Chap. II. Farther proofs of the Laws of Motion. (continued 2)

76. I shall now proceed to the last kind of motion, to be treated on in this place, and shew what line the power of gravity will cause a body to describe, when it is thrown forwards by any force. This was first discovered by the great Galileo, and is the principle, upon which engineers should direct the shot of great guns. But as in this case bodies describe in their motion one of those lines, which in geometry are called conic sections; it is necessary here to premise a description of those lines. In which I shall be the more particular, because the knowledge of them is not only necessary for the present purpose, but will be also required hereafter in some of the principal parts of this treatise.

77. The first lines considered by the ancient geometers were the straight line and the circle. Of these they composed various figures, of which they demonstrated many properties, and resolved divers problems concerning them. These problems they attempted always to resolve by the describing straight lines and circles. For instance, let a square A B C D (fig. 61.) be proposed, and let it be required to make another square in any assigned proportion to this. Prolong one side, as D A, of this square to E, till A E bear the same proportion to A D, as the new square is to bear to the square A C. If the opposite side B C of the square A C be also prolonged to F, till B F be equal to A E, and E F be afterwards drawn, I suppose my readers will easily conceive, that the figure A B F E will bear to the square A B C D the same proportion, as the line A E bears to the line A D. Therefore the figure A B F E will be equal to the new square, which is to be found, but is not it self a square, because the side A E is not of the same length with the side E F. But to find a square equal to the figure A B F E you must proceed thus. Divide the line D E into two equal parts in the point G, and to the center G with the interval G D describe the circle D H E I; then prolong the line A B, till it meets the circle in K; and make the square A K L M, which square will be equal to the figure A B F E, and bear to the square A B C D the same proportion, as the line A E bears to A D.

78. I shall not proceed to the proof of this, having only here set it down as a specimen of the method of resolving geometrical problems by the description of straight lines and circles. But there are some problems, which cannot be resolved by drawing straight lines or circles upon a plane. For the management therefore of these they took into consideration solid figures, and of the solid figures they found that, which is called a cone, to be the most useful.

79. A cone is thus defined by Euclide in his elements of geometry[72]. If to the straight line A B (in fig. 62.) another straight line, as A C, be drawn perpendicular, and the two extremities B and C be joined by a third straight line composing the triangle A C B (for so every figure is called, which is included under three straight lines) then the two points A and B being held fixed, as two centers, and the triangle A C B being turned round upon the line A B, as on an axis; the line A C will describe a circle, and the figure A C B will describe a cone, of the form represented by the figure B C D E F (fig. 63.) in which the circle C D E F is usually called the base of the cone, and B the vertex.

80. Now by this figure may several problems be resolved, which cannot by the simple description of straight lines and circles upon a plane. Suppose for instance, it were required to make a cube, which should bear any assigned proportion to some other cube named. I need not here inform my readers, that a cube is the figure of a dye. This problem was much celebrated among the ancients, and was once inforced by the command of an oracle. This problem may be performed by a cone thus. First make a cone from a triangle, whose side A C shall be half the length of the side B C Then on the plane A B C D (fig. 64.) let the line E F be exhibited equal in length to the side of the cube proposed; and let the line F G be drawn perpendicular to E F, and of such a length, that it bear the same proportion to E F, as the cube to be sought is required to bear to the cube proposed. Through the points E, F, and G let the circle F H I be described. Then let the line E F be prolonged beyond F to K, that F K be equal to F E, and let the triangle F K L, having all its sides F K, K L, L F equal to each other, be hung down perpendicularly from the plane A B C D. After this, let another plane M N O P be extended through the point L, so as to be equidistant from the former plane A B C D, and in this plane let the line Q L R be drawn so, as to be equidistant from the line E F K. All this being thus prepared, let such a cone, as was above directed to be made, be so applied to the plane M N O P, that it touch this plane upon the line Q R, and that the vertex of the cone be applied to the point L. This cone, by cutting through the first plane A B C D, will cross the circle F H I before described. And if from the point S, where the surface of this cone intersects the circle, the line S T be drawn so, as to be equidistant from the line E F; the line F T will be equal to the side of the cube sought: that is, if there be two cubes or dyes formed, the side of one being equal to E F, and the side of the other equal to F T; the former of these cubes shall bear the same proportion to the latter, as the line E F bears to F G.

81. Indeed this 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 cone and 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. Whenever the 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. To any 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 equal in 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. The line A I is called the axis of the parabola M I L, and the point G is called the focus.

85. Now by 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. The geometers 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-fold proportion, 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. This property 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. Now suppose 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 spectator to 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. If the 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 the earth 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 Sir Is. Newton; but has no use in this place. Here it is abundantly sufficient to consider the body as moving in a parabola.

90. The line, 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 by Galileo, and soon after farther improved by his scholar Torricelli; 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 thrown in 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. If the 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. The velocity, 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 will move 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. Thus I have recounted the principal discoveries, which had been made concerning the motion of bodies by Sir Isaac Newton’s predecessors; 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 Sir Isaac Newton has 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 in sensible 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 other bodies, 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. Besides absolute and relative motion are distinguishable by their Effects. One effect of motion is, that bodies, when moved round any center or axis, acquire a certain power, 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 Sir Isaac Newton gives 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, which Des Cartes has 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 rest and 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. Thus the 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.