[Sidenote: How from the description of deficient figures in a parallelogram, any number of mean proportionals may be found out between two given strait lines.] 14. If these deficient figures, which I have described in a parallelogram, were capable of exact description, then any number of mean proportionals might be found out between two strait lines given. For example, in the parallelogram A B C D, (in figure 8) let the three-sided figure of two means be described (which many call a _cubical parabola_); and let R and S be two given strait lines; between which, if it be required to find two mean proportionals, it may be done thus. Let it be as R to S, so B C to B F; and let F E be drawn parallel to B A, and cut the crooked line in E; then through E let G H be drawn parallel and equal to the strait line A D, and cut the diagonal B D in I; for thus we have G I the greatest of two means between G H and G E, as appears by the description of the figure in article 4. Wherefore, if it be as G H to G I, so R to another line, T, that T will be the greatest of two means between R and S. And therefore if it be again as R to T, so T to another line, X, that will be done which was required. In the same manner, four mean proportionals may be found out, by the description of a three-sided figure of four means; and so any other number of means, &c. ------------------------------------------------------------------------ [Illustration: _Vol. 1. Lat. & Eng._ C. XVII. _Fig. 1-8_ ] ------------------------------------------------------------------------ ------- CHAPTER XVIII. OF THE EQUATION OF STRAIT LINES WITH THE CROOKED LINES OF PARABOLAS AND OTHER FIGURES MADE IN IMITATION OF PARABOLAS. 1. To find the strait line equal to the crooked line of a semiparabola.—2. To find a strait line equal to the crooked line of the first semiparabolaster, or to the crooked line of any other of the deficient figures of the table of the 3d article of the precedent chapter. [Sidenote: To find a strait line equal to the crooked line of a semiparabola.] 1. A parabola being given, to find a strait line equal to the crooked line of the semiparabola. Let the parabolical line given be A B C (in figure 1), and the diameter found be A D, and the base drawn D C; and the parallelogram A D C E being completed, draw the strait line A C. Then dividing A D into two equal parts in F, draw F H equal and parallel to D C, cutting A C in K, and the parabolical line in O; and between F H and F O take a mean proportional F P, and draw A O, A P and P C. I say that the two lines A P and P C, taken together as one line, are equal to the parabolical line A B O C.