64.Nowin this pendulum, as all the swings, whether long or short, will be performed in the same time; so the time of each will exactly bear the same proportion to the time required for a body to fall perpendicularly down, through half the length of the pendulum, that is from I to K, as the circumference of a circle bears to its diameter.65.Itmay from hence be understood in some measure, why, when pendulums swing in circular arches, the times of their swings are nearly equal, if the arches are small, though those arches be of very unequal lengths; for if with the semidiameter L K the circular arch O K P be described, this arch in the lower part of it will differ very little from the line C K H.66.Itmay not be amiss here to remark, that a body will fall in this line C K H (fig. 53.) from C to any other point, as Q or R in a shorter space of time, than if it moved through the straight line drawn from C to the other point; or through any other line whatever, that can be drawn between these two points.67.Butas I have observed, that the time, which a pendulum takes in swinging, depends upon its length; I shall now say something concerning the way, in which this length of the pendulum is to be estimated. If the whole ball of the pendulum could be crouded into one point, this length, by which the motion of the pendulum is to be computed, would be the length of the string or rod. But the ball of the pendulum must have a sensible magnitude, and the several parts of this ball will not move with the same degree of swiftness; for those parts, which are farthest from the point, whereon the pendulum is suspended, must move with the greatest velocity. Therefore to know the time in which the pendulum swings, it is necessary to find that point of the ball, which moves with the same degree of velocity, as if the whole ball were to be contracted into that point.68.Thispoint is not the center of gravity, as I shall now endeavour to shew. Suppose the pendulum A B (in fig. 54.) composed of an inflexible rod A C and ball C B, to be fixed on the point A, and lifted up into an horizontal situation. Here if the rod were not fixed to the point A, the body C B would descend directly with the whole force of its weight; and each part of the body would move down with the same degree of swiftness. But when the rod is fixed at the point A, the body must fall after another manner; for the parts of the body must move with different degrees of velocity, the parts more remote from A descending with a swifter motion, than the parts nearer to A; so that the body will receive a kind of rolling motion while it descends. But it has been observed above, that the effect of gravity upon any body is the same, as if the whole force were exerted on the body’s center of gravity[64].Since therefore the power of gravity in drawing down the body must also communicate to it the rolling motion just described; it seems evident, that the center of gravity of the body cannot be drawn down as swiftly, as when the power of gravity has no other effect to produce on the body, than merely to draw it downward. If therefore the whole matter of the body C B could be crouded into its center of gravity, so that being united into one point, this rolling motion here mentioned might give no hindrance to its descent; this center would descend faster, than it can now do. And the point, which now descends as fast, as if the whole matter or the body C B were crouded into it, will be farther removed from the point A, than the center of gravity of the body C B.69.Again, suppose the pendulum A B (in fig. 55.) to hang obliquely. Here the power of gravity will operate less upon the ball of the pendulum, than before: but the line D E being drawn so, as to stand perpendicular to the rod A C of the pendulum; the force of gravity upon the body C B, now it is in this situation, will produce the same effect, as if the body were to glide down an inclined plane in the position of D E. But here the motion of the body, when the rod is fixed to the point A, will not be equal to the uninterrupted descent of the body down this plane; for the bodywill here also receive the same kind of rotation in its motion, as before; so that the motion of the center of gravity will in like manner be retarded; and the point, which here descends with that degree of swiftness, which the body would have, if not hindered by being fixed to the point A; that is, the point, which descends as fast, as if the whole body were crouded into it, will be as far removed from the point A, as before.70.Thispoint, by which the length of the pendulum is to be estimated, is called the center of oscillation. And the mathematicians have laid down general directions, whereby to find this center in all bodies. If the globe A B (in fig. 56.) be hung by the string C D, whose weight need not be regarded, the center of oscillation is found thus. Let the straight line drawn from C to D be continued through the globe to F. That it will pass through the center of the globe is evident. Suppose E to be this center of the globe; and take the line G of such a length, that it shall bear the same proportion to E D, as E D bears to E C. Then E H being made equal to ⅖ of G, the point H shall be the center of oscillation[65]. If the weight of the rod C D is too considerable to be neglected, divide C D (fig. 57) in I, that D I be equal to ⅓, part of C D; and take K in the same proportion to C I, as the weight of the globe A B to the weight of the rod C D. Then having found H, the center of oscillation of the globe, as before, divide I K in I, so that I L shall bear the same proportionto L H, as the line C H bears to K; and L shall be the center of oscillation of the whole pendulum.71.Thiscomputation is made upon supposition, that the center of oscillation of the rod C D, if that were to swing alone without any other weight annexed, would be the point I. And this point would be the true center of oscillation, so far as the thickness of the rod is not to be regarded. If any one chuses to take into consideration the thickness of the rod, he must place the center of oscillation thereof so much below the point I, that eight times the distance of the center from the point I shall bear the same proportion to the thickness of the rod, as the thickness of the rod bears to its length C D[66].72.Ithas 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 thependulum 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.Wemay 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 SirChristopher 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.SirIsaac Newtonhas 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 shallfinish 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 bodymoves, 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 lyingin 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 shallnow 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 greatGalileo, 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.Thefirst 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 anothersquare 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 shallnot 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 coneis thus defined byEuclidein 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.Nowby 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 thecube 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.81.Indeedthis placing a cone to cut through a plane is not a practicable method of resolving problems. But when the geometers had discovered this use of the cone, they applied themselves to consider the nature of the lines, which will be produced by the intersection of the surface of a coneand a plane; whereby they might be enabled both to reduce these kinds of solutions to practice, and also to render their demonstrations concise and elegant.82.Wheneverthe plane, which cuts the cone, is equidistant from another plane, that touches the cone on the side; (which is the case of the present figure;) the line, wherein the plane cuts the surface of the cone, is called a parabola. But if the plane, which cuts the cone, be so inclined to this other, that it will pass quite through the cone (as in fig. 65.) such a plane by cutting the cone produces the figure called an ellipsis, in which we shall hereafter shew the earth and other planets to move round the sun. If the plane, which cuts the cone, recline the other way (as in fig. 66.) so as not to be parallel to any plane, whereon the cone can lie, nor yet to cut quite through the cone; such a plane shall produce in the cone a third kind of line, which is called an hyperbola. But it is the first of these lines named the parabola, wherein bodies, that are thrown obliquely, will be carried by the force of gravity; as I shall here proceed to shew, after having first directed my readers how to describe this sort of line upon a plane, by which the form of it may be seen.83.Toany straight line A B (fig. 67.) let a straight ruler C D be so applied, as to stand against it perpendicularly. Upon the edge of this ruler let another ruler E F be so placed, as to move along upon the edge of the first ruler C D, and keep always perpendicular to it. This being so disposed, let any point, as G, be taken in the line A B, and let a string equalin length to the ruler E F be fastened by one end to the point G, and by the other to the extremity F of the ruler E F. Then if the string be held down to the ruler E F by a pin H, as is represented in the figure; the point of this pin, while the ruler E F moves on the ruler C D, shall describe the line I K L, which will be one part of the curve line, whose description we were here to teach: and by applying the rulers in the like manner on the other side of the line A B, we may describe the other part I M of this line. If the distance C G be equal to half the line E F in fig. 64, the line M I L will be that very line, wherein the plane A B C D in that figure cuts the cone.84.Theline A I is called the axis of the parabola M I L, and the point G is called the focus.85.Nowby comparing the effects of gravity upon falling bodies, with what is demonstrated of this figure by the geometers, it is proved, that every body thrown obliquely is carried forward in one of these lines, the axis whereof is perpendicular to the horizon.86.Thegeometers demonstrate, that if a line be drawn to touch a parabola in any point, as the line A B (in fig. 68.) touches the parabola C D, whose axis is Y Z, in the point E; and several lines F G, H I, K L be drawn parallel to the axis of the parabola: then the line F G will be to H I in the duplicate proportion of E F to E H, and F G to K L in the duplicate proportion of E F to E K; likewise H I to K L in the duplicate proportion of E H to E K. What is to be understood by duplicate or two-foldproportion, has been already explained[73]. Accordingly I mean here, that if the line M be taken to bear the same proportion to E H, as E H bears to E F, H I will bear the same proportion to F G, as M bears to E F; and if the line N bears the same proportion to E K, as E K bears to E F, K L will bear the same proportion to F G, as N bears to E F; or if the line O bear the same proportion to E K, as E K bears to E H, K L will bear the same proportion to H I, as O bears to E H.87.Thisproperty is essential to the parabola, being so connected with the nature of the figure, that every line possessing this property is to be called by this name.88.Nowsuppose a body to be thrown from the point A (in fig. 69.) towards B in the direction of the line A B. This body, if left to it self, would move on with a uniform motion through this line A B. Suppose the eye of a spectator to be placed at the point C just under the point A; and let us imagine the earth to be so put into motion along with the body, as to carry the spectator’s eye along the line C D parallel to A B; and that the eye would move on with the same velocity, wherewith the body would proceed in the line A B, if it were to be left to move without any disturbance from its gravitation towards the earth. In this case if the body moved on without being drawn towards the earth, it would appear to the spectator to be at rest. But if the power of gravity exerted it self on the body, it would appear to the spectatorto fall directly down. Suppose at the distance of time, wherein the body by its own progressive motion would have moved from A to E, it should appear to the spectator to have fallen through a length equal to E F: then the body at the end of this time will actually have arrived at the point F. If in the space of time, wherein the body would have moved by its progressive motion from A to G, it would have appeared to the spectator to have fallen down the space G H: then the body at the end of this greater interval of time will be arrived at the point H. Now if the line A F H I be that, through which the body actually passes; from what has here been said, it will follow, that this line is one of those, which I have been describing under the name of the parabola. For the distances E F, G H, through which the body is seen to fall, will increase in the duplicate proportion of the times[74]; but the lines A E, A G will be proportional to the times wherein they would have been described by the single progressive motion of the body: therefore the lines E F, G H will be in the duplicate proportion of the lines A F, A G; and the line A F H I possesses the property of the parabola.89.Ifthe earth be not supposed to move along with the body, the case will be a little different. For the body being constantly drawn directly towards the center of the earth, the body in its motion will be drawn in a direction a little oblique to that, wherein it would be drawn by the earth in motion, as before supposed. But the distance to the center of theearth bears so vast a proportion to the greatest length, to which we can throw bodies, that this obliquity does not merit any regard. From the sequel of this discourse it may indeed be collected, what line the body being thrown thus would be found to describe, allowance being made for this obliquity of the earth’s action[75]. This is the discovery of SirIs. Newton; but has no use in this place. Here it is abundantly sufficient to consider the body as moving in a parabola.90.Theline, which a projected body describes, being thus known, practical methods have been deduced from hence for directing the shot of great guns to strike any object desired. This work was first attempted byGalileo, and soon after farther improved by his scholarTorricelli; but has lately been rendred more complete by the great Mr.Cotes, whose immature death is an unspeakable loss to mathematical learning. If it be required to throw a body from the point A (in fig. 70.) so as to strike the point B; through the points A, B draw the straight line C D, and erect the line A E perpendicular to the horizon, and of four times the height, from which a body must fall to acquire the velocity, wherewith the body is intended to be thrown. Through the points A and E describe a circle, that shall touch the line C D in the point A. Then from the point B draw the line B F perpendicular to the horizon, intersecting the circle in the points G and H. This being done, if the body be projected directly towards either of these points G or H, it shall fall upon the point B; but with this difference, that, if it be thrownin the direction A G, it shall sooner arrive at B, than if it were projected in the direction A H. When the body is projected in the direction A G; the time, it will take up in arriving at B, will bear the same proportion to the time, wherein it would fall down through one fourth part of A E, as A G bears to half A E. But when the body is thrown in the direction of A H, the time of its passing to B will bear the same proportion to the time, wherein it would fall through one fourth part of A E, as A H bears to half A E.91.Ifthe line A I be drawn so as to divide the angle under E A D in the middle, and the line I K be drawn perpendicular to the horizon; this line will touch the circle in the point I, and if the body be thrown in the direction A I, it will fall upon the point K: and this point K is the farthest point in the line A D, which the body can be made to strike, without increasing its velocity.92.Thevelocity, wherewith the body every where moves, may be found thus. Suppose the body to move in the parabola A B (fig. 71.) Erect A C perpendicular to the horizon, and equal to the height, from which a body must fall to acquire the velocity, wherewith the body sets out from A. If you take any points as D and E in the parabola, and draw D F and E G parallel to the horizon; the velocity of the body in D will be equal to what a body will acquire in falling down by its own weight through C F, and in E the velocity will be the same, as would be acquired in falling through C G. Thus the body moves slowest at the highest point H of the parabola; and at equal distances from this point willmove with equal swiftness, and descend from that highest point through the line H B altogether like to the line A H in which it ascended; abating only the resistance of the air, which is not here considered. If the line H I be drawn from the highest point H parallel to the horizon, A I will be equal to ¼ of B G in fig. 70, when the body is projected in the direction A G, and equal to ¼ of B H, when the body is thrown in the direction A H provided A D be drawn horizontally.93.ThusI have recounted the principal discoveries, which had been made concerning the motion of bodies by SirIsaac Newton’spredecessors; all these discoveries, by being found to agree with experience, contributing to establish the laws of motion, from whence they were deduced. I shall therefore here finish what I had to say upon those laws; and conclude this chapter with a few words concerning the distinction which ought to be made between absolute and relative motion. For some have thought fit to confound them together; because they observe the laws of motion to take place here on the earth, which is in motion, after the same manner as if it were at rest. But SirIsaac Newtonhas been careful to distinguish between the relative and absolute consideration both of motion and time[76]. The astronomers anciently found it necessary to make this distinction in time. Time considered in it self passes on equably without relation to any thing external, being the proper measure of the continuance and duration of all things. But it is most frequently conceived of by us under a relative view to some succession insensible things, of which we take cognizance. The succession of the thoughts in our own minds is that, from whence we receive our first idea of time, but is a very uncertain measure thereof; for the thoughts of some men flow on much more swiftly, than the thoughts of others; nor does the same person think equally quick at all times. The motions of the heavenly bodies are more regular; and the eminent division of time into night and day, made by the sun, leads us to measure our time by the motion of that luminary: nor do we in the affairs of life concern our selves with any inequality, which there may be in that motion; but the space of time which comprehends a day and night is rather supposed to be always the same. However astronomers anciently found these spaces of time not to be always of the same length, and have taught how to compute their differences. Now the time, when so equated as to be rendered perfectly equal, is the true measure of duration, the other not. And therefore this latter, which is absolutely true time, differs from the other, which is only apparent. And as we ordinarily make no distinction between apparent time, as measured by the sun, and the true; so we often do not distinguish in our usual discourse between the real, and the apparent or relative motion of bodies; but use the same words for one, as we should for the other. Though all things about us are really in motion with the earth; as this motion is not visible, we speak of the motion of every thing we see, as if our selves and the earth stood still. And even in other cases, where we discern the motion of bodies, we often speak of them not in relation to the whole motion we see, but with regard to otherbodies, to which they are contiguous. If any body were lying on a table; when that table shall be carried along, we say the body rests upon the table, or perhaps absolutely, that the body is at rest. However philosophers must not reject all distinction between true and apparent motions, any more than astronomers do the distinction between true and vulgar time; for there is as real a difference between them, as will appear by the following consideration. Suppose all the bodies of the universe to have their courses stopped, and reduced to perfect rest. Then suppose their present motions to be again restored; this cannot be done without an actual impression made upon some of them at least. If any of them be left untouched, they will retain their former state, that is, still remain at rest; but the other bodies, which are wrought upon, will have changed their former state of rest, for the contrary state of motion. Let us now suppose the bodies left at rest to be annihilated, this will make no alteration in the state of the moving bodies; but the effect of the impression, which was made upon them, will still subsist. This shews the motion they received to be an absolute thing, and to have no necessary dependence upon the relation which the body said to be in motion has to any other body[77].94.Besidesabsolute and relative motion are distinguishable by their Effects. One effect of motion is, that bodies, when moved round any center or axis, acquire a certainpower, by which they forcibly press themselves from that center or axis of motion. As when a body is whirled about in a sling, the body presses against the sling, and is ready to fly out as soon as liberty is given it. And this power is proportional to the true, not relative motion of the body round such a center or axis. Of this SirIsaac Newtongives the following instance[78]. If a pail or such like vessel near full of water be suspended by a string of sufficient length, and be turned about till the string be hard twisted. If then as soon as the vessel and water in it are become still and at rest, the vessel be nimbly turned about the contrary way the string was twisted, the vessel by the strings untwisting it self shall continue its motion a long time. And when the vessel first begins to turn, the water in it shall receive little or nothing of the motion of the vessel, but by degrees shall receive a communication of motion, till at last it shall move round as swiftly as the vessel it self. Now the definition of motion, whichDes Carteshas given us upon this principle of making all motion meerly relative, is this: that motion, is a removal of any body from its vicinity to other bodies, which were in immediate contact with it, and are considered as at rest[79]. And if this be compared with what he soon after says, that there is nothing real or positive in the body moved, for the sake of which we ascribe motion to it, which is not to be found as well in the contiguous bodies, which are considered as at rest[80]; it will follow from thence, that we may consider the vessel as at restand the water as moving in it: and the water in respect of the vessel has the greatest motion, when the vessel first begins to turn, and loses this relative motion more and more, till at length it quite ceases. But now, when the vessel first begins to turn, the surface of the water remains smooth and flat, as before the vessel began to move; but as the motion of the vessel communicates by degrees motion to the water, the surface of the water will be observed to change, the water subsiding in the middle and rising at the edges: which elevation of the water is caused by the parts of it pressing from the axis, they move about; and therefore this force of receding from the axis of motion depends not upon the relative motion of the water within the vessel, but on its absolute motion; for it is least, when that relative motion is greatest, and greatest, when that relative motion is least, or none at all.95.Thusthe true cause of what appears in the surface of this water cannot be assigned, without considering the water’s motion within the vessel. So also in the system of the world, in order to find out the cause of the planetary motions, we must know more of the real motions, which belong to each planet, than is absolutely necessary for the uses of astronomy. If the astronomer should suppose the earth to stand still, he could ascribe such motions to the celestial bodies, as should answer all the appearances; though he would not account for them in so simple a manner, as by attributing motion to the earth. But the motion of the earth must of necessity be considered, before the real causes, which actuate the planetary system, can be discovered.
64.Nowin this pendulum, as all the swings, whether long or short, will be performed in the same time; so the time of each will exactly bear the same proportion to the time required for a body to fall perpendicularly down, through half the length of the pendulum, that is from I to K, as the circumference of a circle bears to its diameter.65.Itmay from hence be understood in some measure, why, when pendulums swing in circular arches, the times of their swings are nearly equal, if the arches are small, though those arches be of very unequal lengths; for if with the semidiameter L K the circular arch O K P be described, this arch in the lower part of it will differ very little from the line C K H.66.Itmay not be amiss here to remark, that a body will fall in this line C K H (fig. 53.) from C to any other point, as Q or R in a shorter space of time, than if it moved through the straight line drawn from C to the other point; or through any other line whatever, that can be drawn between these two points.67.Butas I have observed, that the time, which a pendulum takes in swinging, depends upon its length; I shall now say something concerning the way, in which this length of the pendulum is to be estimated. If the whole ball of the pendulum could be crouded into one point, this length, by which the motion of the pendulum is to be computed, would be the length of the string or rod. But the ball of the pendulum must have a sensible magnitude, and the several parts of this ball will not move with the same degree of swiftness; for those parts, which are farthest from the point, whereon the pendulum is suspended, must move with the greatest velocity. Therefore to know the time in which the pendulum swings, it is necessary to find that point of the ball, which moves with the same degree of velocity, as if the whole ball were to be contracted into that point.68.Thispoint is not the center of gravity, as I shall now endeavour to shew. Suppose the pendulum A B (in fig. 54.) composed of an inflexible rod A C and ball C B, to be fixed on the point A, and lifted up into an horizontal situation. Here if the rod were not fixed to the point A, the body C B would descend directly with the whole force of its weight; and each part of the body would move down with the same degree of swiftness. But when the rod is fixed at the point A, the body must fall after another manner; for the parts of the body must move with different degrees of velocity, the parts more remote from A descending with a swifter motion, than the parts nearer to A; so that the body will receive a kind of rolling motion while it descends. But it has been observed above, that the effect of gravity upon any body is the same, as if the whole force were exerted on the body’s center of gravity[64].Since therefore the power of gravity in drawing down the body must also communicate to it the rolling motion just described; it seems evident, that the center of gravity of the body cannot be drawn down as swiftly, as when the power of gravity has no other effect to produce on the body, than merely to draw it downward. If therefore the whole matter of the body C B could be crouded into its center of gravity, so that being united into one point, this rolling motion here mentioned might give no hindrance to its descent; this center would descend faster, than it can now do. And the point, which now descends as fast, as if the whole matter or the body C B were crouded into it, will be farther removed from the point A, than the center of gravity of the body C B.69.Again, suppose the pendulum A B (in fig. 55.) to hang obliquely. Here the power of gravity will operate less upon the ball of the pendulum, than before: but the line D E being drawn so, as to stand perpendicular to the rod A C of the pendulum; the force of gravity upon the body C B, now it is in this situation, will produce the same effect, as if the body were to glide down an inclined plane in the position of D E. But here the motion of the body, when the rod is fixed to the point A, will not be equal to the uninterrupted descent of the body down this plane; for the bodywill here also receive the same kind of rotation in its motion, as before; so that the motion of the center of gravity will in like manner be retarded; and the point, which here descends with that degree of swiftness, which the body would have, if not hindered by being fixed to the point A; that is, the point, which descends as fast, as if the whole body were crouded into it, will be as far removed from the point A, as before.70.Thispoint, by which the length of the pendulum is to be estimated, is called the center of oscillation. And the mathematicians have laid down general directions, whereby to find this center in all bodies. If the globe A B (in fig. 56.) be hung by the string C D, whose weight need not be regarded, the center of oscillation is found thus. Let the straight line drawn from C to D be continued through the globe to F. That it will pass through the center of the globe is evident. Suppose E to be this center of the globe; and take the line G of such a length, that it shall bear the same proportion to E D, as E D bears to E C. Then E H being made equal to ⅖ of G, the point H shall be the center of oscillation[65]. If the weight of the rod C D is too considerable to be neglected, divide C D (fig. 57) in I, that D I be equal to ⅓, part of C D; and take K in the same proportion to C I, as the weight of the globe A B to the weight of the rod C D. Then having found H, the center of oscillation of the globe, as before, divide I K in I, so that I L shall bear the same proportionto L H, as the line C H bears to K; and L shall be the center of oscillation of the whole pendulum.71.Thiscomputation is made upon supposition, that the center of oscillation of the rod C D, if that were to swing alone without any other weight annexed, would be the point I. And this point would be the true center of oscillation, so far as the thickness of the rod is not to be regarded. If any one chuses to take into consideration the thickness of the rod, he must place the center of oscillation thereof so much below the point I, that eight times the distance of the center from the point I shall bear the same proportion to the thickness of the rod, as the thickness of the rod bears to its length C D[66].72.Ithas 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 thependulum 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.Wemay 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 SirChristopher 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.SirIsaac Newtonhas 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 shallfinish 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 bodymoves, 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 lyingin 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 shallnow 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 greatGalileo, 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.Thefirst 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 anothersquare 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 shallnot 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 coneis thus defined byEuclidein 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.Nowby 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 thecube 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.81.Indeedthis placing a cone to cut through a plane is not a practicable method of resolving problems. But when the geometers had discovered this use of the cone, they applied themselves to consider the nature of the lines, which will be produced by the intersection of the surface of a coneand a plane; whereby they might be enabled both to reduce these kinds of solutions to practice, and also to render their demonstrations concise and elegant.82.Wheneverthe plane, which cuts the cone, is equidistant from another plane, that touches the cone on the side; (which is the case of the present figure;) the line, wherein the plane cuts the surface of the cone, is called a parabola. But if the plane, which cuts the cone, be so inclined to this other, that it will pass quite through the cone (as in fig. 65.) such a plane by cutting the cone produces the figure called an ellipsis, in which we shall hereafter shew the earth and other planets to move round the sun. If the plane, which cuts the cone, recline the other way (as in fig. 66.) so as not to be parallel to any plane, whereon the cone can lie, nor yet to cut quite through the cone; such a plane shall produce in the cone a third kind of line, which is called an hyperbola. But it is the first of these lines named the parabola, wherein bodies, that are thrown obliquely, will be carried by the force of gravity; as I shall here proceed to shew, after having first directed my readers how to describe this sort of line upon a plane, by which the form of it may be seen.83.Toany straight line A B (fig. 67.) let a straight ruler C D be so applied, as to stand against it perpendicularly. Upon the edge of this ruler let another ruler E F be so placed, as to move along upon the edge of the first ruler C D, and keep always perpendicular to it. This being so disposed, let any point, as G, be taken in the line A B, and let a string equalin length to the ruler E F be fastened by one end to the point G, and by the other to the extremity F of the ruler E F. Then if the string be held down to the ruler E F by a pin H, as is represented in the figure; the point of this pin, while the ruler E F moves on the ruler C D, shall describe the line I K L, which will be one part of the curve line, whose description we were here to teach: and by applying the rulers in the like manner on the other side of the line A B, we may describe the other part I M of this line. If the distance C G be equal to half the line E F in fig. 64, the line M I L will be that very line, wherein the plane A B C D in that figure cuts the cone.84.Theline A I is called the axis of the parabola M I L, and the point G is called the focus.85.Nowby comparing the effects of gravity upon falling bodies, with what is demonstrated of this figure by the geometers, it is proved, that every body thrown obliquely is carried forward in one of these lines, the axis whereof is perpendicular to the horizon.86.Thegeometers demonstrate, that if a line be drawn to touch a parabola in any point, as the line A B (in fig. 68.) touches the parabola C D, whose axis is Y Z, in the point E; and several lines F G, H I, K L be drawn parallel to the axis of the parabola: then the line F G will be to H I in the duplicate proportion of E F to E H, and F G to K L in the duplicate proportion of E F to E K; likewise H I to K L in the duplicate proportion of E H to E K. What is to be understood by duplicate or two-foldproportion, has been already explained[73]. Accordingly I mean here, that if the line M be taken to bear the same proportion to E H, as E H bears to E F, H I will bear the same proportion to F G, as M bears to E F; and if the line N bears the same proportion to E K, as E K bears to E F, K L will bear the same proportion to F G, as N bears to E F; or if the line O bear the same proportion to E K, as E K bears to E H, K L will bear the same proportion to H I, as O bears to E H.87.Thisproperty is essential to the parabola, being so connected with the nature of the figure, that every line possessing this property is to be called by this name.88.Nowsuppose a body to be thrown from the point A (in fig. 69.) towards B in the direction of the line A B. This body, if left to it self, would move on with a uniform motion through this line A B. Suppose the eye of a spectator to be placed at the point C just under the point A; and let us imagine the earth to be so put into motion along with the body, as to carry the spectator’s eye along the line C D parallel to A B; and that the eye would move on with the same velocity, wherewith the body would proceed in the line A B, if it were to be left to move without any disturbance from its gravitation towards the earth. In this case if the body moved on without being drawn towards the earth, it would appear to the spectator to be at rest. But if the power of gravity exerted it self on the body, it would appear to the spectatorto fall directly down. Suppose at the distance of time, wherein the body by its own progressive motion would have moved from A to E, it should appear to the spectator to have fallen through a length equal to E F: then the body at the end of this time will actually have arrived at the point F. If in the space of time, wherein the body would have moved by its progressive motion from A to G, it would have appeared to the spectator to have fallen down the space G H: then the body at the end of this greater interval of time will be arrived at the point H. Now if the line A F H I be that, through which the body actually passes; from what has here been said, it will follow, that this line is one of those, which I have been describing under the name of the parabola. For the distances E F, G H, through which the body is seen to fall, will increase in the duplicate proportion of the times[74]; but the lines A E, A G will be proportional to the times wherein they would have been described by the single progressive motion of the body: therefore the lines E F, G H will be in the duplicate proportion of the lines A F, A G; and the line A F H I possesses the property of the parabola.89.Ifthe earth be not supposed to move along with the body, the case will be a little different. For the body being constantly drawn directly towards the center of the earth, the body in its motion will be drawn in a direction a little oblique to that, wherein it would be drawn by the earth in motion, as before supposed. But the distance to the center of theearth bears so vast a proportion to the greatest length, to which we can throw bodies, that this obliquity does not merit any regard. From the sequel of this discourse it may indeed be collected, what line the body being thrown thus would be found to describe, allowance being made for this obliquity of the earth’s action[75]. This is the discovery of SirIs. Newton; but has no use in this place. Here it is abundantly sufficient to consider the body as moving in a parabola.90.Theline, which a projected body describes, being thus known, practical methods have been deduced from hence for directing the shot of great guns to strike any object desired. This work was first attempted byGalileo, and soon after farther improved by his scholarTorricelli; but has lately been rendred more complete by the great Mr.Cotes, whose immature death is an unspeakable loss to mathematical learning. If it be required to throw a body from the point A (in fig. 70.) so as to strike the point B; through the points A, B draw the straight line C D, and erect the line A E perpendicular to the horizon, and of four times the height, from which a body must fall to acquire the velocity, wherewith the body is intended to be thrown. Through the points A and E describe a circle, that shall touch the line C D in the point A. Then from the point B draw the line B F perpendicular to the horizon, intersecting the circle in the points G and H. This being done, if the body be projected directly towards either of these points G or H, it shall fall upon the point B; but with this difference, that, if it be thrownin the direction A G, it shall sooner arrive at B, than if it were projected in the direction A H. When the body is projected in the direction A G; the time, it will take up in arriving at B, will bear the same proportion to the time, wherein it would fall down through one fourth part of A E, as A G bears to half A E. But when the body is thrown in the direction of A H, the time of its passing to B will bear the same proportion to the time, wherein it would fall through one fourth part of A E, as A H bears to half A E.91.Ifthe line A I be drawn so as to divide the angle under E A D in the middle, and the line I K be drawn perpendicular to the horizon; this line will touch the circle in the point I, and if the body be thrown in the direction A I, it will fall upon the point K: and this point K is the farthest point in the line A D, which the body can be made to strike, without increasing its velocity.92.Thevelocity, wherewith the body every where moves, may be found thus. Suppose the body to move in the parabola A B (fig. 71.) Erect A C perpendicular to the horizon, and equal to the height, from which a body must fall to acquire the velocity, wherewith the body sets out from A. If you take any points as D and E in the parabola, and draw D F and E G parallel to the horizon; the velocity of the body in D will be equal to what a body will acquire in falling down by its own weight through C F, and in E the velocity will be the same, as would be acquired in falling through C G. Thus the body moves slowest at the highest point H of the parabola; and at equal distances from this point willmove with equal swiftness, and descend from that highest point through the line H B altogether like to the line A H in which it ascended; abating only the resistance of the air, which is not here considered. If the line H I be drawn from the highest point H parallel to the horizon, A I will be equal to ¼ of B G in fig. 70, when the body is projected in the direction A G, and equal to ¼ of B H, when the body is thrown in the direction A H provided A D be drawn horizontally.93.ThusI have recounted the principal discoveries, which had been made concerning the motion of bodies by SirIsaac Newton’spredecessors; all these discoveries, by being found to agree with experience, contributing to establish the laws of motion, from whence they were deduced. I shall therefore here finish what I had to say upon those laws; and conclude this chapter with a few words concerning the distinction which ought to be made between absolute and relative motion. For some have thought fit to confound them together; because they observe the laws of motion to take place here on the earth, which is in motion, after the same manner as if it were at rest. But SirIsaac Newtonhas been careful to distinguish between the relative and absolute consideration both of motion and time[76]. The astronomers anciently found it necessary to make this distinction in time. Time considered in it self passes on equably without relation to any thing external, being the proper measure of the continuance and duration of all things. But it is most frequently conceived of by us under a relative view to some succession insensible things, of which we take cognizance. The succession of the thoughts in our own minds is that, from whence we receive our first idea of time, but is a very uncertain measure thereof; for the thoughts of some men flow on much more swiftly, than the thoughts of others; nor does the same person think equally quick at all times. The motions of the heavenly bodies are more regular; and the eminent division of time into night and day, made by the sun, leads us to measure our time by the motion of that luminary: nor do we in the affairs of life concern our selves with any inequality, which there may be in that motion; but the space of time which comprehends a day and night is rather supposed to be always the same. However astronomers anciently found these spaces of time not to be always of the same length, and have taught how to compute their differences. Now the time, when so equated as to be rendered perfectly equal, is the true measure of duration, the other not. And therefore this latter, which is absolutely true time, differs from the other, which is only apparent. And as we ordinarily make no distinction between apparent time, as measured by the sun, and the true; so we often do not distinguish in our usual discourse between the real, and the apparent or relative motion of bodies; but use the same words for one, as we should for the other. Though all things about us are really in motion with the earth; as this motion is not visible, we speak of the motion of every thing we see, as if our selves and the earth stood still. And even in other cases, where we discern the motion of bodies, we often speak of them not in relation to the whole motion we see, but with regard to otherbodies, to which they are contiguous. If any body were lying on a table; when that table shall be carried along, we say the body rests upon the table, or perhaps absolutely, that the body is at rest. However philosophers must not reject all distinction between true and apparent motions, any more than astronomers do the distinction between true and vulgar time; for there is as real a difference between them, as will appear by the following consideration. Suppose all the bodies of the universe to have their courses stopped, and reduced to perfect rest. Then suppose their present motions to be again restored; this cannot be done without an actual impression made upon some of them at least. If any of them be left untouched, they will retain their former state, that is, still remain at rest; but the other bodies, which are wrought upon, will have changed their former state of rest, for the contrary state of motion. Let us now suppose the bodies left at rest to be annihilated, this will make no alteration in the state of the moving bodies; but the effect of the impression, which was made upon them, will still subsist. This shews the motion they received to be an absolute thing, and to have no necessary dependence upon the relation which the body said to be in motion has to any other body[77].94.Besidesabsolute and relative motion are distinguishable by their Effects. One effect of motion is, that bodies, when moved round any center or axis, acquire a certainpower, by which they forcibly press themselves from that center or axis of motion. As when a body is whirled about in a sling, the body presses against the sling, and is ready to fly out as soon as liberty is given it. And this power is proportional to the true, not relative motion of the body round such a center or axis. Of this SirIsaac Newtongives the following instance[78]. If a pail or such like vessel near full of water be suspended by a string of sufficient length, and be turned about till the string be hard twisted. If then as soon as the vessel and water in it are become still and at rest, the vessel be nimbly turned about the contrary way the string was twisted, the vessel by the strings untwisting it self shall continue its motion a long time. And when the vessel first begins to turn, the water in it shall receive little or nothing of the motion of the vessel, but by degrees shall receive a communication of motion, till at last it shall move round as swiftly as the vessel it self. Now the definition of motion, whichDes Carteshas given us upon this principle of making all motion meerly relative, is this: that motion, is a removal of any body from its vicinity to other bodies, which were in immediate contact with it, and are considered as at rest[79]. And if this be compared with what he soon after says, that there is nothing real or positive in the body moved, for the sake of which we ascribe motion to it, which is not to be found as well in the contiguous bodies, which are considered as at rest[80]; it will follow from thence, that we may consider the vessel as at restand the water as moving in it: and the water in respect of the vessel has the greatest motion, when the vessel first begins to turn, and loses this relative motion more and more, till at length it quite ceases. But now, when the vessel first begins to turn, the surface of the water remains smooth and flat, as before the vessel began to move; but as the motion of the vessel communicates by degrees motion to the water, the surface of the water will be observed to change, the water subsiding in the middle and rising at the edges: which elevation of the water is caused by the parts of it pressing from the axis, they move about; and therefore this force of receding from the axis of motion depends not upon the relative motion of the water within the vessel, but on its absolute motion; for it is least, when that relative motion is greatest, and greatest, when that relative motion is least, or none at all.95.Thusthe true cause of what appears in the surface of this water cannot be assigned, without considering the water’s motion within the vessel. So also in the system of the world, in order to find out the cause of the planetary motions, we must know more of the real motions, which belong to each planet, than is absolutely necessary for the uses of astronomy. If the astronomer should suppose the earth to stand still, he could ascribe such motions to the celestial bodies, as should answer all the appearances; though he would not account for them in so simple a manner, as by attributing motion to the earth. But the motion of the earth must of necessity be considered, before the real causes, which actuate the planetary system, can be discovered.
64.Nowin this pendulum, as all the swings, whether long or short, will be performed in the same time; so the time of each will exactly bear the same proportion to the time required for a body to fall perpendicularly down, through half the length of the pendulum, that is from I to K, as the circumference of a circle bears to its diameter.
65.Itmay from hence be understood in some measure, why, when pendulums swing in circular arches, the times of their swings are nearly equal, if the arches are small, though those arches be of very unequal lengths; for if with the semidiameter L K the circular arch O K P be described, this arch in the lower part of it will differ very little from the line C K H.
66.Itmay not be amiss here to remark, that a body will fall in this line C K H (fig. 53.) from C to any other point, as Q or R in a shorter space of time, than if it moved through the straight line drawn from C to the other point; or through any other line whatever, that can be drawn between these two points.
67.Butas I have observed, that the time, which a pendulum takes in swinging, depends upon its length; I shall now say something concerning the way, in which this length of the pendulum is to be estimated. If the whole ball of the pendulum could be crouded into one point, this length, by which the motion of the pendulum is to be computed, would be the length of the string or rod. But the ball of the pendulum must have a sensible magnitude, and the several parts of this ball will not move with the same degree of swiftness; for those parts, which are farthest from the point, whereon the pendulum is suspended, must move with the greatest velocity. Therefore to know the time in which the pendulum swings, it is necessary to find that point of the ball, which moves with the same degree of velocity, as if the whole ball were to be contracted into that point.
68.Thispoint is not the center of gravity, as I shall now endeavour to shew. Suppose the pendulum A B (in fig. 54.) composed of an inflexible rod A C and ball C B, to be fixed on the point A, and lifted up into an horizontal situation. Here if the rod were not fixed to the point A, the body C B would descend directly with the whole force of its weight; and each part of the body would move down with the same degree of swiftness. But when the rod is fixed at the point A, the body must fall after another manner; for the parts of the body must move with different degrees of velocity, the parts more remote from A descending with a swifter motion, than the parts nearer to A; so that the body will receive a kind of rolling motion while it descends. But it has been observed above, that the effect of gravity upon any body is the same, as if the whole force were exerted on the body’s center of gravity[64].
Since therefore the power of gravity in drawing down the body must also communicate to it the rolling motion just described; it seems evident, that the center of gravity of the body cannot be drawn down as swiftly, as when the power of gravity has no other effect to produce on the body, than merely to draw it downward. If therefore the whole matter of the body C B could be crouded into its center of gravity, so that being united into one point, this rolling motion here mentioned might give no hindrance to its descent; this center would descend faster, than it can now do. And the point, which now descends as fast, as if the whole matter or the body C B were crouded into it, will be farther removed from the point A, than the center of gravity of the body C B.
69.Again, suppose the pendulum A B (in fig. 55.) to hang obliquely. Here the power of gravity will operate less upon the ball of the pendulum, than before: but the line D E being drawn so, as to stand perpendicular to the rod A C of the pendulum; the force of gravity upon the body C B, now it is in this situation, will produce the same effect, as if the body were to glide down an inclined plane in the position of D E. But here the motion of the body, when the rod is fixed to the point A, will not be equal to the uninterrupted descent of the body down this plane; for the bodywill here also receive the same kind of rotation in its motion, as before; so that the motion of the center of gravity will in like manner be retarded; and the point, which here descends with that degree of swiftness, which the body would have, if not hindered by being fixed to the point A; that is, the point, which descends as fast, as if the whole body were crouded into it, will be as far removed from the point A, as before.
70.Thispoint, by which the length of the pendulum is to be estimated, is called the center of oscillation. And the mathematicians have laid down general directions, whereby to find this center in all bodies. If the globe A B (in fig. 56.) be hung by the string C D, whose weight need not be regarded, the center of oscillation is found thus. Let the straight line drawn from C to D be continued through the globe to F. That it will pass through the center of the globe is evident. Suppose E to be this center of the globe; and take the line G of such a length, that it shall bear the same proportion to E D, as E D bears to E C. Then E H being made equal to ⅖ of G, the point H shall be the center of oscillation[65]. If the weight of the rod C D is too considerable to be neglected, divide C D (fig. 57) in I, that D I be equal to ⅓, part of C D; and take K in the same proportion to C I, as the weight of the globe A B to the weight of the rod C D. Then having found H, the center of oscillation of the globe, as before, divide I K in I, so that I L shall bear the same proportionto L H, as the line C H bears to K; and L shall be the center of oscillation of the whole pendulum.
71.Thiscomputation is made upon supposition, that the center of oscillation of the rod C D, if that were to swing alone without any other weight annexed, would be the point I. And this point would be the true center of oscillation, so far as the thickness of the rod is not to be regarded. If any one chuses to take into consideration the thickness of the rod, he must place the center of oscillation thereof so much below the point I, that eight times the distance of the center from the point I shall bear the same proportion to the thickness of the rod, as the thickness of the rod bears to its length C D[66].
72.Ithas 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 thependulum 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.Wemay 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 SirChristopher 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.SirIsaac Newtonhas 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 shallfinish 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 bodymoves, 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 lyingin 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 shallnow 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 greatGalileo, 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.Thefirst 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 anothersquare 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 shallnot 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 coneis thus defined byEuclidein 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.Nowby 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 thecube 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.
81.Indeedthis placing a cone to cut through a plane is not a practicable method of resolving problems. But when the geometers had discovered this use of the cone, they applied themselves to consider the nature of the lines, which will be produced by the intersection of the surface of a coneand a plane; whereby they might be enabled both to reduce these kinds of solutions to practice, and also to render their demonstrations concise and elegant.
82.Wheneverthe plane, which cuts the cone, is equidistant from another plane, that touches the cone on the side; (which is the case of the present figure;) the line, wherein the plane cuts the surface of the cone, is called a parabola. But if the plane, which cuts the cone, be so inclined to this other, that it will pass quite through the cone (as in fig. 65.) such a plane by cutting the cone produces the figure called an ellipsis, in which we shall hereafter shew the earth and other planets to move round the sun. If the plane, which cuts the cone, recline the other way (as in fig. 66.) so as not to be parallel to any plane, whereon the cone can lie, nor yet to cut quite through the cone; such a plane shall produce in the cone a third kind of line, which is called an hyperbola. But it is the first of these lines named the parabola, wherein bodies, that are thrown obliquely, will be carried by the force of gravity; as I shall here proceed to shew, after having first directed my readers how to describe this sort of line upon a plane, by which the form of it may be seen.
83.Toany straight line A B (fig. 67.) let a straight ruler C D be so applied, as to stand against it perpendicularly. Upon the edge of this ruler let another ruler E F be so placed, as to move along upon the edge of the first ruler C D, and keep always perpendicular to it. This being so disposed, let any point, as G, be taken in the line A B, and let a string equalin length to the ruler E F be fastened by one end to the point G, and by the other to the extremity F of the ruler E F. Then if the string be held down to the ruler E F by a pin H, as is represented in the figure; the point of this pin, while the ruler E F moves on the ruler C D, shall describe the line I K L, which will be one part of the curve line, whose description we were here to teach: and by applying the rulers in the like manner on the other side of the line A B, we may describe the other part I M of this line. If the distance C G be equal to half the line E F in fig. 64, the line M I L will be that very line, wherein the plane A B C D in that figure cuts the cone.
84.Theline A I is called the axis of the parabola M I L, and the point G is called the focus.
85.Nowby comparing the effects of gravity upon falling bodies, with what is demonstrated of this figure by the geometers, it is proved, that every body thrown obliquely is carried forward in one of these lines, the axis whereof is perpendicular to the horizon.
86.Thegeometers demonstrate, that if a line be drawn to touch a parabola in any point, as the line A B (in fig. 68.) touches the parabola C D, whose axis is Y Z, in the point E; and several lines F G, H I, K L be drawn parallel to the axis of the parabola: then the line F G will be to H I in the duplicate proportion of E F to E H, and F G to K L in the duplicate proportion of E F to E K; likewise H I to K L in the duplicate proportion of E H to E K. What is to be understood by duplicate or two-foldproportion, has been already explained[73]. Accordingly I mean here, that if the line M be taken to bear the same proportion to E H, as E H bears to E F, H I will bear the same proportion to F G, as M bears to E F; and if the line N bears the same proportion to E K, as E K bears to E F, K L will bear the same proportion to F G, as N bears to E F; or if the line O bear the same proportion to E K, as E K bears to E H, K L will bear the same proportion to H I, as O bears to E H.
87.Thisproperty is essential to the parabola, being so connected with the nature of the figure, that every line possessing this property is to be called by this name.
88.Nowsuppose a body to be thrown from the point A (in fig. 69.) towards B in the direction of the line A B. This body, if left to it self, would move on with a uniform motion through this line A B. Suppose the eye of a spectator to be placed at the point C just under the point A; and let us imagine the earth to be so put into motion along with the body, as to carry the spectator’s eye along the line C D parallel to A B; and that the eye would move on with the same velocity, wherewith the body would proceed in the line A B, if it were to be left to move without any disturbance from its gravitation towards the earth. In this case if the body moved on without being drawn towards the earth, it would appear to the spectator to be at rest. But if the power of gravity exerted it self on the body, it would appear to the spectatorto fall directly down. Suppose at the distance of time, wherein the body by its own progressive motion would have moved from A to E, it should appear to the spectator to have fallen through a length equal to E F: then the body at the end of this time will actually have arrived at the point F. If in the space of time, wherein the body would have moved by its progressive motion from A to G, it would have appeared to the spectator to have fallen down the space G H: then the body at the end of this greater interval of time will be arrived at the point H. Now if the line A F H I be that, through which the body actually passes; from what has here been said, it will follow, that this line is one of those, which I have been describing under the name of the parabola. For the distances E F, G H, through which the body is seen to fall, will increase in the duplicate proportion of the times[74]; but the lines A E, A G will be proportional to the times wherein they would have been described by the single progressive motion of the body: therefore the lines E F, G H will be in the duplicate proportion of the lines A F, A G; and the line A F H I possesses the property of the parabola.
89.Ifthe earth be not supposed to move along with the body, the case will be a little different. For the body being constantly drawn directly towards the center of the earth, the body in its motion will be drawn in a direction a little oblique to that, wherein it would be drawn by the earth in motion, as before supposed. But the distance to the center of theearth bears so vast a proportion to the greatest length, to which we can throw bodies, that this obliquity does not merit any regard. From the sequel of this discourse it may indeed be collected, what line the body being thrown thus would be found to describe, allowance being made for this obliquity of the earth’s action[75]. This is the discovery of SirIs. Newton; but has no use in this place. Here it is abundantly sufficient to consider the body as moving in a parabola.
90.Theline, which a projected body describes, being thus known, practical methods have been deduced from hence for directing the shot of great guns to strike any object desired. This work was first attempted byGalileo, and soon after farther improved by his scholarTorricelli; but has lately been rendred more complete by the great Mr.Cotes, whose immature death is an unspeakable loss to mathematical learning. If it be required to throw a body from the point A (in fig. 70.) so as to strike the point B; through the points A, B draw the straight line C D, and erect the line A E perpendicular to the horizon, and of four times the height, from which a body must fall to acquire the velocity, wherewith the body is intended to be thrown. Through the points A and E describe a circle, that shall touch the line C D in the point A. Then from the point B draw the line B F perpendicular to the horizon, intersecting the circle in the points G and H. This being done, if the body be projected directly towards either of these points G or H, it shall fall upon the point B; but with this difference, that, if it be thrownin the direction A G, it shall sooner arrive at B, than if it were projected in the direction A H. When the body is projected in the direction A G; the time, it will take up in arriving at B, will bear the same proportion to the time, wherein it would fall down through one fourth part of A E, as A G bears to half A E. But when the body is thrown in the direction of A H, the time of its passing to B will bear the same proportion to the time, wherein it would fall through one fourth part of A E, as A H bears to half A E.
91.Ifthe line A I be drawn so as to divide the angle under E A D in the middle, and the line I K be drawn perpendicular to the horizon; this line will touch the circle in the point I, and if the body be thrown in the direction A I, it will fall upon the point K: and this point K is the farthest point in the line A D, which the body can be made to strike, without increasing its velocity.
92.Thevelocity, wherewith the body every where moves, may be found thus. Suppose the body to move in the parabola A B (fig. 71.) Erect A C perpendicular to the horizon, and equal to the height, from which a body must fall to acquire the velocity, wherewith the body sets out from A. If you take any points as D and E in the parabola, and draw D F and E G parallel to the horizon; the velocity of the body in D will be equal to what a body will acquire in falling down by its own weight through C F, and in E the velocity will be the same, as would be acquired in falling through C G. Thus the body moves slowest at the highest point H of the parabola; and at equal distances from this point willmove with equal swiftness, and descend from that highest point through the line H B altogether like to the line A H in which it ascended; abating only the resistance of the air, which is not here considered. If the line H I be drawn from the highest point H parallel to the horizon, A I will be equal to ¼ of B G in fig. 70, when the body is projected in the direction A G, and equal to ¼ of B H, when the body is thrown in the direction A H provided A D be drawn horizontally.
93.ThusI have recounted the principal discoveries, which had been made concerning the motion of bodies by SirIsaac Newton’spredecessors; all these discoveries, by being found to agree with experience, contributing to establish the laws of motion, from whence they were deduced. I shall therefore here finish what I had to say upon those laws; and conclude this chapter with a few words concerning the distinction which ought to be made between absolute and relative motion. For some have thought fit to confound them together; because they observe the laws of motion to take place here on the earth, which is in motion, after the same manner as if it were at rest. But SirIsaac Newtonhas been careful to distinguish between the relative and absolute consideration both of motion and time[76]. The astronomers anciently found it necessary to make this distinction in time. Time considered in it self passes on equably without relation to any thing external, being the proper measure of the continuance and duration of all things. But it is most frequently conceived of by us under a relative view to some succession insensible things, of which we take cognizance. The succession of the thoughts in our own minds is that, from whence we receive our first idea of time, but is a very uncertain measure thereof; for the thoughts of some men flow on much more swiftly, than the thoughts of others; nor does the same person think equally quick at all times. The motions of the heavenly bodies are more regular; and the eminent division of time into night and day, made by the sun, leads us to measure our time by the motion of that luminary: nor do we in the affairs of life concern our selves with any inequality, which there may be in that motion; but the space of time which comprehends a day and night is rather supposed to be always the same. However astronomers anciently found these spaces of time not to be always of the same length, and have taught how to compute their differences. Now the time, when so equated as to be rendered perfectly equal, is the true measure of duration, the other not. And therefore this latter, which is absolutely true time, differs from the other, which is only apparent. And as we ordinarily make no distinction between apparent time, as measured by the sun, and the true; so we often do not distinguish in our usual discourse between the real, and the apparent or relative motion of bodies; but use the same words for one, as we should for the other. Though all things about us are really in motion with the earth; as this motion is not visible, we speak of the motion of every thing we see, as if our selves and the earth stood still. And even in other cases, where we discern the motion of bodies, we often speak of them not in relation to the whole motion we see, but with regard to otherbodies, to which they are contiguous. If any body were lying on a table; when that table shall be carried along, we say the body rests upon the table, or perhaps absolutely, that the body is at rest. However philosophers must not reject all distinction between true and apparent motions, any more than astronomers do the distinction between true and vulgar time; for there is as real a difference between them, as will appear by the following consideration. Suppose all the bodies of the universe to have their courses stopped, and reduced to perfect rest. Then suppose their present motions to be again restored; this cannot be done without an actual impression made upon some of them at least. If any of them be left untouched, they will retain their former state, that is, still remain at rest; but the other bodies, which are wrought upon, will have changed their former state of rest, for the contrary state of motion. Let us now suppose the bodies left at rest to be annihilated, this will make no alteration in the state of the moving bodies; but the effect of the impression, which was made upon them, will still subsist. This shews the motion they received to be an absolute thing, and to have no necessary dependence upon the relation which the body said to be in motion has to any other body[77].
94.Besidesabsolute and relative motion are distinguishable by their Effects. One effect of motion is, that bodies, when moved round any center or axis, acquire a certainpower, by which they forcibly press themselves from that center or axis of motion. As when a body is whirled about in a sling, the body presses against the sling, and is ready to fly out as soon as liberty is given it. And this power is proportional to the true, not relative motion of the body round such a center or axis. Of this SirIsaac Newtongives the following instance[78]. If a pail or such like vessel near full of water be suspended by a string of sufficient length, and be turned about till the string be hard twisted. If then as soon as the vessel and water in it are become still and at rest, the vessel be nimbly turned about the contrary way the string was twisted, the vessel by the strings untwisting it self shall continue its motion a long time. And when the vessel first begins to turn, the water in it shall receive little or nothing of the motion of the vessel, but by degrees shall receive a communication of motion, till at last it shall move round as swiftly as the vessel it self. Now the definition of motion, whichDes Carteshas given us upon this principle of making all motion meerly relative, is this: that motion, is a removal of any body from its vicinity to other bodies, which were in immediate contact with it, and are considered as at rest[79]. And if this be compared with what he soon after says, that there is nothing real or positive in the body moved, for the sake of which we ascribe motion to it, which is not to be found as well in the contiguous bodies, which are considered as at rest[80]; it will follow from thence, that we may consider the vessel as at restand the water as moving in it: and the water in respect of the vessel has the greatest motion, when the vessel first begins to turn, and loses this relative motion more and more, till at length it quite ceases. But now, when the vessel first begins to turn, the surface of the water remains smooth and flat, as before the vessel began to move; but as the motion of the vessel communicates by degrees motion to the water, the surface of the water will be observed to change, the water subsiding in the middle and rising at the edges: which elevation of the water is caused by the parts of it pressing from the axis, they move about; and therefore this force of receding from the axis of motion depends not upon the relative motion of the water within the vessel, but on its absolute motion; for it is least, when that relative motion is greatest, and greatest, when that relative motion is least, or none at all.
95.Thusthe true cause of what appears in the surface of this water cannot be assigned, without considering the water’s motion within the vessel. So also in the system of the world, in order to find out the cause of the planetary motions, we must know more of the real motions, which belong to each planet, than is absolutely necessary for the uses of astronomy. If the astronomer should suppose the earth to stand still, he could ascribe such motions to the celestial bodies, as should answer all the appearances; though he would not account for them in so simple a manner, as by attributing motion to the earth. But the motion of the earth must of necessity be considered, before the real causes, which actuate the planetary system, can be discovered.