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.