take the mean of all those in which the effect to be measured is present, and compare it with the mean of the remainder in which the effect is absent, or acts, it may be, in the opposite direction. The difference will then represent the amount of the effect, or double the amount respectively. Thus, in the case of the atmospheric tides, we take the mean of all the observations when the moon was on the meridian, and compare it with the mean of all observations when she was on the horizon. In this case we trust to chance that all other effects will lie about as often in one direction as the other in the drawing of each mean, and will neutralise themselves. It will be a great advantage, however, to be able to decide by theory when each principal disturbing effect is present or absent; for the means may then be so drawn as surely to separate each such effect, leaving only very minor and casual divergences to the law of error. Thus, if there be three principal effects, and we draw means giving respectively the sum of all three, the sum of the first two, and the sum of the last two, then we gain three simple equations, by the solution of which each quantity is determined.

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 different advantages may arise from the comparison.

The

correspondence with theory will seldom or never be absolutely 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

VOL. II.

attributed to various inconsiderable sources of error in the experimental processes. The strict method of procedure is to calculate, if possible, the probable error of the mean of the observed results (vol. i. p. 451), and then observe whether the theoretical result falls within the limits of probable error. If it does, and if, as we may say, 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 way or degree indicates the amount of fixed causes of error. Thus, if the mean results of any two modes of determining a quantity are so far apart that the limits of probable error do not overlap, we may infer the probable existence of some overlooked source of permanent error in one or both modes. We will further consider in a subsequent section the accordance or 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 manner that Newton first attempted to verify his theory of gravitation. He knew approximately the velocity produced in falling bodies at the earth's surface, and if the law of the inverse

square of the distance held true, and the reputed distance of the moon was correct, he could infer that the moon ought to fall towards the earth at the rate of fifteen feet in one minute. Now, the actual divergence of the moon from the tangent of its orbit appeared to amount only to thirteen feet in one minute, and there was a discrepancy of two feet in fifteen, which caused Newton to lay ‘aside at that time any further thoughts of this matter.' Many years afterwards, probably fifteen or sixteen years, Newton obtained more precise data from which he could calculate the size of the moon's orbit, and he then found the discrepancy to be inconsiderable.

His theory of gravitation was then verified so far as the moon was concerned; but this was to him only the beginning of a long course of deductive calculations, each ending in a verification. If the earth and moon attract each other, and also the sun and the earth, similarly there is no reason why the sun and moon should not attract each other. Newton followed out the consequences of this inference, and showed that the moon would not move as if attracted by the earth only, but sometimes faster and sometimes slower. Comparisons with Flamsteed's observations of the moon showed that such was the case. Newton argued again, that as the waters of the ocean are not rigidly attached to the earth, they might attract the moon, and be attracted in return, independently of the rest of the earth. Certain daily motions would then be caused thereby exactly resembling the tides, and there were the tides to verify the fact. It was the almost superhuman power with which he traced out geometrically the consequences of his theory, and submitted them to repeated comparison with experience, which constitutes his pre-eminence over all philosophers b.

The whole progress of physical astronomy has consisted b Elementary Lessons in Logic,' p. 262.

in a succession of predictions grounded on the theory of gravitation as to the inequalities of the planetary movements caused by mutual perturbations. These inequalities are so numerous, so small, and so complicated in character, that it would be an almost hopeless task to attempt to discover them empirically or tentatively by the comparison and classification of observations. But theory pretty easily indicates the period and general nature of the inequality to be detected, and by elaborate calculations even the amount of the effect may be assigned. Thus the inequality arising from the attraction of Venus and the earth was estimated by Sir George Airy to amount to no more than a few seconds at its maximum, while the period is no less than 240 years. Nevertheless, the indirect effects of this inequality upon the moon's motion are considerable, and are entirely verified in the lunar theory. Although prediction by theory is the general rule in physical astronomy, yet the empirical investigation of divergences from theory sometimes discloses effects which had been overlooked, or points out residual effects of unknown origin.

Quantities determined by Theory and not verified.

we may

It will continually happen that we are able, from certain measured phenomena and a correct theory, to determine the amount of some other phenomenon which either be unable to measure at all, or to measure with an accuracy corresponding to that required to verify the prediction. Thus Laplace having worked out an almost complete theory of the motions of Jupiter's satellites on the hypothesis of gravitation, found that these motions were greatly affected by the spheroidal form of Jupiter. Hence from the motions of the satellites, which can be observed with great accuracy owing to the frequent

eclipses and transits, he was able to argue inversely, and assign the ellipticity of the planet's section by theory. The ratio of the polar and equatorial axes thus determined was very nearly that of 13 to 14; and it agrees well with such direct micrometrical measurements of the planet as have been made; but Laplace believed that the theory gave a more accurate result than direct observation could yield, so that the theory could hardly be said to admit of direct verification.

The specific heat of air was believed on the grounds of direct experiment to amount to o 2669, the specific heat of water being taken as unity; but the methods of experiment were open to considerable causes of error. The late Professor Rankine showed in 1850 that it was possible to calculate from the mechanical equivalent of heat, and from other thermodynamic data, what this number should be, and he found for it o 2378. This determination was at the time accepted by him and others as the most satisfactory result, although not verified; subsequently in 1853 Regnault obtained by direct experiment the number O'2377, proving that the prediction had been well grounded.

It will be readily seen that in purely quantitative questions verification will be a matter of degree and probability. A less accurate method of measurement cannot verify the results of a more accurate method, so that if we arrive at a determination of the same physical quantity in several distinct modes it will often become a delicate matter of investigation to decide which result is most reliable, and should be used for the indirect determination of other quantities. For instance, Joule's and Thomson's ingenious experiments upon the thermal phenomena of fluids in motion involved, as one physical constant, the mechanical equivalent of heat; if requisite, c Philosophical Transactions' (1854), vol. cxliv. p. 364.