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ration, and every part of duration in every part of expansion. Such a combination of two distinct ideas is, I suppose, scarce to be found in all that great variety we do or can conceive, and may afford matter to farther speculation.
It has sal idea.
as there is none suggested to the simplest and mind by more ways, so there is none more
most universimple, than that of unity, or one. no shadow of variety or composition in it: every object our senses are employed about, every idea in our understandings, every thought of our minds, brings this idea along with it. . And therefore it is the most. intimate to our thoughts, as well as it is, in its agreement to all other things, the most universal idea we have. For number applies itself to men, angles, actions, thoughts, every thing that either doth exist, or can be imagined
$. 2. By repeating this idea in our minds, Its modes and adding the repetitions together, we come
made by ad
dition. by the complex ideas of the modes of it. Thus by adding one to one, we have the complex idea of a couple ; by putting twelve units together, we have .the complex idea of a dozen ; and so of a score, or a million, or any other number.
$. 3. The simple modes of numbers are Each mode of all other the most distinct ; every the distinct, least variation, which is an unit
, making each combination as clearly different from that which approacheth nearest to it, as the most remote : two being as distinct from one, as two hundred; and the idea of Iwo as distinct from the idea of three, as the magnitude of the whole earth is from that of a mite. This is not so in other simple modes, in which it is not so easy,
nor perhaps possible for us to distinguish 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 distinct ideas of every the least excess in extension? Therefore § 4. The clearness and distinctness of demonstra. each mode of number from all others, even tions in num- those that approach nearest, makes me apt bers the most
to think that demonstrations in numbers, if precise.
they are not more evident and exact than in extension, yet they are more general in their use, and more determinate in their application. Because the ideas of numbers are more precise and distinguishable than in extension, where every equality and excess are not so easy to be observed or measured ; because our thoughts cannot in space arrive at any determined smallness, beyond which it cannot go, as an unit; and therefore the quantity or proportion of any the least excess cannot be discovered: which is clcar 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 birvest to a right one. Names ne
$. 5. By the repeating, as has been said, cessary to the idea of an unit, and joining it to anonumbers. ther 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 give 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 com
prehended prehended 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 bút 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
• Histoire d'un voyage, fait en la terre du Brasil, par Jean de Lery, 6, 20. fil.
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 fignity 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 tirenty 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. 8. Nonillions. Octillions. Septillions. Sextilins. Quintrillions. 857394 102456 345896 457918 493117 Quatrillions. Trillions. Billions.
Millions. Units. 248106 235421 261734 3681 +9 603137
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 invention. Why chil. $. 7. Thus children, either for want of dren number names to mark the several progressions 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 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 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 inarks of the several combinations, from an unit to that number, and that not confusedly, and at randomn, 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 nume. ration will not be attained to. $. 8. This farther is observable in num
Number bers, that it is that which the mind makes
measures all use of in measuring all things that by us measurables. are measurable, which principally are expansion and duration ; and our idea of infinity, even - when applied to those, seems to be nothing but the intinity of number. For what else are our ideas of eternity and immensity, bụt 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 ...pearer the end of the inexhaustible stock of number, VOL I.