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

¶ The ••0. Proposition. If from the power of the diameter of an Icosahedron, be taken away the power tripled of the side of the cube inscribed in the Icosahedron: the power remayning shall be sesquitertia to the power of the side of the I∣cosahedron.

LEt there be taken an Icosahedron ABGD: and l•••• two bases of the cube inscribed in it, ioyned together be EHKL and LKFC: and let the diameter of the cube be FH and the side be EH, and let the diameter of the Icos••h••dron be ••G, and the side be AB. Then I say, that if from the power of the diamet•••• GB, be taken away the power tripled of EH the side of the cube:* 1.1 the power remayning, shall be sesquetertia to the power of AB the side of the Icosahedron. For forasmuch as the centres of inscribed and circum∣scribed figures, are in one & the selfe same poynt, by the ••••rollary of the 21. of the 〈◊〉〈◊〉 the diame∣ters BG and FH shall in one and the selfe same poynt

[illustration]

cutte the one the other into two equall partes: for we haue before by the same corollary taught, that the toppes of equall and like pyramids doo in that poynt concurre, let that poynt be the centre I. Now the an∣gles of the cube, which are at the poyntes F and H are set at the centres of the bases of the Icosahedron by the 11. of the fiuetenth•• Wherefore the line FH shall be perpendicular to both the bases of the Icosahedrō, by the corollary of the assūpt of the 16. of the twelfth. Wherefore the line IB contayneth in power the two lines IH and HB, by the 47. of the first. But the line H∣B, is drawne from the centre of the circle which con∣tayneth the base of the Icosahedron namely, the angle B is placed in the circumference, and the poynt H is the centre. Wh••refore the whole line BG contayneth in power the whole lines FH and the diameter of the circle (namely, the double of the line BH) by the 15•• of the fiueth. But the diameter which is double to the line HB is in power sesquiterti•• to the side of the e∣quilater triangle inscribed in the same circle•• by the corollary of the ••••. of the thirtenth. For it is in pro∣portion to the side•• as the side is to the perpendicular, by the corollary of the 8. of th•• ••ixth. And FH the diameter of the cube, is in power triple to EH the side of the same cube, by the 15•• of the thirtenth. If therefore from the power of the diameter BG, be taken away the power tripled of EH the side of the cube inscribed•• that is•• the power of the line FH: the residue (namely, the power of the diameter of the circle which is duple to the line HB shall be sesquiterti•• to the side of the triangl•• inscribed in that circle: which selfe side is AB the side of the Icosahedron. If therfor•• from the power of the diameter of an Icosahedrō, be takē away the power tripled of the side of the cube inscribed in the Icosahedron, the power remayning shall be s••squitertia ••o the power of the side of the Icosahedron.

A Corollary.

The diameter of the Icosahedron, contayneth in power two lines, namely, the diameter of the cube inscribed, which coupleth the centres of the opposite ba∣ses, and the diameter of the circle which contayneth the base of the Icosahedron. For it was manifest, that BG the diameter contayneth to power the line FH which doupleth the centres, and the double of the line BH, that is, the diameter of the circle contayning the bas•• wherein i•• the centre H••

Notes

• * 1.1

  Demonstra∣tion.