Turning now to the more technical side of the science, it is interesting to notice how this indirect character of most physical measurements determines not only the mode of measurements in general, but their precision as well. A measurement, we say, consists in the comparison of any concrete quantity with a definite portion of the same physical magnitude selected as a unit. In a few instances, the comparison 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 content of a sphere, for instance, is not determined by successive applications 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 appealed 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 psychological impressions do not entail equivalent causes. It is a remarkable fact that practically all exact measurements have been reduced 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