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 4. Proposition. The 4. Probleme. In an Octohedron geuen, to describe a Cube.

SVppose that the Octohedron geuen be ABGDEZ. And let the two pyramids thereof be ABGDE, and ZBGDE.* 1.1 And take the centres of the triangles of the pyramis ABGDE, that is, take the centres of the circles which containe those triangles: and let those centres be the point••s T, I, K, L. And by these centres let there be drawen parall••l lines ••o the sides of the

[illustration]

square BGDE: which parallel ••ig•••• lin•••• let be MTN, NLX, XKO, & OIM.* 1.2 And forasmuch as th••se parallel right lines do (by the 2. of the sixth) cut the equall right lines AB, AG, AD, and AE, proportionally, therfore they concurre in the pointes M, N, X, O. Wherefore the right lines MN, NX, XO, and OM, which subtend equall an∣gles set at the point A, & contained vnd••r ••quall right lines, are equall (by the 4. of the first). And moreouer, seing that they are parallels vnto the lines BG, GD, DE, E••, which make a square, therefore MNXO is also a square, by the 10. of the eleuenth. Wherefore also, by the 15. of the ••ame, the square MNXO is parallel to the squar•• BGDE. For all t••e right lines touch the one the other in the pointes of their sections. From the centres T, I, K, L, drawe these right lines TI, IK, KL, LT•• And drawe the right line AIC. And forasmuch as I is the centre of the equilater triangle ABE, therefore the right line AI being extended, cutteth the right line BE into two equall partes (by the Corollary of the 12. of the thirtenth). And forasmuch as MO is a parallel to BE, therefore the triangle AIO is like to the whole triangle ACE (by the Corollary of the 2. of the sixth). And the right line MO is diuided into two equall partes in the point I (by the 4. of the sixth). And by the same reason may we proue, that the right lines MN, NX, XO, are diuided into two equall

partes in the pointes T, L, K. Wherefore also againe, the bases TI, IK, KL, LT, which sub∣tend the angles set at the pointes M, O, X, N, which angles are right angles, and are contained vnder equall sides, those bases, I say, are equall. And forasmuch as TIM is an Isosceles tri∣angle, therefore the angles set at the base, namely, the angles MTI and MIT, are equal (by the •••• of the first). But the angle M is a right angle: wherefore eche of the angles MIT and MTI, is the halfe of a right angle. And by the same reason the angles OIK & OKI, are equall. Wherefore the angle remayning, namely, TIK,

[illustration]

is a right angle (by the 13. of the first). For the right lines TI and IK are set vpon the line MO. And by the same rea∣son may the rest of the angles, namely, IKL, KLT, LTI, be proued right angles, and they are in one and the self same plaine superficies, namely, MNXO (by the 7. of the eleuēth). Wherefore the right lines which ioyne together the centres of the plaine superficiall triangles which make the solide angle A, do make the square ITKL. And by the same reason may be proued, that the plaine superficiall triangles of the rest of the fiue solide angles of the Octohedron set at the pointes B, G, Z, D, E, do in the centres of their bases receaue squares, So that there are in number sixe squares, for euery Octohe∣dron hath sixe solide angles: and those squares are equall: for their sides do containe equall angles of inclinations con∣tained vnder equall sides, namely, vnder those sides which are drawen from the centre to the side of the equall triangles (by the 2. Corollary of the 18. of the thirtenth). Wherefore ITKLRPVS is a cube (by the 21. definition of the ele∣uenth) and hath his angles in the centres of the bases of the Octohedron, and therefore is inscribed in it (by the first de∣finition of this booke). Wherefore in an Octohedron geuen is described a cube: which was required to be done.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.