In this sense, too, rest is a real state of existence ; -so is no velocity, or perseverance in the same place, no force or perseverance in retaining the previous ve locity ; and so of others. These differ greatly from non-existence. When, in the solution of a problem, we arrive at a quantity of the former description, the determination is real, though of a peculiar kind. But we arrive at an expression of the latter kind only, where the problem is impossible.
From what we have now said, we believe that the general law of continuity is sufficiently manifest ; and it will be hardly necessary for us, as Boscovich has done, to prove that, upon the same principles, the change from one velocity to another never takes place but by passing through all the intermediate veloci• tics. For example, if there was an abrupt passage from the velocity 6, ariiing from all the preceding circumstances affecting the moving body, to the vela city 9, then, at the very instant of the passage, we have a determination to two different velocities, which, as already shown, would be absurd.
It is therefore evident, that the whole velocity of a body, can neither be created nor extinguished in a mo ment : and consequently, in the collision of two bo• dies, there must be a breach of the law of continuity, if the contact takes place with any &terminate dif ference'of their velocities. Let us see what must happen, if no such breach be admitted, since these bodies cannot come into contact with the previous ve• locities. The velocities must begin immediately tc change, either by the increase or diminution of on( or both. The cause of such a change as this, is callec a force; and there must consequently be some fore( acting to produce this effect, even though the bodies have not yet come into contact.
To prevent the breach of continuity, it would b( sufficient that the forces should act only on one a the bodies ; but from the principle of the equalit) of action and reaction, for which there is abundam evidence by induction, we must suppose, that thi force is mutual between them, increasing the velocit) of the one at the same rate as it diminishes that o the other. This is, in fact, producing a sort of op posite velocities, by which, if the bodies were impres sed alone, they would be made to recede. The fore(
is therefore repulsive, and it becomes us to enquirt into the laws by which it is regulated. In the cas( above mentioned, where A, moving with the velocit) 6, is overtaken and hit by B, moving with the velocit; 's 12, it would be enough, that the repulsive power which we have now discovered, should be able to ex tinguish the 6 degrees of difference of velocity, and the actual contact might take place at the very mo ment in which the velocities became equal. But should B, following with 20 degrees of velocity, hit A with 6, the difference or relative velocity is 14 ; and though the repulsive power be equal to the extinction of 6 degrees, there is still a difference of 8 at the time of contact : nay, there must be even a greater difference than 8 degrees ; for the repulsive power will have less time to act than in the former case, and therefore, agreeably to constant experience, it must produce a less effect than before.
The contact of the two bodies would therefore take place, although the difference of their velocities . be even greater than 8 degrees, and we should have the same breach of continuity that we have already demonstrated to be impossible. Nature therefore will provide for this, as for the former case ; and at a distance more minute, an additional force will take away all the 14 degrees of difference even before the coutact takes place. But when we have come thus far, it is evident that there can be no limits assign. cd to the increase of the repulsive power, which acts between the two bodies hindering their approach. It must be equal to the extinction of any velocity, however great. We must therefore admit, that the repulsive forces, as the distances are diminished, in crease ad infinitum : that is, we must admit the ex - istence of the asymptotic arc ED of the curve in Fig. 1st, which exhibits the law of impenetrability, and that the actual contact of bodies or particles is al .together impossible.
This perhaps is not the only asymptotic arc in the curve of forces : There may be others, or a succes sion of them—a circumstance which opens a fertile field of contemplation. But we proceed to those branches of the curve for which we have undoubted evidence.