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. 4.

To a Sphere geuen, to make a cylinder equall, or in any proportion geuen betwene two right lines.

Suppose the geuen Sphere to be A: and the proportion geuen to be that betwene X and Y. I say that a cylinder is to be made, equall to A•• or els in the same proportion to A, that is betwene X to Y. Let a cylinder be made (such one as the Theoreme next before supposed) that shall haue his base equall to the greatest circle in A,* 1.1 and height equall to the diameter of A: Let that cylinder b•• the vpright cy∣linder BC. Le•• the one side of BC, be the right line QC. Deuide QC into three equal part•• of which, let QE containe two, and let the third part be CE. By the point E suppose a plaine (parallel to the bases of BC) to passe through the cylinder BC, cutting the same by the circle DE. I say that the cy∣linder BE is equall to the Sphere A. For seing BC, being an vpright cylinder, is cu•• by a plaine, pa∣rallel to his bases, by construction: therefore as the cylinder DC, is to the cylinder BE, so is the axe of DC, to the axe of BE, by the 13. of this twelfth.* 1.2 Wherefore as the axe is to the axe, so is cylinder to cylinder. But axe is to axe, as side to side, namely, CE to QE, because the axe is parallel to any side

of an vpright cylinder: by the definition of a cylin∣der.

[illustration]

And the circle of the section, is parallel to the bases, by construction. Wherefore in the parallelo∣gramme (made of the axe, and of two semidiame∣ters, on one side parallels, one to the other, being coupled together by a line drawen betwene their endes in their circumferences, which line is the side QC) it is euident, that the axe of BC is cut in like proportion, that the side QC is cut. Wherfore the cylinder DC, is to the cylinder BE, as EC is to QE. Wherefore, by composition, the cylinders DC and BE, that is, whole BC, are to the cylinder BE, as CE and QE (the whole right line QC) are to QE. But by cōstruction, QC is of 3. such partes, as QE containeth 2. Wherefore the cylinder BC, is of 3. such partes, as BE contayneth 2. Wherefore BC the cylinder, is to BE, as 3. to 2: which is sesquialtera proportion. But by the former Theoreme, BC is sesquialtera to the Sphere A: Wherefore, by the 7. of the fift, BE is equall to A. Therefore to a Sphere geuen, we haue made a cylinder equall.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.