72. It has been observed above, that when a pendulum swings in an arch of a circle, as here in fig. 58, the pendulum A B swings in the circular arch C D; if you draw an horizontal line, as E F, from the place whence the pendulum is let fall, to the line A G, which is perpendicular to the horizon: then the velocity, which the pendulum will acquire in coming to the point G, will be the same, as any body would acquire in falling directly down from F to G. Now this is to be understood of the circular arch, which is described by the center of oscillation of the pendulum. I shall here farther observe, that if the straight line E G be drawn from the point, whence the pendulum falls, to the lowest point of the arch; in the same or in equal pendulums the velocity, which the pendulum acquires in G, is proportional to this line: that is, if the pendulum, after it has descended from E to G, be taken back to H, and let fall from thence, and the line H G be drawn; the velocity, which the pendulum shall acquire in G by its descent from H, shall bear the same proportion to the velocity, which it acquires in falling from E to G, as the straight line H G bears to the straight line E G.
[73.] We may now proceed to those experiments upon the percussion of bodies, which I observed above might be made with pendulums. This expedient for examining the effects of percussion was first proposed by our late great architect Sir Christopher Wren. And it is as follows. Two balls, as A and B (in fig. 59.) either equal or unequal, are hung by two strings from two points C and D, so that, when the balls hang down without motion, they shall just touch each other, and the strings be parallel. Here if one of these balls be removed to any distance from its perpendicular situation, and then let fall to descend and strike against the other; by the last preceding paragraph it will be known, with what velocity this ball shall return into its first perpendicular situation, and consequently with what force it shall strike against the other ball; and by the height to which this other ball ascends after the stroke, the velocity communicated to this ball will be discovered. For instance, let the ball A be taken up to E, and from thence be let fall to strike against B, passing over in its descent the circular arch E F. By this impulse let B fly up to G, moving through the circular arch H G. Then E I and G K being drawn horizontally, the ball A will strike against B with the velocity, which it would acquire in falling directly down from I; and the ball B has received a velocity, wherewith, if it had been thrown directly upward, it would have ascended up to K. Likewise if straight lines be drawn from E to F and from H to G, the velocity of A, wherewith it strikes, will bear the same proportion to the velocity, which B has received by the blow, as the straight line E F bears to the straight line H G. In the same manner by noting the place to which A ascends after the stroke, its remaining velocity may be compared with that, wherewith it struck against B. Thus may be experimented the effects of the body A striking against B at rest. If both the bodies are lifted up, and so let fall as to meet and impinge against each other just upon the coming of both into their perpendicular situation; by observing the places into which they move after the stroke, the effects of their percussion in all these cases may be found in the same manner as before.
74. Sir Isaac Newton has described these experiments; and has shewn how to improve them to a greater exactness by making allowance for the resistance, which the air gives to the motion of the balls[67]. But as this resistance is exceeding small, and the manner of allowing for it is delivered by himself in very plain terms, I need not enlarge upon it here. I shall rather speak to a discovery, which he made by these experiments upon the elasticity of bodies. It has been explained above[68], that when two bodies strike, if they be not elastic, they remain contiguous after the stroke; but that if they are elastic, they separate, and that the degree of their elasticity determines the proportion between the celerity wherewith they separate, and the celerity wherewith they meet. Now our author found, that the degree of elasticity appeared in the same bodies always the same, with whatever degree of force they struck; that is, the celerity wherewith they separated, always bore the same proportion to the celerity wherewith they met: so that the elastic power in all the bodies, he made trial upon, exerted it self in one constant proportion to the compressing force. Our author made trial with balls of wool bound up very compact, and found the celerity with which they receded, to bear about the proportion of 5 to 9 to the celerity wherewith they met; and in steel he found nearly the same proportion; in cork the elasticity was something less; but in glass much greater; for the celerity, wherewith balls of that material separated after percussion, he found to bear the proportion of 15 to 16 to the celerity wherewith they met[69].
[75.] I shall finish my discourse on pendulums, with this farther observation only, that the center of oscillation is also the center of another force. If a body be fixed to any point, and being put in motion turns round it; the body, if uninterrupted by the power of gravity or any other means, will continue perpetually to move about with the same equable motion. Now the force, with which such a body moves, is all united in the point, which in relation to the power of gravity is called the center of oscillation. Let the cylinder A B C D (in fig. 60.) whose axis is E F, be fixed to the point E. And supposing the point E to be that on which the cylinder is suspended, let the center of oscillation be found in the axis E F, as has been explained above[70]. Let G be that center: then I say, that the force, wherewith this cylinder turns round the point E, is so united in the point G, that a sufficient force applied in that point shall stop the motion of the cylinder, in such a manner, that the cylinder should immediately remain without motion, though it were to be loosened from the point E at the same instant, that the impediment was applied to G: whereas, if this impediment had been applied to any other point of the axis, the cylinder would turn upon the point, where the impediment was applied. If the impediment had been applied between E and G, the cylinder would so turn on the point, where the impediment was applied, that the end B C would continue to move on the same way it moved before along with the whole cylinder; but if the impediment were applied to the axis farther off from E than G, the end A D of the cylinder would start out of its present place that way in which the cylinder moved. From this property of the center of oscillation, it is also called the center of percussion. That excellent mathematician, Dr. Brook Taylor, has farther improved this doctrine concerning the center of percussion, by shewing, that if through this point G a line, as G H I, be drawn perpendicular to E F, and lying in the course of the body’s motion; a sufficient power applied to any point of this line will have the same effect, as the like power applied to G[71]: so that as we before shewed the center of percussion within the body on its axis; by this means we may find this center on the surface of the body also, for it will be where this line H I crosses that surface.
[76.] I shall now proceed to the last kind of motion, to be treated on in this place, and shew what line the power of gravity will cause a body to describe, when it is thrown forwards by any force. This was first discovered by the great Galileo, and is the principle, upon which engineers should direct the shot of great guns. But as in this case bodies describe in their motion one of those lines, which in geometry are called conic sections; it is necessary here to premise a description of those lines. In which I shall be the more particular, because the knowledge of them is not only necessary for the present purpose, but will be also required hereafter in some of the principal parts of this treatise.
77. The first lines considered by the ancient geometers were the straight line and the circle. Of these they composed various figures, of which they demonstrated many properties, and resolved divers problems concerning them. These problems they attempted always to resolve by the describing straight lines and circles. For instance, let a square A B C D (fig. 61.) be proposed, and let it be required to make another square in any assigned proportion to this. Prolong one side, as D A, of this square to E, till A E bear the same proportion to A D, as the new square is to bear to the square A C. If the opposite side B C of the square A C be also prolonged to F, till B F be equal to A E, and E F be afterwards drawn, I suppose my readers will easily conceive, that the figure A B F E will bear to the square A B C D the same proportion, as the line A E bears to the line A D. Therefore the figure A B F E will be equal to the new square, which is to be found, but is not it self a square, because the side A E is not of the same length with the side E F. But to find a square equal to the figure A B F E you must proceed thus. Divide the line D E into two equal parts in the point G, and to the center G with the interval G D describe the circle D H E I; then prolong the line A B, till it meets the circle in K; and make the square A K L M, which square will be equal to the figure A B F E, and bear to the square A B C D the same proportion, as the line A E bears to A D.
78. I shall not proceed to the proof of this, having only here set it down as a specimen of the method of resolving geometrical problems by the description of straight lines and circles. But there are some problems, which cannot be resolved by drawing straight lines or circles upon a plane. For the management therefore of these they took into consideration solid figures, and of the solid figures they found that, which is called a cone, to be the most useful.
79. A cone is thus defined by Euclide in his elements of geometry[72]. If to the straight line A B (in fig. 62.) another straight line, as A C, be drawn perpendicular, and the two extremities B and C be joined by a third straight line composing the triangle A C B (for so every figure is called, which is included under three straight lines) then the two points A and B being held fixed, as two centers, and the triangle A C B being turned round upon the line A B, as on an axis; the line A C will describe a circle, and the figure A C B will describe a cone, of the form represented by the figure B C D E F (fig. 63.) in which the circle C D E F is usually called the base of the cone, and B the vertex.
80. Now by this figure may several problems be resolved, which cannot by the simple description of straight lines and circles upon a plane. Suppose for instance, it were required to make a cube, which should bear any assigned proportion to some other cube named. I need not here inform my readers, that a cube is the figure of a dye. This problem was much celebrated among the ancients, and was once inforced by the command of an oracle. This problem may be performed by a cone thus. First make a cone from a triangle, whose side A C shall be half the length of the side B C Then on the plane A B C D (fig. 64.) let the line E F be exhibited equal in length to the side of the cube proposed; and let the line F G be drawn perpendicular to E F, and of such a length, that it bear the same proportion to E F, as the cube to be sought is required to bear to the cube proposed. Through the points E, F, and G let the circle F H I be described. Then let the line E F be prolonged beyond F to K, that F K be equal to F E, and let the triangle F K L, having all its sides F K, K L, L F equal to each other, be hung down perpendicularly from the plane A B C D. After this, let another plane M N O P be extended through the point L, so as to be equidistant from the former plane A B C D, and in this plane let the line Q L R be drawn so, as to be equidistant from the line E F K. All this being thus prepared, let such a cone, as was above directed to be made, be so applied to the plane M N O P, that it touch this plane upon the line Q R, and that the vertex of the cone be applied to the point L. This cone, by cutting through the first plane A B C D, will cross the circle F H I before described. And if from the point S, where the surface of this cone intersects the circle, the line S T be drawn so, as to be equidistant from the line E F; the line F T will be equal to the side of the cube sought: that is, if there be two cubes or dyes formed, the side of one being equal to E F, and the side of the other equal to F T; the former of these cubes shall bear the same proportion to the latter, as the line E F bears to F G.