5. The angle in the center, is double to the angle of the periphery standing upon the same base, 20 p iij.
The variety or the example in Euclide is threefold, and yet
the demonstration is but one and the same: As here eai, the angle in the center, shall be prooved to be double to eoi, the angle in the periphery, the right line ou cutting it into two triangles on each side equicrurall; And, by the [17 e vj], at the base equiangles: Whose doubles severally are the angles, eau, of eoa: And iau, of ioa, For seeing it is equall to the two inner equall betweene themselves by the [15 e vj]; it shall be the double of one of them. Therefore the whole eai, is the double of the whole eoi.
The second example is thus of the angle in the center aei: And in the periphery aoi. Here the shankes eo, and ei, by the [28 e iiij], are equall: And by the [17 e vj], the angles at o and i are equall: To both which the angle in the center is equall, by the [15 e vj]. Therefore it is double of the one.
The third example is of the angle in the center, aei, And in the periphery aoi, Let the diameter be oeu. Here the whole angle ieu, by the [15 e vj], is equall to the two inner angles eoi, and eio, which are equall one to another, by the [17 e vj]: And therefore it is double of the one. Item the particular angle aeu, is equall by the [15 e vj], to the angles eoa, and eao, equall also one to another, by the [17 e vj]. Therefore the remainder aei, is the double of the other aoi, in the periphery.
Therefore
6. If the angle in the periphery be equall to the
angle in the center, it is double to it in base. And contrariwise.
This followeth out of the former element: For the angle in the center is double to the angle in the periphery standing upon the same base: Wherefore if the angle in the periphery be to be made equall to the angle in the center, his base is to be doubled, and thence shall follow the equality of them both: S.