Of Space and Time

by Ruđer Bošković

SUPPLEMENTS

§ I.

Of Space and Time^[1]

	The theme; what are the attributes of space?
1. I do not admit perfectly continuous extension of matter; I consider it to be made up of perfectly indivisible points, which are non-extended, set apart from one another by a certain interval, & connected together by certain forces that are at one time attractive & at another time repulsive, depending on their mutual distances. Here it is to be seen, with this theory, what is my idea of space, & of time, how each of them may be said to be continuous, infinitely divisible, eternal, immense, immovable, necessary, although neither of them, as I have shown in a note, have a real nature of their own that is possessed of these properties.
	Real local and temporal modes of existence must of necessity be admitted by every one.
2. First of all it seems clear to me that not only those who admit absolute space, which is of its own real nature continuous, eternal & immense, but also those who, following Leibniz & Descartes, consider space itself to be the relative arrangement which exists amongst necessity things that exist, over and above these existent things ; it seems to me, I say, that all must admit some mode of existence that is real & not purely imaginary ; through which they are where they are, & this mode exists when they are there, & perishes when they cease to be where they were. For, such a space being admitted in the first theory, if the fact that there is some thing in that part of space depends on the thing & space alone ; then, as often as the thing existed, & space, we should have the fact that that thing was situated in that part of space. Again, if, in the second theory, the arrangement, which constitutes position, depended only on the things themselves that have that arrangement ; then, as often as these things should exist, they would exist in the same arrangement, & could never change their position. What I have said with regard to space applies equally to time.
	The name by which this modes is known is immaterial.
3. Therefore it needs must be admitted that there is some real mode of existence, due to which a thing is where it is, & exists then, when it does exist. Whether this mode is called the thing, or the mode of the thing, or something or nothing, it is bound to be beyond our imagination; & the thing may change this kind of mode, having one mode at one time & another at another time.
	Real modes; what real space & real time may be.
4. Hence, for each of the points of matter (to consider these, from which all I say what can be easily transferred to immaterial things), I admit two real kinds of modes of existence, of which some pertain to space & others to time; & these will be called local & temporal modes respectively. Any point has a real mode of existence, through which it is where it is; & another, due to which it exists at the time when it does exist. These real modes of existence are to me real time & space; the possibility of these modes, hazily apprehended by us, is, to my mind, empty space & again empty time, so to speak; in other words, imaginary space & imaginary time.
	Their nature & relations.
5. These several real modes are produced & perish, and are in my opinion quite indivisible, non-extended, immovable & unvarying in their order. They, as well as the positions & times of them, & of the points to which they belong, are real. They afford the foundation of a real relation of distance, which is either a local relation between two points, or a temporal relation between two events. Nor is the fact that those two points of matter have that determinated distance anything essentially different from the fact that they have those determinated modes of existence, which necessarily alter when they change the distance. Those modes which are descriptive of position I call real points of position; & those that are descriptive of time I call instants; & they are without parts, & the former lack any kind of extension, while the latter lack duration; both are indivisible.
	Contiguity of points of space is impossible.
6. Further, a point of matter that is perfectly indivisible & non-extended cannot be contiguous to any other point of matter; if they have no distance from one another, they coincide completely; if they do not coincide completely, they have some distance between them. For, since they have no kind of parts, they cannot coincide partly only; that is, they cannot touch one another on one side, & on the other side be separated. It is but a prejudice acquired from infancy, & born of ideas obtained through the senses, which have not been considered with proper care; these ideas picture masses to us as always being composed of parts at a distance from one another. It is owing to this prejudice that we seem to ourselves to be able to bring even indivisible and non-extended points so close to other points that they touch them & constitute a sort of lengthy series. We imagine a series of little spheres, in fact; & we do not put out of mind that extension, & the parts, which we verbally exclude.
	Given two points, it is possible to add others in the same straight line at equal distances apart; & it is possible to insert others between them; to any extent in either case.
7. Again, where two points of matter are at a distance from one another, another point of matter can always be placed in the same straight line with them, on the far side of either, at an equal distance; & another beyond that, & so on without end, as is evident. Also another point can be placed halfway between the two points, so as to touch neither of them; for, if it touched either of them it would touch them both, & thus would coincide possible to insert with both; hence the two points would coincide with one another & could not be separate points, which is contrary to the hypothesis. Therefore that interval can be divided into two parts; & therefore, by the same argument, those two can be divided into four others, & so on without any end. Hence it follows that, however great the interval between two points, we could always obtain another that is greater; &, however small the interval might be, we could always obtain another that is smaller; &, in either case, without any limit or end.
	The number of points existing in space will always be finite, & the distances between them finite; there is no end to the possible cases.
8. Hence beyond & between two real points of position of any sort there are other real points of position possible; & these recede from them & approach them respectively, without any determinate limit. There will be a real divisibility to an infinite extent of the interval between two points, or, if I may call it so, an endless 'insertibility' of real points. However often such real points of position are interpolated, by real points of matter being interposed, their number will always be finite, the number of intervals intercepted on the first interval, & at the same time constituting that interval, will be finite; but the number of possible parts of this sort will be endless. The magnitude of each of the former will be definite & finite; the magnitude of the latter will be diminished without any limit whatever; & there will be no gap that cannot be diminished by adding fresh points in between; although it cannot be completely removed either by division or by interposition of points.
	Hence, the manner in which we arrive at space that is finite, continuous, necessary, eternal & immovable, by means of an abstract concept.
9. In this way, so long as we conceive as possibles these points of position, we have infinity of space, & continuity, together with infinite divisibility. With existing things there is always a definite limit, a definite number of points, a definite number of intervals; with possibles, there is none that is finite. The abstract concept of possibles, excluding as it does a limit due to a possible increase of the interval, a decrease or a gap, gives us the infinity of an imaginary line, & continuity; such a line has not actually any existing parts, but only possible ones. Also, since this possibility is eternal, in that it was true from eternity & of necessity that such points might exist in conjunction with such modes, space of this kind, imaginary, continuous & infinite, was also at the same time eternal & necessary; but it is not anything that exists, but something that is merely capable of existing, & an indefinite concept of our minds. Moreover, immobility of this space will come from immobility of the several points of position.
	The same things hold for instants of time; as for points, after the first there is no second or last; in time, however, there is but one dimension, while in space there are three.
10. Everything, that has so far been said with regard to points of position, can quite easily in the same way be applied to instants of time; & indeed there is a very great analogy of a sort between the two. For, a point from a given point, or an instant from a given instant, has a definite distance, unless they coincide; & another distance can be found either greater or less than the first, without any limit whatever. In any interval of imaginary space or time, there is a first point or instant, & a last; but there is no second, or last but one. For, if any particular one is supposed to be the second, then, since it does not coincide with the first, it must be at some distance from it; & in the interval between, other possible points or instants intervene. Again, a point is not a part of a continuous line, or an instant a part of a continuous time; but a limit & a boundary. A continuous line, or a continuous time is understood to be generated, not by repetition of points or instants, but by a continuous progressive motion, in which some intervals are parts of other intervals; the points themselves, or the instants, which are continually progressing, are not parts of the intervals. There is but one difference, namely, that this progressive motion can be accomplished in space, not only in a single direction along a line, but in infinite directions over a plane which is conceived from the continuous motion of the line already conceived in the direction of its breadth; & further, in infinite directions throughout a solid, which is conceived from the continuous motion of the plane already conceived. Whereas, in time there will be had but one progressive motion, that of duration; & therefore this will be analogous to a single line. Thus, while for imaginary space there is extension in three dimensions, length, breadth & depth, there is only one for time, namely length or duration only. Nevertheless, in the threefold class of space, & in the onefold class of time, the point & the instant will be respectively the element, from which, by its progression, motion, space & time will be understood to be generated.
	Every point of matter is possessed of the whole of imaginary space, & time, the nature of compenetration.
11. Now here there is one thing that must be carefully noted. Not only when two points of matter exist, & have a distance from one another, do two modes exist which give the foundation of the relation of this distance; & there are two different real points of position, the possibility of which, as conceived by us, will yield two points of imaginary space; & thus, to the infinite number of possible points of matter there will correspond an infinite number of possible modes of existence. But also to any one point of matter there will correspond the infinite possible modes of existing, which are all the possible positions of that point. All of these taken together are sufficient for the possession of the whole of imaginary space; & any point of matter has its own imaginary space, immovable, infinite & continuous; nevertheless, all these spaces, belonging to all points coincide with one another, & are considered to be one & the same. For if we take one real point of position belonging to one point of matter, & associate it with all the real points of position belonging to another point of matter, there is one among the latter, which, if it coexist with the former, will induce a relation of no-distance, which we call compenetration. From this it is clear that, for points which exist, no-distance is not nothing, but a relation induced by some two modes of existence. Any of the others would induce, with that same former point of position, another relation of some determinate distance & position, as we say. Further, those points of position, which induce a relation of no-distance, we consider to be the same; & we consider any of the infinite number of such points belonging to the infinite number of points of matter to be the same; & mean them when we speak of the 'same position.' Moreover this is evidently bound to be true for any pair of points. If now a third point is situated anywhere, it will have some distance & position with respect to the first. If the first is removed, the second can be so situated that it has the same distance & position with respect to the third as the first had. Hence the mode, in which it exists, will be taken to be the same in this case as the mode in which the first point was existing; & if these two modes were existing together, they would induce a relation of no-distance between the first point & the second. All that has been said above with regard to points of space applies equally well to instants of time.
	Several instants belonging to the same point cannot coexist.
12. Now, whether they can coexist is a question that pertains to the relation between points of position & instants of time, whether we consider a single point of matter or several of them. In the first place, several instants of time belonging to the same point of matter cannot coexist; but they must necessarily come one after the other; & similarly, two points of position belonging to the same point of matter cannot be conjoined, but must lie one outside the other; & this comes from the nature of points of this kind, & is essential to them, to use a common phrase.
	Four combinations of space & time for a single point of matter; four worth considering for two points; extraordinary idea of another space situated elsewhere.
13. Next, we have to consider the different kinds of combinations of points of space & instants of time. Any point of matter, if it exists, connects together some point of space & some instant of time; for it is bound to exist somewhere & sometime. Even if it exists alone, it always has its own mode of existence, both local & temporal; & by this fact, if any other point of matter exists, having its own modes also, it will acquire a relation of distance, both local & temporal, with respect to the first. This at least will certainly be the case, if the space belonging to all that exist, or can possibly exist, is common; so that the points of position belonging to the one coincide perfectly with those belonging to the other, each to each. But, what if there are other kinds of things, either different from those about us, or even exactly similar to ours, which have, so to speak, another infinite space, which is distant from this our infinite space by no interval either finite or infinite, but is so foreign to it, situated, so to speak, elsewhere in such a way that it has no com- munication with this space of ours; & thus will induce no relation of distance. The same remark can be made with regard to a time situated outside the whole of our eternity. But such an idea requires an intellect of the greatest power to try to grasp it; & it cannot be admitted by direct consideration, in any way, or at least with difficulty. Hence, omitting altogether such things, or the spaces & times of such things which are no concern of ours, let us consider the things that have to do with us. If therefore, firstly, the same, point of matter connects the same point of space with several instants of time separated from one another by any interval, there will be return to the same place. If, secondly, it connects the point of space to a continuous series of instants of continuous time, there will be rest, which requires a certain continuous time to be connected with the same point of position; without this connection there will be continuous motion, points of position succeeding one another corresponding to instants of time, one after the other. Thirdly, if the same point of matter connects the same instant of time with several points of position distant from one another by some interval, then we shall have replication. Fourthly, if it connects the instant with a continuous series of points of position contained within some continuous interval, we shall have something which several of the Peripatetics admitted, calling it virtual extension; by virtue of which an indivisible particle of matter, quite without parts, could occupy divisible space. There are four other combinations, when several points are considered. That is to say, fifthly, if several points connect the same instant of time with several points of position; in this is involved coexistence. Sixthly, if they connect the same point of space with several instants of time; as would be the case when different points of matter were forced successively into the same position. Seventhly, if they connect the same point of space with the same instant of time; in this is involved compenetration. Eighthly, if they have no instant of time, & no point of space, common to them; as would be the case, if they did not coexist, nor, any of them, occupied the positions that had been occupied by any of the others at any time.
	The relations of these cases to one another of them are possible, & how.
14. Out of these eight cases, the third corresponds to the first, the fourth to the second, the sixth to the fifth, the eighth to the seventh. The third case, namely replication, is usually considered to be naturally impossible. Many think that the fourth case holds good for the rational soul, which they consider to have its seat in some divisible space; for instance, the Peripatetics think that it pervades the whole of the body, other philosophers think it is situated in a certain part of the brain, or in some juice of the nerves; so that, since it is indivisible, the whole of it must be in the whole of the space, & the whole of it in any part of the space. Just in the same way as the same indivisible Divine Nature is as a whole in the whole of space, & as a whole in any part of space, being necessarily present everywhere, & coexisting with & accompanying created things wherever created things are. Others admit this same case for matter, & consider that particles of matter can be extended in a similar manner, as we have said; although they are simple, & although they are devoid of parts, not only parts that are really separated, but also such as are distinct & only separable. I do not consider that this supposition can be entertained, for the reason that, whenever we perceive with our senses matter occupying positions distinct from one another, we see that it is also separable, although we may have to use a very great force; here, parts are separated which were at a distance from one another. Indeed, by no other argument can we exclude replication from Nature, than that we never see any portion of matter, as far as can be perceived by the senses, occupying two positions at the same time. The idea of Virtual extension of matter goes infinitely further beyond the idea of simple replication.
	Rest & return to the same position are infinitely improbable in Nature; hence arises a very great analogy between them.
15. If the second case of rest, & the first case of return to the same position could be obtained naturally, then indeed there would be a certain defect in the analogy between space & time. But it seems to me that I can prove that neither ever happens in Nature; & so they cannot be obtained naturally; this is my argument. If a point of matter at any instant of time is at a certain point of space, & we do not know where it is at some other instant, let us inquire how much more probable it is that it should be somewhere else than at the same point as before. The former will be more probable than the latter in the proportion of the number of all the other points of space to that single point. There are an infinite number of these points in any straight line, the number of lines in any plane is infinite, & the number of planes in the whole of space is infinite. Hence, the number of other points of space is an infinity of the third order; & thus the probability is infinitely greater with an infinity of the third order, when we are concerned with any other particular instant of time. Now let us deal indefinitely with all the instants of infinite time; then the first probability will decrease in proportion as the number of instants increases, at any of which it might at least be possible that the point was in the same place as before. Moreover, there are an infinite number of instants, the infinity being of the same order as that of the number of possible points in an infinite line. Hence, still considering indefinitely all the instants of infinite time, it is infinitely more improbable that the point should be in the same position as before, than that it should be somewhere else. Now consider, not a single point of position occupied at a single particular instant, but any point of position occupied at any indefinite instant; then still the probability of return to any one of these points of position will increase as the number of them increases; & this number, in a time that is also infinite, is an infinity of the same order as the number of lines in any plane. Hence the improbability of this case, in which any particular point of matter returns at some indefinite instant of time to some indefinite point of position, in which it was assumed to be at some other indefinite instant of time, remains an infinity of the first order. Moreover, this, for all points of matter, which are finite in number, will decrease in the finite ratio of this number to infinity (which would not be the case with the usual theory, in which the number of points of matter is taken to be an infinity of the third order). Hence we are still left with an infinite improbability of the return of any indefinitely chosen point of matter to any point of position, occupied at any previous instant of time indefinitely, of a return, I say, taking place at any indefinite instant of subsequent time; hence, such a return must be excluded, without any fear as to error, since it must be considered that an infinite improbability merges into a sort of relative impossibility. This Theory indeed cannot be applied to the ordinary view. Hence, in this way it is clear, in my Theory of points of matter, there must be excluded from Nature both rest, which also we excluded above, & even return to the same point of position in which that point of matter once was situated. Therefore it comes about that all those first four cases will be excluded from Nature, & in them the analogy of time & space will be preserved accurately.
	No point of matter can come into any point of space that was once occupied by any other point; it is only in coexistence, which corresponds to this that the analogy is broken.
16. Finally, if we seek to find whether any point of matter is bound to occupy at some instant a point of position which was occupied by some other point of matter at some other instant, still the improbability will be infinitely infinite. For the number of existing points of matter is finite; & thus, if instead of the return of any point to points of position occupied by itself we consider the return to points that have been occupied by another, the number of cases increases in the ratio of unity to a number of points that is in every case finite, that is to say, in a finite ratio only. Hence, the improbability of the arrival of any point of matter indefinitely taken at a point of space that has been occupied at some time by any other point is still infinite; & this arrival must therefore be taken to be impossible. In this way, indeed, the sixth case, which depended on this return, is excluded; & much more so the seventh case, which involves the simultaneous arrival of a pair of points of matter at any the same point of position, that is to say, compenetration. The eighth case also is excluded for matter; for all things created together as a whole will continually last as a whole, & so will always have a common instant of time.[2] Only the fifth case, in which several points of matter connect the same instant of time with different points of position remains; & this is not only possible, but also necessary for all points of matter, seeing that they coexist. For it cannot be the case that the seventh & the eighth are excluded, unless straightway, on that very account, the fifth is included, as will be easily seen on consideration. Therefore in this point the analogy fails, namely, in that several points of matter can connect different points of space with the same instant of time, which is the fifth case; whereas it is impossible for the same point of space to be connected with several instants of time, which is the third case. This defect is necessarily induced by the exclusion of the seventh & eighth cases; for if either of the latter is included, the fifth might be excluded; just as if it were possible for points of matter, which had been created together, & do not perish, not to coexist; for then the same instant of time would in no way be connected with different points of position.
	Which of the cases are possible through Divine Omnipotence; use of the theorem give above on impenetrability.
17. At least six of the seven cases seem to be possible through Divine Omnipotence, that is to say, omitting the virtual extension of matter, about which there may be possibly some doubt; for in this case there must exist at the same time an absolutely infinite number of those real points of position; & this is impossible, if an existing thing that is infinite in number is contradictory in the modes. Moreover, since all points of position can exist one after another, arranged along any line, for instance, in continuous motion, & so can also all instants of continuous time, one after another in the duration of any thing, there will be reason for doubt as to whether all those points of position can also exist at the same time. This is a matter upon which I dare not make a definite statement. All I say is that this theory of mine with regard to the nature of space & continuity completely avoids all the chief difficulties that are obstacles in other theories; & that it is very suitable for the explanation of everything in connection with this matter. I will also add the remark that, if the arrival of any point of matter at a point of position, at which any point of matter has arrived at any instant, is excluded, & along with it compenetration is thus excluded, then real impenetrability of matter must necessarily follow, which will be of great service to us in our tenth book[3]. That is, unless repulsive forces prevent such a thing, any perfectly free mass will permeate through any other mass, without there being any danger of a collision of one point with another. Here there would be an apparent compenetration similar to the penetration of light through crystals, oils through wood, & marble, without any real compenetration of the points. In denser masses, & those endowed with a smaller velocity, the repulsive forces for the most part prevent further motion without any impact; & this also excludes sensible as well as apparent compenetration. In very tenuous masses moving with very great velocities, as rays of light propagated through homogeneous substances, or through other rays, the very slight inequality of the actions, derived from the unequal distances of the circumjacent points, will be prevented by the high velocity; & perfectly free progress will take place in all directions without any danger of collisions. This removes altogether the greatest & only real difficulty in the idea of the propagation of light by means of a substance that is emitted & travels forward. But I have now said quite enough upon this matter.

§ II.

Of Space & Time, as we know them

	We cannot obtain and absolute knowledge of local modes of existence; nor yet of absolute distances or magnitudes.
18. We have spoken, in the preceding Supplement, of Space & Time, as they are in themselves; it remains for us to say a few words on matters that pertain to them, in so far as they come within our knowledge. We can in no direct way obtain a knowledge through the senses of those real modes of existence, nor can we discern one of them from another. We do indeed perceive, by a difference of ideas excited in the mind by means of the senses, a determinate relation of distance & position, such as arises from any two local modes of existence; but the same idea may be produced by innumerable pairs of modes or real points of position; these induce the relations of equal distances & like positions, both amongst themselves & with regard to our organs, & to the rest of the circumjacent bodies. For, two points of matter, which anywhere have a given distance & position induced by some two modes of existence, may somewhere else on account. of two other modes of existence have a relation of equal distance & like position, for instance if the distances exist parallel to one another. If those points, we, & all the circumjacent bodies change their real positions, & yet do so in such a manner that all the distances remain equal & parallel to what they were at the start, we shall get exactly the same ideas. Nay, we shall get the same ideas, if, while the magnitudes of the distances remain the same, all their directions are turned through any the same angle, & thus make the same angles with one another as before. Even if all these distances were diminished, while the angles remained constant, & the ratio of the distances to one another also remained constant, but the forces did not change owing to that change of distance; then if the scale of forces is correctly altered, that is to say, that curved line, whose ordinates express the forces; then there would be no change in our ideas.
	The motion, if any, common to us & the Universe could not come within our knowledge; nor could we know it, if it were increased in any ratio, or diminished, as a whole.
19. Hence it follows that, if the whole Universe within our sight were moved by a parallel motion in any direction, & at the same time rotated through any angle, we could never be aware of the motion or the rotation. Similarly, if the whole region containing the room in which we are, the plains & the hills, were simultaneously turned round by some approximately common motion of the Earth, we should not be aware of such a motion; for practically the same ideas would be excited in the mind. Moreover, it might be the case that the whole Universe within our sight should daily contract or expand, while the scale of forces contracted or expanded in the same ratio; if such a thing did happen, there would be no change of ideas in our mind, & so we should have no feeling that such a change was taking place.
	Since, of our position & that of everything we see is changed, our ideas are not changed; therefore we can ascribe no motion to ourselves or to anything else.
20. When either objects external to us, or our organs change their modes of existence in such a way that that first equality or similitude does not remain constant, then indeed the ideas are altered, & there is a feeling of change; but the ideas are the same exactly, whether the external objects suffer the change, or our organs, or both of them unequally. In every case our ideas refer to the difference between the new state & the old, & not to the absolute change, which does not come within the scope of our senses. Thus, whether the stars move round the Earth, or the Earth & ourselves move in the opposite direction round them, the ideas are the same, & there is the same sensation. We can never perceive absolute changes; we can only perceive the difference from the former configuration that has arisen. Further, when there is nothing at hand to warn us as to the change of our organs, then indeed we shall count ourselves to have been unmoved, owing to a general prejudice for counting as nothing those things that are nothing in our mind; for we cannot know of this change, & we attribute the whole of the change to objects situated outside of ourselves. In such manner any one would be mistaken in thinking, when on board ship, that he himself was motionless, while the shore, the hills & even the sea were in motion.
	The manner in which we are to judge of the equality of two things from their equality with a third; there never can be congruence in length, any more than there can be in time; the matter is to be inferred from causes.
21. Again, it is to be observed first of all that from this principle of the unchangeability of those things, of which we cannot perceive the change through our senses, there comes forth the method that we use for comparing the magnitudes of intervals with one another; here, that, which is taken as a measure, is assumed to be unchangeable. Also we make use of the axiom, things that are equal to the same thing are equal to one another; & from this is deduced another one pertaining to the same thing, namely, things that are equal multiples, or submultiples, of each, are also equal to one another; & also this, things that coincide are equal. We take a wooden or iron ten-foot rod; & if we find that this is congruent with one given interval when applied to it either once or a hundred times, & also congruent to another interval when applied to it either once or a hundred times, then we say that these intervals are equal. Further, we consider the wooden or iron ten-foot rod to be the same standard of comparison after translation. Now, if it consisted of perfectly continuous & solid matter, we might hold it to be exactly the same standard of comparison; but in my theory of points at a distance from one another, all the points of the ten-foot rod, while they are being transferred, really change the distance continually. For the distance is constituted by those real modes of existence, & these are continually changing. But if they are changed in such a manner that the modes which follow establish real relations of equal distances, the standard of comparison will not be identically the same; & yet it will still be an equal one, & the equality of the measured intervals will be correctly determined. We can no more transfer the length of the ten-foot rod, constituted in its first position by the first real modes, to the place of the length constituted in its second position by the second real modes, than we are able to do so for intervals themselves, which we compare by measurement. But, because we perceive none of this change during the translation, such as may demonstrate to us a relation of length, therefore we take that length to be the same. But really in this translation it will always suffer some slight change. It might happen that it underwent even some very great change, common to it & our senses, so that we should not perceive the change; & that, when restored to its former position, it would return to a state equal & similar to that which it had at first. However, there always is some slight change, owing to the fact that the forces which connect the points of matter, will be changed to some slight extent, if its position is altered with respect to all the rest of the Universe. Indeed, the same is the case in the ordinary theory. For no body is quite without little spaces interspersed within it, altogether incapable of being compressed or dilated; & this dilatation & compression undoubtedly occurs in every case of translation, at least to a slight extent. We, however, consider the measure to be the same so long as we do not perceive any alteration, as I have already remarked.
	Conclusion reached; the difference between ordinary people & philosophers in the matter of judgment.
22. The consequence of all this is that we are quite unable to obtain a direct knowledge of absolute distances; & we cannot compare them with one another by a common standard. We have to estimate magnitudes by the ideas through which we recognize them; & to take as common standards those measures which ordinary people think suffer no change. But philosophers should recognize that there is a change; but, since they know of no case in which the equality is destroyed by a perceptible change, they consider that the change is made equally.
	Although, when the ten-foot rod is moved in position, those modes that constitute the relations of the interval are also altered, yet equal intervals are reckoned as same for the reasons stated.
23. Further, although the distance is really changed when, as in the case of the translation of the ten-foot rod, the position of the points of matter is altered, those real modes which constitute the distance being altered; nevertheless it the change takes place in such a way that the second distance is exactly equal to the first, we shall call it the same, & say that it is altered in no way, so that the equal distances between the same ends will be said to be the same distance & the magnitude will be said to be the same : & this is defined by means of these equal distances, just as also two parallel directions will be also included under the name of the same direction. In what follows we shall say that the distance is not changed, or the direction, unless the magnitude of the distance, or the parallelism, is altered.
	The same observations apply equally to Time; but in it, it is well known, even to ordinary people, that the same temporal interval cannot be translated for the purpose of comparing two intervals; it is because of this that they fall into error with regard to space.
24. What has been said with regard to the measurement of space, without difficulty can be applied to time; in this also we have no definite & constant measurement. We obtain all that is possible from motion; but we cannot get a motion that is perfectly uniform. We have remarked on many things that belong to this subject, & bear upon the nature & succession of these ideas, in our notes. I will but add here, that, in the measurement of time, not even ordinary people think that the same standard measure of time can be translated from one time to another time. They see that it is another, consider that it is an equal, on account of some assumed uniform motion. Just as with the measurement of time, so in my theory with the measurement of space it is impossible to transfer a fixed length from its place to some other, just as it is impossible to transfer a fixed interval ot time, so that it can be used for the purpose of comparing two of them by means of a third. In both cases, a second length, or a second duration is substituted, which is supposed to be equal to the first; that is to say, fresh real positions of the points of the same ten-foot rod which constitute a new distance, such as a new circuit made by the same rod, or a fresh temporal distance between two beginnings & two ends. In my Theory, there is in each case exactly the same analogy between space & time. Ordinary people think that it is only for measurement of space that the standard of measurement is the same; almost all other philosophers except myself hold that it can at least be considered to be the same from the idea that the measure is perfectly solid & continuous, but that in time there is only equality. But I, for my part, only admit in either case the equality, & never the identity.

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1. This & the following section are to be found in the Philosophiæ Recentior, by Benedict Stay, Vol. I, 6, 7.
2. This case also would never happen, if the duration were not something continuously permanent; in place of it, we should have to admit a kind of, so to speak, skipping existence; that is to say, as if any point of matter (and the same thing applies to all created entities) existed only in indivisible instants remote from one another, and in all intermediate instants possible did not exist at all. Coexistence, in this case, would be infinitely improbable, the argument being nearly the same, as in the case of the arrival of one point of matter at a point of space in which some other point had once been. In this case too, there would be no real continuum even in motion; different velocities could be explained much more easily; it would be much more evident in what way the very short life of an insect can be equivalent to the longest of lives, by means of the same number of existences coming in between the first & last instants. Indeed the exclusion of any coexistence would carry away with it all immediate physical influence altogether, & determinations; indeed, a continually fresh creation, & other inadmissible things of that sort, would be obtained.
3. The reference is to Stay's "Philosophy," in which that most refined & learned author expounds my Philosophy. On what I have said above, I have plucked the fruit of the theorem, in which, in Art. 360 of this work, I dealt with impenetrability, & the apparent compenetration that would result, if there were no mutual forces.