§ 1.: Of the Source of Metaphysics.

In presenting a branch of knowledge as science, it is necessary to be able to define with precision its distinguishing characteristic, that which it possesses in common with no other branch, and which is therefore special to itself; when this is not the case the boundaries of all sciences run into one another, and no one of them can be thoroughly treated of, according to its own nature.

Now this speciality may consist in the distinction of its object, of its sources of cognition, of its mode of cognition, or lastly, of several if not all these points taken together, on which the idea of a possible science and of its territory primarily rests.

Firstly, as regards the sources of metaphysical knowledge, the very conception of the latter shows that these cannot be empirical. Its principles (under which not merely its axioms, but also its fundamental conceptions are included) must consequently never be derived from experience; since it is not physical but metaphysical knowledge, i.e., knowledge beyond experience, that is wanted. Thus neither external experience, the source of physical science proper, nor internal experience, the groundwork of empirical psychology, will suffice for its foundation. It consists, then, in knowledge à priori, that is, knowledge derived from pure understanding and pure reason.

But in this, there is nothing to distinguish it from pure mathematics; it must be defined, therefore, as pure philosophical knowledge; respecting the meaning of which expression, I must refer the reader to the Critique of Pure Reason, (Bohn’s Ed. p. 435,) where the distinction between these two modes of the Reason’s use are clearly and exhaustively expounded. So much as to the sources of metaphysical knowledge.

§ 2.: Of the Mode of Cognition1 that can alone be termed Metaphysical.

§ 3.: Observation on the universal division of Judgments into Analytic and Synthetic.

This division is in view of the Critique of human understanding indispensable, and deserves therefore to be classic in this department; though I am not aware of any other in which it has any important use. And here I also find the cause why dogmatic philosophers who looked for the sources of metaphysical judgments in metaphysics itself (rather than outside of it, in the laws of the pure Reason in general), have always neglected this division, that seems so naturally to offer itself, and like the celebrated Wolff, or the acute Baumgarten, who followed in his steps, have sought the proof of the principle of sufficient reason, which is obviously synthetic, in that of contradiction. On the other hand, I can trace already in “Locke’s Essays on the Human Understanding” a notion of this division. For in the third chapter of the fourth book, (Chap. III. § 9 et seq.,) after he has spoken of the connection of different presentations in judgments, and of their sources, one of which he places in identity or contradiction (analytic judgments), and the other in the existence of presentations in a subject (synthetic judgments), he confesses, § 10, that our knowledge (à priori) of the last is very limited, amounting almost to nothing. But there is so little that is definite and reduced to rule in what he says respecting this kind of knowledge, that one cannot wonder that nobody, strange to say, not excepting Hume, was induced thereby to institute investigations into the Edition: current; Page: [17] class of propositions in question. For universal yet definite principles like these, are not easily learnt from other men, to whom they have been only dimly discernible. One must, first of all, have come upon them through one’s own reflection, and one will then find them elsewhere, in places where otherwise they would certainly not have been discovered; since not even the authors knew that such an idea lay at the foundation of their own remarks. Those who do not think for themselves, possess notwithstanding the sharpness of insight to detect everything after it has already been shown them, in what has previously been said, where no one could before see it.

§ 4.

Were metaphysics actually present as a science, one might say: Here is metaphysics, you only require to learn it, and it will convince you permanently and irresistibly of its truth. In that case the present question would be unnecessary, and there would only remain one which would more concern a testing of our acuteness, than a proof of the existence of the thing itself; namely, the question, How is it possible, and how is the Reason to set about attaining it? Unfortunately, in this case, human Reason is not in such a happy position. There is no single book that can be shown, like for instance Euclid, of which it can be said: This is metaphysics, herein is to be found the chief end of the science, the knowledge of a Supreme Being and of a future world, demonstrated upon principles of the pure Reason. It is possible, doubtless, to bring forward many propositions that are apodictically certain, and that have never been contested; but these are in their entirety analytic, and concern more the materials and the elements of construction, than the extension of knowledge, which is our special object in the present case. But even when synthetic propositions are produced (such as the principle of sufficient Edition: current; Page: [18] Reason), which though they have never been proved from mere Reason, that is, à priori, as they ought to have been, are willingly admitted; even then, whenever it is attempted to make use of them for the main purpose, one is landed in such unstable and doubtful assertions, that it has always happened that one system of metaphysics has contradicted another, either in respect of the assertions themselves or their proofs, and has thus destroyed all claim to a lasting recognition. The very attempts made to establish the science have without doubt been the primary cause of the scepticism that so early arose, a mode of thought in which the Reason treats itself with such violence, that it would never have arisen but from the latter’s utter despair of satisfying its chief aspirations. For long before man began methodically to question Nature, he interrogated his own isolated Reason, already practised, in a measure by common experience; because Reason is always present, while the laws of Nature generally require to be laboriously sought out. And so metaphysics floated to the surface like foam, and like foam, too, no sooner was it gathered up than it dissolved, while another mass of it appeared upon the scene which some were always found eager to grasp; while others, instead of seeking to penetrate the cause of the phenomenon in question, thought themselves wise in laughing at the futile exertions of the former.

The essential feature distinguishing pure mathematical knowledge from all other knowledge à priori, is that it does not proceed from conceptions themselves, but always through the construction of conceptions. (Critique, p. 435.) Since, therefore, in its propositions it must pass out of the conception to that containing the corresponding intuition, these can and ought never to arise from the dissection of conceptions, that is, analytically; in other words, they are, in their entirety, synthetic.

I cannot refrain from remarking on the disadvantage resulting to philosophy from a neglect of this simple and apparently insignificant observation. Hume, indeed, feeling it a task worthy of a philosopher, cast his eye over the whole field of pure knowledge à priori in which the human understanding claims such extensive possession. He, however, inconsiderately severed from it an Edition: current; Page: [19] entire, and indeed the most important, province, namely, that of pure mathematics, under the impression that its nature, and, so to speak, its constitution, rested on totally different principles, that is, solely on the principle of contradiction; and although he did not make such a formal and universal division of propositions as is here done by me, or under the same name, yet it was as good as saying, pure mathematics contains simply analytic judgments, but metaphysics, synthetic judgments à priori. Now in this he made a great mistake, and this mistake had decidedly injurious consequences on his whole conception. For if he had not made it, he would have extended his question respecting the origin of our synthetic judgments far beyond his metaphysical conception of causality, and comprehended therein the possibility of mathematics à priori; for he must have regarded this as equally synthetic. But in the latter case he could, under no circumstances, have based his metaphysical propositions on mere experience, as he would then have been obliged to have subordinated the axioms of pure mathematics themselves to experience, a proceeding for which he was much too penetrating.

The good company into which metaphysics would then have been brought must have ensured it against contemptuous treatment; for the strokes aimed at the latter must have also hit the former, and this neither was nor could have been his intention. The result must have been to lead the acute man to considerations similar to those with which we are now occupied, but which must have gained infinitely by his inimitable style.

Essentially metaphysical judgments are, in their entirety, synthetic. We must distinguish between judgments belonging to metaphysics from metaphysical judgments proper. Among the former are comprised many that are analytic, but they only furnish the means for metaphysical judgments, these forming the entire purpose of the science, and being all synthetic. For when conceptions belong to metaphysics, as, for instance, that of substance, the judgments arising from their dissection belong also to metaphysics; e.g., substance is that which only exists as subject, &c., and many more similar analytic judgments, by Edition: current; Page: [20] means of which an en leavour is made to approach the definition of the conception. Since, however, the analysis of a pure conception of the understanding (such as those metaphysics contains) cannot proceed differently from the analysis of any other conception (even an empirical one) not belonging to metaphysics (e.g., air is an elastic fluid, the elasticity of which is not destroyed by any known degree of cold), it follows that the conception but not the analytic judgment, is properly metaphysical. The science in question has something special and peculiar in the production of its cognitions à priori, which must be distinguished from what it has in common with all other cognitions of the understanding; so, for instance, the proposition, “all that is substance in things is permanent,” is a synthetic and properly metaphysical judgment.

When the conceptions à priori constituting the materials of metaphysics have been previously collected according to fixed principles, the dissection of these conceptions is of great value. They can be then presented as a special department (as it were a philosophia definitiva), containing solely analytic propositions relating to metaphysics, though quite distinct from the synthetic, which constitute metaphysics itself. For, indeed, these analyses have nowhere any important use, except in metaphysics, that is, in reference to the synthetic propositions, to be generated from these dissected conceptions.

The conclusion drawn in this section is then, that metaphysics is properly concerned with synthetic propositions à priori, and that these alone constitute its purpose, but that, in addition to this, it requires frequent dissections of its conceptions, or analytic judgments, the procedure in this respect being only the same as in other departments of knowledge, where conceptions are sought to be made plain by analysis. But the generation of knowledge à priori, as much in intuition as in conceptions, in fine, synthetic propositions à priori in philosophical cognitions, make up the essential content of metaphysics.

Wearied, then, of the dogmatism that teaches us nothing, as well as of the scepticism that promises us nothing, not even the rest of a permissible ignorance, led on by the importance of the knowledge we need, rendered mistrustful Edition: current; Page: [21] by a long experience, of all we believe ourselves to possess, or that offers itself in the name of pure Reason, there only remains one critical question, the answer to which must regulate our future procedure—Is metaphysics possible at all? But this question must not be answered by sceptical objections to particular assertions of any actual system of metaphysics (for we do not admit any at present), but from the, as yet, only problematical conception of such a science.

In the ‘Critique of Pure Reason,’ I went synthetically to work in respect of this question, in instituting researches into the pure Reason itself, and in this source endeavoured to determine the elements, as well as the laws of its pure use, according to principles. The task is difficult, and demands a resolute reader, gradually to think out a system, having no datum other than the Reason itself, and which, therefore, without supporting itself on any fact, seeks to unfold knowledge from its original germs. Prolegomena should, on the contrary, be preparatory exercises, designed more to show what has to be done, to realise a science as far as is possible, than to expound one. They must, therefore, rely on something known as trustworthy, from which we may with confidence proceed, and ascend to its sources, as yet unknown to us, and the discovery of which will not only explain what we already knew, but at the same time exhibit to us a range of many cognitions, all arising from these same sources. The methodical procedure of Prolegomena, especially of those destined to prepare a future system of metaphysics, will therefore be analytic.

Now it fortunately happens that, although we cannot accept metaphysics as a real science, we may assert with confidence that certain pure synthetic cognitions are really given à priori, namely, pure mathematics and pure natural science, for both contain propositions, partly apodictically certain through mere Reason, and partly recognised by universal consent as coming from experience, and yet as completely independent of it.

We have, then, at least some uncontested, synthetic knowledge à priori, and do not require to ask whether this is possible, since it is actual, but only—How it is possible, in order to be able to deduce from the principle, rendering possible what is already given, the possibility of all the rest.

§ 5.

We have already seen the important distinction between analytic and synthetic judgments. The possibility of analytic propositions can be very easily conceived, for they are based simply on the principle of contradiction. The possibility of synthetic propositions à posteriori, i.e., of such as are derived from experience, requires no particular explanation, for experience is nothing more than a continual adding together (synthesis) of perceptions. There remains, then, only synthetic propositions à priori, the possibility of which has yet to be sought for, or examined, because it must rest on other principles than that of contradiction.

But we do not require to search out the possibility of such propositions, that is, to ask whether they are possible, for there are enough of them, actually given, and with unquestionable certainty; and as the method we are here following is analytic, we shall assume at the outset that such synthetic but pure knowledge from the Reason, is real; but thereupon we must investigate the ground of this possibility and proceed to ask—How is this knowledge possible? in order that, from the principles of its possibility, we may be in a position to determine the conditions, the scope, and limits of its use. The proper problem, on which everything turns, when expressed with scholastic precision, will accordingly stand thus—How are synthetic propositions à priori possible?

In the above, for the sake of popularity, I have expressed the question somewhat differently, namely, as an inquiry after knowledge from pure Reason, which I could do on this occasion without detriment to the desired insight. For as we are here simply concerned with metaphysics and its sources, I hope, after the above remarks, readers will constantly bear in mind that, when we here speak of knowledge from pure Reason, we invariably refer to synthetic and never to analytic knowledge.1 Edition: current; Page: [23] Upon the solution of this problem, the standing or falling of metaphysics, in other words, its very existence, entirely lepends. Let any one lay down assertions, however plausible, with regard to it, pile up conclusions upon conclusions to the point of overwhelming, if he has not been able first to answer satisfactorily the above question, I have a right to say: It is all vain, baseless philosophy, and false wisdom. You speak through pure Reason, and claim to create à priori cognitions, inasmuch as you pretend not merely to dissect given conceptions but new connections which do not rest on the principle of contradiction, and which you think you conceive quite independently of all experience. How do you arrive at them, and how will you justify yourself in such pretensions? To appeal to the concurrence of the general common sense of mankind you cannot be allowed, for that is a witness whose reputation rests only on vulgar report.

• • Quodcunque ostendis mihi sic, incredulus odi.
• • (All that thou thus showest me, I disbelieve and hate.)
  • Horat.

But indispensable as is the answer to this question, it is at the same time no less difficult, and although the chief cause why men have not long ago endeavoured to answer it, lies in the fact of its never having occurred to them that anything of the kind could be asked; there is a second cause, in that the satisfactory answer to this one question demands a more persistent, a deeper Edition: current; Page: [24] and more laborious reflection than the most diffuse work, on metaphysics, the first appearance of which has given promise of immortal fame to its author. And every thoughtful reader, on attentively considering the requirement of this problem, frightened at the outset by its difficulty, would regard it as insoluble; and indeed, were it not for the actual existence of such pure synthetic cognitions à priori, as altogether impossible. This happened in the case of David Hume, although he did not place the problem before him in such generality by far as is here done, and as must be done if the answer is to be decisive for the whole of metaphysics. For how is it possible, said the acute man, that when a conception is given me, I can pass out of it, and connect it with another, which is not contained in the former, and indeed in such a manner as if it necessarily belonged to it? Only experience can present us with such connections (this he concluded from the difficulty which he mistook for an impossibility), and all this imagined necessity, or, what is the same thing, knowledge assumed to be à priori, is nothing but a long habit of finding something true, and thence of holding the subjective necessity for objective. If the reader complains of the difficulty and trouble I shall give him in the solution of this problem, let him only set about the attempt to solve it in an easier way. He will then perhaps feel obliged to one who has undertaken for him the labour of such deep research, and rather show some surprise at the facility with which the solution has been able to be given, when the nature of the subject is taken into account. It has cost years of trouble to solve this problem in its whole universality (in the sense in which mathematicians use this word, namely, as sufficient for all cases), and to be able finally to present it in analytic form, such as the reader will here find.

All metaphysicians are therefore solemnly and lawfully suspended from their occupations, till they shall have adequately answered the question—How are synthetic cognitions à priori possible? for in their answer alone consists the credentials they must produce, if they have aught to bring us in the name of pure Reason; in default of this, they can expect nothing else, than to be rejected, without Edition: current; Page: [25] any further inquiry as to their productions, by sensible people who have been so often deceived.

If, on the other hand, they carry on their business not as a science, but as an art of wholesome persuasion, suitable to the general common sense of mankind, this calling cannot in fairness be denied them. In that case they will only use the modest language of a rational belief; they will admit that it is not allowed them even to conjecture, much less to know, anything, respecting that which lies beyond the boundaries of all possible experience, but merely to assume (not indeed for speculative use, for this they must renounce, but for purely practical purposes) what is possible and even indispensable for the direction of the understanding and will, in life. In this way alone can they possibly carry the reputation of wise and useful men, and they will do so the more in proportion as they renounce that of metaphysicians. For the object of the latter is to be speculative philosophers, and inasmuch as when we are concerned with judgments à priori, bare probabilities are not to be relied on (for what on its assumption is known à priori, is thereby announced as necessary), it cannot be allowed them to play with conjectures, but their assertions must be either science, or they are nothing at all.

It may be said that the whole transcendental philosophy which necessarily precedes all metaphysics is itself nothing more than the full solution in systematic order and completeness of the question here propounded, and that therefore as yet we have no transcendental philosophy. For what bears its name is properly a part of metaphysics, but the former science must first constitute the possibility of the latter, and must therefore precede all metaphysics. Considering, then, that a complete and in itself entirely new science, and one respecting which no aid is to be derived from other sciences, is necessary before a single question can be adequately answered, it is not to be wondered at if the solution of the same is attended with trouble and difficulty, and even perhaps with some degree of obscurity.

As we now proceed to this solution according to analytic method, in which we presuppose that such cognitions from pure Reason are real, we can only call to our aid two sciences of theoretic knowledge (with which alone we are Edition: current; Page: [26] here concerned), namely, pure mathematics and pure natural science, for only these can present to us objects in intuition, and therefore (if a cognition à priori should occur in them) show their truth or agreement with the object in concreto, i.e., their reality; from which to the ground of their possibility we can proceed on the analytic road. This facilitates the matter very much, as the universal considerations are not merely applied to facts but even start from them, rather than as in synthetic procedure, being obliged to be derived, wholly in abstracto, from conceptions.

But from these real and at the same time well-grounded pure cognitions à priori, to rise to a possible one such as we are seeking, namely, to metaphysics as a science, we must needs embrace under our main question that which occasions it, to wit, the naturally given, though as regards its truth not unsuspicious, knowledge à priori lying at its foundation, and the working out of which, without any critical examination of its possibility, is now usually called metaphysics—in a word, the natural tendency to such a science; and thus the transcendental main question, divided into four other questions, will be answered step by step:—

• 1. How is pure mathematics possible?
• 2. How is pure natural science possible?
• 3. How is metaphysics in general possible?
• 4. How is metaphysics as a science possible?

It will be seen, that although the solution of these problems is chiefly meant to illustrate the essential contents of the Critique, it has nevertheless something special, which is of itself worthy of attention, namely, to seek the sources of given sciences in the Reason, in order to investigate and measure this, their faculty of knowing something à priori, by means of the act itself. In this way the particular science itself must gain, if not in respect of its content, at least as regards its right employment, and while it throws light on the higher question of its common origin, at the same time give occasion to better elucidating its own nature.

§ 6.

Here is a great and established branch of knowledge, already of remarkable compass, and promising unbounded extension in the future, carrying with it a thorough apodictic certainty, i.e., absolute necessity, and thus resting on no empirical grounds, but being a pure product of the Reason, besides thoroughly synthetic. “How is it possible for the human Reason to bring about such a branch of knowledge entirely à priori?” Does not this capacity, as it does not and cannot stand on experience, presuppose some ground of knowledge à priori, lying deep-hidden, but which might reveal itself through these its effects, if their first beginnings were only diligently searched for?

§ 7.

But we find that all mathematical knowledge has this speciality, that it must present its conception previously in intuition, and indeed à priori, that is, in an intuition that is not empirical but pure, without which means it cannot make a single step; its judgments therefore are always intuitive, whereas philosophy must be satisfied with discursive judgments out of mere conceptions; for though it can explain its apodictic doctrines by intuition, these can never be derived from such a source. This observation respecting the nature of mathematics, itself furnishes us with a guide as to the first and foremost condition of its possibility, namely, that some pure intuition must be at its foundation, wherein it can present all its conceptions in concreto and à priori at the same time, or as it is termed, construct them. If we can find out this pure intuition together with its possibility, it will be readily explicable how synthetic propositions à priori are possible in pure mathematics, and therefore, also, how Edition: current; Page: [28] this science is itself possible. For just as empirical intuition enables us, without difficulty, to extend synthetically in experience the conception we form of an object of intuition, by new predicates, themselves afforded us by intuition, so will the pure intuition, only with this difference: that in the last case the synthetic judgment à priori is certain and apodictic, while in the first case it is no more than à posteriori and empirically certain, because the latter only contains what is met with in chance empirical intuition, but the former what is necessarily met with in the pure intuition, inasmuch as being intuition à priori, it is indissolubly bound up with the conception before all experience or perception of individual things.

§ 8.

But the difficulty seems rather to increase than to diminish by this step. For the question is now: How is it possible to intuite anything à priori? Intuition is a presentation, as it would immediately depend on the presence of the object. It seems therefore impossible to intuite originally à priori, because the intuition must then take place without either a previous or present object to which it could refer, and hence could not be intuition. Conceptions are indeed of a nature that some of them, namely, those containing only the thought of an object in general, may be very well formed à priori, without our being in immediate relation to the object (e.g., the conceptions of quantity, of cause, &c.), but even these require a certain use in concreto, i.e., an application to some intuition, if they are to acquire sense and meaning, whereby an object of them is to be given us. But how can intuition of an object precede the object itself?

§ 9.

Were our intuition of such a nature as to present things as they are in themselves, no intuition à priori would take place at all, but it would always be empirical. For what is contained in the object in itself, I can only know when it is given and present to me. It is surely then inconceivable how the intuition of a present thing should Edition: current; Page: [29] enable me to know it as it is in itself, seeing that its properties cannot pass over into my presentative faculty. But granting the possibility of this, the said intuition would not take place à priori, that is, before the object was presented to me, for without it no ground of connection between my presentation and the object could be imagined; in which case it must rest on inspiration (Eingebung). Hence there is only one way possible, by which my intuition can precede the reality of the object and take place as knowledge à priori, and that is, if it contain nothing else but that form of sensibility which precedes in my subject all real impressions, by which I am affected by objects. For, that objects of sense can only be intuited in accordance with this form of sensibility, is a fact I can know à priori. From this it follows, that propositions merely concerning the form of sensible intuition, will be valid and possible for all objects of sense; and conversely, that intuitions possible à priori, can never concern other things than objects of our sense.

§ 10.

Hence, it is only by means of the form of sensuous intuition that we can intuite things à priori, but in this way we intuite the objects only as they appear to our senses, not as they may be in themselves; an assumption absolutely necessary if synthetic propositions à priori are to be admitted as possible, or in the event of their being actually met with, if their possibility is to be conceived and defined beforehand.

Now, such intuitions are space and time, and these lie at the basis of all the cognitions and judgments of pure mathematics, exhibiting themselves at once as apodictic and necessary. For mathematics must present all its conceptions primarily in intuition, and pure mathematics in pure intuition, i.e., it must construct them. For without this it is impossible to make a single step, so long, that is to say, as a pure intuition is wanting, in which alone the matter of synthetic judgments à priori can be given; because it cannot proceed analytically, that is, by the dissection of conceptions, but is obliged to proceed synthetically. Edition: current; Page: [30] The pure intuition of space constitutes the basis of geometry—even arithmetic brings about its numerical conceptions by the successive addition of units in time; but above all, pure mechanics can evolve its conception of motion solely with the aid of the presentation of time. Both presentations, however, are mere intuitions; for when all that is empirical, namely, that belongs to feeling, is left out of the empirical intuitions of bodies and their changes (motion), space and time still remain over, and are therefore pure intuitions, lying à priori at the foundation of the former. For this reason, they can never be left out, but being pure intuitions à priori, prove that they are the bare forms of our sensibility, which must precede all empirical intuition, i.e., the perception of real objects, and in accordance with which objects can be known à priori, though only as they appear to us.

§ 11.

The problem of the present section is therefore solved. Pure mathematics is only possible as synthetic knowledge à priori, in so far as it refers simply to objects of sense, whose empirical intuition has for its foundation a pure intuition à priori (that of time and space), which intuition is able to serve as a foundation, because it is nothing more than the pure form of sensibility itself, that precedes the real appearance of objects, in that it makes them in the first place possible. Yet this faculty of intuiting à priori does not concern the matter of the phenomenon, i.e., that which is feeling (Empfindung) in the latter, for this constitutes the empirical element therein; but only its form, space and time. Should anybody cast the least doubt on the fact that neither of them are conditions of things in themselves, but only dependent on their relation to sensibility. I should be glad to be informed how he deems it possible to know à priori, and therefore before all acquaintance with the things, that is, before they are given us, how their intuition must be constructed, as is here the case with space and time. Yet this is quite conceivable, as soon as they both count for nothing more than formal determinations of our sensibility, and the objects merely as phenomena, for in that case the form of the phenomenon, Edition: current; Page: [31] that is, the pure intuition, can be conceived as coming from ourselves, in other words, as à priori.

§ 12.

To contribute something to the explanation and confirmation of the above, we have only to consider the ordinary and necessary procedure of geometricians. All the proofs of complete likeness between two given figures, turn at last upon the fact of their covering each other; in other words, of the possibility of substituting one, in every point, for the other, which is obviously nothing else but a synthetic proposition resting on immediate intuition. Now this intuition must be given pure and à priori, for otherwise the proposition in question could not count as apodictically certain, but would possess only empirical certainty. We could only say in that case, it has been always so observed, or it is valid so far as our perception has hitherto extended. That complete space, itself no boundary of a further space, has three dimensions, and that no space can have more than this number, is founded on the proposition that not more than three lines can bisect each other at right angles in a single point. But this proposition cannot be presented from conceptions, but rests immediately on intuition, and indeed on pure à priori intuition, because it is apodictically certain that we can require a line to be drawn out to infinity (in indefinitum), or that a series of changes (e.g., spaces passed through by motion) shall be continued to infinity, and this presupposes a presentation of space and time, merely dependent on intuition, namely, so far as in itself, it is bounded by nothing, for from conceptions it could never be concluded. Pure intuitions à priori, then, really lie at the foundation of mathematics, and these make its synthetic and apodictically valid propositions possible, and hence our transcendental deduction of conceptions in space and time explains at the same time the possibility of pure mathematics, which without such a deduction, and without our assuming that “all which can be given to oursenses (the outer in space, the inner in time) is only intuited by us, as it appears to us, and not as it is in itself,” might indeed be conceded, but could in nowise be understood

§ 13.

Those who are unable to free themselves from the notion, that space and time are real qualities (Beschaffenheiten) appertaining to the things in themselves, may exercise their wits on the following paradoxes, and when they have in vain attempted their solution, may suppose, being freed from their prejudices at least for a few moments, that perhaps the degradation of space and time to the position of mere forms of our sensible intuition, may have some foundation.

When two things are exactly alike [equal] in all points that can be cognised in each by itself (i.e., in all respecting quantity or quality), it must follow, that one can in all cases and relations be put in the place of the other, without this substitution occasioning the least cognisable difference. This indeed applies to plane figures in geometry; but there are many spherical figures, which in spite of this complete internal agreement exhibit in their external relations an agreement falling short of admitting one to be put in the place of the other.

For instance, two spherical triangles on opposite hemispheres, having an arc of the equator as a common base, are perfectly equal both in respect of their sides and their angles, so that in neither of them, if separately and at the same time completely described, would anything be found which was not equally present in the other; and yet notwithstanding this, one cannot be put in the place of the other, i.e., on the opposite hemisphere, and herein consists the internal difference of both triangles, that no understanding can indicate as internal, but which reveals itself only by means of the external relation in space. I will now adduce some more ordinary cases taken from common life.

What can more resemble my hand or my ear, and be in all points more like, than its image in the looking-glass? And yet I cannot put such a hand as I see in the glass in the place of its original; for when the latter is a right hand, the one in the glass is a left hand, and the image of the right ear is a left one, which can never take the place of the former. Now, here there are no internal differences Edition: current; Page: [33] that could be imagined by any understanding. And yet the differences are internal, so far as the senses teach us, for the left hand cannot, despite all equality and similarity, be enclosed within the same bounds as the right (they are not congruent); the glove of one hand cannot be used for the other. What then is the solution? These objects are not presentations of things as they are in themselves, and as the pure understanding would cognise them, but they are sensuous intuitions, i.e., phenomena, the possibility of which rests on the relations of certain unknown things in themselves to something else, namely, to our sensibility. Now, space is the form of the outward intuition of these, and the inward determination of every space is only possible through the determination of outward relations to the whole space, of which each [separate] space is a part (i.e., by its relation to the outward sense); in other words, the part is only possible through the whole, which though it could never be the case with things in themselves, namely, with objects of the mere understanding, can very well be so with mere phenomena. Hence we can render the difference of similar and equal, though incongruent things (e.g., spirals winding opposite ways1) intelligible by no single conception, but only by the relation of the right and left hands, which refers immediately to intuition.

§ 14.

Nature is the existence of things, in so far as it is determined according to universal laws. If Nature signified the existence of things in themselves, we could never know it either à priori or à posteriori. Not à priori, for how shall we know what applies to things in themselves? since this can never be done by the dissection of our conceptions (analytic propositions). For what I want to know, is not what is contained in my conception of a thing (for that concerns its logical nature), but what in the reality of the thing is superadded to this conception, by which the thing itself is determined outside my conception. My understanding and the conditions under which alone it can connect the determination of things in their existence, prescribes no rules for the things in themselves; these do not conform themselves to my understanding, but my understanding conforms itself to them. They must therefore be previously given me, in order for these determinations to be discovered in them; and in this case they would not be known à priori.

But à posteriori such a knowledge of the nature of things in themselves would be equally impossible. For if experience is to teach me laws to which the existence of things is subordinated, these must, in so far as they concern things in themselves, of necessity also apply to them outside my experience. Now experience teaches me, indeed, what exists and how it exists, but never that it exists necessarily in such a manner and no other. It can never, therefore, teach the nature of things in themselves.

§ 15.

We are nevertheless really in possession of a pure natural science, which à priori and with all the necessity requisite to apodictic propositions, puts forward laws to which Nature is subordinated. I only require here to Edition: current; Page: [42] call to witness that propaedeutic, which, under the title of universal natural science, precedes all physical scienco based on empirical principles. Therein we find mathematics applied to phenomena, also those discursive principles (from conceptions) constituting the philosophical part of pure natural knowledge. But the latter also contains much that is not pure, and independent of the sources of experience, as the conception of motion, of impenetrability (on which the empirical conception of matter rests), of inertia and others, which prevent its being called a perfectly pure natural science. Besides, it is only concerned with the objects of the external sense, and thus furnishes no example of a pure natural science in its strictest meaning; for this would have to bring Nature generally under universal laws, irrespective of whether it concerned the object of the outer or of the inner sense of physical science, or of psychology. But among the principles of the above universal physical science are to be found some that really possess the universality we require, as the proposition that substance continues and is permanent, and that all which happens is at all times previously determined by a cause, according to fixed laws. These are really universal natural laws, existing completely à priori. There is then in fact a pure natural science, and now the question arises—how is it possible?

§ 16.

The word Nature further assumes another meaning, which defines the object, whereas in the above meaning the mere regularity of the existence of the determinations of things generally, is denoted. Nature considered materialiter is the sum-total of all the objects of experience. With this we are alone concerned at present, for things which could never be objects of an experience were they to be known according to their nature, would necessitate us to form conceptions, to which meaning could never be given in concreto (in any example from a possible experience), and of the nature of which we should be obliged to make conceptions alone, whose reality, that is, whether they really referred to objects or were mere figments of thought, could never be decided. With that which cannot Edition: current; Page: [43] be an object of experience, the knowledge of which would be hyperphysical, or anything like it, we have here nothing at all to do, but only with the natural knowledge whose reality can be confirmed by experience, notwithstanding its being à priori possible, and preceding all experience.

§ 17.

The formal in Nature, in this narrower signification, is then the regularity of all the objects of experience, and in so far as they are known à priori, their necessary regularity. But it has been just demonstrated that the laws of Nature can never be known à priori in objects, in so far as they are considered not as the objects of a possible experience but as things in themselves. We are not here concerned with things in themselves (the qualities of which we put on one side), but merely with things as the objects of a possible experience, and the sum-total of which is properly what we call Nature. And I now ask, whether, if the question be as to the possibility of a cognition of Nature à priori, it would be better to formulate the problem, as follows: How is it possible to cognise à priori the necessary regularity of things as objects of experience? or, How is the necessary regularity of experience itself in respect of all its objects, generally [possible to be cognised à priori]?

Seen in its true light, the solution of the problem, whether presented in the one or in the other form, in respect of the pure cognition of Nature (which constitutes the real point of the question) is in the end altogether the same. For the subjective laws under which alone an experiential cognition of things is possible, are valid also of those things as objects of a possible experience (though not indeed as things in themselves; but the latter we are not here considering). It is quite the same, then, whether I say: Without the law—that on an event being perceived, it must invariably be referred to something preceding it, upon which it follows according to a universal rule—a judgment of perception can never avail as experience; or whether I express myself thus: Everything that experience teaches us, happens, must have a cause.

It is, however, advisable to choose the first formula. For as we can have a knowledge à priori and before all Edition: current; Page: [44] given objects, of those conditions under which alone an experience in respect of them is possible, but never of what laws, they, without reference, to a possible experience, are subordinated to, in themselves; we shall not be able to study the nature of things à priori, otherwise than by investigating the conditions and universal (although subjective) laws, under which such a knowledge is alone possible (in respect of mere form), as experience, and in accordance therewith determine the possibility of things as objects of experience. Were I to choose the second mode of expression and seek the conditions à priori under which Nature is possible as an object of experience, I should easily be led into misunderstanding, and fancy I had to explain Nature as a thing in itself, and I should then be fruitlessly involved in endless endeavours to seek laws for things of which nothing is given me.

We shall here, therefore, be simply concerned with experience, and the universal and à priori given conditions of its possibility, and thence determine Nature as the complete object of all possible experience. I think it will be understood, that I do not refer to the rules for the observation of a nature already given, which presuppose experience, or how through experience we can arrive at the laws of Nature, for these would not then be laws à priori, and would give no pure science of Nature; but how the conditions à priori of the possibility of experience are at the same time the sources from which all the universal laws of Nature must be derived.

§ 18.

We must first of all observe then, that, although all the judgments of experience are empirical, i.e., have their ground in the immediate perception of sense, yet on the other hand all empirical judgments are not judgments of experience, but that beyond the empirical, and beyond the given sensuous intuition generally, special conceptions must be superadded, having their origin entirely à priori in the pure understanding, under which every perception is primarily subsumed, and by means of which only it can be transformed into experience.

Empirical judgments, in so far as they have objective validity, are judgments of experience; but those which are merely subjectively valid I call judgments of perception. The last require no pure conception of the understanding; but only the logical connection of perception in a thinking subject. But the first demand, above the presentations of sensuous intuition, special conceptions originally generated in the understanding, which make the judgment of experience objectively valid.

All our judgments are at first mere judgments of perception; they are valid simply for us, namely, for our subject. It is only subsequently that we give them a new reference, namely, to an object, and insist that they shall be valid for us always, as well as for every one else. For when a judgment coincides with an object, all judgments must both coincide with the same object and with one another, and thus the objective validity of the judgment of experience implies nothing more than the necessary universal validity of the same. But, on the other hand, when we see reason to hold a judgment of necessity universally valid (which never hinges on the perception itself, but on the pure conception of the understanding under which the perception is subsumed), we are obliged to regard it as objective, i.e., as expressing not merely the reference of the perception to a subject but a quality of the object; for there would be no reason why the judgments of other persons must necessarily coincide with mine, if it were not that the unity of the object to which they all refer, and with which they coincide, necessitates them all agreeing with one another.

§ 19.

Objective validity and necessary universality (for every one) are therefore exchangeable notions, and although we do not know the object in itself, yet when we regard a judgment as at once universal and necessary, objective validity is therewith understood. We cognise in this judgment the object (though it remain unknown what it is in itself) by the universal and necessary connection of Edition: current; Page: [46] given perceptions, and as this is the case with all objects of sense, judgments of experience owe their objective validity not to the immediate cognition of the object (for this is impossible), but merely to the condition of universality in the empirical judgment, which, as has been said, never rests on empirical, or on any sensuous conditions, but on a pure conception of the understanding. The object in itself always remains unknown; but when through the conception of the understanding, the connection of the presentations given to our sensibility by the latter is determined as universally valid, the object is determined by this relation, and the judgment is objective.

We will explain this; that the room is warm,1 the sugar sweet, the wormwood bitter, are merely subjectively valid judgments. I do not expect that I shall always, or that every other person, will find them as I do now. They only express a reference of two sensations to the same subject, namely, myself, and that only in my present state of perception, and are not therefore valid of objects. I call these judgments of perception. With judgments of experience the case is altogether different. What experience teaches me under certain circumstances, it must teach me at all times, and every other person as well; its validity is not limited to the subject or to the state of the latter at a particular time. I pronounce, therefore, all such judgments to be objectively valid. For instance when I say—the air is elastic, this judgment is immediately a judgment of perception, since I only refer the feelings in my senses to one another. If I insist it shall be called a judgment of experience, I expect this connection to stand under a condition making it universally valid. I insist, Edition: current; Page: [47] that is, that I at all times and every other person, shall necessarily so combine the same perceptions, under the same circumstances.

§ 20.

We must therefore dissect experience, in order to see what is contained in this product of sense and understanding, and how the judgment of experience itself is possible. The intuition of which I am conscious, namely, perception (perceptio), which merely belongs to the senses, lies at its foundation. But secondly, judgment (which pertains solely to the understanding) also belongs to it. This [act of] judgment may be twofold; firstly, I may simply compare the perceptions in a particular state of my own consciousness; or secondly, I may combine them in a consciousness in general. The first judgment is a simple judgment of perception, and has therefore only subjective validity, being the mere connection of perceptions in my mental state, without reference to the object. Hence it is not sufficient for experience, as is commonly imagined, to compare perceptions and to connect them in a consciousness by means of the judgment. No universality and necessity in the judgment can arise therefrom, by means of which alone it can be objectively valid, and experience.

There is another and quite a different judgment presupposed, before perception can become experience. The given intuition must be subsumed under a conception determining the form of the judgment generally in respect of the intuition, connecting the empirical consciousness of the last in a consciousness in general, and thereby obtaining universality for the empirical judgment; such a conception is a pure à priori conception of the understanding, that does nothing but determine for an intuition the general manner in which it can serve for judgment. Should the conception be that of cause, it determines the intuition subsumed under it in respect of judgment generally; for instance, in the case of air, that in respect of expansion, it stands in the relation of antecedent to consequent, in a hypothetical judgment. The conception of cause is then a pure conception Edition: current; Page: [48] of the understanding, entirely distinct from all possible perception, and only serves to determine that presentation contained under it, in respect of judgment generally, in short, to make a universally valid judgment possible.

Now, before a judgment of perception can become a judgment of experience, it is first of all necessary that the perception be subsumed under these conceptions of the understanding. For instance, air belongs to the conception of causes, which determines the judgment regarding its extension, as hypothetical.1 In this way, the extension is represented not merely as belonging to my perception of air in my particular state, or in many of my states, or in a particular state of the perception of others, but as necessarily belonging thereto; and the judgment, the air is elastic, becomes universally valid, and therefore a judgment of experience, preceded by certain judgments, which subsume the intuition of air under the conception of cause and effect, and thereby the perceptions, not merely with respect to one another in my subject, but relatively to the form of judgment generally (here the hypothetical), and thus make the empirical judgment universally valid.

If we dissect all our synthetic judgments, in so far as they are valid objectively, we shall find that they never consist of mere intuitions, connected (as is commonly believed) through comparison in a judgment, but that they would be impossible were there not beyond the conceptions drawn from experience, a pure conception of the understanding, under which the former conceptions are subsumed, and in this way only, connected in an objectively valid judgment. Even the judgments of pure Edition: current; Page: [49] mathematics in their simplest axioms, are not excepted from this condition. The axiom, the straight line is the shortest way between two points, presupposes that the line be subsumed under the conception of quantity, which is assuredly no intuition, but has its seat in the understanding, and serves to determine the intuition (the line) in the reference of the judgment that may be made regarding it, in respect of its quantity, namely, of plurality (as judicia plurativa),1 inasmuch as it is thereby understood that in a given intuition, many homogeneous parts are contained.

§ 21.

In order to demonstrate the possibility of experience, in so far as it rests on pure à priori conceptions of the understanding, we must first present what belongs to judgment generally, and the various momenta of the understanding in the same, in a complete table, for the pure conceptions of the understanding, which are nothing more than conceptions of intuitions in general, in so far as these are determined in themselves by one or other of these momenta of judgment, that is, are necessarily and universally valid, must run exactly parallel to them [viz., these momenta]. In this way, the axioms à priori of the possibility of all experience as an objectively valid empirical cognition, are precisely determined. For they are nothing but propositions, subsuming all perception (in accordance with certain universal conditions of perception), under the above pure conceptions of the understanding.

§ 21a.

In order to grasp the preceding in a single notion, it is necessary to remind the reader that we are not here speaking of the origin of experience, but of that which lies within it. The first belongs to empirical psychology, and would exist without the second, which belongs to the critique of cognition, and especially to that of the understanding, and can never be sufficiently developed.

Experience consists of intuitions, belonging to sensibility, and of judgments which are entirely the work of the understanding. But the judgments the understanding constructs merely out of sensuous intuitions, are not, by far, judgments of experience. For in the one case the judgment simply connects the perceptions, as they are given in sensuous intuition; but in the other, the judgments must say what experience generally contains, and not what the mere perception, the validity of which is purely subjective, contains. The judgment of experience must add something to a judgment, over and above the sensuous intuition, and the logical connection of the same (after it has been made universal by comparison), something that determines the synthetic judgment, as at once necessary and thereby universally valid; and this can be nothing else but that conception which presents the intuition as determined in itself, in respect to one form of judgment rather than another, i.e., a conception of that synthetic unity of intuitions, which can only be presented through a given logical function of the judgment.

§ 22.

The sum of the above is this: the business of the senses is to intuite, that of the understanding to think. But to think is to unite presentations in a consciousness. This union is either merely relative to the subject, and is contingent and subjective, or is given unconditionally, and is necessary or objective. The union of presentations in a consciousness is judgment. Thinking, then, is the same as judging, or referring presentations to judgments in general. Hence judgments are either entirely subjective Edition: current; Page: [52] when presentations are solely referred to a consciousness in one subject, and are therein united, or they are objective when they are united in a consciousness in general, that is, are necessarily united therein. The logical momenta of all judgments are so many possible modes of uniting presentations in a consciousness. But if they serve as conceptions, they are conceptions of the necessary union of the same in a consciousness, and therefore principles of objectively valid judgments. This union in a consciousness is either analytic by identity, or synthetic by the combination and addition of different presentations to one another. Experience consists in the synthetic connection of phenomena (perceptions) in a consciousness, in so far as this is necessary. Hence pure conceptions of the understanding are those under which all perceptions must be previously subsumed, before they can serve as judgments of experience, in which the synthetic unity of perceptions is presented as necessary and universal.1

§ 23.

Judgments, considered merely as the union of given presentations in a consciousness, are rules. These rules, in so far as they present the union as necessary, are rules à priori, and in so far as there are none beyond them from which they can be derived, they are axioms. Since, then, in respect of the possibility of all experience, when viewed as the mere form of thought, there are no conditions of the judgments of experience beyond those Edition: current; Page: [53] which bring the phenomena in the various forms of their intuition under the pure conceptions of the understanding which make the empirical judgment objectively valid, these must be the à priori axioms of all possible experience.

The axioms of possible experience are at the same time the universal laws of Nature as known à priori. And thus the problem contained in our present second question—How is pure natural science possible? is solved. For the systematic character required by the form of a science is met with here in completeness, since beyond the above-named formal conditions of all judgments in general, that is, of all the general rules to be found in logic, there are none possible, and these constitute a logical system; while the conceptions founded upon them, containing the conditions à priori of all synthetic and necessary judgments, [constitute] in the same way a transcendental system, and finally the axioms, by means of which all phenomena are subsumed under these conceptions, [constitute] a physiological1 system, i.e., a system of nature, preceding all empirical knowledge of nature, rendering this in the first place possible, and therefore to be properly termed the universal and pure natural science.

§ 24.

The first of the above physiological2 axioms subsumes all phenomena, as intuitions in space and time, under the conception of quantity, and is so far a principle of the application of mathematics to experience. The second subsumes the properly empirical, namely, the feeling, which denotes the reality of intuitions, not precisely under the conception of quantity, because feeling is no intuition, contained in space and time, although it places its corresponding object in both. But between reality (presentation of feeling) and zero, i.e., the complete emptiness of intuition in time, there is a difference which has a quantity. For between each given degree of light Edition: current; Page: [54] and darkness, between each degree of heat, and complete coldness, each degree of weight and of absolute lightness, each degree of the containing of space and of totally empty space, progressively smaller degrees can be thought of, and similarly between consciousness and complete unconsciousness (psychological darkness) continually smaller [degrees] exist. Hence no perception is possible that would prove an absolute void; for instance, no psychological darkness that could be viewed otherwise than as a consciousness, which is but surpassed by another stronger consciousness, and the same in all cases of feeling. In this way the understanding can even anticipate feelings which constitute the proper quality of empirical presentations (phenomena), by means of the axiom that they all (that is, the real of every phenomenon) have a degree, and this is the second application of mathematics (mathesis intensorum) to natural science.

§ 25.

As regards the relation of phenomena, and indeed simply as to their existence, the determination of this relation is not mathematic but dynamic, and can never be valid objectively, and therefore adequate to an experience, if it be not subordinated to principles à priori rendering the cognition of experience regarding them in the first place possible. Hence phenomena must be subsumed under the conception of substance, which lies at the foundation of all determination of existence as a conception of the thing itself; or secondly, in so far as a succession, that is, an event, is met with among the phenomena, under the conception of an effect in reference to cause; or in so far as co-existence is to be cognised objectively, that is, through a judgment of experience, under the conception of community (reciprocal action); and these principles à priori lie at the foundation of objectively valid although empirical judgments, that is, the possibility of experience in so far as it is to connect the existence of objects in Nature. These principles are the particular laws of Nature, which may be termed dynamic.

There belongs, finally, to the judgments of experience Edition: current; Page: [55] the cognition of the agreement and connection, not so much of phenomena among one another in experience, as of their relation to experience generally, which unites either their agreement with the formal conditions cognised by the understanding or their coherence with the material of sense and of perception, or both, in one conception, and consequently contains possibility, reality and necessity, according to universal natural laws, thereby constituting the physiological doctrine of method, the distinction between truth and hypotheses, and the limits of the reliability of the latter.

§ 26.

Although the third table of the principles drawn from the nature of the understanding on the critical method, shows a completeness in itself, which raises it far above every other that has been vainly attempted or may be attempted in the future [to be drawn] from the nature of the thing itself, in a dogmatic way, inasmuch as therein all synthetic axioms à priori have been produced in accordance with a principle, that is, the possibility of judgment in general, which constitutes the essence of experience, in reference to the understanding, in such a manner that one may be certain there are no more such axioms (a satisfaction never to be obtained from the dogmatic method)—yet this is by far not its greatest service.

Attention must be paid to the ground of proof, which discovers the possibility of this knowledge à priori, and limits at the same time all such axioms by a condition, that must never be overlooked, if they are not to be misunderstood, and extended farther in use than the original sense attached to them by the understanding will admit of: namely, that they only contain the conditions of possible experience in general, in so far as it is subordinated to laws à priori. Thus I do not say that things in themselves contain a quantity, their reality, a degree, their existence, connection of accidents in a substance, &c.; for this no one can prove, because such a synthetic connection is simply impossible out of mere conceptions, where all Edition: current; Page: [56] reference to sensuous intuition on the one hand, and all connection of the same in a possible experience on the other, is wanting. The essential limitation of conceptions in these axioms is, therefore, that all things only stand under the above-mentioned conditions à priori as objects of experience.

From this there follows, in the second place, a special and peculiar mode of proof of the foregoing: that the axioms in question do not refer directly to phenomena and their relation, but to the possibility of experience of which phenomena constitute the matter but not the form, i.e., to objective and universally valid synthetic propositions, wherein judgments of experience are distinguished from mere judgments of perception. This happens in that the phenomena as mere intuitions, taking in a portion of space and time, are subordinated to the conception of quantity, which unites the manifold in the same synthetically in accordance with à priori rules; and that in so far as the perception contains feeling as well as intuition, between which and zero, namely, its total disappearance, a progression by diminution always takes place, the real of the phenomena must have a basis, seeing that in itself it takes in no portion of space or time.1 But this progression towards it [viz., reality] from empty time or space, is only possible in time. Consequently, although feeling as the quality of empirical intuition can never be known à priori in respect of that wherein it is specifically distinguished from other feelings, it can nevertheless be distinguished in a possible experience generally, as quantity of perception intensively [distinct] from every Edition: current; Page: [57] other of the same kind; which means the application of mathematics to Nature in respect of the sensuous intuition, by which the former is given us, and by which it becomes in the first place possible and definite.

But the reader must give the greatest attention to the mode of proof of the principles coming under the name of analogies of experience. For inasmuch as these do not, like the principles of the application of mathematics to natural science generally, concern the generation of intuitions, but the connection of their existence in an experience, this can be nothing but the determination of existence in time according to necessary laws, under which alone they are objectively valid, and therefore experience. Thus the proof of synthetic unity does not turn on the connection of things in themselves, but of perceptions, and even of these, not in respect of their content but of their determination in time, and of the relation of existence thereto, according to universal laws. These universal laws contain, therefore, the necessity of the determination of existence in time generally (consequently, according to a rule of the understanding, à priori) when the empirical determination in the relative time is to be objectively valid, that is, experience. I cannot enter further into the matter here, in Prolegomena, than to recommend the reader who has been long accustomed to regard experience as a mere empirical aggregation of perceptions, and hence does not reflect that it greatly exceeds the sphere of these, that it gives, namely, to empirical judgments, universal validity, and that for this a pure unity of the understanding is necessary to precede à priori, [to recommend him] to give attention to this distinction of experience from a mere aggregrate of perceptions, and to judge the manner of proof from this point of view.

§ 27.

It is here the place to raze Hume’s doubt from its foundation. He maintained justly that we can in nowise discern through the Reason the possibility of causation, namely, the reference of the existence of one thing to the existence of some other thing posited by the former. I Edition: current; Page: [58] may add to this, that we can just as little discern the conception of subsistence, i.e., the necessity contained therein, that a subject must lie at the basis of the existence of a thing, and itself be no predicate of any other thing. [I would say even] that we can form no conception of the possibility of such a thing (though we can point out examples of its use in experience). In the same way this inconceivability attaches even to the community of things, since it is not discernible how, from the state of one thing, a consequence can be drawn as to the state of some totally different thing, external to it, and vice versâ; and how substances of which each has its own separate existence, are necessarily dependent on one another. At the same time, I am far from regarding these conceptions as merely borrowed from experience, and the necessity, that is presented in them, as fictitious and mere illusion, induced in us by long custom. I have, rather, sufficiently shown that both they and the axioms deduced from them, subsist à priori before all experience, and possess indubitable objective correctness, though unquestionably only in respect of experiences.

§ 28.

Although I cannot have the slightest notion of such a connection of things in themselves as of their existing as substances, working as causes, or being able to stand in community with other [substances] as parts of a real whole, I can still less conceive such properties in phenomena as phenomena, because these conceptions contain nothing that lies in the phenomena, but something the understanding alone can conceive. We have, then, from such a connection of presentations in our understanding, and, indeed, in judgments generally, a similar conception, namely, that presentations cohere in one kind of judgments, as subject with reference to predicate, in another as cause with reference to effect, in a third as parts together making up a complete possible cognition. Further, we cognise à priori, that without the presentation of an object, in respect of one or the other of these momenta, to be considered as something definite, we could have no cognition that could be valid of objects, and if we occupied ourselves Edition: current; Page: [59] with the object in itself, there would be no single mark possible, by which I could cognise whether it was determined in respect of one or of another cogitated moment, i.e., whether it cohered under the conception of substance, or of cause, or (in relation to other substances) of reciprocity, for of the possibility of such a connection of existence I should have no conception. But it is not the question, how things in themselves, but how cognition of experience of things in respect of cogitated momenta of judgments generally, is defined, that is, how things as objects of experience can and should be subsumed under the above conceptions of the understanding. And hence it is clear, that I fully recognise not only the possibility, but also the necessity, of subsuming all phenomena under these conceptions, namely, of using them as axioms of the possibility of experience.

§ 29.

Let us now attempt a solution of Hume’s problematical conception (his crux metaphysicorum), namely, the conception of Cause. Firstly, there is given me, à priori, by means of Logic, the form of a conditioned judgment generally, one cognition as antecedent and another as consequent. But it is possible that in the perception, a rule of the relation may be met with, which will say, that on [the occurrence of a] given phenomenon another always follows (though not conversely), and this would be a case in which to make use of the hypothetical judgment, and to say, for instance, if a body be illumined long enough by the sun, it will become warm. There is certainly no necessity of connection here, in other words, no conception of cause. But I continue: if the above proposition, which is a mere subjective connection of perception, is to be a proposition of experience, it must be regarded as necessary and universally valid; but such a proposition would run: Sun is through its light the cause of heat. The above empirical rule is now looked upon as law, and indeed, not alone as valid of phenomena, but valid of them in relation to a possible experience, which requires thoroughly, and therefore necessarily, valid rules. I perfectly understand, then, the conception of Cause, as a Edition: current; Page: [60] conception necessarily belonging to the mere form of experience, and its possibility as a synthetic union of perceptions, in a consciousness in general; but the possibility of a thing in general as a cause I do not understand, because the conception of cause does not refer at all to things, but only indicates the condition attaching to experience, namely, that this can be only an objectively valid knowledge of phenomena, and their sequence in time, in so far as the antecedent can be united to the consequent according to the rule of hypothetical judgments.

§ 30.

Hence the pure conceptions of the understanding have no meaning whatever, when they quit the objects of experience and refer to things in themselves (noumena). They serve, as it were, to spell out phenomena, that these may be able to be read as experience. The axioms arising from their relation to the world of sense, only serve our understanding for use in experience. Beyond this, are only arbitrary combinations, destitute of objective reality, and the possibility of which can neither be known à priori, nor their reference to objects be confirmed, or even made intelligible by an example, because all examples are borrowed from some possible experience, and consequently the objects of those conceptions are nothing but what may be met with in a possible experience.

This complete solution of Hume’s problem, although it turns out to be contrary to the opinion of its originator, preserves for the pure conceptions of the understanding their origin à priori, and for the universal laws of Nature their validity as laws of the understanding, but in such a manner that their use is limited to experience, because their possibility has its basis, solely, in the reference of the understanding to experience; not because they are derived from experience, but because experience is derived from them, which completely reversed mode of connection never occurred to Hume.

The following result of all previous researches follows from the above investigations: “All synthetic axioms à priori are nothing more than principles of possible experience,” Edition: current; Page: [61] and can never be referred to things in themselves, but only to phenomena as objects of experience. Hence pure mathematics no less than pure natural science can never refer to anything more than mere phenomena, and only present that which either makes experience in general possible, or which, inasmuch as it is derived from these principles, must always be able to be presented in some possible experience.

§ 31.

And thus we have at last something definite to hold by in all metaphysical undertakings, which hitherto, bold enough, but always blind, have pursued all things without distinction. Dogmatic thinkers have never let it occur to them, that the goal of their endeavours should be extended such a short way from them, and even those most confident in their imagined common sense have started with conceptions and principles of the mere Reason, legitimate and natural, it is true, but intended merely for use in experience, [in search of] spheres of knowledge, for which they neither knew nor could know of any definite boundaries, because they had neither reflected nor could reflect on the nature or even the possibility of any such pure understanding.

Many a naturalist of the pure Reason (by which I understand he who ventures to decide in questions of metaphysics, without any science) might well profess that what has been here put forward with so much preparation, or if he will have it so, with tediously pedantic pomp, he has long ago not merely conjectured but known and penetrated, by the prophetic spirit of his common sense, namely, “that with all our Reason, we can never pass beyond the field of experiences.” But he must confess, notwithstanding, when questioned seriatim as to his principles of Reason, that amongst these there are many to be found not drawn from experience, and therefore valid, independently thereof, and à priori. How then, and on what grounds, will he hold the dogmatist and himself in limits, who use these conceptions and principles outside all possible experience, simply because they are recognised as independent of it? And even this adept of common sense, Edition: current; Page: [62] in spite of all his pretended, cheaply acquired, wisdom, is not proof against wandering, unobserved, beyond the objects of experience into the field of chimeras. He is, indeed, in the ordinary way, deeply enough involved therein, although by the use of popular language, by putting everything forward as probability, reasonable supposition or analogy, he gives some colour to his groundless assumptions.

§ 32.

From the earliest ages of philosophy, investigators of the pure Reason have postulated, beyond the sensible essences (phenomena) which constitute the world of sense, special essences of the understanding (noumena) which are supposed to constitute a world of understanding; and since they held appearance and illusion [Erscheinung und Schein] for the same thing, which in an undeveloped epoch is to be excused, ascribed reality to the intelligible essence alone.

In fact, when we regard the objects of sense, as is correct, as mere appearances, we thereby at the same time confess that a thing in itself lies at their foundation, although we do not know it, as it is constituted in itself, but only its appearance, that is, the manner in which our senses are affected by this unknown something. The understanding then, by accepting appearances, admits also the existence of things in themselves, and we may even say that the presentation of such essences as lie at the basis of appearances, in short, mere essences of the understanding, is not only admissible, but unavoidable.

Our critical deduction does not by any means exclude such things (noumena), but rather limits the principles of æsthetic, in so far that these should not be extended to all things, whereby everything would be changed into mere appearance, but that they should only be valid of objects of a possible experience. Essences of the understanding are hereby admitted only by the emphasising of this rule, which admits of no exception, that we know nothing definite whatever of these pure essences of the understanding, neither can we know anything of them, because our pure conceptions of the understanding no less Edition: current; Page: [63] than our pure intuitions, concern nothing but objects of a possible experience, in short, mere essences of sense, and as soon as we leave these, the above conceptions have not the least significance remaining.

§ 33.

There is indeed something seductive about our pure conceptions of the understanding, as regards temptation to a transcendent use; for so I name that which transcends all possible experience. Not only do our conceptions of substance, force, action, reality, &c., which are entirely independent of experience containing no phenomenon of sense, really seem to concern things in themselves (noumena); but what strengthens this supposition is, that they contain a necessity of determination in themselves, to which experience can never approach. The conception of cause contains a rule, according to which from one state another follows in a necessary manner; but experience only teaches us that often, or at most usually, one state of a thing follows upon another, and can therefore acquire neither strict universality nor necessity.

Hence these conceptions of the understanding seem to have far too much significance and content for mere use in experience to exhaust their entire determination, and the understanding builds in consequence, unobserved, by the side of the house of experience, a much more imposing wing, which it fills with sheer essences of thought, without even noticing that it has overstepped the legitimate bounds of its otherwise correct conceptions.

§ 34.

There were two important, and indeed altogether indispensable, although exceedingly dry investigations necessary, that have been undertaken in the Critique (p. 107), in the first of which it was shown that the senses do not furnish the pure conceptions of the understanding in concreto, but only the schema for their use, and that the object which conforms to it is only to be met with in experience as the [common] product of the understanding, Edition: current; Page: [64] and the materials of sense. In the second investigation (Critique, p. 178) it is shown, that—notwithstanding the independence of our pure conceptions of the understanding and principles of experience, even to the apparently greater range of their use—nothing whatever could be conceived through them outside the field of experience, because they can do nothing but determine the merely logical form of judgment in respect of given intuitions. But since, beyond the field of sensibility, no intuition is given, these pure conceptions become totally void of meaning, masmuch as they can in no way be presented in concreto. Consequently, all these noumena together with their sum-total, an intelligible world,1 are nothing but presentations of a problem, the subject of which in itself is indeed possible, but the solution of which is, by the nature of our understanding, utterly impossible, since our understanding is no faculty of intuition, but is merely the connection of given intuitions in an experience, and must comprise therefore all objects for our conceptions; but apart from these, all conceptions which cannot be supported by an intuition, must be without meaning.

§ 35.

The imagination may perhaps be forgiven, if it sometimes dreams, and fails to keep itself carefully within the limits of experience; for certainly it is invigorated and strengthened by a free flight like this, and it is always easier to moderate its boldness than to stimulate its languor. But for the understanding, which ought to think, to dream instead, can never be forgiven, as it is our only support in setting bounds to the fantasies of the imagination, where this is necessary.

It begins, however, very innocently and modestly. First of all, it reduces the elementary cognitions inhering in it before all experience, but having their application, notwithstanding, in experience, to their pure state. Gradually it lets fall these limits; and what is there then to hinder it, seeing that the understanding has taken its principles quite freely from itself? First of all, it is led to newly invented powers in Nature, soon after to essences outside Nature, in a word, to a world for whose fitting-up we can never fail in material, because by a fruitful imagination this will always be richly procured, and although not substantiated by experience, will yet never be confuted by it. This is the reason why young thinkers are so fond of metaphysics, treated in a genuinely dogmatic manner, and sacrifice to it their time and talents which might be otherwise useful.

But it is of no avail attempting to moderate these fruitless attempts of the pure Reason, by all manner of cautions as to the difficulty of the solution of such deeply-hidden questions, lamentations over the limits of our Reason, and by lowering assertions to mere conjectures. For if their impossibility be not clearly shown, and the self-knowledge of the Reason be not [raised to] a true science, in which the field of its right use is separated from that of its nugatory and fruitless use, so to speak, with geometrical certainty, these vain endeavours will never be completely laid aside.

§ 36.: How is Nature itself possible?

This question, which is the highest point the transcendental philosophy can ever touch, and to which it must also, as its boundary and completion, be directed, properly comprises two questions.

Firstly: How is Nature, in its material signification, namely, as intuition, as the sum-total of phenomena—how is space, time, and that which fills them both, namely, the object of feeling in general—possible? The answer is, by means of the construction of our sensibility, in accordance Edition: current; Page: [66] with which, it is affected in a special manner by objects, in themselves unknown and entirely distinct from these appearances. This answer has been given in the book itself in the Transcendental Æsthetic, but in these Prolegomena in the solution of the first general question.

Secondly: How is Nature in its formal signification—as the sum-total of the rules to which all phenomena must be subordinated, if they are to be thought of as connected in an experience—possible? The answer cannot but be: It is only possible by means of the construction of our understanding, in accordance with which all the above presentations of sensibility are necessarily referred to a consciousness, and whereby the special manner of our thought (namely, by rules), and by means of these, experience (which is to be wholly distinguished from a knowledge of things in themselves) is possible. This answer has been given in the book itself in the Transcendental Logic, but in these Prolegomena in the course of the solution of the second general question.

But how this special property of our sensibility itself, or of our understanding together with the necessary apperception lying at its basis, and at that of all thought, is possible, will not admit of any further solution or answer, because we invariably require it for all answers and for all thought of objects.

There are many laws of Nature that we can only know by means of experience, but regularity in the connection of phenomena, i.e., Nature in general, we can never learn through experience, because experience itself requires such laws, and these lie at the foundation of its possibility à priori. The possibility of experience in general is at once the universal law of Nature, and the axioms of the one are at the same time the laws of the other. For we know nothing of Nature otherwise than as the sum-total of phenomena, namely, of presentations in us, and hence can derive the law of their connection in no other way than from the principles of the same connection in ourselves; in other words, from the conditions of necessary union in a consciousness, which constitutes the possibility of experience.

Even the main proposition, worked out through the whole of this section, that universal natural laws are to Edition: current; Page: [67] be known à priori, of itself leads to the further proposition, that the highest legislation of Nature must lie in ourselves, namely, in our understanding, and that we must seek its universal laws, not in Nature, by means of experience; but conversely, must seek Nature, as to its universal regularity, solely in the conditions of the possibility of experience lying in our sensibility and understanding. For how would it otherwise be possible to know these laws à priori if they be not rules of analytic knowledge, but actually synthetic extensions of the same? Such a necessary agreement of the principles of possible experience with the laws of the possibility of Nature can only occur from one of two causes; either the laws are borrowed from Nature by means of experience, or conversely, Nature is derived from the laws of the possibility of experience generally, and is entirely the same thing as the purely formal regularity of the latter. The first supposition contradicts itself, for the universal laws of Nature can and must be known à priori (i.e., independently of all experience), and be posited as the basis of the empirical use of the understanding; so that only the second [hypothesis] remains to us.1

But we must distinguish the empirical laws of Nature, which always presuppose particular perceptions, from the pure or universal natural laws, which without any particular perceptions at their foundation, merely contain the conditions of their necessary union in an experience; and in respect of the last, Nature and possible experience are the same thing. Hence, as in this, the legitimacy rests on the necessary connection of phenomena in an experience, in other words, on the original laws of the understanding (without which we could cognise no object of the sensuous world whatever), it sounds at first singular, but is none the less certain, when I say in respect of the latter: The Edition: current; Page: [68] understanding draws its laws (à priori) not from Nature, but prescribes them to it.

§ 37.

We will illustrate this apparently daring proposition by an instance, showing that laws, which we discover in objects of sensuous intuition, especially when they are cognised as necessary, are held by ourselves to be such as the understanding has placed them, although in all other respects they may resemble the natural laws we attribute to experience.

§ 38.

If we consider the properties of the circle, by which the figure unites in itself so many arbitrary determinations of space, in a universal rule, one cannot do otherwise than attribute a nature to this geometrical thing. Two lines, for instance, which intersect one another and the circle, it matters not how they may be drawn, are yet always so regular that the rectangle under the segments of the one line is equal to that under the segments of the other. Now I ask, “Does this law lie in the circle or in the understanding?” in other words, does this figure contain independently of the understanding the ground of this law in itself, or does the understanding impose the law that chords cut one another in geometrical proportion, upon it, inasmuch as it has itself constructed the figure according to its own conceptions, namely, the equality of radii? We soon perceive when we follow the proofs of this law, that it can only be derived from the condition the understanding places at the foundation of the construction of this figure, namely, the equality of radii. If we extend the conception, in order to pursue still farther the unity of the manifold properties of geometrical figure under common laws, and consider the circle as a conic section, subordinated to the same fundamental conditions of construction as other conic sections, we find that all chords that intersect within the ellipse (parabola and hyperbola) always intersect, so that the rectangles under their segments, though not indeed equal, yet stand in the same ratio to one another. If we proceed still farther, Edition: current; Page: [69] namely, to the fundamental doctrines of physical astronomy, a physical law of mutual attraction is seen extended over the whole of material nature, whose rule is, that it decreases inversely as the square of the distance from each attracting point, that is, as the spherical surfaces, in which this power diffuses itself, increase; and this seems to lie necessarily in the nature of things themselves, and therefore is usually enunciated as cognisable à priori. However simple the sources of this law may be, as they rest merely on the relations of spherical surfaces of different radii, the consequences are so valuable, as regards the manifold nature of its agreement and regularity, that not only all possible orbits of the heavenly bodies [are described] in conic sections, but such a relation of them among one another follows, that no law of attraction could be conceived as suitable for a world-system, other than that of the inverse square of the distance.

Here then is Nature resting on laws which the understanding cognises à priori, and indeed mainly on universal principles of the determination of space. Now I ask: Do these natural laws lie in space, and does the understanding learn them by merely seeking to investigate the abundant meaning contained therein, or do they lie in the understanding and in the manner in which this determines space according to the conditions of synthetic unity, on which all these conceptions hinge? Space is something so uniform, and as regards all particular properties so indefinite, that certainly no one will seek for any wealth of natural laws in it. On the other hand that which determines space to the circular form, to the figure of the cone or of the sphere, is the understanding in so far as it contains the ground of the unity of its construction. The mere universal form of intuition called space, is the substratum of all particular objects of definable intuitions, and in this certainly lies the condition of its possibility and variety. But the unity of objects is determined simply by the understanding, according to conditions that lie in its own nature, and the understanding is thus the source of the universal order of Nature, since it comprehends all phenomena under its own laws; and thereby it first constructs experience (according to its Edition: current; Page: [70] form) à priori, by means of which all that is to be known through experience becomes necessarily subordinated to its laws. For we have nothing to do with the nature of things in themselves, which is as independent of the conditions of our sensibility as of those of the understanding, but with Nature as the object of a possible experience; and the understanding, while making this possible, [insists] that the world of sense be either no object of experience at all, or else, a Nature.

§ 39.: APPENDIX TO PURE NATURAL SCIENCE. Of the System of the Categories.

There can be nothing more desired by a philosopher than that the variety of conceptions or principles he had previously had presented to him in a scattered manner through the use he had made of them in concreto, should be deduced from one principle à priori, and should be all united in this manner in one cognition. Formerly he only believed that those things which remained over, after a certain abstraction, and which by comparison with one another seemed to constitute a particular kind of cognitions, were completely collected; but this was only an aggregate. Now he knows that exactly so many, neither more nor less, can constitute the mode of cognition, and sees the necessity of their division, which is a comprehension; and thus, for the first time, he has a system.

To search out conceptions from common cognitions, having no particular experience at their bases, and at the same time occurring in all cognition of experience, of which they constitute, as it were, the mere form of connection, presupposes no greater reflection or more insight than to search out in a language rules for the real use of words in general, and thus to get together the elements of a grammar. Indeed, both investigations are very nearly related, even if we are unable to give a reason why each language has precisely this and no other formal construction, and still less why exactly so many, neither Edition: current; Page: [71] more nor less, of such formal determinations of the same, generally, are to be found.

Aristotle collected ten such pure elementary cognitions under the name of categories.1 To these, which were also called predicaments, he saw himself, subsequently, obliged to add five post-predicaments,2 which yet lay partly in the former (as prius, simul, motus); but this rhapsody could but serve, and be admired, as a hint for future investigators, rather than be valid as a regularly developed idea; hence in more advanced [stages] of philosophy it has been rejected as altogether useless. On investigation of the pure elements (containing nothing empirical) of the human cognition, I first succeeded, after long reflection, in distinguishing and separating with confidence the elementary conceptions of sensibility (space and time) from those of the understanding, Under these circumstances, the 7th, 8th, and 9th categories were excluded from the list. The remainder could be of no use to me, because there was no principle at hand by which the understanding could be fully gauged, and all its functions, from which its pure conceptions arise, be defined completely and with precision.

In order to find out such a principle, I looked about me for an act of the understanding containing all the rest, and distinguishing itself, only through different modifications or momenta, in bringing the manifold of presentation under the unity of thought generally, and I then found this act of the understanding to consist in judgment. There lay already before me the entire, although not altogether faultless, work of the logicians, whereby I was placed in a position to present a complete table of the pure functions of the understanding that were indefinite as regards the whole object-world. I finally referred these functions of judgment to objects generally, or rather to the conditions determining judgments as objectively valid, and there Edition: current; Page: [72] arose pure conceptions of the understanding, respecting which I could be without doubt that they alone, and only so many of them, neither more nor less, could constitute our whole cognition of things from mere understanding. I called them, as was suitable, by their old name of categories; in doing which, however, I reserved to myself the right to add in their entirety, under the name of predicables, all conceptions to be derived from these—whether by connection with one another, or with the pure form of the phenomenon (space and time), or with their matter so far as it is not empirically determined (object of feeling, generally), as soon as a system of transcendental philosophy, in furtherance of which I was now occupied with a Critique of the Reason itself, should be constructed.

But that which is essential in this system of categories, and distinguishes it from the old rhapsody which proceeded without any principle, and that which alone entitles it to be counted as philosophy, consists in that by its means the true significance of the pure conceptions of the understanding and the conditions of their use can be clearly defined. For it is evident that they are only logical functions in themselves, and as such do not constitute the least conception of an object in themselves, but require sensuous intuition at their foundation. And hence they serve only to determine in respect of the same empirical judgments that are otherwise undetermined and indifferent as regards all functions of judgment; to procure for them thereby universality, and by means of them to make judgments of experience generally, possible.

Such an insight into the nature of the categories, at the same time limiting them to use in experience, never occurred either to their first originator or to any one after him. But without this insight (which exactly depends on their derivation or deduction) they are quite purposeless, and a miserable list of names without explanation or rule of use. Had anything of the kind ever entered into the minds of the ancients, without doubt the whole study of the cognition of the pure Reason, which under the name of metaphysics has through long centuries ruined many a good head, would have come down to us in quite a different form, and would have enlightened the human understanding Edition: current; Page: [73] instead of, as has actually happened, [causing it] to exhaust itself in obscure and vain subtleties, and making it unfruitful for true science.

This system of categories makes all treatment of any object of the pure Reason itself systematic, and affords an indubitable direction or clue how and to what point in the investigation every metaphysical consideration, if it is to be complete, must be reduced; for it exhausts all the momenta of the understanding, under which every other principle must be brought. It is thus that the table of conceptions has arisen, of whose completeness we can only be assured by means of the system of categories.1 And even in the division of these conceptions destined to transcend the physiological use of the understanding (Critique, pp. 207 and 257), it is always the same clue, which, because it must be always carried through the same fixed points, determined à priori in the human understanding, invariably forms a closed circle, leaving no doubt remaining that the object of a pure conception of the understanding or of the Reason, in so far as it is to be weighed philosophically and according to principles à priori, can be completely known in such a manner. I have not been able even to omit from this derivation, to make use of the Edition: current; Page: [74] most abstract of ontological divisions, namely, the manifold distinction of conceptions of something and nothing, and accordingly to construct a regular and necessary table (Critique, p. 207).

This system, like every true system based on a universal principle, shows its inestimable utility, in that all foreign conceptions, which might otherwise creep in between the above pure conceptions of the understanding, are excluded, and its place given to every cognition. Those conceptions which under the name of conceptions of reflection, I had reduced to a table, on the clue of the categories, mingle themselves, in an ontology without favour or just claim, under the pure conceptions of the understanding, although the latter are conceptions of the connection [of the object] and thereby of the object itself; but the former are the mere comparison of previously given conceptions, and have therefore an altogether different nature and use: by my legitimate division1 they are saved from this confusion. But the utility of the above separate table of the categories will be seen much more clearly, when, as we are now about to do, we separate the table of the transcendental conceptions of the Reason which are of quite a different nature and origin from the former conceptions of the understanding, and must consequently have a form other than the latter. This necessary separation has never yet taken place in any system of metaphysics, where ideas of the Reason and conceptions of the understanding intermingle, without distinction, as though they were members of one family—a state of confusion which in the absence of a special system of categories could never be avoided.

How is Metaphysics possible at all?

General Remark on the Transcendental Ideas.

§ 57.

After all the very clear proofs we have above given, it would be absurd for us to expect to cognise more in any object than what belongs to its possible experience, or to lay claim to the least knowledge of anything whatever which would determine its constitution in itself, unless we assume it to be an object of possible experience. For wherewith shall we effect this determination, inasmuch as time, space, and all the conceptions of the understanding, and still more the conceptions derived from empirical intuition or perception in the sense-world would neither have nor could have any other use than merely to make experience possible, and when if we leave out this condition from the pure conceptions of the understanding, they determine no object whatever, and have no significance anywhere [?].

But it would be a still greater absurdity for us not to admit things in themselves at all, or to wish to give out our experience for the only possible mode of the cognition of objects, in other words, our intuition in space and time for the only possible intuition, and our discursive Edition: current; Page: [100] understanding for the model of every possible understanding, thereby wishing principles of the possibility of experience to be held for the universal conditions of things in themselves.

Our principles, which limit the use of the Reason to possible experience, might accordingly become transcendent, and the limits of our Reason be given out for the limits of things themselves, of which Hume’s Dialogues may serve as an example, if a careful Critique of the boundaries of our Reason did not keep watch on its empirical use, and set a limit to its pretensions. Scepticism originally arose from metaphysics and its anarchical (Polizeilosen) dialectic. At first, to favour the empirical use of the understanding, it might well give out for nugatory and deceptive all that exceeded this; but gradually, as it became evident that the very same principles which we make use of in experience are à priori, and that they led unobserved, and as it seemed with the same right, still farther than experience reaches, a doubt began to be thrown on the principles of experience themselves. Now as to these there is no danger, for herein a healthy understanding will always assert its rights; but there arose a special confusion in science, which could not determine how far, and why only thus far and no farther, the Reason is to be trusted; but this confusion can only be got rid of, and any future relapse prevented, by a formal limitation of the use of our Reason, derived from principles. It is true we cannot form any definite conception of what things in themselves, beyond all possible experience, may be. But we are nevertheless not free to withdraw ourselves wholly from the inquiry as to these; for experience never fully suffices for the Reason; it thrusts us ever farther and farther back for the answer to this question, and leaves us as regards its complete solution dissatisfied; as any one can see from the dialectic of the pure Reason, which on this account has its valid subjective ground. Who can tolerate [the circumstance] that by the nature of our soul we can attain to the clear consciousness of the subject, and to the conviction that its phenomena cannot be explained materialistically without asking what the soul really is, and if no empirical conception suffices [to explain] Edition: current; Page: [101] this, at least assuming a conception of the Reason (of a simple immaterial essence) merely for the above purpose, although we cannot demonstrate its objective reality in any way? Who can satisfy himself in all cosmological questions, as to the size and duration of the world, of freedom or natural necessity, with mere empirical knowledge, since, begin it as we will, every answer given according to the fundamental laws of experience, gives birth to a new question, just as much requiring an answer, and thereby clearly exposing the inadequacy of all physical modes of explanation for the satisfaction of the Reason? Finally, who in the face of the thoroughgoing contingency and dependence of all that he can assume and think according to empirical principles, does not see the impossibility of taking his stand on these, and does not feel himself necessarily impelled, in spite of all prohibition against losing himself in transcendent ideas, to seek rest and satisfaction beyond all conceptions he can verify by experience, in that of a Being, of whom the possibility of the idea in itself cannot indeed be apprehended, but which cannot be refuted, because it is a mere being [essence] of the understanding, and without which the Reason must remain for ever unsatisfied.

Boundaries (with extended beings) always presuppose a space, met with, outside a certain definite place, and enclosing it. Limits do not require this, being mere negations affecting a quantity, so far as it has no absolute completeness. Our Reason, however, sees around it as it were a space for the cognition of things in themselves, although it can never have definite conceptions of them, being limited to phenomena.

As long as the cognition of the Reason is homogeneous, no definite boundaries can be conceived therein. In mathematics and natural science the human Reason recognises indeed limits but no boundaries, i.e., [it recognises] that something exists outside itself, to which it can never attain, but not that it can itself be anywhere terminated in its inner progress. The extension of our views in mathematics and the possibility of new inventions reaches to infinity; and the same can be said of the discovery of new qualities in Nature, and of new forces and laws, through continued Edition: current; Page: [102] experience and the union of the same by the Reason. But, at the same time, it cannot be mistaken that there are limits here, for mathematics refers only to phenomena, and what cannot be an object of sensuous intuition, such as the conceptions of metaphysics and morals, lies wholly outside its sphere, [in a region] to which it can never lead, and which does not at all require it. There is, then, a continuous progress and approach to these sciences, and as it were a point or line of contact. Natural science will never discover for us the inner [nature] of things, namely, that which is not phenomenon, but which can still serve as the highest ground of the explanation of phenomena. But it does not require this for its physical explanations; nay, if such were offered it from another source (e.g., the influence of immaterial beings), it ought to reject it, and on no account to bring it into the course of its explanations, but invariably to base these on that which pertains to experience as object of sense, and which can be brought into connection with our real perceptions, and empirical laws.

But metaphysics leads us to boundaries in the dialectical attempts of the pure Reason (which are not commenced arbitrarily or rashly, but to which the nature of the Reason itself urges us), and the transcendental ideas, as we cannot have intercourse with them, and as they will never allow themselves to be realised, serve, not only to show us the actual boundaries of the use of the pure Reason, but also the way to determine them. And this is also the end and use of this natural disposition of our Reason, which has given birth to metaphysics as its pet child, whose generation, like that of everything else in the world, is not to be ascribed to chance, but to an original germ, wisely organised for great ends. For metaphysics is, perhaps more than any other science, rooted in us in its fundamental features by Nature herself, and can by no means be regarded as the product of a voluntary choice or as chance extension in the progress of experiences (from which it is wholly divided).

The Reason, though all its conceptions and laws of the understanding are adequate in the sense-world, does not find any satisfaction for itself in them, for it is deprived of all hope of a complete solution by questions recurring ad infinitum. The transcendental ideas which have this Edition: current; Page: [103] completion for an object are such problems of the Reason. It sees clearly that the sense-world cannot contain the completeness [required], and therefore just as little can those conceptions which serve simply to the understanding of the same, namely, space and time, and all that we have adduced under the name of pure conceptions of the understanding. The sense-world is nothing but a chain of phenomena, connected according to universal laws, and has therefore no subsistence for itself, being not properly the thing in itself, and only being necessarily referable to that which contains the ground of this phenomenon, to essences that cannot be cognised merely as phenomena but as things in themselves. Only in the cognition of these can Reason hope to see its desire for completeness in the progress from the conditioned to its conditions, once for all satisfied.

We have above (§§ 33, 34) assigned the limits of the Reason in respect of all cognition of mere essences of thought. Now, as the transcendental ideas make the progress up to these necessary, and have thus led us, as it were, to the contact of the full space (of experience) with the void of which we know nothing (to the noumena), we can determine the boundaries of the pure Reason. For in all boundaries there is something positive (for instance surface is the boundary of corporeal space and yet is itself a space; line, a space which is the boundary of the surface; point, the boundary of the line, but still [occupying] a position in space), while, on the other hand, limits contain mere negations. The limits assigned in the paragraph cited, are not sufficient, after we have found that something lies beyond them (although we can never know what this may be in itself). For the question is now, what is the attitude of our Reason in this connection of that which we know, with that which we do not know, and never can know? Here is a real connection of the known with a wholly unknown (and something that will always remain unknown), and even if in this the unknown should not become in the least [degree] more known—which is indeed not to be expected—the conception of this connection must be able, notwithstanding, to be determined and reduced to distinctness.

We are obliged, then, to think of an immaterial essence, an intelligible world, and a highest of all beings (mere noumena), because only in these, as things in themselves, does the Reason meet with the completeness and satisfaction it can never hope for from the derivation of phenomena from their homogeneous ground, because they really refer to something distinct from the latter (and therefore wholly heterogeneous), inasmuch as phenomena always presuppose a thing in itself, and indicate this, [it matters not] whether we may know it more closely or not.

But as we can never know these essences of the understanding as to what they may be in themselves, that is, determinately, but are obliged nevertheless to assume such in relation to the sense-world, and to connect them with it through the Reason, we shall be at least able to cogitate this connection by means of such conceptions as express its relation to the sense-world. For if we cogitate the essence of the understanding, through nothing but pure conceptions of the understanding, we really cogitate thereby nothing definite, and our conception is consequently without meaning; if we cogitate it through qualities borrowed from the sense-world, then it is no longer an essence of the understanding, but is conceived as one of the phenomena, and belongs to the sense-world. We will take an instance from the conception of the Supreme Being.

The deistic conception is an entirely pure conception of the Reason, which, however, only represents a thing containing all reality, without our being able to determine a single one of its [qualities], because for this an instance would have to be borrowed from the sense-world, in which case I should always have to do with an object of sense, and not with something completely heterogeneous, and which cannot be an object of sense. For instance, I attribute understanding to it; but I have no conception whatever of any understanding but of one like my own, namely, of one to which intuitions must be given through the senses, and which occupies itself with reducing these under rules of the unity of the consciousness. But then the elements of my conception would always lie in the phenomenon; yet I was necessitated by the inadequacy of Edition: current; Page: [105] the phenomena to pass beyond this, to the conception of a being in no way dependent on phenomena, or bound up with them, as conditions of its determination. If, however, I sever the understanding from the sensibility in order to have a pure understanding, nothing remains over but the mere form of thought without intuition, by means of which I can cognise nothing determinate as object. For this purpose I should have to conceive another understanding which intuited objects, but of which I have not the least conception, because the human understanding is discursive and can only cognise through universal conceptions. But I am also involved in contradiction if I attribute will to the Supreme Being. For I have this conception only in so far as I derive it from my inner experience, and thereby from the dependence of my satisfaction from objects whose existence we require; but at the foundation of this lies sensibility, which wholly contradicts the pure conception of the Supreme Being. The objections of Hume to Deism are weak, touching no more than the proofs, and never the proposition of the deistic assertion itself. But as regards Theism, which must be arrived at by a closer determination of our, there [viz., in Deism], merely transcendent conception of the Supreme Being, they are very strong, and, according as the conception is constructed, in certain (indeed in all ordinary) cases are irrefragable. Hume always insists, that through the mere conception of an original being, to whom we can attribute none but ontological predicates (eternity, omnipresence, omnipotence) we really think nothing definite, but that qualities expressing an object in concreto must be superadded. It is not enough to say it is Cause, but [we must also say] what is the nature of its causality, as, whether [it operates] through understanding and will; and at this point his attacks on the thing itself, namely, on Theism, commence, whereas before he had only stormed the grounds of proof of Deism, which does not carry any especial danger with it. His dangerous arguments refer entirely to anthropomorphism, which he holds to be inseparable from Theism, and to make it contradictory in itself; while if this be left out, [Theism itself] would also fall, and nothing would remain but a Deism wherewith Edition: current; Page: [106] nothing could be done, which could not avail us for anything, and could not serve as a foundation for religion and morals. If this inevitability of anthropomorphism were certain, the proofs of the existence of a Supreme Being might be what one liked, and all conceded, yet the conception of this Being would never be able to be determined by us, without involving ourselves in contradictions.

But if with the injunction to avoid all transcendent judgments of the pure Reason, we connect the apparently contradictory injunction to proceed to conceptions lying outside the field of its immanent (empirical) use, we shall be aware that both may subsist together, but only on the exact boundary of all admissible use of the Reason; for this belongs as much to the field of experience as to that of essences of thought, and we shall be taught thereby, at the same time, how the above remarkable ideas serve simply, for the determination of the boundaries of the human Reason; namely, on the one hand not to extend cognition of experience in an unbounded manner, so that nothing but mere world remains for us to cognise, and on the other hand not to pass beyond the boundaries of experience, or to seek to judge of things outside the latter as things in themselves.

But we keep to this boundary when we limit our judgment to the relation the world may have to a Being, whose conception itself lies outside all the cognition of which we are capable within the world. For in this case, we do not attribute to the Supreme Being any of the qualities in themselves by which we cogitate objects of experience, and thereby avoid the dogmatic anthropomorphism; but we apply the relations of the same to the world, and thereby allow ourselves a symbolical anthropomorphism, which as a matter of fact only concerns the language and not the object.

When I say we are obliged to regard the world as though it were the work of a supreme understanding and will, I do not really say more than—as a watch, a ship, a regiment is related to the artisan, shipbuilder or general, so is the sense-world (or all that which constitutes the foundation of this sum-total of phenomena) [related] to Edition: current; Page: [107] the unknown, that I cognise, not indeed according to what it is in itself, but according to what it is for me, namely, in respect of the world, of which I am a part.

§ 58.

Such a cognition as this is one according to analogy, which does not signify an imperfect resemblance of two things, as the word is commonly taken [to mean], but a perfect resemblance of two relations between totally dissimilar things.1 By means of this analogy a, for us, adequately defined conception of the Supreme Being remains, although we have left out everything that could determine it simply, and in itself; for we define it as regards the world, and therefore as regards ourselves, and more is not necessary for us. The attacks Hume makes on those who would define this conception absolutely, in that they borrow the materials from themselves and from the world, do not affect us; and moreover he cannot reproach us that there remains nothing over, after the objective anthropomorphism of the conception of the Supreme Being is taken away.

For at the outset, let the deistic conception of an original Being be conceded us as a necessary hypothesis (as Hume does in his Dialogues, in the person of Philo against Cleanthes), in which the original Being is Edition: current; Page: [108] conceived through purely ontological predicates, of substance, cause, &c. This must be done, because the Reason is impelled in the sense-world by mere conditions, which are themselves again conditioned, without the possibility of any satisfaction; it can also be very well done, without lapsing into anthropomorphism, which transfers predicates from the sense-world to a Being quite distinct from the world, inasmuch as these predicates [in our case] are mere categories, affording no definite [conception at all], and hence no conception of it limited to conditions of the sensibility. Nothing can hinder us, therefore, from predicating of this Being a causality through Reason in respect of the world, and so from passing over to Theism without being obliged to attribute to it this Reason, as a quality attaching to it in itself. For as regards the first point, the only possible way of pursuing the use of the Reason in respect of all possible experience in the sense-world, to its highest extent and in thorough agreement with itself, is when a supreme Reason is assumed as a cause of all connections in the world. Such a principle must be throughout advantageous to it, and can never injure it in its natural use. But secondly, the Reason is not transferred as a quality to the original Being in itself, but only in its relation to the sense-world, and thus anthropomorphism is altogether avoided. For here, only the cause of the form of Reason everywhere met with in the world is considered, and to the Supreme Being, so far as it is the ground of this form of Reason in the world, Reason is attributed, but only on the principle of analogy, i.e., in so far as this expression [viz., Reason] indicates what the, to us, unknown ultimate cause of the world has wherewith to determine all things therein, in the highest degree, in accordance with Reason. In this we take care to make use of the quality of Reason, not by its means to conceive God, but [rather] the world, as it is necessary to have the greatest possible use of the Reason in respect of the latter [determined] according to a principle. We confess thereby that the Supreme Being, as to what it may be in itself, is entirely impenetrable to us, and is even unthinkable in a definite manner, and hence we are prevented from making Edition: current; Page: [109] any transcendent use of our conceptions, derived from the Reason as an efficient cause (by means of the will), for determining the divine nature, by qualities that are only borrowed from human nature, and thus from losing ourselves in gross or chimerical conceptions; but, on the other hand, [we are prevented] from inundating the view of the world, [attained] by our conceptions of the human Reason as transferred to God, with hyperphysical modes of explanation, and thus from degrading it, from its proper destination according to which it ought to be a study of mere Nature through the Reason, and not a presumptuous derivation of its phenomena from a supreme Reason. The expression suited to our feeble conceptions will be that we conceive the world as though it orginated from a supreme Reason, as to its reality and as to its inward determination, by which we partly recognise the constitution belonging to it, the world itself, though without presuming to wish to define its cause in itself; and partly, on the other hand, place the ground of this constitution in the relation of the supreme Cause to the world ([viz.] to the form of Reason in the world), without finding the world adequate for this purpose by itself.1

In this way the difficulties seeming to oppose Theism vanish, in that to the principle of Hume, not to push the use of the Reason dogmatically beyond all possible experience, another principle is united, completely overlooked by Hume, namely, not to mistake the field of possible experience for that which bounds itself in the eye of our Reason. Critique of Reason here signifies the true middle path between the dogmatism Hume combated, and the scepticism he would have introduced in its place, a middle path which is unlike other middle paths that attempt to determine themselves as it were mechanically (by taking something from one and something Edition: current; Page: [110] from another), and by which no one is taught a better way, but one, such as can be determined accurately, according to principles.

§ 59.

I have made use of the metaphor of a boundary at the commencement of this observation, in order to fix the limits of the Reason in respect of its appropriate use. The sense-world contains merely phenomena, which are not things in themselves, yet the understanding must assume the latter (noumena), for the very reason that it recognises the objects of experience for mere phenomena. In our Reason both are alike included, and the question is: How does the Reason proceed in determining both fields? Experience, which contains all that belongs to the sense-world, is not bounded by itself; it only attains from one conditioned to another conditioned. That which shall bound it must lie wholly outside it, and this is the field of pure essences of the understanding. But this is for us a blank space, in so far as the determination of the nature of these essences of the understanding is concerned, and thus, when we have to do with dogmatically defined conceptions, we cannot pass beyond the field of possible experience. But as a boundary is itself something positive, belonging as much to what is within as to the space without a given content, so it is a really positive cognition, in which the Reason merely participates, by extending itself to this boundary, in such wise, that it does not attempt to go beyond the boundary, because it finds a blank space before it, wherein it can indeed cogitate forms to things, but cannot cogitate things themselves. But the bounding of the field of experience by something otherwise unknown to it, is a cognition remaining to the Reason in this standpoint, whereby it is not enclosed within the sense-world, neither is it left dreaming [schwärmend] outside it, but limits itself, as befits the knowledge of a boundary, to the relation of that which lies outside the same, to that which is within it.

Natural theology is a conception of this nature, at the Edition: current; Page: [111] boundary of the human Reason, inasmuch as it sees itself necessitated to look beyond to the idea of the Supreme Being (and in a practical connection, also, to that of an intelligible world), not in order to determine anything in respect of this mere essence of the understanding, in other words, anything outside the world of sense, but to guide itself for its own use within the latter, according to principles of the greatest possible unity (theoretically as well as practically). And for this purpose it makes use of the reference of the same to an independent Reason as the cause of all these connections, thereby not merely inventing a being, but inasmuch as outside the world something must necessarily exist (anzutreffen sein) which only the understanding cogitates, determining it [viz., this being] in the above manner, although only on the principle of analogy.

In this way our original proposition remains, which is the result of the whole Critique: “that our Reason can never teach more by its principles à priori than simply objects of possible experience, and even of these no more than what can be cognised in experience.” But this limitation does not prevent it from leading us to the objective boundary of experience, namely, the reference to something which is not itself object of experience, but is nevertheless the highest ground of all experience, without however teaching us anything respecting this in itself, but only with reference to its [viz., the Reason’s] own complete use as directed to its highest end, within the field of possible experience. But this is also all the use that can be reasonably expected or even wished, as concerns it, and with this we have cause to be content.

§ 60.

Thus we have fully exhibited metaphysics according to its subjective necessity, as it is really given in the natural disposition of the human Reason, and indeed in what constitutes its essential purpose. We have found in the course of this investigation, that such a merely natural use of such a disposition of our Reason involves us in extravagant dialectical conclusions, partly apparently, and partly really, conflicting [with one another], if no Edition: current; Page: [112] discipline bridles it and keeps it within limits, which is only possible by means of scientific criticism. And, in addition, [we have found] this fallacious metaphysics to be dispensable to the promotion of the knowledge of Nature, and even prejudicial to it. It always remains, notwithstanding, a task worthy of research, to find out the natural ends aimed at by this disposition in our Reason to transcendent conceptions, since everything in Nature must have been originally designed for some useful purpose.

Such an investigation is here out of place; I confess, moreover, that all I here say respecting the primary ends of Nature is only conjecture, but which may be permitted me in this case, as the question does not concern the objective validity of metaphysical judgments, but refers merely to the natural disposition to the latter, and thus lies outside the system of metaphysics, in that of anthropology.

When I compare all transcendental ideas whose content constitutes the special problem of the natural, pure Reason, compelling it to leave the mere contemplation of Nature and to pass beyond all possible experience, and in this endeavour to produce the thing (be it knowledge or nonsense) called metaphysics, I believe myself to have discovered that this natural disposition is intended to free our conceptions from the chains of experience and the limits of the mere contemplation of Nature, in so far that it may at least see a field opened before it, containing mere objects for the pure Reason, which cannot be arrived at by any sensibility. The purpose is not, indeed, to occupy ourselves speculatively with these objects, (because we can find no firm ground for our feet), but because practical principles, without finding such a space before them for their necessary expectation and hope, could not expand themselves to the universality, the Reason indispensably requires, from a moral point of view.

Now, I find that the psychological idea, however little may be the insight I obtain by its means into the pure nature of the human soul, which is raised above all conceptions of experience, at least sufficiently shows me the inadequacy of the latter, and thereby preserves me from Edition: current; Page: [113] materialism as being a psychological conception of no avail for the explanation of Nature, and besides, as narrowing the Reason in its practical aspect. In the same way the cosmological ideas, by the obvious inadequacy of all possible knowledge of Nature to satisfy the Reason in its justifiable inquiries, serve to keep us from the Naturalism which proclaims Nature for self-sufficing. Finally, as all natural necessity in the sense-world is invariably conditioned, inasmuch as it always presupposes dependence of things on one another, and, as unconditioned necessity must be sought for in the unity of a Cause separate from the sense-world, (but the causality of which, if it were mere Nature, could yet never render comprehensible the existence of the contingent as its consequence;) [this being so,] the Reason frees itself by means of the theological idea from fatalism, as well from that of a blind natural necessity in the coherence of Nature, without a first principle, as in the causality of this principle itself, and leads to the conception of a cause through freedom, in other words, a supreme intelligence. Thus the transcendental ideas serve, if not to instruct us positively, at least to do away with the audacious assertions of materialism, naturalism, and fatalism, which narrow the field of the Reason, and thereby to procure a place for moral ideas outside the region of speculation; and this, as it seems to me, will in some measure explain the above natural disposition.

The practical utility a merely speculative science may have, lies outside the boundaries of this science, and hence can be merely viewed as a scholium, and, like all scholia, not as forming a part of the science itself. At the same time, this reference lies at least within the boundaries of philosophy, especially of that which draws from the sources of pure Reason, where the speculative use of the Reason in metaphysics must have a necessary unity with its practical use in morals. Hence the unavoidable dialectic of the pure Reason in metaphysics must be considered as natural disposition—not merely as an illusion requiring to be resolved, but as a natural institution, as concerns its end—deserving, if possible, to be explained, Edition: current; Page: [114] although this task, being supererogatory, cannot in justice be claimed of metaphysics proper.

As a second scholium, more related to the content of metaphysics, the solution of the problems must be regarded which are discussed in the Critique from pp. 410 to 432. For certain principles of Reason are there expounded, determining the order of Nature, or rather the understanding, which is to seek out her laws through experience, à priori. They seem to be constitutive and legislative in respect of experience, whereas they arise from mere Reason, which cannot be regarded like the understanding as a principle of possible experience. Now whether this agreement rests upon the fact that just as Nature is not itself dependent on the phenomena or their source, the sensibility, but only on the relation of the latter to the understanding; so the thorough-going unity of its use, for the sake of a complete possible experience (in a system), can only pertain to this understanding in its relation to the Reason—whether experience, in other words, stand mediately under the legislation of the Reason—[is a question which] may be further considered by those who desire to investigate the nature of the Reason, apart from its use in metaphysics, and to construct a systematic history of Nature upon general principles. This question I have indeed noticed as important in the book itself, although I have not attempted its solution.1

And thus I conclude the analytical solution of the problem I had myself proposed—How is metaphysics at all possible? having proceeded from that in which its use is really given, at least in its consequences, to the grounds of its possibility.

Explanation I.

Matter is the movable in space; space, which is itself movable, is termed material or relative space; that in which all motion must in the last resort be conceived (which is therefore itself absolutely immovable), is termed pure or absolute space.

Explanation II.

Motion of a thing is the change of the external relations of the same to a given space.

Explanation III.

Rest is the permanent present (præsentia perdurabilis) in the same place; permanent is that which exists throughout a time, i.e. lasts.

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 Edition: current; Page: [156] 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 Edition: current; Page: [157] 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.

Explanation IV.

To Construct the conception of a composite motion means to present à priori in intuition a motion so far as it arises from two or more given [motions] united in one movable.

Explanation V.

The composition of motion is the presentation of the motion of a point as bound together in one with two or more motions of the same.

Proposition 1.

The composition of two motions of one and the same point, can only be conceived by one of them being presented in absolute space, but, instead of the other, a motion of an equal velocity in the contrary direction of the relative space [being presented] as identical with it.

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.

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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 Edition: current; Page: [164] 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.

Explanation I.

Matter is the movable, in so far as it fills a space. To fill a space means to resist everything movable, which endeavours by its motion to press into a certain space. A space that is not filled is an empty space.

Proposition 1.

Matter fills a space, not by its mere existence, but by a special moving force.

Explanation II.

Attractive force is that moving force whereby a matter may be the cause of the approach of others to itself (or, which is the same thing, whereby it opposes the retreat of others from itself).

Repulsive force is that whereby a matter can be the cause of repelling others from itself (or, which is the same thing, whereby it resists the approach of others to itself). The latter we shall also sometimes term driving, and the former, drawing force.

Proposition 2.

Matter fills its spaces by the repulsive forces of all its parts, i.e., by its own force of extension, which has a definite degree, beyond which smaller or larger [degrees] can be conceived to infinity.

Explanation III.

A matter penetrates another in its motion when it completely abolishes the space of its extension by compression.

Explanation IV.

The impenetrability of matter, resting on resistance, which increases proportionately to the degree of the compression, I term relative; but that which rests on Edition: current; Page: [175] the assumption that matter, as such, is capable of no compression at all, is termed absolute impenetrability. The filling of space with absolute impenetrability may be termed mathematical; that with merely relative [impenetrability] dynamical filling of space.

Explanation V.

Material substance is that in space, which for itself, namely, separated from all else existing outside it in space, is movable. The motion of a part of matter whereby it ceases to be a part, is separation. The separation of the parts of a matter is physical division.

Proposition 4.

Matter is divisable to infinity into parts, of which each is again matter.

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 Edition: current; Page: [178] 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 Edition: current; Page: [179] 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.

Proposition 5.

The possibility of matter requires a force of attraction, as its second essential fundamental force.

Proposition 6.

By mere attraction, without repulsion, no matter is possible.

Explanation 6.

Contact in the physical sense is the immediate action and reaction of impenetrability. The action of one matter upon another outside contact is action at a distance (actio in distans). This action at a distance, which is also possible without a medium between matters lying within oneanother, is called immediate action at a distance, or the action of matter on another [matter] through empty space.

Proposition 7.

The attraction essential to all matter is an immediate effect of it on other matter, through empty space.

Explanation 7.

A moving force, by which matters can directly act upon one another only in the common surface of contact, I call a superficial force; but that whereby one matter can directly act on the parts of the other beyond the surface of contact, a penetrative force.

Proposition 8.

The original attractive force, on which the possibility of matter itself as such rests, extends itself directly throughout the universe to infinity, from every part of the same to every other part.

General Note to the Dynamics.

If we review all [our] discussions on the above, we shall observe that the following things have been taken into consideration: Firstly, the real in space (otherwise called the solid) in its filling through the force of repulsion; Secondly, what, in respect of the first, as the proper object of our external perception, is negative, namely, the force of attraction, by which, so far as may be, all space is penetrated, [or], in other words, the solid, is wholly abolished; Thirdly, the limitation of the first force by the second, and the thence resulting determination of the degree of a filling of space; [we shall observe] therefore that the quality of matter has been thoroughly dealt with, under the heads of reality, negation, and limitation, in so far as they belong to a metaphysical dynamics.

General Observation on Dynamics.

The universal principle of the Dynamics of material nature, that all [that is] real in the objects of our external sense, that, namely, which is not mere determination of space (place, extension and figure), must be regarded as moving force; by which, therefore, the so-called solid, or absolute impenetrability, is banished from natural science as an empty conception, and in its stead a repulsive force is posited; while the true and immediate attraction is defended against all the sophistries of a metaphysics that misunderstands itself, and is explained as a fundamental force necessary even to the possibility of the conception of matter. Now from this the consequence arises, that space, should it be found necessary, could be assumed as throughout, and at the same time in different degrees, filled even without distributing empty mediate spaces within the matter. For according to the originally varying degree of the repulsive forces on which is founded the first property of matter, namely, that of filling a space, its relation to the original attraction (whether of each matter for itself, or to the united attraction of all matter in the universe) is conceived as infinitely diverse, inasmuch as attraction rests on the mass of matter in a given space Edition: current; Page: [200] while its expansive force [rests] on the degree in which it fills it [viz., the space], which can be specifically very different (as for instance the same quantity of air, in the same volume, exhibits greater or less elasticity, according to its higher or lower temperature). The general ground of this is that by true attraction all parts of matter act directly on all parts of other matter, but through expansive force only those on the surface of contact, owing to which it is the same, whether behind this, much or little of the matter exists. From the above, however, a great advantage for Natural Science arises, by its being relieved of the burden of having to manufacture a world from fullness and emptiness, merely according to fancy, and being able rather to conceive all spaces as full, and yet as filled in varying amount, by which empty space at least loses its necessity, and is relegated to the rank of an hypothesis; whereas otherwise, under the pretext of being a necessary condition to the explanation of the varying degree of the filling of space, it might lay claim to the title of a principle.

With all this the advantage of a methodically-employed metaphysic to the detriment of equally metaphysical principles, but such as have not been subjected to the test of criticism, is apparently only negative. But indirectly, notwithstanding, the field of the investigator of Nature is extended, since the conditions, by which it previously limited itself, and whereby all original forces of motion were philosophised away, now lose their validity. But one must guard against going beyond what the universal conception of a matter in general renders possible, and seeking to explain its particular or specific definition and variety à priori. The conception of matter is reduced to mere moving forces, and this could not be expected to be otherwise, seeing that in space no activity—no change—can be thought of, except as motion. But who can comprehend the possibility of fundamental forces? They can only be assumed, if they inevitably belong to a conception of which it is demonstrable that it is a fundamental conception which cannot be deduced from any other (as that of the filling of space), and of this [nature] is the force of repulsion, Edition: current; Page: [201] and the opposing force of attraction, [considered] generally. We can indeed judge of this, their connection and consequences well enough à priori, whatever their relations among each other may be conceived to be, provided they do not contradict themselves; but [must] not lay claim to assume either of them as real, because to the admissibility of constructing an hypothesis, it is indispensably requisite that the possibility of what is assumed be quite certain, while with fundamental forces, their possibility can never be comprehended. And in this, the mathematico-mechanical mode of explanation has an advantage over the metaphysico-dynamical, which cannot be taken from it—namely, that from a completely homogeneous material, through the manifold form of the parts, by means of empty mediate spaces interspersed, it can accomplish a great specific multiplicity of matters, in density no less than in mode of action (if foreign forces be superadded). For the possibility of the forces, as wel as of the empty mediate spaces, admit of demonstration with mathematical evidence; on the other hand, if the matter itself be transformed into fundamental forces (to define the laws of which, à priori, we are not in a position, and still less to indicate confidently a multiplicity of the same, sufficient for the explanation of the specific variety of matter), all means are wanting for the construction of this conception of matter, and for presenting as possible, in intuition, what we conceived in general. But a mere mathematical physics, pays for the foregoing advantage doubly on the other side, in that it first of all lays at its foundation an empty conception (that is, absolute impenetrability), and secondly that it must give up all the proper forces of matter, in addition to its original configuration of the fundamental matter and interspersion of empty spaces, and, after having called forth the need for explanation, must concede more freedom to the imaginative faculty in the field of philosophy—[and concede it] indeed as legitimate claim—than is consistent with the caution of the latter.

Instead of an adequate explanation of the possibility of matter and its specific variety, from the fundamental forces, which I am unable to furnish, I shall, as I hope, present the momenta to which its specific variety must Edition: current; Page: [202] admit of being reduced, completely in its totality à priori (although [I cannot] conceive its possibility in the same way). The observations inserted between the definitions will explain their application.

1. A body in a physical signification, is a matter between definite boundaries (which therefore has a figure). The space between these boundaries considered as to its size, is the content of space (volume). The degree of the filling of a space of definite content is termed density. Otherwise the expression dense is used absolutely, for that which is not hollow (bladdery, perforated). In this sense there is an absolute density in the system of absolute impenetrability, if a matter contains no empty mediate spaces. According to this conception of the filling of space comparisons are instituted, and one matter containing less emptiness within itself is called denser than another, till at last, that in which no part of the space is empty is termed perfectly dense. The latter expression can only be made use of, on the mere mathematical conception of matter, for in the dynamical system of a simply relative impenetrability there is no maximum or minimum of density, and any matter however thin can equally be termed fully dense if it wholly fill its space, without containing empty mediate spaces; in other words, if it be a continuum and not an interruptum; but it is in comparison with another [matter], less dense in a dynamical sense, if, although it fill its space wholly, it does not do so in an equal degree. Yet even in the latter system, it is awkward to conceive a relation of matters according to their density, unless they are represented as specifically homogeneous among one another, so that one can be generated from the other merely by mutual pressure. As now, the latter does not appear to be absolutely requisite to the nature of all matter in itself, no comparison can properly be made between heterogeneous matters in respect of their density, as for instance, between water and quicksilver, although this is commonly done.

II. Attraction, in so far as it is merely conceived as active in contact, is called cohesion [zusammenhang]. It is demonstrated by very good experiments, that the same force, called cohesion in contact, is found active at a very small distance; but attraction is only called cohesion, Edition: current; Page: [203] in so far as I think of it only in contact, in accordance with common experience by which it is hardly perceived at small distances. Cohesion is commonly assumed as an altogether universal property of matter, not because we are led to it through the mere conception of a matter, but because experience presents it everywhere. But this universality must not be understood collectively, as though every matter, through this kind of attraction, acted at the same time on every other [matter] in the universe—in the same way as gravitation—but merely disjunctively, namely on one or the other, it does not signify what kind of matters they may be, that come in contact with it. For this reason, and since this attraction, as is demonstrable on various grounds, is not a penetrating but only a superficial force, inasmuch as it is not itself regulated on all sides according to the density—since to complete strength of cohesion a preceding state of fluidity of the matters and their subsequent solidification is requisite, and the closest contact of broken but hard matters in the same surfaces, with which they previously firmly cohered (as for instance a looking-glass where there is a crack), do not any longer admit the degree of attraction which they received on solidifying after their fluid [state—for this reason] I hold this attraction in contact to be no fundamental force of matter, but only a derivative one; of which more hereafter. A matter whose parts, notwithstanding their strong cohesion among one another, can be impelled by every moving force—be it never so small—past one another, is fluid. But parts of a matter are impelled past one another, if, without diminishing the quantum of contact, they are obliged to change [places] among one another. Parts, in other words, matters, are separated if their contact is not merely changed with others but destroyed, or its quantum diminished. A firm—better a solid—body (corpus rigidum) is that whose parts cannot be impelled past one another by every force, and which consequently resist impulsion with a certain degree of force.

The obstacle to the impulsion of matters past one another is friction.

The resistance to separation of matters in contact is cohesion. Fluid matters, therefore, suffer no friction in their division; but where this is met with, the matters Edition: current; Page: [204] are assumed as solid, in greater or less degree, of which the smallest is termed adhesiveness (viscositas), at least in its lesser parts. The solid body is brittle, if its parts cannot be impelled past one another without breaking, in other words when its cohesion cannot be changed without being at the same time destroyed. The distinction between fluid and solid matters is very incorrectly placed in the different degree of the cohesion of their parts. For to call a body fluid does not depend on the degree of its resistance to rupture, but only on [its resistance] to the impulsion of its parts past one another. The former may be as great as one chooses, but the latter is always in a fluid matter = 0. Let us contemplate a drop of water. If a molecule within the same be drawn on one side, by never so great an attraction of the neighbouring parts, touching it, it will be drawn exactly as much toward the opposite side, and as the attractions reciprocally abolish their effects, the molecule is just as easily movable as if it existed in empty space. The force namely, which is to move it, has no cohesion to overcome, but only the so-called inertia which it would have to overcome with all matter, even if it did not cohere at all. A small microscopical animalcule would therefore move itself as easily within this drop as if there were no cohesion to overcome. For in reality it has not any cohesion of the water to abolish, nor to diminish its contact within itself, but only to change it. But conceive this animalcule as wanting to work its way through the outer surface of the drop; it is then first to be observed, that the reciprocal attraction of the parts of this drop of water cause them to move themselves, until they have attained the greatest contact among one another, in other words, the smallest contact with empty space, that is, have constituted a globular form. If now, the said insect be endeavouring to work its way beyond the surface of the drop, it must change this globular form, and consequently effect more contact of the water with the empty space and hence less contact of the parts among one another, that is, diminish its cohesion; and now for the first time the water resists it through its cohesion, though [even now] not within the drop, for here the contact of the parts among one another is in no way Edition: current; Page: [205] lessened, but only changed in their contact with other parts, in other words, not separated, but only shifted. One may therefore, and indeed for similar reasons, apply to this microscopical animalcule, what Newton says of the lightray; that it cannot be repelled through dense matter, but only through empty space. It is thus clear that the increase of the cohesion of the parts of a matter does not in the least affect its fluidity. Water coheres in its parts much more strongly than is commonly believed, when an experiment with a metal plate drawn off from the surface of the water is relied upon, which decides nothing, because the water does not split in the whole surface of the original contact, but from a much smaller surface resulting from the shifting of its parts, just as a stick of soft wax when a weight is suspended at the end, becomes gradually thinner, and is then torn off from a much smaller surface than the original one. What, however, is quite decisive with respect to our conception of fluidity is this, that fluid matters can be explained as those of which every point seeks to move itself in all directions with the same force, with which it is impressed towards any one [in particular]; a property, upon which the first law of hydro-dynamics rests, but which can never be attributed to an aggregation of smooth and at the same time solid particles, as a very slight removal of its pressure according to the laws of composite motion will show, and thereby prove the originality of the property of fluidity. If now the fluid matter should suffer the least hindrance to impulsion, in other words the smallest friction, this would grow with the strength of the pressure with which the parts were pressed against one another, and finally a pressure would obtain, by which the parts of this matter would not admit of impulsion past one another, by every small force. For instance, in a bent tube, [composed] of two pieces, of which the one may be as wide as one chooses, the other as narrow as one chooses, provided it is not a mere hair-tube—if one supposes both pieces to be some hundred feet high, the fluid matter in the narrow one would stand just as high as that in the wide, according to the laws of hydrostatics. But because the pressure on the bottom of the tubes, and hence on the part uniting both Edition: current; Page: [206] these tubes (which stand in communication), can be conceived as in proportion to the heights increasingly greater to infinity, so, if the least friction between the parts of the fluid took place, a height of the tubes must be able to be found, by which a small quantity of water, poured into the narrow one, would not move that in the wide one out of its place, in short, [by which] the column of water in the latter would come to stand higher than that in the former, inasmuch as the lower parts, with such great pressure against one another, would not any longer admit of impulsion, by so small a moving force as the added weight of water—[a cohesion] which is opposed to experience, and even to the conception of the fluid. The same may be said if, instead of pressure by weight, the cohesion of the parts be posited, it matters not how great it may be. The second definition of fluidity cited, upon which the fundamental law of hydrostatics rests, namely, that it is the property of a matter by which every part of the same endeavours to move itself towards all sides with the same force with which it is impressed in a given direction, follows from the first definition, if the fundamental principle of universal dynamics be combined with it, that all matter is originally elastic, since it must endeavour to extend itself—that is (if the parts of a matter admit of being impelled past one another by every force without hindrance, as is actually [the case] with fluids), to move itself—towards all sides of the space in which it is compressed, with the same force with which the pressure in any [given] direction, whichever it may be, is exercised. There are therefore properly only the solid matters (the possibility of which requires another ground of explanation beside the cohesion of the parts), to which friction can be attributed, and the friction already presupposes the property of solidity. But why certain matters, although possessing not a larger, it may be even a smaller, force of cohesion, than fluid [matters], resist notwithstanding so powerfully the shifting of their parts, as not to admit of separation otherwise than by the abolition of the cohesion of all parts at once in a given surface, whereby the appearance of a pre-eminent cohesion is afforded—in short, how rigid bodies are possible—is still Edition: current; Page: [207] an unsolved problem, in spite of the ease with which ordinary natural science believes itself to dispose of it.

3. Elasticity (spring-force) is the capacity of a matter, to reassume its size or shape [which has been] altered by another moving force, on the cessation of the latter. It is either expansive or attractive elasticity; the former in order after compression to assume the previously greater [volume], the latter in order after expansion [to assume] the previously smaller volume. The attractive elasticity, as the expression itself shows, is obviously derived. An iron wire stretched by weights appended, springs, if the connection is cut, back into its [original] volume. By virtue of this attraction, which is the cause of its cohesion (or with fluid matters, [as?] when the heat is suddenly withdrawn from quicksilver), their matter hastens to assume again the previous smaller volume. The elasticity which consists in rehabilitation of the previous figure, is always attractive, as in a bent sword-blade, where the parts on the convex side which are forced back, seek to recover their former proximity, and in the same way a small drop of quicksilver may be called elastic. But the expansive elasticity may be original or it may be derivative. Thus the air has a derivative elasticity, by means of the matter of heat which is most intimately united with it, and the elasticity of which is perhaps original. On the other hand, the fundamental material of the fluid which we term air, must nevertheless as matter generally already have elasticity in itself, which may be called original. Of what kind a perceived elasticity may be, is not possible to decide with certainty in cases as they arise.

4. The effect of moved bodies on one another through the communication of their motion is termed mechanical; but that of matters, in so far as they change the combination of their parts reciprocally by their own forces while at rest, is termed chemical. This chemical influence is termed solution [auflosung] in so far as it has for its effect the separation of the parts of a matter; (mechanical division, as for instance a wedge driven between the parts of a matter, is thus, since the wedge does not act by its own force, entirely different from chemical [division]); but that which has for its effect the severance of two matters resolved by one another, is [chemical] Edition: current; Page: [208] analysis. The solution of specifically distinct matters by one another, in which no part of the one is met with, that is not united with a part of the other specifically distinct from it in the same proportion as the whole, is absolute solution, and may also be termed chemical penetration. Whether the resolving forces really discoverable in nature, are capable of effecting a complete solution may remain undiscussed. Here the question is only whether such admit of being conceived. Now it is obvious that so long as the parts of a resolved matter are still particles (moleculœ), a solution of them is not less possible than of the larger, indeed that this must really proceed, if the resolving force continue, until there is no part left, that is not compounded of the medium of solution and the matter to be resolved in the proportion in which they each stand to one another in the whole. As, then in such a case, there can be no part of the volume of the solution, not containing a part of the resolving medium, this must also, as a continuum, completely fill the volume. In the same way, as there can be no part of this volume of solution, that does not contain a proportional part of resolved matter, this must also, as a continuum, fill the whole space, constituting the volume of the mixture. But when two matters, each of them, entirely fill one and the same place, they penetrate one another; hence a perfect chemical solution would be a penetration of the matter, which nevertheless would be wholly distinguished from the mechanical, inasmuch as by the latter it would be conceivable that with the greater approach of moved matters, the repulsive force of the one might entirely counterbalance that of the other, and one or both reduce its extension to nothing. On the contrary, here, the extension remains, only that the matters [are] not outside, but within one another, i.e. occupy by intersusception (as it is usually termed) together a space equal to the sum of their densities. Against the possibility of this perfect solution, and hence of chemical penetration, it is difficult to allege anything, although it involves a complete division to infinity, for this in the present case contains no contradiction, as the solution takes place continuously throughout time; in other words, through an infinite series of moments, Edition: current; Page: [209] with acceleration; by the division moreover, the sums of the outer surfaces of the matters yet to be divided, grow, and as the resolving force acts continuously, the whole solution may be completed in an assignable time. The incomprehensibility of such a chemical penetration of two matters is to be ascribed to the score of the incomprehensible [nature] of the divisibility to infinity of every continuum, generally. If we depart from this complete solution we must assume it to extend only to certain small particles of the matter to be resolved, which swim in the medium of solution at fixed distances from each other, without our being able to assign the least ground why these particles, as they are still divisible matters, may not in the same way be resolved. For that the medium of solution does not act farther, may always, in nature, so far as experience teaches be true enough; but the question here is of the possibility of a resolving force, which may resolve this particle, and every other that remains over, till the solution is completed. The volume occupied by the solution may be equal to the sum of the spaces occupied by the mutually resolving matters before the mixture, or [it may be] smaller or larger, according to the relation in which the attractive forces stand to the repulsions. They constitute in solution, each for itself and both combined, an elastic medium. This alone, will afford a sufficient reason why the resolved matter does not by its weight separate itself again from the resolving medium. For the attraction of the latter, as it occurs with equal strength toward all sides, abolishes its resistance, and to assume any adhesiveness in the fluid, does not harmonise with the great force exercised by such resolved matters, as for instance, acids diluted with water, on metallic bodies, on which they do not merely rest, as must happen if they simply swam in their medium, but which separate themselves from each other with great attractive force, and diffuse themselves in the whole space of the vehicle. Admitting, moreover, that art has no chemical forces of solution of this kind, capable of effecting a complete solution, in its power, nature might still exhibit them in its vegetal and animal operations and thereby perhaps generate matters, which although indeed mixed, no art could Edition: current; Page: [210] again separate. This chemical penetration might even be met with, where one of the two matters might not be severed by the other, and in a literal sense resolved; as for instance, heat-matter penetrates bodies, since if it only distributed itself in their empty mediate spaces, the solid substance itself would remain cold, since it could not absorb any of it. In the same way, an apparently free passage of certain matters through others could be conceived in such a manner as that of magnetic matter, without preparing for it, to this end, open pores and empty mediate spaces, in all, even the densest matters. But this is not the place to point out hypotheses for special phenomena, but only the principle according to which they are all to be judged. Everything that relieves us of the necessity of having recourse to empty spaces, is a real gain to natural science. For these give far too much freedom to the imagination, to supply the want of accurate knowledge of nature by fancy. Absolute vacuity and absolute density are, in natural science, much the same as blind chance and blind fate in metaphysical science, namely, stumbling-blocks for the investigating reason, by which, either fancy occupies its place, or it is lulled to rest on the pillow of occult qualities.

But as concerns the procedure in natural science in respect of the most important of all its problems, namely, the explanation of a possible specific variety of matters [extending] to infinity, one can only strike out two ways: the mechanical, by the union of the absolutely full with the absolutely empty, or a dynamical way, opposed to it, by explaining all varieties of matters through the mere variety in the combination of the original forces of repulsion and attraction. The first has, as the materials of its deduction, atoms and the void [emptiness]. An atom is a small portion of matter physically indivisible. A matter is physically indivisible, whose parts cohere with a force, capable of being overpowered by no discoverable moving force in Nature. An atom, in so far as it is specifically distinguished from others by its figure, is called a primal body. A body whose moving force depends on its figure is called a machine. The mode of explanation of the specific variety of matters by the construction and composition of their smallest parts Edition: current; Page: [211] as machines is mechanical natural philosophy, but that which derives the specific variety of matter from matters not as machines, that is, mere tools of external moving forces, but from the moving forces of attraction and repulsion originally belonging to them, may be called dynamical natural philosophy. The mechanical mode of explanation, as it is the most available in mathematics, has, under the name of the atomistic or corpuscular philosophy, always retained its reputation and influence on the principles of natural science, with little change from old Demokritos to Descartes, and even our own times. It consists essentially in the presupposition of the absolute impenetrability of the primitive matter, in the absolute homogeneity of this matter, differences only being admitted in the figure, and in the absolute unconquerability of the cohesion of the matter of these fundamental bodies themselves. Such were the materials for the generation of specifically different matters, in order not only to have at hand an unchangeable, and at the same time variously-formed fundamental material for the unchangeableness of species and kinds, but, also from the form of these primal parts, as machines (to which nothing more than an externally impressed force was wanting), to explain the several effects of nature mechanically. The first and most important credential of this system rests, however, on the pretended unavoidable necessity of employing empty spaces for the specific distinction of the density of matters which were assumed as distributed within the matters and between the said particles in [such] proportion as was found necessary, for the sake of some phenomena so large, that the filled part of the volume, even of the densest matter, would be well nigh as nothing, against the empty. In order, now, to introduce a dynamical mode of explanation (which is far more suited and more advantageous to experimental philosophy, inasmuch as it leads directly to the discovery of the proper moving forces of matters and their laws, while it limits the freedom of assuming empty mediate spaces and fundamental bodies of definite figures, neither of which admit of definition or discovery by any experiments) it is by no means necessary to forge new hypotheses, but merely to refute the postulate of the mechanical mode of explanation [namely] that it is Edition: current; Page: [212] impossible to conceive a specific distinction of the density of matters without the intermixture of empty spaces, by the mere citation of a way in which this admits of being conceived without contradiction. For if the postulate in question, on which the mere mechanical mode of explanation stands, be only first declared invalid, as a fundamental principle, it is self-evident that it must not be adopted as a hypothesis in natural science, so long as a possibility remains of conceiving the specific distinction of densities without any mediate spaces. But this necessity rests upon [the fact] that matter does not (as mere mechanical investigators of nature assume) fill its space by absolute impenetrability, but by repulsive force, which has its degree, that may be different in different matters, and as it has nothing in itself, in common with the attractive force, which is regulated by the quantity of the matter, it may be originally different in degree, in different matters with the same attractive force; and consequently the degree of extension of these matters may with the same quantity of matter, and conversely, the quantity of matter with the same volume—i.e., density—admit of very great original specific differences. In this way we should not find it impossible to conceive a matter (as, for instance, the ether is represented), which wholly filled its space, without any void, and yet with incomparably less quantity of matter, at an equal volume, than any bodies which we can subject to our experiments. The repulsive force in ether must, in relation to its proper attractive force, be conceived as incomparably greater than in any other matter known to us. And the only [reason] why we merely assume it, because it can be conceived, is as a foil to a hypothesis (that of empty spaces), which is alone supported by the pretension, that such [viz., matter] does not admit of being conceived without empty spaces. Besides this, no law whatever of the attractive or repulsive force may be risked on à priori conjectures, but everything, even the universal attraction as cause of gravity must, together with its laws, be inferred from data of experience. Still less may such be attempted with chemical affinities, otherwise than by way of experiment. For it lies generally beyond the horizon of our Reason, to comprehend Edition: current; Page: [213] original forces à priori as to their possibility; all natural philosophy consists rather in the reduction of given forces in appearance diverse, to a small number of forces and powers, adequate to the explanation of the effects of the former, but which reduction only extends to fundamental forces, beyond which our Reason cannot proceed. And thus, metaphysical research, behind what lies at the foundation of the empirical conception of matter, is only useful for the purpose of leading natural philosophy so far as is possible to the investigation of dynamical grounds of explanation, as these alone admit the hope of definite laws, and consequently of a true rational coherence of explanations.

This is all that metaphysics can ever accomplish to the construction of the conception of matter—in other words, for the application of mathematics to natural science, in respect of properties whereby matter fills its space in definite amount—namely, to regard these properties as dynamical and not as unconditioned original positions, such for instance, as a mere mathematical treatment would postulate.

The well-known problem as to the admissibility of empty spaces in the world may furnish the conclusion. The possibility of this does not admit of dispute. For to all forces of matter space is requisite, and, as it also contains the conditions of the laws of its diffusion, is necessarily pre-supposed before all matter. Thus, attractive force is attributed to matter, in so far as it occupies a space around itself by attraction, without, at the same time, filling it, which, therefore, even where matter is active, may be conceived as empty, because it is not active by repulsive forces, and hence does not fill it. But, to assume empty spaces as real, no experience, inference from [experience], or hypothesis necessary to its explanation, can justify us. For no experience gives us any but comparatively empty spaces to cognise, which can be perfectly explained, from the property of matter, as filling its space by an expansive force, greater or progressively smaller to infinity, in all possible degrees, without requiring empty spaces.

Explanation 1.

Matter is the movable, in so far as it is something having a moving force.

Explanation 2.

The quantity of the matter is the multitude of the movable in a definite space. This, in so far as all its parts may be considered as at the same time active (moving) in their motion is termed the mass, and it is said a matter acts in mass when all its parts are moved in the same direction, exercising, at the same time, their moving force, outside themselves. A mass of definite figure is called a body in a mechanical sense). The quantity of motion (mechanically estimated) is that which is estimated at once, by the quantity of the moved matter and its velocity; phoronomically it consists merely in the degree of the velocity.

Proposition 1.

The quantity of the matter may be estimated in comparison, with every other, only by the quantity of motion at a given velocity.

Proposition 2.

First law of mechanics.—With all changes of corporeal nature, the quantity of the matter remains, on the whole, the same, unincreased and undiminished.

Proposition 3.

Second law of mechanics.—All change of matter has an external cause. (Every body remains in its state of rest or motion in the same direction and with the same velocity, if not compelled by an external cause to forsake this state.)

Proposition 4.

Third mechanical law.—In all communication of motion, action and reaction are always equal to one another.

Demonstration.

(From universal metaphysics the proposition must be borrowed, that all external action is reciprocal action. In this place it only has to be shown in order to remain within the bounds of mechanics that this reciprocal action (actio mutua) is at the same time reaction (reactio); but, without doing violence to the completeness of the insight, the above metaphysical law of reciprocity nevertheless cannot be left out here. All active relations of matters in space, and all changes of these relations, in so far as they can be causes of certain effects, must always be conceived as reciprocal, that is since all change of the same is motion, no motion of a body, with reference to an absolutely-resting [one] which would be thereby set in motion, can be conceived; but the latter must rather be conceived as only relatively-resting in respect of the space, to which it is referred, but together with this space as moved in the Edition: current; Page: [224] opposite direction with the same quantity of motion in absolute space, as the moved [body] has against it, in the same space. For the change of relation (in other words, the motion) is completely reciprocal between both; by as much as the one body approaches every part of the other, by so much the other approaches every part of the first. And because here the question is not as to the empirical space surrounding both bodies, but only of the line lying between them (inasmuch as these bodies are considered simply in mutual relation, according to the influence, which the motion of the one can have on the change of state of the other, by abstraction of all relation to empirical space), their motion will be regarded as merely determinable in absolute space, in which each of the two bodies must have an equal share of the motion attributed to the one in relative space, since there is no ground for ascribing more to one of them than to the other. On this footing the motion of a body, A, against another, resting, B, with regard to which it may be moving if reduced to absolute space—that is, as the relation of active causes merely referred to one another—is so considered that each has an equal share in the motion, which in the phenomenon is attributed to the body A alone. This cannot occur otherwise, than by the velocity attributed to the body A in the relative space, being distributed between A and B in inverse proportion to the masses, to A only what belongs to it in absolute space, to B, on the other hand, the relative, in addition, in which it rests, in the opposite direction, whereby the same phenomenon of motion is completely retained, the effect in the reciprocity of both bodies being constructed in the following manner:

Let a body A be in motion with a velocity = AB in respect of the relative space towards the body B, which in respect of the same space is resting. Let the velocity AB be divided into two parts, Ac and Bc, which are related to one another inversely as the masses B and A. Conceive A as moved with the velocity Ac, in absolute Edition: current; Page: [225] space, but B with the velocity Bc, in the opposite direction, together with the relative space; both motions are then opposite and equal to one another, and as they reciprocally destroy one another, both bodies are translated with reference to one another, that is, in absolute space, into [a state of] rest. B, however, was in motion with the velocity Bc in the direction BA, which is exactly opposed to that of the body A, namely AB, together with the relative space. If then the motion of the body B is destroyed by impact, the motion of the relative space is not therefore also destroyed. Thus, after the impact, the relative space moves in respect of both bodies A and B (which now rest in absolute space) in the direction BA with the velocity Bc, or, which is the same thing, both bodies move after the impact with equal velocity, Bd = Bc, in the direction of the impacting AB. According to the foregoing, however, the quantity of the motion of the body B in the direction and with the velocity Bc, and hence also that in the direction Bd with the same velocity, is equal to the quantity of the motion of the body A with the velocity and in the direction Ac. Consequently the effect, namely, the motion Bd, which maintains the body B by impact in relative space, and therefore the action of the body A with the velocity Ac, is always equal to the reaction Bc. Since this law (as mathematical mechanics teaches) suffers no alteration, when instead of the impact of a resting, an impact of the same body in the same way on a moved body is assumed; similarly as the communication of motion by impact, is only distinguished from that by traction by the direction in which the matters resist one another in their motion, it follows that in all communication of motion action and reaction are always equal to one another (that no impact can communicate the motion of a body to another except by means of an equal counter-impact, no pressure except by means of an equal counter-pressure, and in the same way no traction except by means of an equal counter-traction).*

General Observation on Mechanics.

The communication of motion only takes place by means of such moving forces, as inhere in a matter at rest (impenetrability and attraction). The action of a moving force on a body in one moment is its solicitation, the velocity acquired by the latter through solicitation, in so far as it increases in equal proportion to the time, is the moment of acceleration. (The moment of acceleration must therefore only contain an infinitely small velocity, as otherwise the bodies would attain through this an infinite velocity in a given time, which is impossible. The possibility of acceleration generally moreover, rests, through a continuous moment of the same, on the law of inertia.) The solicitation of matter through expansive force (e.g., a compressed air that bears a weight) occurs always with a finite velocity; but the velocity impressed thereby on another body (or withdrawn from it) can only be infinitely small; for the former is only a superficial force, or, which is the same thing, the motion of an infinitely small quantum of matter, which must occur consequently with finite velocity in order to be equal to the motion of a body Edition: current; Page: [231] of finite mass with infinitely small velocity (a weight). On the other hand attraction is a penetrating force, by virtue of which, a finite quantum of matter exercises moving force on a similarly finite quantum of another [matter]. The solicitation of attraction must therefore be infinitely small, because it is equal to the moment of acceleration (which must always be infinitely small), while with repulsion, where an infinitely small portion of matter is to impress a moment on a finite [portion] this is not the case. No attraction admits of being conceived with a finite velocity without the matter being obliged to penetrate itself by its own attractive force. For the attraction, which a finite quantity of matter exercises on [another] finite with a finite velocity, must be superior to every finite velocity, whereby matter reacts through its impenetrability, but only with an infinitely small portion of the quantity of its matter, on all points of the compression. If attraction is only a superficial force, as cohesion is conceived, the opposite of this would follow. But it is impossible, so to conceive it, if it is to be true attraction (and not mere external compression).

An absolutely hard body would be one whose parts attracted one another so strongly, that they could not be separated by any weight, nor altered in their position with regard to one another. Now, since the parts of the matter of such a body would have to attract one another with a moment of acceleration, which would be infinite as against that of gravity, but finite as to the mass thereby driven, resistance by impenetrability as expansive force, since it always occurs with an infinitely small quantity of matter, would have to take place with more than finite velocity of solicitation, that is, the matter would seek to extend itself with infinite velocity which is impossible. Thus an absolutely hard body, that is, one which would oppose in one moment a resistance on impact, to a body moved with finite velocity equal to the whole of its force, is impossible. Consequently, a matter exercises by its impenetrability or cohesion only an infinitely small resistance in one moment, to the force of a body in finite motion. Hence follows the mechanical law of continuity (lex continui mechanica), namely: in no body is the state of Edition: current; Page: [232] rest or motion—and in the latter, velocity or direction—changed by impact, in one moment, but only in a certain time, through an infinite series of intermediate states whose difference from one another is smaller than the first and last. A moved body that strikes against a matter, is not brought to rest by its resistance at once, but only by continuous retardations, or that which was at rest only [set in] motion by continuous acceleration, or from one degree of velocity into another according to the same rule. In the same way, the direction of its motion in [a body] that describes an angle, is only changed by means of all possible intermediate directions, that is, by means of motion in a curved line (which law for a similar reason, can be also extended to the change of the state of a body by attraction). This lex continui is based on the law of the inertia of matter, while, on the other hand, the metaphysical law of continuity in all change (internal as well as external) must be extended universally, and hence would be based on the mere conception of a change in general, as quantity, and on the generation of the same (which must necessarily proceed continuously in a certain time, like time itself), and thus has no place here.

Explanation.

Matter is the movable, in so far as it can be an object of experience as such.

Proposition 1.

The rectilinear motion of a matter is, in respect of an empirical space, as distinguished from the opposite motion of the space, a merely possible predicate. The same [thing] conceived in no relation to a matter outside it, that is, as absolute motion, is impossible.

Demonstration.

Whether [in the case of] a body moved in relative space, this latter be described as resting, or conversely, as moved with equal velocity in an opposite direction, and the former as resting, there is no statement as to what belongs to the object, but only to its relation to the subject, in other words, to the phenomenon and not to experience. For if the spectator place himself in the same space as resting, he terms the body moved; but if he place himself (at least in thought) in another space enclosing this, in respect of which the body is, in the same way, resting, then the relative space is termed “moved.” In experience, therefore (a cognition, determining validly the object for all phenomena), there is no difference whatever between the motion of the body in relative space, or the rest of the body in absolute, and the equal and opposite motion of the relative, space. Now the presentation of an object by one of its two predicates—which, in respect of the object, are equivalent, and only as regards the subject and its mode of presentation distinguished from one another—is not its determination according to a disjunctive, but merely an alternative judgment according to choice (of which Edition: current; Page: [235] the first of two objectively opposed predicates, one with the exclusion of its contrary, but the other of objectively equivalent indeed, but subjectively opposed judgments without excluding the contrary of the object, in other words, by mere choice)—one is assumed for the determination of the same [viz., the object].* This means: by the conception of motion as object of experience, it is in itself undetermined, and therefore equivalent, whether a body is conceived as moved in relative space or the space in respect of the body. Now that which, in respect of two mutually opposed predicates, is in itself undetermined, is so far merely possible. Hence the rectilinear motion of a matter in empirical space, as distinguished from the equal opposite motion of the space, is in experience a merely possible predicate, which was the first [point].

Further, since a relation, in other words a change of the same, namely, motion, can only be an object of experience in so far as both correlates are objects of experience—but pure space, also called, in contradistinction to the relative (empirical), absolute space, is no object of experience and nothing at all—therefore rectilinear motion, without reference to anything empirical—that is, absolute motion—is simply impossible;—which was the second [point.]

Proposition 2.

The circular motion of a matter as distinguished from the opposite motion of the space, is a real predicate of the same; while, on the other hand, if the opposite motion of a relative space be taken, instead of the motion of the body, there is no real motion of the latter, but [should it be regarded as such] a mere illusion.

Proposition 3.

In every motion of a body, whereby it is moving in respect of another, an opposite and equal motion of the latter is necessary.

General Observation on Phenomenology.

There are thus three conceptions noticeable here, whose employment in universal natural science is unavoidable, and whose exact definition is for this reason necessary, although not so easy and comprehensible: firstly, the conception of motion in relative (movable) space; secondly, the conception of motion in absolute (immovable) space; thirdly, the conception of relative motion generally, as distinguished from absolute [motion.] The conception of Edition: current; Page: [238] absolute space is laid at the foundation of all [these]. But how do we come by this singular conception, and on what rests the necessity of its employment?

It can be no object of experience; for space without matter is no object of perception, and yet is a necessary conception of the Reason, and therefore nothing but a a mere idea. For in order that motion may be given even as phenomenon, an empirical presentation of space in respect of which the movable has to change its relation is required. But space, which is to be perceived, must be material, and therefore in accordance with the conception of a matter generally, itself movable. Now, in order to conceive it as moved, one has only to conceive it as contained in a space of greater compass, and to assume the latter as resting. But this admits of being treated similarly as regards a still more extended space, and so on to infinity, without ever attaining through experience to an immovable (immaterial) space, in respect of which any matter could have absolute motion or rest attributed to it; but the conception of these relational determinations will have to be constantly changed, according as the movable is considered as in relation to one or the other of these spaces. Now, as the condition of regarding anything as resting or moved is always again and again conditioned to infinity in relative space, it thence appears: firstly, that all motion or rest is merely relative, and that neither can be absolute, i.e., that matter can merely be conceived in relation to matter as moved or resting, but not in respect of mere space without matter; in other words, that absolute motion, such, namely, as is conceived without any reference of one matter to another, is simply impossible: secondly, [it will appear] that for this very reason no conception of motion or rest, in relative space, valid for every phenomenon, is possible, but that a space must be conceived, in which the latter itself can be thought of as moved, but whose determination does not depend on any other empirical space, and hence is not again conditioned, that is, an absolute space to which all relative motions may be referred, and in which everything empirical is movable; [and this] in order that all motions of the material in the same can be valid as merely relative to one another, as alternatively-reciprocal,* Edition: current; Page: [239] but none as absolute motion or rest (since, inasmuch as one is called moved, the other, with reference to which our former is moved, may be similarly conceived as absolutely resting). Absolute space is then necessary, not as a conception of a real object, but as a mere idea which is to serve as a rule, for considering all motion therein as merely relative, and all motion and rest must be reduced to absolute space if the phenomenon of the same is to be transformed into a definite conception of experience (which combines all phenomena).

In this way the rectilinear motion of a body in relative space, is reduced to absolute space, which does not fall within the range of the senses if I conceive the body, as at rest in itself, and this presentation as that which gives precisely the same phenomenon, whereby all possible phenomena of rectilinear motions, which a body may happen at the same time, to possess, are reduced to the conception of experience, which unites them together (namely, to that of merely relative motion and rest).

Circular motion, inasmuch as, according to the second proposition, even without reference to the external empirically given space, it can be given as real motion in experience, seems to be really absolute motion. For the relative in Edition: current; Page: [240] respect of external space (for instance, the motion of the earth on its axis, relative to the heavenly bodies), is a phenomenon, in place of which, the opposite motion of this space (the heavens), in the same time, can be posited as fully equivalent to the former, but which, according to this proposition, can never in experience be put in the place of the former; and therefore the above circular motion cannot be regarded as externally relative, which sounds as though this kind of motion were assumed as absolute.

But it is to be observed that the question is here of the true (real) motion, which does not appear as such—which therefore, were we content to judge according to empirical relations of the space, might be regarded as rest—in other words, the question is of the true motion as distinguished from the illusive, but not of it as absolute motion in contradistinction to the relative; and hence circular motion, although it exhibits in the phenomenon, no change of place, that is, no phoronomic [change] of the relation of the moved to empirical space, exhibits, nevertheless, a continuous dynamic change of the relation of matter in its space, demonstrable by experience; for instance, it shows a constant diminution of the attraction by an effort to retreat, as the effect of circular motion, and thereby decisively indicates its distinction from illusion. For instance, one can conceive the earth as turned about its axis in infinite empty space, and demonstrate this motion by experience, although neither the relation of the parts of the earth among one another, or to the space outside it, is changed phoronomically, i.e., in the phenomenon. For, as regards the first, nothing changes its place upon or in the earth as empirical space; and with reference to the second, which is quite empty, no externally changed relation, and therefore no phenomenon of a motion can take place. But if I suppose a deep cavern tending towards the centre of the earth, and dropping a stone into it, find that although at every distance from the centre, the gravity is always directed thereto, the falling stone nevertheless, continuously reverts from its upright position, from west to east, I conclude that the earth is from evening to morning turned about its axis. Or, if I withdraw the stone from the surface of the earth, and it does not remain over the Edition: current; Page: [241] same point of the surface, but moves itself from east to west, I shall still infer the foregoing motion of the earth on its axis, and both perceptions are a sufficient proof of the reality of this motion, for which the change of relation to external space (the starry heaven) is inadequate as it is mere phenomenon, which may proceed from two actually opposed causes, and which is not a cognition deducible from the ground of explanation of all phenomena of this change, that is, experience. But that this motion, although no change of relation to empirical space, is nevertheless no absolute motion, but continuous change of the relation of matters to one another, and while conceived in absolute space, is really only relative and for this very reason, alone true motion; this rests on the conception of the reciprocally continuous retreat of each part of the earth (outside the axis) from every other [part], situated opposite to it in the diameter, at equal distance from the centre. For this motion is real in absolute space, in that thereby the retreat from the distance in question, when gravity in itself would attract to the body, and indeed without any dynamical repulsive cause (as may be seen from the instances chosen in Newton’s Princ. Phil. Nat., p. 10, Edit. 1711),* is continuously replaced by real motion inclosed within the moved matter (namely, the centre of the same), but not having reference to the external space.

As to the case of the third proposition, it does not require, in order to show the truth of the reciprocally opposed and equal motion of two bodies even without reference to the empirical space, [to exhibit] the active dynamical influence (of gravity or of a distended string) given through experience, which is necessary in the second case, but the mere dynamical possibility of such an influence as property Edition: current; Page: [242] of matter (repulsion or attraction) since the motion of the one carries with it, at the same time, the opposite and equal motion of the other, and indeed from mere conceptions of a relative motion, if it be considered in absolute space, i.e. according to truth; and it is, therefore, like all that is adequately demonstrable from mere conceptions a law of absolutely necessary counter-motion.

There is no absolute motion, even where a body is conceived as moved in respect of another in empty space; the motion of both being here, not relative to the space surrounding them, but only to that between them, which alone determines their external relation to each other, considered as abstract space, and is thus in its turn, only relative. Hence, absolute motion would be only that accruing to a body without relation to any other matter. But such would be the rectilinear motion of the universe, i.e. the system of all matter. For so long as any other matter existed outside of a matter, even though separated by empty space, the motion would still be relative. For this reason every proof of a law of motion, having as its result, that its contrary would necessarily imply a rectilinear motion of the whole universe as its consequence, is an apodictic demonstration of its truth; simply because absolute motion would thence ensue, which is quite impossible. Of this kind is the law of antagonism in all community of matter through motion. For every deviation from the same would move the common centre of gravity of all matters, in short, the whole universe, from its place, while on the contrary this would not happen if one regarded the latter as turned on its axis, a motion always possible to be conceived, although so far as one can see, there would be no use in assuming it.

The different conceptions of empty space also have their reference to the different conceptions of motion and moving forces. Empty space in a phoronomic sense, also termed absolute space, ought not properly to be called empty space; for it is only the idea of a space, in which I abstract from all particular matter, making it an object of experience, in order to conceive therein, the material, or every empirical space, as movable, and the motion not merely as on one side absolute, but as Edition: current; Page: [243] mutually relative predicate. Hence it is nothing belonging to the existence of things, but merely to the determination of the conception, and in so far no empty space exists. Empty space, in a dynamic sense, is that which is not filled, i.e., in which nothing else movable resists the penetration of the movable, consequently in which no repulsive force acts, and it may be either the empty space within the world (vacuum mundanum), or, if the latter be conceived as bounded, empty space outside the world (vacuum extramundanum); the first moreover, either as distributed (vacuum disseminatum), which constitutes only one portion of the volume of the matter, or as continuous empty space (vacuum coacervatum, which separates bodies, for instance, the heavenly bodies, from one another), a distinction which, inasmuch as it rests on the difference of places, assigned to empty space in the universe, is not essential, but is used in various ways; firstly, in order to deduce the specific difference of density, and secondly, in order [to deduce] the possibility of a movement in the universe, free from all external resistance. That empty space in the first sense is not necessary to be assumed, has already been shown in the general remark on dynamics; but that it is impossible can by no means be demonstrated from its conception alone, according to the principle of contradiction. Yet, even if no merely logical ground for its rejection be present, a universal physical ground for banishing it from natural science exists, namely, that of the possibility of the composition of a matter generally, if the latter [question] were only better understood. For if attraction, which is assumed for the explanation of the cohesion of matter, be only apparent, not real, attraction—but as it were the effect of a compression, by external matter (the ether) existing throughout the universe, which is itself brought to this pressure, by a universal and original attraction, namely, gravitation, an opinion supported by many reasons—empty space within matters would then, although not logically, be nevertheless dynamically, and hence physically, impossible, since every matter would expand of itself, in the empty spaces assumed within the same (as nothing would then resist its expansive force), and they Edition: current; Page: [244] would thus be always filled. An empty space outside the world, would, if by this expression be understood all the principal attractive matters (the large heavenly bodies), be impossible, for the same reason, for in proportion as the distance from these increased, the attractive force on the ether (which encloses all the above bodies, and impelled by them maintains in their density by compression), would diminish in inverse proportion, and the latter itself, would diminish in density to infinity, though it would nowhere leave the space entirely empty. Meanwhile, it need surprise no one that in this rejection of empty space, we are proceeding quite hypothetically; for its assumption fares no better. Those who venture to decide this moot question dogmatically, whether they do so affirmatively or negatively, support themselves in the end on mere metaphysical assumptions, as may be seen from the dynamics; but it was at least necessary to show here, that this could not decide in the problem in question. Thirdly, as concerns empty space in a mechanical sense, this is continuous emptiness within the universe, in order to procure free motion for the heavenly bodies. It is easily seen, that the possibility or impossibility of this rests, not on metaphysical grounds, but on the hardly disclosed secrets of nature, as to the way in which matter sets limits to its own expansive force. Notwithstanding this, if that be admitted which has been said in the general observation on dynamics, as to the possibly greater expansion to infinity of specifically different matters, with the same quantity of matter (as regards its weight) an empty space might indeed be then unnecessary to assume, even for the sake of the free and lasting motion of the heavenly bodies, as the resistance, even in entirely filled spaces, might then be assumed to be as small as one liked.

And so ends the metaphysical doctrine of body with emptiness and therefore incomprehensibility, and the reason has the same fortune in all other attempts, where it strives to reach principles of the ultimate grounds of Edition: current; Page: [245] things, inasmuch as its nature is such, that it can never comprehend anything except in so far as it is determined under given conditions; consequently, since it can neither rest at the conditioned nor can make the unconditioned comprehensible, when thirst for knowledge stimulates it, to grasp the absolute totality of all conditions, nothing remains for it but to turn back from objects, upon itself, in order that instead of the ultimate boundaries of things, it may investigate and determine the ultimate boundaries of the capacity pertaining to itself.