tion are present together.

With the Method of Residues we

do not directly observe what happens when all the causes except the one in question are present together. We only calculate it from what we know of the way in which they act when they are present separately.

The Method of Residues is attended in practice by three dangers.

The first danger is that in making our subtraction we may overlook the combined' or heteropathic effect of some of the causes which we subtract, and thus attribute too much

to the remaining causes. For example, three persons, A, B, and C, are in a room from which we hear the sounds of a violent disturbance. We know by previous inductions that A's disposition is quiet and peaceable; we know the same about B; and so we conclude that C is responsible for the disturbance. And yet we may be wrong, for however quiet and peaceable A and B may be in themselves, there may be something about them—some trait of disposition or some old misunderstanding—that makes a conflict almost inevitable when the two are together.

The second danger to which we are exposed in using this Method of Residues is that of overlooking some circumstance that is really present, and of thus attributing an entirely false value to the presence of something else. It may be that the man whose income we were inquiring about a little while ago has an allowance from his grandfather, of which he says nothing, and that the iron company is a source not of income but of expense; and it may be that the man suspected of peculation is getting an immense royalty, that his employer knows nothing about, from a patent.

The third danger is that even when the data are all correct there may be a blunder somewhere in our calculations. If the bookkeeper has made a serious mistake in his addition or subtraction, all the reasoning by which we prove that a certain transaction 'must' have been responsible for a gain or loss of such an amount is worse than useless.

Of course the way to make sure that we have not been misled by any of these blunders is to try the residual cause by itself and see if it really does produce the precise effect indicated by the calculations. Often it does not, and where this is the case and the calculations are all correct it indicates that there is still another residual cause or group of causes to be looked for. Often, of course, direct experiment is impossible, and then we have to get along as best we can with the abstract calculations.

In spite of its difficulties "the Method of Residues is one of the most important among our instruments of discovery. Of all the methods of investigating laws of nature, this is the most fertile in unexpected results: often informing us of sequences in which neither the cause nor the effect were sufficiently conspicuous to attract of themselves the attention of observers. The agent C [i.e., the residual cause] may be an obscure circumstance, not likely to have been perceived unless sought for, nor likely to have been sought for until attention had been awakened by the insufficiency of the obvious causes to account for the whole of the effect.

And

c [the residual effect] may be so disguised by its intermixture with a and b [the effects whose causes are already known] that it would scarcely have presented itself spontaneously as a subject of separate study.

The Method of Residues is, like all the rest, a method of exhaustion; for we cannot be sure that a given residual effect is due, or at least partly due, to a certain residual antecedent until we are sure that that residual antecedent is the only one present that could have any influence on the effect.

Often it is impossible to use any of the methods already discussed, at least without modification, simply because it is impossible to find or to make cases in which Concomitant all of several possible causes are not present. Suppose that we want to know what makes a wheel stop turning after a while, or a pendulum stop swing

Method of

Variations.

* Mill: Bk. III, Chap. VIII, Sec. 5.

ing, or a sleigh stop sliding along a smooth and level road. It may be the nature of all material things to stop moving and come to rest, or it may be the presence of the earth that makes them do so, or it may be friction or some resisting influence exerted by the air. How are we to tell? We cannot experiment for the sake of comparison with things that are not material; we cannot get away from the earth; we cannot create conditions in which there is absolutely no friction and absolutely no air or other surrounding medium. We cannot eliminate any of these possible causes. How then can we choose between them?

Though we cannot wholly eliminate any of them, we can introduce changes in some of them, and if we find that the variation of any condition is accompanied by a variation in the result in question, then we can be sure that that condition has some causal connection with the result. We know that the presence of the air has something to do with the movements of bodies, because we know that things stop moving sooner when the wind is against them than when there is no wind at all, and sooner when there is no wind at all than when it is moving with them. If the air exerted no influence, the direction in which it moved could make no difference either. This proves that air exerts an influence, at least when it moves. That stationary air also exerts an influence and that this influence tends to make a thing stop moving can also be proved; for a pendulum will swing or a wheel will turn longer in a box from which the air has been partially exhausted than in one from which it has not, and the more nearly the air is exhausted the longer the motion will continue. So much for the influence of the air; but how about friction? We cannot make any contrivance in which there is no friction at all; but every one knows what happens when we diminish it. The more slippery we make a surface the further things will slide upon it, and the more we diminish friction in wheels and pendulums by lubricants or special bearings the longer they will keep on moving.

If friction did not affect the continuance of a movement, a vast number of influences that agree in nothing else but the diminution of friction would not all agree in prolonging the movement. Thus we can use the Method of Concomitant Variations to prove that the resistance of a surrounding medium and friction help at least to make things stop moving.

Suppose

To take another illustration of the same method. that things weighed on an extremely delicate spring balance seem lighter and lighter, or that pendulums swing more and more slowly, the farther we take them on a mountain or in a balloon, away from the centre of the earth, no matter which side of the earth we may be on or where the earth may be in its orbit. If the experiments are tried with proper precautions, the distance from the centre of the earth is the only condition that is varied alike in all of them, and the continual variation in this one respect is invariably accompanied by a corresponding variation in the downward pull of everything we try, whether we measure that pull by its effect upon the spring in the balances or by its effect upon the rate at which the pendulum swings. From this we have a right to conclude that the nearness of the earth, and therefore the earth itself, has at least something to do with the tension exerted by a weight upon a spring and with the swinging of a pendulum-and perhaps with the general tendency of things to fall.

Mill's canon for the method is as follows:

66

Whatever phenomenon varies in any manner whenever some other phenomenon varies in some particular manner is either a cause or an effect of that phenomenon, or is connected with it through some fact of causation."

The last clause in the canon is intended to cover cases where two phenomena have corresponding variations, not because one causes the other, but because they both depend, at least to some extent, upon some third variable which perhaps has not been observed at all. If we could be sure

that there were only two variables, we could be sure that one of them was the cause, or at least a part of the cause, of the other. Because we cannot be sure that one of the observed variables is the cause of the other until we are sure that there is no third variable to play this part the Method of Concomitant Variations is at bottom, like all the other inductive methods, one of exhaustion.

So much we can infer according to the Method of Concomitant Variations if we take account of only the mere fact of change. If we know and take account of its amount also, we can often go further and find out with more or less certainty how much the one variable has to do with the other: whether it is the complete cause or only a part of the cause, and what is the mathematical relation between the cause and the effect. For example, if we know that the income of a certain agent always increases as his sales increase and diminishes as they diminish, we may infer from this that at least a part of his income is derived from a commission on sales; but that is all. On the other hand, if our information is more definite, and we know that when the sales are doubled or quadrupled the income is also doubled or quadrupled, we shall probably be correct in inferring that the whole income is derived from a commission on sales, and of course we can tell what the commission is. If we know that when the monthly sales amount to $10,000 the income is $280, when they amount to $20,oco it is $480, and when they amount to $40,000 it is $880, we shall be able to guess that the agent has a salary of $80 a month and a commission of two per cent on his sales.

In reasoning of this sort, however, there are two things about which we must be very careful. In the first place, where there are only a few data they are often consistent with any one of several laws of rela

Cautions.

tion, and we cannot be absolutely certain that the one we

happen to hit upon is correct.

To take the case just given:

the figures would be explained just as well if we supposed