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 6. Proposition. The 6. Probleme. In an Octohedron geuen, to inscribe a trilater equilater Pyramis.

SVppose tha•• the Octohedron where•••• the Tetrahedron is required to be ins••ri••••••, be ABGDEI. Take 〈…〉〈…〉 bases of the Octo••••dron, that is,* 1.1 〈…〉〈…〉 close in the lowe•••• triangle BGD, namely, AE••, HED, IGD: and let the fourth be AIB, which is opposite to the lowest trian∣gle before put, namely, to EGD. And take the centres of those fower bases, which let be the pointes H, C, N, ••. And vpon the triangle HCN erecte a pyramis HCNL. Now ••orasmuch as these two bases of the Octohedron, namely, AGE and ABI are set vpon the right lines EG and BI which are opposite the one to the other•• in the square GEBI of the Octohedron, from the poin•• A dra••e by the centres of the bases, namely, by the centres H, L, perpendicular lines AHF, ALK, cutting the lines EG and BI 〈◊〉〈◊〉 two equall partes in the pointes F, K (by the Corollary of the 1•• of the thirtenth). Wherfore

[illustration]

a right line drawen frō the point F to the point K,* 1.2 shall be a parallel and equall to the sides of the Octohedron, namely, to •••• and GI (by the 33. of the first). And the right line HL which cutteth the 〈…〉〈…〉 AF, AK, proportionally (for AH and AL are drawen from the centres of equall circles to the circumferences) is a pa∣rallel to the right line FK (by the 2. of the sixth) and also to the sides of the Octohedron, name∣ly, to E•• and IG (by the 9. of the eleuenth). Wherefore as the line AF is to the line AH, so is the line FK to the line HL (by the 4. of the sixth): For the triangles AFK and AHL are like (by th•• Corollary of the 2. of the sixth). But the line AF is in sesquialter proportion to the line AH: (for the side EG maketh HF the halfe of the right line AH, by the Corollary of the 12. of the thirtenth). Wherfore FK or GI the side of the Octohedron, is sesquialter to the righ••∣line HL. And by the same reason may we proue that the sides of the Octohedron are sesquialter to the rest of the right lines which make the pyramis HNCI, namely, to the right line•• ••, N, NC, CI, LN, and CH: wherefore those right lines are equall, and therefore the triangle•• which are described of them, namely, the triangles HCN, HNL, NCL, and CHL. which make the pyramis HNCL, are equall and equilater. And forasmuch as the angles of the same pyramis, namely, the angles H, C, N, L, do end in the centres of the bases of the Octohedron, therefore it is inscribed ••o the same Octohedron, by the first definition of this booke. Wherefore in an Octohedron ••euen, is inscribed a tril••ter equila∣•••••• ••••••amis•• which was required to ••e don••.

A Corollary.

The bases of a Pyramis inscribed in an Octohedron, are parallels to the bases of the Octohedron. For forasmuch as the sides of the bases of the Pyramis touching the one the other, are parallels to the sides of the Octohedron which also touch the one the other, as for example, HL was proued to be a parallel to GI, and LC to DI, therefore, by the 15. of the eleuenth, the plaine superficies which is drawen by the lines HL and LC, is a parallel to the plaine superficies drawen by the lines GI and DI. And so likewise of the rest.

Second Corollary.

A right line ioyning together the centres of the opposite bases of the Octo∣hedron, is sesquialter to the perpendicular line drawen from the angle of the in∣scribed

pyramis to the base thereof. For forasmuch as the pyramis and the cube which containeth it do in the selfe same pointes end their angles (by the 1. of this booke): therefore they shall both be inclosed in one and the selfe same Octohedron (by the 4. of this booke). But the diame∣ter of the cube ioyneth together the centres of the opposite bases of the Octohedron, and therefore is the diameter of the Sphere which containeth the cube and the pyramis inscribed in the cube (by the 13. and 14. of the thirtenth): which diameter is sesquialter to the perpendicular which is drawen from the angle of the pyramis to the base thereof: for the line which is drawen from the centre of the sphere to the base of the pyramis, is the sixth part of the diameter (by the 3. Corollary of the 13. of the thir∣tenth). Wherefore of what partes the di••meter containeth sixe, of the same partes the perpendicular containeth fower.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.