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. 8.

Euery Sphere, to the cube, made of his diameter, is (in maner) as 11. to 21.

As vpon the first and second propositiōs of this booke, I began my additions with the circle (be∣ing the chiefe among playne figures) and therein brought manifold considerations, about circles: as of the proportion betwene their circumferences and their diameters of the content or Area of circles: of the proportion of circles to the squares described of their diameters: & of circles to be geuen in al pro••portions, to other circles: with diuerse other most necessary problemes (whose vse is partly there speci∣fied): So haue I in the end of this booke, added some such Problemes & Theoremes•• about the sphere (being among solides the chiefe) as of the same, either in it selfe considered, or to cone and cylinder, compared (by reason of superficies, or soliditie, in the hole, or in part••) such certaine knowledge demon¦stratiue may arise, and such mechanical exercise thereby be deuised, that (sure I am) to the sincere & true student great light, ayde, and comfortable courage (farther to wade) will enter into his hart; and to the Mechanicall, witty, and industrous deuiser, new maner of inuentions, & executions in his workes will (with small trauayle for fete application) come to his perceiueraunce and vnderstanding. Therefore, e∣uen a, manifolde speculations & practises may be had with the circle, his quantitie being not knowne in any kinde of smallest certayne measure: So likewise of the sphere many Problemes may be executed and his precise quantitie, in certaine measure, not determined, or knowne: yet, because, both one of the first (humane) occasiōs of inuenting and stablishing this Arte, was measuring of the earth (and therfore called Geometria, that is, Earthmeasuring), and also the chiefe and generall end (in deede) is mea∣sure: and measure requireth a determination of quantitie in a certayne measure by nūber expressed: It was nedefull for Mechanicall earthmeasures, not to be ignorant of the measure and contents of the circle, neither of the sphere his measure and quantitie, as neere as sense can imagine or wish. And (in very deede) the quantitie and measure of the circle, being knowne, maketh not onely, the cone and cy∣linder, but also the sphere his quantitie to be as precisely knowne, and certayne. Therefore seing in re∣spect of the circles quantitie (by Archimedes specified) this Theoreme is noted vnto you: I wil, by order, vpon that (as a supposition) inferre the conclusion of this our Theoremes.

Suppose a sphere to be

[illustration]

signified by A•• whose diame∣ter let be RS. To RS, let a line equall be taken, which let be TV: of TV, by the 46. of the first, describe a square. Let that square be TY. With in TY let a circle be inscri∣be••: by the 2. of the fourth, which cir••l•• suppose to be OZW. That OZW is e∣quall to the greatest circle in the sphere A conteyned, it is euident by the diameter, e∣qual to TV. If vpō the square TY, as a base, be erected, a pa¦rallelipipedō rectāgle, whose h••ith is equall to TV, it is e∣uident that that parallelipi∣pedon is a cube. Which let be done: and that cube pro∣duced, let be noted by TX. Likewise, if vpon the circle OZW, as a base, and of the heith equall to the line TV, a cylinder be erected, it is ma¦nifest that the cylinder hath his base equall to the greatest circle, in the sphere A, con∣teined: & heith, a line equall to the diameter of the same sphere A. Which cylinder let be produced and noted by Z∣M. I say now that the sphere A, is to the cube TX, (in ma∣ner) as the number 11. is to the number 21. For seing the cube TX, was produced of his base, (the square TY), be∣ing brought into the heith of a line equall to TV: & like∣wise seing the cylinder ZM,

is produced, of his base (the circle OZW) being brought into a line•• equal to the said TV: it followeth, seing their heithes is all one, that the cube TX shall be to the cylinder ZM, as the base of TX, (which is the square TY) is to the base of ZM, that is the circle OZW. But the square TY, is to the circle OZW, as the number 14. is to the nūber 11. (in maner), by Archimedes demōstratiō: wherfore, the cube TX is to the cylinder ZM, as the number 1••, is to the number 11. (well nere). And by my third Theoreme (here added) the cylinder ZM, is to the Sphere A, in sesquialtera proportion: that is, as 3. to 2. Where∣fore the cylinder ZM, hauing the same 11. equall partes (which he conteyneth in respect of the cube T∣X, being 14. of the same partes) deuided into 3. equall portions, euery one of those portions is 3⅔. And allowing to the Sphere A, two of those portions: it is euident, that the Sphere A shall be 71/••; such partes as are 14. in the cube TX: and 11. in the cylinder ZM. Wherefore the Sphere A, is to the cube TX, as 71/•• to 14. The fraction being reduced, maketh ••••/••: and the number 14. being brought to the same name, and denomination of thirds, maketh 4••/••. Put away now theyr common denominator•• and then remay∣neth, for the Sphere A, 22. such partes, as the cube TX hath 4••. And then depressing them, to the smal∣lest termes: for the Sphere A, you shall haue 11. such partes as the cube TX conteyneth 21. Wherefore euery Sphere, to the cube made of his diameter is, as 11. to 21. which was requisite to be demonstrated.

Note. 1.

Wherfore if you deuide the

[illustration]

one side (as TQ) of the cube TX into 21. equall partes, and where 11. partes do end, recke∣ning from T, suppose the point P: and by that point P, imagine a plaine (passing parallel to the opposite bases) to cut the cube TX: and therby, the cube TX, to be deuided into two rectan∣gle parallelipipedons, namely, TN, and PX: It is manifest, * 1.1 TN, to be equall to the Sphere A, by construction: and the 7. of the fift.

Note. 2.

Secondly, the whole quan∣titie, of the Sphere A,* 1.2 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,* 1.3 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

  A rectangle parallelipipe∣don geu••n, e∣quall to a Sphere geuen.
• * 1.2

  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.
• * 1.3

  Farther vse of Sphericall Geometrie.