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 Probleme. 9.

A Sphere being geuen, to geue an other Sphere, to whose Sphericall superficies, the superficies Sphe∣ricall of the Sphere geuen shall haue any proportion, betwene two right lines geuen.

Suppose A, to be a sphere geuen, and the proportion geuen, to be that, which is betwene the right lines X and Y. I say that a sphere is to be g••uen to whose sphericall superficies, the superficies sphericall of A, shall haue that proportion which X hath to Y. Let the greatest circle, conteyned in A the sphere be the circle BCD.* 1.1 And by the probleme of my additions, vpon the second proposition of this booke, as X is to Y, so let the circle BCD be to an other circle found, let that other circle be EFG: and his dia∣meter EG. I say that

[illustration]

the sphericall superfici∣es of the sphere A, hath to the sphericall super∣ficies of the sphere, whose greatest circle is EFG, (or his equall) that proportion, which X hath to Y.* 1.2 For (by con¦struction) BCD is to E∣FG, as X is to Y: and by the theoreme next be∣fore•• as BCD is to ••F∣G, so is the spherical su¦perficies of A (whose greatest circle is BCD, by supposition) to the sphericall superficies, of the sphere, whose grea∣test circle is EFG: wherefore, by the 11. of the fifth as X is to Y: So is the sphericall su∣perficies of A, to the sphericall superficies of the sphere, whose greatest circle is EFG: wherefore, the sphere whose diame∣ter is EG, (the diameter also of EFG) is the sphere, to whose sphericall superficies, the sphericall su∣perficies of the sphere A, hath that proportion which X hath to Y. A sphere being geuen therefore, we haue geuen an other sphere, to whose sphericall superficies, the superficies sphericall of the sphere geu•• hath any proportion geuen, betwene two right lines: which ought to be done.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.