When a distinct want of accordance is found to exist between the results of theory and direct measurement, interesting questions arise as to the mode in which we can account for this discordance. The ultimate explanation of the discrepancy may be accomplished in at least four ways as follows:—
(1) The direct measurement may be erroneous owing to various sources of casual error.
(2) The theory may be correct as far as regards the general form of the supposed laws, but some of the constant numbers or other quantitative data employed in the theoretical calculations may be inaccurate.
(3) The theory may be false, in the sense that the forms of the mathematical equations assumed to express the laws of nature are incorrect.
(4) The theory and the involved quantities may be approximately accurate, but some regular unknown cause may have interfered, so that the divergence may be regarded as a residual effect representing possibly a new and interesting phenomenon.
No precise rules can be laid down as to the best mode of proceeding to explain the divergence, and the experimentalist will have to depend upon his own insight and knowledge; but the following recommendations may be made.
If the experimental measurements are not numerous, repeat them and take a more extensive mean result, the probable accuracy of which, as regards casual errors, will increase as the square root of the number of experiments. Supposing that no considerable modification of the result is thus effected, we may suspect the existence of more deep-seated sources of error in our method of measurement. The next resource will be to change the size and form of the apparatus employed, and to introduce various modifications in the materials employed or the course of procedure, in the hope (p. [396]) that some cause of constant error may thus be removed. If the inconsistency with theory still remains unreduced we may attempt to invent some widely different mode of arriving at the same physical quantity, so that we may be almost sure that the same cause of error will not affect both the new and old results. In some cases it is possible to find five or six essentially different modes of arriving at the same determination.
Supposing that the discrepancy still exists we may begin to suspect that our direct measurements are correct, and that the data employed in the theoretical calculations are inaccurate. We must now review the grounds on which these data depend, consisting as they must ultimately do of direct measurements. A comparison of the recorded data will show the degree of probability attaching to the mean result employed; and if there is any ground for imagining the existence of error, we should repeat the observations, and vary the forms of experiment just as in the case of the previous direct measurements. The continued existence of the discrepancy must show that we have not attained to a complete acquaintance with the theory of the causes in action, but two different cases still remain. We may have misunderstood the action of those causes which we know to exist, or we may have overlooked the existence of one or more other causes. In the first case our hypothesis appears to be wrongly chosen and inapplicable; but whether we are to reject it will depend upon whether we can form another hypothesis which yields a more accurate accordance. The probability of an hypothesis, it will be remembered (p. [243]), is to be judged, in the absence of à priori grounds of judgment, by the probability that if the supposed causes exist the observed result follows; but as there is now little probability of reconciling the original hypothesis with our direct measurements the field is open for new hypotheses, and any one which gives a closer accordance with measurement will so far have better claims to attention. Of course we must never estimate the probability of an hypothesis merely by its accordance with a few results only. Its general analogy and accordance with other known laws of nature, and the fact that it does not conflict with other probable theories, must be taken into account, as we shall see in the next book. The requisite condition of a good hypothesis, that it must admit of the deduction of facts verified in observation, must be interpreted in the widest manner, as including all ways in which there may be accordance or discordance. All our attempts at reconciliation having failed, the only conclusion we can come to is that some unknown cause of a new character exists. If the measurements be accurate and the theory probable, then there remains a residual phenomenon, which, being devoid of theoretical explanation, must be set down as a new empirical fact worthy of further investigation. Outstanding residual discrepancies have often been found to involve new discoveries of the greatest importance.
Accordance of Measurements of Astronomical Distances.
One of the most instructive instances which we can meet, of the manner in which different measurements confirm or check each other, is furnished by the determination of the velocity of light, and the dimensions of the planetary system. Roemer first discovered that light requires time to travel, by observing that the eclipses of Jupiter’s satellites, although they occur at fixed moments of absolute time, are visible at different moments in different parts of the earth’s orbit, according to the distance between the earth and Jupiter. The time occupied by light in traversing the mean semi-diameter of the earth’s orbit is found to be about eight minutes. The mean distance of the sun and earth was long assumed by astronomers as being about 95,274,000 miles, this result being deduced by Bessel from the observations of the transit of Venus, which occurred in 1769, and which were found to give the solar parallax, or which is the same thing, the apparent angular magnitude of the earth seen from the sun, as equal to 8″·578. Dividing the mean distance of the sun and earth by the number of seconds in 8m. 13s.3 we find the velocity of light to be about 192,000 miles per second.