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 21. Probleme. The 21. Proposition. In euery one of the regular solides to inscribe a Sphere.

IN the 13. of th•• thirtenth and th•• other 4. propositio•••• following,* 1.1 i•• was declared that ••he •••• regular solides•• ••re so conta••••ed in a sphere, that ••ight lin•••• drawne from the cen∣•••••• o•• the 〈…〉〈…〉 of 〈◊〉〈◊〉 solide in∣scribed, are equall. Which right lines therefore make pyramids, whose ••oppes are the centre of the sphere, or of the solide, and the bas•••• ••••e cu•••••• one of the bases of those solides. And 〈…〉〈…〉 solide ••quall and like the one to the other, and described in equall circles: those cir••les shall cutte the sphere: for the angles which touch the circumference of the circle, touch also the superficies of the sphere. Wherefore perpē∣diculars drawne from the centre of the sphere to the bases, or to the playne superficieces of the equall circles, are equall, by the corollary of the assumpt of the 1••. of the twelfth. Wherefore making the cen∣tre the 〈◊〉〈◊〉 of the sphere which 〈◊〉〈◊〉 the solide, and th•• space some one of the equall perpen∣dicular••, d••scrib•• a sphere, and it shall touch euery one of the bases of 〈◊〉〈◊〉 solide: 〈…〉〈…〉∣perficies of the sphere passe beyond those bases: when as those p••••pe••diculars 〈…〉〈…〉 are drawne from the centre to the bases, by the 3. corollary of the sa•••• ••••••umpt. Wher••fore ••e haue i•• euery one of the regular bodies inscribed a sphere: which regular bo•••••• are in number one i•• 〈◊〉〈◊〉 by the corollary of the 1••. of the 〈◊〉〈◊〉.

A Corollary.

The regular figures inscribed in spheres, and also the spheres circumscri∣bed about them, or contayning them, haue one and the selfe same centre. Namely, their pyramids, the ••ngles of whose b••ses touch the super•••••••••••• of th•• ••••here, doo from those angles cause equall right lines to be draw•••• to one and ••he selfe 〈◊〉〈◊〉 poyn•• making the top•••••• of the pyra∣mid•• in the same poynt: and therefore they 〈…〉〈…〉 th•• c••••tres of the spheres in the selfe same toppes when 〈◊〉〈◊〉 the right lines drawne from those angles to the cro••••ed superficies, wh••rein are 〈◊〉〈◊〉 the an∣gles of the bases of the pyramid••, are equall••

An adue••••••sment of Flussas••

Of these solides, onely the Octohedron receaueth the other solides inscribed one with 〈…〉〈…〉 other. For the Octohedron contayneth the Icosahedron inscribed in it: and the same Icosahedron contayneth the Dodecahedron inscribed in the same Icosahedron: and the same dodecahedron contayneth the cube inscribed in the same Octohedron, and 〈…〉〈…〉 ••••r••••mscribeth the Pyra∣mis inscribed in the sayd Octohedron. But this happ••∣neth not in the other solides.

Notes

• * 1.1

  This propositi∣on Campane hath, & is the last also in or∣der of the 15. booke with him.