are no more distant from his knowledge, no farther removed from his fight, than the prefent; they all lie under the fame view; and there is nothing which he cannot make exist each moment he pleafes; for the existence of all things depending upon his good pleasure, all things exift every moment that he thinks fit to have them exist. To conclude, expanfion and duration do mutually embrace and comprehend each other, every part of space being in every part of duration, and every part of duration in every part of expanfion. Such a combination of two diftinct ideas is, I fuppose, scarce to be found in all that great variety we do or can conceive, and may afford matter to farther fpeculation.

CHAP. XVI.

OF NUMBER.

A

§1. Number the fimplest and most universal Idea. MONGST all the ideas we have, as there is none fuggested to the mind by more ways, fo there is none more fimple than that of unity, or one. It has no fhadow of variety or compofition in it; every object our fenfes are employed about, every idea in our underftandings, every thought of our minds, brings this idea along with it; and therefore it is the moft intimate to our thoughts, as well as it is in its agreement to all other things, the most univerfal idea we have; for number applies itfelf to men, angels, actions, thoughts, every thing that either doth exift or can be imagined.

$2. Its Modes made by Addition.

By repeating this idea in our minds, and adding the repetitions together, we come 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 of a fcore, or a million, or any other number.

§ 3. Each Mode diftin&t.

THE fimple modes of number are of all other the most diflint, every the leaft variation, which is an unit, making each combination as clearly different from that which approacheth nearest to it as the moft remote, two being as diftinct from one as two hundred, and the idea of two as diftinct from the idea of three as the magnitude of the whole earth is from that of a mite. This is not fo in other fimple modes, in which it is not fo cafy, nor perhaps poffible for us to diftinguish befwixt 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 diftin&t ideas of every the least excefs in extenfion?

§4. Therefore Demonftrations in Numbers the mat

precife.

THE clearness and diftinctness of each mode of number from all others, even thofe that approach nearest, makes me apt to think that demonftrations in numbers, if they are not more evident and exact than in extenfion, yet they are more general in their ufe, and more determinate in their application, because the ideas of numbers are more precife and diftinguifhable than in extenfion, where every equality and excefs are not fo eafy to be obferved or meafured, becaufe our thoughts cannot in fpace arrive at any determined fmallness, beyond which it cannot go, as an unit, and therefore the quantity or proportion, of any the leaft excefs cannot be discovered; which is clear otherwife in number, where, as has been faid, or is as diftinguifhable from go as from, goco, though 91 be the next immediate excefs to go. But it is not fo in extenfion, where whatfoever is more than just a foot or an inch, is not diftinguishable 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 affign an angle which fhall be the next biggest to a right one.

$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 last collective idea which he had of any number, and give a name to it, may count or have ideas for feveral 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 in one idea, a new or diftinct name or fign whereby to know it from those before and after, and diftinguish it from every fmaller 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, especially where the combination is made up of any great multitude of units, which put together without a name or mark to distinguish that precife collection, will hardly be kept from being a heap in confufion.

§6. Names neceffary to Numbers.

THIS I think to be the reafon why fome Americans I have spoken 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 scanty, 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; so 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 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.

Nonilions. Octilions. 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 diftinguishing 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 Lery, c. 20. 207-282.

felves, and more plainly fignified to others, I leave it to be confidered. This I mention only to show 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 ftore 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 feveral 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 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 stand 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 distinguish 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 neceflary to diftinct numeration will not be attained to.