauvenet's 

 "Manual of Spherical and Practical Astronomy," which 

 contains a full and excellent discussion of the methods of 

 treating numerical observations. 3 



Very difficult questions sometimes arise when one or 

 more results of a method of experiment diverge widely 

 from the mean of the rest. Are we or are we not to ex- 

 clude them in adopting the supposed true mean result of 

 the method? The drawing of a mean result rests, as I 



1 Baily, Account of Flamsteed, p. 56. 



2 Gould's Astronomical Journal, Cambridge, Mass., vol. ii. p. 161. 



3 Philadelphia (London, Triibner) 1863. Appendix, vol. ii. p. 558. 



