continually tend, and the result we are in want of in order to satisfy the problem, are one and the same.

The above remarks merely concern the theory of the process. The actual process of calculation may of course be complicated. In the case of the clock-face it involved the determination of that fraction towards which the series +114+1728+ &c. if continued indefinitely, would approximate without limit. In the present case it involves the determination of the position of that line towards which the chord PQ will similarly approach.

It is here that the so-called infinitely small quantities make their appearance. The direction of the above chord may be determined by its tangent, viz. by the ratio QN: NP. These, remember, are always finite quantities, and always determine the chord and not the tangent. As Q approaches P, these of course become smaller and smaller. What line will they ultimately, not give, for they must always continue to give a varying result, but, as I prefer to say, indicate, as being the only line towards which they are approaching without any limit of nearness? Certainly the tangent. This is commonly expressed, in the language which Berkeley and so many others have found so hard to follow, by saying that the ratio of QN to NP, when both vanish, gives the desired direction of the tangent. Take again the following examples. Suppose first we want to determine the area of the triangle ABC. We should natu

rally do it directly by saying that it was half the area of the rectangle AC. CB. But as in the case of the clock-face we may also get at the same result by a more circuitous process. We may divide the triangle up into slips such as PMNQ. Then, instead of determining the triangle itself we may give the sum of all the pieces such as QRMN. Now no such sum of parallelograms will ever make up the triangle; they will always be too small, however many they be. But they will indicate the area of that triangle, and this with perfect accuracy.

For the area towards which such a sum of parallelograms indefinitely approaches is none other than that of the triangle we want.

In the above example we had a case where the employment of limits, and the machinery they involve, was quite unnecessary. Take now the following.

Suppose we want to determine the area of the curve RSO. Here we have no such simple resource as in the case of the triangle. Our only plan is to conceive the area divided up into a multitude of strips such as PMNQ. We then begin by taking the sum of all the parallelograms such as PTNM. As before, this is not the result we want, and never could yield it directly. But it will indicate that result, for the only area towards which such a sum will approach without limit is the area of the curve in question.

If this were a work on mathematics we should have to give a detailed explanation of the process. It may seem a very circuitous way, instead of determining an area directly, to determine the limit towards which another, and composite area, is continuously tending. But it so happens that the latter result (generally speaking) is attainable and the former is not.

Beginners are so apt to confound a process with a result; in the sense of supposing that when the former can never be completely carried out, and would if suspended at any point yield an erroneous result, the final or indicated result must also be to some small extent erroneous, that it will be worth giving one more illustration. Suppose I have to pay a sum of £1. There are a variety of ways of proceeding to assign this sum. I may do it simply and directly, by, say, two payments of 10s. each. But again I may do it by beginning with 10s., then giving 5s., then 2s. 6d., and so on; and saying that the sum to be paid is that which is indicated by the sum

of such a series continued indefinitely. Or I may adopt a quite different plan by saying that the number of pounds is that towards which the ratio of the number of heads and tails in the throws of a perfectly fair coin tends to approach. Or finally I may say that it is expressed by the ratio of the chord to the arc of a circle as these are made continually smaller.

In all these cases the same number, viz. unity, is assigned, but it is assigned in very different ways. In the first case it is given directly by a simple and finite operation. In the other three cases it is indicated rather than given. We indicate it by a proposed process which from the nature of the case can never be completed, and which is such that if we trust to it as assigning the result directly, would be distinctly erroneous. For, since the process can never be completed, any result which it directly yields is too little or too great. But the result which it aims at, or indicates, is perfectly definite, and in each case precisely accurate. This is the only result with which we are really concerned, the process by which it is indicated. being a mere scaffolding necessary for its attainment. The general method of the Differential Calculus is of this kind; and inasmuch as it deals almost exclusively with the limiting ratios of magnitudes which are supposed to diminish without limit, it partakes of the nature of the last of these three examples.

CHAPTER XXI.

-EXPLANATION AND VERIFICATION.

NOTHING is more frequent, both in science and in common life, than a demand for an 'Explanation' of some fact, or class of facts, or law. The attitude towards the phenomena of nature which prompts this demand, is one of immemorial antiquity. Science, in fact, has taken the conception from common life, doing nothing more than defining it, widening its scope, and making its conditions as stringent as possible.

However far back into the past we try to project ourselves, we cannot, of course, reach a time in which a very great amount of experience has not been already reduced to order and become accepted and established. The earliest dawn of self-consciousness and enquiry presupposes ages of semi-conscious and unenquiring dependence upon the order of nature; for no life could continue to exist without such dependence. The earliest time therefore from which we can suppose ourselves to start is that in which patches, so to say, of the warp and woof of nature have already been woven together, just sufficient to yield a certain amount of firm texture,—but in which these are mere patches, surrounded by an enormous mass of skeins in loose but inextricable confusion. Take, as a case in point, the attitude, as we may conceive it, of some unusually intelligent savage. He finds himself surrounded by myriads of isolated facts. These are quite enough to prompt the curiosity; and the questions, so incessantly put by a young child, seem to show that, as things now are, mere curiosity is an abundant motive for enquiry. But one would suppose that the primitive motive must have been one of a far more serious and urgent character. An isolated fact may merely arouse the inquisitiveness of a modern child, for his conduct is controlled by the care of those who are more experienced; but to the man who has to

face the consequences of his own actions it means danger, for it may at any moment result in injury or death. His position may be compared to that of some stranger who has wandered into a gigantic foundry and workshop. He can hardly touch anything without a risk of being burnt: he does not know where he can stand without being knocked down: at any moment he may be crushed by a steam-hammer, blinded by a spark, or swept away by a revolving band.

If this is at all a fair illustration of the position of man in an unfamiliar or 'unexplained' world, it is obvious what sort of want he must experience. This is, in a word, the want of order or of interconnexion between the facts which surround him. This, I consider, points to the primary and familiar signification of explanation, or at any rate corresponds to that desired improvement in our intellectual position which we afterwards come to designate by the name of explanation. If the reader will recall what was said in an earlier chapter about the nature of Causation in its primitive and popular sense, he will see that 'explanation' almost exactly covers the same ground as what we there designated as 'uniformity' in its widest sense. By Uniformity, as there stated, I understand any kind of order whatever, any arrangement of the things which enables us to anticipate without actual experience. It is in fact the objective counterpart of inferribility. In this sense any of the innumerable questions, put by the child or the grown person-What is that? Why does this happen?—are demands for an explanation; and for all purposes of a merely practical kind are reasonably answered by the suggestion of any rule or generalization which brings the facts in question into some relation with other facts already known, or, at any rate, better known.

When we take this view of the embarrassment out of which the desire for explanation springs, and of the simplest means at hand for practically removing it, we see that many forms of proffered explanation which the popular mind accepts, and which the scientific mind scornfully rejects, have much to be said in their favour. Remember that the only really important requirement, in the very early stages of scientific development, is to link the fact in question to other facts, and we shall see that few of the commonly proposed explanations fail in some measure to do this.