e should have supposed that, mysterious as the connection may be between the external world and our impressions of it, the possibility of really infinite external space would be admitted by any one, unless he held the metaphysical system of Berkeley, which denies the necessity of any external substratum of our conceptions, and substitutes the direct agency of the Creator ; and we should have thought it impossible to maintain the necessary finitude of matter, without also maintaining the same of real external space. Nevertheless, to show how differently these subjects strike different persons, we quote the following from a dissertation of an eminent writer :—" Every real, existing, material body must enjoy that indefeasible attribute of body—namely, definite place. Now place is defined by direction and distance from a fixed points Every body, therefore, which does exist, exists at a certain definite distance from us, and at no other, either more or less. The distance of every individual body in the universe from us is therefore necessarily admitted to be finite." Now it will hardly be denied that the space which a body fills is as real and existent as the body itself, and this whether so occupied or not. Leave out the word material in the above, and for " body " read " part of space," and the argument remains as good as before, ending with a denial of the infinity of space. Every assignable body is at a finite distance from us ; but this is an identical proposition contained in the meaning of the word assignable. lint who is therefore to deny the following ? Name any distance, how ever great ; matter exists at still greater distances.
If we estimate the reality of a conception by its necessity, which is what we do when we settle the pre-eminence of space and time among our ideas, then it is certain that the conception of infinity is as real as that of space or time, being essentially united with them. Many mathematicians try to deny this, and substitute various modes of speaking to avoid the introduction of the idea. It is true that the notion of infinite is one which it is difficult to use without falling into error; a very good reason for avoiding it until the understanding has been well practised in mathematical deduction, but none for denying its existence. Why say that the notion of infinity arises from our not being able to assign a limit, when we know that we feel something more positive ; when we are as certain as we are of any right to use the words can and cannot, that there cannot be a limit, to either space or time ? Those who examine the views of different writers on the first principles of science see a great variety of modes of expression on this point, but a uniform practical use of nothing more than the denial of finitude, accompanied by the mere expression of incapacity to attain infinity ; resolutely coupled, in many cases, with a determination not to allow any words capable of expressing the absolute notion of infinity which actually is before the thoughts. Now it should be the object of elementary writing, while guarding the avenues to error which branch in all directions from an improper use of the word infinite, to acknow ledge the existence of the idea, and to make a gradual preparation for its correct and legitimate use. Both infinitely small and infinitely
great ought to become terms which may be employed without fear ; and the student who has been trained to the natural and healthy use of all his notions will in the end succeed better than the one who has had sonic of them tied up from the beginning because they are some what difficult or somewhat unsafe to use at firsts So soon as an attempt is made to. fetter one branch of thought, the effect Ls sum to be immediately felt in othere. The infinite divisibility of space hi a truth of the same sort as its Infinite extension. Matter may not be divisible without end, and the truths of modern chemistry would seem to show that there are ultimate particles inseparable by chemical, and still less by mechanical, means. But there is a solvent which everyone has it In his power to apply to space; it is the intuitive conviction that every portion of it, however small, except that ultimate notion which is called a point, is divisible into parts, which are them. selves divisible into parts; a process which may be continued without end. Now, a person who trifles with the notion of infinite extension, and persuades himself that lie has not the idea, will probably end by denying infinite diminution ; and as motion, however small it may be, requires the succession of positions answering to an unlimited sepa ration of the time of motion into parts, the next step will be to deny the Infinite divisibility of time, and the possibility of motion, an com monly conceived. Change of place will be imaginel to be physically impossible, If it be asserted that between the first and last positions there have been an infinite number of others ; and the mind will be driven, in order to avoid the notion of infinity, into a sort of opinion that motion is a very large (but finite) number of annihilations and re-creations : annihilation in one spot, and re-creation a little farther on, without anything intermediate. This is no imaginary case ; and it seems to us that when this theory of motion is once attained, nature has taken a satirical revenge for the attempt to smother her conceptions.
The errors which arise from the improper use of the notion of infinite, lie mostly in the idea that all that is proved of finite space or time must necessarily be true of the infinite. We pass over the error that all infinites must be equal, as being that of the merest beginner ; there are enough remaining to claim great caution. The process adopted in the article INFINITE is perhaps the best way of habituating the young mind to the rigorous attainment of results, provided only that the understanding be duly apprised that such a course of pro ceeding is not pursued because there is not infinity, but because there is, and because the notion, though inevitable, is not easily used. The road need not be carried over any unsafe foundation ; but that is no reason why the quicksand and the marsh should be left out of the map.