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

¶An Assumpt added by Flussas.

If a Sphere be cut of a playne superficies, the common section of the superficieces, shall be the cir∣cumference of a circle.

Suppose that the sphere ABC be cut by the playne superficies AEB, and let the centre of the sphere be the poynt D.* 1.1 And from the poynt D, let there be drawne vnto the playne superficies AEB a perpendicular line (by the 11. of the eleuenth) which let be the line DE. And from the poynt E draw in the playne superficies AEB vnto the common section of the sayd superficies and the sphere, lines how many so euer, namely, EA and EB. And draw these lines DA and DB. Now forasmuch as the right angles DEA and DEB are equall (for the line DE is e∣rected

[illustration]

perpendicularly to the playne superficies).* 1.2 And the right lines DA & DB which subtend those angles are by the 12. defini••ion of the eleuenth equall: which right lines moreouer (by the 47. of the first) do contayne in power the squares of the lines DE, EA, and DE, EB: if therfore from the squares of the lines DE, EA, and DE, EB ye take away the square of the line DE which is cōmon vnto them, the residue namely the squares of the lines EA & EB shall be equal. Wherfore also the lines EA and EB are equall. And by the same reason may we proue, that all the right lines drawne from the poynt E to the line which is the cōmon section of the superficies of sphere and of the playne su∣perficies, are equall. Wherefore that line shall be the cir∣cumference of a circle, by the 15. definition of the first. * 1.3 But if it happen the plaine superficies which cutteth the sphere, to passe by the centre of the sphere, the right lines drawne from the centre of the sphere to their common section, shal•• be equall by the 12. definition of the eleuenth. For that common section is in the superficies of the sphere. Wherefore of neces••itie the playne superficies comprehended vnder that line of the common section shall be a circle, and his centre shall be one and the same with the centre of the sphere.

Iohn Dee.

Euclide hath among the definition of solides omitted certayne, which were easy to conceaue by a kinde of Analogie. As a segment of a sphere, a sector of a sphere, the vertex, or toppe of the segment of a sphere: with such like. But that (if nede be) some farthe•• light may be geuē, in this figure next before,* 1.4 vn∣ders••and a segment of the sphere ABC to be that part of the sphere contayned betwen the circle AB, (whose center is E) and the sphericall superficies AFB. To which (being a lesse segment) adde the cone ADB (whose base is the former circle: and toppe the center of the sphere) and you haue DAFB a sector of a sphere, or solide sector (as I call it). DE extended to F, sheweth the top or vertex of the seg∣ment, to be the poynt F and EF is the altitude of the segment sphericall. Of segmentes, some are grea∣ter thē the halfe sphere, some are lesse. As before ABF is lesse, the remanent, ABC is a segment greater then the halfe sphere.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  The circles so made: or so con∣sidered in the sphere, are cal∣led the greatest circles: All o∣ther, not ha∣uing the center of the sphere, to be their center also•• are called lesse circles.
• * 1.4

  Note these de∣scriptions.