A View of Sir Isaac Newton's Philosophy

by Henry Pemberton

13. Thus I have given the sum of what has been written concerning the effects of percussion, when two bodies freely in motion strike directly against each other; and the results here set down, as the consequence of our reasoning from the laws of motion, answer most exactly to experience. A particular set of experiments has been invented to make trial of these effects of percussion with the greatest exactness. But I must defer these experiments, till I have explained the nature of pendulums[49]. I shall therefore now proceed to describe some of the appearances, which are caused in bodies from the influence of the power of gravity united with the general laws of motion; among which the motion of the pendulum will be included.

[14.] The most simple of these appearances is, when bodies fall down merely by their weight. In this case the body increases continually its velocity, during the whole time of its fall, and that in the very same proportion as the time increases. For the power of gravity acts constantly on the body with the same degree of strength: and it has been observed above in the first law of motion, that a body being once in motion will perpetually preserve that motion without the continuance of any external influence upon it: therefore, after a body has been once put in motion by the force of gravity, the body would continue that motion, though the power of gravity should cease to act any farther upon it; but, if the power of gravity continues still to draw the body down, fresh degrees of motion must continually be added to the body; and the power of gravity acting at all times with the same strength, equal degrees of motion will constantly be added in equal portions of time.

15. This conclusion is not indeed absolutely true: for we shall find hereafter[50], that the power of gravity is not of the same strength at all distances from the center of the earth. But nothing of this is in the least sensible in any distance, to which we can convey bodies. The weight of bodies is the very same to sense upon the highest towers or mountains, as upon the level ground; so that in all the observations we can make, the forementioned proportion between the velocity of a falling body and the time, in which it has been descending, obtains without any the least perceptible difference.

16. From hence it follows, that the space, through which a body falls, is not proportional to the time of the fall; for since the body increases its velocity, a greater space will be passed over in the same portion of time at the latter part of the fall, than at the beginning. Suppose a body let fall from the point A (in fig. 8.) were to descend from A to B in any portion of time; then if in an equal portion of time it were to proceed from B to C; I say, the space B C is greater than A B; so that the time of the fall from A to C being double the time of the fall from A to B, A C shall be more than double of A B.

17. The geometers have proved, that the spaces, through which bodies fall thus by their weight, are just in a duplicate or two-fold proportion of the times, in which the body has been falling. That is, if we were to take the line D E in the same proportion to A B, as the time, which the body has imployed in falling from A to C, bears to the time of the fall from A to B; then A C will be to D E in the same proportion. In particular, if the time of the fall through A C be twice the time of the fall through A B; then D E will be twice A B, and A C twice D E; or A C four times A B. But if the time of the fall through A C had been thrice the time of the fall through A B; D E would have been treble of A B, and A C treble of D E; that is, A C would have been equal to nine times A B.

[18.] If a body fall obliquely, it will approach the ground by slower degrees, than when it falls perpendicularly. Suppose two lines A B, A C (in fig. 9.) were drawn, one perpendicular, and the other oblique to the ground D E: then if a body were to descend in the slanting line A C; because the power of gravity draws the body directly downwards, if the line A C supports the body from falling in that manner, it must take off part of the effect of the power of gravity; so that in the time, which would have been sufficient for the body to have fallen through the whole perpendicular line A B, the body shall not have passed in the line A C a length equal to A B; consequently the line A C being longer than A B, the body shall most certainly take up more time in passing through A C, than it would have done in falling perpendicularly down through A B.

19. The geometers demonstrate, that the time, in which the body will descend through the oblique straight line A C, bears the same proportion to the time of its descent through the perpendicular A B, as the line it self A C bears to A B. And in respect to the velocity, which the body will have acquired in the point C, they likewise prove, that the length of the time imployed in the descent through A C so compensates the diminution of the influence of gravity from the obliquity of this line, that though the force of the power of gravity on the body is opposed by the obliquity of the line A C, yet the time of the body’s descent shall be so much prolonged, that the body shall acquire the very same velocity in the point C, as it would have got at the point B by falling perpendicularly down.

[20.] If a body were to descend in a crooked line, the time of its descent cannot be determined in so simple a manner; but the same property, in relation to the velocity, is demonstrated to take place in all cases: that is, in whatever line the body descends, the velocity will always be answerable to the perpendicular height, from which the body has fell. For instance, suppose the body A (in fig. 10.) were hung by a string to the pin B. If this body were let fall, till it came to the point C perpendicularly under B, it will have moved from A to C in the arch of a circle. Then the horizontal line A D being drawn, the velocity of the body in C will be the same, as if it had fallen from the point D directly down to C.

[21.] If a body be thrown perpendicularly upward with any force, the velocity, wherewith the body ascends, shall continually diminish, till at length it be wholly taken away; and from that time the body will begin to fall down again, and pass over a second time in its descent the line, wherein it ascended; falling through this line with an increasing velocity in such a manner, that in every point thereof, through which it falls, it shall have the very same velocity, as it had in the same place, when it ascended; and consequently shall come down into the place, whence it first ascended, with the velocity which was at first given to it. Thus if a body were thrown perpendicularly up in the line A B (in fig. II.) with such a force, as that it should stop at the point B, and there begin to fall again; when it shall have arrived in its descent to any point as C in this line, it shall there have the same velocity, as that wherewith it passed by this point C in its ascent; and at the point A it shall have gained as great a velocity, as that wherewith it was first thrown upwards. As this is demonstrated by the geometrical writers; so, I think, it will appear evident, by considering only, that while the body descends, the power of gravity must act over again, in an inverted order, all the influence it had on the body in its ascent; so as to give again to the body the same degrees of velocity, which it had taken away before.