duration as we suppose equal to, and would have admitted, 7640 annual revolutions of the sun, moving at the rate it now does. And thus likewise we sometimes speak of place, distance, or bulk, in the great INANE, beyond the confines of the world, when we consider so much of that space as is equal to, or capable to receive, a body of any assigned dimensions, as a cubic foot; or do suppose a point in it, at such a certain distance from any part of the universe.
8. They belong to all finite beings.
WHERE and WHEN are questions belonging to all finite existences, and are by us always reckoned from some known parts of this sensible world, and from some certain epochs marked out to us by the motions observable in it. Without some such fixed parts or periods, the order of things would be lost, to our finite understandings, in the boundless invariable oceans of duration and expansion, which comprehend in them all finite beings, and in their full extent belong only to the Deity.
And therefore we are not to wonder that we comprehend them not, and do so often find our thoughts at a loss, when we would consider them, either abstractly in themselves, or as any way attributed to the first incomprehensible Being. But when applied to any particular finite beings, the extension of any body is so much of that infinite space as the bulk of the body takes up. And place is the position of any body, when considered at a certain distance from some other. As the idea of the particular duration of anything is, an idea of that portion of infinite duration which passes during the existence of that thing; so the time when the thing existed is, the idea of that space of duration which passed between some known and fixed period of duration, and the being of that thing. One shows the distance of the extremities of the bulk or existence of the same thing, as that it is a foot square, or lasted two years; the other shows the distance of it in place, or existence from other fixed points of space or duration, as that it was in the middle of Lincoln’s Inn Fields, or the first degree of Taurus, and in the year of our Lord 1671, or the 1000th year of the Julian period. All which distances we measure by preconceived ideas of certain lengths of space and duration,—as inches, feet, miles, and degrees, and in the other, minutes, days, and years, &c.
9. All the Parts of Extension are Extension, and all the Parts of Duration are Duration.
There is one thing more wherein space and duration have a great conformity, and that is, though they are justly reckoned amongst our SIMPLE IDEAS, yet none of the distinct ideas we have of either is without all manner of composition: it is the very nature of both of them to consist 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 lengths which it is acquainted with. On the other side, the ordinary smallest measure we have of either is looked on as an unit in number, when the mind by division would reduce them into less fractions. Though on both sides, both in addition and division, either of space or duration, when the idea under consideration becomes very big or very small, its precise bulk becomes very obscure and confused; 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 resolved. Such a small part in 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 further 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 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.
11. Duration is as a Line, Expansion as a Solid.
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 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; an 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 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 or 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 further 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 further speculation.
CHAPTER XVI.
IDEA 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, 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; everything 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; 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. Each Mode distinct.
The SIMPLE MODES of NUMBER 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, 91 is as distinguishable from 90 as from 9000, though 91 be the next immediate excess to 90. 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, 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 gave 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. Another reason for the necessity of names to numbers.
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 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 necessaries of a needy, simple life, unacquainted either with trade or mathematics, had no words in it to stand for 1000; 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 5; 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 further 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 see all these following figures in one continued line, as the marks of one number: v. g.
Nonillions. 857324 Octillions. 162486 Septillions. 345896 Sextillions. 437918 Quintrillions. 423147 Quartrillions. 248106 Trillions. 235421 Billions. 261734 Millions. 368149 Units. 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.
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, 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, annexed in their distinct orders, and the dependence of so long a train of numeral progressions, 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 distinct 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 progress in numbering can go no further. So that to reckon right, it is required, (1) That the mind distinguish carefully two ideas, which are different one from another only by the addition or subtraction of ONE unit: (2) That it retain in memory the names or marks of the several combinations, from an unit to that number; and that not confusedly, 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 confused idea of multitude, but the ideas necessary to distinct numeration will not be attained to.
8. Number measures all Measurables.
This further is observable in number, that it is that which the mind makes use of in measuring 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 Immensity, but the repeated additions of certain ideas of imagined parts of duration and expansion, with the infinity of number; in which we can come to no end of addition? For such an inexhaustible stock, number (of all other our ideas) most clearly furnishes us with, as is obvious to every one. For let a man collect into one sum as great a number as he pleases, this multitude how great soever, lessens not one jot the power of adding to it, or brings him any nearer the end of the inexhaustible stock of number; where still there remains as much to be added, as if none were taken out. And this ENDLESS ADDITION or ADDIBILITY (if any one like the word better) of numbers, so apparent to the mind, is that, I think, which gives us the clearest and most distinct idea of infinity: of which more in the following chapter.
CHAPTER XVII.
OF INFINITY.
1. Infinity, in its original Intention, attributed to Space, Duration, and Number.
He that would know what kind of idea it is to which we give the name of INFINITY, cannot do it better than by considering to what infinity is by the mind more immediately attributed; and then how the mind comes to frame it.
FINITE and INFINITE seem 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 those things which have parts, and are capable of increase or diminution by the addition or subtraction of any the least part: and such are the ideas of space, duration, and number, which we have considered in the foregoing chapters. It is true, that we cannot but be assured, that the great God, of whom and from whom are all things, is incomprehensibly infinite: but yet, when we apply to that first and supreme Being our idea of infinite, in our weak and narrow thoughts, we do it primarily in respect to his duration and ubiquity; and, I think, more figuratively to his power, wisdom, and goodness, and other attributes which are properly inexhaustible and incomprehensible, &c. For, when we call THEM infinite, we have no other idea of this infinity but what carries with it some reflection on, and imitation of, that number or extent of the acts or objects of God’s power, wisdom, and goodness, which can never be supposed so great, or so many, which these attributes will not always surmount and exceed, let us multiply them in our thoughts as far as we can, with all the infinity of endless number. I do not pretend to say how these attributes are in God, who is infinitely beyond the reach of our narrow capacities: they do, without doubt, contain in them all possible perfection: but this, I say, is our way of conceiving them, and these our ideas of their infinity.
2. The Idea of Finite easily got.
Finite then, and infinite, being by the mind looked on as MODIFICATIONS of expansion and duration, the next thing to be considered, is,—HOW THE MIND COMES BY THEM. As for the idea of finite, there is no great difficulty. The obvious portions of extension that affect our senses, carry with them into the mind the idea of finite: and the ordinary periods of succession, whereby we measure time and duration, as hours, days, and years, are bounded lengths. The difficulty is, how we come by those BOUNDLESS IDEAS of eternity and immensity; since the objects we converse with come so much short of any approach or proportion to that largeness.
3. How we come by the Idea of Infinity.
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 feet; and by the addition of a third, three feet; and so on, without ever coming to an end of his additions, whether of the same 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 whichever of these he takes, and how often soever 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 pleases, he has no more reason to stop, nor is one jot nearer the end of such addition, than he was at first setting out: the power of enlarging his idea of space by further additions remaining still the same, he hence takes the idea of infinite space.
4. Our Idea of Space boundless.
This, I think, is the way whereby the mind gets the IDEA of infinite space. It is a quite different consideration, to examine whether the mind has the idea of such a boundless space ACTUALLY EXISTING; since our ideas are not always proofs of the existence of things: but yet, since this comes here in our way, I suppose I may say, that we are APT TO THINK that space in itself is actually boundless, to which imagination the idea of space or expansion of itself naturally leads us. For, it being considered by us, either as the extension of body, or as existing by itself, without any solid matter taking it up, (for of such a void space we have not only the idea, but I have proved, as I think, from the motion of body, its necessary existence,) it is impossible the mind should be ever able to find or suppose any end of it, or be stopped anywhere in its progress in this space, how far soever it extends its thoughts. Any bounds made with body, even adamantine walls, are so far from putting a stop to the mind in its further progress in space and extension that it rather facilitates and enlarges it. For so far as that body reaches, so far no one can doubt of extension; and when we are come to the utmost extremity of body, what is there that can there put a stop, and satisfy the mind that it is at the end of space, when it perceives that it is not; nay, when it is satisfied that body itself can move into it? For, if it be necessary for the motion of body, that there should be an empty space, though ever so little, here amongst bodies; and if it be possible for body to move in or through that empty space;—nay, it is impossible for any particle of matter to move but into an empty space; the same possibility of a body’s moving into a void space, beyond the utmost bounds of body, as well as into a void space interspersed 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 same, 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, nowhere find any bounds, any end; and so must necessarily conclude it, by the very nature and idea of each part of it, to be actually infinite.
5. And so of Duration.
As, by the power we find in ourselves of repeating, as often as we will, any idea of space, we get the idea of IMMENSITY; so, 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 it is another question, quite different from our having an IDEA of eternity, to know whether there were ANY REAL BEING, whose duration has been eternal. And as to this, I say, he that considers something now existing, must necessarily come to Something eternal. But having spoke of this in another place, I shall say here no more of it, but proceed on to some other considerations of our idea of infinity.
6. Why other Ideas are not capable of Infinity.
If it be so, that our idea of infinity be got from the power we observe in ourselves of repeating, without end, our own ideas, it may be demanded,—Why we do not attribute infinity to other ideas, as well as those of space and duration; since they may be as easily, and as often, repeated in our minds as the other: and yet nobody ever thinks of infinite sweetness or infinite whiteness, though he can repeat the idea of sweet or white, as frequently as those of a yard or a day? To which I answer,—All the ideas that are considered as having parts, and are capable of increase by the addition of an equal or less parts, afford us, by their repetition, the idea of infinity; because, with this endless repetition, there is continued an enlargement of which there CAN be no end. But for other ideas it is not so. For to the largest idea of extension or duration that I at present have, the addition of any the least part makes an increase; but to the perfectest idea I have of the whitest whiteness, if I add another of a less equal whiteness, (and of a whiter than I have, I cannot add the idea,) it makes no increase, and enlarges not my idea at all; and therefore the different ideas of whiteness, &c. are called degrees. For those ideas that consist of part are capable of being augmented by every addition of the least part; but if you take the idea of white, which one parcel of snow yielded yesterday to our sight, and another idea of white from another parcel of snow you see to-day, and put them together in your mind, they embody, as it were, all run into one, and the idea of whiteness is not at all increased and if we add a less degree of whiteness to a greater, we are so far from increasing, that we diminish it. Those ideas that consist not of parts cannot be augmented to what proportion men please, or be stretched beyond what they have received by their senses; but space, duration, and number, being capable of increase by repetition, leave in the mind an idea of endless room for more; nor can we conceive anywhere a stop to a further addition or progression: and so those ideas ALONE lead our minds towards the thought of infinity.
7. Difference between infinity of Space, and Space infinite.
Though our idea of infinity arise from the contemplation of quantity, and the endless increase the mind is able to make in quantity, by the repeated additions of what portions thereof it pleases; yet I guess we cause great confusion in our thoughts, when we join infinity to any supposed idea of quantity the mind can be thought to have, and so discourse or reason about an infinite quantity, as an infinite space, or an infinite duration. For, as our idea of infinity being, as I think, AN ENDLESS GROWING IDEA, but the idea of any quantity the mind has, being at that time TERMINATED in that idea, (for be it as great as it will, it can be no greater than it is,)—to join infinity to it, is to adjust a standing measure to a growing bulk; and therefore I think it is not an insignificant subtilty, if I say, that we are carefully to distinguish between the idea of the infinity of space, and the idea of a space infinite. The first is nothing but a supposed endless progression of the mind, over what repeated ideas of space it pleases; but to have actually in the mind the idea of a space infinite, is to suppose the mind already passed over, and actually to have a view of ALL those repeated ideas of space which an ENDLESS repetition can never totally represent to it; which carries in it a plain contradiction.
8. We have no Idea of infinite Space.
This, perhaps, will be a little plainer, if we consider it in numbers.
The infinity of numbers, to the end of whose addition every one perceives there is no approach, easily appears to any one that reflects on it. But, how clear soever this idea of the infinity of number be, there is nothing yet more evident than the absurdity of the actual idea of an infinite number. Whatsoever POSITIVE ideas we have in our minds of any space, duration, or number, let them be ever so great, they are still finite; but when we suppose an inexhaustible remainder, from which we remove all bounds, and wherein we allow the mind an endless progression of thought, without ever completing the idea, there we have our idea of infinity: which, though it seems to be pretty clear when we consider nothing else in it but the negation of an end, yet, when we would frame in our minds the idea of an infinite space or duration, that idea is very obscure and confused, because it is made up of two parts, very different, if not inconsistent. For, let a man frame in his mind an idea of any space or number, as great as he will; it is plain the mind RESTS AND TERMINATES in that idea, which is contrary to the idea of infinity, which CONSISTS IN A SUPPOSED ENDLESS PROGRESSION. And therefore I think it is that we are so easily confounded, when we come to argue and reason about infinite space or duration, &c. Because the parts of such an idea not being perceived to be, as they are, inconsistent, the one side or other always perplexes, whatever consequences we draw from the other; as an idea of motion not passing on would perplex any one who should argue from such an idea, which is not better than an idea of motion at rest. And such another seems to me to be the idea of a space, or (which is the same thing) a number infinite, i. e. of a space or number which the mind actually has, and so views and terminates in; and of a space or number, which, in a constant and endless enlarging and progression, it can in thought never attain to. For, how large soever an idea of space I have in my mind, it is no larger than it is that instant that I have it, though I be capable the next instant to double it, and so on in infinitum; for that alone is infinite which has no bounds; and that the idea of infinity, in which our thoughts can find none.
9. Number affords us the clearest Idea of Infinity.
But of all other ideas, it is number, as I have said, which I think furnishes us with the clearest and most distinct idea of infinity we are capable of. For, even in space and duration, when the mind pursues the idea of infinity, it there makes use of the ideas and repetitions of numbers, as of millions and millions of miles, or years, which are so many distinct ideas,—kept best by number from running into a confused heap, wherein the mind loses itself; and when it has added together as many millions, &c., as it pleases, of known lengths of space or duration, the clearest idea it can get of infinity, is the confused incomprehensible remainder of endless addible numbers, which affords no prospect of stop or boundary.
10. Our different Conceptions of the Infinity of Number contrasted with those of Duration and Expansion.
It will, perhaps, give us a little further light into the idea we have of infinity, and discover to us, that it is NOTHING BUT THE INFINITY OF NUMBER APPLIED TO DETERMINATE PARTS, OF WHICH WE HAVE IN OUR MINDS THE DISTINCT IDEAS, if we consider that number is not generally thought by us infinite, whereas duration and extension are apt to be so; which arises from hence,—that in number we are at one end, as it were: for there being in number nothing LESS than an unit, we there stop, and are at an end; but in addition, or increase of number, we can set no bounds: and so it is like a line, whereof one end terminating with us, the other is extended still forwards, beyond all that we can conceive.
But in space and duration it is otherwise. For in duration we consider it as if this line of number were extended BOTH ways—to an unconceivable, undeterminate, and infinite length; which is evident to anyone that will but reflect on what consideration he hath of Eternity; which, I suppose, will find to be nothing else but the turning this infinity of number both ways, a parte ante and a parte post, as they speak. For, when we would consider eternity, a parte ante, what do we but, beginning from ourselves and the present time we are in, repeat in our minds ideas of years, or ages, or any other assignable portion of duration past, with a prospect of proceeding in such addition with all the infinity of number: and when we would consider eternity, a parte post, we just after the same rate begin from ourselves, and reckon by multiplied periods yet to come, still extending that line of number as before. And these two being put together, are that infinite duration we call ETERNITY which, as we turn our view either way, forwards or backward appears infinite, because we still turn that way the infinite end of number, i.e. the power still of adding more.
11. How we conceive the Infinity of Space.
The same happens also in space, wherein, conceiving ourselves to be, as it were, in the centre, we do on all sides pursue those indeterminable lines of number; and reckoning any way from ourselves, a yard, mile, diameter of the earth or orbis magnus,—by the infinity of number, we add others to them, as often as we will. And having no more reason to set bounds to those repeated ideas than we have to set bounds to number, we have that indeterminable idea of immensity.