nor perhaps poffible for us to diftinguish betwixt two approaching ideas, which yet are really different. For who will undertake to find a difference between the white of this paper, and that of the next degree to it; or can form diftinct ideas of every the least excess in extenfion?

-Therefore demonftra

tions in num

bers the most precife.

§. 4. The clearness and diftinctness of each mode of number from all others, even thofe that approach nearest, makes me apt to think that demonstrations in numbers, if they are not more evident and exact than in extenfion, yet they are more general in their use, and more determinate in their application. Because the ideas of numbers are more precife and diftinguishable than in extenfion, where every equality and excefs are not so easy to be obferved or measured; because our thoughts cannot in fpace arrive at any determined fmallnefs, beyond which it cannot go, as an unit; and therefore the quantity or proportion of any the leaft excefs cannot be difcovered: which is clear otherwise in number, where, as has been faid, ninety-one is as diftinguishable from ninety, as from nine thousand, though ninety-one be the next immediate excefs to ninety. But it is not fo in extenfion, where whatfoever is more than juft a foot or an inch, is not diftinguishable from the ftandard of a foot or an inch; and in lines which appear of an equal length, one may be longer than the other by innumerable parts; nor can any one affign an angle, which shall be the next biggest to a right one.

Names neceffary to

§. 5. By the repeating, as has been faid, the idea of an unit, and joining it to anonumbers. ther 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 several collections of units, diftinguished one from another, as far as he hath a series of names for following numbers, and a memory to retain that feries, with their feveral names: all numeration being but still the adding of one unit more, and giving to the whole together, as comprehended

prehended in one idea, a new or diftinct name or fign, whereby to know it from thofe before and after, and diftinguish it from every fmaller or greater multitude of units. So 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 fubtracting an unit from each collection, retreat and deffen 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 neceffary than in any other fort of ideas. For without fuch names or marks we can hardly well make ufe 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 diftinguifh that precife collection, will hardly be kept from being a heap in confufion.i

§. 6. 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 one thoufand; nor had any diftinct idea of that number, though they could reckon very well to twenty. 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 one thoufand; 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 coulă not number: which inability, I fuppofe, proceeded from their want of names. The Tououpinambos had no names for numbers above five; any number beyond that they made out by fhowing their fingers, and the fingers of others who were prefent *. And I doubi

+ Hiftoire d'un voyage, fait en la terre du Brafil, par Jean de Lery, 20.401

not

not but we ourselves 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 millions of millions of millions, &c. it is hard to go beyond eighteen, or at most four and twenty decimal progreffions, without confufion. But to fhow how much diftinct names conduce to our well reckoning, or having useful ideas of numbers, let us fet all thefe 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 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, these, and perhaps a great many more figures in progreffion, might not eafily be counted dif tinctly, and ideas of them both got more eafily to ourfelves, 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.

Why chil- §. 7. Thus children, either for want of dren number names to mark the feveral progreffions of not earlier. 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 twenty. And fome, through the default of their memories, who cannot retain the feveral com binations of numbers, with their names annexed in their distinct 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 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 progress in numbering can go no farther. So that to reckon right, it is required, I. 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 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 disturbed, and there will remain only the confused idea of multitude, but the ideas neceffary to diftinct numeration will not be attained to.

Number measures all meafurables.

§. 8. This farther is obfervable in numbers, that it is that which the mind makes use 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 else are our ideas of eternity and immenfity, but the repeated additions of certain ideas of imagined parts of duration and expan-' fion, 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 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 inexhauftible ftock of number, VOL. I. where

where still there remains as much to be added, as if none were taken out. And this endlefs 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.

Infinity, in its original intention, at

S. I.

tributed to

E that would know what kind of 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.

tion and

number.

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 defignation only to those things which have parts, and are capable of increase or diminution, by the addition or fubtraction of any the least 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, wisdom, 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

multi