8 If the diameter of a circle circumscribed about a quinquangle be rationall, it is irrationall unto the side of the inscribed quinquangle, è 11. p xiij.
So before the segments of a right line proportionally cut were irrationall.
The other triangulates hereafter multiplied from the ternary, quaternary, or quinary of the sides, may be inscribed into a circle by an inscript triangle, quadrate, or quinquangle. Therefore by a triangle there may be inscribed a triangulate of 6. 12, 24, 48, angles: By a quadrate, a triangulate of 8. 16, 32, 64, angles. By a quinquangle, a triangulate of 10, 20, 40, 80. angles, &c.
9 The ray of a circle is the side of the inscript sexangle. è 15 p iiij.
A sexangle is inscribed by an inscript equilaterall triangle, by halfing of the three angles of the said triangle: But it is done more speedily by the ray or semidiameter of the circle, sixe times continually inscribed. As in the circle given, let the diameter be ae; And upon the center o, with the ray ie, let the periphery uio, be described: And from the points o and u, let the diameters be oy, and us; These knit both one with another, and also with the diameter ae shall inscribe an equilaterall sexangle into the circle given, whose side shal be equal to the ray of the same circle. As eu, is equal to ui, because they both equall to the same ie, by the [29 e, iiij]. There fore eiu, is an equilater triangle: And likewise eio, is an equilater. The angles also in the center are ⅔ of one rightangle: And therefore they are equall. And by the [14. e v], the angle sio, is ⅓. of two rightangles: And by the [15. e v]. the angles at the toppe are also equall. Wherefore sixe are equall: And therefore, by the [7 e xvj]. and [32. e, xv], all the bases are equall, both betweene themselves, and as was even now made manifest, to the ray of the circle given. Therefore the sexangle inscript by the ray of a circle is an
equilater; And by the [1 e xvij]. equiangled.
Therefore