The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed

Of the nature of an Icosahedron.

Fiue triangles of an Icosahedron, do make a solide angle, the bases of which [ 1] triangles make a pentagon. If therfore from the opposite bases of the Icosahedron

be taken the other pentagon by them described, these pentagons shall in such sort cut the diameter of the Icosahedron which coupleth the forsaid opposite angles, that that part which is contained betweene the plaines of those two pentagons, shalbe the greater segment: and the residue which is drawne from the plaine to [ 2] the angle shall be the lesse segment. If the opposite angles of two bases ioyned to∣gether, be coupled by a right line, the greater segment of that right line is the side [ 3] of the Icosahedron. A line drawne from the centre of the Icosahedron to the an∣gles, is in power quintuple, to halfe that line which is takē betwene the pentagōs, or of the halfe of that line which is drawne from the centre of the circle which cō∣tayneth [ 4] the foresaid pentagon: which two lines are therefore equall. The side of the Icosahedron contayneth in power either of them, and also the lesse segment, [ 5] namely, the line which falleth from the solide angle to the pentagon. The diame∣ter of the Icosahedron contayneth in power the whole line, which coupleth the opposite angles of the bases ioyned together, and the greater segment thereof, [ 6] namely, the side of the Icosahedron. The diameter also is in power quintuple to the line which was taken betwene the pentagons, or to the line which is drawne from the centre to the circumference of the circle which containeth the pentagon [ 7] cōposed of the sides of the Icosahedron. The dimetient contayneth in power the right line which coupleth the centres of the opposite bases of the Icosahedron, [ 8] and the diameter of the circle which contayneth the base. Moreouer the sayd di∣metient contayneth in power the diameter of the circle, which contayneth the pentagon, and also the line which is drawne from the centre of the same circle to the circumference: That is, it is quintuple to the line drawne from the centre to [ 9] the circumference. The line which coupleth the centres of the opposite bases, con¦tayneth in power the line which coupleth the centres of the next bases, and also the rest of that line of which the side of the cube inscribed in the Icosahedron is [ 10] the greater segment. The line which coupleth the middle sections of the opposite sides, is triple to the side of the dodecahedron inscribed in it. Wherefore if the [ 11] side of the Icosahedron, and the greater segment thereof be made one line, the third part of the whole, is the side of the dodecahedron inscribed in the Icosahe∣dron.