The side of the Icosahedrum is had by this meanes.
14. If a right line equall to the axis of the sphearicall, and to it from the end of the perpendicular be knit unto the center, a right line drawne from the cutting of the
periphery unto the said end shall be the side of the Icosahedrum.
As here let the Axis ae; be the diameter of the circle aue, and ai, equall to the same axis, and perpendicular from the end, be knit unto the center, by the right line io: A right drawne from the section u, unto a, shall be the side of the Icosahedrum. From u, let the perpendicular uy, be drawne: Here the two triangles iao, & uyo, are equiangles by the [13 e, vij]. Therfore by the [12 e, vij]. as ia, is unto ao: so is uy, unto yo. But ia, is the double of the said ao: Therefore uy, is the double of the same yo: Therefore by the [14 e, xij], it is of quadruple power unto it: And therefore also uy, and yo, that is, by the [9 e xij], uo, that is againe by the [28 e, iiij], ao, is of quintuple power to yo. But yo, is lesser than ao, that is, than oe: Let therefore os, be cut off equall to it. Now as the halfe of ao, is of quintuple valew to the halfe of yo: so the double ae, is of quintuple power to the double ys. Therefore, by the [18 e xxv]. seeing that the diagony ae, is of quintuple power to ys; the said ys, shall be the side of the sexangle inscribed into a circle, circumscribing the quinquangle of the Icosahedrum. But the perpendicular uy, is equall to ys; because each of them is the double of yo. Wherefore uy, is the side of the sexangle. But ay, is the side of the Decangle: For it is equall to se: Because if from equall rayes ao, and oe, you take equall portions oy, and os: There shall remaine equall, ya, and se. And the Diagony of an Icosahedrum by the [16 e xxv], is compounded of the side of the sexangle, continued at each end with the side of the decangle. Wherefore ay, is the side of the decangle. Lastly, ua, whose power is as much as the sides of the
sexangle and decangle, by the [15. e, xviij], shall be the side of an Icosahedrum.
15 Of the five ordinate bodies inscribed into the same spheare, the tetrahedrum in respect of the greatnesse of his side is first, the Octahedrum, the second; the Cube, the third; the Icosahedrum, the fourth; and the Dodecahedrum, the fifth.
As it will plainely appeare, if all of them be gathered into one, thus. For ai, the side of the Tetrahedrum, subtendeth a greater periphery than ao, the side of the Octahedrum; And ao, a greater than ie, the side of the Cube; because it subtendeth but the halfe: And ie, greater than ue, the side of the Icosahedrum: And ue, greater than ye, the side of Dodecahedrum.
The latter, Euclide doth demonstrate with a greater circumstance. Therefore out of the former figures and demonstrations, let here be repeated, The sections of the axis first into a double reason in s: And the side of the sexangle rl: And the side of the Decangle ar, inscribed into the same circle, circumscribing the quinquangle of an icosahedrum: And the perpendiculars is, and ul.