felves, and more plainly fignified to others, I leave it to be confidered. This I mention only to fhow how neceffary diftinct 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 feveral progreffions of numbers, or not having yet the faculty to collect fcattered ideas into complex ones, and range them in a regular order, and fo retain them in their memories, as is neceffary to reckoning, do not begin to number very early, nor proceed in it very far or fteadily, till a good while after they are well furnifhed with good store of other ideas; and one may often obferve them difcourfe and reafon pretty well, and have very clear conceptions of feveral other things, before they can tell 20; and fome, through the default of their memories, who cannot retain the feveral combinations of numbers, with their names annexed in their diftinct orders, and the dependence of fo long a train of numeral progreffions, and their relation one to another, are not able all their lifetime 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 diftinct name or fign of every one of them as they ftand marked in their order; for wherever this fails, a gap is made, the chain breaks, and the progrefs in numbering can go no farther: So that to reckon right, it is required, 1. That the mind diftinguish carefully two ideas which are different one from another only by the addition or fubtraction of one unit. 2. That it retain in memory the names or marks of the feveral combinations from an unit to that number, and that not confufedly and at random, but in that exact order that the numbers follow one another; in either of which, if it trips, the whole bufinefs of numbering will be disturbed, and there will remain only the confufed idea of multitude, but the ideas neceflary to diftinct numeration will not be attained to.

$8. Number Measures all Meafurable.

THIS farther is obfervable in number, that it is that which the mad makes ufe 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 thofe, feems to be nothing but the infinity of number; for what else are our ideas of eternity and immenfity, 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 fuch an inexhaustible stock, number, of all other our ideas, moft clearly furnishes us with, as is obvious to every one. For let a man collect into one fum as great a number as he pleases, this multitude, how great foever, leffens not one jot the power of adding to it, or brings him any nearer the end of the inexhauftible ftock of number, where ftill 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, fo 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.

§ 1. Infinity, in its original Intention, attributed to Space,

HE

Duration, and Number.

E that would know what kind of idea it is to which we give the name of infinity, cannot do it better than by confidering to what infinity is by the mind more immediately attributed, and then how the mind comes to frame it.

Finite and infinite feem 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 thofe things which have parts, and are capable of increafe or dimi

nution by the addition or fubtraction of any the leaft part; and fuch are the ideas of fpace, duration, and number, which we have confidered in the foregoing chapters. It is true that we cannot but be affured that the great God, of whom and from whom are all things, is incomprehenfibly infinite; but yet, when we apply to that firft and fupreme Being our idea of infinite in our weak and narrow thoughts, we do it primarily in refpect of his duration and ubiquity, and, I think, more figuratively to his power, wildom, and goodnefs, and other attributes, which are properly inexhaustible and incomprehenfible, &c.; for when we call them infinite, we have no other idea of this infinity but what carries with it fome reflection on and intimation of that number or extent of the acts or objects of God's power, wifdom, and goodnefs, which can never be fuppofed fo great or fo many, which thefe attributes will not always. furmount and exceed, let us multiply them in our thoughts as far as we can, with all the infinity of endlefs number. I do not pretend to fay how thefe attributes are in God, who is infinitely beyond the reach of our narrow capacities; they do, without doubt, contain in them all poffible perfection; but this, I fay, is our way of conceiving them, and thefe our ideas of their infinity.

2. The Idea of Finite eafily got. FINITE, then, and infinite, being by the mind looked on as modifications of expanfion and duration, the next thing to be confidered is, How the mind comes by them. As for the idea of finite, there is no great difficulty; the obvious portions of extenfion that affect our fenfes carry with them into the mind the idea of finite; and the ordinary periods of fucceffion, 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 immenfity, fince the objects which we converfe with come fo much fhort of any approach or proportion to that largenefs.

§ 3. How we come by the Idea of Infinity. EVERY one that has any idea of any ftated lengths of

fpace, as a foot, finds that he can repeat that idea, and joining it to the former, make the idea of two feet, and by the addition of a third, three feet, and fo on, without ever coming to an end of his additions, whether of the fame idea of a foot, or, if he pleafes, 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 whichfoever of these he takes, and how often foever he doubles, or any otherwife multiplies it, he finds, that after he has continued his doubling in his thoughts, and enlarged his idea as much as he pleafes, he has no more reason to stop, nor is one jot nearer the end of fuch addition, than he was at firft fetting out. The power of enlarging his idea of fpace by farther additions remaining ftill the fame, 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 confideration to examine whether the mind has the idea of fuch a boundless space actually exifling, fince our ideas are not always proofs of the existence of things; but yet, fince this comes here in our way, I fuppofe I may fay, that we are apt to think that space in itself is actually boundlefs, to which imagination the idea of fpace or expanfion of itself naturally lead us; for it being confidered by us either as the extenfion of body, or as exifting by itself, without any solid matter taking it up (for of fuch a void space we have not only the idea, but I have proved, as I think, from the motion of body, its neceffary existence), it is impoflible the mind fhould be ever able to find or fuppofe any end of it, or be ftcpped any where in its progrefs in this fpace, how far foever it cxtends its thoughts. Any bounds made with body, even adamantine walls, are fo far from putting a ftop to the mind in its farther progrefs in fpace and extenfion, that it rather facilitates and enlarges it; for fo far as that body reaches, fo far no one can doubt of extenfion; and when we are come to the utmost extremity of body, what is there that can there put a ftop, and fatisfy

the mind that it is at the end of fpace, when it perceives it is not, nay, when it is fatisfied that body itfelf can move into it? For if it be neceffary for the motion of body that there should be an empty space, though ever fo little, here amongst bodies, and if it be poffible for body to move in or through that empty Ipace, nay, it is impoffible for any particle of matter to move but into an empty space, the fame poffibility of a body's moving into a void space, beyond the utmost bounds of body, as well as into a void space interfpersed amongst bodies, will always remain clear and evident; the idea of empty pure fpace, whether within or beyond the confines of all bodies, being exactly the fame, differing not in nature, though in bulk, and there being nothing to hinder body from moving into it; fo that wherever the mind places itself by any thought, either amongst or remote from all bodies, it can in this uniform idea of space no where find any bounds, any end, and fo muft neceffarily conclude it, by the very nature and idea of each part of it, to be actually infinite.

$5. And fo 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 immenfity, fo by being able to repeat the idea of any length of duration we have in our minds, with all the endlefs addition of number, we come by the idea of eternity; for we find in ourselves we can no more come to an end of fuch 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 whofe duration has been eternal. And as to this, I fay, he that confiders fomething now exifting, muft neceffarily come to fomething eternal. But having fpoke of this in another place, I fhall fay here no more of it, but proceed on to fome other confiderations of our idea of infinity.

6. Why other Ideas are not capable of Infinity. If it be fo that our idea of infinity be got from the power we obferve in ourfelves of repeating without end