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 a Dodecahedron.

[ 1] The diameter of a dodecahedron contayneth in power the side of the dode∣cahedron, and also that right line, vnto which the side of the dodecahedron is the lesse segment, and the side of the cube inscribed in it is the greater segmēt: which line is that which subtendeth the angle of the inclination of the bases, contayned [ 2] vnder two perpendiculars of the bases of the dodecahedron. If there be taken two bases of the dodecahedron distant the one from the other by the length of one of the sides, a right line coupling their centres, being diuided by an extreame and meane proportion, maketh the greater segment the right line which coupleth the [ 3] centres of the next bases. If by the centres of fiue bases set vppon one base, be drawne a playne superficies, and by the centres of the bases which are set vpon the opposite base be drawne also a playne superficies, and then be drawne a right line coupling the centres of the opposite bases, that right line is so cut, that eche of his partes set without the playne superficies, is the greater segment of that part which [ 4] is contayned betwene the playnes. The side of the dodecahedron is the greater

segment of the line which subtendeth the angle of the pentagon. A perpendicular [ 5] line drawne from the centre of the dodecahedron to one of the bases, is in power quintuple to half the line which is betwene the playnes: And therfore the whole [ 6] line which coupleth the centres of the opposite bases, is in power quintuple to the whole line which is betwene the sayd playnes. The line which subt••deth the [ 7] angle of the base of the dodecahedrō, together with the side of the base, are in power quintuple to the line which is drawne from the cētre of the circle, which contayneth the base, to the circumference. A secti∣on [ 8] of a sphere contayning three bases of the dodecahedron taketh a third part of the diameter of the sayd sphere. The side of the dodecahedron, and the line which [ 9] subtendeth the angle of the pentagon, are e∣quall to the right line which coupleth the middle sections of the opposite sides of the dodecahedron.