Explained Results of Measurement.
The second class of measured phenomena contains those which, after being determined in a direct and purely empirical application of measuring instruments, are afterwards shown to agree with some hypothetical explanation. Such results are turned to their proper use, and several advantages 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 inconsiderable 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 that 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 phenomenon, 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