*

a correct notion of it, when we exhibit a solid body—the locality at which it ceases to exist marks the termination of the space in which it is contained; and thus form is exhibited. Of the extension of space, we become readily cognizant by combining our notions of infinitude with those of termination. When we look at a surface, we do not mentally reflect on space as existing only at that surface, but as bounded by it, and extending to some other term. And although the opposite surface comes not to the eye, yet it readily presents itself to the mind by the aid which our conceptions of infinite space afford. The composition of space is represented by the composition of matter. As a substance may be broken up into a number of parts, so may a space into a number of spaces; and as the parts of the body make up neither more nor less than the whole, so the division of magnitude does not alter its amount.† Inseparable from composition is divisibility. I am not about to follow Barrow through his dissertation on this subject, which, in his way of treating it, refers not to space but to matter. with the division of magnitude alone. observe, 1. That it is impossible to otherwise than that it may be potentially divided in any way. There is no adherence of parts, no connection of atoms in space; the idea of it as simple extension necessarily produces that of its divisibility. 2. That this hypothesis of infinite divisibility does not require us to admit any conception of an infinitude of parts, leading to * Compare Arist. Phys. Ausc. iv. 2, with Plato, Timæus, and Philebo, p. 26-7. Barrow, Lect. ix.

Aristotle de gen. et corrup. i. 2,

Our concern is And here, I may conceive of space

such absurdities as Lucretius points out, and Barrow combats. For, by infinite divisibility, we understand that separation may be conceived to take place where one pleases, conformably to the doctrine of infinitude delivered by Aristotle at the end of the 3d Chap. of the Phys. Ausc.

Hitherto we have contemplated magnitude as fixed in position. The next thing to be reflected on is potential mobility. Here, again, the senses aid in calling up, if not in directing our conceptions. A body may be transferred in place relatively to another. By this transference its dimensions are not altered. The space occupied

:

by it, then, is the same as before the change is only à change of place. The antarctic circle cannot actually be made to coincide with the arctic, but the idea or phantasm of it may. Whether it remain fixed in place or be moved, its amount of space remains unaltered, and the mind takes cognizance of this amount, not in reference to its position, but abstractedly. And thus can it reflect on forms of space transferred to other localities. Hence comes mutual penetrability by the combination of ideas of fixed space with movable. A magnitude is mentally carried to the place occupied by another, and the two fill the same space. This affection of magnitude has certainly no parallel in the senses.

The last and most important thing to be reflected on is comparison, of which the elements are equality, inequality, &c. The principle of this method is mutual penetrability or superposition. One magnitude is shewn to be equal to another, by the fact, that the two fill the same space or lie one on the other.

In Euclid's elemen

tary books, no other species of comparison is ever alluded to-nor is there any attempt to disguise the direct use of this method. Some modern writers affect to consider this sort of demonstration inelegant.* But it must be observed that no other mode, which is legitimate, exists. Others which have been suggested are loose and unsatisfactory. Besides they merely place the argument one step lower down, leaving it to the metaphysician to investigate the elements of equality, which he can only do by Euclid's method. It will be requisite in the sequel to enter more fully on the doctrine of superposition. I propose to defer it until I have discussed the nature of the elements on which demonstration is based.

A great deal of confusion exists in what has been written on this subject, arising from the fact that Aristotle and others have attempted to arrange under the same category all sciences of a like construction. Now, in reality, geometry differs in some essential points from physics. Any branch of the latter requires hypotheses as to the nature of a probable existence, the former demands only communications as to the quality of a conception. The principles of Optics, for instance, are very different in this respect from those of Geometry. One of these which I referred to in my last lecture, "that light may be represented by right lines," is neither the necessary result of experience, nor is it an intuitive perception. Sciences like this, from their want of universality, are based, of necessity, on hypotheses. And this is really, if I do not mistake, the meaning of Aristotle,† which has *T. Simpson, Elem. of Geom. p. 250. Leslie, Supp. Encyc. Brit. and Geom

+ Anal. Post i. 2.

been much misunderstood by writers, ancient and modern.* For we do not find Aristotle entertaining wrong notions relative to the nature of a definition, nor, indeed, so far as I know, of an axiom either. He says in the place quoted, "that an hypothesis is that which declares a thing to be or not to be, taking one part or another of a contradiction." As when Galileo,† for instance, assumes that bodies tend towards the centre of the earth :-Since they may or may not do so, his admitting that they do is an hypothesis. Now this differs altogether from one of the principles of geometry, and the two must not be confounded. For the ideas of the things about which geometry is conversant are universal and necessary. What is required, then, in the first place, is not an hypothesis according to which we are to view the affections of space, but a series of definitions, to serve as the instruments of communicating ideas, in order that it may be known about what we are arguing. It is not that two persons may or can conceive differently of a straight line or a circle, but that such a definition be sought out as shall express what is meant, with sufficient clearness to call up the idea of it in the mind.

And accordingly Aristotle in the same treatise divides his principles differently, with an especial reference to geometry. I am, however, quite aware that the word definition is sometimes applied to designate something very different from what we have supposed it to do. In order to avoid confusion, therefore, I deem it right to

* Proclus, ii. 8. Barrow, Lecture viii.

+ And Huygens after him, De Causa Gravitatis,

Anal. Post, i. 10.

distinguish two kinds of definitions: 1 Definitions proper : 2. Definitions consequential.

1. A definition proper is the communication of an idea in exact terms.

2. A definition consequential is the product of axioms expressing, not the nature, but some affections of the thing defined. Thus, when it has been found that a certain class of things possess a certain property-that property itself, or something connected with it, is made the basis of a definition, under which all things which possess the property are ranged. For example, when it has been proved that all the three angles of a rectilinear triangle are equal to two right angles, this property may be taken as the definition of a species of figure. "Every figure, (rectilinear or not,) the sum of whose angles is equal to two right angles, is called," &c.* In applying this definition, we should reason generally, but descend to par ticulars by admitting that when the figure is rectilinear it is three-sided. I should have preferred that the term defining property were applied in such cases, but retain the term definition in accordance with the language of others.

In the present Lecture, we shall be conversant only with the former class of definitions, and may, consequently, drop the distinctive epithet.

Much of the confusion which has existed in the minds of modern writers has arisen from their imperfect perception of the nature of a definition. Had they attended to the statement given by Aristotle,† "A definition is a phrase which signifies what a thing is," we should not +. Top. i. 5.

Anal. Post. i. 10.