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an endless repetition can never totally represent to it; which carries in it a plain contradiction. 6. 8. This, perhaps, will be a little

We have no plainer, if we consider it in numbers. The

idea of infi. infinity of numbers, to the end of whose nite space. addition every one perceives there is no approach, easily appears to any one that reflects on it: but how clear soever this idea of the infinity of number be, there is nothing yet more evident, than the absurdity of the actual idea of an infinite number. Whatsoever positive ideas we have in our minds of any space, duration, or number, let them be ever so great, they are still finite; but when we suppose an inexhaustible remainder, from which we remove all bounds, and wherein we allow the mind an endless progression of thought, without ever compleating the idea, there we have our idea of infinity; which though it seems to be pretty clear when we consider nothing else in it but the negation of an end, yet when we would frame in our minds the idea of an intinite space or duration, that idea is

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obbcure and confused, because it is made up of two parts, very different, if not inconsistent. For let a man frame in his mind an idea of any space or number, as great as he will : it is plain the mind rests and terminates in that idea, which is contrary to the idea of infinity, which consists in a supposed endless progression. And therefore I think it is, that we are so easily confounded, when we come to argue and reason about infinite space or duration, &c. Because the parts of such an idea not being perceived to be, as they are, inconsistent, the one side or other always perplexes, whatever consequences we draw from the other; as an idea of notion not passing on would perplex any one, who should argue from such an idea, which is not better than an idea of motion at rest; and such another seems to me to be the idea of a space, or (which is the same thing) a number infinite, i. e. of a space or number which the mind actually has, and so views and terminates in; and of a space or num ber, which in a constant and endless enlarging and pro, gressiori, it can in thought never attain to. For how large soever an idea of space I have in my mind, it is

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no larger than it is that instant that I have it, though I be capable the next instant to double it, and so on it i infruitum: for that alone is infinite which has no bounds; and that the idea of infinity, in which our thoughts c n find none. Numer af.

5.9. But of all other ideas, it is numfords u the

ber, as I have said, which I think for. clearest idea nishes 11$ with the clearest and most distinct of infinity. idea of infinity we are capable of. l'or even in spec and duration, wh n the mind pursues the idea of intinity, it there makes use of the ideas and repetitions of numbers, as of millions and millions of miles, or years, which are so many distinet ideas, kept best by number from running into a confused heap, wherein the mind loses itself; and when it has added together as many millions, &c. as it pleasts, of known lengths of space or duration, the clearest idea it can get of intinity, is the confused incomprehensible remainder of endless addible numbers, which affords no prospect of stop or boundary Our different

§. 10. It will, perhaps, give us a little conception of farther light into the idea we have of infithe infinityof nity, and discover to us that it is nothing number, du- but the infinity of number applied to deration, and expansion.

terminate parts, of which we have in our

minds the distinct ideas, if we consider, that number is not generally thought by us infinite, whereas duration and extension are apt to be so; which arises from hence, that in number we are at one end as it were: for there being in number nothing less than an unit, we there stop, and are at an end; but in addition or increase of number, we can set no bounds, And so it is like a line, whereof one end terminating with us, the other is extended still forwards beyond all that we can conceive; but in space and duration it is otherwise. For in duration we consider it, as if this line of number were extended both ways to an unconceivable, undeterminate, and infinite length; which is evident to any one that will but reflect on what consideration he hath of eternity; which, I suppose, "he will find to be nothing else, but the turning this infinity of

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number both ways, à parte ante and à parte post, as they speak. For when we would consider eternity, à parte ante, what do we but, beginning from ourselves and the present tine we are in, repeat in our minds the ideas of years, or ages, or any other assignable portion of duration past, with a prospect of proceeding in such addition, with all the infinity of number? and when we would consider eternity, à parte post, we just after the same rate begin from ourselves, and reckon by multiplied periods yet to come, still extending that line of number, as before. And these two being put together, are that infinite duration we call eternity : which, as we turn our view either way, forwards or backwards, appears infinite, because we still turn that way the infinite end of number, i. e. the power still of adding more.

$. 11. The same happens also in space, wherein conceiving ourselves to be as it were in the centre, we do on all sides pursue those indeterminable lines of number; and reckoning any way from ourselves, a yard, mile, diameter of the earth, or orbis magnus, by the infinity of number, we add others to them as often as we will; and having no more reason to set bounds to those repeated ideas than we have to set bounds to number, we have that indeterminable idea of immensity. §. 12. And since in any bulk of matter

Infinite di. our thoughts can never arrive at the utmost

visibility. divisibility, therefore there is an apparent infinity to us also in that, which has the infinity also of number; but with this difference, that, in the former considerations of the infinity of space and duration, we only use addition of numbers; whereas this is like the division of an unit into its fractions, wherein the mind also can proceed in infinitum, as well as in the former additions ; it being indeed but the addition still of new numbers: Though in the addition of the one we can have no more the positive idea of a space infinitely great, than, in the division of the other, we can have the idea of a body infinitely little; our idea of infinity being, as I may say, a growing or fugitive idea, still in a boundless progression, that can stop nowhere. 3 oils as vias

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§. 13. Though it be hard, I think, to No positive find any one so absurd as to say, he has the idea of infi. nity.

positive idea of an actual infinite nuinber;

the infinity whereof lies only in a power still of adding any combination of units to any former number, and that as long and as much as one will; the like also being in the infinity of space and duration, which power leaves always to the mind room for endless additions; yet there be those who imagine they have positive ideas of infinite duration and space. It would, I think, be enough to destroy any such positive idea of infinite, to ask him that has it, whether he could add to it or no; which would easily show the mistake of such a positive idea. We can, I think, have no positive idea of any space or duration which is not made up, and commensurate to repeated numbers of feet or yards, or days and years, which are the common measures, whereof we have the ideas in our minds, and whereby we judge of the greatness of this sort of quantities. And therefore, since an infinite idea of space or duration must needs be made up of infinite parts, it can have no other infinity than that of number, capable still of farther addition ; but not an actual positivo idea of a number infinite. For, I think, it is evident that the addition of finite things together (as are all lengths, whereof we have the positive ideas) can never otherwise produce the idea of infinite, than as number does; which consisting of additions of finite units one to another, suggests the idea of infinite, only by a power we find we have of still increasing the sum, and adding more of the same kind, without coming one jot nearer the end of such progression.

$. 14. They who would prove their idea of infinite to be positive, seem to me to do it by a pleasant argument, taken from the negation of an end; wbich being negative, the negation of it is positive. He that considers that the end is, in body, but the extremity or superficies of that body, will not perhaps be forward to grant that the end is a bare negative: and he that perceives the end of his pen is black or white, will be apt do think that the end is something more than a pure

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negation. Nor is it, when applicd to duration, the bare negation of existence, but inore properly the last moment of it.

But it they will have the end to be nothing but the bare negation of existence, I am sure they cannot deny but the beginning is the first instant of being, and is not by any body conceived to be a bare negation ; and therefore by their own argument, the idea of eter. nal, à parte ante, or of a duration without a beginning is but a negative idea.

§. 13. The idea of infinite has, I confess, something of positive in all those tive, what things we apply to it. When we would negative, in think of infinite space or duration, we at our idea of

infinite. first step usually inake soine very large idea, as perhaps of millions of ages, or miles, which possibly we double and multiply several times. All that we thus amass together in our thoughts is positive, and the assemblage of a great number of positive ideas of space or duration.

But what still remains beyond this, we have no more a positive distinct notion of, than a mariner has of the depth of the sea; where having let down a large portion of his sounding-line, he reaches no bottom: whereby he knows the depth to be so many fathoins, and more; but how much the more is, he hath no distinct notion at all: And could he always supply new line, and find the plummet always sink, without ever stopping, he would be something in the posture of the mind reaching after a complete and positive idea of infmity. In which case let this line be ten, or one thousand fathoms long, it equally discovers what is beyond it; and gives only this confused and comparative idea, that this is not all

, but one may yet go farther. So much as the mind comprehends of any space, it has a positive idea of; but in endeavouring to make it infinite, it being always enlarging, always advancing, the idea is still imperfect and inconiplete. So much space as the mind takes a view of in its contemplation of greatness, is a clear picture, and positive in the understanding: but infinite is still greater 1. Then the idea of so much is positive and clear. %. The idea of greater is also clear, but it is

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