Kant's Prolegomena and Metaphysical Foundations of Natural Science/The Metaphysical Foundations of Natural Science/Second Division



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.


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

Proposition 1.

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


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


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

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.


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

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.


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

Note 1.

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

Note 2.

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

Explanation III.

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


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

have occasion to make some mention.

Proposition 3.

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


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


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

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 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.

Observation 1.

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

Observation 2.

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

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.


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

Proposition 4.

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


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

Observation 1.

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

But by the above demonstration, the monadologist is entirely deprived of this resort. For, thereby it is clear, that in a filled space there can be no point that does not itself resist repulsion on all sides in the same way as it is repelled; in other words, as a reacting subject, in itself movable, existing outside every other repulsive point; and hence that the hypothesis of a point filling a space by its mere driving force, and not by means of other equal repulsive forces, is impossible. In order to make this, and thereby also the demonstration of the previous proposition apparent, one must assume that is the place

of a monad in space, that is the diameter of the sphere of its repulsive force, and therefore that is its semi-diameter; so between , 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], , point is possible to be indicated (in accordance with the infinite divisibility of space). Now, if resist that which seeks to impress itself on , must resist both the points and . For if this were not so, they would approach one another with impunity; consequently and would meet in the point , i.e. the space would be penetrated. Something must thus exist in that resists the impression of and , and thus repels the monad as much as it is repelled by it. As now, repulsion is a movement, is something movable in space; in other words, matter, and the space between and , could not be filled by the sphere of the activity of a single monad, neither could the space between and , and so on to infinity.

When mathematicians conceive the repulsive forces of the parts of elastic mailers 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 distance's from one another, because their elasticity stands in inverse proportion to the spaces in winch 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 tho above, all contact can be conceived as an infinitely small cases where a larger or smaller space is to be conceived as entirely tilled by the same quantity of matter, that is, by an identical quantum of repulsive forces. By an infinitely divisible [thing], therefore, no real distance of parts, which, with all extension of the space of the whole, always constitute a continuum, may be assumed, although the possibility of this extension can only be made comprehensible under the idea of an infinitely small distance.

Observation 2.

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

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

Proposition 5.

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


Impenetrability, as the fundamental quality of matter, whereby it first reveals itself as something real in the space of our external senses, is nothing but the capacity of extension in matter (proposition). Now an essentially moving force, by which parts of matter recede from one another, cannot, firstly, be limited by itself, because matter is rather impelled thereby to extend the space it fills continuously; secondly, it cannot be fixed by space alone, at a certain boundary of extension—for though space may contain the ground of [the fact] that with the increase of the volume of a matter extending itself, the extending distance, which, moreover, must necessarily happen in force will become weaker in inverse proportion—yet, inasmuch as smaller degrees of every moving force are possible to infinity, it cannot contain the ground for their ever ceasing. Matter then, by its repulsive force alone (which contains the ground of its impenetrability), and if no other opposing force contradicted this, would be held within no boundaries of extension, that is, would dissipate itself to infinity, and no assignable quantity of matter would be met with in any assignable space. With merely repulsive forces of matter, all spaces would consequently be empty, in other words no matter would properly speaking exist at all. To the existence of all matters, forces opposed to the extending [forces], in other words, compressive forces, are requisite. But these again cannot be sought for originally, in the opposition of another matter, for it requires, in order that it may be matter, itself a compressive force. An original force of matter, working in an opposite direction to the repulsive, in other words [a force] of approach, that is, an attractive force must be assumed. Now as this attractive force belongs to the possibility of a matter, as matter generally, consequently precedes all distinctions of the same, it must not be ascribed merely to a special species [of matter], but to every matter generally and originally. An original attraction then belongs to all matter as a fundamental force pertaining to its essence.


With this transition from one property of matter to another specifically different from it, which yet equally belongs to the conception of matter, although it is not contained therein, the attitude of our understanding must be more closely considered. If attractive force be itself originally requisite to the possibility of matter, why do we not equally make use of it with impenetrability as the primary sign of a matter? why is the last immediately given with the conception of a matter, while the first is not thought in the conception, but only attributed to it, by inference? That our senses do not allow us to perceive attraction so immediately as repulsion and the resistance of impenetrability, does not sufficiently solve the difficulty. For if we had such a faculty, it is easy to comprehend that our understanding would none the less choose the filling of space, in order to indicate thereby the substance in space, namely, matter, just as in this filling, or, as it is otherwise called, solidity, the characteristic of matter as a thing distinct from space, is posited. Attraction, it matters not how well we might feel it, could never reveal to us a matter of definite volume and figure, nor anything beyond the endeavour of our organ to approach a point outside us (the central point of the attracting body). For the attractive force of all parts of the earth can affect us, neither more nor otherwise, than if it were wholly concentrated in its central point, and it were this alone that influenced our sense; similarly with the attraction of a mountain, and of every stone, &c. We should acquire thereby no definite conception of any object in space, as neither figure nor size, nor even the place where it exists, could fall within our senses. The mere direction of the attraction would be able to be perceived as in weight; the attracting point would be unknown, and I do not see how it could be arrived at, through conclusions, without the perception of matter, in so far as it fills space. It is hence clear, that the first application of our conceptions of quantity to matter, by which it is primarily possible for us to transform our external perceptions into the experiential conception of a matter as object generally, is only founded on its property of filling space, which by means of the sense of feeling, procures for us the size and figure of an extended, and therewith a conception of a definite object in space which must be laid at the foundation of all else that one can predicate of any [particular] thing. This is undoubtedly the reason why, with what are the clearest proofs otherwise, that attraction must belong to the fundamental forces of matter, equally as much as repulsion, one is so unwilling to admit it, or to concede any other moving forces but those of impact and pressure (both by means of impenetrability). For that whereby space is filled is substance, it is said, and this is correct enough. But as substance only reveals its existence to us by sense, whereby we perceive its impenetrability, namely by feeling—and therefore only in reference to contact, whose beginning (in the approach of one matter to another) is termed impact, but its continuation pressure—it seems as though the immediate effect of one matter on another could never be anything else but pressure or impact, the only two influences we can immediately feel; while on the other hand attraction, which can give us either no feeling at all, or at least no definite object of it, becomes difficult for us to conceive as fundamental force.

Proposition 6.

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


Attractive force is the moving force of matter, whereby it compels another [matter] to approach it; consequently, when it is met with, between all parts of matter, the matter seeks by means of it to diminish the distance of its parts from one another, and therefore the space that they together occupy. Now nothing can hinder the effect of a moving force, except another moving force opposed thereto, but this [force] that is opposed to it is repulsive force. Thus, without repulsive forces, and by mere approach, all parts of matter would approach one another without hindrance and diminish the space that they occupy. As now, in the case assumed, there is no distance of parts, in which a greater approach through attraction is rendered impossible by a repulsive force, they would move towards one another until no distance existed between them; that is, they would coalesce in a mathematical point, and the space would be empty; in other words, without any matter. Matter is accordingly impossible by mere attractive forces, without repulsive.


That property, on which the inner possibility of a thing rests as its condition, is an essential element therein. Hence repulsive force belongs just as much to the essence of matter as attractive force; and the one cannot be separated

from the other in the conception of matter.


As no more than two moving forces in space, repulsion and attraction, can ever be conceived, it was previously necessary—to prove the union of both in the conception of a matter generally a priori—that each should be considered separately, in order to see what taken singly they could achieve in the presentation of a matter. It is evident now that as well when we lay neither of them at the basis, as when we assume merely one of them, space always remains empty, and no matter exists therein.

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.


Contact, in a mathematical signification, is a common boundary of two spaces, and is hence neither within the one nor the other space. Straight lines therefore cannot touch one another, but when they have a point in common, it belongs as much within the one as the other of these lines, when they are produced, that is, cut one another. But circle and straight line, circle and circle, touch each other in a point, surfaces in a line, and bodies in surfaces. Mathematical contact therefore is laid at the basis of the physical, but does not alone constitute it; in order that the latter may arise, a dynamical relation must be superadded in thought, and that, not of the attractive, but of the repulsive forces, namely, those of impenetrability. Hence physical contact is the reciprocal action of repulsive forces in the common boundary of two matters.

Proposition 7.

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


The original attractive force itself contains the ground of the possibility of matter as that thing which fills a space in a definite degree, in other words of the very possibility of a physical contact. Hence, it must precede this, and its effect must consequently be independent of the condition of the contact. Now, the effect of a moving force is independent of all contact—independent even of the filling of space between the moving and the moved, that is, it must take place without the space between them being filled up, and, therefore, as an effect through empty space. The original and essential attraction of all matter is then an immediate effect of the same upon another [matter] through empty space.

Observation 1.

That the possibility of fundamental forces should be made conceivable is a quite impossible demand: for they are called fundamental forces, precisely because they cannot be deduced from any other, that is, cannot be conceived. But the original attractive force is not one whit more inconceivable than the original repulsion. It does not so immediately obtrude itself on the senses as impenetrability, in affording us conceptions of definite objects in space. Hence, while it is not felt, but only to be inferred, it has the appearance of a deduced force, just as though it were only a hidden play of moving forces [produced by] repulsion. More closely considered, [however,] we see that it cannot be further deduced from any source, least of all from the moving force of matters, through their impenetrability, as its effect is precisely the opposite of the latter. The commonest objection to immediate effect at a distance is, that a matter cannot directly operate where it is not. If the earth directly influences the moon to approach it, the earth acts upon a thing many thousand miles removed from it, and nevertheless [acts] immediately, even though the space between it and the moon were regarded as entirely empty. For, although matter may exist between two bodies, this does not affect the attraction. It acts, therefore, directly, in a place where it is not; something, to all appearance, contradictory. But it is so far from being contradictory, that one might rather say: everything in space acts on another [thing] in a place where the acting [thing] is not. For if it acted in the place where it was itself, the thing on which it acted would not be outside it; for outside signifies presence in a place, where the other is not. If earth and moon touched one another, the point of contact would be a place where neither earth nor moon existed, for they would be removed from one another by the sum of their diameters. In the point of contact, moreover, no portion, either of the earth or of the moon would exist, for this point lies at the boundary of either filled space, which constitutes no portion either of the one or of the other. Thus, that matters cannot act upon each other at a distance is as much as to say they cannot act immediately upon one another, without the intervention of the forces of impenetrability. Now this would be as much as though I were to assert, that the repulsive forces were the only ones by means of which matters could be operative, or they were at least the necessary conditions under which alone matters could act upon one another, which would declare the force of attraction either wholly impossible or always dependent on the action of repulsive forces; but both are assertions without any foundation. The confusion of the mathematical contact of spaces and physical [contact] through repulsive forces constitutes tho ground of this misunderstanding. To attract immediately outside contact, means to approach one another according to a constant law, without the force of repulsion containing the condition thereto, which must admit of being conceived just as well as directly to repel one another, that is to fly from one another according to a constant law, without the attractive force having any share therein. For the two moving forces are wholly different in kind, and there is not the least reason for making one dependent on the other, or denying its possibility without the intervention of the other.

Observation 2.

Except from attraction, no motion can arise on contact, for contact is the reciprocal action of impenetrability, which restrains all motion. Some immediate attraction nrust thus be found apart from contact, in other words, at a distance; for otherwise, even the pressing and impulsive forces, which produce the effort to approach, as they act in an opposite manner to the repulsive force of matter, could have no cause at least originally inherent in the nature of matter. That attraction which takes place without the intervention of repulsive forces may be termed the true attraction, that which proceeds in the other manner the apparent. For properly, the body which another is striving to approach, exercises no attractive force whatever on the latter, because this has been driven towards it from elsewhere by impact. But even these apparent attractions must, at last, have a true one at their basis, because matter made up only of pressure or impact, instead of attraction would not even be matter without attractive forces (proposition 5), and consequently the mode of explaining all phenomena of approach by merely apparent attraction moves in a circle. It is commonly held that Newton did not find it necessary to his system to assume an immediate attraction of matters, but with the strictest abstinence of pure mathematics, left the physicists perfect freedom, in this particular, to explain its possibility as they might find good, without mixing up his propositions with their play of hypotheses. But how could he base the proposition that the "universal attraction of bodies, exercised by them equidistantly on every side is proportioned to the quantity of their matter, if he did not assume that all matter exercised this force of motion simply as matter, and by its essential property? For although, indeed, between two bodies, whether homogeneous or not, as to matter, if one draws the other, the mutual approach (according to the law of the equality of reciprocal action) must always occur in inverse proportion to the quantity of the matter, this law only constitutes a principle of mechanics, hut not of dynamics, i.e., it is a law of motions, following from attractive forces, not the proportion of attractive forces themselves, and applying generally, to all moving forces. If, therefore, a magnet he attracted by another similar magnet, and again by the same magnet enclosed in a wooden box double its weight, in the latter case this will impart more relative motion to the first [magnet] than in the former, although the wood, which increases the quantity of its matter, adds nothing to its attractive power, and proves no magnetic attraction of the box. Newton says (cor. 2, prop. 6, lib. III., Princip. Phil. Nat.): "If the æther or any other body existed without weight, it would, inasmuch as it differs from any other matter in nothing but in form, be capable of being transformed little by little through a gradual change of this form into a matter of the same kind as that which has the greatest weight; and conversely, this latter, by a gradual change of its form, might lose all its weight, which is contrary to experience," etc. Thus he did not even exclude the æther (much less other matters) from the law of attraction. What kind of matter, then, could remain for him, by the mere impact of which the approach of bodies to one another could be regarded as merely apparent attraction? One cannot, therefore, adduce the great founder of the theory of attraction as our precursor, if one takes the liberty of substituting for the true attraction which lie maintained, a false one, and for assuming the necessity of an impulse through impact, in order to explain the phenomena of approach. He justly made abstraction of all hypotheses, in solving the problem, as to the cause of the universal attraction of matter; for this problem is physical or metaphysical, but not mathematical, and although in the preface to the second edition of his Optics, he says: ne quis gravitatem inter essentiales corporum proprietates me habere existimet, quæstionem unam de ejus causa investiganda subjeci, one can easily see that the dislike his contemporaries, and perhaps he himself, had to the conception of an original attraction, made him at issue with himself. For he could not say, unconditionally, that the attractive forces of two planets —for instance, Jupiter and Saturn—which they show in the equal distances of their satellites (whose mass is unknown), is proportioned to the quantity of the matter of these heavenly bodies, if he did not assume that they attracted other matter merely as matter—in other words, according to a universal property of the same.

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 contract, a penetrative force.


The repulsive force, by means of which matter fills a space, is a merely superficial force. For the parts touching each other mutually limit each other's sphere of action, and the repulsive force cannot move any more distant part, except by means of those lying between, and an immediate effect of a matter, passing straight through these, on another, by means of the forces of extension, is impossible. An attractive force, on the contrary, by means of which a matter occupies a space, without filling it, by which therefore it acts on other distant [matters] through empty space, and whose action thus posits no matter intervening [would have] no limits. Now it is thus that the original attraction which makes matter itself possible, must be conceived, and which is hence a penetrative force, and for this reason alone always proportioned to the quantity of the matter.

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.


Because the original attractive force pertains to the essence of matter, it belongs to every part of the same, to act directly at a distance. Now let it be granted, there is a distance beyond which it does not extend, this limitation of the sphere of its activity would rest either on the matter lying within this sphere, or merely on the size of the space, in which the influence was extended. The first does not take place; for this attraction is a penetrative force, and acts directly at a distance, in spite of all intervening matters, through each space as an empty space. The second, in the same way, does not take place. For inasmuch as every attraction is a moving force, having a cause, beyond which smaller can be conceived to infinity; so, in the greater distance, a cause would indeed lie, for diminishing the degree of attraction in inverse proportion, to the amount of the diffusion of the force but never for completely destroying it. As then there is nothing that anywhere limits the sphere of the activity of the original attraction of any part of matter, it extends itself beyon d all assignable limits to every other matter, in other words, [extends itself] throughout the universe, to infinity.

Note 1.

From this original attractive force, as a penetrative [force] exercised by all matter upon all other matter—and therefore in proportion to the quantity of the same, extending to all possible regions of its activity—in combination with its opposite, namely, repulsive force, the limitation of the latter, in other words, the possibility of a space filled in a definite degree, can be deduced; and thus the dynamic conception of matter as the movable, filling its space can (in a definite degree) be constructed. But to tiiis. one requires a law of relation, as well of the original attraction as of repulsion at different distances of matter, and of its parts from one another, which, as it rests simply on the difference of direction of these two forces (since a point is driven either to approach others or to recede from them), and on the size of the space, in which these forces diffuse themselves at different distances, is a task belonging to pure mathematics, and with which metaphysics is no longer concerned, not even as regards the responsibility of constructing the conception of matter in this way, in the event of its non-success. For it is responsible only for the correctness of the elements of construction vouchsafed to our cognition of pure Reason, but for the inadequacy and the limits of our Reason, in its working out, it is not responsible.

Note 2.

As all given matter must fill its space with a definite degree of repulsive force, in order to constitute a definite material thing, only an original attraction in conflict with the original repulsion can make a definite degree of the filling of space, in other words, matter, possible. This is so, whether the former results from the proper attraction of the parts of the compressed matter amongst each other, or from their union with the attraction of all matter.

The original attraction is proportional to the quantity of the matter, and extends to infinity. Thus the filling of a space by matter, definite as to amount, can in the end only be effected by the infinitely extending attraction of the same, and every matter [must be] distributed according to the amount of its repulsive force.

The effect of the universal attraction, which all matter exercises directly upon all [matter] and at all distances, is termed gravitation; the endeavour to move itself in the direction of the greater gravitation is weight. The effect of the thorough going repulsive force of the parts of each given matter is termed its original elasticity. This and weight therefore, constitute the only discoverable à priori universal characteristics of matter, the former in internal, the latter in external relations; for on their mutual bases the possibility of matter itself, rests; cohesion (zusammenhang), when explained as the reciprocal attraction of matter, limited simply to the condition of contact, does not belong to the possibility of matter in general, and cannot therefore be cognised as bound up with it à priori. This characteristic would hence not be metaphysical but physical, and thus would not belong to the present subject of consideration.

Observation 1.

I cannot forbear adding a small preliminary observation, for the sake of any attempt that may perhaps be made toward such a possible construction.

1.It may be said of every force, immediately working at different distances, and which is limited in respect of the degree whereby it exercises moving force, on every given point at a certain distance, only by the size of the space over which it has to diffuse itself in order to act upon this point; that in all spaces over which it is diffused, however small or great they may be, it always constitutes an equal quantum; but that the degree of its effect on the particular point in this space always stands in inverse proportion to the space in which it has had to diffuse itself, in order to act upon it [viz. the point]. So, for instance, light diffuses itself from a luminous point on all sides, in discs that increase with the square of the distance, and the quantum of the luminosity is in all these infinitely increasing discs on the whole the same; whence follows, that an equal part assumed in these discs, must be, in point of degree, so much the less luminous as the surface diffusion of the same quantity of light is greater; and so with all other forces, according to the laws of which they must diffuse themselves either in superficial or corporeal space, in order to act according to their nature on distant objects. It is better to represent the diffusion of a moving force from one point at all distances in the ordinary way, [not?] for instance [as?] in optics, by rays diverging in a circle from a central point. For as lines drawn in this way can never fill the space through which they pass, nor therefore the surface which they touch, it, matters not how many of them may be drawn or supposed—this being the inevitable consequence of their divergence—they give occasion to troublesome inferences, and these to hypotheses, which can easily be avoided if merely the size of the whole disc be taken into consideration, as uniformly illumined by the same quantity of light, and of course the degree of its luminosity, in every place, as assuming an inverse proportion to the size of the whole; and similarly with every other diffusion of a force, through spaces of different sizes.

2.If the force he an immediate attraction at a distance, the direction of the attraction must still less be represented as rays going out from the attracting point, but rather as coalescing from all points of the surrounding disc (the diameter of which is the given distance) at the attracting point. For the line of direction of the movement to this point, which is its cause and goal, assigns the terminus a quo, whence the lines must begin, namely from all points of the surface, from which they take their direction to the attracting middle-point, and not conversely; for the size of the surface alone determines the number of lines; the middle point leaves them undetermined.[1]

3.If the force be an immediate repulsion, so that a point (in merely mathematical presentation) fills a space dynamically, and the question is, according to what law of infinitely small distances (here equivalent to contact) an original repulsive force (the limitation of which consequently rests merely with the space in which it is diffused) acts at different distances, this force can still less be rendered apparent by divergent repulsive rays from the assumed repellant points, although the direction of the motion has it for a terminus a quo, because the space in which the force must be diffused, in order to act at a distance, is a corporeal space, which is to be conceived as filled. The manner in which this is done, how, namely a point can fill a space corporeally by moving force, that is dynamically, is certainly capable of no further mathematical demonstration, but, it is impossible for rays diverging from a point to render conceivable the repelling force of a corporeally-filled space. The repulsion, at various infinitely small distances, of these mutually repelling points, we could simply estimate in inverse proportion to the corporeal spaces which fill each of these points dynamically; in other words, as the cube of their distances from one another, without our being able to construct them.

4. Thus the original attraction of matter would act in inverse proportion to the square of the distance at all distances, the original repulsion in inverse proportion to the cube at infinitely small distances, and by such an action and reaction of both fundamental forces, matter as a definite degree of the filling of space would be possible; for, insomuch as the repulsion increases in greater degree with approach of the parts than the attraction, the limits of approach beyond which by given attraction no greater is possible, in other words the degree of compression which constitutes the amount of the intensive filling of space, is also determined.

Observation 2.

I readily see the difficulty of this mode of explaining the possibility of a matter in general, which consists in that, if a point cannot directly drive another by its repulsive force, without at the same time filling the whole corporeal space, up to the given distance by its force, this, as it seems to follow, must contain several repulsive points, which contradicts the assumption, and was above refuted (proposition 4) under the name of a sphere of repulsion of the simple in space. But there is a distinction to be made between the conception of a real space, that can be given, and the mere idea of a space, simply conceived for the determination of the relations of given spaces, but which is in reality no space. In the case cited of a supposed physical monadology, there ought to be real spaces, to be filled from a point dynamically, namely, by repulsion, for they [the monads] existed as points, before any possible generation of matter from them, and defined by the proper sphere of their activity, the portion of the space to be filled, which could belong to them. In the hypothesis in question, therefore, the matter cannot be regarded as infinitely divisible and as quantum continuum; for the parts, directly repelling one another, have notwithstanding a determinate distance from one another (the sum of the diameter of the sphere of their repulsion) [while] on the contrary, when we, as really happens, think of matter as continuous quantity, no distance whatever of the directly repelling parts obtains, and consequently, no increasing or diminishing sphere of its immediate activity. Matters however can be expanded or compressed (like the air), and in this case we conceive a distance of their nearest parts as capable of increasing or diminishing. But because the nearest parts of a continuous matter touch one another, whether they are farther expanded or compressed, the distances from one another are conceived as infinitely small, and this infinitely small space, as filled in a greater or less degree by its force of repulsion. The infinitely small mediate space is not however distinguishable from contact, and thus it is only the idea of space, which serves to render intuitable the expansion of matter as continuous quality, but whether it is really thus cannot be conceived. When, therefore, it is said: the repulsive forces of the parts of matter immediately driving one another, stand in inverse proportion to the cube of their distances, this only signifies that they stand in inverse proportion to the corporeal spaces that are conceived between parts immediately touching one another notwithstanding, and where distance must for this reason be termed infinitely small, in order that it may be distinguished from all real distance. Hence we must not from the difficulties of the construction of a conception, or rather, from its misapplication, cast any slur on the conception itself; for in that case it would touch the mathematical presentation of the proportion, with which the attraction occurs at different distances, no less than that whereby each point in an expanding or compressed whole of matter, directly repels the other. The universal law of dynamics would in either case be this: the effect of the moving force, exercised from one point upon every other outside it, is in inverse proportion to the space in which the same quantity of moving force has had to expand itself, in order to act directly upon this point at the determinate distance.

From the law that the parts of matter originally repel one another in inverse cubic proportion to their infinitely small distances, a quite different law of their extension and compression must necessarily follow to that of Mariotte [in respect] of the air; for this proves repulsive forces of its nearest parts, which stand in inverse proportion to their distances, as Newton demonstrates. (Princ. Phil. Lat., Lib. II., Propos. 23, Schol.) But the expansive force of the latter also cannot be regarded as the effect of originally repulsive forces, but rests on heat, which compels the proper constituents [viz. the molecules] of the air (to which moreover real distances from each other may be conceded) to fly from one another, not as a matter interpenetrating them, but, to all appearance through their vibrations. But that these vibrations of the parts nearest one another must communicate a repulsive force, standing in inverse proportion to their distances, may be made readily comprehensible by the laws of the communication of motion through the vibration of elastic matters.

I may explain that I do not wish the present exposition of the law of an original repulsion to be regarded as necessarily belonging to the object of my metaphysical treatment of matter, nor the latter (for which it is enough, to have presented the filling of space as dynamic property) to be mixed up with the disputes and doubts which might affect the former.

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,

  1. It is impossible to represent surfaces at given distances as wholly filled by the action of lines spreading out from a point in the form of rays, whether of luminosity or attraction. Thus, by such diverging rays of light, the inferior luminosity of a distant surface would merely rest on the fact that between the luminous there remain non-luminous places, and these so much the larger the farther the surfaces are removed. Euler's hypothesis avoids this inconvenience, but has certainly so much the greater difficulty in rendering the rectilinear motion of the light conceivable. But this difficulty arises from an easily avoidable mathematical conception of light-matter as a mass of globules, which according to their variously oblique arrangement, as regards the direction of the impact, would produce a lateral motion of light; whereas nothing prevents us from conceiving this matter as originally and in every sense fluid, instead of as divided into fixed globules. If the mathematician wishes to render intuitable the diminution of light by increasing distance, he makes use of rays spreading in a circle, in order to exhibit on the disc of its diffusion the size of the space, in which the same quantity of light is to be uniformly diffused between these circle-rays, in short, the diminution of the degree of luminosity; but he does not intend these rays to be regarded as the only [places of] luminosity, as though there were always places devoid of light, to be met with between them, these increasing with the distance. If one wishes to conceive each of these places as throughout luminous, the same quantity of luminosity which covers the smaller must be conceived as in equal proportion in the larger, and therefore, in order to indicate the rectilinear direction, they must be drawn from the surface and all its points to the luminous straight lines. The effect and its quantity must be previously fixed, and the cause indicated in accordance therewith. The same applies to rays of attraction, if one chooses to call them so, and indeed to all directions of forces, which are to fill a space, be it even a corporeal one, from a point.