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.