Yet no incontestable general law has been established. Several functions have been proposed to express the elastic force of the vapour as depending on the temperature. The first form is that of Young, namely F = (a + b t)TM, in which a, b, and m are unknown quantities to be determined by observation. Roche proposed, on theoretical grounds, a complicated formula of an exponential form, and a third form of function is that of Biot, as follows-log F a + bat + c ßt. = a + bat + c ßt. I mention these formulæ, because they well illustrate the feeble. powers of empirical inquiry. None of the formulæ can be made to correspond closely with experimental results, and the two last forms correspond almost equally well. There is very little probability that the real law has been reached, and it is unlikely that it will be discovered except by deduction from mechanical theory. Much ingenious labour has been spent upon the discovery of some general law of atmospheric refraction. Tycho Brahe and Kepler commenced the inquiry: Cassini first formed a table of refractions, calculated on theoretical grounds Newton entered into some profound investigations upon the subject: Brooke Taylor, Bouguer, Simpson, Bradley, Mayer, and Kramp successively attacked the question, which is of the highest practical importance as regards the correction of astronomical observations. Laplace next laboured on the subject without exhausting it, and Brinkley and Ivory have also treated it. The true law is yet undiscovered. A closely connected problem, that regarding the relation between the pressure and elevation in different strata of the atmosphere, has received the attention of a long succession of physicists and was most carefully investigated by Laplace. Yet no invariable and general law has been detected. The same may be said concerning the law of human mortality; abundant statistics on this subject are available, and many hypotheses more or less satisfactory have been put forward as to the form of the curve of mortality, but it seems to be impossible to discover more than an approximate law. It may perhaps be urged that in such subjects no single invariable law can be expected. The atmosphere may be 1 Jamin, Cours de Physique, vol. ii. p. 138. divided into several variable strata which by their unconnected changes frustrate the exact calculations of astronomers. Human life may be subject at different ages to a succession of different influences incapable of reduction under any one law. The results observed may in fact be aggregates of an immense number of separate results each governed by its own separate laws, so that the subjects may be complicated beyond the possibility of complete resolution by empirical methods. This is certainly true of the mathematical functions which must some time or other be introduced into the science of political economy. Simple Proportional Variation. When we first treat numerical results in any novel kind of investigation, our impression will probably be that one. quantity varies in simple proportion to another, so as to obey the law y = mx + n. We must learn to distinguish carefully between the cases where this proportionality is really, and where it is only apparently true. In considering the principles of approximation we found that a small portion of any curve will appear to be a straight line. When our modes of measurement are comparatively rude, we must expect to be unable to detect the curvature. Kepler made meritorious attempts to discover the law of refraction, and he approximated to it when he observed. that the angles of incidence and refraction if small bear a constant ratio to each other. Angles when smal are nearly as their sines, so that he reached an approximate result of the true law. Cardan assured, probably as a mere guess, that the force required to sustain a body on an inclined plane was simply proportional to the angle of elevation of the plane. This is approximately the case when the angle is small, but in reality the law is much more complicated, the power required being proportional to the sine of the angle. The early thermometer-makers were unaware whether the expansion of mercury was proportional or not to the heat communicated to it, and it is only in the present century that we have learnt it to be not so. We now know that even gases obey the law of uniform expansion by heat only in an approximate manner. Until reason to the contrary is shown, we should do well to look upon every law of simple proportion as only provisionally true. Nevertheless many important laws of nature are in the form of simple proportions. Wherever a cause acts in independence of its previous effects, we may expect this relation. An accelerating force acts equally upon a moving and a motionless body. Hence the velocity produced is in simple proportion to the force, and to the duration of its uniform action. As gravitating bodies never interfere with each other's gravity, this force is in direct simple proportion to the mass of each of the attracting bodies, the mass being measured by, or proportional to inertia. Similarly, in all cases of "direct unimpeded action," as Herschel has remarked, we may expect simple proportion to manifest itself. In such cases the equation expressing the relation may have the simple form y = mx. A similar relation holds true when there is conversion of one substance or form of energy into another. The quantity of a compound is equal to the quantity of the elements which combine. The heat produced in friction is exactly proportional to the mechanical energy absorbed. It was experimentally proved by Faraday that "the chemical power of the current of electricity is in direct proportion to the quantity of electricity which passes." When an electric current is produced, the quantity of electric energy is simply proportional to the weight of metal dissolved. If electricity is turned into heat, there is again simple proportion. Wherever, in fact, one thing is but another thing with a new aspect, we may expect to find the law of simple proportion. But it is only in the most elementary cases that this simple relation will hold true. Simple conditions do not, generally speaking, produce simple results. The planets move in approximate circles round the sun, but the apparent motions, as seen from the earth, are very various. All those motions, again, are summed up in the law of gravity, of no great complexity; yet men never have been, and never will be, able to exhaust the complications of action and reaction arising from that law, even among a small number of planets. Preliminary Discourse, &c., p. 152. We should be on our guard against a tendency to assume that the connection of cause and effect is one of direct proportion. Bacon reminds us of the woman in Æsop's fable, who expected that her hen, with a double measure of barley, would lay two eggs a day instead of one, whereas it grew fat, and ceased to lay any eggs at all. It is a wise maxim that the half is often better than the whole. CHAPTER XXIII. THE USE OF HYPOTHESIS. IF the views upheld in this work be correct, all inductive investigation consists in the marriage of hypothesis and experiment. When facts are in our possession, we frame an hypothesis to explain their relations, and by the success of this explanation is the value of the hypothesis to be judged. In the invention and treatment of such hypotheses, we must avail ourselves of the whole body of science already accumulated, and when once we have obtained a probable hypothesis, we must not rest until we have verified it by comparison with new facts. We must endeavour by deductive reasoning to anticipate such phenomena, especially those of a singular and exceptional nature, as would happen if the hypothesis be true. Out of the infinite number of experiments which are possible, theory must lead us to select those critical ones which are suitable for confirming or negativing our anticipations. This work of inductive investigation cannot be guided by any system of precise and infallible rules, like those of deductive reasoning. There is, in fact, nothing to which we can apply rules of method, because the laws of nature must be in our possession before we can treat them. If there were any rule of inductive method, it would direct us to make an exhaustive arrangement of facts in all possible orders. Given the specimens in a museum, we might arrive at the best classification by going systematically through all possible classifications, and, were we endowed with infinite time and patience, this would be an effective method. It is the method by which the first simple steps |