language, or for which he hath names, though not perhaps of more. For the several simple modes of numbers, being in our minds but so many combinations of units, which have no variety, nor are capable of any other difference but more or less, names or marks for each distinct combination seem more necessary than in any other sort of ideas. For without such names or marks we can hardly well make use of numbers in reckoning, especially where the combination is made up of any great multitude of units; which put together without a name or mark, to distinguish that precise collection, will hardly be kept from being a heap in confusion.

§ 6. This I think to be the reason why some Americans I have spoken with (who were otherwise of quick and rational parts enough), could not, as we do, by any means count to one thousand, nor had any distinct idea of that number, though they could reckon very well to twenty; because their language being scanty, and accommodated only to the few necessaries of a needy simple life, unacquainted either with trade or mathematics, had no words in it to stand for one thousand; so that when they were discoursed with of those great numbers, they would show the hairs of their head to express a great multitude which they could not number; which inability, I suppose, proceeded from their want of names. The Tououpinambos had no names for numbers above five; any number beyond that they made out by showing their fingers, and the fingers of others who were present*. And I doubt not but we ourselves might distinctly number in words a great deal farther than we usually do, would we find out but some fit denomination to signify 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

* Histoire d'un voyage, fait en la terre du Brasil, par Jean de Lery, c. 20. 387.

twenty decimal progressions, without confusion. But to show how much distinct names conduce to our well reckoning, or having useful ideas of numbers, let us set all these following figures in one continued line, as the marks of one number; v. g.

423147

Nonillions. Octillions. Septillions. Sextillions. Quintillions. 857324 162486 345896 437918 Quatrillions. Trillions. Billions. Millions. 248106 235421 261734 368149

Units. 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 second six figures.) In which way it will be very hard to have any distinguishing notions of this number: but whether, by giving every six figures a new and orderly denomination, these, and perhaps a great many more figures in progression, might not easily be counted distinctly, and ideas of them both got more easily to ourselves, and more plainly signified to others, I leave it to be considered. This I mention only to show how necessary distinct names are to numbering, without pretending to introduce new ones of my in

vention.

Why chil

dren num ber not earlier.

$7. Thus children, either for want of names to mark the several progressions of numbers, or not having yet the faculty to collect scattered ideas into complex ones, and range them in a regular order, and so retain them in their memories, as is necessary to reckoning; do not begin to number very early, nor proceed in it very far or steadily, till a good while after they are well furnished with good store of other ideas; and one may often observe them discourse and reason pretty well, and have very clear conceptions of several other things, before they can tell twenty. And some, through the default of their memories, who cannot retain the several combinations of numbers, with their names an

nexed in their distinct orders, and the dependence of so long a train of numeral progressions, and their relation one to another, are not able all their life-time 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 distinct name or sign of every one of them, as they stand 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, 1. That the mind distinguish carefully two ideas, which are different one from another only by the addition or subtraction 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 business of numbering will be disturbed, and there. will remain only the confused idea of multitude, but the ideas necessary to distinct numeration will not be attained to.

Number measures all

measura

bles.

§ 8. This farther is observable in numbers, that it is that which the mind makes. use 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 those, seems 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 such an inexhaustible stock, number (of all other our ideas), most clearly furnishes us with, as is obvious to every one. For let a man collect into one sum as great a number as he pleases, this multitude, how great soever, lessens not one jot the power of adding to it, or brings him any nearer the end of the inexhaustible stock of num

ber, 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, so 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.

Infinity, in its original intention, attributed to

tion, and number.

CHAPTER XVII.

Of Infinity.

1. HE that would know what kind of idea it is to which we give the name of infinity, cannot do it better than by conspace, dura- sidering to what infinity is by the mind more immediately attributed, and then how the mind comes to frame it. Finite and infinite seem 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 subtraction of any the least part; and such are the ideas of space, duration, and number, which we have considered in the foregoing chapters. It is true, that we cannot but be assured, that the great God, of whom and from whom are all things, is incomprehensibly infinite: but yet when we apply to that first and supreme Being our idea of infinite, in our weak and narrow thoughts, we do it primarily in respect of his duration and ubiquity; and, I think, more figuratively to his power, wisdom, and goodness, and other attributes, which are properly inexhaustible and incomprehensible, &c. For, when we call them infinite, we have no other idea of this infinity, but what carries with it some

reflection on, and imitation of, that number or extent of the acts or objects of God's power, wisdom, and goodness, which can never be supposed so great or so many, which these attributes will not always surmount 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 possible perfection: but this, I say, is our way of conceiving them, and these our ideas of their infinity. § 2. Finite then, and infinite, being by The idea of the mind looked on as modifications of finite easily expansion and duration, the next thing to got. be considered is, how the mind comes by them. As for the idea of finite, there is no great difficulty. The obvious portions of extension that affect our senses, carry with them into the mind the idea of finite; and the ordinary periods of succession, 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 immensity, since the objects we converse with come so much short of any approach or proportion to that largeness. § 3. Every one that has any idea of any stated 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 on, without ever coming to an end of his addition, whether of the same idea of a foot, or if he pleases of doubling it, or any other idea he has of any length, as as a mile, or diameter of the earth, or of the orbis magnus: for whichsoever of these he takes, and how often soever he doubles, or any otherwise multiplies it, he finds that after he has continued his doubling in his thoughts, and enlarged his idea as much as he pleases, he has no more reason to stop, nor is one jot nearer the end of such addition, than he was at first setting

VOL. I.

How we

come by the idea of

infinity.

Р