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

The 10. Theoreme. The 10. Proposition. Euery cone is the third part of a cilinder, hauing one and the selfe same base and one and the selfe same altitude with it.

SVppose that there be a cone hauing to his base the circle ABCD; and let there be a cilinder hauing the selfe same base, and also the same altitude that the cone hath. Then I say that the cone is the third part of the cilinder, that is, that the cilinder is in treble proportion to the cone. For if the cilinder be not in treble proportion to the cone, then the cilinder is either in greater proportions then triple to the cone, or els in lesse. First let it be in greater then triple.* 1.1 And describe (by the 6. of the fourth) in the circle ABCD a [ 1] square ABCD. Now the square ABCD,

[illustration]

is greater then the halfe of the circle ABC∣D. For if about the circle ABCD, we de∣scribe a square, the square described in the circle ABCD is the halfe of the square described about the circle. And let there be Parallelipipedon prismes described vpon those squares,* 1.2 equall in altitude with the cilinder. But prismes are in that proporti∣on the one to the other, that their bases are (by the 32. of the eleuenth, and 5. Co∣rollary of the 7. of this booke). Wherefore the prisme described vpon the square A∣BCD is the halfe of the prisme described vpon the square that is described about the circle. Now the clinder is lesse then the

prisme which is made of the square described abou•• the circle ABCD, being equal in altitude with it, for it contayneth it. Wherfore the prisme described vpon the square ABC∣D and being equall in altitude with the cylinder, is greater then half the cylinder. Deuide (by the 30. of the third) the circumferences AB, BC, CD and DA into two equall parts in the points E, F, G, H, And draw these right lines AE, EB, BF, FC, CG, GD, DH & HA. Wherfore euery one of these triangles AEB, BFC, CGD and DHA is greater then halfe of that segment of the circle ABCD which is described about it, as we haue before in the 2. proposition declared. Describe vpon euery one of these triangles AEB, BFC, CGD, and DHA a prisme of equall altitude with the cylinder. Wherefore euery one of these prismes so described is greater then the halfe part of the segment of the cylinder that is set vpon the sayd segments of the circle. For if by the pointes E, F, G, H, be drawen parallell lines to the lines AB, BC, CD and DA, and then be made perfect the parallelogrammes made by those parallell lines, and moreouer vpon those parallelogrāmes be erected parallelipipedons equall in altitude with the cylinder, the prismes which are described vpon eche of the triangles AEB, BFC, CGD, and DHA are the halfes of euery one of those parallelipipedons. And the segments of the cylinder are lesse then those parallelipipedons so described. Wherefore also euery one of the prismes which are described vpon the triangles AEB, BFC, CGD and DHA is greater then the halfe of the segment of the cylinder set vpon the sayd segment. Now therefore deuiding euery one of the circumferences remaining into two equall partes, and drawing right lines, and raysing vp vpon euery one of these triangles prismes equall in altitude with the cylinder, and doing this continually, we shall at the length (by the first of the tenth) leaue certaine segments of the cylinder which shalbe lesse then the excesse whereby the cylinder excedeth the cone more then thrise. Let those segments be AE, EB, BF, FC, CG, GD, DH and HA. Wherfore the prisme remayning, whose base is the poligonon ••igure AEBFCGDH, and altitude the selfe

[illustration]

same that the cylinder hath, is greater then the cone taken three tymes.* 1.3 But the prisme whose base is the poligonon figure AEBFCGDH and altitude the selfe same that the cylinder hath, is treble to the pyramis whose base is the poligonon figure AEBFCGDA and altitude the selfe same that the cone hath, by the corollary of the 3. of this booke. Wherfore also the pyramis whose base is the poligonon figure AEBFCGDH and toppe the self same that the cone hath, is greater then the cone which hath to his base the circle ABCD. But it is also lesse, for it is contayned of it which is impossible. Wherefore the cylinder is not in greater proportion then triple to the cone.

I say moreouer that the cylinder is not in lesse proportion then triple to the cone•• For if it be possible let the cylinder be in lesse proportion then triple to the cone. Wherefore by con∣uersion, the cone is greater then the third part of the cylinder. Describe now (by the sixth of the fourth) in the circle ABCD a square ABCD. Wherefore the square ABCD is grea∣ter then the halfe of the circle ABCD vpon the square ABCD describe a pyramis hauing one & the selfe same altitude with the cone. Wherfore the pyramis so described is greater thē halfe of the cone. (For if as we haue before declared we describe a square about the circle, the square ABCD is the halfe of the square described about the circle, and if vppon the squares be described parallelipipedons equall in altitude with the cone, which solides are also

called prismes, the prisme or parallelipipedon described vpō the square ABCD is the halfe of the prisme which is described vpō the square described about the circle, for they are the one to the other in that proportiō that their bases are (by the 32. of the eleuēth, & 5. corollary of the 7. of this booke.) Wherfore also their third parts are in the self same proportion (by the 15. of the fift). Wherfore the pyramis whose base is the square ABCD is the halfe of the pyramis set vpon the square described about the circle. But the pyramis set vpon the square described a∣bout the circle is greater then the cone whome it comprehendeth. Wherfore the pyramis whose base is the square ABCD, and altitude the self same that the cone hath; is greater then the halfe of the cone.) Deuide (by the 30. of the third) euery one of the circumferences AB, BC, CD, and DA into two equall partes in the pointes E, F, G, and H: and drawe these right lines AE, EB, BF, FC, CG, GD, DH, and HA. Wherefore euery one of these triangles AEB, BFC, CGD, and DHA is greater then the halfe part of the segment of the circle described about it. Vppon euery one of these triangles AEB, BFC, CGD, and DHA de∣scribe a pyramis of equall altitude with the cone and after the same maner euery one of those pyramids so described is greater then the halfe part of the segment of the cone set vpon the segment of the circle. Now therefore diuiding (by the 30, of the third) the circumferences re∣maining into two equall parts, & drawing right lines & raysing vp vpon euery one of those triangles a pyramis of equall altitude with the cone, and doing this continually, we shal at the length (by the first of the tenth) leaue certayne segmentes of the cone, which shalbe lesse then the excesse whereby the cone excedeth the third part of the cylinder. Let those segmentes be AE, EB, BF, FC, CG, GD, DH, and HA. Wherefore the pyramis remayning, whose base is the poligonō figure AEBFCGDH and altitude the self same with the cone, is grea∣ter then the third part of the cylinder. But the pyramis whose base is the poligonon figure AEBFCGDH and altitude the self same with the cone, is the third part of the prisme whose base is the poligonō figure AEBFCGDH and altitude the self same with the cylin∣der. Whefore * 1.4 the prisme whose base is the poligonon figure AEBFCGDH, and altitude the self same with the cylinder, is greater then the cylinder whose base is the circle ABCD. But it is also lesse, for it is contayned of it, which is impossible. Wherfore the cylinder is not in lesse proportion to the cone then in treble proportion. And it is proued that it is not in grea∣ter proportion to the cone then in treble proportion, wherefore the cone is the third part of the cylinder. Wherfore euery cone is the third part of a cylinder, hauing one & the self same base, and one and the selfe same altitude with it: which was required to be demonstrated.

¶ Added by M. Iohn Dee.

¶ A Theoreme. 1.

The superficies of euery vpright Cylinder, except his bases, is equall to that circle whose semidia∣meter is middell proportionall betwene the side of the Cylinder, and the diameter of his base.

¶ A Theoreme. 2.

The superficies of euery vpright, or Isosceles Cone, except the base, is equall to that circle, whose semidiameter is middell proportionall betwene the side of that Cone, and the semidiameter of the cir∣cle: which is the base of the Cone.

My entent in additions is not to amend Euclide•• Method, (which nedeth little adding or none at all). But my desire is somwhat to furnish you, toward a more general art Mathematical thē Euclides Elemēts,* 1.5 (remayning in the termes in which they are written) can sufficiently helpe you vnto. And though Eu∣clides Elementes with my Additions, run not in one Methodicall race toward my marke: yet in the meane space my Additions either geue light, where they are annexed to Euclides matter, or geue some ready ayde, and shew the way to dilate your discourses Mathematicall; or to inuent and practise thinges

Mechanically. And (in deede) if more leysor had happened, many more straunge matters Mathemati∣call had, (according to my purpose generall) bene presently published to your knowledge: but want of due leasour cau••eth you to want, that, which my good will toward you, most hartely doth wish you.

As concerning the two Theoremes here annexed, their veritie, is by Archimedes, in his booke of the Sphere and Cylinder manifestly demonstrated, and at large: you may therefore boldly trust to them, and vse them, as suppositions, in any your purposes: till you haue also their demōstrations. But if you well remember my instructions vpon the first proposition of this booke, and my other addition, vpon the second, with the suppositions how a Cylinder and a Cone are Mathematically produced, you will not neede Archimedes demonstration: nor yet be vtterly ignoraunt of the solide quantities of this Cylin¦der and Cone here compared: (the diameter of their base, and heith being knowne in any measure) nei∣ther can their croked superficies remayne vnmeasured. Whereof vndoubtedly great pleasure and com∣moditie may grow to the sincere student, and precise practiser.

Notes

• * 1.1

  Constr〈7 letters〉〈7 letters〉
• * 1.2

  Parallelipipe∣dons called Prismes.
• * 1.3

  By this it is manifest that Euclide com∣prehended si∣ded Columnes also vnder the name of a Prisme.
• * 1.4

  A prisme hauing for his base a po∣ligonon figure as we haue often before noted vnto you.
• * 1.5

  Note: M. Dee his chiefe purpose in his additions.