We have [already] remarked the distinction between concrete, and abstract, terms; and explained wherein the difference of their signification consists. We have 27 also observed, that though in very many cases, the concrete term, and the abstract term, are different words, as good and goodness, true and truth, there are many others in which the concrete and abstract terms are the same; and this is the case, unhappily, with the word Truth itself, which is used in the concrete sense, as well as the abstract. Thus we call a proposition, a Truth; in which phrase, the word Truth, means “True Proposition;” and in this sense we talk of eternal truths, meaning, Propositions, always true. “Property,” is another word, which is sometimes concrete, sometimes abstract. Thus, a man calls his horse, his field, his house, his property. In such phrases the word is concrete. He also says, he has a property in such and such things. In these phrases, it is abstract.
Of this ambiguity, the word Line is an instance. It is applied as well to what we call a physical line, as to what we call a mathematical line. In the first case, it is a concrete, or connotative term; in the second case, it is an abstract or non-connotative term. Let us then conceive clearly the two meanings. The purest idea of a physical line, is that which we have already formed; the aggregate of particle after particle, in the direction of a radius. When this aggregate of particles in this order is called a line, the word, line, is connotative; it marks or notes the direction, but it also marks or connotes the particles; it means the particles and the direction both; it is, in short, the concrete term. When it is used as the abstract term, the connotation is left out. It marks the direction without marking the particles.
It is here necessary to call to mind, that abstract 28 terms derive their meaning wholly from their concretes; and that by themselves they have absolutely no meaning at all. I know a green tree, a sweet apple, a hard stone, but greenness without something green, hardness without something hard, are just nothing at all.
The same, in its abstract sense, is the case with line, though we have not words by which we can convey the conception with equal clearness. If we had an abstract term, separate from the concrete, the troublesome association in question would have been less indissoluble, and less deceptive. If we had such a word as Lineness, or Linth, for example, we should have much more easily seen, that our idea is the idea of the physical line; and that linth without a line, as breadth without something broad, length without something long, are just nothing at all.[8]
[8] This conception of a geometrical line, as the abstract, of which a physical line is the corresponding concrete, is scarcely satisfactory. An abstract name is the name of an attribute, or property, of the things of which the concrete name is predicated. It is, no doubt, the name of some part, some one or more, of the sensations composing the concrete group, but not of those sensations simply and in themselves; it is the name of those sensations regarded as belonging to some group. Whiteness, the abstract name, is the name of the colour white, considered as the colour of some physical object. Now I do not see that a geometrical line is conceived as an attribute of a physical object. The attribute of objects which comes nearest to the signification of a geometrical line, is their length: but length does not need any name but its own; and the author does not seem to mean that a geometrical line is the same thing as length. He seems to have fallen into the mistake of confounding an abstract with an ideal. The line which is meant in all the theorems of geometry I take to be as truly concrete as a physical line; it denotes an object, but one purely imaginary; a supposititious object, agreeing in all else with a physical line, but differing from it in having no breadth. The properties of this imaginary line of course agree with those of a physical line, except so far as these depend on, or are affected by, breadth. The lines, surfaces, and figures contemplated by geometry are abstract, only in the improper sense of the term, in which it is applied to whatever results from the mental process called Abstraction. They ought to be called ideal. They are physical lines, surfaces, and figures, idealized, that is, supposed hypothetically to be perfectly what they are only imperfectly, and not to be at all what they are in a very slight, and for most purposes wholly unimportant, degree.—Ed.
29 What are, then, the sensations, the ideas of which, in close association, we mark by the word line?
Though it appears to all men that they see position, length, breadth, distance, figure; it is nevertheless true, that what appear, in this manner, to be sensations of the eye, are Ideas, called up by association. This is an important phenomenon, which throws much light upon the darker involutions of human thought.
The sensations, whence are generated our ideas of synchronous order, are from two sources; they are partly the sensations of touch, and partly those of which we have spoken under the name of muscular sensations, the feelings involved in muscular action.[9]
[9] In attaining the ideas of synchronous order, which is another name for Space, or the Extended World, sight is a leading instrumentality. It is by sight more than by any other sense that we get somewhat beyond the strict limits of the law of the successiveness of all our perceptions. Although we can distinctly see only a limited spot at one instant, we can couple with this a vague perception of an adjoining superficies. This is an important sign of co-existence, as contrasted with succession, and enters with various other signs into the very complex notion of the author’s synchronous order, otherwise called the Simultaneous or Co-existing in Space.—B.
30 A line, we have said, is an order of particles, contiguous one to another, in the direction of a radius from one particle. Let us begin from this one particle, and trace our sensations. One particle may be an object of touch; it may be felt, as we call it, and nothing more; it may, at the same time, give the sensation of resistance, which we have already described as a feeling seated in the muscles, just as sound is a feeling in the ear. Resistance, is force applied to force. What we feel, is the act of the muscle. Without that, no resistance. This state of consciousness is, in reality, what we mark by the name. It is, at the same time, a state of consciousness not a little obscure; because we habitually overlook many of the sensations of which it is composed; because it is, in itself, very complex; and because it is entangled with a number of extraneous associations.