Names neceffa ry to Numbers. §. 6. This I think to be the Reason why fome Americans, I have fpoken with, (who were otherwife of quick and rational Parts enough) could not, as we do, by any means, count to 1000; nor had any distinct 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 Mathematicks, had no Words in it to ftand for 1000; fo that when they were dif courfed with of thofe greater Numbers, they would fhew 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 fhewing 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 ufually 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 moft four and twenty decimal Progreffions, without Confufion. But to fhew how much diftinct Names conduce to our well reckoning, or having useful Ideas of Numbers, let us fet all these following Figures, as the Marks of one Number: v. g. Nonilions. Octilions. Septilions. Sextilions. 857324. 162486. 345896. Hiftoire d'un Voyage fait en la Terre du Brafil, par Jean de Lery, C. 20. 307 382 Quintilions. 437916. 423147. Millions. Units. Quartilions. Trilions. Bilions. 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 diftinguifhing Notions of this Number: But whether, by giving every fix Figures a new and orderly Denomination, these, and perhaps a great many more Figures, in Progreffion, might not eafily be counted diftinctly, and Ideas of them both got more easily to ourfelves, and more plainly fignified to others, I leave it to be confidered. This I mention only, to fhew how neceffary L 3 diftin&t number not earlier. distinct Names are to Numbering, without pretending to introduce new ones of my Invention. Why Children §. 7. 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 steadily, '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 Dependance 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 Series of Numbers. For he that will count Twenty, or have any Idea of that Number, must know that Nineteen went before, with the diftinct Name or Sign 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 diftinguish carefully two Ideas, which are different one from another only by the Addition or Substraction 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 Business of Numbering will be disturbed, and there will remain only the confufed Idea of Multitude, but the Ideas necessary to distinct Numeration will not be attained to. Number meafures all Meafurables. 6.8. This farther is obfervable in Number, That it is that which the Mind makes ufe of in meafuring all Things, that by us are measurable, which principally are Expansion and Duration; and our Idea of Infinity, even when applied to those, seems to be nothing but the Infinity of Number. For what else are our Ideas of Eternity and Immenfity, but the repeated Additions of certain Ideas of imagined Parts of Duration and Expanfion, with the Infinity of Number, in which we can come to no End of Addition? For fuch an inexhauftible Stock, Number (of all other other our Ideas) moft clearly furnishes us with, as is obvious to every one. For let a Man collect into one Sum as great a Number as he pleafes, this Multitude, how great foever, leffens not one Jot the Power of adding to it, or brings him any nearer the End of the inexhauftible Stock of Number, where ftill there remains as much to be added, as if none were taken out. And this endlefs Addition, or Addibility (if any one like the Word better) of Numbers, fo apparent to the Mind, is that, I think, which gives us the clearest and moft diftinct Idea of Infinity: Of which more in the following Chapter. §. I. "H Η Of INFINITY. E that would know what kind of Idea it is, to which we give the Name of Infinity, cannot do it better, than by confidering to what Infinity is by the Mind more immediately attributed, and then how the Mind comes to frame it. Infinity, in its original Intention attributed to Space, Duration, and Number. Finite and Infinite feem to me to be looked upon by the Mind as the Modes of Quantity, and to be attributed primarily in their first Designation only to thofe Things which have Parts, and are capable of Increase or Diminution, by the Addition or Subtraction of any the least Part: And fuch are the Ideas of Space, Duration, and Number, which we have confidered in the foregoing Chapters. 'Tis true that we cannot but be affured, that the great GOD, of whom, and from whom are all Things, is incomprehenfibly infinite. But yet, when we apply to that firft and fupreme Being our Idea of Infinite, in our weak and narrow Thoughts, we do it primarily in respect of his Duration and Ubiquity; and, I think, more figuratively to his Power, Wisdom, and Goodress, and other Attributes, which are properly inexhaustible and incomprehenfible, &c. For when we call them infinite, we have no other Idea of this Infinity, but what carries with it fome Reflection on, and Intimation of that Number or Extent of the Acts or Objects of God's Power, Wifdom and Goodnefs, which can never be fuppofed fo great, or fo many, L 4 which which these Attributes will not always furmount and exceed, let us multiply them in our Thoughts as far as we can, with all the Infinity of endlefs Number. I do not pretend to fay how these Attributes are in GOD, who is infinitely beyond the Reach of our narrow Capacities: They do, without doubt, contain in them all poffible Perfection: But this, I say, is our way of conceiving them, and these our Ideas of their Infinity. The Idea of Finite eafily got. §. 2. Finite then, and Infinite, being by the Mind looked on as Modifications of Expansion and Duration, the next Thing to be confidered is, How the Mind comes by them. As for the Idea of Finite, there is no great Difficulty. The obvious Portions of Extenfion, that affect our Senfes, carry with them into the Mind the Idea of Finite: And the ordinary Periods of Succeffion, whereby we measure Time and Duration, as Hours, Days, and Years, are bounded Lengths. The Difficulty is, how we come by thofe boundless Ideas of Eternity and Immenfity, fince the Objects, which we converfe with, come fo much fhort of any Approach or Proportion to that Largeness. How we come by the Idea of Infinity. §. 3. Every one, that has any Idea of any stated Lengths of Space, as a Foot, finds that he can repeat that Idea; and joining it to the former, make the Idea of two Foot; and by the Addition of a third, three Foot; and fo on, without ever coming to an End of his Addition, whether of the fame Idea of a Foot, or if he pleases of doubling it, or any other Idea he has of any Length, as a Mile, or Diameter of the Earth, or of the Orbis Magnus: For which foever of these he takes, and how often foever he doubles, or any otherwise multiplies it, he finds, that after he has continued his doubling in his Thoughts, and enlarged his Idea as much as he pleafes, he has no more Reason to ftop, nor is one Jot nearer the End of such Addition, than he was at firft fetting out. The Power of enlarging his Idea of Space by farther Additions, remaining still the fame, he hence takes the Idea of infinite Space. Our Idea of | Space boundless. §. 4. This, I think, is the Way whereby the Mind gets the Idea of infinite Space. 'Tis a quite different Confideration to examine, whether the Mind has the Idea of fuch a boundless Space actually exifting, fince our Ideas are not always Proofs of the Exiftence of Things; but yet, fince this comes here in our way, I fuppofe I may fay, that we are apt to think, that that Space in itself is actually boundlefs; to which Imagination the Idea of Space or Expansion of itself naturally leads us. For it being confidered by us, either as the Extenfion of Body, or as exifting by itself, without any folid Matter taking it up, (for of fuch a void Space we have not only the Idea, but I have proved, as I think, from the Motion of Body, its neceffary Exiftence) it is impoffible the Mind should be ever able to find or fuppofe any End of it, or be stopped any where in its Progress in this Space, how far foever it extends its Thoughts. Any Bounds made with Body, even Adamantine Walls, are fo far from putting a Stop to the Mind in its farther Progrefs in Space and Extenfion, that it rather facilitates and enlarges it: For fo far as that Body reaches, fo far no one can doubt of Extenfion; and when we are come to the utmost Extremity of Body, what is there, that can there put a Stop, and fatisfy the Mind that it is at the End of Space, when it perceives it is not; nay, when it is fatisfied that Body itself can move into it? For if it be neceffary for the Motion of Body, that there fhould be an empty Space, though ever fo little, here amongst Bodies; and if it be poffible for Body to move in or through that empty Space; nay, it is impoffible for any Particle of Matter to move but into an empty Space; the fame Poffibility of a Body's moving into a void Space, beyond the utmoft Bounds of Body, as well as into a void Space interfperfed amongst Bodies, will always remain clear and evident: The Idea of empty pure Space, whether within or beyond the Confines of all Bodies, being exactly the fame, differing not in Nature, though in Bulk; and there being nothing to hinder Body from moving into it: So that wherever the Mind places itself by any Thought, either amongst or remote from all Bodies, it can in this uniform Idea of Space no where find any Bounds, any End; and fo muft neceffarily conclude it, by the very Nature and Idea of each Part of it, to be actually infinite. S. 5. As by the Power we find in ourselves of And fo of Durepeating, as often as we will, any Idea of Space, ration. we get the Idea of Immenfity; fo by being able to repeat the Idea of any Length of Duration we have in our Minds, with all the endless Addition of Number, we come by the Idea of Eternity. For we find in ourselves, we can no more come to an End of such repeated Ideas, than we can come to the End of Number, which every one perceives he cannot. But here again, 'tis another Question quite different from our having an Idea of Eternity, to know whether there were any |