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position*: it is the very nature of both of them toconsist of parts: but their parts being all of the same kind, and without the mixture of any other idea, hinder them not from having a place amongst simple ideas. Could the mind, as in number, come to so small a part of extension or duration, as excluded divisibility, that would be, as it were, the indivisible unit, or idea; by repetition of which it would make its more enlarged ideas of extension and duration. But since the mind is not able to frame an idea of any space without parts; instead thereof it makes use of the common measures, which by familiar use, in each country, have imprinted themselves on the memory (as inches and feet; or cubits and parasangs; and so seconds, minutes, hours, days, and years in duration:) the mind makes use, I say, of such ideas as these, as simple ones; and these are the component parts of larger ideas, which the mind, upon occasion, makes by the addition of such known
* It has been objected to Mr. Locke, that if space consists of parts, as it is confessed in this place, he should not have reckoned it in the number of simple ideas: because it seems to be inconsistent with what he says elsewhere, that a simple idea is uncompounded, and contains in it nothing but one uniform appearance or conception of the mind, and is not distinguishablft*nto different ideas. It is farther objected, that Mr. Locke has not given in the eleventh chapter of the second book, wbeN> he begins to speak of simple ideas, an exact definition of what he understands by the word simple ideas. To these difficulties Mr. Lacks answers thus: To begin with the last, he declares, that he has not treated his subject in an order perfectly scholastic, having not had much familiarity with those sort of books during the writing of his, and not remembering at all the method in which they are written; and therefore his readers ought not to expect definitions regularly placed at the beginning of each new subject. Mr. Locke contents himself to employ the principal terms that he uses, so that from his use of them the reader may easily comprehend what he means by them. But with respect to the term simple idea, behasbad the good luck to define that in the place cited in the objection; and therefore there is no reason to supply that defect. The question then is to know, whether the idea of extension agrees with this definition? which will effectually agree to it, if it be understood in the sense which Mr.Locke had principally in his view: for that composition which he designed to exclude in that definition, was a composition of different ideas in the mind,and not a composition of the same kind in a thing whoseessence consists in having parts of the same kind, where you can never come to a part entirely exempted from this composition. So 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. AH 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 §' Duration, and time which is a part has never of it, is the idea we have of perishing distwopartsto- tance, of which no two parts exist toge8ansion all Sether» DU' foMow eacn other in succession; together? as expansion is the idea of lasting distance, all whose parts exist together, and are not capable of succession. And therefore though wecannot 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 i 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 say of man I say of all finite beings j 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 du
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. AMONGST all the ideas we have, Numberthe as there is none suggested to the mind simplest and by more ways, so there is none more most universimple, than that of unity, or one. It has sal ldeano 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 come made by by the complex ideas of the modes of it. add,tIonThus 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 ^ 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 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 §. ^ne c^earness and distinctness of demonstra- each mode of number from all others, even tions in those that approach nearest, makes me apt numbersthe to think that demonstrations in numbers, if most precise. afe no^. 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, cessaryto 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 j any number beyond that they made out by showing their fingers, and the fingers of others who were present*. And I doubt