or for which he hath names, though not, perhaps, of more. For the several simple modes of numbers, being in our minds but so many combinations of units, which have no variety, nor are capable of any other difference but more or less, names or marks for each distinct combination seem more necessary than in any other sort of ideas. For without such names or marks, we can hardly well make use of numbers in reckoning, especially where the combination is made up of any great multitude of units; which put together without a name or mark, to distinguish that precise collection, will hardly be kept from being a heap in confusion.

§ 6. This I think to be the reason why some Americans I have spoken with (who were otherwise of quick and rational parts enough), could not, as we do, by any means, count to one thousand; nor had any distinct idea of that number, though they could reckon very well to twenty. Because their language being scanty, and accommodated only to the few necessaries of a needy simple life, unacquainted either with trade or mathematics, had no words in it to stand for one thousand; so that when they were discoursed with of those great numbers, they would shew the hairs of their head, to express a great multitude, which they could not number; which inability, I suppose, proceeded from their want of names. The Tououpinambos had no names for numbers above five; any number beyond that, they made out by shewing their fingers, and the fingers of others who were present.* And I doubt not but we ourselves might distinctly number in words, a great deal farther than we usually do, would we find out but some fit denominations to signify them by; whereas in the way we take now to name them, by millions of millions of millions, &c., it is hard to go beyond eighteen, or at most four-and-twenty, decimal progressions, without confusion. But to shew how much distinct names conduce to our well reckoning, or having useful ideas of numbers, let us set all these following figures in one continued line, as the marks of one number: v. g.

The ordinary way of naming this number in English, will be the often repeating of millions, of millions, of millions, of millions, of millions, of millions, of millions, of millions (which is the denomination of the second six figures). In which way, it will be very hard to have any distinguishing notions of this number: but whether, by giving every six figures a new and orderly denomination, these, and perhaps a great many more, figures, in progression, might not easily be counted distinctly, and ideas of them both got more easily to ourselves, and more plainly signified to others, I leave it to be considered. This I mention only to shew how necessary distinct names are to numbering, without pretending to introduce new ones of my invention.

§ 7. Why children number not earlier. Thus children, either for want of names to mark the several progressions of numbers, or not having yet the faculty to collect scattered ideas into complex ones, and

* Histoire d'un Voyage, fait en la terre du Brasil, per Jean de Lery, c. 20.381.

range them in a regular order, and so retain them in their memories, as is necessary to reckoning, do not begin to number very early, nor proceed in it very far or steadily, until a good while after they are well furnished with good store of other ideas; and one may often observe them discourse and reason pretty well, and have very clear conceptions of several other things, before they can tell twenty. And some, through the default of their memories, who cannot retain the several combinations of numbers, with their names annexed in their distinct orders, and the dependance of so long a train of numeral progressions, and their relation to one another, are not able, all their life-time, to reckon, or regularly go over, any moderate series of numbers. For he that will count twenty, or have any idea of that number, must know, that nineteen went before, with the distinct name or sign of every one of them, as they stand marked in their order; for wherever this fails, a gap is made, the chain breaks, and the progress in numbering can go no farther. So that to reckon right, it is required, 1, That the mind distinguishes carefully two ideas, which are different one from another, only by the addition or subtraction of one unit. 2, That it retain in memory the names or marks of the several combinations from an unit to that number; and that not confusedly, and at random, but in that exactorder, that the numbers follow one another; in either of which, if it trips, the whole business of numbering will be disturbed, and there will remain only the confused idea of multitude; but the ideas necessary to distinct numeration, will not be attained to.

§8. Number measures all measurables.-This farther is observable in number, that it is that which the mind makes use of in measuring all things that by us are measurable, which principally are expansion and duration; and our idea of infinity, even when applied to those, seems to be nothing but the infinity of number. For what else are our ideas of eternity and immensity, but the repeated additions of certain ideas of imagined parts of duration and expansion, with the infinity of number in which we can come to no end of addition? For such an inexhaustible stock, number (of all other ideas) most clearly furnishes us with, as is obvious to every one. For let a man collect into one sum, as great a number as he pleases, this multitude, how great soever, lessens not one jot the power of adding to it, or brings him any nearer the end of the inexhaustible stock of number, where still there remains as much to be added, as if none were taken out. And this endless addition, or addibility (if any one like the word better) of numbers, so apparent to the mind, is that, I think, which gives us the clearest and most distinct idea of infinity of which, more in the following chapter.

CHAP. XVII.

OF INFINITY.

§ 1. Infinity, in its original intention, attributed to space, duration, and number. He that would know what kind of idea it is to which we give the name of infinity, cannot do it better than by considering to

what infinity is by the mind more immediately attributed, and then how the mind comes to frame it.

Finite and infinite, seem to me to be looked upon by the mind as the modes of quantity; and to be attributed primarily, in their first designation, only to those things which have parts, and are capable of increase or diminution, by the addition or subtraction of any the least part; and such are the ideas of space, duration, and number, which we have considered in the foregoing chapters. It is true, that we cannot but be assured, that the great God, of whom, and from whom, are all things, is incomprehensibly infinite. But yet, when we apply to that first and supreme Being, our idea of infinite, in our weak and narrow thoughts, we do it primarily in respect to his duration and ubiquity; and, I think, more figuratively to his power, wisdom, and goodness, and other attributes, which are properly inexhaustible and incomprehensible, &c. For when we call them infinite, we have no other idea of this infinity, but what carries with it some reflection on, and intimation of, that number or extent of the acts or objects of God's power, wisdom, and goodness, which can never be supposed so great, or so many, which these attributes will not always surmount and exceed, let us multiply them in our thoughts as far as we can, with all the infinity of endless number. I do not pretend to say how these attributes are in God, who is infinitely beyond the reach of our narrow capacities: they do, without doubt, contain in them all possible perfection: but this, 1 say, is our way of conceiving them, and these our ideas of their infinity. § 2. The idea of finite easily found.-Finite, then, and infinite, being by the mind looked on as modifications of expansion and duration, the next thing to be considered, is, how the mind comes by them. As for the idea of finite, there is no great difficulty. The obvious portions of extension that affect our senses, carry with them into the mind the idea of finite and the ordinary periods succession, whereby we measure time and duration, as hours, days, and years, are bounded lengths. The difficulty is, how we come by those boundless ideas of eternity and immensity, since the objects we converse with come so much short of any approach or proportion to that largeness.

s. How we come by the idea of infinity.-Every one that has any idea of any stated lengths of space, as a foot, finds that he can repeat the idea; and joining it to the former, make the idea of two feet; and by the addition of a third, three feet; and so on, without ever coming to an end of his addition, whether of the same idea of a foot, or if he pleases of doubling it, or any other idea he has of any length, as a mile, or diameter of the earth, or of the orbis magnus; for whichsoever of these he takes, and how often soever he doubles, or any otherwise multiplies it, he finds, that after he has continued this doubling in his thoughts, and enlarged his idea as much as he pleases, he has no more reason to stop, nor is one jot nearer the end of such addition, than he was at first setting out; the power of enlarging his idea of space by farther additions, remaining still the same, he hence takes the idea of infinite

space.

§ 4 Our idea of space, boundless.-This, I think, is the way whereby the mind gets the idea of infinite space. It is a quite different cou

sideration to examine, whether the mind has the idea of such a boundless space actually existing, since our ideas are not always proof of the existence of things; but yet, since this comes here in our way, I suppose 1 may say, that we are apt to think that space in itself is actually boundless; to which imagination, the idea of space or expansion of itself naturally leads us. For it being considered by us either as the extension of body, or as existing by itself, without any solid matter taking it up (for of such a void space we have not only the idea, but I have proved, as I think, from the motion of body, its necessary existence), it is impossible the mind should be ever able to find or suppose any end of it, or be stopped any where in its progress in this space, how far soever it extends its thoughts. Any bounds made with body, even adamantine walls, are so far from putting a stop to the mind in its farther progress in this space and extension, that it rather facilitates than enlarges it! for so far as that body reaches, so far no one can doubt of extension; and when we are come to the utmost extremity of body, what is there that can there put a stop, and satisfy the mind that it is at the end of space, when it perceives it is not; nay, when it is satisfied that body itself can move into it? For if it be necessary for the motion of the body that there should be an empty space, though ever so little, here amongst bodies; and it be possible for body to move in or through that empty space; nay, it is impossible for any particle of matter to move but into an empty space; the same possibility of a body's moving into a void space, beyond the utmost bounds of body, as well as into a void space interspersed amongst bodies, will always remain clear and evident, the idea of empty pure space, whether within, or beyond the confines of all bodies, being exactly the same, differing not in nature, though in bulk ; and there being nothing to hinder body from moving into it. So that wherever the mind places itself by any thought, either amongst, or remote from all bodies, it can, in this uniform idea or space, no where find any bounds, any end; and so must necessarily conclude it, by the very nature and idea of each part of it, to be actually infinite.

5. And so of duration.-As by the power we find in ourselves of repeating, as often as we will, any idea of space, we get the idea of immensity; so, by being able to repeat the idea of any length of duration we have in our minds, with all the endless addition of number, we come by the idea of eternity. For we find in ourselves, we can no more come to the end of such repeated ideas, than we can come to the end of number, which every one perceives he cannot. But here again it is another question, quite different from our having an idea of eternity, to know whether there were any real being, whose duration has been eternal. And as to this, I say, he that considers something now existing, must necessarily come to something eternal. But having spoke of this in another place, I shall here say no more of it, but proceed on to some other considerations of our idea of infinity.

§6. Why other ideas are not capable of infinity.-If it be so, that our idea of infinity be got from the power we observe in ourselves, of repeating without end our own ideas, it may be demanded, “Why we do not attribute infinity to other ideas, as well as those of space and duration; since they may be as easily, and as often, repeated in our

minds as the other; and yet nobody ever thinks of infinite sweetness or infinite whiteness, though he can repeat the idea of sweet or white, as frequently as those of a yard or a day?" To which I answer, all the ideas that are considered as having parts, and are capable of increase by the addition of any equal or less parts, afford us, by their repetition, the idea of infinity; because, with this endless repetition, there is continued an enlargement, of which there can be no end. But in other ideas it is not so; for to the largest idea of extension or duration, that I at present have, the addition of any of the least part, makes an increase; but to the perfectest idea I have of the whitest whiteness, if I add other of a less or equal whiteness (and of a whiter than I have, I cannot add the idea), it makes no increase, and enlarges not my idea at all; and, therefore, the different ideas of whiteness, &c., are called degrees. For those ideas that consist of parts, are capable of being augmented by every addition of the least part; but if you take the idea of white, which one parcel of snow yielded yesterday to your sight, and another idea of white, from another parcel of snow you see to-day, and put them together in your mind, they embody, as it were, and run into one, and the idea of whiteness is not at all increased; and if we add a less degree of whiteness to a greater, we are so far from encreasing, that we diminish it. Those ideas that consist not of parts, cannot be augmented to what proportion men please, or be stretched beyond what they have received by their senses; but space, duration, and number, being capable of increase by repetition, leave in the mind an idea of an endless room for more; nor can we conceive any where a stop to a farther addition or progression, and so those ideas alone lead our minds towards the thought of infinity.

§7. Difference between infinity of space, and space infinite.Though our idea of infinity arise from the contemplation of quantity, and the endless increase the mind is able to make in quantity, by the repeated additions of what portions thereof it pleases; yet I guess we cause great confusion in our thoughts, when we join infinity to any supposed idea of quantity the mind can be thought to have, and so discourse or reason about an infinite quantity, viz., an infinite space, or an infinite duration. For our idea of infinity being, as I think, an endless growing idea, by the idea of any quantity the mind has, being at that time terminated in that idea (for be it as great as it will, it can be no greater than it is), to join infinity to it, is to adjust a standing measure to a growing bulk; and, therefore, I think it is not an insignificant subtilty, if I say, that we are carefully to distinguish between the idea of the infinity of space, and the idea of a space infinite. The first is nothing but a supposed endless progression of the mind, over what repeated ideas of space it pleases; but to have actually in the mind the idea of a space infinite, is to suppose the mind already passed over, and actually to have a view of all those repeated ideas of space which an endless repetition can never totally represent to it; which carries in it a plain contradiction.

8. We have no idea of infinite space.-This perhaps will be a little plainer, if we consider it in numbers. The infinity of numbers, to the end of whose addition every one perceives there is no approach, easily