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

FIRST DIVISION.
METAPHYSICAL FOUNDATIONS OF PHORONOMY.
Explanation I.
Matter is the movable in space; space, which is itself movable, is termed material or relative space; that in which all motion must in the last resort be conceived (which is therefore itself absolutely immovable), is termed pure or absolute space.
Observation 1.
As in Phoronomy nothing is to be discussed but motion, its subject, namely matter, has here no other quality attributed to it than movability. It can therefore itself be valid for one point so far, and in Phoronomy we abstract from all internal construction, hence also, from the quantity of the movable, and concern ourselves only with motion, and what can be regarded as quantity therein (velocity and direction). If the expression body is sometimes used here, it occurs only to anticipate in a measure the application of the principles of Phoronomy to the following more definite conceptions of matter, in order that the exposition may be less abstract and more comprehensible.
Observation 2.
If I am to explain the conception of matter not by a predicate, applying to it as object, but only by the relation to the faculty of knowledge, in which the presentation can be primarily given me, matter is every object of the external sense, and this would be its mere metaphysical explanation. But space would be simply the form of all external sensuous intuition (whether this accrued to the external ​object we call matter in itself, or remained merely in the construction of our sense, a point which does not enter into the present question). Matter, in contradistinction to form, would be that which in external intuition, is an object of feeling, and consequently the properly empirical of sensible and outward intuition, because it cannot be given at all à priori In all experience something must be felt, and this is the real of sensuous intuition. In consequence, space, in which we are to institute experience respecting motions, must be capable of being felt, that is, of being indicated by that which can be felt, and this, as the sum-total of all objects of experience, and itself an object of the same, is called empirical space. Now this, as material, is itself movable; but a movable space, if its movement is to be able to be perceived, presupposes again an enlarged material space in which it is movable, and this again another, and so on to infinity.
Thus all motion that is an object of experience is merely relative; the space in which it is perceived is a relative space, which again moves itself perhaps in an opposite direction, in a space further enlarged, and therefore the matter moved in reference to the first may be termed at rest in relation to the second; and these alterations of the conception of motion go forward with the alteration of the relative space to infinity. To assume an absolute space, that is, one which, because it is not material, can be no object of experience as given fur itself, means assuming something which, neither in itself nor in its consequences (motion in absolute space), can be perceived, for the sake of the possibility of experience, which nevertheless must always exist without it. Absolute space is in itself nothing and no object at all, but signifies merely every other relative space that I can at any time conceive outside the given space, and that I can extend beyond each given space to infinity; one that includes the [given space], and in which I can assume it as moved. But since I have the enlarged, although still material, space only in thought, nothing is known to me of the matter indicating it. I abstract from this, and it is conceived, therefore, as a pure, non-empirical and absolute space, with which I can compare, and in which I ​can conceive as movable, each empirical space, and therefore, which is itself always regarded as immovable. To constitute it a real thing means confounding the logical universality of any space, with which I can compare each empirical [space] as being included in it with a physical universality of real compass, and misunderstanding the reason in its idea.
I may observe in conclusion that as the movability of an object in space cannot be known à priori and without the teaching of experience, it could not for the same reason be counted in the Critique of pure Reason amongst the pure conceptions of the understanding, and this conception as empirical could only find a place in a natural science, as applied metaphysics, which occupies itself with a conception given through experience, although according to principles à priori.
Explanation II.
Motion of a thing is the change of the external relations of the same to a given space.
Observation 1.
I have already laid the conception of matter at the basis of the conception of motion; but, as I wished to determine the latter independently of the conception of extension, and thus could consider matter only in one point, I had to admit the use of the common explanation of motion as change of place. Now that the conception of matter is to be explained universally, and therefore as applicable to moved bodies, this definition is inadequate, for the place of every body is a point. If one wishes to determine the distance of the moon from the earth, one wishes to know the distance of their places, and to this end one does not measure from any point of the surface, or of the interior of the earth, to any point of the moon at pleasure, but takes the shortest line from the central point of the one to the central point of the other, and therefore, in each of these bodies there is only one point that constitutes its place. Now a body may move without changing its place, as the earth in turning on its axis; but its relation to external ​space changes notwithstanding, for it presents for instance its different sides to the moon in the course of the twentyfour hours, from which all kinds of transformative effects result on the earth. Only of a movable, i.e., physical point can one say: motion is always a change of place. It might be objected against this explanation that internal motion (e.g., fermentation) is not included therein; but the thing which one speaks of as in motion must so far be regarded as unity. That matter, as, for instance, a cask of beer, is in motion signifies something different to the beer in the cask being in motion. The motion of a thing is not one and the same with motion in this thing; but the question is here only of the former. The application of this conception to the latter case is afterwards easy.
Observation 2.
Motions may be circular (without change of place) or progressive, and these again may either enlarge the space or be motions limited to a given space. Of the first kind are rectilinear, or even non-rectilinear, [motions] that do not return in upon themselves. Of the second are those that return in upon themselves. The latter are again either circular or oscillating motions. The first cover the same space always in the same direction; the second alternatingly in an opposite direction, like a swaying pendulum. To both belong trembling (motus tremulus), which, though not a progressive motion of a body, is nevertheless a reciprecative motion of a matter, which does not change its place on the whole thereby, as the vibrations of a bell that has been struck, or the tremblings of air set in motion by sound. I merely make mention of these different kinds of motion in a Phoronomy, because with all that are not progressive the word velocity is generally used in another sense than with the progressive, as the following observation shows.
Observation 3.
In every motion direction and velocity are the two ​momenta for consideration, when one abstracts from all other qualities of the movable. I presuppose here the ordinary definition of both; but that of direction has sundry limitations. A body moved in a circle changes its direction continuously, so that, until its return to the point from which it started, all is comprised in a surface of merely possible directions, and yet one says it moves itself always in the same direction, as, for instance, the planet from evening to morning.
But what is the side, in this case, towards which the motion is directed? A question related to the one: Upon what does the internal distinction of spirals, otherwise similar and even equal, rest, but of which one species winds to the right, and the other to the left; or the winding of the kidney-bean, and of the hop, of which the one runs round its pole like a corkscrew, or as sailors express it against the sun, and the other with the sun? This is a conception that allows itself to be constructed indeed, but as conception does not admit of being made plain by universal marks in the discursive mode of cognition. In the things themselves (e.g., in those rare cases of the human subject where on dissection all the parts agree according to physiological rules with other human subjects, only that all the viscera are found displaced, either to the right or the left, against the usual order) there can be no imaginable difference in the internal consequences, and yet there is a real mathematical and indeed internal difference, whereby two circular movements, differing in direction but in all other respects alike, notwithstanding their not being completely identical, nevertheless correspond. I have elsewhere shown that as this difference, though it must be given in intuition, does not admit of 1 icing brought to clear conceptions, and therefore intelligibly explained (dari, non intelligi), it affords a good substantiating ground of proof for the proposition: that space generally, belongs, not to the qualities or relations of the things in themselves, for this would necessarily have to admit of reduction to objective conceptions, but merely to the subjective form of our sensible intuition of things or ​relations, which, as to what they may be in themselves, must remain wholly unknown. But this is a deviation from our present business in which we must necessarily treat space as a quality of the things we have in consideration, namely, corporeal entities, because these themselves are merely phenomena of the external sense, and only require to be explained as such in this place. As concerns the conception of velocity, this expression acquires in use a variable meaning. We say: the earth moves more rapidly on its axis than the sun, because it does so in a shorter time, although the motion of the latter is much more rapid. The circulation of the blood of a small bird is much more rapid than that of a man, although the streaming motion in the former has, without doubt less velocity; and so with the vibrations of elastic matters. The shortness of the time of return, whether of a circulating or oscillating motion, constitutes the ground of this employment, in which, if otherwise misunderstanding be avoided, there is no harm done. For the mere increase in the hurry of return, without increase of spacial velocity, has special and very important effects in nature, of which, in the circulation of the juices of animals, perhaps not enough notice has been taken. In Phoronomy we use the word velocity merely in a spacial signification: $\textstyle C={\frac {S}{T}}$.
Explanation III.
Rest is the permanent present (praesentia perdurabilis) in the same place; permanent is that which exists throughout a time, i.e. lasts.
Observation.
A body, which is in motion, is in every point of the line it passes over—a moment. The question remains, whether it rests therein, or moves. Without doubt the ​latter, one will say; for, only in so far as it moves is it present in this point. But let us assume the motion in this way:
that the body describes the line $A\;B$ forwards and backwards, from $B$ to $A$, with uniform velocity in suchwise that, since the moment it is in $B$ is common to both motions, the motion from $A$ to $B$ is described in half a second, that from $B$ to $A$ also in half a second, but both together in a whole second, so that not the smallest portion of time has been expended on the presence of the body in $B$; in this way, without the least increase of these motions, the latter, which took place in the direction $B\;A$, can be changed into that in the direction $B\;a$, which lies in a straight line with $A\;B$, and hence the body, while it is in $B$, must be regarded not as at rest, but as moved. It would have therefore also to be considered as moved in the first motion, returning in upon itself in the point $B$, which is impossible; because, in accordance with what has been assumed, it is only a moment that belongs to the motion $A\;B$, and at the same time to the equal motion $B\;A$, which is opposed to the former one and conjoined with it in one and the same moment of complete lack of motion; consequently if this constitutes the conception of rest, in the uniform motion $A\;a$, rest of the body must also be proved in every point (e.g., in $B$), which contradicts the above assertion. Again, let the line $A\;B$ be represented as over the point $A$ perpendicularly, so that a body rising from $A$ to $B$, after having lost its motion through gravity in the point $B$, would fill back again from $B$ to $A$. Now I ask whether the body in $B$ is to be considered as moved or at rest? Without doubt, it will be said, at rest; because all previous motion has been taken from it, after it has reached this point, and a uniform motion back is as yet to follow, consequently is not present, and the lack of motion, it will be added, is rest. In the first case, however, of an assumed uniform motion, the motion $B\;A$ could not commence otherwise, than by the motion $A\;B$ having previously ceased, and ​that from $B$ to $A$ being non-existent, and consequently there being in $B$ a lack of all motion, whereby, according to the usual explanation, rest would have to be assumed; but we may not assume it, because at a given velocity, no body may be conceived as at rest in any point of its uniform motion. Upon what, then, is the assumption of rest based in the second case, since this rising and falling is only separated by a moment? The ground lies in the latter motion not being conceived as uniform with the given velocity, but as being at first uniformly delayed, and afterwards uniformly accelerated, in suchwise that the velocity in point $B$ is not delayed, wholly, but only up to a certain degree, smaller than any velocity that can be given, by which, if instead of falling back, the line of its fall $B\;A$ were placed in the direction $B\;a$; in other words, the body were conceived as still rising, it would, as with a mere moment of velocity (the resistance of gravity being set aside), pass over, in any given time, however great, a space smaller than any space that could be given, and therefore its place (for any possible experience) would not change to all eternity. In consequence of this, it assumes a state of lasting presence in the same place, that is, of rest, although owing to the continuous action of gravity, that is, of the change of this state, the latter is immediately abolished. To be in a permanent state and to persist therein (if nothing else shifts it) are two distinct conceptions, of which one does no violence to the other. Thus rest cannot be explained through the lack of motion, which, as = o, does not admit of being constructed at all, but must be explained by permanent presence in the same place, and as this conception is constructed by the presentation of a motion with infinitely small velocity, throughout a finite time, it can be used for the subsequent application of mathematics to natural science.
Explanation IV.
To Construct the conception of a composite motion means to present à priori in intuition a motion so far as it arises from two or more given [motions] united in one movable.
Observation.
For the construction of conceptions, it is requisite that the condition of their presentation should not he borrowed from experience, and thus that they should not presuppose certain forces, the existence of which can only be deduced from experience, or, in short, that the condition of the construction should not be itself a conception incapable of being given a priori in intuition; as for instance, that of cause and effect, action and resistance, &c. It is here especially to be observed that Phoronoiny is throughout, primarily construction of motions in general as quantities, and that, as it has for its subject, matter merely as something movable, and of which no quantity therefore comes into consideration, it has to determine these motions alone as quantities (as concerns their velocity as well as their direction, and indeed their combination) a priori. For thus much must be established entirely a priori and intuitionally, for the sake of applied mathematics. For the rules of the connection of motions through physical causes, that is forces, never admit of being fundamentally expounded before the principles of their composition generally are previously laid down mathematically as a foundation.
Principle 1.
Every motion, as object of a possible experience, may be viewed, at pleasure, as motion of a body in a space that is at rest, or as rest of the body, and motion of the space in the opposite direction with equal velocity.
Observation.
In order to make an experience of the motion of a body it is requisite that not only the body but also the space in which it moves should be objects of external experience, or in other words, material. An absolute motion, therefore, that is, in reference to a non-material space, ​is unsuited to any experience whatever, and hence for use, nothing (even if one were willing to admit absolute space to be something in itself). But even in all relative motion the space itself, because it is assumed as material, may again be conceived as resting or moved. The first happens when, beyond the space in reference to which I regard a body as moved, there is no more extended space given, that includes it (as when in the cabin of a ship I see a ball moved on the table); the second, when, outside this space there is another space given, that includes it (as, in the case mentioned, the bank of the river), since I can view the nearest space (the cabin) with respect to the latter as moved and the body itself as at rest. As thus it is absolutely impossible to determine of an empirically given space, it matters not how extended it may be, whether, with respect to a still greater space enclosing it, it be itself moved or not, it must be wholly the same for all experience, and for every consequence drawn from experience, whether I choose to regard a body as moved or at rest, and the space as moved in the opposite direction with an equal velocity. Once more : as absolute space is nothing for any possible experience, the conceptions are the same whether I say a body moves with respect to this given space, in this direction, with this velocity, or whether I conceive it as at rest, and ascribe all this [motion] to the space, but in an opposite direction. For every conception is wholly of the same kind as the latter, of whose distinction froro. the former no instance is possible, and only with reference to the connection we wish to give it in the understanding is it different.
We are, moreover, not in a position to postulate a fixed point, in any experience, in reference to which it could be defined what motion and rest mean absolutely; for everything given us in this way is material, and hence movable, and (as we know of no extreme boundary of possible experience in space) it may be really moved without our being able to perceive this motion. Of this motion of a body in empirical space I can assign one portion of the given velocity to the body, the other to the space, but in the opposite direction, and the whole possible experience as concerns the consequences of these two combined ​motions is wholly the same whether conceived of the body alone as moved with the whole velocity or (conceiving it) as at rest, and the space as moved with the same velocity in the opposite direction. I assume here all motions as rectilinear. For as concerns the non-rectilinear it is not in all respects the same, whether I am at liberty to regard the body as moved (e.g., the earth in its daily rotation), and the surrounding space (the starry heaven) as resting, or the latter as moved and the former as resting; but we shall treat of this more particularly in the sequel. Thus in Phoronomy, where I consider the motion of a body only in relation to the space (on the rest or motion of which it has no influence at all), it is quite undetermined and arbitrary whether any or all, or how much, of the velocity of the given motion I attribute to the one or to the other.
Farther on in mechanics where a moved body is to be considered in real relation to other bodies, in the space of its motion, this will not be any longer so entirely indifferent, as will be demonstrated in its proper place.
Explanation V.
The composition of motion is the presentation of the motion of a point as bound together in one with two or more motions of the same.
Observation.
In Phoronomy, as I can cognise the matter by no other property but that of movability, and can consider it itself therefore only as a point, the motion can only be viewed as description of a space, yet so that I do not merely pay attention to the space described, as in geometry, but also to the time [involved] therein; in other words, to the velocity with which a point describes the space. Phoronomy is thus the pure doctrine of the quantity (mathesis) of motions. The definite conception of a quantity is the conception of the generation of the presentation of an object through the composition of the homogeneous. Now, as motion is nothing homogeneous, but again motion ​Phoronomy is a doctrine of the composition of the motions of the same point according to its direction and velocity i.e., the presentation of a single motion as one that comprises within it two or perhaps several motions in one, at the same time, in the same point, so far as they together constitute one, that is, are one with this motion, but not in so far as they produce the latter as causes produce their effects. In order to find the motion arising from the composition of several—as many as one likes—one has only, as with the production of all quantities, first to seek out those that are compounded under given conditions, of two; and thereupon combine this with a third, etc. In consequence the doctrine of the composition of all motions is reducible to that of two. But two motions, of one and the same point that are present at the same point may be distinguished in a double manner, and as such be combined in a triple way therein. Firstly, they occur at the same time either in one and the same line, or in different lines; the latter are motions enclosing an angle. Those that occur in one and the same line are either contrary to one another in direction or maintain the same direction. As all these motions are contemplated as taking place alone, there results immediately from the relation of the lines, that is, of the spaces of motion described in equal time, the relation of velocity. Thus there are three cases:—1. As two motions (it matters not whether of equal or unequal velocities) combined in one body in the same direction, are to constitute a resultant compound motion; 2. As Two motions of the same point (of equal or unequal velocity), combined in contrary directions, are, through their composition, to constitute a third motion in the same line; 3. Two motions of a point, with equal or unequal velocities, but in different lines, enclosing an angle, are considered as compounded.
Proposition 1.
The composition of two motions of one and the same point, can only be conceived by one of them being presented in absolute space, but, instead of the other, a motion of an equal velocity in the contrary direction of the
Demonstration.
First Case.—Two motions in the same line and direction arrive at the same time in one and the same point.
Let two velocities, $AB$ and $ab$, be presented as contained in one velocity of the motion. Let these velocities be assumed, for the time, as equal, $AB=ab$; in this case I assert they cannot be presented at once in the same point, in one and the same space (whether absolute or relative). For, because the lines $AB$ and $ab$, denoting the velocities, are properly spaces, passed over in equal times, the composition of these spaces $AB$ and $ab=BC$, and, therefore, the Line $AC$, as the sum of the spaces, cannot but express the sum of both velocities. But the parts $AB$ and $BC$ do not, individually, present the velocity $=ab$; for they are not passed over in the same time as $ab$. Thus, the double line $AC$, which is traversed in the same time as the line $ab$, does not represent the double velocity of the latter, as was required. Hence the composition of two velocities in one direction in the same space does not admit of being sensuously presented.
On the contrary, if the body $A$ be presented as moved in absolute space with the velocity $AB$, and I give to the relative space, a velocity $ab=AB$ in addition, in the contrary direction $ba=CB$; this is the same as though I distributed the latter velocity to the body in the direction $AB$ (axiom 1). But the body moves itself, in this case, in the same time through the sum of the lines $AB$ and $BC=2ab$, in which it would have traversed the line $ab=AB$ only, and yet its velocity is conceived as the sum of the two equal velocities $AB$ and $ab$, which is what was required.
Second Case.—Two motions in exactly contrary directions are united in one and the same point.
​Let $AB$ be one of these motions, and $AC$ the other in the opposite direction, the velocity of which we assume here to be equal to that of the first; in this case the very idea of representing two such motions, at the same time, in one and the same space, and in one and the same point, in short, the case of such a composition of motions would itself be impossible, which is contrary to the assumption.
On the other hand, let the motion $AB$ be conceived as in absolute space, and instead of the motion $AC$ in the same absolute space, let the contrary motion $CA$ of the relative space [be conceived] with the same velocity, which (according to axiom 1) is equal to the motion $AC$, and may thus be entirely substituted for it; in this case two exactly opposite and equal motions of the same point, at the same time, may be very well presented. Now, as the relative space is moved with the same velocity $CA=AB$ in the same direction with the point $A$, this point, or the body, present therein, does not change its place in respect of the relative space; i.e., a body moved in two exactly contrary directions with equal velocity, rests, or generally expressed, its motion is equal to the difference of the velocities in the direction of the greater (which admits of being easily deduced from what has already been demonstrated).
Third Case.—Two motions of the same point are presented as combined according to directions that enclose an angle.
The two given motions are $AB$ and $AC$, whose velocity and directions are expressed by these lines, but the
$BAC$
$AB$
$AC$
$AB$
$AC$
On the other hand, let the motion $AC$ be taken as proceeding in absolute space, but instead of the motion $AB$, the motion of the relative space in the opposite direction. Let the line $AC$ be divided into three equal parts, $AE,EF,FG$. Now, while the body $A$ in absolute space passes over the line $AE$, the relative space, and therewith the point $E$, passes over the space $Ee=MA$; while the body passes over the two parts together $=AF$, the relative space and therewith the point $F$, describes the line $Ff=NA$; while, finally, the body passes over the whole line $AC$, the relative space, and therewith the point $C$ describes the line $Cc=BA$. All this is the same as though the body $A$ had passed over in these three divisions of time, the lines $Em$, $Fn$ and $CD=AM,AN,AB$, and in the whole time in which it passes over $AC$, had passed over the line $CD=AB$. It is therefore at the last moment in the point $D$, and in the whole time gradually in all points of the diagonal line $AD$, which expresses the direction as well as the velocity of the compound motion.
Observation 1.
Geometrical construction demands that one quantity should be identical with the other, or two quantities in composition, with a third, not that they should pronuce the third as causes, which would be mechanical con​mechanical construction. Complete similarity and equality, in so far as they can only be cognised in intuition, is congruity. All geometrical construction of complete identity rests on congruity. This congruity of two motions combined with a third (in short, the motu composito itself) can never take place, when the two former are presented in one and the same space, i.e. relative [space]. Hence all attempts to demonstrate the above proposition in its three cases, have always been mechanical solutions only, inasmuch, namely, as though moving causes by which a given motion was combined with another, were made to produce a third, the proofs that the former were the same as the latter, and as such, admitted of being presented in pure intuition a priori [were not given].
Observation 2.
When, for instance, a velocity $AB$ is termed double, nothing else can be understood thereby, but that it consists of two simple and equal [velocities] $AB$ and $BC$, (see Fig. 1). But if a double velocity be explained by saying that it is a motion by which a doubly great space is passed over in the same time, something is here assumed which is not necessarily implied, namely, that two equal velocities may be combined in the same way as two equal spaces, for it is not in itself obvious that a given velocity consists of smaller [velocities]; and in the same way that a rapidity consists of slownesses as a space does of smaller [spaces]. For the parts of the velocity are not outside one another, as the parts of the space; and if the former are to be considered as quantity, the conception of their quantity, as it is intensive, must be constructed in a different manner to that of the extensive quantity of space. But this construction is possible in no other way than by the mediate composition of two equal motions, one of which is that of the body, the other that of the relative space in the contrary direction, but which, for this reason, is completely identical with an equal motion of the body in the previous direction. For in the same direction two equal velocities would not admit of being compounded in one body, except through external moving causes; for ​instance, a ship carrying the body with one of these velocities, while another movable force, immovably bound up with the ship, impresses upon the body the second velocity, which is equal to the previous one. In this it must always be presupposed that the body maintains itself in free motion with the first velocity when the second enters; but this is a natural law of moving forces, which cannot come into consideration when the question is simply how the conception of veh city is constructed as a quantity; so much as to the addition of velocities to one another. But when the question is of the subtraction of one from the other, this latter is easily conceivable, if the possibility of a velocity, as quantity by addition, has once been admitted; yet this conception cannot be so easily constructed, for to this end two contrary motions must be combined in one body; and how is this to happen? Immediately, namely, in respect of the same resting space, it is impossible to conceive of two equal motions in contrary directions in the same body; but the idea of the impossibility of these two motions in one body is not the conception of its rest, but of the impossibility of the construction of this composition of contrary motions, which is nevertheless assumed in the proposition as possible. Now this construction is not otherwise possible, than by the combination of the motion of the body with the motion of the space as has been demonstrated. Finally, as concerns the composition of two motions, whose direction encloses an angle, they do not admit of being conceived in a body, in reference to one and the same space, if one of them be not affected by an external continuous inflowing force (for instance, a vessel bearing the body onward), while the other maintains itself unaltered, or generally [expressed]: one must have as a basis, moving forces, and the production of a third movement from two combined forces, but this, although the mechanical carrying out of that which contains a conception, is not its mathematical construction, which has only to render inimitable what the object is (as quantum), not, how it may be transformed by nature or art, by means of sundry implements and forces. The composition of motions, in order to determine their relation to others as quantity, must take place ​according to the rules of congmity, which is only possible, in all three cases, by means of the motion of the space that is congruous with one of the two given motions, whereby both are congruous with the compound [motion].
Observation 3.
Thus Phoronomy, not as pure doctrine of motion, but as pure doctrine of the quantity of motion, in which matter is conceived by no other quality but that of mere movability, contains nothing but this single proposition, carried out in the three cases adduced, of the composition of motion, and indeed of the possibility of rectilinear motion alone, not of curvilinear; for, because in the latter the motion is continuously changed in direction, a cause of this motion, which cannot be merely space, must be brought to bear. That only the single case in which the directions of the same enclose an angle, is usually understood by the designation compound motion, does some detriment to the principle of the division of a pure philosophical science generally, although not to physics: for, as concerns the latter, all the three cases treated in the above proposition admit of being adequately presented in the third alone. For when the angle enclosing the two given motions is conceived as infinitely small, it contains the first [case]; but if it be conceived as only divided in an infinitely small degree from a single straight line, it contains the second case; so that, in the proposition already stated respecting composite motion, all three cases mentioned by us, are capable of being given as in a universal formula. But in this way one could not learn to comprehend the qualitative doctrine of motion in its parts a priori, which in many respects is also useful.
If any one cares to connect the three parts in question of the universal Phoronomic proposition with the scheme of the subdivision of all pure conceptions of the understanding, here, especially with that of the conception of quantity, he will observe: that, as the conception of a quantity always contains that of the composition of the homogeneous, the doctrine of the composition of motions ​is at the same time the pure doctrine of quantity therein; and indeed that in all three momenta furnished by space, the unity of line and direction, the plurality of directions in one and the same line, and finally the totality of directions as well as of lines, according to which the motion can take place, it contains the determination of all possible motion as quantum, although its quantity (in a movable point) consists merely in velocity. This observation only has its uses in transcendental philosophy.