OF ARTS AND SCIENCES. 349 



hypothesis which leads to improbable results is itself improbable to 

 just that extent. 



It is not likely that a series of observations of any great extent was 

 ever made, in which some were not rejected on account of the magni- 

 tude of the deviations from the arithnietio mean. The object of the 

 Criterion is to systematize this species of rejection, and reduce it to 

 a form of exact computation. Wherever it has been applied, I believe 

 that it has been found to accord with the best judgment of observers 

 and comiiuters. This fact has been to me an agreeable surprise, which 

 has more than balanced the unfavorable criticism, having its founda- 

 tion in misconception. It has been a surprise, because the theory was 

 altogether a priori, and independent of comparison with observation. 



I might add concerning the charge of inconsistency, that it would 

 seem to be almost equally applicable to the case where two sets of 

 observations made by different observers, and the arithmetical mean of 

 each set differing materially from the arithmetical mean of the whole, 

 the difference is explained by personal' equation. The argument for 

 the hypothesis of personal equation must rigidly assume the same form 

 with that by which the Criterion is established. 



There might be a doubt as to the reality of such abnormal observa- 

 tions with their obscure sources of error. I am frank to admit that 

 in most cases science will detect the system of the sources of error, 

 and free the observations from them. But even such a case as the 

 familiar one of writing down a wrong figure must be included among 

 these sources, and is evidently insusceptible of correction ; whereas, if 

 it be sufficiently large, it will be eliminated by the Criterion. Another 

 case, which must often have occurred in transit observations by eye 

 and ear, depends upon the erroneous mental sub-division of the record, 

 of which I have given the analysis in another place. I can point out 

 many observations which were manifestly erroneous on this account, 

 and which it would be too vast a lalior to undertake to rectify. Before 

 the judgment of tlie Criterion all such errors disappear, if they are 

 large enough to be of serious injury. 



That some of the observations wliich are not rejected may be ab- 

 normal, notwithstanding the sraallness of their errors, must be admitted. 

 This possibility was fully recognized in the geometrical development 

 which was given in the Astronomical Journal ; and I am not awai'e 

 that there has been any criticism adverse to the mathematics of that 

 article. 



To Hon. C. P. Patterson, 



Superintendent U. S. Coast Survey. 



