As here, with ae, one side of the triangle aei, two third parts of the halfe periphery are imployed: For with one side one third of the whole eu, is imployed: Therefore eu, is the other third part, that is, the sixth part of the whole periphery. Therefore the inscript eu, is the ray of the circle, by the [9 e]. Now the power of the diameter aou, by the [14 e xij]. is foure times so much as is the power of the ray, that is, of eu: And by [21. e xvj], and [9 e xij], ae, and eu, are of the same power; take away eu, and the side ae, shall be of treble power unto the ray.
13 If the side of a sexangle be cut proportionally, the greater segment shall be the side of the decangle.
Pappus lib. 5. ca. 24. & Campanus ad 3 p xiiij. Let the ray ao, or side of the sexangle be cut proportionally, by the [3 e xiiij]: And let ae, be equall to the greater segment. I say that ae, is the side of the decangle. For if it be moreover continued with the whole ray unto i, the whole aei, shall by the [4 e xiiij]. be cut proportionally: and the greater segment ei, shal be the same ray. For the if the right line iea, be cut proportionally, it shall be as ia, is unto ie, that is to oa, to wit, unto the ray: so ao, shal be unto ae. Therefore, by the [15. e vij]. the triangles iao, and oae, are equiangles: And the angle aoe, is equall to the angle oia. But the angle uoe, is foure times as great as the angle aoe: for it is equall to the two inner at a, and e, by the [15 e vj]: which are equall between themselves, by the [10 e v]. and by the [17 e vj]. And therefore it is the double of
aeo, which is the double, for the same cause, of aio, equall to the same aoe. Therefore uoe, is the quadruple of the said aoe. Therefore ue, is the quadruple of the periphery ea. Therefore the whole uea, is the quintuple of the same ea: And the whole periphery is decuple unto it. And the subtense ae, is the side of the decangle.
Therefore
14 If a decangle and a sexangle be inscribed in the same circle, a right line continued and made of both sides, shall be cut proportionally, and the greater segment shall be the side of a sexangle; and if the greater segment of a right line cut proportionally be the side of an hexagon, the rest shall be the side of a decagon. 9. p xiij.
The comparison of the decangle and the sexangle with the quinangle followeth.
15 If a decangle, a sexangle, and a pentangle be inscribed into the same circle the side of the pentangle shall in power countervaile the sides of the others. And if a right line inscribed do countervaile the sides of the sexangle and decangle, it is the side of the pentangle. 10. p xiiij.
