inscript perpendicularly, it is the diamiter of the Circle, and the middest of it is the center. 1 p iij.
As let the Inscript ae, cut the inscript iu perpendicularly: dividing it into two equall parts in o. I say that the one inscript thus halfing the other, is the diameter of the Circle: And that the middest of it is the center thereof: As in the circle, let the Inscript is, cut the inscript ae, and that perpendicularly dividing into two equall parts in o. I say that iu, thus dividing ae, is the Diameter of the Circle: And y, the middest of the said iu, is the Center of the same.
The cause is the same, which was of the [5 e xj]. Because the inscript cut into halfes is for the side of the inscribed rectangle, and it doth subtend the periphery cut also into two parts; By the which both the Inscript and Periphery also were in like manner cut into two equall parts: Therefore the right line thus halfing in the diameter of the rectangle: But that the middle of the circle is the center, is manifest out of the [7 e v], and [29 e iiij].
Euclide, thought better of Impossibile, than he did of the cause: And thus he forceth it. For if y be not the Center, but s, the part must be equall to the whole: For the Triangle aos, shall be equilater to the triangle eos. For ao, oe, are equall by the grant: Item sa, and se, are the rayes of the circle: And so, is common to both the triangles. Therefore by the [1 e vij], the angles on each side at o are equall; And by the [13 e v], they are both right angles. Therefore soe is a right angle; It is therefore equall by the grant, to the right angle yoe, that is, the part is equall to the whole, which is impossible. Wherefore s is not the Center. The same will fall out of any other points whatsoever out of y.
Therefore
8. If two right lines doe perpendicularly halfe two
inscripts, the meeting of these two bisecants shall be the Center of the circle è 25 p iij.
As here ae, and io, let them cut into halfes the right lines uy, and ys. And let them meete, that they cut one another in r. I say r is the center of the circle ayoseiu. For before, at the [6], and [7 e], it was manifest that the Center was in the Diameter. And in the meeting of the diameters. [Therefore two manner of wayes is the Center found; First by the middle of the diameter: And then againe by the concourse, or meeting of the diameters, in the middest of the lines halfed or cut into two equall portions.] Here is no neede of the meeting of many diameters, one will serve well enough.