554 THE PRINCIPLES OF SCIENCE. [CHAP. 



atmospheric tide was thus proved to amount to between 

 003 and '004 inch. 1 



Theory yields the greatest possible assistance in applying 

 the method of means. For if we have a great number of 

 empirical measurements, each representing the joint effect 

 of a number of causes, our object will be to take the mean 

 of all those in which the effect to be measured is present, 

 and compare it with the mean of the remainder in which 

 the effect is absent, or acts in the opposite direction. The 

 difference will then represent the amount of the effect, or 

 double the amount respectively. Thus, in the case of the 

 atmospheric tides, we take the mean of all the observations 

 when the moon was on the meridian, and compare it with 

 the mean of all observations when she was on the horizon. 

 In this case we trust to chance that all other effects will 

 lie about as often in one direction as the other, and will 

 neutralise themselves in the drawing of each mean. It is 

 a great advantage, however, to be able to decide by theory 

 when each principal disturbing effect is present or absent ; 

 for the means may then be drawn so as to separate each 

 such effect, leaving only minor and casual divergences to 

 the law of error. Thus, if there be three principal effects, 

 and we draw means giving respectively the sum of all 

 three, the sum of the first two, and the sum of the last 

 two, then we gain three simple equations, by the solution 

 of which each quantity is determined. 



Explained JResidts of Measurement. 



The second class of measured phenomena contains those 

 which, after being determined in a direct and purely empi- 

 rical application of measuring instruments, are afterwards 

 shown to agree with some hypothetical explanation. Such 

 results are turned to their proper use, and several advan- 

 tages may arise from the comparison. The correspondence 

 with theory will seldom or never be precise ; and, even if 

 it be so, the coincidence must be regarded as accidental. 



If the divergences between theory and experiment be 

 comparatively small, and variable in amount and direction, 

 they may often be safely attributed to inconsiderabla 



1 Grant's History of Physical Astronomy, p. 162. 



