determined by so many conditions, that a description of the physical condition, in terms of magnitude, would not even yield that approximate index to the resultant sensation which is afforded by saying of the water that it is, for example, 120° Fahrenheit. Then again the sensation by its intrinsic force is apt to overpower all other considerations. The physical condition of my tooth is of no importance to anyone else; and even to myself the resultant ache,-thanks to the mechanical skill of modern dentistry, in the way of finding substitutes for our own teeth,-is almost the only thing of any great significance. Owing to such causes as these there is really nothing to divert us from attempting to measure the mere sensation, if we think of introducing the conception of measurement at all.

When we come to look at the characteristics of the sensation, from this point of view, a little more closely, the first thing to strike the attention seems to me to be this. We can get as far as the notion of a standard tolerably easily; the difficulty is in proceeding from this to the notion of a unit. Something was said about this distinction in the last chapter, and it is easily applied to the case before us. A standard is simply an instance, arbitrarily selected, to which we can bring other similar instances for comparison, so as to say whether they are equal to, or greater or less than the standard. I might take my toothache as it was last night at 10 o'clock as such a standard, and by simple comparison I am able to say confidently that the pain is not now as bad as it was then. I might also set up this standard for permanent reference, by resolving that whenever the ache came up to this standard I would give up work and settle down over the fire. In the language of the Act of Parliament which fixes the pound avoirdupois we might decree that that particular ache “shall be and be denominated the imperial standard" of such aches for the future.

The defects of such a standard are too obvious to require much pointing out. For one thing it is purely private and personal, and I have no means of comparing my standard with that which others may set up. Of course we all of us make an assumption which really involves such a comparison. Every theory of Ethics upon which we can act involves it to some extent, and the Utilitarian Theory, in its common form of

Hedonism, is largely founded upon such a comparison. Unless we took it for granted that others suffer and enjoy in the same sort of way, and to the same sort of degree, as ourselves, we could not know what was our duty towards them; and if there are any duties towards animals the notion of such a comparison must be carried a great way further.

As Ethics, however, is not generally studied quantitatively, or, if it is, has to decide by averages and large generalizations, considerations of this kind can practically be evaded there. But when as here, the question before us is as to the existence of a standard, the difficulty becomes very serious. In every case where we are able to employ a common standard, we shall find that we have more than one line, so to say, of sensible appeal and verification. How do we recognize our pound weight again? Certainly not by its attribute of weight alone. If a number of people were successively sent into a dark room where there were a number of weights to handle, they would not find it easy to ascertain afterwards whether they had handled the same one; and if they did succeed even approximately in coming to an agreement it would only be because a long course of reliance upon other appeals had given them a tolerably accurate notion of the standard. Our means of certainly identifying the standard involve a miscellaneous appeal by various senses to various physical properties. We know the object again by sight and by touch; its size and shape are what we remember: it may be labelled: and as a final security in the case of the State standard, we keep it under lock and key.

In the case of the toothache we have no such resources at hand. Not even the dentist will tell us whether a given tooth is at the moment aching or not; still less how severely it is aching. If it were suggested to take 'writhing' point or 'screaming out' point, by analogy with blood heat or boiling point on old-fashioned thermometers, we should of course be falling back on an objective standard, and on one not at all more trustworthy than the physical condition of decay of the organ. Turn it which way we will we find it impossible to set up anything approaching to a standard which shall be common. to others as well as to ourselves.

Moreover the same causes which make it theoretically impossible to compare correctly the sensations of one person with

those of another, make it practically impossible to compare our own sensations at different times. Accordingly a standard selected in the arbitrary way suggested would be of but little avail even to ourselves. If I could recall the ache of last night I might compare it with that of the present moment; but how am I to know that I have recalled the right one? Only by a process of comparison between what is felt and what is imagined, a comparison which is not strictly trustworthy over short intervals of time, and which deteriorates as the time increases. If I could see and handle the ache as I can see and handle the pound weight, and knew that,—as Berkeley's hypothesis seems in consistency to demand,—this ache had merely been somewhere else in the meanwhile, just as the pound weight was when I was not lifting or handling it, I should have what I wanted. But when, as here, the sensation to be measured is itself the sole means of identification, an all-important guarantee is wanting.

As regards then the mere conception of psychical standards there does not seem to be any intrinsic difficulty. It is as easy to select them here as it is in the department of physics, and on the same general principles. Only unfortunately there are insuperable difficulties in the way of our making any such use of them in the former case as we can in the latter. As regards the employment of Units however the case seems different. It is by no means easy to get a clear conception of what we want in this respect, but when the preliminary difficulties are surmounted, we seem to be in possession of a sort of system of units which, though of a very peculiar nature, do really adapt themselves to a certain kind of application.

The two main difficulties arise in the following way. The essential thing about any unit clearly is that it should be strictly determined. If it be a length, for instance, we must know exactly where to fix the two ends which determine that length. Now mere intensity, the quality we propose to consider, and that in respect of which there is likely to be most trouble,―resembles a length: it is, so to say, a quantity of one dimension. And in regard to nearly (but not quite) all our sensations there is no difficulty in securing one of the two fixed points, one of the ends, as we may call it, of our unit,— which are needed in order to mark off a definite length. That

is, we can in most instances start from a clearly realized zero, by our own clear experience of what it is to have none of that sensation. It is the other end which gives us some trouble; for we nowhere find any kind of natural break or stopping place, or anything analogous to the graduation on a scale. Begin with zero, and suppose the intensity gradually increased; we never come to any point at which it seems convenient to mark off a stage for subsequent subdivision; nor do we ever come to a final stop from having actually reached the end of that growing magnitude.

The other main difficulty arises when we try to compound sensations. We saw, in the last chapter, that it was essential to the conception of a unit that a bigger magnitude should be in some intelligible way built up of those which are smaller: on no other supposition can we say that the thing to be measured "contains" the unit such and such a number of times. A plank ten feet in length can be divided into ten pieces of one foot, and each of these pieces can be considered separately and the aggregate be put together and shown to be equal to the original. Similarly, under due explanation, when we are dealing with time and with mass. Any assigned magnitude of these elements admits of analysis and combination. Each portion is similar to the whole, and the aggregate of the parts will,-for the purpose in question, that is, of measurement, for we are not here taking account of any other physical characteristics,-be perceptibly equivalent to the whole. We showed also that the same conditions hold good in other directions, in which at first sight they might seem to be lacking. We are able, for instance, in a perfectly intelligible sense, to compound and resolve velocities, and to say how many times our unit is "contained" in a given velocity.

Can we then be said, in any analogous way, to be able to break up a mere feeling into parts, to consider these separately, and to re-compound them into a whole? It may perhaps be answered off-hand that if I am suffering from two bad teeth which are apt to ache from time to time, and which seem to me about equally bad when they are at work, that then when they both ache at the same moment the pain must be twice as intense. To such an answer there are two objections.

In

the first place we are thus ignoring the distinction between extensive and intensive increments of sensation. This distinction may be readily illustrated as follows. Put a finger into hot water, and there are two ways of increasing the pain, viz. by either plunging the finger deeper in, or by making the water hotter. The latter is the true intensive addition which we want to measure, whereas the case of the two aching teeth corresponds rather to the former. And, in the second place, to lay it down that the pain was rendered twice as bad (or whatever adjective we use) would be to beg the whole question at issue. We are then making an inference, and judging what the pain ought to be, instead of deciding by direct consciousness what it is. This would be analogous to deciding how hot we feel by looking at a thermometer. The increment of sensation, and the measure of that increment, must be decided somehow by direct appeal to our own feelings. The unit must be a really subjective one.

The device adopted by Fechner, in order to surmount these difficulties, is very ingenious. On his method, what we may be said theoretically' to adopt as our unit is the minimum. difference of sensation which is just perceptible. In order to understand what this means, recur to the toothache, and conceive another such ache just a little less bad, but by the smallest difference which is distinctly perceptible as a difference. Or, to vary the phraseology, conceive a number of such aches successively applied (supposing we can retain their intensity in our memory), and that we select the worst of them

1 I mean that this is the principle underlying his methods, and, as I conceive, the only one which is universally applicable. As every modern psychologist knows, the formula he adopts is y = k log which slightly departs in appearance from the above description. In the first place we thus abandon the conception of small and discrete differences, and adopt the "limits" of the Differential Calculus. And in the second place, when we compare two sensations in respect of magnitude, we do this, not directly by successive applications of our subjective unit, but indirectly through comparison of the objective stimuli required in order to produce them. In practice this is the only convenient or available method, but it could not be applied to a case like that of the toothache,-which was purposely chosen for this purpose,—where we have no means of measuring the magnitude of the stimulus. If I wanted to find a toothache which should be double a given ache, I could not employ the mathematical formula, because I have no notion as to what ẞ is in this case.