Demonstration.

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 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.

Observation.

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.

Demonstration.

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.

Note.

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.

Observation.

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 à 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.

Observation.

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.

Demonstration.

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 the 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 must 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, but 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 be 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 he 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.

Note.

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 1 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.

Demonstration.

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 beyond 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 this, 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 be 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 he 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.

Observation.

Now this is the third definition of a matter; the mere dynamical conception could also regard matter as in rest; the moving force, which was then taken into consideration, concerned merely the filling of a particular space, without our being permitted to regard the matter which filled it, as itself moved. Repulsion was thus an original moving force to impart motion; in mechanics, on the contrary, the force of a matter, set in motion, is considered as [present] in order to communicate this motion to another. But it is clear that the movable would have no moving force through its motion if it did not possess original moving forces, whereby it is active before all proper motion, in every place in which it exists, and that no matter would impress uniform motion upon another matter, the motion of which lay in the path of the straight line before it, if both did not possess original laws of repulsion; nor that it could compel another by its motion, to follow it in the straight line (that it could drag it after it), if both did not possess attractive forces. Thus, all mechanical laws presuppose dynamical, and a matter as moved can have no moving force, except by means of its repulsion or attraction, upon which, and with which, it acts directly in its motion, and thereby communicates its own motion to another. It will be observed that I do not make further mention here of the communication of motion by attraction—for instance, as if a comet of stronger attractive capacity than the earth, in passing by the latter, should drag it after it—but only of the mediation of repulsive forces, in other words, of pressure (as by means of a distended spring), or by impact, since, without this, the application of the laws of the one to those of the other is only different in the line of direction, but otherwise the same in both cases.

Demonstration.

Matter is divisible to infinity; consequently none of its quantity can be determined directly by a multitude of its parts. For if this occur in the comparison of the given matter, with a homogeneous one, in which case the quantity of the matter is proportional to the quantity of the volume, this is opposed to the requirements of the proposition [which says], it is to be estimated in comparison with every other (even specifically different) [matter]. Thus matter can be neither indirectly nor directly estimated in comparison with every other matter, so long as abstraction is made of its own motion. Consequently, no other universally valid measure of it remains, but the quantity of its motion. But in this, the difference of the motion, which rests on the different quantity of the matter, can only be given when the velocity is assumed as equal among the compared matters, therefore, c.

Note.

The quantity of the motion of bodies is in compound proportion to the quantity of its matter and its velocity, i.e., it is the same whether I make the quantity of the matter of a body doubly as great, and retain the velocity, or whether I double the velocity and retain the mass. For the definite conception of a quantity is only possible through the construction of the quantum. But this is, in respect of the conception of the quantity, nothing but the composition of the equivalent; and consequently the construction of the quantity of a motion is the composition of many motions equivalent to each other. Now it is the same thing, according to the phoronomic propositions, whether I impart to a movable a certain degree of velocity, or to many equal movables all the smaller degrees of velocity, produced by the given velocity being divided by the multitude of the movable. Hence arises, at first, an apparently phoronomic conception of the quantity of a motion, as compounded of many motions outside one another, but yet as a whole united in a movable point. If now this point be conceived as something possessing moving force by its motion, there arises the mechanical conception of the quantity of the motion. But in phoronomy it is not practicable to conceive of a motion as compounded of many parts outside one another, because the movable, since it is conceived as without any moving force, gives no distinction in real quantity of the motion, no matter with how many others of its kind it be compounded, beyond that which consists merely in the velocity. As the quantity of the motion of a body to that of another, so is related also the quantity of its effect, the whole effect being understood thereby. Those who assumed merely the size of a space filled with resistance (e.g., the height to which a body can rise with a given velocity against gravitation or the depth to which the same [body] can penetrate into soft matters) as the measure of the whole effect, brought forward another law of moving forces with real motions, namely, that of compound relation, from [the law] of the quantity of the matters and of the squares of their velocities; but they overlooked the quantity of the effect in the given time, in which the body traverses its space with less velocity, and this can alone be the measure of a motion exhausted by a given uniform resistance. Hence no difference can obtain between living and dead forces, if moving forces are considered mechanically, that is, as those such as bodies possess, in so far as they are themselves moved, it matters not whether the velocity of their motion be finite or infinitely small (mere effort towards motion). One might far more suitably indeed call those forces with which matter (even when abstraction is wholly made of its own proper motion, or even effort to move itself), acts on others; in other words, the original moving forces of dynamics, dead forces, and all mechanical [forces], that is, forces moving by their own motion, living forces, regard not being given to the difference of velocity, the degree of which may be infinitely small; always supposing that these designations of dead and living forces deserve to be retained at all.

Observation.

In order to avoid diffuseness, we will condense the explanation of the preceding three paragraphs into one observation.

That the quantity of the matter can only be conceived as the multitude of the movable (outside one another), as the definition expresses it, is a remarkable and fundamental proposition of universal mechanics. For it is indicated thereby, that matter can have no other quantity than that which consists in the multitude of the manifold outside one another; consequently no degree of moving force with given velocity that would be independent of this multitude, and which could be conceived as merely intensive quantity, which would certainly be the case if the matter consisted of monads, whose reality in every connection must have a degree, that might be greater or smaller, without depending on a multitude of parts external to one another. As to that which concerns the conception of mass in the same explanation it cannot be regarded, as is usually [done], as the same as the quantity. Fluid matters can act by their own motion in mass, and they can also act in flux. In the so-called water-hammer the water in striking acts in mass, that is, with all its parts at the same time; the same occurs in water which has been enclosed in a vessel, and which presses by its weight upon the scale on which it stands. On the other hand, the water of a mill-stream acts on the paddle of the water-wheel that strikes it, not in mass, that is, at the same time with all its parts that rush against it, but only successively. If therefore, in this case, the quantity of the matter that is moved with a certain velocity, and that has moving force, is to be determined, one must first of all seek the body of the water, that is, such quantity of matter, that when it acts in mass with a certain velocity (by its weight) can produce the same effect. Hence by the word mass is generally understood the quantity of the matter of a solid body (the vessel, in which a fluid is enclosed, taking the place of its solidity). Finally, as concerns the proposition, together with the appended note, there is nothing strange that according to the former, the quantity of the matter has to be estimated by the quantity of the motion with given velocity, while according to the latter, on the contrary, the quantity of the motion (of a body, for that of a point, consists only in the degree of the velocity) at the same velocity, by the quantity of the moved matter, though this seems to revolve in a circle, and to promise no definite conception of either the one or the other. This supposed circle would indeed be real if it were a reciprocal deduction of two identical conceptions from one another. It contains, however, on the one side only the explanation of a conception, and on the other its application to experience. The quantity of the movable in space is the quantity of the matter; but this quantity of the matter (the multitude of the movable), demonstrates itself in experience only by the amount of the motion, at equal velocity (e.g. by equilibrium.)

It remains yet to be observed, that the quantity of matter is the quantity of substance in the movable; consequently, not the amount of a given quality of the same (of repulsion or attraction, as has been said in the dynamics), and that the quantum of the substance is here nothing else than what is signified by the multitude of the movable, which constitutes matter. For only this multitude of the moved can with the same velocity give a difference in the amount of the motion. But that the moving force a matter possesses in its own motion can alone prove the quantity of the substance, rests on the conception of the latter as the ultimate subject (that is no further predicate of another) in space, which for this reason can have no other quantity, but that of the multitude of the homogeneous outside one another. But as the proper motion of matter is a predicate which determines its subject (the movable), and in a matter, as a multitude of the movable, indicates the plurality of the moved subjects (at equal velocity in the same kind)—while with dynamical properties, whose quantity may be also the quantity of the effect of a single subject (e.g. a [single] molecule of air may have more or less elasticity), this is not the case—it is clear that the quantity of the substance in a matter can only be estimated mechanically, that is, by the amount of its motion, and not dynamically, by the amount of its original moving forces. In the same way the original attraction, as the cause of universal gravitation can afford a measure of the quantity of matter and its substance (as really happens in the comparison of matters by weighing), although in this case, not proper motion of the attracting matter, but a dynamical measure, namely attractive force, seems to be laid at the foundation. But inasmuch as with this force the effect of a matter occurs with all its parts, directly on all parts of another, and thus (at equal distances) is obviously proportioned to the multitude of the parts, and the attracting body itself thereby imparts a velocity of its own motion (by the resistance of the attracted [body]), which, in similar external circumstances, is exactly proportioned to the multitude of its parts, [for this reason] the estimate takes place here, [also] as a matter of fact, mechanically, although only indirectly so.

Demonstration.

(From universal metaphysics the proposition is laid at the foundation, that with all changes of nature, no substance can either arise or be annihilated, and here it is only demonstrated what is substance in matter.) In every matter the movable in space is the ultimate subject of all the accidents inhering in matter, and the multitude of this movable outside one another the quantity of the substance. Thus the amount of the matter as substance, is nothing other than the multitude of the substances of which it consists. Hence the quantity of the matter cannot be increased or diminished except by new substance arising or being annihilated. Now, with all change of matter, substance never arises or is destroyed; thus the quantity of matter is thereby neither increased nor diminished, but remains always the same as a whole, that is, so that somewhere in the world it continues [to exist], although this or that [particular] matter may by the addition or subtraction of its parts be increased or diminished.

Observation.

The essential, characterising substance in this demonstration, which is only possible in space and according to the conditions of the same, consequently as object of the external sense, is that its amount cannot be increased or diminished, without substance arising or being annihilated; therefore as any quantity of a merely possible object in space must consist of parts outside one another, these, if they are real (something movable) must be necessarily substances. That, on the contrary, which is considered as object of the internal sense may have a quantity as substance, not consisting of parts outside one another, whose parts are therefore not substances, whose origination or annihilation therefore need not be the origination or annihilation of a substance, and hence whose increase or diminution is possible, notwithstanding the principle of the permanence of substance. Thus consciousness, in other words, the clearness of the presentations of my soul, and in consequence of this also, the faculty of consciousness, apperception, and therewith even the substance of the soul, has a degree that may be greater or smaller, without, to this end any substance requiring to arise or to be annihilated. But because with the gradual diminution of this faculty of apperception, a total disappearance of the same could not but finally result, the substance of the soul would still be subjected to a gradual destruction, even were it of simple nature, inasmuch as this disappearance of its fundamental force could not result through division (separation of substance from a composite), but, as it were, by extinction, and even this not in a moment, but by the gradual failing of its degree, from whatever cause arising. The ego, the universal correlate of apperception and itself merely a thought, indicates as a mere prefix, a thing of undefined signification, namely, the subject of all predicates without any condition distinguishing this presentation of the subject from a something generally, in short, substance, of which no conception of what it is [is conveyed] through this expression. On the contrary, the conception of a matter as substance is the conception of the movable in space. It is no wonder therefore, if permanence of substance can be proved of the latter, but not the former, since with matter it follows from its conception, namely, as being the movable, which is only possible in space, that that which possesses quantity in it, contains a plurality of the real outside one another, in other words of substances, and consequently its quantity can only be diminished by division, which is no disappearance, and even the latter would be impossible in this case according to the law of permanence. The thought I is on the contrary, no conception, but only inward perception; from it therefore nothing whatever can be deduced (except the complete distinction of an object of the internal sense from that which is merely conceived as object of external sense), and consequently not the permanence of the soul as substance.

Demonstration.

(From universal metaphysics the proposition that all change has a cause, is laid at the foundation; here it only remains to be proved of matter, that its change must always have an external cause. ) Matter, as mere object of the external sense, has no determinations but those of external relation in space, and hence is subject to no change except through motion. In respect of this, a change of one motion with another, or of the same with rest, and conversely, a cause of the same though this, must be traceable (according to principles of metaphysics). But this cause cannot be internal, for matter has no absolutely internal determinations and grounds of determination. Hence all change of a matter is based upon external causes (i.e., a body continues, c.).

Observation.

This mechanical law can only be called the law of inertia ( lex inertiæ ); the law that every action has an equal reaction opposed to it, cannot bear this name. For the latter says what matter does, but the former, only what it does not do, which is better adapted to the expression inertia. The inertia of matter is and means nothing but its lifelessness, as matter in itself. Life means the capacity of a substance, to act from an internal principle, determining a finite substance to change, and a material substance to rest or motion, as change of its state. Now we know no other internal principle of a substance to change its state but desire, and no other internal activity whatever but thought, with that which depends upon it, feeling of pleasure or pain, and impulse or will. But these grounds of determination and action in no wise belong to the presentations of the external sense, and thus not to the determinations of matter as matter. Thus all matter as such is lifeless. The proposition of inertia says so much and no more. If we seek the cause of any change of matter whatsoever in life, we shall have to seek it at once in another substance, distinct from matter, although bound up with it. For in natural knowledge it is necessary, first of all, to know the laws of matter as such, and to clear them from the admixture of all other efficient causes, before connecting them therewith, in order to distinguish how each acts for itself alone. On the law of inertia (next to that of the permanence of substance) the possibility of a natural science proper entirely rests. The opposite of the first, and therefore the death of all natural philosophy, would be hylozoism. From the same conception of inertia as that of mere lifelessness, it follows of itself, that it does not signify a positive effort to maintain its state. Only living beings can be termed inert in this latter sense, inasmuch as they have a conception of another state, which they dread and strive against with all their might.

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 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 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). *

Note 1.

From the above there follows, the natural, and for universal mechanics, not unimportant law, that every body, however great its mass may be, must be movable by the impact of every other, however small its mass or velocity may be. For to the motion of A in the direction AB, there corresponds necessarily an equal opposite motion of B in the direction BA. Both motions destroy one another in absolute space by impact. But thereby both bodies retain a velocity Bd = Bc in the direction of the striking [one]; consequently the body B is movable by even the smallest force of impact.

Note 2.

This, then, is the mechanical law of the equality of action and reaction, which is based upon [the fact] that no communication of motion takes place except in so far as a community of these motions is pre-supposed, and thus that no body strikes another, which is at rest in respect of itself, but that if it be so in respect of the space, it is only in so far as together with this space it is moved in equal degree, but in contrary direction to the motion, falling to the relative share of the former, [both together] giving the quantity of the motion to be attributed to the former, in absolute space. For no motion which is [conceived as] moving in respect of another body, can be absolute; but if it be relative in respect of the latter, there is no relation in space that is not reciprocal and equal. But there is yet another, namely, a dynamical law of the action and reaction of matters not in so far as one communicates its motion to another, but imparts it to the latter originally, and by its resistance at the same time produces it in itself. This may be readily demonstrated in a similar way. For if the matter A attract the matter B, it compels the latter to approach it, or, which is the same thing, the former resists the force with which the latter strives to retreat. But inasmuch as it is the same thing whether B retreats from A or A from B, this resistance is at the same time a resistance that the body B exercises against the body A in so far as it strives to retreat, and hence traction and countertraction are equal to one another. In the same way, if A repel the matter B, A resists the approach of B. But it is the same thing whether B approaches A, or A B, for B resists just as much the approach from A, hence pressure and counter-pressure are always equal to one another.

Observation 1.

This, then, is the construction of the communication of motion, which at the same time carries with it as its necessary condition the law of the equality of action and reaction, which Newton did not trust himself to prove à priori, but for which we appealed to experience, and for the sake of which others introduced into natural science a special force of matter under the name force of inertia ( vis inertiæ ) first invented by Kepler, and thus, in the end, also deduced it from experience; while finally others again placed it in the conception of a mere communication of motion which they regarded as a gradual transference of the motion of one body into the other, whereby the moving sacrificed precisely as much as it imparted to the moved until it impressed the latter no longer (when, namely, it had arrived at equality of velocity in the direction of it). * In this way all reaction, that is, all really reacting force of the one struck against the striking [body], (such for instance as would be possible to distend a spring) is abolished; and besides that it fails to prove what is really meant by the law referred to, in nowise explains the communication of motion itself, as to its possibility. For the word transference of motion from one body to another explains nothing, and if one is unwilling to take it, so to speak literally ([as being] opposed to the principle, accidentia non migrant e substantiis in substantias ) as though motion were poured from one body into the other, as water from one glass into the other, the problem is, how to make this possibility—the explanation of which rests precisely on the same ground, whence the law of the equality of action and reaction is derived—comprehensible. One cannot conceive how the motion of a body A is necessarily connected with the motion of another B, except that forces are conceived in both, as accruing to them before all motion (dynamically)—as for instance repulsion—and it can be proved, that the motion of the body A through approach towards B, with the approach of B towards A, and if B be regarded as at rest, its motion together with its space towards A, are necessarily connected, in so far as the bodies with their (original) moving forces, are merely considered in motion as relative to one another. This latter can be thereby fully comprehended à priori [viz.] that whether the body B in respect of empirically cognisable space be resting or moved, it must be regarded as necessarily moved in respect of the body A, and [moved] in an opposite direction; since otherwise, no influence thereof on the repulsive force of both would take place, without which no mechanical action whatever of matters on one another, i.e. no communication of motion by impact is possible.

Observation 2.

The designation force of inertia ( vis inertiæ ) must thus, in spite of the eminence of its founder’s name, be entirely banished from natural science,—not only because it carries with it a contradiction in expression, or because the law of inertia (lifelessness) might thereby be easily confounded with the law of reaction in every communicated motion, but principally—because thereby the mistaken conception of those, insufficiently acquainted with the mechanical laws, would be maintained and strengthened according to which the reaction of bodies, of which we are speaking under the name force of inertia, consists in the motion being thereby swallowed up, diminished or destroyed, without the mere communication of motion being effected, in that, namely, the moving body would have to apply a part of its motion to overcoming the inertia of the resting [one] (which would be pure loss), and with the remaining portion only, could set the latter in motion; but if nothing remained, would not be able by its impact to bring the latter into motion on account of its great mass. A motion can resist nothing except opposite motion of another, but, in nowise its rest. Here therefore inertia of matter, that is mere incapacity to move of itself, is not the cause of a resistance. The expression force of inertia used to designate a special and quite peculiar force, merely in order to resist without being able to move a body, would be a word without any significance. The three laws of universal mechanics might be more suitably designated, the law of the subsistence, the inertia, and the reaction of matters ( lex subsistentiæ, inertiæ et antagonismi ) by all changes of the same. That these, in other words, the entire propositions of the present science, exactly answer to the categories of substance, causality and community, in so far as these conceptions are applied to matter, requires no further elucidation.

Observation.

Motion, like all that can be presented through sense, is only given as phenomenon. In order that its presentation may become experience, it requires in addition, that something should be conceived through the understanding, namely, as to the way in which the presentation inheres in the subject, not the definition of an object through the same. Thus the movable, as such, is an object of experience, when a certain object (here a material thing) is conceived as defined in respect of the predicate of motion. But motion is change of relation in space. Hence, firstly there are always two correlates here, to one no less than to the other of which, change is attributed in the phenomenon, and either the one or the other can be termed moved inasmuch as it is indifferent to both, or secondly, of which one must, in experience be conceived as moved to the exclusion of the other, or thirdly of which both must necessarily be conceived through Reason as moved at the same time. In the phenomenon, which contains nothing but the relation in motion (as to its change), there are none of these determinations, but when the movable, as such, i.e. as to its motion, is to be conceived as determined, namely, for the sake of a possible experience, it is necessary to indicate the conditions, by which the object (matter) would have to be determined in this or that manner, by the predicate of motion. Here, the question is not of the transformation of illusion into truth, but of phenomenon into experience. For with illusion the understanding is always engaged with its own judgment determining an object—although it is in danger of mistaking the subjective for objective—but in the phenomenon no judgment of the understanding is to be met with; and this is necessary to be remembered, not only here, but in the whole of philosophy, because, otherwise, when we are concerned with phenomena, and this expression is taken as identical in signification with that of illusion, misunderstanding will always arise.

Observation.

This proposition determines the modality of the motion in respect of Phoronomy.

Demonstration.

The circular motion is (like every non-rectilinear [motion]) a continuous change of the rectilinear, and as this is itself a continuous change of relation in respect of external space, the circular motion is a change of the change of these external relations in space, and consequently a continuous arising of new motions; since, now, according to the law of inertia, a motion, in so far as it arises, must have an external cause, while the body, in every point of this circle, is endeavouring, according to the same law, to proceed in the straight line touching the circle, which motion works against the above external cause, every body in circular motion demonstrates by its motion a moving force. Now the motion of the space as distinguished from that of the body is merely phoronomic, and has no moving force. As a consequence, the judgment, that here, either the body or the space is moved in an opposite direction, is a disjunctive judgment, by which, if the one member, the motion of the body, be posited, the other, namely, that of the space, is excluded. Hence the circular motion of the body, as distinguished from the motion of the space, is a real motion, and consequently the latter, even though as phenomenon it coincide with the former, nevertheless, in the complex of all phenomena, that is, of possible experience, contradicts it, and hence is nothing but mere illusion.

Observation.

This proposition determines the modality of motion in respect of Dynamics; for a motion, which cannot take place without the influence of a continuously active external moving force, proves indirectly or directly original moving forces of matter, either of attraction or repulsion. For the rest, Newton’s scholium to the definitions with which he introduces his Princ. Philos. Nat. Math., towards the end, may be referred to, on the present subject, from which it will appear, that the circular motion of two bodies round a common centre (hence, also the motion of the earth on its axis), even in empty space, and thus without any comparison being possible through experience, with external space, may nevertheless be cognised by means of experience, in short, that a motion which is a change of external relation in space can be given empirically, although this space itself is not empirically given, and is no object of experience—a paradox deserving to be solved.

Demonstration.

According to the third law of mechanics (Proposition 4) the communication of the motion of a body is only possible through the community of its original moving forces, and these only through reciprocal and equal motion. The motion of both is then real. But as the reality of this motion does not rest (as in the second proposition) on the influence of external forces, but follows immediately and inevitably from the conception of the relation of the moved in space, to every other [thing] thereby movable, the motion of the latter is necessary.

Observation.

This proposition determines the modality of motion in respect of mechanics; that, for the rest, these three propositions determine the motion of matter in respect of its possibility, reality, and necessity, in short, in respect of all the three categories of modality, is sufficiently obvious of itself.