A View of Sir Isaac Newton's Philosophy

by Henry Pemberton

22. After the same manner, if the body were thrown upwards in the oblique straight line C A (in fig. 9.) from the point C, with such a degree of velocity as just to reach the point A; it shall by its own weight return again through the line A C by the same degrees, as it ascended.

[23.] And lastly, if a body were thrown with any velocity in a line continually incurvated upwards, the like effect will be produced upon its return to the point, whence it was thrown. Suppose for instance, the body A (in fig. 12.) were hung by a string A B. Then if this body be impelled any way, it must move in the arch of a circle. Let it receive such an impulse, as shall cause it to move in the arch A C; and let this impulse be of such strength, that the body may be carried from A as far as D, before its motion is overcome by its weight: I say here, that the body forthwith returning from D, shall come again into the point A with the same velocity, as that wherewith it began to move.

[24.] It will be proper in this place to observe concerning the power of gravity, that its force upon any body does not at all depend upon the shape of the body; but that it continues constantly the same without any variation in the same body, whatever change be made in the figure of the body: and if the body be divided into any number of pieces, all those pieces shall weigh just the same, as they did, when united together in one body: and if the body be of a uniform contexture, the weight of each piece will be proportional to its bulk. This has given reason to conclude, that the power of gravity acts upon bodies in proportion to the quantity of matter in them. Whence it should follow, that all bodies must fall from equal heights in the same space of time. And as we evidently see the contrary in feathers and such like substances, which fall very slowly in comparison of more solid bodies; it is reasonable to suppose, that some other cause concurs to make so manifest a difference. This cause has been found by particular experiments to be the air. The experiments for this purpose are made thus. They set up a very tall hollow glass; within which near the top they lodge a feather and some very ponderous body, usually a piece of gold, this metal being the most weighty of any body known to us. This glass they empty of the air contained within it, and by moving a wire, which passes through the top of the glass, they let the feather and the heavy body fall together; and it is always found, that as the two bodies begin to descend at the same time, so they accompany each other in the fall, and come to the bottom at the very same instant, as near as the eye can judge. Thus, as far as this experiment can be depended on, it is certain, that the effect of the power of gravity upon each body is proportional to the quantity of solid matter, or to the power of inactivity in each body. For in the limited sense, which we have given above to the word motion, it has been shown, that the same force gives to all bodies the same degree of motion, and different forces communicate different degrees of motion proportional to the respective powers[51]. In this case, if the power of gravity were to act equally upon the feather, and upon the more solid body, the solid body would descend so much slower than the feather, as to have no greater degree of motion than the feather: but as both bodies descend with equal swiftness, the degree of motion in the solid body is greater than in the feather, bearing the same proportion to it, as the quantity of matter in the solid body to the quantity of matter in the feather. Therefore the effect of gravity on the solid body is greater than on the feather, in proportion to the greater degree of motion communicated; that is, the effect of the power of gravity on the solid body bears the same proportion to its effect on the feather, as the quantity of matter in the solid body bears to the quantity of matter in the feather. Thus it is the proper deduction from this experiment, that the power of gravity acts not on the surface of bodies only, but penetrates the bodies themselves most intimately, and operates alike on every particle of matter in them. But as the great quickness, with which the bodies fall, leaves it something uncertain, whether they do descend absolutely in the same time, or only so nearly together, that the difference in their swift motion is not discernable to the eye; this property of the power of gravity, which has here been deduced from this experiment, is farther confirmed by pendulums, whose motion is such, that a very minute difference would become sufficiently sensible. This will be farther discoursed on in another place[52]; but here I shall make use of the principle now laid down to explain the nature of what is called the center of gravity in bodies.

[25.] The center of gravity is that point, by which if a body be suspended, it shall hang at rest in any situation. In a globe of a uniform texture the center of gravity is the same with the center of the globe; for as the parts of the globe on every side of its center are similarly disposed, and the power of gravity acts alike on every part; it is evident, that the parts of the globe on each side of the center are drawn with equal force, and therefore neither side can yield to the other; but the globe, if supported at its center, must of necessity hang at rest. In like manner, if two equal bodies A and B (in fig. 13.) be hung at the extremities of an inflexible rod C D, which should have no weight; these bodies, if the rod be supported at its middle E, shall equiponderate; and the rod remain without motion. For the bodies being equal and at the same distance from the point of support E, the power of gravity will act upon each with equal strength, and in all respects under the same circumstances; therefore the weight of one cannot overcome the weight of the other. The weight of A can no more surmount the weight of B, than the weight of B can surmount the weight of A. Again, suppose a body as A B (in fig. 14.) of a uniform texture in the form of a roller, or as it is more usually called a cylinder, lying horizontally. If a straight line be drawn between C and D, the centers of the extreme circles of this cylinder; and if this straight line, commonly called the axis of the cylinder, be divided into two equal parts in E: this point E will be the center of gravity of the cylinder. The cylinder being a uniform figure, the parts on each side of the point E are equal, and situated in a perfectly similar manner; therefore this cylinder, if supported at the point E, must hang at rest, for the same reason as the inflexible rod above-mentioned will remain without motion, when suspended at its middle point. And it is evident, that the force applied to the point E, which would uphold the cylinder, must be equal to the cylinder’s weight. Now suppose two cylinders of equal thickness A B and C D to be joined together at C B, so that the two axis’s E F, and F G lie in one straight line. Let the axis E F be divided into two equal parts at H, and the axis F G into two equal parts at I. Then because the cylinder A B would be upheld at rest by a power applied in H equal to the weight of this cylinder, and the cylinder C D would likewise be upheld by a power applied in I equal to the weight of this cylinder; the whole cylinder A D will be supported by these two powers: but the whole cylinder may likewise be supported by a power applied to K, the middle point of the whole axis E G, provided that power be equal to the weight of the whole cylinder. It is evident therefore, that this power applied in K will produce the same effect, as the two other powers applied in H and I. It is farther to be observed, that H K is equal to half F G, and K I equal to half E F; for E K being equal to half E G, and E H equal to half E F, the remainder H K must be equal to half the remainder F G; so likewise G K being equal to half G E, and G I equal to half G F, the remainder I K must be equal to half the remainder E F. It follows therefore, that H K bears the same proportion to K I, as F G bears to E F. Besides, I believe, my readers will perceive, and it is demonstrated in form by the geometers, that the whole body of the cylinder C D bears the same proportion to the whole body of the cylinder A B, as the axis F G bears to the axis E F[53]. But hence it follows, that in the two powers applied at H and I, the power applied at H bears the same proportion to the power applied at I, as K I bears to K H. Now suppose two strings H L and I M extended upwards, one from the point H and the other from I, and to be laid hold on by two powers, one strong enough to hold up the cylinder A B, and the other of strength sufficient to support the cylinder C D. Here as these two powers uphold the whole cylinder, and therefore produce an effect, equal to what would have been produced by a power applied to the point K of sufficient force to sustain the whole cylinder: it is manifest, that if the cylinder be taken away, the axis only being left, and from the point K a string, as K N, be extended, which shall be drawn down by a power equivalent to the weight of the cylinder, this power shall act against the other two powers, as much as the cylinder acted against them; and consequently these three powers shall be upon a balance, and hold the axis H I fixed between them. But if these three powers preserve a mutual balance, the two powers applied to the strings H L and I M are a balance to each other; the power applied to the string H L bearing the same proportion to the power applied to the string I M, as the distance I K bears to the distance K H. Hence it farther appears, that if an inflexible rod A B (in fig. 15.) be suspended by any point C not in the middle thereof; and if at A the end of the shorter arm be hung a weight, and at B the end of the longer arm be also hung a weight less than the other, and that the greater of these weights bears to the lesser the same proportion, as the longer arm of the rod bears to the shorter; then these two weights will equiponderate: for a power applied at C equal to both these weights will support without motion the rod thus charged; since here nothing is changed from the preceding case but the situation of the powers, which are now placed on the contrary sides of the line, to which they are fixed. Also for the same reason, if two weights A and B (in fig. 16.) were connected together by an inflexible rod C D, drawn from C the center of gravity of A to D the center of gravity of B; and if the rod C D were to be so divided in E, that the part D E bear the same proportion to the other part C E, as the weight A bears to the weight B: then this rod being supported at E will uphold the weights, and keep them at rest without motion. This point E, by which the two bodies A and B will be supported, is called their common center of gravity. And if a greater number of bodies were joined together, the point, by which they could all be supported, is called the common center of gravity of them all. Suppose (in fig. 17.) there were three bodies A, B, C, whose respective centers of gravity were joined by the three lines D E, D F, E F: the line D E being so divided in G, that D G bear the same proportion to G E, as B bears to A; G is the center of gravity common to the two bodies A and B; that is, a power equal to the weight of both the bodies applied to G would support them, and the point G is pressed as much by the two weights A and B, as it would be, if they were both hung together at that point. Therefore, if a line be drawn from G to F, and divided in H, so that G H bear the same proportion to H F, as the weight C bears to both the weights A and B, the point H will be the common center of gravity of all the three weights; for H would be their common center of gravity, if both the weights A and B were hung together at G, and the point G is pressed as much by them in their present situation, as it would be in that case. In the same manner from the common center of these three weights, you might proceed to find the common center, if a fourth weight were added, and by a gradual progress might find the common center of gravity belonging to any number of weights whatever.

26. As all this is the obvious consequence of the proposition laid down for assigning the common center of gravity of any two weights, by the same proposition the center of gravity of all figures is found. In a triangle, as A B C (in fig. 18.) the center of gravity lies in the line drawn from the middle point of any one of the sides to the opposite angle, as the line B D is drawn from D the middle of the line A C to the opposite angle B[54]; so that if from the middle of either of the other sides, as from the point E in the side A B, a line be drawn, as E C, to the opposite angle; the point F, where this line crosses the other line B D, will be the center of gravity of the triangle[55]. Likewise D F is equal to half F B, and E F equal to half F C[56]. In a hemisphere, as A B C (fig. 19.) if from D the center of the base the line D B be erected perpendicular to that base, and this line be so divided in E, that D E be equal to three fifths of B E, the point E is the center of gravity of the hemisphere[57].

27. It will be of use to observe concerning the center of gravity of bodies; that since a power applied to this center alone can support a body against the power of gravity, and hold it fixed at rest; the effect of the power of gravity on a body is the same, as if that whole power were to exert itself on the center of gravity only. Whence it follows, that, when the power of gravity acts on a body suspended by any point, if the body is so suspended, that the center of gravity of the body can descend; the power of gravity will give motion to that body, otherwise not: or if a number of bodies are so connected together, that, when any one is put into motion, the rest shall, by the manner of their being joined, receive such motion, as shall keep their common center of gravity at rest; then the power of gravity shall not be able to produce any motion in these bodies, but in all other cases it will. Thus, if the body A B (in fig. 20, 21.) whose center of gravity is C, be hung on the point A, and the center C be perpendicularly under A (as in fig. 20.) the weight of the body will hold it still without motion, because the center C cannot descend any lower. But if the body be removed into any other situation, where the center C is not perpendicularly under A (as in fig. 21.) the body by its weight will be put into motion towards the perpendicular situation of its center of gravity. Also if two bodies A, B (in fig. 22.) be joined together by the rod C D lying in an horizontal situation, and be supported at the point E; if this point be the center of gravity common to the two bodies, their weight will not put them into motion; but if this point E is not their common center of gravity, the bodies will move; that part of the rod C D descending, in which the common center of gravity is found. So in like manner, if these two bodies were connected together by any more complex contrivance; yet if one of the bodies cannot move without so moving the other, that their common center of gravity shall rest, the weight of the bodies will not put them in motion, otherwise it will.

[28.] I shall proceed in the next place to speak of the mechanical powers. These are certain instruments or machines, contrived for the moving great weights with small force; and their effects are all deducible from the observation we have just been making. They are usually reckoned in number five; the lever, the wheel and axis, the pulley, the wedge, and the screw; to which some add the inclined plane. As these instruments have been of very ancient use, so the celebrated Archimedes seems to have been the first, who discovered the true reason of their effects. This, I think, may be collected from what is related of him, that some expressions, which he used to denote the unlimited force of these instruments, were received as very extraordinary paradoxes: whereas to those, who had understood the cause of their great force, no expressions of that kind could have appeared surprizing.

29. All the effects of these powers may be judged of by this one rule, that, when two weights are applied to any of these instruments, the weights will equiponderate, if, when put into motion, their velocities will be reciprocally proportional to their respective weights. And what is said of weights, must of necessity be equally understood of any other forces equivalent to weights, such as the force of a man’s arm, a stream of water, or the like.

30. But to comprehend the meaning of this rule, the reader must know, what is to be understood by reciprocal proportion; which I shall now endeavour to explain, as distinctly as I can; for I shall be obliged very frequently to make use of this term. When any two things are so related, that one increases in the same proportion as the other, they are directly proportional. So if any number of men can perform in a determined space of time a certain quantity of any work, suppose drain a fish-pond, or the like; and twice the number of men can perform twice the quantity of the same work, in the same time; and three times the number of men can perform as soon thrice the work; here the number of men and the quantity of the work are directly proportional. On the other hand, when two things are so related, that one decreases in the same proportion, as the other increases, they are said to be reciprocally proportional. Thus if twice the number of men can perform the same work in half the time, and three times the number of men can finish the same in a third part of the time; then the number of men and the time are reciprocally proportional. We shewed above[58] how to find the common center of gravity of two bodies, there the distances of that common center from the centers of gravity of the two bodies are reciprocally proportional to the respective bodies. For C E in fig. 16. being in the same proportion to E D, as B bears to A; C E is so much greater in proportion than E D, as A is less in proportion than B.

31. Now this being understood, the reason of the rule here stated will easily appear. For if these two bodies were put in motion, while the point E rested, the velocity, wherewith A would move, would bear the same proportion to the velocity, wherewith B would move, as E C bears to E D. The velocity therefore of each body, when the common center of gravity rests, is reciprocally proportional to the body. But we have shewn above[59], that if two bodies are so connected together, that the putting them in motion will not move their common center of gravity; the weight of those bodies will not produce in them any motion. Therefore in any of these mechanical engines, if, when the bodies are put into motion, their velocities are reciprocally proportional to their respective weights, whereby the common center of gravity would remain at rest; the bodies will not receive any motion from their weight, that is, they will equiponderate. But this perhaps will be yet more clearly conceived by the particular description of each mechanical power.