And
4. It is greater than the halfe of the circumscribed Circle.
Because the circumscribed quadrate, which is his double, is greater than the whole circle.
For the inscribing or other multangled odde-sided figures we must needes use the helpe of a triangle, each of whose angles at the base is manifold to the other: In a Quinquangle first, that which is double unto the remainder, which is thus found.
5. If a right line be cut proportionally, the base of that triangle whose shankes shall be equall to the whole line cut, and the base to the greater segment of the same, shall have each of the angles at base double to the
remainder: And the base shall be the side of the quinquangle inscribed with the triangle into a circle. 10, and 11. p iiij.
Here first thou shalt take for the fabricke or making of the Triangle, for the ray the right line ae by the [3 e xiiij], cut proportionally in o: A circle also shalt thou make upon the center a, with the ray ae: And then shalt thou by the [6 e xv], inscribe a right line equall to the greater segment: And shalt knit the same inscript with the whole line cut with another right line. This triangle shall be your desire. For by the [17 e vj], the angles at the base ei are equall, so that looke whatsoever is prooved of the one, is by and by also prooved of the other. Then let oi be drawne; And a Circle, by the [8 e xvij], circumscribed about the triangle aoi. This circle the right line ei, shall touch, by the [27 e xv]. Because, by the grant, the right line ae, is cut proportionally, therefore the Oblong of the secant and outter segment, is equall to the quadrate of the greater segment, to which by the grant, the base ei, is equall. Here therefore the angle aie is the double of the angle at a: because it is equall to the angles aio, and oai, which are equall betweene themselves. For by the [27 e xvj] it is equall to the angle oai in the alterne segment. And the remainder aio, is equall to it selfe. Therefore also the angle aei, is equall to the same two angles, because it is equall to the angle aie. But the outter angle eoi, is equall to the same two, by the [15 e vj]. Therefore the angles ioe and oei (because they are equall to the same) they are equall betweene themselves. Wherefore by the [17 e vj], the sides oi and ei are equall. And there also ao and oi: And the angles oai & oia are equall by the [17 e vj]. Wherefore seeing
that to both the angle aie is equall, it shall be the double of either of the equalls.
