Let A C and A D be drawn. Seeing, therefore, the angles C E D and C B D together taken are equal to twice the angle C A D (as has been demonstrated in the 5th article); and the angle C A D twice taken is quadruple to the angle C E D; the angles C E D and C B D together taken will also be equal to the angle C E D four times taken; and therefore if the angle C E D be taken away on both sides, there will remain the angle C B D on one side, equal to the angle C E D thrice taken on the other side; which was to be demonstrated. Coroll. Therefore a point being given within a circle, there may be drawn two lines from it to the circumference, so as their reflected lines may meet in the circumference. For it is but trisecting the angle C B D, which how it may be done shall be shown in the following chapter. ------------------------------------------------------------------------ [Illustration: _Vol. 1. Lat. & Eng._ C. XIX. _Fig. 1-10_ ] ------------------------------------------------------------------------ ------- CHAPTER XX. OF THE DIMENSION OF A CIRCLE, AND THE DIVISION OF ANGLES OR ARCHES. 1. The dimension of a circle never determined in numbers by Archimedes and others.—2. The first attempt for the finding out of the dimension of a circle by lines.—3. The second attempt for the finding out of the dimension of a circle from the consideration of the nature of crookedness.—4. The third attempt; and some things propounded to be further searched into.—5. The equation of the spiral of Archimedes with a strait line.—6. Of the analysis of geometricians by the powers of lines. [Sidenote: The dimension of a circle never determined in numbers by Archimedes and others.]