A View of Sir Isaac Newton's Philosophy

by Henry Pemberton

18. Now this experience is alone sufficient to shew to us the whole of this law of motion.

19. Since we find, that the same power will always produce the same change in the motion of any body, whether that body were before moving with a swifter or slower motion; the change wrought in the motion of a body depends only on the power applied to it, without any regard to the body’s former motion: and therefore the degree of motion, which the body already possesses, having no influence on the power applied to disturb its operation, the effects of the same power will not only be the same in all degrees of motion of the body; but we have likewise no reason to doubt, but that a body perfectly at rest would receive from any power as much motion, as would be equivalent to the effect of the same power applied to that body already in motion.

20. Again, suppose a body being at rest, any number of equal powers should be successively applied to it; pushing it forward from time to time in the same course or direction. Upon the application of the first power the body would begin to move; when the second power was applied, it appears from what has been said, that the motion of the body would become double; the third power would treble the motion of the body; and so on, till after the operation of the last power the motion of the body would be as many times the motion, which the first power gave it, as there are powers in number. and the effect of this number of powers will be always the same, without any regard to the space of time taken up in applying them: so that greater or lesser intervals between the application of each of these powers will produce no difference at all in their effects. Since therefore the distance of time between the action of each power is of no consequence; without doubt the effect will still be the same, though the powers should all be applied at the very same instant; or although a single power should be applied equal in strength to the collective force of all these powers. Hence it plainly follows, that the degree of motion, into which any body will be put out of a state of rest by any power, will be proportional to that power. A double power will give twice the velocity, a treble power three times the velocity, and so on. The foregoing reasoning will equally take place, though the body were not supposed to be at rest, when the powers began to be applied to it; provided the direction, in which the powers were applied, either conspired with the action of the body, or was directly opposite to it. Therefore if any power be applied to a moving body, and act upon the body either in the direction wherewith the body moves, so as to accelerate the body; or if it act directly opposite to the motion of the body, so as to retard it: in both these cases the change of motion will be proportional to the power applied; nay, the augmentation of the motion in one case, and the diminution thereof in the other, will be equal to that degree of motion, into which the same power would put the body, had it been at rest, when the power was applied.

21. Farther, a power may be so applied to a moving body, as to act obliquely to the motion of the body. And the effects of such an oblique motion may be deduced from this observation; that as all bodies are continually moving along with the earth, we see that the visible effects of the same power are always the same, in whatever direction the power acts: and therefore the visible effects of any power upon a body, which seems only to be at rest, is always to appearance the same as the real effect would be upon a body truly at rest. Now suppose a body were moving along the line A B (in fig. 2.) and the eye accompanied it with an equal motion in the line C D equidistant from A B; so that when the body is at A, the eye shall be at C, and when the body is advanced to E in the line A B, the eye shall be advanced to F in the line C D, the distances A E and C F being equal. It is evident, that here the body will appear to the eye to be at rest; and the line F E G drawn from the eye through the body shall seem to the eye to be immoveable; though as the body and eye move forward together, this line shall really also move; so that when the body shall be advanced to H and the eye to K, the line F E G shall be transferred into the situation K H L, this line K H L being equidistant from F E G. Now if the body when at E were to receive an impulse in the direction of the line F E G; while the eye is moving on from F to K and carrying along with it the line F E G, the body will appear to the eye to move along this line F E G: for this is what has just now been said; that while bodies are moving along with the earth, and the spectator’s eye partakes of the same motion, the effect of any power upon the body will appear to be what it would really have been, had the body been truly at rest, when the power was applied. From hence it follows, that when the eye is advanced to K, the body will appear somewhere in the line K H L. Suppose it appear in M; then it is manifest, from what has been premised at the beginning of this paragraph, that the distance H M is equal to what the body would have run upon the line E G, during the time, wherein the eye has passed from F to K, provided that the body had been at rest, when acted upon in E. If it be farther asked, after what manner the body has moved from E to M? I answer, through a straight line; for it has been shewn above in the explication of the first law of motion, that a moving body, from the time it is left to it self, will proceed on in one continued straight line.

22. If E N be taken equal to H M and N M be drawn; since H M is equidistant from E N, N M will be equidistant from E H. Therefore the effect of any power upon a moving body, when that power acts obliquely to the motion of the body, is to be determined in this manner. Suppose the body is moving along the straight line A E B, if when the body is come to E, a power gives it an impulse in the direction of the line E G, to find what course the body will afterwards take we must proceed thus. Take in E B any length E H, and in E G take such a length E N, that if the body had been at rest in E, the power applied to it would have caused it to move over E N in the same space of time, as it would have employed in passing over E H, if the power had not acted at all upon it. Then draw H L equidistant from E G, and N M equidistant from E B. After this, if a line be drawn from E to the point M, where these two lines meet, the line E M will be the course into which the body will be put by the action of the power upon it at E.

23. A mathematical reader would here expect in some particulars more regular demonstrations; but as I do not at present address my self to such, so I hope, what I have now written will render my meaning evident enough to those, who are unacquainted with that kind of reasoning.

24. Now as we have been shewing, that some actual force is necessary either to put bodies out of a state of rest into motion, or to change the motion, which they have once received; it is proper here to observe, that this quality in bodies, whereby they preserve their present state, with regard to motion or rest, till some active force disturb them, is called the vis inertiae of matter: and by this property, matter, sluggish and unactive of it self, retains all the power impressed upon it, and cannot be made to cease from action, but by the opposition of as great a power, as that which first moved it. By the degree of this vis inertiae, or power of inactivity, as we shall henceforth call it, we primarily judge of the quantity of solid matter in each body; for as this quality is inherent in all the bodies, upon which we can make any trial, we conclude it to be a property essential to all matter; and as we yet know no reason to suppose, that bodies are composed of different kinds of matter, we rather presume, that the matter of all bodies is the same; and that the degree of this power of inactivity is in every body proportional to the quantity of the solid matter in it. But although we have no absolute proof, that all the matter in the universe is uniform, and possesses this power of inactivity in the same degree; yet we can with certainty compare together the different degrees of this power of inactivity in different bodies. Particularly this power is proportional to the weight of bodies, as Sir Isaac Newton has demonstrated[44]. However, notwithstanding that this power of inactivity in any body can be more certainly known, than the quantity of solid matter in it; yet since there is no reason to suspect that one is not proportional to the other, we shall hereafter speak without hesitation of the quantity of matter in bodies, as the measure of the degree of their power of inactivity.

25. This being established, we may now compare the effects of the same power upon different bodies, as hitherto we have shewn the effects of different powers upon the same body. And here if we limit the word motion to the peculiar sense given to it in philosophy, we may comprehend all that is to be said upon this head under one short precept; that the same power, to whatever body it is applied, will always produce the same degree of motion. But here motion does not signify the degree of celerity or velocity with which a body moves, in which sense only we have hitherto used it; but it is made use of particularly in philosophy to signify the force with which a body moves: as if two bodies A and B being in motion, twice the force would be required to stop A as to stop B, the motion of A would be esteemed double the motion of B. In moving bodies, these two things are carefully to be distinguished; their velocity or celerity, which is measured by the space they pass through during any determinate portion of time; and the quantity of their motion, or the force, with which they will press against any resistance. Which force, when different bodies move with the same velocity, is proportional to the quantity of solid matter in the bodies; but if the bodies are equal, this force is proportional to their respective velocities, and in other cases it is proportional both to the quantity of solid matter in the body, and also to its velocity. To instance in two bodies A and B: if A be twice as great as B, and they have both the same velocity, the motion of A shall be double the motion of B; and if the bodies be equal, and the velocity of A be twice that of B, the motion of A shall likewise be double that of B; but if A be twice as large as B, and move twice as swift, the motion of A will be four times the motion of B; and lastly, if A be twice as large as B, and move but half as fast, the degree of their motion shall be the same.