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

A further construction of the dodecahedron after Flussas.

Forasmuch as it hath bene proued that the pentagon BVZCW is equilater and equiangle and toucheth one of the sides of the cube.* 1.1 Let vs show also by what meanes vpon eche of the 12. sides of the cube may in like sort be applyed pentagons ioyning one to the other, and composing the 12. bases of the dodecahedron. Draw in the former figure these right lines AI, ID, IL, ctK. Now forasmuch as the line PL was in the point ct diuided like vnto the lines PH, ON. or OX, and vpon the pointes T, P, ct, were erected perpendicular lines equall vnto the line OY, and the rest: namely, vnto the greater segmēt: and the lines T W and ct I were proued parallels, therefore the lines WI and Tct are parallels, by the 7. of the eleuenth, and 33. of the first. Wherefore also, by the 9. of the eleuenth, the lines WI and DC are parallels. Wherefore by the 7. of the eleuenth CWID is a playne superficies. And the triangle AID is a playne superficies, by the 2. of the eleuenth. Now it is manifest that the right lines ID, & IA are equall to the right line WC. For the right lines AL & ••ct (which are equall to the right lines BH, & HT) do make the subtēded lines Act and BT equall by the 4. of the first. And agayne forasmuch as the lines BT and TW contayne a right angle BTW, as also doo the right lines Act and ctI contayne the right an∣gle ActI (for the right lines WT, and Ict are erected perpendicularly vnto one and the selfe same playne ABCD by supposition). And the squares of the lines BT and TW are equall to the squares of the lines Act, and ctI (for it is proued that the line BT is equall to the line Act, and the line TW to the line ctI). And vnto the squares of the lines BT and TW is equall the square of the line BW, by the 47. of the first: likewise by the same vnto the squares of the lines Act and ctI is equall the square of the line AI. Wherefore the square of the line BW is equall to the square of the line AI, wherefore also the line BT is equall to the line AI. And by the same reason are the lines ID and WC equall to the same lines. Now forasmuch as the lines AI and ID, and the lines AL and LD are equall, and the base IL is common to them both, the angles ALI and DLI shalbe equall, by the 8. of the first: and therefore they are right angles, by the 10. diffinition of the first. And by the same reason are the angles WHB, and W∣HC right angles. And forasmuch as the two lines HT and TW are equall to the two lines Lct and ct∣I, and they contayne equall angles, that is, right angles by supposition, therefore the angles WH∣T, and ILct, are equall by the 4. of the first. Wherefore the playne superficies AID is in like sort incli∣ned to the playne superficies ABCD, as the playne superficies BWC is inclined to the same playne A∣BCD, by the 4. diffinition of the eleuenth. In like sort may we proue that the playne WCDI is in like sort inclined to the playne ABCD, as the playne BVZC is to the playne EBCF. For that in the trian∣gles YOH and ctPK which consist of equall sides (eche to his correspondent side), the angles YHO, and ctKP, which are the angles of the inclination, are equall. And now if the right line ctK be extended to the point a, and the pentagon CWIDa be made perfect, we may, by the same reason, proue that that playne is equiangle and equilater, that we proued the pentagon BVZCW to be equaliter and e∣quiangle. And likewise if the other playnes BWIA and AID be made perfect, they may be proued to be equall and like pentagons and in like sort situate, and they are set vpon these common right lines B∣W, WC, WI, AI, and ID. And obseruing this methode, there shall vpon euery one of the 12. ••id••s of the cube be set euery one of the 12. pentagons which compose the dodecahedron.

Notes

• * 1.1

  Draw in the former figure these lines, ctA, ctL, c••D.