of physical science. Only a few instances of each class can be given here. Empirical Measurements. Under the first head of purely empirical measurements, which have not been brought under any theoretical system, may be placed the great bulk of quantitative facts recorded by scientific observers. The tables of numerical results which abound in books on chemistry and physics, the huge quartos containing the observations of public observatories, the multitudinous tables of meteorological observations, which are continually being published, the more abstruse results concerning terrestrial magnetism—such results of measurement, for the most part, remain empirical, either because theory is defective, or the labour of calculation and comparison is too formidable. In the Greenwich Observatory, indeed, the salutary practice has been maintained by the present Astronomer Royal, of always reducing the observations, and comparing them with the theories of the several bodies. The divergences from theory thus afford material for the discovery of errors or of new phenomena; in short, the observations have been turned to the use for which they were intended. But it is to be feared that other establishments are too often engaged in merely recording numbers of which no real use is made, because the labour of reduction and comparison with theory is too great for private inquirers to undertake. In meteorology, especially, great waste of labour and money is taking place, only a small fraction of the results recorded being ever used for the advancement of the science. For one meteorologist like Quetelet, Dove, or Baxendell, who devotes himself to the truly useful labour of reducing other people's observations, there are hundreds who labour under the delusion that they are advancing science by loading our book-shelves with numerical tables. It is to be feared, in like manner, that almost the whole bulk of statistical numbers, whether commercial, vital, or moral, is of little scientific value. Purely empirical measurements may have a direct practical value, as when tables of the specific gravity, or strength of materials, assist the engineer; the specific gravities of mixtures of water with acids, alcohols, salts, &c., are useful in chemical manufactories, customhouse gauging, &c. ; observations of rainfall are requisite for questions of water supply; the refractive index of various kinds of glass must be known in making achromatic lenses; but in all such cases the use made of the measurements is not scientific but practical. It may be asserted, that no number which remains isolated, and uncompared by theory with other numbers, is of scientific value. Having tried the tensile strength of a piece of iron in a particular condition, we know what will be the strength of the same kind of iron in a similar condition, provided we can ever meet with that exact kind of iron again; but we cannot argue from piece to piece, nor lay down any laws exactly connecting the strength of iron with the quantity of its impurities. Quantities indicated by Theory, but Empirically Measured. In many cases we are able to foresee the existence of a quantitative effect, on the ground of general principles, but are unable, either from the want of numerical data, or from the entire absence of any mathematical theory, to assign the amount of such effect. We then have recourse to direct experiment to determine its amount. Whether we argued from the oceanic tides by analogy, or deductively from the theory of gravitation, there could be no doubt that atmospheric tides of some amount must occur in the atmosphere. Theory, however, even in the hands of Laplace, was not able to overcome the complicated mechanical conditions of the atmosphere, and predict the amounts of such tides; and, on the other hand, these amounts were so small, and were so masked by far larger undulations arising from the heating power of the sun, and from other meteorological disturbances, that they would probably have never been discovered by purely empirical observations. Theory having, however, indicated their existence and their periods, it was easy to make series of barometrical observations in places selected so as to be as free as possible from casual fluctuations, and then, by the suitable application of the method of means, to detect the small effects in question. The principal lunar atmospheric tide was thus proved to amount to between 003 and 004 inch.1 Theory yields the greatest possible assistance in applying the method of means. For if we have a great number of empirical measurements, each representing the joint effect of a number of causes, our object will be to 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 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, and will neutralise themselves in the drawing of each mean. It is 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 drawn so as to separate each such effect, leaving only 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 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 1 Grant's History of Physical Astronomy, p. 162. 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 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 thus verified as 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, there is reason to expect that the sun and moon should 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. Comparison 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 resembling the tides would then be caused, and there were the tides to verify the reasoning. It was the extraordinary power with which Newton traced out geometrically the consequences of his theory, and submitted them to repeated comparison with experience, which constitutes his pre-eminence over all physicists. Quantities determined by Theory and not verified. 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 we may 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 a 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. The motions of the satellites can be observed with great accuracy owing to their frequent eclipses and transits, and from these motions he was able to argue inversely, and assign the ellipticity of the planet. 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 obser |