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. If frō the angles of the base of a * 1.1 Pyramis, be drawne to the opposite sides, right lines cutting the sayd sides by an extreme and meane proportion: they shall containe the bise of the Icosahedron inscribed in the Pyramis, which base shalbe inscribed in an equilater triangle, whose angles cut the sides of the base of the Pyramis by an extreme and meane proportion.

* 1.2SVppose that ABG be the base of a pyramis, in which let be inscribed an equilater triangle FKH, which is done by deuiding the sides into two equal partes. And in ••his triangle let there be inscribed the base of the Icosahedrō inscribed in the pyramis: which is described by deuiding the sides FK, KH, HF, by an extreme & meane proportiō in the poynts C, D, E, by the 19. of the fiuetēth. Againe let the sides of the pyramis, namely, AB, BG, and GA be deuided by an extreme and me••ne proportion in the poynts I, M, L, by the 30. of the sixth. And drawe these right lines AM, BL, GI.* 1.3 Then I say that those lines describe the triangle CDE of the Icosahedron. For forasmuch as the lines BG and FH are parallels, by the 2. of the sixth: by the point D let the line ODN be drawne parallel to either of the lines BG & FH. Wherfore the triangle HDN shalbe like to the triangle HKG, by the corollary of the 2. of the sixth. Wherfore either of these lines DN and NH shall be equal to the line DH, the greater segment of the line KH or FH. And forasmuch as the line FO is a parallel to the line HK, and the line OD to the line FH•• the line OD shall be equal to the whole line FH in the pa∣rallelogramme FODH, by the 34. of the

[illustration]

••irst. Wherefore as the whole line FH is to the grea••er segment FE, so shall the lines equal to them be, namely, the line OD to the line DN, by the 7. of the fifth. Wherfore the line ON is deuided by an extreme and meane proportion in the poynt D, by the 2. of the fourtenth. But the triangles AOD, AFE, and ABM, are like the one to the other, and so also are the triangles ADN, AEH, and AMG, by the corollary of the secōd of the sixth•• Wherefore as FE is to EH, so is OD to DN, and BM to MG. Whe••fore the line AM cutting the lines FH and ON, lyke vnto the line BG in the pointes E, D, M, describeth ED the side of the triangle of the Icosahedron ECD, which is descri∣bed in the sections E, C, D, by suppositiō. And by the same r••ason the lines BL and GI shall describe the other sides EC and CD of the same triangle. By the point E, let there be drawne to GI a parallel line PEQ. Now forasmuch as the lines BM and FE are parallels, the line AM is in the poynt E, cut like to the line AB in the poynt F, by the 2. of the sixth. Wherefore the line AE is equal to the line EM: and vnto the line EM also are equal either of the lines GD and DI: which ••re cut l••ke vnto the forsaid lines. Againe forasmuche as in the triangle ADI the lines DI and EP are parallels, as the line DI is to the line EP, so is the line AD to the line AE: but as the line AD is to the line AE, so is the line DG to the line EQ by the 2. of the sixth: wherefore as the line DI is to the line EP, so is the line DG to the line EQ: and alternately as the line DI is to the line DG, so is the line EP to the line EQ: but the lines DI and IG are equal: wherfore also the lines EP and EQ are equal. And forasmuch as the line AH is equal to the line FH, whose greater segmēt is the line HN•• therfore the whole line AN, is deuided by an extreme and meane proportion in the poynt H, by the ••. of the thirtenth. But as the line AN is to the line AH, so is the line AD to the line AE, by the 2. of sixth (for the line•• FH and ON are parallel••:) and againe as the line AD is to the line AE, so (by the same) is the line AG to the line AQ, and the line AI to the line AP: for the lines PQ, and GI are parallels: Wherefore the lines AG and AI are deuided by an extreme and meane proportion in the points Q & P: & the line AQ shalbe the greater segmēt of the line AG or AB. And forasmuch as the whol•• line AG is to the greater segment AQ, as the greater segment AI is to the residue AP: the line A•• shalbe the lesse segment of the whole line A•• or AG. Wherfore the li•••• PEQ (which by the poynt E passeth parallelwise to the line GI) cutteth the lines AG and BA by an extreme and meane proportion in the poynts Q and P. And by the same reason the line ••R (which by the poynt C, passeth parallelwise to the line AM) shall fall vpon the sections P and R: so also shal the line RQ (which by the poynt D passeth parallelwise to the line BL) fall vpo•• the sections RQ. Wherefore either of the lines PE and EQ shalbe equal to the line CD, in the parallelogrammes PD, and QC, by the 34. of the first. And forasmuch as the lines PE and EQ are equal, the lines PC, CR, RD and DQ shalbe likewise equal. Wh••rfore the triangle PRQ i•• ••quilater, and cutteth the sides of the base of the pyrami•• in the poyntes P, Q, R, by an extreme and meane proportion. And in it is inscribed the base ECD of the Ico∣sahedron contained in the for••sayd pyramis. If therefore from the angles of the base of a pyramis, be drawne to the opposite sid••s, right lines cutting the sayde sides by an extreme and meane proportion: they shall containe the base of the Icosahedron inscribed in the pyramis, which base shall be inscribed in an equilater triangle, whose angles cut the sides of the base of the pyramis by an extreme & meane propo••tion.

¶ A Corollary.

The side of an Icosahedron inscribed in an Octohedron, is the greater segment of the line, which being drawen from the angle of the base of the Octohedron cutteth the opposite side by an extreame and meane proportion. For, by the 16. of the fiuetenth, FKH is the base of the Octohedron, which containeth the base of the I∣cosahedron CDE: vnto which triangle FKH, the triangle HKG is equall, as hath bene proued. By the point H draw vnto the line ME a parallel line HT, cutting the line DN in the point S. Wherefore ES, DT, and ET, are parallelogrammes: and therefore the lines EH and MT are equall: and the lines EM and HT are like cut in the pointes D and S, by the 34. of the first. Wherefore the greater segment of the line HT is the line HS, which is equall to ED the side of the Icosahedron. But (by the 2. of the sixth) the line TK is cut like to the line HK by the parallel DM. And therefore (by the 2. of the fourtenth) it is di∣uided by an extreme and meane proportion. But the line TM is equall to the line EH. Wherefore also the line TK is equall to the line EF or DH. Wherefore the residues EH and TG are equall. For the whole lines FH and KG are equall. Wherefore KG the side of the triangle HKG is in the point T diui∣ded by an extreme and meane proportion in the point T, by the right line HT, and the greater segment thereof is the line ED the side of the Icosahedron inscribed in the Octohedron, whose base is the trian∣gle HKG (or the triangle FKH which is equall to the triangle HKG) by the 16. of the fiuetenth.

Notes

• * 1.1

  By a Pyra∣mis vnder∣stand a Te∣trahedron throughout all this booke.
• * 1.2

  Construction.
• * 1.3

  Demonstra∣tion.