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

This being proued, now let there be drawne a Circle comprehending both the Pentagon of a Dodecahedron, and the triangle of an Icosahedron, being both described in one and the selfe same Sphere.

* 1.1LEt the circle be ABC. And in it describe as before, two sides of an equilater pentagon, namely BA and AC•• and draw a right line from the point B to the point C: and take the centre of the circle and let the same be E. And from the point A to the point E draw a right line AE: and extend the line AE to the point F. And let it cut the line BC in the point K. And let the line AE be do∣ble to the line EG, & let the line CK be treble to the line CH, by the .9. of the sixth. And frō the point G raise vp (by the .11. of the first) vnto the line AF a perpendicular line GM: and extend the line GM directly to the point D. Wherfore the line MD is the side of an equi∣liter triāgle, by the corollary of the .1••. of the thirtenth: draw these right lines AD and AM. Wherfore ADM is an equilater triangle.* 1.2 And for as much as that which is contained vn∣der the lines AG and BH is equal to the pentagon (by the former assump••) and that which

is cōtained vnder the lines AG and GD is equal

[illustration]

to the triangle ADM: therefore as that which is contained vnder the lines AG and HB is to that which is contained vnder the lines AG and GD, so is the pentagon to the triāgle. But as that which is contained vnder the lines BH & AG is to that which is contained vnder the lines AG and GD, so is the line BH to the line DG (by the .1. of the sixth) wherefore (by the .15. of the fifth) as 12. such lines as BH is, are to .20 such lines as DG is, so are 12. pentagons to 20. triangles, that is the su∣perficies of the Dodecahedron, to the superficies of the Icosahedron. And 12. suche lines as BH is, are equall to tenne suche lines as BC is (for the line HB is quintuple to the line HC): and the line BC is sextuple to the line CH•• Wherfore six such lines as BH is, are equal to fiue such lines as BC are: and in the same pro∣portion are their doubles: and 20. such lines as the line DG is, are equal to .10. such lines as the line DM is: for the line DM is double to the line DG. Wherfore as 10. such lines as BC is, are to 10. such lines as DM is, that is, as the line BC is to the line DM, so is the superficies of the Dodecahedron to the superficies of the Icosahedron. But the line BC is the side of the cube, and the line DM the side of the Icosahedron: wherefore (by the 11. of the fifth) as the superficies of the Dodecahedron is to the superficies of the Icosahedron, so is the line BC to the line DM, that is, the side of the cube to the side of the Icosahedron.

Notes

• * 1.1

  Construction pertaining to the second d••∣monstration of the 4. propositiō.
• * 1.2

  Second demon∣stration o•• the 4. proposition.