« ZurückWeiter »
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 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.
Number the simplest and
§ 1. AMONGST all the ideas we have, as there is none suggested to the mind by more ways, so there is none more simple most unithan that of unity, or one. It has no versal idea. 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, angels, 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 made by adcome by the complex ideas of the modes dition... 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
li 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. The clearness and distinctness of Therefore each mode of number from all others, even
tions in those that approach nearest, makes me apt
numbers the to think that demonstrations in numbers, most precise. 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. Names ne § 5. By the repeating, as has been said, cessary to the idea of an unit, and joining it to annumbers. other 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 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 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 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
. * Histoire d'un voyage, fait en la terre du Brasil, par Jean de Lery, c. 20. 181.
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. Quintillions.
857324 162486 345896 437918 423147 Quatrillions. Trillions. Billions. 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 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 num ber 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 an