CHAPTER XXV. ACCORDANCE OF QUANTITATIVE THEORIES AND EXPERIMENTS. In the preceding chapter we found that facts may be classed under four heads as regards their connexion with theory, and our powers of explanation or prediction. The facts hitherto considered were generally of a qualitative rather than a quantitative nature; but when we look exclusively to the quantity of a phenomenon, and the various modes in which we may estimate or establish its amount, almost the same system of classification will hold good. There will, however, be five possible cases : (1) We may directly and empirically measure a phenomenon, without being able to explain why it should have any particular quantity, or to connect it by theory with other quantities. (2) In a considerable number of cases we can theoretically predict the existence of a phenomenon, but may be unable to assign its amount, except by direct measurement, or to explain the amount theoretically when thus ascertained. (3) We may measure a quantity, and afterwards explain it as related to other quantities, or as governed by known quantitative laws. (4) We may predict the quantity of an effect on theoretical grounds, and afterwards confirm the prediction by direct measurement. (5) We may indirectly predict or determine the quantity of an effect without being able to verify it by experi ment. These various classes of quantitative facts might be illustrated by an almost infinite number of interesting points in the history of physical science. Philosophical prophecies especially serve to show the mastery which is Sometimes attained over the secrets of nature, and to convince the least intelligent of the value of theory. 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 compiled and printed, 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 recognised theories of motion of the several bodies. The divergences from theory thus afford a constant supply of 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 reduc tion and comparison with theory, in detail, is far too great for private inquirers to undertake. In meteorology, especially, an enormous waste of labour and money is taking place, only a very 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 are under the delusion that they are advancing science by merely loading our book-shelves with numerical tables. Purely empirical measurements may often indeed 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 guaging, &c. ; observations of rain-fall 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 probably be asserted with truth, that no number which remains entirely isolated, and uncompared by theory with other numbers, is of any really 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, or lay down any laws exactly connecting the strength of iron with the quantity of its impurities. It is to be feared that almost the whole bulk of statistical numbers, whether commercial, vital, or moral, is at present, and probably will long continue, of little scientific value. madhị Tay, but Empirically In many as we are able to foresee the existence of a Maske efet in the grind of general principles Hem de want of numerical data, the tire a sense of any mathematical theory, to the amount of such effect. We then have recourse direct experiment to determine its amount. Whether we argued from the talle tiles by analogy, or more generally fr the thery of gravitation, there could be 1, 1 uit that atmospheric tides of some amount, depending on the apparent heights of the sun and the moon, 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 amount 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, 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 001 inch". Theory, in fact, 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 Grant's History of Physical Astronomy,' p. 162. 1 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. |