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

Note. 2.

Secondly, the whole quan∣titie, of the Sphere A,* 1.1 being cō∣tayned in the rectangle paralle∣lipipedon TN, you may easilie transforme the same quantitie, into other parallelipipedons rectangles, of what height, and of what parallelogramme base you list: by my first and second Problemes vpon the 34. of this booke. And the like may you do, to any assigned part of the Sphere A: by the like meanes deuiding the parallelipipedon TN: as the part assigned doth require. As if a third, fourth, fifth, or sixth, part of the Sphere A, were to be had in a parallelipipedon, of any parallelogra••••e base assigned, or of any heith assigned: then deuiding TP, in∣to so many partes (as into 4. if a fourth part be, to be transformed: or into fiue, if a fifth part, be to be transformed &c.) and then proceede, ••s you did with cutting of TN, from TX. And that I say of paral∣lelipipedons, may in like sort (by my ••••yd two problemes, added to the 34. of this booke) be done in any sided columnes, pyramids, and prisme••: so th•••• in pyramids and some prismes you vse the cautions ne∣cessary, in respect of their quan〈…〉〈…〉odyes hauing parallel, equall, and opposite bases: whose partes 〈…〉〈…〉re in their propositions, is by Eu∣clide demonstrated. And finally, 〈…〉〈…〉 additions, you haue the wayes and orders how to geue to a Sphere, or any segme•••• o•• the same, Cones, or Cylinders equall, or in any proportion be∣twene two right lines, geuen: with many other most necessary speculations and practises about the Sphere. I trust that I haue sufficiently ••raughted your imagination, for your honest and profitable stu∣die herein, and also geuen you rea••••, ••••tter, whe•••• with to s••••p the mouthes of the malycious, igno∣rant,

and arrogant, despisers of the most excellent discourses, trauayles, and inuentions mathematicall. Sting aswel the heauenly spheres, & sterres their sphericall soliditie,* 1.2 with their conue••e spherical super∣ficies, to the earth at all times respecting, and their distances from the earth, as also the whole earthly Sphere and globe it selfe, and infinite other cases, concerning Spheres or globes, may hereby with as much ease and certainety be determined of, as of the quantitie of any bowle, ball, or bullet, which we may gripe in our handes (reason, and experience, being our witnesses): and without these aydes, such thinges of importance neuer hable of vs, certainely to be knowne, or attayned vnto.

Here ende M. Iohn Dee his additions vpon the last proposition of the twelfth booke.

Notes

• * 1.1

  To a Sphere, or to any part of a Sphere assig∣ned: as a third, fourth, fifth &c to geue a paral∣lelipipedon e∣quall.

  Sided Columes Pyramids, and prismes to be geuen equall to a Sphere, or to any certayne part thereof.

  To a Sphere or any segment, or sector of the same, to geue a cone or cylinder equall or in any proportion as∣signed.
• * 1.2

  Farther vse of Sphericall Geometrie.