lf, mt. And let them be knit first one with another, by the lines nj, jv, vf, ft, tn. Secondarily, with the angles of the first quinquangle, by the lines ne, ej, ji, iv, of, fu, ut, ta, an. The sixth perpendicular from the center d, let it be bg, the ray dc, continued at each end with the side of the decangle, cg, and db, tied together about with the perpendiculars, as by the lines ng, tg: Beneath with the angles of the first quinquangle, as by the lines be, bi, and in other places in like manner, and let all the plaines be made up. This say I, is an Icosaedrum; And is comprehended of 20. triangles, both equilaters and equall. First, the tenne middle triangles, leaving out the perpendiculars, that they are equilaters and equall, one shall demonstrate, as nat. For mt and yu, because they are perpendiculars, they are also, by the [6 e xxj]. parallells: And by the grant, equall. Therefore by the [27 e, v], nt, is equall to ym, the side of the quinquangle. Item na, by the [6 e xij]. is of as great power, as both the shankes ny, and ya, that is, by the construction, as the sides of the sexangle and decangle: And, by the converse of the [15. e xviij]. it is the side of the quinquangle. The same shall fall out of ot. Wherefore nat, is an equilater triangle. The same shall fall out of the other nine middle triangles, nae, nej, eji, jiv, ivo, vof, fou, fut, uta, tan.
In like manner also shall it be proved of the five upper triangles, by drawing the right lines dy and cn which as afore (because they knit together equall parallells, to wit, dc, and yn) they shall be equall. But dy, is the side of a sexangle: Therefore cn, shall be also the side of a sexangle: And cg, is the side of a decangle: Therefore an, whose power is equall to both theirs by the [9 e xij]. shall by the converse of the [15 e xviij], be the side of a quinquangle: And in like manner gt, shall be concluded to be the side of a quinquangle. Wherefore ngt, is an equilater: And the foure other shall likewise be equilaters.
The other five triangles beneath shall after the like manner be concluded to be equilaters. Therefore one shall be for all, to wit, ibe, by drawing the raies di, and de. For ib,
whose power, as afore, is as much as the sides of the sexangle, and decangle, shall be the side of the quinquangle: And in like sort be, being of equall power with de, and do, the sides of the sexangle and decangle, shall be the side of the quinquangle. Wherefore the triangle ebi, is an equilater: And the foure other in like manner may be shewed to be equilaters. Therefore all the side of the twenty triangles, seeing they are equall, they shall be equilater triangles: And by the [8 e, vij]. equall.
17 The diagony of an icosaedrū is irrational unto the side.
This is the fourth example of irrationality, or incommensurability. The first was of the Diagony and side of a square or quadrate. The second was of the segments of a line proportionally cut. The third of the Diameter of a circle and the side of a quinquangle.
And
18 The power of the diagony of an icosaedrum is five times as much as the ray of the circle.
For by the [13 e, xviij]. the line continually made of the side of the sexangle and decangle is cut proportionally, and the greater segment is the side of the sexangle: As here. Let the perpendicular ae, be cut into two equall parts in i. Then eo, that is the lesser segment continued with the halfe of the greater, that is, with ie. it shall by the [6 e xiiij], be of power five times so great as is the power of the same halfe. Therefore seeing that io, the halfe of the diagony is of power fivefold to the halfe: the whole diagony shall be of power fivefold to the whole cut.