not but we ourfelves might diftinctly number in words a great deal farther than we ufually do, would we find out but fome fit denomination to fignify them by ; whereas in the way we take now to name them by mil lions of millions of millions, &c. it is hard to go beyond eighteen, or at moft four and twenty decimal progreffions, without confufion. But to fhow how much diftinét names conduce to our well reckoning, or hav ing ufeful ideas of numbers, let us fet all thefe following figures in one continued line, as the marks of one number; v. g.

Nonillions. Octillions. Septillions. Sextillions. Quintrillions. 857324 162486 345896

Quatrillions. Trillions.

248106 235421

437918

423147.

Billions.

Millions.

Units.

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 mil lions, of millions, (which is the denomination of the fecond fix figures.) In which way, it will be very hard to have any distinguishing notions of this number: but whether, by giving every fix figures a new and orderly denomination, thefe, and perhaps a great many more figures in progreffion, might not eafily be counted diftinctly, and ideas of them both got more easily to our felves, and more plainly fignified to others, I leave it to be confidered. This I mention only to fhow how necef Lary diftinct names are to numbering, without pretend❤ ing to introduce new ones of my invention.

not earlier.

Why chil§. 7. Thus children, either for want of dren number 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 furnished 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 twenty. 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 life-time to reckon, or regularly go over any moderate feries 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 several combinations, from an unit to that number; and that not confusedly, 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 difturbed, and there will remain only the confufed idea of multitude, but the ideas neceffary to diftinct numeration will not be attained to.

Number measures all measurables.

§. 8. This farther is obfervable in numbers, that it is that which the mind makes ufe of in measuring all things that by us are measurable, which principally are expanfion and duration; and our idea of infinity, even when applied to thofe, feems to be nothing but the infinity of number. For what elfe are our ideas of eternity and immenfity, but the repeated additions of certain ideas of imagined parts of duration and expanfion, with the infinity of number, in which we can come to no end of addition? For fuch an inexhaustible ftock, number (of all other our ideas) most clearly furnishes us with, as is obvious to every one. For let a man collect into one fum as great a number as he pleafes, this multitude, how great foever, leffens not one jot the power of adding to it, or brings him any nearer the end of the inexhaustible stock of number, VOL. I. where

O

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 diftinct idea of infinity: af which more in the following chapter.

CHA P. XVII.

Of Infinity.

§. 1. HE that would know what kind of

Infinity, in S. its original intention, attributed to

tion and

number.

idea it is to which we give the

name of infinity, cannot do it better, than fpace, dura- 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 those things which have parts, and are capable of increase or diminution, 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 to his duration and ubiquity; and, I think, more figuratively to his power, wifdom, and goodness, and other attributes, which are properly inexhauftible 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 imitation of, that number or extent of the acts or objects of God's power, wifdom, and goodness, which can never be fuppofed fo great or fo many, which these attributes will not always furmount 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 poffible perfection: but this, b fay, is our way of conceiving them, and these our ideas of their infinity.

The idea of finite eafily

got.

As for The ob

§. 2. 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. the idea of finite, there is no great difficulty. vious 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 we converfe with, come fo much fhort of any approach or proportion to that largenefs.

How we

come by the idea of infinity.

on, with

$. 3. Every one that has any idea of any ftated lengths of space, 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 so out ever coming to an end of his addition, whether of the fame 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 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 ftop, nor is one jot nearer the end of fuch addition, than he was at first fetting out. The power of enlarging his idea of space by farther additions remaining ftill the fame, he hence takes the idea of infinite space.

§. 4. 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

O 2

our idea of fpace boundlefs.

a bound

a boundless space actually exifting, 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 itfelf is actually boundlefs; to which imagination, the idea of space or expanfion of itself naturally leads us. For it being confidered by us, either as the extenfion of body, or as exifting by itself, without any folid 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 exiftence) it is impoffible the mind fhould be ever able to find or fuppofe any end of it, or be stopped any where in its progrefs in this space, how far foever it extends its thoughts. Any bounds made with body, even adamantine walls, are fo far from putting a top to the mind in its farther progress 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 that it is not; nay, when it is fatisfied that body itself 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 space; 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 interfperfed amongft 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. So that wherever the mind places itfelf 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 con-clude it, by the very nature and idea of each part of it, to be actually infinite.