fused; and it is the number of its repeated additions, or divisions, that alone remains clear and distinct, as will easily appear to any one, who will let his thoughts loose in the vast expansion of space, or divisibility of matter. Every part of duration, is duration too; and every part of extension, is extension, both of them capable of addition or division in infinitum. But the least portions of either of them, whereof we have clear and distinct ideas, may perhaps be fittest to be considered by us, as the simple ideas of that kind, out of which our complex modes of space, extension, and duration, are made up, and into which they can again be distinctly revolved. 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. Their parts inseparable.—Expansion and duration have this farther agreement, that though they are both 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 a 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. §. 11. Duration is as a line, expansion as a solid.-But yet 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 expansion, is beyond my comprehension; and, perhaps, for us, who have understandings and comprehensions suited to our own preservation, 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 expansion; 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. §. 12. Duration has never two parts together, expansion altogether.-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 passed, he can never recal; and what is yet to come, he cannot make present. 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; 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. CHAPTER XVI.. OF NUMBER. §. 1. Number, the simplest and most universal idea.-Amongst all the ideas we have, 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 object our senses are employed about; every idea in our understandings; every thought of our minds, bring 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, angels, actions, thoughts, every thing that either doth exist, 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; but 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. Each mode distinct.—The simple modes of numbers are of all other the most distinct; every the 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 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? §. 4. Therefore demonstrations in numbers the most precise.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 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, of 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 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 numbers; 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 combinations of units, which have no variety, nor are many 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 denominations 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 set all these following figures in one continued line, as the marks of one number: v. g. Nonillions. Octillions. Septillions. Sextillions. Quintrillions. 857324. 162486. 345896. 437918. 423147. 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. §. 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, until a * Histoire d'un Voyage, fait en la terre du Brasil, par Jean de Lery, c. 20, 307 |