(3) It must be noticed that the standard thus selected can hardly claim to be called objective; or rather, if the distinction may be admitted, it is formally but not materially objective. By this is meant that I cannot convey to any one else a correct notion of what my actual standard is, of any particular sensation, but I can put him in possession of a rule by which he can determine the corresponding sensation for himself. If, for instance, two persons were suffering from toothaches which they respectively designate by the numbers 15 and 20, on the method of estimation here under consideration, each would be able to form some notion of how much the other is suffering on that other's scale he knows, say, that his friend is better or worse, as the case may be, than he was yesterday. But there is no intelligible sense in which one of the two can say that he is suffering more or less than the other; at least these measurements give us no assurance of the fact.

So far as this is concerned, any two persons are relatively to each other in the same sort of position as would be occupied by (say) an Englishman and a Frenchman who could each speak the language of the other, but who did not know what was the area respectively of an acre and a hectare. Each could explain whether his crop was greater or less than it was last year, and each can translate his own measures into terms of his own experience, but he has no means of comparing the fertility of the land cultivated by the other with that of the land which he cultivates himself. This state of things is of course inevitable in every case where we attempt intercommunication about our simple feelings, but it makes itself naturally most felt when the question is one of measuring such feelings.

(4) The remaining characteristic of such a scheme of measurement which deserves notice becomes of less importance than it would otherwise possess, after the admissions we have just made. It is obvious that we cannot avoid a multiplicity of temperature. Some people cannot rid themselves of the notion that it ought to be twice as hot' when the thermometer stands at 90° as when it stands at 45°. We have to point out to them that what has been doubled is not our sensation of heat, nor even the heat itself, but only the number of degrees on the scale, or the corresponding length of the mercury column. We can, no doubt, speak of 'doubling' a given quantity of heat, but this leads to a very different enquiry from that of temperature as shown by a thermometer.

units. In fact every distinct sensation involves a distinct unit, namely the least perceptible difference in that kind of sensation. We saw, in the last chapter, that primitive measures of weight, volume, length, &c. had started from a bewildering variety and multiplicity; but that Law and Science were at work, doing what they could to introduce one universal system for each class of measurable elements or abstractions. And such a step is very important, because we are perpetually having to compare things of different kinds in respect of their magnitude; and, what is more, not only to compare them but physically to compound, divide, and adjust them to each other. A lens made in Paris on a scale of metres may have to be fitted into a tube made in Birmingham on one of feet; fruit may be bought by the kilogramme in one country and sold by the pound in another. In other words, where physical division and composition and adjustment are required, and where accordingly we work with a unit of fixed magnitude, it becomes exceedingly important that the unit should if possible be a universal one. But where we cannot cut a thing up into pieces, and join these pieces together, or actually lay one thing alongside another, 'measurement' becomes a very subordinate matter, and such a unit as we have described above will answer our purpose very fairly.

CHAPTER XX.

ON CERTAIN FOUNDATIONS OF MATHEMATICS.

THE DATA OF GEOMETRY.

SUCH discussion of the foundations of geometrical reasoning as has hitherto found a place in Treatises on Induction or on Philosophy generally, has mostly been directed into one channel. What has been discussed has been mainly the origin and nature of the certainty we feel about the axioms and conclusions:-have these been derived from within or from without? do they differ in kind, in respect of the certainty we feel, from the axioms and conclusions of other branches of knowledge? and so forth1. Of late years the dispute has shifted to somewhat new ground. In the pre-evolutionary stage, when the ultimate test was considered to be that of simple introspection, and the question actually in dispute was little more than this, -Can the certainty we now feel have grown up by association in the life of the individual? it was very different. The dispute was by comparison a narrow one, and the issue not very doubtful. At least we cannot regard the arguments of J. S. Mill (the most distinguished of the recent supporters of nonevolutionary empiricism) in his System of Logic, as really satisfactory. Now, however, it is very different. The matter cannot reasonably be discussed, as a sort of side issue, in a single chapter of a work on Inductive reasoning. Nor is it necessary that we should discuss it at all. Formerly, the mere

1 Recognition of this topic seems inevitably called for here, if only because most of the works which the reader is likely to have studied contain some such discussion at this stage. But at the risk of destroying any trust the philosophical reader may entertain about my right to touch these questions at all, I must frankly admit that I cannot claim to have arrived at any confident judgment upon the matter. Fortunately, as above remarked, such a confident or final judgment is not necessary for present logical purposes.

admission that the certainty we feel, is, so far as introspection can furnish a test, absolute certainty; that is, that by no effort can we conceive or imagine the facts being otherwise, was an admission which at once committed us to one side or the other in the debate, or was at least a potent factor in so doing. And since the question was forced upon our notice, it was but reasonable that we should be called upon to give an answer one way or the other. But now we are presumably all agreed about the subjective or individual certainty. No one supposes that any possible selection of experience to which an infant could be subjected would avail to instil geometrical axioms different from those which we all recognize.

What it is here proposed to discuss is something rather different from this'. Our enquiry does not so much refer to the origin of geometrical axioms as to the nature of the geometrical subject-matter, namely the lines and surfaces with which we there deal. To what extent, and in what way, do these differ from the elements of experience from which we start in our ordinary inductive reasonings? And how are we to account for the peculiar discussions and difficulties to which they have given rise?

I. The first point then which we propose to discuss here is the nature of the subject-matter of geometry: that is, to enquire what exactly are the surfaces, lines, and points with which the geometer has to deal in his investigations. The best short answer that can be given is, that, as regards their general nature, they are abstractions from things which actually present themselves to us as existing; and that, as regards their quality, they represent improvements or simplifications upon things which exist; such improvements and simplifications being

1 The discussions in this chapter may be passed over without serious loss by the non-mathematical reader. But the various difficulties involved are so widely felt by those who read mathematics intelligently, that some attempt to solve them logically ought to be made. I am only too well aware that many advanced modern mathematicians,-especially those who have reached the enviable position of accepting the absolutely infinite,--will reject such attempts at smoothing our difficulties as are here made. Still what has brought help to one mind may be offered without impertinence (if expressed without dogmatism) to other minds. One young Cambridge student at least, many years ago, found the various problems here discussed bristling with difficulties which seemed to be contemptuously ignored by his teachers and his text books.

conceived as carried out to perfection. These remarks will need some little explanation and illustration.

In saying above that the surfaces, lines and points of geometry are abstractions, the doctrine which we most prominently have in view as one to be rejected, is that which is presumably the prevalent unscientific opinion, in accordance with which these elements are regarded as being a sort of entities which can exist apart. Such an opinion is probably greatly encouraged by the large resort necessarily made to the process of tracing our geometrical figures upon paper when working out problems. Even Mill seems to suppose that this is the only alternative open to the geometers, when he maintains that it is impossible for us to conceive such a thing as a perfect line or point. Any such impossibility rests upon the common habit of regarding the lines and points as being, so to say, substantive entities which we must try to conceive as existing by themselves, instead of being attributes, that is, boundaries, of something else, namely of solids. Presumably a very thin sheet of paper, a very fine thread, a very small particle, are taken respectively as first approximations, and then the effort is made to fine these down as far as possible so as to obtain what we want. To draw a straight line' from one position to another is to conceive an exceedingly thin spider's thread stretched between them. Or perhaps, in accordance with an ingenious suggestion which has been offered, in order to attain to a due conception of the geometrical line, we are recommended to picture to ourselves the finest conceivable thread, and then to imagine the central line down the middle of this thread. It need hardly be remarked that any attempt in this direction must be a failure. Do what we will in the way of refinement, any plane arrived at in such a way continues to have both an upper and an under side, namely to be a lamina: the line continues to have a surface all round it, namely to be a cylinder of some sort; and the point continues to have a surface all over it, that is, to be a closed solid of some figure or other.

The soundest way of securing what we want is to begin with the solid. This must have an outside or surface; it is with this outside or surface that the geometer deals, and it is this which, when perfectly flat, forms his 'plane'. We must regard it, not as an indefinitely thin lamina or sheet, but simply as the