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 Theoreme. 7.

To any segment, or portion of a Sphere, that cone

[illustration]

i•• equall, which hath that circle to his base, which is the b••se of the segmēt, and heith, a right line, which vnto the heith of the segmēt hath that proportiō, which the semidiameter of the Sphere together with the heith of the other segment remay••••••g hath to the heith of the same other segment remayn••ng.

This is well demonstrated by Archi••••des & there∣fore nedeth no inuention of myne,* 1.1 to confirme the same: and for that the sayd demonstration is ouer long here to be added, I will refere you thether for the de∣monstration: and here supply that which to Archimedes demonstration shall geue light, and to your farther spe∣culation and practise, shal be a great ayde and direction. Suppose K to be a sphere: & the greatest circle K in cō∣teyned, let be ABCE, and his diameter BE, & cēter D. Let the sphere K, be cutte by a playne superficies, per∣p••ndicularly erected vpon the sayd greatest circle AB∣CE: let the section be the circle about AC: And let the

segmentes of the sphere be the one that wherein is A∣BC,

[illustration]

whose ••oppe is B•• and the other let be that where∣in is AEC and his toppe, let be E: I say that a cone which hath his base the circle about AC, & held a line which to BF (the heith of the segment, whose toppe is B,) hath that proportion that a line compo••ed of DE, the semidiameter of the sphere, and EF (the heith of the other remayning segment, whose toppe is E) hath to EF, (the heith of that other segment remayning), is equall to the segment of the sphere K, whose toppe is B. To make this cone, take my easy order thus. Frame your worke for the find••ng of the fourth proportionall line•• by making EF the first and a line composed of D∣E and EF, the second•• and the third, let be BF: then by the 12. of the six••h, let the fourth proportionall line be found: which let be FG•• vpon F the center of the base of the segment, whose toppe is B, erect a line perpendi∣cular equall to FG found and drawe the lines GA and GC: and so make perfect the cone GAC. I say, that the cone GAC, is equall to the segment (of the sphere K) whose toppe is B. In like maner, for the other segmēt whose toppe is E, to finde the heith due for a cone equal to it: by the order of the Theoreme you must thus frame your lines: let the first be BF: the second DB and BF, composed in one right line, and the third must be EF: where by the 12. of the sixth, finding the fourth, it shall be the heith, to rere vpon the base, (the circle a∣bout AC,) to make an vpright cone, equall to the seg∣ment, whose toppe is E.

Notes

• * 1.1

  Lib 2 prop 2. de Sphe••a & Cylindr••.