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 7. Proposition. The 7. Probleme. In a dodecahedron geuen, to inscribe an Icosahedron.

SVppose that the dodecahedron geuen, be ABCDE. And let the centres of the circles which cōtayne sixe bases of the same dodecahedron be the polnes L, M, N, P, Q, O.* 1.1 And draw these right lines OL, OM, ON, OP, OQ, and moreouer these right lines LM, MN, NP, PQ, QL. And now forasmuch as equall and equilater pentagons are contay∣ned in equall circles, therefore perpendicular lines drawne from their centres to the sides shall be equall (by the 14. of the third), and shall diuide the sides of the dodecahedron into two e∣quall partes (by the 3. of the same). Wherefore the foresayde perpendicular lines shall co••outre in the point of the section,* 1.2 wherein the sides are diuided

[illustration]

into two equall partes, as LF and MF doo. And they also containe equall angles, namely, the in∣clination of the bases of the dodecahedron, (by the 2. corollary of the 18. of the thirtenth). Wher∣fore the right lines LM, MN, NP, PQ, QL, and the rest of the right lines which ioyne together two centres of the bases, and which subtende the equall angles ••ontayned vnder the sayd equall perpendicular lines, are equall the one to the o∣ther (by the 4. of the first). Wherefore the trian∣gles OLM, OMN, ONP, OPQ, OQL, and the rest of the triangles which are set at the cen∣tres of the pentagons, are equilater and equall. Now forasmuch as the 12. pentagons of a dodeca∣hedron containe 60. plaine superficiall angles, of which 60. eue••y ••hre make one solide angle of the dodecahedron, it followeth that a dodecahedron hath 20. solide angles: but eche of those solide an∣gles is subtēded of ech of the triangles of the Ico∣sahedron, namely, of ech of those triangles which ioyne together the centres of the pentagōs which make the solide angle, as we haue before proued. Wherefore the 20, equall and equilater triangles which subtende the 20. solide angles of the dodecahedron, and haue their sides which are drawne from the centres of the pentagons common, doo make an Icosahedron (by the 25. diffinition of the e∣leuenth): and it is inscribed in the dodecahedron geuen (by the first diffinition of this booke) for that the angles thereof doo all at one time touch the bases of the dodecahedron. Wherefore in a dodecahe∣dron geuen•• i•• inscribed an Icosahedron: which was required to be done.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.