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.