which it cannot go, as an unit; and therefore the quantity or proportion of any the least excess cannot be discovered: which is clear otherwise in number, where, as has been said, ninety-one is as distinguishable from ninety as from nine thousand, though ninety-one be the next immediate excess to ninety. But it is not so in extension, where, whatsoever is more than just a foot or an inch, is not distinguishable from the standard 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 assign an angle, which shall be the next biggest to a right one.

5. Names necessary to Numbers. By the repeating, as has been said, the idea of an unit, and joining it to another unit, we make thereof one collective idea, marked by the name two: and whosoever can do this, and proceed on, still adding one more to the last collective idea which he had of any number, and gave a name to it, may count, or have ideas for several collections of units, distinguished one from another, as far as he hath a series of names for following numbers, and a memory to retain that series, 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 distinct name or sign, whereby to know it from those before and after, and distinguish it from every smaller or greater multitude of units. So that he that can add one to one, and so to two, and so go on with his tale, taking still with him the distinct names belonging to every progression; and so again, by subtracting an unit from each collection, retreat and lessen them; is capable of all the ideas of numbers within the compass of his 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 greater 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 further 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 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 see all these following figures in one continued line, as the marks of one number, v. g.,

Nonil- Octil- Septil- Sextil- Quintril- Quartril- Tril

lions. lions. lions. lions.. lions. lions. lions. Billions. Millions. Units. 857324 162486 345896 437918 423147 248106 235421 261734 368149 623137

The ordinary way of naming this number in English, will

* Histoire d'un Voyage, fait en la Terre du Bresil, par Jean de Lery, chap. xx. pp. 307–382.

Nearly all systems of arithmetic are founded on the decimal progression, obtained at first by counting the fingers, and proceeding after the manner of the Tououpinambos. The value of the several systems of arithmetic has been discussed by La Place, in a discourse delivered to the Normal School:- "Vous concevez, par les principes métaphysiques sur lesquels est fondé notre système de numération, que rien n'obligeoit de s'en tenir à dix caractères; on pouvait en employes plus ou moins. Il parait très-probable que le nombre des doigts est ce qui a determiné l'arithmétique décimale. Les hommes primitivement ont compté par leurs doigts jusqu'â dix: mais de ce que cette arithmétique était bonne dans l'enfance des societés, est-elle maintenant la meilleure? C'est ce que nous allons examiner." (Arithmétique d'Emile, p. 460. Lausanne, 1823.)--ED.

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 invention.

7. Why Children number not earlier.—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 annexed 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 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 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 further. 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.

8. Number measures all Measurables.-This further 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 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, 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.

CHAPTER XVII.

OF INFINITY.

1. Infinity, in its original Intention, attributed to Space, Duration, and Number.- HE that would know what kind of idea it is to which we give the name of infinity, cannot do it better than by considering 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 to 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. The Idea of Finite easily got.-Finite, then, and infinite, being by the mind looked on as modifications of expansion and duration, the next thing to 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. How we come by the Idea of Infinity.—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 Îength, 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,