asmuch as it was a representation of that particular space only, which human sense-perception was adequate to reveal.

But if these initial and unwarranted assumptions, of an a priori form of perception, and of a separation between sense-perception and thought as two independent psychological functions, are dropped, and recourse is had to experience alone, we come to a very different conclusion. If to experience alone without assumptions we put the question, why the number three is fixed upon as the only possible number of the dimensions of space, the true answer is readily forthcoming. For then we have to ask in the first place, What we mean by space; what is the analysis of that complex percept, or if you like concept; in short, how do we define it? And in this way of approaching the question, analysis shows, as we have seen in detail in Book I., that it is a complex percept built up by perception and thought combined, and is of such a nature that three dimensions, and three only, are necessary and sufficient to define it, while, at the same time, being infinite, no other space can be conceived beyond it. Farther than this, or in contravention of this, thought alone has no power to go; its powers have already been employed in building up, and at the same time conceiving, the complex percept Space, and providing it with a definition. The supposed antagonism between perception and thought as independent functions is at an end, and turns out to be in reality a cooperation; whereas thought itself becomes its own. antagonist, and involves itself in contradiction, when it attempts to set up the possibility of

CH. I.

§ 6. The Conception of Infinity.

so-called n-dimensional space in contravention of its own conclusions, arrived at in combination with perception. It is self-contradictory to subsume space under itself, as a particular under a general term. The space of perception and thought combined is incapable of being generalised, that is, converted into a logical universal. The next highest general term, under which space falls as a particular case, is not space again, but continuity; time being the only other positively known continuum, of equal rank with it in point of abstraction, belonging to the same head.

Leaving the consideration of so-called space of n dimensions, I turn once more to those particular configurations or modes of configurations of space, which are known as the creations and objects of non-Euclidean geometries. These stand on a very different footing from that of so-called n-dimensional space. They involve, so far at least as I can venture to form an opinion, no contradictory conceptions, but are perfectly valid as ideal constructions introduced into, or as it were carved out of, space, considered as in itself indifferent to any configuration of divisions or boundaries demarcating one portion of it from another. Nevertheless, they do not share with the space which may be considered as the creation and object of Euclidean geometry the property of being co-extensive with that infinite space, which is of necessity the field which Geometry has to cover.

When, therefore, we put the question, with regard to these particular non-Euclidean spaces, whether they or any of them can in reality be "our" space, the answer must be No. The reason

BOOK II.

CH. I.

§ 6.

The

for which answer is, not that they do not correspond to, or are incompatible with, the laws of physical Nature, or that physical phenomena are Conception not conformable to their ideal constructions (for Infinity. this, it would seem, we have no sufficient means of ascertaining), but that they fail to satisfy that conception of space which our thought educes from the data of sense-perception, I mean the conception of it as infinite in all directions. "Our" space must be that in which thought and sense-perception combine to show, that we actually live and move; and this, I have tried to prove, can be no other than that infinite Vacuity, with which the physical world of Matter may possibly turn out to be conceivable as co-extensive (a point on which it is not here the place to express an opinion), but which it cannot possibly be conceived to transcend. The ideal determinations of space, not the laws of physical matter or force occupying it, are the object-matter of pure or abstract Geometry. The determinations which matter and force introduce into space are an object-matter, the nature and laws of which pure Geometry may be applied to ascertain, precisely because its own determinations, being ideal, can be ascertained in abstraction from them.

§ 7. Enough perhaps for the purpose of the present work has now been said of time, space, and abstract motion, with the sciences of measurement to which they give rise, Calculation, Geometry, and Kinematic; sciences which, as compared to those which deal with objects in the concrete, may be classed as purely mathematical. It remains to consider Matter itself, that concrete physical

§ 7. Matter and

Force

BOOK II.
CH. I.

$ 7. Matter and Force.

existent, with the nature and laws of which the less abstract positive sciences are concerned; the events which take place in and between material things constituting that order of real conditioning, which is the object-matter of scientific analysis. We have seen what matter is on the subjective side, or as known by our objective thought of it, in Book I. Let us now see what it is on its objective side, or as a physical existent possessing reality in that fullest sense of the term which is distinguished as the reality of real conditions.

Now the first thing which has to be noted in matter as a physical existent is, that it is an instance of the inseparability of distinct elements, analogous to that inseparability of formal and material elements, which we have already found to be universal in perceptual experience. Matter as a real condition also shows an inseparability of distinct elements, but elements of another kind. That solid tangibility which we call matter includes force, and cannot exist as matter without it. Similarly there is no such thing as force, unless it be inherent in or exerted by matter. Neither matter exists without force, nor force without matter. Try to imagine free or pure force, I mean free or pure from matter, and as it were in vacuo;—it cannot be done. At every attempt to do it, some substance or agent possessing and exerting it is supplied in thought, and this substance or agent is thought of as analogous to matter. Or take matter, and try to imagine it pure from force. Again you fail. Its very coherence or consistence as matter, or solid tangibility, is force. It is force whereby it exists as solid tangibility. And this holds good of the

CH. I.

§ 7. Matter

and

Force.

ultimate atoms or particles of matter, supposing supposing BOOK II. them to be assumed as separable existents, just as much as it holds good of matter in any other form, whether in that of a configuration of atoms, with aggregation of molecules, or in that of a continuous medium filling their interstices, and extending indefinitely into space beyond them, or in that of a continuous medium, in which rotatory or other motions are the ultimate constituents of whatever can be called material bodies.

Force, then, is one of the inseparable constituent elements in matter. But what is the other, or others if more than one? Our previous analysis shows that there are two and two only; one the element of time-duration which matter, like all existents without exception, must occupy if it exists at all, and the other the element of spatial extension in three dimensions, though this is not so wholly simple and unanalysable as time. In all

cases where matter is perceived or thought of as perceivable, what we so perceive or think of is some portion or portions of three-dimensional space, each coherent within itself, and offering resistance to touch or pressure from without. Wherever three-dimensional space can be tactually felt or thought of as being felt, we have matter presented or thought of as real; and any portion of it which can be so felt is said to be occupied by the matter which coherently fills it. Nothing further is needed to satisfy our conception or experience of it. The space occupied, and the felt occupancy of it, or what is the same thing, three-dimensional extension and force, are thus the two inseparable,