extension, and duration, are made up, and into which they can again be distinctly resolved. Such a small part of duration may be called a moment, and is the time of one idea in our minds in the train of their ordinary succession there. The other, wanting a proper name, I know not whether I may be allowed to call a sensible point, meaning thereby the least particle of matter or space we can discern, which is ordinarily about a minute, and to the sharpest eyes seldom less than thirty seconds of a circle, whereof the eye is the centre. §. 10. Expansion and duration have this Their parts farther agreement, that though they are both infeparable, considered by us as having parts, yet their parts are not separable one from another, no not even in thought: though the parts of bodies from whence we take our measure of the one, and the parts of motion, or rather the succession of ideas in our minds, from whence we take the measure of the other, may be interrupted and separated; as the one is often by rest, and the other is by sleep, which we call rest too.

Duration is as a line, expansion as a

solid.

§. 11. But there is this manifest difference between them, that the ideas of length, which we have of expansion, are turned every way, and so make figure, and breadth, and thickness; but duration is but as it were the length of one straight line, extended in infinitum, not capable of multiplicity, variation, or figure; but is one common measure of all existence whatsoever, wherein all things, whilst they exist, equally partake. For this present moment is common to all things that are now in being, and equally comprehends that part of their existence, as much as if they were all but one single being; and we may truly say, they all exist in the same moment of time. Whether angels and spirits have any analogy to this, in respect to expanfion, is beyond my comprehension: and perhaps for us, who have understandings and comprehensions suited to our own pres servation, and the ends of our own being, but not to the reality and extent of all other beings; it is near as hard to conceive any existence, or to have an idea of any real being, with a perfect negation of all manner of expan

sion; as it is to have the idea of any real existence, with a perfect negation of all manner of duration; and therefore what spirits have to do with space, or how they communicate in it, we know not. All that we know is, that bodies do each singly possess its proper portion of it, according to the extent of solid parts; and thereby exclude all other bodies from having any share in that particular portion of space, whilst it remains there.

Duration has

never two parts together, expan sion all to

gether.

§. 12. Duration, and time which is a part of it, is the idea we have of perishing distance, of which no two parts exist together, but follow each other in succession; as expansion is the idea of lasting distance, all whose parts exist together, and are not capable of succession. And therefore though we cannot conceive any duration without succession, nor can put it together in our thoughts, that any being does now exist to-morrow, or possess at once more than the present moment of duration; yet we can conceive the eternal duration of the Almighty far different from that of man, or any other finite being. Because man comprehends not in his knowledge, or power, all past and future things; his thoughts are but of yesterday, and he knows not what to-morrow will bring forth. What is once past he can never recall; and what is yet to come he cannot make present. What I What I say of man I say of all finite beings; who, though they may far exceed man in knowledge and power, yet are no more than the meanest creature, in comparison with God himself. Finite of any magnitude holds not any proportion to infinite. God's infinite duration being accompanied with infinite knowledge, and infinite power, he sees all things past and to come; and they are no more distant from his knowledge, no farther removed from his sight, than the present: they all lie under the same view; and there is nothing which he cannot make exist each moment he pleases. For the existence of all things depending upon his good-pleasure, all things exist every moment that he thinks fit to have them exist. To conclude, expansion and duration do mutually embrace and comprehend each other; every part of space being in every part of duration,

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.

§. 1.

CHA P. XVI.

Of Number.

MONGST all the ideas we have,

Number the simplest and.

most univer. sal idea.

as there is none suggested to the mind by more ways, so there is none more simple, than that of unity, or one. It has no shadow of variety or composition in it: every ob ject 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.

made by ad

dition.

§. 2. By repeating this idea in our minds, Its modes 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 so of a score, or a million, or any other number.

Each mode

§. 3. The simple modes of numbers are 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 two 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

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

demonstra

tions in numbers the most precise.

4. The clearness and distinctness of each mode of number from all others, even those that approach nearest, makes me apt to think that demonstrations in numbers, if 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 clear otherwise in num→ ber, 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.

Names ne

§. 5. By the repeating, as has been said, cessary to the idea of an unit, and joining it to ano, numbers. 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

of units.

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 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

* Histoire d'un voyage, fait en la tèrre du Brasil, par Jean de Lery,

not