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. 2.

A Sphere being geuen, and a circle, to re••re an vpright Cone, vpon that circle (as a base) equall to the Sphere geuen: or in any proportion betwene two right lines assigned.

Suppose the Sphere geuen, to be Q: and the circle geuen to be C. By the first Probleme make an vpright cone equall to Q the Sphere geuen: which cone suppose to be A•• and (by the 2. Probleme of my additions vpon the second of this twelfth booke) as C the circle ••euen, is to the base of A, so let

the height of A, be to a line found: which let be D.

[illustration]

Then it is euident, that the cone, which hath for his base C, the circle geuen, and height the line D, last found, shall be equall to Q the Sphere geuen: which cone let be F. For, by construction, F hath his base and height in reciprokall proportion with the cone A, made equall to Q the Sphere geuen: Whe••fore by the 15. of this twelfth, and 7. of the fifth, this vp∣right cone F, reared vpon C, the circle geuen, is e∣quall to Q, the Sphere geuen: which thing the Probleme first required.

And the second part of this Probleme is thus per∣formed. Suppose the proportion geuen to be that which is betwene X & Y.* 1.1 Then, as X is to Y, so let an other right line found, be to the h••ight of F: which line let be G. For this G, the found height (by con∣struction) being to the height of F, as X is to Y, doth cause this cone (which let be M) vpon C, the circle geuē (or an other to it equall) duely reared, to be vn∣to the cone F, as X is to Y, by the 14. of this twelfth. But F is proued equall to the Sphere geuen: Wher∣fo••e M, is to the Sphere geuen, as X is to Y. And M, is ••eared vpon the circle geuen: or his equall. Wher∣fo••e, a Sphere being geuen, & a circle, we haue rea∣red an vp••ight cone, vpon that geuen circle (as a base) equall to the Sphere geuen: or in any propor∣tion, betwene two right lines assigned: which was required to be done.

Notes

• * 1.1

  The second part of the Probleme.