§ 52.

The above is the most remarkable phenomenon of the human Reason, of which no instance can be shown in any other sphere. If, as generally happens, we regard the phenomena of the world of sense as things in themselves; if we assume the principles of their connection as universal of things in themselves and not merely as principles valid of experience, as is usual and indeed unavoidable without our Critique; then an unexpected conflict arises, never to be quelled in the ordinary dogmatic way, because both theses and antitheses can be demonstrated by equally evident, clear and irresistible proofs—for I pledge myself as to the correctness of all these proofs—and the Reason thus sees itself at issue with itself, a state over which the sceptic rejoices, but which must plunge the critical philosopher into reflection and disquiet.

§ 52 b.

One may bungle in metaphysics in many ways, without any danger of being detected in fallacy. For if we only do not contradict ourselves, which is quite possible in synthetic propositions, even though they may be purely invented, we can never in such cases (the conceptions we connect, being mere ideas, which as to their whole content can never be given in experience) be refuted by experience. For how should we decide by experience whether the world exists from eternity, or has a beginning? or whether matter is infinitely divisible, or consists of simple parts? Such conceptions cannot be given in any, even the largest possible experience, and therefore the fallacy of the propositions maintained or denied cannot be discovered by that test.

The only possible case in which the Reason could reveal against its will its secret dialectic, fallaciously given out by it as dogmatic, would be, if it grounded an assertion on a universally admitted axiom, and from another, equally conceded, drew a precisely opposite conclusion, with the greatest logical accuracy. This case is here realised, and indeed in respect of four natural ideas of the Reason whence four assertions on the one hand, and just as many counter-assertions on the other, arise, each as a correct consequence from universally admitted premises, and thereby reveal the dialectical illusion of the pure Reason in the use of these principles, which must otherwise have been for ever hidden.

Here then is a decisive attempt, which must necessarily disclose to us the fallacy lying hidden in the assumptions of the Reason. 1 Of two mutually contradictory propositions, both cannot be false, unless the conception at their basis be itself contradictory. For instance, two propositions, a square circle is round and a square circle is not round, are both false. For as regards the first, it is false that the [figure] mentioned is round, because it is square, but it is also false that it is not round, or that it is square, because it is a circle. For in this consists the logical mark of the impossibility of a conception, that under the same assumption two contradictory propositions would be equally false; in other words, because no middle can be conceived between them, nothing at all is cogitated by that conception.

§ 52 c.

Now, a contradictory conception like the foregoing lies at the basis of the two first antinomies, which I call mathematical, because they are concerned with the addition or division of things similar in Nature; and thence I explain how it happens that thesis and antithesis are alike false.

When I speak of objects in time and space, I do not speak of things in themselves, because of these I know nothing, but only of things in the phenomenon, in other words, of experience, as the special mode of the cognition of objects, which is alone vouchsafed to man. I must not say that what I think in space or in time exists in itself in space and time apart from this my thought; for I should then contradict myself, because space and time, together with the phenomena in them, are nothing existing in themselves and apart from my presentations, but are themselves only modes of presentations, and it is obviously contradictory to say that a mere mode of our presentation exists outside our presentation. The objects of sense exist then only in experience; and to give them a special substantive existence for themselves, apart from or before the latter, is equivalent to imagining that experience can be present without or before experience.

Now, when I inquire as to the size of the world in space and time, it is for all my conceptions just as impossible to say, it is infinite, as it is finite. For neither of them can be contained in experience, because experience is neither possible respecting an infinite space, or an infinite time, or the boundary of the world by an empty space or a previous empty time; these [things] are only ideas. Hence as regards either one or the other kind of determinate quantity, it must lie in the world itself, separate from all experience. But this contradicts the conception of a world of sense, which is only a content of experience, whose reality and connection takes place in presentation, namely, in experience, because it is not a thing in itself, but is itself nothing but a mode of presentation. It follows from the above, that, as the conception of a self-existent world is in itself contradictory, the solution of the problem as to its size will be always fallacious, no matter whether it be affirmatively or negatively attempted.

The same applies to the second antinomy, which concerns the division of phenomena. For these are mere presentations, and the parts exist merely in their presentation, and therefore in their division; in other words, in a possible experience in which they are given, and they only extend as far as the latter reaches. To assume that a phenomenon, for instance, that of body, contains all parts in itself, before all experience, to which nought but possible experience can ever attain, is equal to giving to a mere appearance, which can exist only in experience, a special existence preceding experience, or to say that mere presentations are there before they are met with in the faculty of presentation, which contradicts itself; and so, consequently, does every solution of this misunderstood problem, whether it be maintained that bodies consist of infinitely many parts, or of a finite number of simple parts.

§ 53.

In the first class of antinomy (the mathematical), the fallacy of the assumption consisted in that what is self-contradictory (namely, phenomenon and thing in itself) was represented as capable of union in one idea. But as regards the second, or dynamical class of antinomy, the fallacy of the assumption consists in that what is capable of union is represented as contradictory, and consequently, as in the first case, both contradictory assertions were false; so here, where they are opposed to one another merely through misunderstanding, both may be true.

The mathematical connection necessarily presupposes homogeneity in the connected (in the conception of quantity), while the dynamical by no means requires this. Where the quantity of the extended is concerned, all the parts must be homogeneous, both with each other and with the whole; whereas in the connection of cause and effect, although homogeneity may also be met with, it is not necessary. For the conception of causality, by means of which a thing is posited by something quite distinct therefrom, at least does not require it. If the objects of the sense-world were taken for things in themselves, and the above-cited laws of Nature for laws of things in themselves, the contradiction would be un avoidable. In the same way, if the subject of freedom were presented like other objects as mere appearance, the contradiction would be equally unavoidable; for the same thing would be at once affirmed and denied of the same kind of object in the same sense. But if natural necessity be referred merely to phenomena, and freedom merely to things in themselves, no contradiction arises, in assuming or admitting both kinds of causality, however difficult or impossible it may be to render the latter kind comprehensible.

In the phenomenon, every effect is an event, or something that happens in time; a determination of the causality of its cause (a state of the same), must precede it, upon which it follows according to a uniform law. But this determination of the cause to causality must also be something that takes place, or happens. The cause must have begun to act, otherwise between it and the effect, no succession in time could be conceived. The effect would always have existed, as well as the causality of the cause. Thus, among phenomena, the determination of the cause to the effect must also have arisen, and therefore be just as much as its effect, an event which, in its turn, must have a cause, and so on; and consequently, necessity must be the condition according to which the efficient causes are determined. If, on the other hand, freedom be a characteristic of certain causes of phenomena, it must, as regards the latter as events, be a faculty of beginning them from itself ( sponte ), i.e., without the causality of the causes themselves having begun, and hence another ground would be necessary to determine its beginning. In that case, however, the cause, as to its causality, must not be subject to time determinations of its state; that is, it must not be phenomenon, but it must be regarded as a thing in itself, and its effects only, as phenomena. 1 If one can conceive such an influence of the essences of the understanding on phenomena without contradiction, though necessity would attach to all connection of cause and effect in the sense-world, yet of the cause which is itself no phenomenon, although it lies at the foundation of the latter, freedom would be admitted. Thus Nature and Freedom can be attributed without contradiction to the same thing, at one time as phenomenon, at another, as thing in itself.

We have a faculty within us, not only standing in connection with its subjective determining grounds, which are the natural causes of its actions, and in so far the faculty of a being, belonging to phenomena, but also referable to objective grounds, though these are merely ideas, in so far as they can determine this faculty; and this connection is expressed by ought. The above faculty is termed Reason, and when we contemplate a being (man) simply according to this subjectively determining Reason, it cannot be regarded as an essence of sense, but the quality thought of is the quality of a thing in itself, of the possibility of which, namely, the ought of that which has never happened, and yet the activity of which can be the determination and cause of actions, whose effect is phenomenal in the sense-world, of this we can form no conception whatever. At the same time, the causality of the Reason as concerns its effects in the sense-world would be freedom, so far as objective grounds, which are themselves ideas, are regarded as determining these effects. For its action would then depend not on subjective, and therefore on time-conditions, nor on natural laws, serving to determine these, since grounds of the Reason in general would furnish the rule for actions according to principles, without the influence of circumstances, time, or place.

What I adduce here, is merely meant as an instance for the sake of intelligibility, and does not necessarily belong to our question, which must be decided from mere conceptions, independently of the qualities we meet with in the real world.

I can say now without contradiction, that all actions of rational beings, inasmuch as they are phenomena, met with in any experience, are subject to necessity; but precisely the same actions, with reference to the rational subject, and its capacity of acting according to mere Reason, are free. For what is demanded by necessity? Nothing more than the determinability of every event in the sense-world according to uniform laws; in other words, a reference to Cause in the phenomenon, whereby the thing in itself, lying at its foundation, and its causality, remains unknown. But I say: the natural law subsists alike, whether the rational being [acting] from Reason, and hence through freedom, be the cause of the effects in the sense-world, or whether these are determined by other grounds than those of Reason. For in the first case, the action happens according to maxims, whose effect in the phenomenon will be always in accordance with uniform laws; in the second case, if the action does not happen according to principles of the Reason, it is subordinated to the empirical laws of the sensibility, and in both cases the effects are connected according to uniform laws; more than this we do not require to [constitute] natural necessity, nay, more we do not know respecting it. But in the first case, Reason is the cause of these natural laws, and is hence free; in the second case, the effects follow the mere natural laws of the sensibility, because the Reason exercises no influence upon them; the Reason, however, is not on this account itself determined by the sensibility (which is impossible), and is consequently in this case also free. The freedom does not hinder the natural law of the phenomena, any more than the latter interferes with the freedom of the practical use of the Reason, which stands in connection with things in themselves as determining grounds.

In this way, the practical freedom, namely, that by which the Reason has causality, according to objective determining grounds, is saved, without natural necessity being curtailed in the least, in respect of the same effects as phenomena. The above may also be serviceable as an explanation of what we had to say regarding transcendental freedom, and its union with natural necessity (in the same subject, but not taken in the same connection). For as to this, every beginning of the action of a being, from objective causes, so far as its determining grounds are concerned, is always a first beginning, although the same action in the series of phenomena is only a subaltern beginning, necessarily preceded by a state of the cause determining it, and itself determined by a [state] immediately preceding; so that without falling into contradiction with the laws of Nature, we may conceive of a faculty in rational beings, or in beings generally, in so far as their causality is determined in them, as things in themselves, by which a series of states is begun of themselves. For the relation of the action to objective grounds of the Reason is no relation in time; here, what determines the causality does not precede the action according to time, because such determining grounds [as these] do not present a reference of the objects to sense, or, in other words, to causes in the phenomenon, but to determining causes, as things in themselves, which are not subordinated to time-conditions. Hence, the action may be viewed with regard to the causality of the Reason as a first beginning, but at the same time, as regards the series of the phenomena, as a merely subordinate beginning, and without contradiction, in the former aspect as free, and in the latter, inasmuch as it is merely phenomenon, as subordinate to natural necessity.

As concerns the fourth antinomy, it is solved in the same manner as is the conflict of the Reason with itself, in the third. For if the cause in the phenomenon be only distinguished from the cause of the phenomena, so far as they can be considered as things in themselves, both propositions can subsist beside one another, namely, that no cause takes place anywhere in the sense-world (according to similar laws of causality) whose existence is absolutely necessary; while, on the other hand, this world may be connected with a necessary being as its cause, though of another kind, and according to other laws; the incompatibility of the above two propositions simply resting on the misunderstanding by which what is merely valid of phenomena is extended to things in themselves, both being mixed up in one conception.

§ 54

This is the arrangement and solution of the whole antinomy, in which the Reason finds itself involved, in the application of its principles to the sense-world, and of which even this (the mere arrangement) would be itself a considerable service to the knowledge of the human Reason, even though the solution of the conflict should not fully satisfy the reader, who has here a natural illusion to combat, which has only recently been presented to him as such, and which he has previously regarded as true. For one consequence of this is inevitable, namely, that seeing it is quite impossible to get free of this conflict of the Reason with itself, so long as the objects of the sense-world are taken for things in themselves, and not for what they are in reality, namely, mere phenomena, the reader is necessitated thereby again to undertake the deduction of all our knowledge à priori, and its examination as given by me, in order to come to a decision in the matter. I do not require more [than this] at present; for if he has but first penetrated deeply enough into the nature of the pure Reason, the conceptions by which the solution of this conflict of the Reason is alone possible, will be already familiar to him, without which circumstance I cannot expect full credit even from the most attentive reader.

§ 55.: III. Theological Idea (Critique, p. 350).

The third transcendental idea, which furnishes material to the most important, but, when merely conducted speculatively, to the exaggerated (transcendent) and thereby dialectical use of the Reason, is the ideal of the pure Reason. The Reason does not here, as with the psychological and cosmological ideas, start from experience, and is not, by a [progressive] raising ( Steigerung ) of the grounds, misled into an endeavour to contemplate the series in absolute completeness, but wholly breaks therewith, and from mere conceptions of what would constitute the absolute completeness of a thing in general, and consequently by means of the idea of a most perfect original being, descends to the determination of the possibility, and thereby also to the reality, of all other things. For this reason, the mere assumption of a being, which although not given in the series of experience, is nevertheless conceived for the sake of experience, to render comprehensible the connection, order, and unity of the latter, that is, the Idea is more easily distinguishable from the conceptions of experience [in the present] than in the foregoing cases. The dialectical illusion therefore arising from our holding the subjective conditions of our thought for the objective conditions of things themselves, and a necessary hypothesis for the satisfaction of our Reason for a dogma, may be easily exposed to view; and hence I have nothing further to recall on the assumptions of the transcendental theology, for what the Critique has said on this point is comprehensible, clear, and decisive.

§ 56.

The objects given us through experience are in many respects incomprehensible, and there are many problems to which the natural law leads us, when it is carried to a certain height, (though always in accordance with these laws,) which can never be solved; as for instance, how it is that substances attract one another. But, if we entirely leave Nature, or in the progress of its connection overstep all possible experience, and thereby immerse ourselves in mere ideas, we cannot then say that the object is incomprehensible, and that the nature of things places insoluble problems before us; for we have in that case, nothing whatever to do with Nature or with given objects, but merely with conceptions, having their origin simply in our Reason, and with mere essences of thought, in respect of which all problems arising from the conception of the same, can be solved, because the Reason can and must certainly give a complete account of its own procedure. 1 As the psychological, cosmological, and theological ideas, are simply conceptions of the Reason, not capable of being given in any experience, so the problems which the Reason in respect thereof places before us, are not propounded by the objects, but by mere maxims of the Reason for its own satisfaction, and must be capable of being adequately answered in their totality, which is effected by showing them to be principles [designed] to bring the use of our understanding to thorough agreement, completeness and synthetic unity, and which are in so far valid merely of experience, but of the whole of the latter.

Now, although an absolute whole of experience is impossible, the idea of a whole of knowledge according to principles in general, is what alone can procure a particular kind of unity, namely, that of a system, without which our knowledge is nothing but a patchwork, and cannot be used for the highest end (which is always the system of all ends); by this I understand not merely the practical, but also the highest end of the speculative use of the Reason.

The transcendental ideas express, then, the specific destiny of the Reason, namely, as being a principle of the systematic unity of the use of the understanding. But when this unity of the mode of cognition be viewed as though it depended upon the object of cognition; when we hold that which is merely regulative for constitutive, and persuade ourselves that we can extend our cognition by means of these ideas, far beyond all possible experience in a transcendent manner, notwithstanding that they merely serve to bring experience as nearly as possible to completeness, i.e., to limit its progress by nothing which cannot belong to experience—then this is a simple misunderstanding in judging the special destiny of our Reason and its principles, and a dialectic, partly confusing the use of the Reason in experience, and partly making the Reason to be at issue with itself.

Observation 1.

As in Phoronomy nothing is to be discussed but motion, its subject, namely matter, has here no other quality attributed to it than movability. It can therefore itself be valid for one point so far, and in Phoronomy we abstract from all internal construction, hence also, from the quantity of the movable, and concern ourselves only with motion, and what can be regarded as quantity therein (velocity and direction). If the expression body is sometimes used here, it occurs only to anticipate in a measure the application of the principles of Phoronomy to the following more definite conceptions of matter, in order that the exposition may be less abstract and more comprehensible.

Observation 2.

If I am to explain the conception of matter not by a predicate, applying to it as object, but only by the relation to the faculty of knowledge, in which the presentation can be primarily given me, matter is every object of the external sense, and this would be its mere metaphysical explanation. But space would be simply the form of all external sensuous intuition (whether this accrued to the external object we call matter in itself, or remained merely in the construction of our sense, a point which does not enter into the present question). Matter, in contradistinction to form, would be that which in external intuition, is an object of feeling, and consequently the properly empirical of sensible and outward intuition, because it cannot be given at all à priori. In all experience something must be felt, and this is the real of sensuous intuition. In consequence, space, in which we are to institute experience respecting motions, must be capable of being felt, that is, of being indicated by that which can be felt, and this, as the sum-total of all objects of experience, and itself an object of the same, is called empirical space. Now this, as material, is itself movable; but a movable space, if its movement is to be able to be perceived, presupposes again an enlarged material space in which it is movable, and this again another, and so on to infinity.

Thus all motion that is an object of experience is merely relative; the space in which it is perceived is a relative space, which again moves itself perhaps in an opposite direction, in a space further enlarged, and therefore the matter moved in reference to the first may be termed at rest in relation to the second; and these alterations of the conception of motion go forward with the alteration of the relative space to infinity. To assume an absolute space, that is, one which, because it is not material, can be no object of experience as given for itself, means assuming something which, neither in itself nor in its consequences (motion in absolute space), can be perceived, for the sake of the possibility of experience, which nevertheless must always exist without it. Absolute space is in itself nothing and no object at all, but signifies merely every other relative space that I can at any time conceive outside the given space, and that I can extend beyond each given space to infinity; one that includes the [given space], and in which I can assume it as moved. But since I have the enlarged, although still material, space only in thought, nothing is known to me of the matter indicating it. I abstract from this, and it is conceived, therefore, as a pure, non-empirical and absolute space, with which I can compare, and in which I can conceive as movable, each empirical space, and therefore, which is itself always regarded as immovable. To constitute it a real thing means confounding the logical universality of any space, with which I can compare each empirical [space] as being included in it with a physical universality of real compass, and misunderstanding the reason in its idea.

I may observe in conclusion that as the movability of an object in space cannot be known à priori and without the teaching of experience, it could not for the same reason be counted in the Critique of pure Reason amongst the pure conceptions of the understanding, and this conception as empirical could only find a place in a natural science, as applied metaphysics, which occupies itself with a conception given through experience, although according to principles à priori.

Observation 1.

I have already laid the conception of matter at the basis of the conception of motion; but, as I wished to determine the latter independently of the conception of extension, and thus could consider matter only in one point, I had to admit the use of the common explanation of motion as change of place. Now that the conception of matter is to be explained universally, and therefore as applicable to moved bodies, this definition is inadequate, for the place of every body is a point. If one wishes to determine the distance of the moon from the earth, one wishes to know the distance of their places, and to this end one does not measure from any point of the surface, or of the interior of the earth, to any point of the moon at pleasure, but takes the shortest line from the central point of the one to the central point of the other, and therefore, in each of these bodies there is only one point that constitutes its place. Now a body may move without changing its place, as the earth in turning on its axis; but its relation to external space changes notwithstanding, for it presents for instance its different sides to the moon in the course of the twenty-four hours, from which all kinds of transformative effects result on the earth. Only of a movable, i.e., physical point can one say: motion is always a change of place. It might be objected against this explanation that internal motion (e.g., fermentation) is not included therein; but the thing which one speaks of as in motion must so far be regarded as unity. That matter, as, for instance, a cask of beer, is in motion signifies something different to the beer in the cask being in motion. The motion of a thing is not one and the same with motion in this thing; but the question is here only of the former. The application of this conception to the latter case is afterwards easy.

Observation 2.

Motions may be circular (without change of place) or progressive, and these again may either enlarge the space or be motions limited to a given space. Of the first kind are rectilinear, or even non-rectilinear, [motions] that do not return in upon themselves. Of the second are those that return in upon themselves. The latter are again either circular or oscillating motions. The first cover the same space always in the same direction; the second alternatingly in an opposite direction, like a swaying pendulum. To both belong trembling ( motus tremulus ), which, though not a progressive motion of a body, is nevertheless a reciprecative motion of a matter, which does not change its place on the whole thereby, as the vibrations of a bell that has been struck, or the tremblings of air set in motion by sound. I merely make mention of these different kinds of motion in a Phoronomy, because with all that are not progressive the word velocity is generally used in another sense than with the progressive, as the following observation shows.

Observation 3.

In every motioa direction and velocity are the two momenta for consideration, when one abstracts from all other qualities of the movable. I presuppose here the ordinary definition of both; but that of direction has sundry limitations. A body moved in a circle changes its direction continuously, so that, until its return to the point from which it started, all is comprised in a surface of merely possible directions, and yet one says it moves itself always in the same direction, as, for instance, the planet from evening to morning.

But what is the side, in this case, towards which the motion is directed? A question related to the one: Upon what does the internal distinction of spirals, otherwise similar and even equal, rest, but of which one species winds to the right, and the other to the left; or the winding of the kidney-bean, and of the hop, of which the one runs round its pole like a corkscrew, or as sailors express it against the sun, and the other with the sun? This is a conception that allows itself to be constructed indeed, but as conception does not admit of being made plain by universal marks in the discursive mode of cognition. In the things themselves (e.g., in those rare cases of the human subject where on dis ection all the parts agree according to physiological rules with other human subjects, only that all the viscera are found displaced, either to the right or the left, against the usual order) there can be no imaginable difference in the internal consequences, and yet there is a real mathematical and indeed internal difference, whereby two circular movements, differing in direction but in all other respects alike, notwithstanding their not being completely identical, nevertheless correspond. I have elsewhere shown 1 that as this difference, though it must be given in intuition, does not admit of being brought to clear conceptions, and therefore intelligibly explained ( dari, non intelligi ), it affords a good substantiating ground of proof for the proposition: that space generally, belongs, not to the qualities or relations of the things in themselves, for this would necessarily have to admit of reduction to objective conceptions, but merely to the subjective form of our sensible intuition of things or relations, which, as to what they may be in themselves, must remain wholly unknown. But this is a deviation from our present business, in which we must necessarily treat space as a quality of the things we have in consideration, namely, corporeal entities, because these themselves are merely phenomena of the external sense, and only require to be explained as such in this place. As concerns the conception of velocity, this expression acquires in use a variable meaning. We say: the earth moves more rapidly on its axis than the sun, because it does so in a shorter time, although the motion of the latter is much more rapid. The circulation of the blood of a small bird is much more rapid than that of a man, although the streaming motion in the former has, without doubt less velocity; and so with the vibrations of elastic matters. The shortness of the time of return, whether of a circulating or oscillating motion, constitutes the ground of this employment, in which, if otherwise misunderstanding be avoided, there is no harm done. For the mere increase in the hurry of return, without increase of spacial velocity, has special and very important effects in nature, of which, in the circulation of the juices of animals, perhaps not enough notice has been taken. In Phoronomy we use the word velocity merely in a spacial signification: C = S/T. 1

Observation.

A body, which is in motion, is in every point of the line it passes over—a moment. The question remains, whether it rests therein, or moves. Without doubt the latter, one will say; for, only in so far as it moves is it present in this point. But let us assume the motion in this way:

that the body describes the line A B forwards and backwards, from B to A, with uniform velocity in suchwise that, since the moment it is in B is common to both motions, the motion from A to B is described in half a second, that from B to A also in half a second, but both together in a whole second, so that not the smallest portion of time has been expended on the presence of the body in B; in this way, without the least increase of these motions, the latter, which took place in the direction B A, can be changed into that in the direction B a, which lies in a straight line with A B, and hence the body, while it is in B, must be regarded not as at rest, but as moved. It would have therefore also to be considered as moved in the first motion, returning in upon itself in the point B, which is impossible; because, in accordance with what has been assumed, it is only a moment that belongs to the motion A B, and at the same time to the equal motion B A, which is opposed to the former one and conjoined with it in one and the same moment of complete lack of motion; consequently if this constitutes the conception of rest, in the uniform motion A a, rest of the body must also be proved in every point (e.g., in B ), which contradicts the above assertion. Again, let the line A B be represented as over the point A perpendicularly, so that a body rising from A to B, after having lost its motion through gravity in the point B, would fall back again from B to A. Now I ask whether the body in B is to be considered as moved or at rest? Without doubt, it will be said, at rest; because all previous motion has been taken from it, after it has reached this point, and a uniform motion back is as yet to follow, consequently is not present, and the lack of motion, it will be added, is rest. In the first case, however, of an assumed uniform motion, the motion B A could not commence otherwise, than by the motion A B having previously ceased, and that from B to A being non-existent, and consequently there being in B a lack of all motion, whereby, according to the usual explanation, rest would have to be assumed; but we may not assume it, because at a given velocity, no body may be conceived as at rest in any point of its uniform motion. Upon what, then, is the assumption of rest based in the second case, since this rising and falling is only separated by a moment? The ground lies in the latter motion not being conceived as uniform with the given velocity, but as being at first uniformly delayed, and afterwards uniformly accelerated, in suchwise that the velocity in point B is not delayed wholly, but only up to a certain degree, smaller than any velocity that can be given, by which, if instead of falling back, the line of its fall B A were placed in the direction B a; in other words, the body were conceived as still rising, it would, as with a mere moment of velocity (the resistance of gravity being set aside), pass over, in any given time, however great, a space smaller than any space that could be given, and therefore its place (for any possible experience) would not change to all eternity. In consequence of this, it assumes a state of lasting presence in the same place, that is, of rest, although owing to the continuous action of gravity, that is, of the change of this state, the latter is immediately abolished. To be in a permanent state and to persist therein (if nothing else shifts it) are two distinct conceptions, of which one does no violence to the other. Thus rest cannot be explained through the lack of motion, which, as = o, does not admit of being constructed at all, but must be explained by permanent presence in the same place, and as this conception is constructed by the presentation of a motion with infinitely small velocity, throughout a finite time, it can be used for the subsequent application of mathematics to natural science.

Observation.

For the construction of conceptions, it is requisite that the condition of their presentation should not be borrowed from experience, and thus that they should not presuppose certain forces, the existence of which can only be deduced from experience, or, in short, that the condition of the construction should not be itself a conception incapable of being given à priori in intuition; as for instance, that of cause and effect, action and resistance, c. It is here especially to be observed that Phoronomy is throughout, primarily construction of motions in general as quantities, and that, as it has for its subject, matter merely as something movable, and of which no quantity therefore comes into consideration, it has to determine these motions alone as quantities (as concerns their velocity as well as their direction, and indeed their combination) à priori. For thus much must be established entirely à priori and intuitionally, for the sake of applied mathematics. For the rules of the connection of motions through physical causes, that is forces, never admit of being fundamentally expounded before the principles of their composition generally are previously laid down mathematically as a foundation.

Principle 1.

Every motion, as object of a possible experience, may be viewed, at pleasure, as motion of a body in a space that is at rest, or as rest of the body, and motion of the space in the opposite direction with equal velocity.

Observation.

In order to make an experience of the motion of a body it is requisite that not only the body but also the space in which it moves should be objects of external experience, or in other words, material. An absolute motion, therefore, that is, in reference to a non-material space, is unsuited to any experience whatever, and hence for use, nothing (even if one were willing to admit absolute space to be something in itself). But even in all relative motion the space itself, because it is assumed as material, may again be conceived as resting or moved. The first happens when, beyond the space in reference to which I regard a body as moved, there is no more extended space given, that includes it (as when in the cabin of a ship I see a ball moved on the table); the second, when, outside this space there is another space given, that includes it (as, in the case mentioned, the bank of the river), since I can view the nearest space (the cabin) with respect to the latter as moved and the body itself as at rest. As thus it is absolutely impossible to determine of an empirically given space, it matters not how extended it may be, whether, with respect to a still greater space enclosing it, it be itself moved or not, it must be wholly the same for all experience, and for every consequence drawn from experience, whether I choose to regard a body as moved or at rest, and the space as moved in the opposite direction with an equal velocity. Once more: as absolute space is nothing for any possible experience, the conceptions are the same whether I say a body moves with respect to this given space, in this direction, with this velocity, or whether I conceive it as at rest, and ascribe all this [motion] to the space, but in an opposite direction. For every conception is wholly of the same kind as the latter, of whose distinction from the former no instance is possible, and only with reference to the connection we wish to give it in the understanding is it different.

We are, moreover, not in a position to postulate a fixed point, in any experience, in reference to which it could be defined what motion and rest mean absolutely; for everything given us in this way is material, and hence movable, and (as we know of no extreme boundary of possible experience in space) it may be really moved without our being able to perceive this motion. Of this motion of a body in empirical space I can assign one portion of the given velocity to the body, the other to the space, but in the opposite direction, and the whole possible experience as concerns the consequences of these two combined motions is wholly the same whether conceived of the body alone as moved with the whole velocity or (conceiving it) as at rest, and the space as moved with the same velocity in the opposite direction. I assume here all motions as rectilinear. For as concerns the non-rectilinear it is not in all respects the same, whether I am at liberty to regard the body as moved (e.g., the earth in its daily rotation), and the surrounding space (the starry heaven) as resting, or the latter as moved and the former as resting; but we shall treat of this more particularly in the sequel. Thus in Phoronomy, where I consider the motion of a body only in relation to the space (on the rest or motion of which it has no influence at all), it is quite undetermined and arbitrary whether any or all, or how much, of the velocity of the given motion I attribute to the one or to the other.

Farther on in mechanics where a moved body is to be considered in real relation to other bodies, in the space of its motion, this will not be any longer so entirely indifferent, as will be demonstrated in its proper place.

Observation.

In Phoronomy, as I can cognise the matter by no other property but that of movability, and can consider it itself therefore only as a point, the motion can only be viewed as description of a space, yet so that I do not, merely pay attention to the space described, as in geometry, but also to the time [involved] therein; in other words, to the velocity with which a point describes the space. Phoronomy is thus the pure doctrine of the quantity ( mathesis ) of motions. The definite conception of a quantity is the conception of the generation of the presentation of an object through the composition of the homogeneous. Now, as motion is nothing homogeneous, but again motion Phoronomy is a doctrine of the composition of the motions of the same point according to its direction and velocity i.e., the presentation of a single motion as one that comprises within it two or perhaps several motions in one, at the same time, in the same point, so far as they together constitute one, that is, are one with this motion, but not in so far as they produce the latter as causes produce their effects. In order to find the motion arising from the composition of several—as many as one likes—one has only, as with the production of all quantities, first to seek out those that are compounded under given conditions, of two; and thereupon combine this with a third, etc. In consequence the doctrine of the composition of all motions is reducible to that of two. But two motions of one and the same point that are present at the same point may be distinguished in a double manner, and as such be combined in a triple way therein. Firstly, they occur at the same time either in one and the same line, or in different lines; the latter are motions enclosing an angle. Those that occur in one and the same line are either contrary to one another in direction or maintain the same direction. As all these motions are contemplated as taking place alone, there results immediately from the relation of the lines, that is, of the spaces of motion described in equal time, the relation of velocity. Thus there are three cases:—1. As two motions (it matters not whether of equal or unequal velocities) combined in one body in the same direction, are to constitute a resultant compound motion; 2. As Two motions of the same point (of equal or unequal velocity), combined in contrary directions, are, through their composition, to constitute a third motion in the same line; 3. Two motions of a point, with equal or unequal velocities, but in different lines, enclosing an angle, are considered as compounded.

Demonstration.

First Case. —Two motions in the same line and direction arrive at the same time in one and the same point.

Let two velocities, AB and ab, be presented as contained in one velocity of the motion. Let these velocities be assumed, for the time, as equal, AB = ab; in this case I assert they cannot be presented at once in the same point, in one and the same space (whether absolute or relative). For, because the lines AB and ab, denoting the velocities, are properly spaces, passed over in equal times, the composition of these spaces AB and ab = BC, and, therefore, the line AC, as the sum of the spaces, cannot but express the sum of both velocities. But the parts AB and BC do not, individually, present the velocity = ab; for they are not passed over in the same time as ab. Thus, the double line AC, which is traversed in the same time as the line ab, does not represent the double velocity of the latter, as was required. Hence the composition of two velocities in one direction in the same space does not admit of being sensuously presented.

On the contrary, if the body A be presented as moved in absolute space with the velocity AB, and I give to the relative space, a velocity ab = AB in addition, in the contrary direction ba = CB; this is the same as though I distributed the latter velocity to the body in the direction AB (axiom 1). But the body moves itself, in this case, in the same time through the sum of the lines AB and BC = 2 ab, in which it would have traversed the line ab = AB only, and yet its velocity is conceived as the sum of the two equal velocities AB and ab, which is what was required.

Second Case. —Two motions in exactly contrary directions are united in one and the same point.

Let AB be one of these motions, and AC the other in the opposite direction, the velocity of which we assume here to be equal to that of the first; in this case the very idea of representing two such motions, at the same time, in one and the same space, and in one and the same point, in short, the case of such a composition of motions would itself be impossible, which is contrary to the assumption.

On the other hand, let the motion AB be conceived as in absolute space, and instead of the motion AC in the same absolute space, let the contrary motion CA of the relative space [be conceived] with the same velocity, which (according to axiom 1) is equal to the motion AC, and may thus be entirely substituted for it; in this case two exactly opposite and equal motions of the same point, at the same time, may be very well presented. Now, as the relative space is moved with the same velocity CA = AB in the same direction with the point A, this point, or the body, present therein, does not change its place in respect of the relative space; i.e., a body moved in two exactly contrary directions with equal velocity, rests, or generally expressed, its motion is equal to the difference of the velocities in the direction of the greater (which admits of being easily deduced from what has already been demonstrated).

Third Case. —Two motions of the same point are presented as combined according to directions that enclose an angle.

The two given motions are AB and AC, whose velocity and directions are expressed by these lines, but the angle, enclosed by the latter, by BAC (it matters not whether it be a right angle, as in this case, or any other angle). If these two motions are to occur, at the same time, in the directions AB and AC, and indeed in the same space, they would not be able to occur, at the same time, in both these lines AB and AC, but only in lines running parallel to these. It would have, therefore, to be assumed, that one of these motions effected a change in the other (namely, the deviation from the given course), although the directions remained the same on either side. But this is contrary to the assumption of the proposition, which indicates by the word composition, that both the given motions are contained in a third, and must therefore be one with this, and not that, by one changing the other, a third is produced.

On the other hand, let the motion AC be taken as proceeding in absolute space, but instead of the motion AB, the motion of the relative space in the opposite direction. Let the line AC be divided into three equal parts, AE, EF, FG. Now, while the body A in absolute space passes over the line AE, the relative space, and therewith the point E, passes over the space Ee = MA; while the body passes over the two parts together = AF, the relative space and therewith the point F, describes the line Ff = NA; while, finally, the body passes over the whole line AC, the relative space, and therewith the point C describes the line Cc = BA. All this is the same as though the body A had passed over in these three divisions of time, the lines Em, Fn and CD = AM, AN, AB, and in the whole time in which it passes over AC, had passed over the line CD = AB. It is therefore at the last moment in the point D, and in the whole time gradually in all points of the diagonal line AD, which expresses the direction as well as the velocity of the compound motion.

Observation 1.

Geometrical construction demands that one quantity should be identical with the other, or two quantities in composition, with a third, not that they should produce the third as causes, which would be mechanical construction. Complete similarity and equality, in so far as they can only be cognised in intuition, is congruity. All geometrical construction of complete identity rests on congruity. This congruity of two motions combined with a third (in short, the motu composito itself) can never take place, when the two former are presented in one and the same space, i.e. relative [space]. Hence all attempts to demonstrate the above proposition in its three cases, have always been mechanical solutions only, inasmuch, namely, as though moving causes by which a given motion was combined with another, were made to produce a third, the proofs that the former were the same as the latter, and as such, admitted of being presented in pure intuition à priori [were not given].

Observation 2.

When, for instance, a velocity AB is termed double, nothing else can be understood thereby, but that it consists of two simple and equal [velocities] AB and BC, (see Fig. 1). But if a double velocity be explained by saying that it is a motion by which a doubly great space is passed over in the same time, something is here assumed which is not necessarily implied, namely, that two equal velocities may be combined in the same way as two equal spaces, for it is not in itself obvious that a given velocity consists of smaller [velocities]; and in the same way that a rapidity consists of slownesses as a space does of smaller [spaces]. For the parts of the velocity are not outside one another, as the parts of the space; and if the former are to be considered as quantity, the conception of their quantity, as it is intensive, must be constructed in a different manner to that of the extensive quantity of space. But this construction is possible in no other way than by the mediate composition of two equal motions, one of which is that of the body, the other that of the relative space in the contrary direction, but which, for this reason, is completely identical with an equal motion of the body in the previous direction. For in the same direction two equal velocities would not admit of being compounded in one body, except through external moving causes; for instance, a ship carrying the body with one of these velocities, while another movable force, immovably bound up with the ship, impresses upon the body the second velocity, which is equal to the previous one. In this it must always be presupposed that the body maintains itself in free motion with the first velocity when the second enters; but this is a natural law of moving forces, which cannot come into consideration when the question is simply how the conception of velocity is constructed as a quantity; so much as to the addition of velocities to one another. But when the question is of the subtraction of one from the other, this latter is easily conceivable, if the possibility of a velocity, as quantity by addition, has once been admitted; yet this conception cannot be so easily constructed, for to this end two contrary motions must be combined in one body; and how is this to happen? Immediately, namely, in respect of the same resting space, it is impossible to conceive of two equal motions in contrary directions in the same body; but the idea of the impossibility of these two motions in one body is not the conception of its rest, but of the impossibility of the construction of this composition of contrary motions, which is nevertheless assumed in the proposition as possible. Now this construction is not otherwise possible, than by the combination of the motion of the body with the motion of the space as has been demonstrated. Finally, as concerns the composition of two motions, whose direction encloses an angle, they do not admit of being conceived in a body, in reference to one and the same space, if one of them be not affected by an external continuous inflowing force (for instance, a vessel bearing the body onward), while the other maintains itself unaltered, or generally [expressed]: one must have as a basis, moving forces, and the production of a third movement from two combined forces, but this, although the mechanical carrying out of that which contains a conception, is not its mathematical construction, which has only to render intuitable what the object is (as quantum), not, how it may be transformed by nature or art, by means of sundry implements and forces. The composition of motions, in order to determine their relation to others as quantity, must take place according to the rules of congruity, which is only possible, in all three cases, by means of the motion of the space that is congruous with one of the two given motions, whereby both are congruous with the compound [motion].

Observation 3.

Thus Phoronomy, not as pure doctrine of motion, but as pure doctrine of the quantity of motion, in which matter is conceived by no other quality but that of mere movability, contains nothing but this single proposition, carried out in the three cases adduced, of the composition of motion, and indeed of the possibility of rectilinear motion alone, not of curvilinear; for, because in the latter the motion is continuously changed in direction, a cause of this motion, which cannot be merely space, must be brought to bear. That only the single case in which the directions of the same enclose an angle, is usually understood by the designation compound motion, does some detriment to the principle of the division of a pure philosophical science generally, although not to physics: for, as concerns the latter, all the three cases treated in the above proposition admit of being adequately prsented in the third alone. For when the angle enclosing the two given motions is conceived as infinitely small, it contains the first [case]; but if it be conceived as only divided in an infinitely small degree from a single straight line, it contains the second case; so that, in the proposition alrady stated respecting composite motion, all three cases mentioned by us, are capable of being given as in a universal formula. But in this way one could not learn to comprehend the qualitative doctrine of motion in its parts à priori, which in many respects is also useful.

If any one cares to connect the three parts in question of the universal Phoronomic proposition with the scheme of the subdivision of all pure conceptions of the understanding, here, especially with that of the conception of quantity, he will observe: that, as the conception of a quantity always contains that of the composition of the homogeneous, the doctrine of the composition of motions is at the same time the pure doctrine of quantity therein; and indeed that in all three momenta furnished by space, the unity of line and direction, the plurality of directions in one and the same line, and finally the totality of directions as well as of lines, according to which the motion can take place, it contains the determination of all possible motion as quantum, although its quantity (in a movable point) consists merely in velocity. This observation only has its uses in transcendental philosophy.

Observation.

This is the dynamical explanation of the conception of matter. It presupposes the Phoronomic, but adds thereto a property that is related as cause to an effect, namely, the capacity of resisting a motion within a certain space. This could not come into consideration in the foregoing science, even when we had to do with the motions of one and the same point in opposite directions. This filling of space keeps a certain space free from the intrusion of any other movable when the motion of the latter is directed to any place within this space. On what the resistance of matter on all sides rests, and what it is, now remains to be investigated. But it may be already seen from the above explanation, that matter is not here considered as resisting when it is driven from its place, and thus as itself moved (this case will hereafter come into consideration as mechanical resistance), but only when the mere space of its own extension is to be diminished. The expression is used to occupy space, namely, to be immediately present in all its points, in order to indicate thereby the extension of a thing in space. But inasmuch as it is not defined in this conception, what effect, or whether any effect at all, arises from this presence, whether in resisting others that are attempting to press into it, or whether it signifies merely a space without matter, in so far as it is a sumtotal of several spaces, just as one may say of every geometrical figure, “it occupies a space” (it is extended); or even whether there be something in space necessitating another movable to penetrate deeper into the same (attracting others); because, I say, by the conception of the occupying of a space, all this is undetermined; so, to fill a space is a closer definition of the conception to occupy a space.

Demonstration.

The penetration into a space (in the moment of commencement this is called the endeavour to penetrate) is a motion. The resistance to motion is the cause of its diminution, and also its change into rest. Now nothing can be connected with any motion, as lessening or destroying it but another motion of the same movable in the opposite direction (phoronomic proposition). Thus the resistance offered by a matter in the space which it fills, to all impression of another [matter], is a cause of the motion of the latter in the opposite direction; but the cause of a motion is called moving force. Thus matter fills its space by moving force and not by its mere existence.

Observation.

Lambert and others called the property of matter, by which it fills a space, solidity (a rather ambiguous expression), and maintained that we must assume it in everything which exists (substance), at least in the outer world of sense. According to their notions, the presence of something real in space, must carry with it this resistance by its very conception, in other words according to the principle of contradiction; and must exclude the coexistence of anything else, in the space of its presence. But the principle of contradiction does not preclude any matter from advancing, in order to penetrate into a space in which another [matter] exists. Only when I attribute to that which occupies a space, a power of repelling everything externally movable which approaches it, do I understand how it involves a contradiction, that in the space which a thing occupies, another [thing] of the same kind should penetrate. Here the mathematician has assumed something as a first datum of the construction of the conception of a matter, which itself does not admit of being further constructed. Now he can begin his construction of a conception from any datum he pleases, without committing himself again to the further explanation of this datum; but he is nevertheless not thereby permitted to explain the former as something wholly incapable of any mathematical construction, in order by this means to prevent a return to the first principles of natural science.

Note.

These are the only two moving forces of matter admiting of being conceived. For all motion which one matter can impress upon another, as in this respect each of them is only considered as a point, must always be regarded as distributed in the straight line between two points. But in this straight line only two kinds of motion are possible, one, by which the above points recede from one another, and a second by which they approach one another. But the force which is the cause of the first motion is called repulsive force, and that of the second attractive force. Thus, only these two kinds of forces, as such, to which all the forces of motion in material nature must be reduced, are capable of being conceived.

Demonstration.

Matter fills a space only by moving force (proposition 1), this being such as to resist the impression, that is, the approach of others. Now this is a repulsive force (explanation II.). Thus matter fills its space, and indeed all the parts thereof, by repulsive forces only, because otherwise a part of its space would not be filled (against the assumption), but would only be enclosed. But the force of an extended by virtue of the repulsion of all its parts is a force of extension (expansive). Thus matter fills its space by its own force of extension; which was the first point. Beyond every given force a greater must be conceived, for that beyond which there is no greater possible would be one, whereby, in a finite time, an infinite space would be passed over (which is impossible). Further, beyond every given moving force a smaller must be able to be conceived (for the smallest would be that, by the infinite addition of which to itself, throughout any given time, no finite velocity could be generated, but this signifies the lack of all moving force). Thus below every given degree of a moving force, a smaller must always be able to be given; which is the second [ point ]. The force of extension, therefore, whereby all matter fills its space, has its degree, which is never the greatest or smallest; but beyond which, greater as well as smaller, may be found to infinity.

Note 1.

The expansive force of a matter is termed elasticity. Now as the former is the basis on which the filling of space, as an essential property of all matter, rests, this elasticity must be termed original; seeing that it cannot be derived from any other property of matter. All matter is accordingly originally elastic.

Note 2.

Because beyond every extending force a greater moving force can be found, which might work against it, and would thus diminish the space it is seeking to extend; in which case the latter would be termed a compressive force; so for every matter a compressive force must be able to be found, capable of driving it from every space it fills into a narrower space.

Observation.

When, in the sucker of an air-pump that is filled with air, the piston is driven nearer the bottom, the air-matter is compressed. Now if this compression could be carried so far that the piston completely touched the bottom (without the least amount of air escaping), the air-matter would be penetrated; for the matters, between which it is, leaving no superfluous room for it, it would exist between the bottom and the piston, without occupying a space. This penetrability of matter by external compressive forces, if one were willing to assume, or even conceive, such, would be termed mechanical. I have reasons for distinguishing by such a limitation, this penetrability of matter from another [kind], the conception of which is perhaps just as impossible as that of the present, and of which I may hereafter have occasion to make some mention.

Proposition 3.

Matter can be compressed to infinity, but it can never be penetrated, by a matter, it does not signify how great its pressing force.

Demonstration.

An original force, by which a matter seeks to extend itself on all sides over a given space occupied by it, must, enclosed in a smaller space, be greater, and compressed into an infinitely small space, be infinite. Now, for any given extensive force of matter, a greater compressive force may be found that compels it into a smaller space, and so on to infinity; which was the first [point]. But for the penetration of a matter, a compression into an infinitely small space, and therefore an infinitely compressive force, is required, which is impossible. Hence, a matter cannot be penetrated by the compression of any other [matter]; which is the second [point].

Observation.

I have, at the commencement of this demonstration, assumed that an extending force, the more it is narrowed, must operate so much the more strongly in the opposite [direction]. Now this would not apply to all kinds of elastic forces, [including those] that are merely derivative: but with matter possessing essential elasticity, in so far as it is matter in general, filling a space, it may be postulated. For expansive force exercised from all points towards all sides, constitutes its very conception. But the same quantum of expanding forces, brought into a narrower space, must, in every point of the latter, repel so much the more strongly, in inverse proportion to the smallness of the space in which a given quantum of force diffuses its activity.

Observation 1.

According to the mere mathematical conception of impenetrability (which assumes no moving force as originally inherent in the matter), no matter is capable of compression, except in so far as it contains within itself empty spaces. Matter, therefore, as matter, resists all impression unconditionally and by absolute necessity. But according to our explanation of this property, impenetrability rests on a physical basis; for the extensive force renders it primarily possible, as an extended that fills its space. But as this force has a degree that overpowers, and hence diminishes the space of extension, that is, can be impressed upon the same up to a certain degree, by a given compressive force, but only in such wise that the entire penetration, inasmuch as it would require an endless compressive force, is impossible; [therefore] the filling of space must be regarded only as relative impenetrability.

Observation 2.

Absolute impenetrability is, indeed, neither more nor less than a qualitas occulta For we ask the cause, why matters in their motion cannot penetrate one another; and receive the answer: because they are impenetrable. The appeal to repulsive force is free from this objection. For although this likewise cannot be explained further, according to its possibility, and hence must be admitted as a fundamental force, it nevertheless gives a conception of an active cause and its laws, in accordance with which the effect, namely, the resistance in the filled space, may be estimated according to its degrees.

Observation.

The conception of a substance signifies the ultimate subject of existence, namely, that which does not itself belong, as mere predicate to the existence of another. Now matter is the subject of all that, in space, which can be counted [as belonging] to the existence of things; for outside it, no subject would be able to be conceived, but space itself; and this is not a conception containing anything existent, but merely the necessary conditions of the external relation of possible objects to our sense. Matter then, as the movable in space, is substance therein. But just in the same way are all its parts substances, in so far as one can say of them that they are subjects, and not merely predicates of other matters; and hence must again themselves be termed matter. But they are themselves subjects, if they are something movable existing in space, and hence not in combination with other adjacent parts. The independent motion of matter, then, or any of its parts, is a demonstration at once, that this movable, and every movable part of it, is substance.

Demonstration.

Matter is impenetrable by its own original force of extension (proposition 3); but this is only the result of the repulsive forces of each point in a space filled with matter. Now the space that is filled by matter is mathematically divisible to infinity; that is, its parts can be distinguished to infinity, although they cannot be moved, and consequently cannot be separated (according to demonstrations of geometry). But in a space filled with matter, every part contains the same repulsive force, to counteract all other forces, on all sides; in other words, to drive them back, and in the same way to be driven back by them, that is, to be moved to a distance from them. Hence, every part of a space filled with matter is, movable in itself, and consequently separable from those remaining, as material substance, by physical division. So far, then, as the mathematical divisibility of space filled by a matter reaches, thus far does the possibility of the physical division of the substance that fills it, reach. But the mathematical division extends to infinity, and consequently also the physical; that is, all matter, is divisible to infinity, and indeed to parts, of which each is itself again material substance.

Observation 1.

By the demonstration of the infinite divisibility of space, that of matter has not, by a long way, been proved, if it has not previously been established, that in every part of space material substance exists, that is, that parts in themselves movable are to be met with. For if a monadologist wished to assume that matter consisted of physical points, each of which (for this reason) had no movable parts, but nevertheless, filled a space by mere repulsive force, he would still be able to admit that this space, although not the substance acting in it (in other words, the sphere of the latter’s activity, though not the acting movable subject itself), could be divided by the division of its spaces. He would thus compound matter of physical by indivisible parts, and yet allow it to occupy space in a dynamical manner.

But by the above demonstration, the monadologist is entirely deprived of this resort. For, thereby it is clear, that in a filled space there can be no point that does not itself resist repulsion on all sides in the same way as it is repelled; in other words, as a reacting subject, in itself movable, existing outside every other repulsive point; and hence that the hypothesis of a point filling a space by its mere driving force, and not by means of other equal repulsive forces, is impossible. In order to make this, and thereby also the demonstration of the previous proposition apparent, one must assume that A is the place of a monad in space, that ab is the diameter of the sphere of its repulsive force, and therefore that aA is its semi-diameter; so between a, where the impression of an external monad in space, occupying the sphere in question, is understood, and the central point of the latter [viz., the sphere], A, a point c is possible to be indicated (in accordance with the infinite divisibility of space). Now, if A resist that which seeks to impress itself on a, c must resist both the points A and a. For if this were not so, they would approach one another with impunity; consequently A and a would meet in the point c, i.e. the space would be penetrated. Something must thus exist in c that resists the impression of A and a, and thus repels the monad A as much as it is repelled by it. As now, repulsion is a movement, c is something movable in space; in other words, matter, and the space between A and a, could not be filled by the sphere of the activity of a single monad, neither could the space between c and A, and so on to infinity.

When mathematicians conceive the repulsive forces of the parts of elastic matters in their greater or lesser compression, as increasing or diminishing in a certain proportion to their distances from one another (for instance, that the smallest parts of the air repel each other in inverse proportion to their distances from one another, because their elasticity stands in inverse proportion to the spaces in which they are compressed), one would wholly mistake their meaning and misapply their language were one to attribute to the conception in the object itself, what [nevertheless] necessarily belongs to the process of the construction of a conception. For, according to the above, all contact can be conceived as an infinitely small distance, which, moreover, must necessarily happen in distance where a larger or smaller space is to be conceived as entirely filled by the same quantity of matter, that is, by an identical quantum of repulsive forces. By an infinitely divisible [thing], therefore, no real distance of parts, which, with all extension of the space of the whole, always constitute a continuum, may be assumed, although the possibility of this extension can only be made comprehensible under the idea of an infinitely small distance.

Observation 2.

Mathematics can indeed, in its internal employment, be quite indifferent to the chicane of a mistaken metaphysics, and rest in the certain possession of its evident assertions of the infinite divisibility of space, no matter what objections a sophistry, clinging to mere conceptions, may throw in its way; but in the application of its propositions, which apply to space, to substance, which fills it, it must rely on a test according to mere conceptions; in other words, on metaphysics. The above proposition is itself a proof of this. For it does not follow necessarily that matter is physically divisible to infinity, although it is so in a mathematical connection, every part of space being again a space, and hence always including within itself parts external to one another; but this cannot prove that in every possible part of this filled space, there is substance, which, consequently, separated from all the rest, exists as in itself, movable; something has been wanting then hitherto, to the mathematical demonstration, without which it can have no certain application to Natural Science, and this defect has been obviated in the proposition above given. But as concerns the remaining attacks of metaphysics on the at present physical proposition, of the infinite divisibility of matter, the mathematician must entirely resign himself to the philosopher, who, apart from this, through these objections, betakes himself into a labyrinth, out of which it is difficult for him to find his way, even in questions immediately concerning him, and hence has enough to do on his own account, without the mathematician mixing himself up in the business. If, namely, matter be infinitely divisible, then (concludes the dogmatic metaphysician), it consists of an infinite number of parts; for a whole must originally contain within itself all the parts into which it can be divided, in their entirety. But the latter proposition is also indubitably certain of every whole as a thing in itself, and, therefore, although one cannot admit matter, or even space, to consist of infinitely many parts (inasmuch as it is a contradiction to think of an infinite number, the conception of which itself implies that it can never be conceived as fully ended), one must resolve either to defy the geometrician by saying space is not infinitely divisible, or to irritate the metaphysician [by saying], space is no property of a thing in itself, and hence, matter is no thing in itself, but the mere phenomenon of our external sense generally, just as space is its essential form.

The philosopher now finds himself in a strait between the horns of a dangerous dilemma. To deny the first proposition, that space is divisible to infinity, is a vain undertaking, for mathematics does not admit of being reasoned away; but yet to regard matter as a thing in itself, in other words, space as property of the thing in itself, and to deny the above proposition, is one and the same thing. He sees himself thus necessitated to depart from this assertion, however common and suited to the common understanding it may be; but of course only under the condition, that in the event of his reducing matter and space to the phenomenon (hence the latter [viz. space] to the form of our external sensuous intuition, and so [constituting] both, not things in themselves, but only subjective modes of the presentation to us, of objects in themselves unknown), he should be helped out of the difficulty as to the infinite divisibility of matter, while it yet does not consist of infinitely many parts. This latter easily admits of being conceived by the Reason, although impossible to construct and render intuitable. For of that which is only real by its being given in presentation, there is not more given than is met with in the presentation, that is, so far as the progressus of presentations reaches. Thus we can only say of phenomena, the division of which goes on to infinity, that there exist so many of the parts of the phenomenon, as we give of them, that is, as far as we can ever subdivide. For the parts, as belonging to the existence of a phenomenon exist only in thought, namely, in their division itself. Now though the division proceeds to infinity, it is never given as infinite, and hence it does not follow that the divisible contains an infinite number of parts in itself and outside our presentation merely because its division is infinite. For it is not the thing, but only its presentation, whose division could be continued to infinity, and in the object that is unknown in itself, which has also a cause, and yet can be never completed and consequently fully given, it proves no real infinite number, for this would be an express contradiction. A great man who has perhaps contributed more than any one else to maintain the reputation of mathematics in Germany, has more than once turned aside metaphysical claims to upset the propositions of geometry relative to the infinite divisibility of space with the well-grounded observation, that space only belongs to the phenomenon of external things; but he has not been understood. The proposition was taken as though he meant: space appears to us, otherwise it is a thing or relation of things in themselves, but the mathematician considers it only as it appears. Instead of this he ought to have been understood [as meaning] that space is no quality appertaining to anything outside our senses, but only to the subjective form of our sensibility, under which objects of our external sense, unknown to us as to their construction in themselves, appear to us, this appearance being termed matter. By the foregoing misunderstanding, space was always conceived as a quality [existing] independently, outside our faculty of presentation, but which the mathematician only thought of according to common conceptions, that is, confusedly (for so appearance [phenomenon] is commonly explained); it ascribed the mathematical proposition of the infinite divisibility of matter, a proposition presupposing the highest clearness in the conception of space, to a confused presentation of space, which the geometrician laid at his foundation. In this way, it remained open to the metaphysician to compound space of points, and matter of simple parts, and thus in his opinion to bring clearness into the conception. The ground of the confusion lies in a misunderstood monadology, which does not belong to the explanation of natural phenomena, but is a platonic conception of the world, carried out by Leibnitz. This is correct in itself, in so far as it [the world] is regarded, not as object of sense, but as thing in itself; but is nevertheless a mere object of the understanding, though it lies at the foundation of the phenomena of sense. The composite of things in themselves must consist in the simple; for the parts must here be given before all composition. But the composite in the phenomenon consists not of the simple, because in the phenomenon, which can never be given otherwise than as composite (extended), the parts can only be given through division, and thus not before the composite, but in it. Hence Leibnitz’s opinion, so far as I understand, [did not consist] in explaining space by the arrangement of simple entities side by side, but rather in [regarding it] as corresponding to a merely intelligible, for us unknown, world by its side, and maintained nothing more than what has elsewhere been shown, namely, that space, together with matter of which it is the form, comprises, not the world of things in themselves, but only the phenomenon of this [world], and is itself only the form of our sensuous intuition.