more simple than that of unity: every thought of our mind brings this idea with it, for number applies itself to every thing that either doth exist or can be imagined. By repeating this idea, we come by the complex ideas of the modes of it; thus by putting twelve units together we have the complex idea of a dozen.

The simple modes of number are of all other the most distinct, two being as distinct from one as two hundred. This is not so in other simple modes; for who will undertake to find a difference between the white of this paper and that of the next degree to it?

The distinctness of each mode of number makes me think that demonstrations in numbers, if not more exact than in extension, are more determinate in their application, because the ideas of numbers are more distinguishable than in extension. Number 91 is as distinguishable from 90, to which it is the next excess, as it is from 9000; but in extension of lines, which appear of an equal length, one may be longer than the other by innumerable parts; nor can any one assign an angle which shall be the next biggest to a right

one.

By repeating the idea of a unit we make a collective idea, marked by the name two; and whoever can proceed, still adding one to the last collective idea, and give a name to it, may have ideas for collections of units as far as he has names for numbers, and memory to retain them. So that he who can add one to one and so to two, and go on taking distinct names to every progression, and again by subtracting, can retreat and lessen them, is capable of all the ideas of numbers within the compass of his language, though not of more; for without names and marks we cannot use numbers in reckoning, which being put together without a name, will be hardly kept from being a heap of confusion. This I think to be the reason why some Americans could not count to 1000, although

they could reckon very well to 20; because their language had no word to stand for 1000, and in order to express a great multitude, they would show the hairs of their head. The Tououpinambos had no names for numbers above 5; any number above that they made out by showing their fingers, and those of others who were present; and we ourselves might number farther than we do, would we but find out fit denominations. To show how much distinct names conduce to well reckoning, let us set the following figures as the marks of one number:

Nonilions. Octilions. Septilions. Sextilions. Quintilions. 623,137. 857,324.

Quatrilions.

432,147.

162,486.

345,896.

437,916.

The ordinary way of naming this number is the repeating of millions, of millions, of millions, &c. &c. in which way it will be hard to have any distinguishing notion of this number; but by giving every six figures a denomination, a great many more may be easily counted, and more plainly signified to others.

Thus children, either for want of names, or not having the faculty to collect scattered ideas into complex ones, do not begin to number very early, and they have clear conceptions of several other things before they can count 20. And some, through default of memory to retain the combinations, are not able all their life-time to reckon any moderate series of numbers for to reckon right it is required-1. that the mind distinguishes two ideas, which differ only by the addition or subtraction of a unit; 2. that it retain in memory the names of the several combinations in exact order, without which there will remain only the confused idea of multitude, but the ideas necessary to distinct numeration will not be attained.

This farther is observable in number, that the mind makes use of it in measuring all things, and our idea of

infinity seems to be nothing but the infinity of number. For let a man collect into one sum as great a number as he pleases, he still has the power of adding to it; and this capacity of endless addition is that which gives us the clearest idea of infinity.

CHAPTER XVII.

Of Infinity.

He that would know what the idea of infinity is, must consider to what infinity is attributed, and how the mind comes to frame the idea.

Finite and infinite seem to be looked on as modes of quantity, and to be attributed primarily to whatever is capable of increase or diminution; such are the ideas of space, duration, and number. We cannot but be assured that the great God is infinite; but when we apply to him our idea of infinite, we do it primarily, in respect of his duration and ubiquity, but more figuratively to his power, wisdom, and goodness, which are properly inexhaustible and incomprehensible: for when we call them infinite, we intimate that the number of their acts and objects can never be supposed so great, that these attributes will not always exceed them. I do not pretend to say how these attributes are in God, but these are our ideas of their infinity.

Finite and infinite being modifications of expansion and duration, it is next to be considered how the mind comes by them. The portions of extension that affect our senses, and the ordinary measures of duration, as hours, days, and years, carry with them the idea of finite. The difficulty is how we come by the idea of infinity.

Every one who has an idea of any length of space can repeat that idea without ever coming to an end of his additions; for how often soever he increases it, he finds that the power of enlarging this idea of space still remains the same; hence he has the idea of infi

nite space. It is a different consideration to examine whether the mind has the idea of such boundless space actually existing, since our ideas are not proof of existence; but I may say that we are apt to think that space itself is actually boundless, to which imagination the idea of space itself naturally leads us: for it being considered either as the extension of body, or as existing by itself, without solid matter taking it up, it is impossible that the mind should be able to suppose any end to it. So far as body reaches, no one 'can doubt of extension; and when we are come to the extremity of body, what is there that can satisfy the mind that it is at the end of space, when it is satisfied that body itself can move into it? For if it be impossible for matter to move but into empty space, the same possibility of a body's moving into space beyond the bounds of body, as well as into space interspersed among bodies, will always remain clear, the idea of space, whether within or beyond the confines of body, being exactly the same. So that wherever the mind places itself either amongst or remote from all bodies, it can, in this uniform idea of space, no where find any bounds, and must necessarily conclude it to be infinite.

As by repeating any idea of space we get the idea of immensity, so by repeating any lengths of duration we come by the idea of eternity: for we can no more come to the end of such repeated ideas than we can come to the end of number. But it is another question to know whether there were any real Being whose duration has been eternal. Having spoken of this in another place, I shall proceed to some other considerations of our idea of infinity.

If our idea of infinity be got by repeating our own ideas, it may be demanded why we do not attribute infinity to other ideas as well as those of space and duration? yet nobody ever thinks of infinite sweetness or infinite whiteness. I answer, all the ideas

that are capable of increase by addition of equal or less parts afford us the idea of infinity: to the largest idea of extension or duration that I at present have, the addition of any part makes an increase; but to the most perfect idea I have of whiteness, if I add another less or equal, (and a greater I cannot have) it makes no increase, and therefore the different ideas of whiteness, &c. are called degrees. If you take the idea of white which a parcel of snow yielded yesterday, and another idea of white from snow seen today, and put them together, your idea of whiteness is not at all increased. And if we add a less degree of whiteness to a greater, we diminish it.

Though our idea of infinity arise from the contemplation of quantity and the endless increase the mind is able to make in quantity, yet we cause great confusion when we join infinity to any supposed idea of quantity, and so discourse about an infinite space or an infinite duration for our idea of infinity being a growing idea, and our idea of a quantity being terminated in that idea, to join infinity to it is to adjust a standing measure to a growing bulk. There is a distinction therefore between the idea of the infinity of space and the idea of a space infinite. The first is an endless progression, but the latter supposes the mind to have passed over those repeated ideas of space which endless repetition can never totally represent; which is a plain contradiction.

This will be plainer if we consider it in numbers. The infinity of numbers easily appears to any one that reflects; but there is nothing more evident than the absurdity of the actual idea of an infinite number. Whatever ideas we have of any space, duration, or number, they are finite: but when we suppose an inexhaustible remainder from which we remove all bounds, we have our idea of infinity; which seems clear when we consider it but the negation of an end; yet when we would frame an idea of an in