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. 3.

Euery Cylinder which hath his base, the greatest Circle in a Sp••er••, & heith equall to the diame∣ter of that Sphere, is Sesquialtera to that Sphere. Also the superficies of that Cylinder, with his two bases is Sesquilatera to the superficies of the Sphere: and without his two bases, is equall to the superfi∣cies of that Spher••

Suppose, a sphere to be signified by A, and an vpright cylinder hauing to his base a circle equall to the greatest circle in A contayned, and his heith equall to the diameter of A, let be signified by FG. I say that FG, is sesquialter to A: Secondly I say that the croked cylindricall super••icies of FG, together [ 1] with the superfici••ces of his two opposit bases, is sesquialtera to the whole superficies sphericall of A. [ 2] Thirdly I say that the cylindricall superficies of FG, omitting his two opposite bases, is equall to the superficies of the spere A. Let the base of FG, be the circle FLB: whose center, sup••ose M, and [ 3] diameter FB. And the axe of the same FG, let be, MH. Which is his heith (for we suppose the cy∣linder to be vpright): and suppose H, to be his toppe or vertex. Forasmuch as, by supposition MH is equall to the diameter of A. Let MH be deuided into two equall partes in the point N, by a playne su∣perficies passing by the point N, and being parallell to the opposit bases of FG. By the thirtenth of this twelfth booke, it then foloweth, that the cylinder FG, is also deuided into two equall parts: being cy∣linders: which two equall cylinders let be IG, and FK: the axe of IG suppose to be HN: and of FK the axe to be NM. And for that, FG, is

[illustration]

an vpright cylinder, and at the poynt N, cut by a playne Superficies parallell to his opposite bases, the common se∣ction of that playne superficies and the cylind••r FG, must be a * 1.1 circle, equall, to his base FLB, and haue his center, the point N. Which circle, let be IO∣K: And seing that FLB is, by supposi∣tion, equall to the greatest circle in A, IOK, also, shall be equall to the grea∣test circle, in A, contained: Also, by rea∣son MH, is by supposition, equal to the diameter of A: and NH, by constru∣ctiō, half of MH, it is manifest that N∣H is equall to the semidiameter of A. If, therefore, you suppose a cone to haue the circle IOK to hi•• base: and NH to his heith, the sphere A, shall be to that Cone, quadrupla, by the 2. Theoreme. Let that cone be HIOK. Wherefore A, is quadrupla to HIOK. And the Cylinder IG hauing the same base, with HIOK (the circle IOK) and the same heith, (the right line NH) is triple to the cone HIOK by the 10. of this twelfth booke. But to IG, the whole cylinder FG, is double, as is proued: Wherefore FG, is triple and triple, to the cone HIOK, that is, sextuple. And A is proued quadrupla to the same HIOK. Wherefore FG is to HIOK, as 6. to 1: and A, is to HIOK, as 4. to 1: * 1.2 Therfore FG is to A, as 6, to 4: which in the least termes, is, as 3 to 2. but 3 to 2, is the termes of sesquialtera proportion. Wherefore the cylinder FG, is to A sesquialtera in proportion. Secondly, forasmuch as the superficies of a cylinder (his two oppo∣site bases excepted) is equall to that circle whose semidiameter is middell proportionall betwene the side of the cylinder, and the diameter of his base: (as vnto the 10. of this booke, I haue added.) But of FG, the side BG, being parallell and equall to the axe MH, must also be equall to the diameter of A. And the base FLB, being (by supposition) equall to the greatest circle in A contained, must haue his di∣ameter (FB) equal to the sayd diameter of A. The middle proportional therfore betwene BG and FB, being equall eche to other, shal•• be a line, equall to either of them.

[As i•• 〈◊〉〈◊〉 set BG and FB together, as one line, and vpon that line composed, as a diameter make a semicircl•• and from the center, to the circumference draw a lin•• perpendicular to the sayd diameter:* 1.3 by the 〈◊〉〈◊〉 of the sixth, that perpendicular, is middel proportional betwene FB and BG, the semidiame∣ters•• and he him selfe also a semidiameter: and therfore by the definition of a circle, equall to FB, and likewise, to BG.] And a circle, hauing his semidiameter, equall to the diameter FB, is quadruple to the circle FLB. [For the square of euery whole line, is quadruple to the squ••re of his halfe line, as may

be proued by the 4. of the second: and by the second of this twelfth, circles are one to the other, as the squares of their diameters, are.]

Wherfore the superficies cylindricall of FG, alone, is quadrupla to his base FLB. But if a certayne quantity be dupla to one thing, and an other, quadrupla to the same one thing, those two quantities together are sextupla to the same one thing. Therefore seing the base, oppo∣site to FLB, (being equall to to FLB)

[illustration]

added to FLB, maketh that cōpound, double to FLB: that double added to the cylindricall super••icies of FG, doth make a superficies sextupla to FLB. And the superficies of A, is quadrupla to the same FLB, by the first Theo∣reme. Therefore the cylindricall super∣ficies of FG, with the superficieces of his two bases is to the superficies FLB, as 6 to 1, and the superficies of A to F∣LB, is as 4 to 1. Wherfore the cylindri∣call superficies of FG, & his two bases, together, are to the superficies of A, as 6 to 4: that is, in the smallest termes, as 3 to 2. Which is proper to ses••uialtera proportion. Thirdly, it is already made euident that the superficies cylindrical, of FG (onely by it self) [ 3] is quadrupla to FLB. And also it is proued, that the superficies of the sphere A, is quadrupla to the same FLB, Wherefore by the 7. of the fifth, the cylindricall superficies of FG, is equall to the superfi∣cies of A. Therfore, euery cylinder, which hath his base the greatest circle in a sphere, and heith equal to the diameter of that sphere, is sesquialtera to that spere: Also the superficies of that cylinder with his two bases, is sesquialtera, to the superficies of the sphere: and without his two bases is equall to the su∣perficies of the sphere: which was to be demonstrated.

Notes

• * 1.1

  This may ea∣sely be demon∣strated, as in th•• 17. proposition the section of a sphere was pro∣ued to be a cir∣cle.
• * 1.2

  For taking a∣way all doubt, this, a•• a Lem∣ma, afterward is dem••strated.
• * 1.3

  A Lemma (as it were) presently de∣monstrated.