The fourth conclusion is a corollary of the third. For nothing is necessarily permanent in local movement, except that which constitutes its essence. Now, its essence lies in this: that it must evolve extension at the rate and in the direction determined by the momentum of which it is the exponent. Therefore the permanent determination of which we are speaking is nothing else than the virtuality of the momentum itself as developing into extension. And since the momentum by which the moving body is animated has a determinate intensity and direction, which virtually contains a determinate velocity and direction of movement, it follows that the permanent determination in question consists in the actual tendency of movement to evolve uniformly and in a straight line—uniformly, because velocity is the form of movement, and the velocity determined by the intensity of the actual momentum is actually one; in a straight line, because the actual momentum being one, it gives but one direction to the movement, which therefore will be straight in its tendency. Whence we conclude that it is of the essence of local movement to have an actual tendency to evolve uniformly in a straight line.
Some will object that local movement may lack both uniformity and straightness. This is quite true, but it does not destroy our conclusion. For, as movement is always in fieri, and exists only by infinitesimal instants in which it is impossible to admit more than one velocity and one direction, it remains always true that within every instant of its existence the movement is straight and uniform, and that in every such instant it tends to continue in the same direction and at the same rate—that is, with the velocity and direction it actually possesses. This velocity and direction may, of course, be modified in the following instant; but in the following instant, too, the movement will tend to evolve uniformly and in a straight line suitably to its new velocity and direction. Whence it is manifest that, although in the continuation of the movement there may be a series of different velocities and directions, yet the tendency of the movement is, at every instant of its existence, to extend uniformly in a straight line. This truth is the foundation of dynamics.
Our fifth conclusion is sufficiently evident from what we have just said. For, whatever be the intensity and direction of the movement, its determination to extend uniformly in a straight line is not interfered with.
Our last conclusion has no need of explanation. For, since the affections of local movement are the result of new momentums impressed on the subject it is plain that they are intrinsic modes characterizing a movement individually different from the movement that preceded. The tendency to evolve uniformly in a straight line remains unimpaired, as we have shown; but the movement itself becomes entitatively—viz., quantitatively—different.
Remarks.—Local movement is divided into uniform and varied. Uniform movement we call that which has a constant velocity. For, as velocity is the form of movement, to say that a movement is uniform is to say that it has but one velocity in the whole of its extension. We usually call “uniform” all movement whose apparent velocity is constant; but, to say the truth, no rigorously uniform movement exists in nature for any appreciable length of time. In fact, every element of matter lies within the sphere of action of all other elements, and is continually acted on, and continually receives new momentums; the evident consequence of which is that its real movement must undergo a continuous change of velocity. Hence rigorously uniform movement is limited to infinitesimal time.
Varied movement is that whose rate is continually changing. It is divided into accelerated and retarded; and, when the acceleration or the retardation arises from a constant action which in equal times imparts equal momentums, the movement is said to be uniformly accelerated or retarded.
Epilogue.—The explanation we have given of space, duration, and movement suffices, if we are not mistaken, to show what is the true nature of the only continuous quantities which can be found in the real order of things. The reader will have seen that the source of all continuity is motive power and its exertion. It is such an exertion that engenders local movement, and causes it to be continuous in its entity, in its local extension, and in its duration. In fact, why is the local movement continuous in its entity? Because the motive action strengthens or weakens it by continuous infinitesimal degrees in each successive infinitesimal instant, thus causing it to pass through all the degrees of intensity designable between its initial and its final velocity. And again: why is the local movement continuous in its local extension? Because it is the property of an action which proceeds from a point in space and is terminated to another point in space, to give a local direction to the subject in which the momentum is received; whence it follows that the subject under the influence of such a momentum must draw a continuous line in space. Finally, why is the local movement continuous in its duration? Because, owing to the continuous change of its ubication, the subject of the movement extends its absolute when from before to after, in a continuous succession, which is nothing but the duration of the movement.
Hence absolute space and absolute duration, which are altogether independent of motive actions, are not formally continuous, but only supply the extrinsic reason of the possibility of formal continuums. It is matter in movement that by the flowing of its ubi from here to there actually marks out a continuous line in space, and by the flowing of its quando from before to after marks out a continuous line in duration. Thus it is not absolute space, but the line drawn in space, that is formally extended from here to there; and it is not absolute duration, but the line successively drawn in duration, that is formally extended from before to after.
With regard to the difficulties which philosophers have raised at different times against local movement we have very little to say. An ancient philosopher, when called to answer some arguments against the possibility of movement, thought it sufficient to reply: Solvitur ambulando—“I walk; therefore movement is possible.” This answer was excellent; but, while showing the inanity of the objections, it took no notice of the fallacies by which they were supported. We might follow the same course; for the arguments advanced against movement are by no means formidable. Yet we will mention and solve three of them before dismissing the subject.