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 Corollary.

Now also I say that besides the fiue foresayd solides there can not be described any other solide cōprehēded vnder figures equilater & equiangle the one to the other.* 1.1 For of two trian∣gles, or of any two other playne superficieces can not be made a solide angle (for, that is cōtrary to the diffinition of a solide angle). Vnder three triangles is contayned the solide angle of a pyramis: vnder fower, the solide angle of an octohedrō: vnder fiue, the solide angle of an Ico∣sahedrō: of sixe, equilater & equiangle triangles set to one point can not be made a solide an∣gle. For forasmuch as the angle of an equilater triangle is two third partes of a right angle, the sixe angles of the solide shalbe equall to fower right angles, which is impossible. For euery solide angle is (by the 21. of the eleuēth) contayned vnder playne angles lesse thē fower right angles. And by the same reason can not be made a solide angle contained vnder more thē sixe playne superficiall angles of equilater triangles. Vnder three squares is contained the angle of a cube. Vnder fower squares it is impossible that a solide angle should be contayned: for then

agayne it should be contayned vnder fower right angles. Wherefore much lesse can any solide angle be contayned vnder more squares then fower. Vnder three equilater and equiangle pen∣tagons is contayned the solide angle of a dodecahedron. But vnder fower it is impossible. For forasmuch as the angle of a pentagon is a right angle and the fift part more of a right angle, the fower angles shalbe greater then fower right angles: which is impossible. And ther∣fore much lesse can a solide angle be composed of more pentagons then fower. Neither can a solide angle be contayned vnder any other equilater and equiangle figures of many angles, for that that also should be absurd. For the more the sides increase, the greater are the angles which they contayne, and therfore the farther of are the superficiall angles contayned of those sides from composing of a solide angle. Wherefore besides the foresayd fiue figures there can not be made any solide figure contayned vnder equall sides and equall angles: which was re∣quired to be proued.

Notes

• * 1.1

  That there can be no other so∣lide besids these fiue, contayned vnder equilater and equiangle bases.