ON THE STUDY OF PHYSICS. 63 



Turning now to the more technical side of the science, it is inter- 

 esting to notice how this indirect character of most physical measure- 

 ments determines not only the mode of measurements in general, but 

 their precision as well. A measurement, we say, consists in the com- 

 parison of any concrete quantity with a definite portion of the same 

 physical magnitude selected as a unit. In a few instances, the com- 

 parison is direct, as in the determination of a length by a divided scale, 

 but in the great majority of cases the numerical measure of a quantity 

 is computed by the aid of a relation between other magnitudes which 

 may be more directly, or, at least, more simply, measured. The con- 

 tent of a sphere, for instance, is not determined by successive applica- 

 tions of the unit-cube to the enclosed space, but by first measuring its 

 diameter with calipers and then calculating the volume by the known 

 geometrical relation between the two. Or, to cite another illustration, 

 the direct comparison of a given velocity with the assumed unit of 

 velocity would be a troublesome thing, involving, if they were not very 

 nearly equal in amount, the repeated subdivision of the one or the 

 multiplication of the other. To avoid this, we define the measure of 

 velocity to be the distance traversed per second, and the measurement 

 may then be effected by the simpler process of measuring separately 

 a distance and a time. 



In making comparisons, one of the senses must ultimately be ap- 

 pealed to as the judge of the coincidence of two values, but in forming 

 this judgment apparatus is introduced of such sort that the comparison 

 shall contain the least amount of personal bias or subjective impression, 

 thus eliminating as far as possible the psychological element, since the 

 thing desired is a physical equality rather than a psychological one. 

 The former must indeed involve some form of the latter, but equal psy- 

 chological impressions do not entail equivalent causes. It is a re- 

 markable fact that practically all exact measurements have been re- 

 duced to the judgment of the coincidence of two lines by the sense of 

 sight. This universal preference of the eye is probably due not so 

 much to the greater freedom of this sense from illusive deception, as 

 to its unique relation to geometrical space. Various of the other 

 senses are able to distinguish and even to compare degrees or amounts 

 of differences in the sensations peculiar to them, i. e., they are able 

 to estimate a kind of interval or difference in these sensations. The ear 

 in connection with memory, is able to distinguish an interval of time 

 between two successive taps as small as one one hundredth of a second, 

 which is perhaps ten times as well as the eye can do with successive 

 flashes. In another kind of sensation peculiar to hearing, namely 

 pitch, the ear without the aid of beats easily distinguishes sounds whose 

 frequencies are in a smaller ratio than 25/24. Similarly the muscular 

 sense will under proper conditions distinguish an increase in weight of 



