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. 12.

To cut a Sphere geuen, into two such segmentes, that the Sphericall superficieces of the segment•••• shall haue one to the other, any proportion geuen between two right lines.

Suppose F to be a Sphere geuen: and the proportion geuen, to be that, which is betwene GH and HI. I say, that the Sphere F, is to be cut into two such segmentes, that the Sphericall superficies of those segmentes, shall haue that proportion, one to the other, which, the right line GH, hath to the right line HI.* 1.1 Suppose ABCE to be a greatest circle, in the Sphere F, contained: and his diameter, to be AB. Deuide ABinto two such partes, as GI is di∣uided

[illustration]

into, in the point H (by the 10. of the sixt) Let those partes be AD, and DB. So that, as GH is to HI, so is AD to DB. By the point D, let a plaine superficies passe, cutting the Sphere F, and the diameter AB: So, that vnto that cutting plaine, the diameter AB, be perpendi∣cular: and the Sphere also thereby by deuided into two seg∣mentes, whose cōmon base suppose to be the circle CE, hauing the center, the point D: and the toppe of the one to be the point A, and the toppe of the other to be the point B: and the segmentes them selues, to be noted by EAC, and EBC: Drawe from the two toppes, A and B, to C (a point in the circumference of their common base) two right lines AC and BC. I say now, that the Sphericall superficies of the segment EAC, hath to the Sphericall superficies of the segment EBC, the same proportion,* 1.2 which GH hath to HI. For, forasmuch as circles haue that proportion, one to the other, that the squares of their diameters haue one to the other (by the 2. of this twelfth).* 1.3 And the squares of theyr semidi••me∣ters, have the same proportion one to the other, which the squares of theyr diameters haue. [For like partes haue

that proportion one to the other, that the whole magnitudes•• whose like partes they are, haue the one to the other: by the 15. of the fift.

But the square of euery diameter is quadruple to the the square of his semidiameter: as hath often before•• bene proued: therefore, circle haue•• one to an other, that proportion, that the squares of their semidiameters haue one to the other].

Wherefore, seing AC and BC are semidiameters of two circles, whereof eche is equall to the Sphericall superficies of the seg∣mentes, betwene whose toppes and circumference of their base, they are drawen: by the 4. Theoreme of these additions: it followeth that both those circles whose semidiameters they are: and also those Sphericall superficieces, which are equall to those circles, haue the one to the other, the same propor∣tion, which the square of AC hath to the square of BC. But AC is drawen betwene the circumference of the base, and toppe of the segment Sphericall, EAC, by construction: and likewise BC is drawen betwene the 〈◊〉〈◊〉 and the 〈…〉〈…〉 of the base Sphericall segments ••BC, by construction: Wherefore the Sphericall superficies of the segment EAC, is to the Sphericall superficies of the seg∣ment EBC, as the square of AC is to the square of BC. But the square of AC is to the square of BC, as AD is to DB: by the Corollary of the Probleme of my additions vpon the second of this twelfth: And AD is to DB, as GH is to HI: by construction. Wherefore the Sphericall superficies of the seg∣ment EAC, is to the Sphericall superficies of the segment EBC, as GH is to HI. We haue therfore, cut the Sphere geuen, into two such segmentes, that the Sphericall superficieces of the segmentes, haue one to the other any proportion geuen betwene two right lines: which was to be done.

¶ A Corollary. 1.

Here it app••areth demonstrated, that, circles are one to the other, as the squares of their semidi∣ameter are, one to the other.

Wherby (as occasion shall serue) you may, by force of the former argument, vse other like partes of the diameter, as well as halues.

¶ A Corollary. 2.

It is also euident, that the Sphericall superficieces of the two segment••s of any Sphere, to whose common base, the diameter (passing to their two topp••s) is perpendicular, haue that proportion the o••e to the other, that the portions of the sayd diameter•• haue the one to the other: that superficies and that portion of the diameter on the one side of the common base, being compared to that superficies, and that portion of the diameter•• on the other side of the common base.

¶ A Corollary. 3.

It likewise euidently followeth, that the two Sphericall superficieces of two segmentes of a Sphere: which two segmentes are equall to the Sphere, are in that proportion the one to the other, that their axes (perpendicularly erected to their bases) are in, one to the other: where soeuer in the Sphere those seg∣mentes be taken.

I say that the Sphericall superficies of the segment

[illustration]

CAE, and the Sphericall superficies of the segment FGH, hauing their axes AD and GI (perpendicular to their bases): are in proportion one to the other, as AD is to GI: if the segment of the Sphere contai∣ning CAE with (the segment of the same Sphere) FGH be equall to the whole Sphere. For seing the diameter (of axe) AD, extended to the other pole or toppe, opposite to A (which opposite toppe, let be Q) doth make with the segment CAE, the comple∣ment of the whole Sphere:* 1.4 and by supposition, the segment FGH, whith the segment CAE, are equall to the whole Sphere: Wherefore from equall, taking CAE (the segment common) remayneth the seg∣ment CQE, equall to the segment FGH. And ther∣by, Axe, Base, Solitie, and superficies Sphericall of the segment FGH, must (of ncessitie) be equall to

the Axe, Base, Soliditie, and superficies Sphericall of the segment CQE: Wherefore, by the second Corollary here, and the 7. of the fift, our conclusion is inferred, the superficies Sphericall, of the seg∣ment CAE, to be, to the superficies Sphericall of the segment FGH, as AD is to GI.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstratiō
• * 1.3

  This in maner of a Lemm•• is presently pro∣ued.
• * 1.4

  Note here of Axe, base, & soliditie, more then I nede to bring any far¦ther proofe for.