xxv.] ACCORDANCE OF THEORIES. 555 



sources of error in the experimental processes. The strict 

 method of procedure is to calculate the probable error of 

 the mean of the observed results (p. 387), and then observe 

 whether the theoretical result falls within the limits of 

 probable error. If it does, and if the experimental results 

 agree as well with theory as they agree with each other, 

 then the probability of the theory is much increased, and 

 we may employ the theory with more confidence in the 

 anticipation of further results. The probable error, it 

 should be remembered, gives a measure only of the effects 

 of incidental and variable sources of error, but in no degree 

 indicates the amount of fixed causes of error. Thus, if the 

 mean results of two modes of determining a quantity are 

 so far apart tbat the limits of probable error do not overlap, 

 we may infer the existence of some overlooked source of 

 fixed error in one or both modes. We will further consider 

 in a subsequent section the discordance of measurements. 



Quantities determined by Theory and verified by 

 Measurement. 



One of the most satisfactory tests of a theory consists in 

 its application not only to predict the nature of a pheno- 

 menon, and the circumstances in which it may be observed, 

 but also to assign the precise quantity of the phenomenon. 

 If we can subsequently apply accurate instruments and 

 measure the amount of the phenomenon witnessed, we have 

 an excellent opportunity of verifying or negativing the 

 theory. It was in this manner that Newton first attempted 

 to verify his theory of gravitation. He knew approximately 

 the velocity produced in falling bodies at the earth's surface, 

 and if the law of the inverse square of the distance held 

 true, and the reputed distance of the moon was correct, he 

 could infer that the moon ought to fall towards the earth at 

 the rate of fifteen feet in one minute. Now, the actual 

 divergence of the moon from the tangent of its orbit ap- 

 peared to amount only to thirteen feet in one minute, and 

 there was a discrepancy of two feet in fifteen, which caused 

 Newton to lay " aside at that time any further thoughts of 

 this matter." Many years afterwards, probably fifteen or 

 sixteen years, Newton obtained more precise data from 



