the two sets of instances differ, is the effect or the cause or an indispensable part of the cause of the phenomenon". It is clear that before any such statement as this can be accepted we must take for granted a considerable amount of extraneous information, i.e. information not contained within the terms of the problem itself. We must take it for granted that we are talking only of relevant circumstances, and that we have a fair knowledge of what are relevant circumstances. In any literal sense of the terms it is absurd to speak of two instances agreeing only in the presence or absence of such and such characteristics; for any two instances whatever that we might happen to select could not fail to agree in many points of presence, and must agree in simply indefinitely numerous points of absence. It must be understood, I presume, that the absences' are to be confined to those circumstances which have been actually present in the affirmative instances, or were at any rate set before us at the outset as elements of the problem to be reckoned with.

Jevons' rendering, it must be remembered that he is quoting from, and closely following, Mill here,—does not seem any better in this respect. His statement is as follows (Elementary Lessons, p. 246):

"The Joint Method of Agreement and Difference is similarly represented by,

Here the presence of A is followed, as in the simple Method of Agreement, by a; and the absence of A, in circumstances differing from the previous ones, is followed by the absence of a. Hence there is a very high probability that A is the cause of a"

I cannot understand this rendering. If the absence of a

letter here is meant to indicate, as is presumably the case,the known absence of the corresponding cause, then it is clear that the members of the second set of instances have a great deal more in common than the mere absence of A. For B, C, D,... are also absent from them all; in fact the whole second set of instances agree in displaying throughout the entire absence of every element in the first. So that Mill's verbal statement of the Method, or Principle underlying it, is widely departed from. And if the omission of a letter symbol does not mean the exclusion of the corresponding Cause, how do we know that A is absent from the second set of instances?

Surely, to quote Jevons' words, what we want is not "the absence of A in circumstances differing from the previous ones", but its absence under circumstances as much as possible resembling them. The introduction of a set of letters taken from the latter half of the alphabet, for our negative instances, suggests the selection of phenomena widely remote from those appearing in the affirmative instances. Nothing having been said as to the limitation of the field over which they are sought, it would be scarcely a parody of the procedure to say that if a stood for the smoking of a chimney and A, B, C, D, E... for the tight fitting of the door and window, the north wind, fog, &c.; that PQ indicated that the price of stocks was high, and RS that a parliamentary election was in progress.

Surely what we want is something of the following kind, Let a be some phenomenon in regard to which eleven antecedents, viz. A to K are to be taken account of, and suppose that we have collected the following sets of affirmative and negative instances:

It is clear that A is the only element, of the given lot of eleven, which is always present in the former set of instances, and the only one which is always absent in the latter set. If we knew for certain that there could be only one cause, then clearly A is that one. So much indeed is established by the affirmative instances. What the negative instances do is to disprove one

after another of the alternative causes other than A. It might be that A was not a cause at all, but that B, D, F, and H had respectively been at work in producing a in the four cases in question. The negative instances disprove this, however; and since they take account of every one of the ten letters,--or cause symbols,-B to K, and show that no one of these is operative, we are led to conclude that A alone is the cause which we are seeking.

In the above example we have supposed that the negative instances cover the whole ground after A has been omitted, but of course in practice it is to the last degree unlikely that we should be able to get more than a small proportion of them.

The general conclusion then to be drawn in reference to the nature, value and certainty of these methods seems to be the following. We must always assume a considerable amount of preliminary information as to the nature and limits of the field over which the cause is to be sought. That is, the claimants to that post must be supposed to be finite in number, and to have all had their names previously submitted to us, so that we have merely the task of choosing amongst their respective qualifications. In fact we must assume more than this; for unless the possible causes are extremely few in number,—far fewer than they can well be in any physical enquiry,—so that all their combinations can be taken into account, we must take it for granted that we have some indications given to us as to which are the serious claimants whose qualifications only have to be carefully tested. If more than a very few are introduced the combinations to which they give rise soon become quite unmanageable in number.

Having got the possible causes thus before us we proceed. to apply our methods to them. We begin with the Method of Agreement, helping this out as far as possible by Exclusions, -i.e. by the so-called Joint Method. The latter serves to clear the ground, by eliminating a certain number of the admissible causes; and then the former will give a presumption in favour of one or more of the remaining ones. These latter we must then proceed to test by direct experiment or by deduction. If the subject is one which admits of the test we may appeal to the Method of Difference. If not, we must try what we can do by deductions from already known and admitted generalizations.

CHAPTER XVIII.

STANDARDS AND UNITS.

(1) PHYSICAL.

Up to the present point our discussion has been mainly, if not entirely, of a qualitative kind: we have now to take the important step which carries us on to quantitative considerations. The fact of being able to take this step is the main characteristic which distinguishes the scientific from the merely practical estimate of things. The well known 'Four Methods' of Inductive Enquiry involve no appreciation of quantity; and therefore, as here maintained, they belong essentially to the popular attitude. In three of them we take no account whatever of anything but the complete presence or absence of the quality under consideration; and in the remaining one we merely consider somewhat vaguely whether there is more or less of the effect when more or less of the cause is present. Any help of this kind, it need hardly be pointed out again, carries us but a little way forward. Very few things or agencies are to be found, at any time or place, entirely present or entirely absent and, when they are partially present, the guidance of our conduct in life will often seriously depend upon our being able to estimate how much of them is present.

Almost any example will serve to show how urgently important are these quantitative considerations in order to supplement the results of the merely inductive or logical enquiry. Suppose the question were raised as to what is the cause why an iron girder of a bridge does not remain always of the same length, but is observed to expand and contract from time to time. This is an x for which we are asked to determine the A. The Four Methods, duly employed, will serve well enough to ascertain the cause: that is, they will suffice to show us that it

is not, say, rain and drought, but heat and cold which cause the expansion and contraction of the iron. So far good. But to the engineer of course it is all-important to be able to say with some accuracy how much expansion corresponds to any assigned increase of temperature. He has found by experience, let us suppose, that a tubular bridge in England requires a margin of 'play' of one inch in 200 feet: how much more must he allow for a similar bridge which he is going to construct in India ? In cases such as these success or failure may depend upon an exact quantitative estimate.

Accurate measurement, in fact, forms the essence of true science. Our conclusions may be very useful without this; but until we have made out the quantities' we have not completed our work. Indeed we might almost say that the extension of science from time to time is correspondent to the discovery of fresh measurable elements in nature; and that, within the limits of such extent at any given time, our progress is correspondent to the improvements made in the accuracy of measuring those elements.

The question may of course be raised whether considerations of this kind rightly belong to a treatise on Inductive Logic. My opinion is that they lie just on the borders of our subject; in the sense that the leading principles of measurement ought to be discussed here, but that all details, and all account of the practical devices to be adopted to aid our efforts, are best relegated to the special physical sciences. I shall therefore steer between the entire omission of the subject, as by Mill, and its rather too practical discussion (at least for a logical treatise) as by Jevons, in his Principles of Science.

All measurement involves the conception of a Standard or Unit. These words are often used synonymously, I apprehend; but there is a real distinction towards which they point, and to which it seems best to make them correspond. By a standard nothing more is meant than a fixed point of reference: a sort of typical specimen alongside of which other things may be brought and with which they may be compared. It forms the basis of measurement, but it does not carry us any further. The Unit is a standard which admits of being applied again and again to the same thing so as to actually measure it. Take an example. The boiling point (due precautions being taken) is a