sidering the principles of approximation we found that a small portion of any curve will appear to be a straight line. Whenever our modes of measurement are comparatively rude, we must expect to be unable to detect the curvature. Thus Kepler made meritorious attempts to discover the law of refraction, and he slightly approximated to it when he observed that the angles of incidence and refraction if small bear a constant ratio to each other. Angles when small are very nearly as their sines, so that he reached an approximate result of the true law. Cardan assumed, 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 very small, and it becomes true again when the angle is a right angle; 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 quite unaware whether the expansion of mercury was exactly 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 some reason to the contrary is shown, we should do well to look upon every law of simple proportion as only provisionally true.

Nevertheless, there are many of the most important laws of nature which are in the form of simple proportions. Wherever a uniform cause acts in independence of its previous effects, we may expect this relation. Thus, an accelerating force acts equally upon a moving and a motionless body. Hence the velocity produced is always in simple proportion to the force, and also 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 Sir J. Herschel has remarked, we may expect simple proportion to manifest itself. In such cases the equation expressing the relation may have the still simpler form y=mx.

A similar simple relation holds true wherever there is a conversion of one substance or form of energy into another. The quantity of chloride of silver is proportional to the quantity either of chlorine or silver. The amount of 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. It is only among the most elementary causes and effects 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 so various, that men have not believed in such a simple view of the matter for more than about two centuries and a half. All those motions, again, are summed up in the law of gravity, of no great complexity, yet men never have, and never can be, able to exhaust the complications of action and reaction, even among a small number of planets. We should be on our guard against a tendency to assume that Preliminary Discourse,' &c. p. 152.

K

the connexion 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 thereby grew fat, and ceased to lay any eggs at all.

CHAPTER XXIII.

THE USE OF HYPOTHESIS.

IF the views of induction upheld in this work be correct, all inductive investigation consists in a marriage of hypothesis and experiment. When facts are already in our possession, we frame an hypothesis to explain their mutual relations, and by the success or non-success of this explanation is the value of the hypothesis to be entirely judged. In the framing and deductive treatment of such hypotheses, we must avail ourselves of the whole body of scientific truth 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. By deductive reasoning and calculation, we must endeavour to anticipate such new phenomena, especially those of a singular and exceptional nature, as would necessarily happen if the hypothesis be true. Out of the infinite number of observations and experiments which are possible at every moment, theory must lead us to select those few 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 to be treated must be in our possession before we can treat them. If, indeed, there were any single rule of

inductive method, it would direct us to make an exhaustive arrangement of facts in all possible orders. Given a certain number of specimens in a museum, we might arrive at the best possible classification by going systematically through all possible classifications, and, were we endowed with infinite time and patience, this would be an effective method. It doubtless is the method by which the first few simple steps are taken in every incipient branch of science. Before the dignified name of science is applicable, some coincidences will chance to force themselves upon the attention. Before there was a science of meteorology, or any comprehension of the true conditions of the atmosphere, all observant persons learned to associate a peculiar clearness of the atmosphere with coming rain, and a colourless sunset with fine weather. Knowledge of this kind is called empirical, as seeming to come directly from experience; and there is doubtless a considerable portion of our knowledge which must always bear this character.

We may be obliged to trust to the casual detection of coincidences in those branches of knowledge where we are deprived of the aid of any guiding notions; but a very little reflection will show the utter insufficiency of haphazard experiment, when applied to investigations of a complicated nature. At the best, it will be the simple identity, or partial identity, of classes, as illustrated in pp. 146-154 of the first volume, which can be thus detected. It was pointed out that, even when a law of nature involves only two circumstances, and there are one hundred distinct circumstances which may possibly be connected, there will be no less than 4950 pairs of circumstances between which a coincidence may exist. When a law involves three or more circumstances, the possible number of coincidences becomes vastly greater still. When considering, again, the subject