¶The 12. Proposition. * 1.1If a cube be contayned in a sphere: the square of the diameter doubled, is e∣quall to all the superficieces of the cube taken together. And a perpendicu∣lar line drawne from the centre of the sphere to any base of the cube, is e∣quall to halfe the side of the cube.
* 1.2FOr forasmuch as (by the 15. of the thirtenth) the diameter of the sphere is in power triple to the side of th•• cube: therefore the square of the diameter doubled is sex∣tuple to the base of the same cube. But the sextuple of the power of one of the sides contayneth the whole superficies of the cube 〈◊〉〈◊〉 or the cube is composed of sixe square superficieces (by the 2••. diffinition of th•• eleuenth) whose sides therefore are equall: wherefore the square of the di••meter ••oubled is equall to the whole super∣ficies of the cube. And forasmuch as the diameter of the cube, and the line which falleth perpendicularly vpō the opposite bases of the cube,* 1.3 do cut the one the other into two equall partes in the centre of the sphere which containeth the cube (by the 2. corollary of the 15. of the thirtenth) and the whole right line which coupleth the centres of the opposite bases, is equall to the side of the cube by the 33. of the first, for it coupleth the equall and parallel semidiameters of the bases: therefore the halfe thereof shall be equall to the halfe of the side of the cube by the 15. of the fifth. If therefore a cube be contayned in a sphere: the square of the diameter doubled is equall to all the superficieces of the cube taken together. And a perpendicular line drawne from the centre of the sphere to any base of the cube, is equall to halfe the side of the cube: which was required to be prou••d.
¶ A Corollary.
If two thirds of the power of the diameter of the sphere be multiplyed into the perpendicular line equall to halfe the side of the cube,* 1.4 there shall be produced a solide equall to the solide of the cube. For it is before manifest that two third partes of the power of the diameter of the sphere are equall to two bases of the cube. If therefore vnto eche of those two thirds be applyed halfe the altitude of the cube, they shall make eche of those solides equall to halfe of the cube, by the 31. of the eleuenth: for they haue equall bases. Wherefore two of those solides are equall to the whole cube.
You shall vnderstand (gentle reader) that Campane in his 14. booke of Euclides Ele∣mentes hath 18. propositiōs with diuers corollaries following of them. Some of which propositions and corollaries I haue before in the twelfth and thirtenth bookes added out of Flussas as corollaries (which thing also I haue noted on the side of those corol∣laries, namely, with what proposition or corollary of Campanes 14. booke they doo agree). The rest of his 18. propositions and corollaries are contained in the twelue for∣mer propositions and corollaries of this 14. booke after Flussas: where ye may see on the side of eche proposition and corollary with what proposition and corollary of Campanes they agree. But the eight propositions following together with their corol∣laries, Flussas hath added of him selfe, as he him selfe affirmeth.