$5. Names neceffary to Numbers.

By the repeating, as has been faid, of the idea of an unit, and joining it to another unit, we make thereof one collective idea, marked by the name two; and whofoever can do this, and proceed on, ftill adding one more to the laft collective idea which he had of any number, and give a name to it, may count or have ideas for feveral collections of units, diftinguifhed one from another, as far as he hath a series of names for following numbers, and a memory to retain that feries, with their several names; all numeration being but still the adding of one unit more, and giving to the whole together, as comprehended in one idea, a new or diftinct name or fign whereby to know it from thofe before and after, and diftinguish it from every smaller or greater multitude of units; fo that he that can add one to one, and fo to two, and fo go on with his tale, taking ftill with him the diftinct names belonging to every progreffion, and fo again, by abftracting an unit from each collection, retreat and leffen them, is capable of all the ideas of numbers within the compafs of his language, or for which he hath names, though not perhaps of more; for the feveral fimple modes of numbers being in our minds but fo many combinations of units, which have no variety, nor are capable of any other difference, but more or lefs names or marks for each diftinct combination, feem more neceflary than in any other fort of ideas; for without fuch names or marks we can hardly well make use of numbers in reckoning, efpecially where the combination is made up of any great multitude of units, which put together without a name or mark to diftinguish that precife collection, will hardly be kept from being a heap in confusion. $6. Names neceffary to Numbers.

THIS I think to be the reason why fome Americans I have fpoken with (who were otherwife of quick and rational parts enough) could not, as we do, by any means count to 1000, nor had any diftinct idea of that number, though they could reckon very well to 20, because their language being fcanty, and accommodated

only to the few neceffaries of a needy fimple life, unacquainted either with trade or mathematics, had no words in it to ftand for 1000; fo that when they were difcourfed with of thofe greater numbers, they would fhow the hairs of their head, to exprefs a great multitude which they could not number; which inability, I fuppofe, proceeded from their want of names. *The Tououpinambos had no names for numbers above 5; any number beyond that, they made out by fhowing their fingers, and the fingers of others who were prefent: And I doubt not but we ourselves might diftinctly number in words a great deal farther than we usually do, would we find out but fome fit denominations to fignify 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-andtwenty decimal progreffions, without confufion. But, to fhow how much diftinct names conduce to our well reckoning, or having ufeful ideas of numbers, let us fet all thefe following figures in one continued line as the marks of one number; v. g.

Nonilions. Ocilions. Septilions. Sextilions. Quintilions: 857324. 162486. 345896. 437916. 423147. Quartilions. Trilions. Bilions. Millions. Units. 248106. 235421. 261734. 368149. 623137.

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 fecond fix figures), in which way it will be very hard, to have any diftinguithing 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 easily be counted diftinctly, and ideas of them both got more eafily to our

*Hiftoire d'un voyage fait en la terre du Brafil, par Jean de Lety, C. 20. 207-282.

felves, and more plainly fignified to others, I leave it to be considered. 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 scattered 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 20; and fome, through the default of their memories, who cannot retain the several 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 ferics 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 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 diftinguith 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 difturbed, and there will remain only the confufed idea of multitude, but the ideas necefiary to diflinct numeration will not be attained to.

$8. Number Measures all Meafurable.

THIS farther is obfervable in number, that it is that which the mind makes ufe of in meafuring 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 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 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, leflens 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, fo apparent to the mind, is that, I think, which gives us the clearest and most diftinct idea of infinity; of which more in the following chapter.

CHAP. XVII.

OF INFINITY.

§ 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 firft defignation only to thofe things which have parts, and are capable of increase or dimi

nution by the addition or fubtraction of any the least part; and fuch are the ideas of space, 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 first 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 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 intimation 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 so 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 easily get. 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 fucceflion, 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 converse with come fo much fhort of any approach or proportion to that largeness.

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