then, they might have been used to predict or to correct that most important constant. But if other more direct methods of experiment give the mechanical equivalent of heat with superior accuracy, then the experiments on fluids will be turned to a better use in detecting and assigning various quantities relating to the theory of fluids. We will further consider questions of this kind in succeeding sections.

There are of course many quantities assigned on theoretical grounds which we are quite unable to verify with corresponding accuracy. The thickness of a film of gold leaf, the average depths of the oceans, the velocity of a star's approach to or regression from the earth as inferred from spectroscopic data, or other quantities indirectly determined (see vol. i. pp. 345-349), might be cases in point; but many others might be quoted where direct verification seems impossible. Newton and many subsequent physicists have accurately measured the lengths of light undulations, and by several distinct methods we learn the velocity with which light travels. Since an undulation of the middle green is about five ten-millionths of a metre in length, and travels at the rate of nearly 300,000,000 of metres per second, it necessarily follows that about 600,000,000,000,000 undulations must strike in one second the retina of an eye which perceives such light. But how are we to verify such an astounding calculation by directly counting pulses which recur six hundred billions of times in a second?

Discordance of Theory and Experiment.

When a distinct want of accordance is found to exist between the results of theory and direct measurement, several interesting questions may arise as to the mode in which we can account for this discordance. The ultimate

explanation of the discrepancy may be accomplished in any one of at least four distinct ways, as follows:

(1) The direct measurement may be erroneous owing to various sources of casual error.

(2) The theory may be correct so 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 general recommendations may perhaps be made.

In the first place, if the experimental measurements are not numerous, repeat them and take a more extensive mean result, the probable accuracy of which, as regards freedom from casual errors of experiment, 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 some more deepseated and constant source 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 in the course of procedure, in the hope, as before explained (vol. i. p. 462), that some cause of constant error may thus be removed. If the inconsistency with theory still re

mains 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 well begin to suspect that our direct measurements are correct, but 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 various recorded results 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 really 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 do 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 any other hypothesis which yields a more accurate accordance. The probability of an hypothesis, it will be remembered (vol.`i. p. 279), is to be judged entirely by the probability that if the supposed causes exist the observed result follows; but as there is now very 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 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 any 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 possible 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 deliberate investigation. As a matter of fact these 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 could meet, as regards 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 in travelling, by observing that the eclipses of Jupiter's satellites, although they of course occur at fixed moments of absolute time, are visible at different moments in different parts of the earth's orbit, according to the distance of the earth and Jupiter. The time occupied by light in traversing the mean semidiameter 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 what is the same thing, the apparent size of the earth as seen from the sun, as equal to 8" 578. Now, dividing the mean distance of the sun and earth by the number of seconds in 8m. 138.3 we find the velocity of light to be about 192,000 miles per second.

apparent change in the

Nearly the same result was obtained in an apparently very different manner. The aberration of light is the direction of a ray of light owing to the composition of its motion with that of the earth's motion round the sun. If we know the amount of aberration and the mean velocity of the earth we can very simply estimate that of light which is thus found to be 191,102 miles (166,072 geographical miles) per second. Now this determination depends upon an entirely new physical quantity, that of aberration, which is ascertained by direct observation of the stars, so that the close accordance of the estimates of the velocity of light as thus arrived at by different methods might seem to leave little room for doubt, the difference being less than one per cent.

Nevertheless, experimentalists were not satisfied until they had succeeded in actually measuring the velocity of light by direct experiments performed upon the earth's surface. Fizeau, by a rapidly revolving toothed wheel, estimated the velocity at 195,920 miles per second. As this result differed by about one part in sixty from estimates previously accepted, there was thought to be room for further investigation. The revolving mirror, previously used by Mr. Wheatstone in measuring the velocity of electricity, was now applied in a more refined manner by Fizeau and by Foucault to determine the velocity of light. The latter physicist finally came to the startling