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 3. Probleme. The 15. Proposition. To make a solide called a cube, and to comprehend it in the sphere geuen, namely, that Sphere wherein the former two solides were comprehend••d•• and to proue that the diameter of the sphere, is in power treble to the side of the cube.

TAke the diameter of the sphere geuen, namely, AB, and diuide it in the point C•• So that let the line AC be double to the line BC by the 9. of the sixt. And vpon the

line AB describe a semicircle ADB. And (by the 11. of the first) from the p••ynt C r••yse vp vnto the line AB a perpēdicular line CD. And draw a right lin•• DB. And describe a squar•• EFGH, hauing euery one of his sides equall

[illustration]

to the line DB, And from the pointes E, F, G, H, rayse vp (by the 12. of the eleuenth) vnto the playne superficies of the square EFGH perpendicular lines EK, FL, GM, and HN: and let euery one of the lines EK, FL, GM, and HN be put equall to one of the lines EF, FG, GH, or HE, which are the sides of the square, and draw these right lines KL, LM, MN, and NK.* 1.1 Wherfore there is made a cube namely FN which is contayned vnder six e∣quall squares.* 1.2 Now it is required to compre∣hend the same cube in the sphere geuen, and to proue that y^e diameter of the sphere is in power ble to the side of the cube.* 1.3 Draw these right lines KG and EG. And forasmuch as the an∣gle KEG is a right angle, for that the line K∣E is erected perpendicularly to the playne su∣perficies E••, and therefore also to the right line EG, by the 2. diffinitiō of the eleuenth, wherefore a semicircle described vpon the line KG shall * 1.4 passe by the poynt E. Agayne for∣asmuch as the line FG is erected perpendicu∣larly to either of these lines FL and FE, by the diffinition of a square, & by the 2. diffini∣tion of the eleuenth, therefore the line FG is erected perpendicularly to the playne superficies FK, by the 4. of the eleuenth. Wherefore if we draw a right line from the point F to the point K, the line GF shall be erected perpendicularly to the line KF, by the 2. diffinition of the ele∣uenth. And by the same reason agayne a semicircle described vpon the line GK shall passe also by the point F. And likewise shall it passe by the rest of the pointes of the angles of that cube. If now the diameter KG abiding fixed the semicircle be turned round about vntill it returne into the selfe same place from whence it began first to be moued, the cube shalbe comprehēded in a sphere. I say also that it is comprehended in the sphere geuen.

* 1.5For forasmuch as the line GF is equall to the lin•• ••E and the angle F is a right an∣gle, therefore the square of the line EG is by the 47. of the first double to the square of the line ••F. But the line EF is equall to the line EK. Wherefore the square of the line EG is double to the square of the line EK. Wherfore the squares of EG and EK that is the square of the line GK, by the 47. of the first, are treble to the square of the line EK. And forasmuch as the line AB is treble to the line BC, but as the line AB is to the line BC, so is the square of the line AB to the square of the line BD, by the corollaries of the 8. and 20. of the sixt. Wherefore the square of the line AB is treble to the square of the line BD. And it is proued that the square of the line GK is treble to the square of the line KE, and the line KE is put equall to the line BD. Wherefore the line KG is also equall to the line AB. And the line AB is the diameter of the sphere geuen. Wherefore the line KG is equall to the diameter of the sphere geuen. Wherfore the cube is cōprehēded in the sphere geuen: and it is also proued that the diameter of the Sphere is in power treble to the side of the cube: which was required t•• ••e doone, and to be proued.

An other demonstration after Flussas.

Suppose that the diameter of the Sphere geuen in the former Propositions, be the line A••. And let the center be the point C, vpon which describe a semicircle ADB. And from the diameter AB cut of a third part BG, by the 9. of the sixt. And from the point G raise vp vnto the line AB a perpendi∣cular line DG, by the 11. of the first. And draw these right lines DA, DC, and DB. And vnto the right line DB put an equall right line ZI:

[illustration]

and vpon the line ZI describe a square EZIT. And frō the pointes E, Z, I, T, erecte vnto the superficies EZIT per∣pendicular lines EK, ZH, IM, TN (by the 12. of the eleuenth): and put euery one of those perpendicular lines equall to the line ZI. And drawe these right lines KH, HM, MN, and NK, ech of which shall be equall and parallels to the line ZI, and to the rest of the lines of the square, by the 33. of the first. And moreouer they shall containe equall angles (by the 10. of the eleuenth): and therefore the angles are right angles, for that EZIT is a square: wherfore the rest of the bases shall be squares. Wherfore the solide EZITKHMN being cōtained vnder 6. equall squares, is a cube, by the 21. definition of the eleuenth. Extend by the opposite sides KE and MI of the cube, a plaine KEI∣M: and againe by the other opposite sides NT and HZ, extend an other plaine HZTN. Now forasmuch as ech of these plaines deuide the solide into two equall partes, namely, into two Prismes equall and like (by the 8. defi∣nition of the eleuenth): therfore those plaines shall cut the cube by the cen∣tre, by the Corollary of the 39. of the eleuenth. Wherefore the cōmon secti∣on of those plaines shall passe by the centre. Let that common section be the line LF. And forasmuch as the sides HN and KM of the superficieces KE∣IM and HZTN do diuide the one the other into two equall partes, by the Corollary of the 34. of the first, and so likewise do the sides ZT and EI: therefore the common section LF is drawen by these sections, and diuideth the plaines KEIM and HZTN into two equall partes, by the first of the sixt: for their b••ses are equall, and the altitude is one and the ••ame, namely, the altitude of the cube. Wherefore the line LF shall diuide into two equall partes the diameters of his plaines, namely, the right lines KI, EM, ZN, and NT, which are the diameters of the cube. Wher∣fore those diameters shall concurre and cut one the other in one and the selfe same poynt, let the same be O. Wherfore the right lines OK, OE, OI, OM, OH, OZ, OT, and ON, shall be equ••ll the on•• to the other, for that they are the halfes of the diameters of equall and like rectangle parallelogrāmes. Wherefore making the centre the point O, and the space any of these lines OE, or OK. &c. a Sphere described, shall passe by euery one of the angles of the cube, namely, which are at the pointes E, Z, I, T, K, H, M, N, by the 12. definition of the eleuenth, for that all the lines drawen from the point O to the an∣gles of the cube are equall. But the right line EI containeth in power the two equall right lines EZ, and ZI, by the 47. of the first. Wherefore the square of the line EI is double to the square of the line ZI. And forasmuch as the right line KI subtendeth the right angle KEI (for that the right line KE•• is erected perpendicularly to the plai••e ••uperficies of the right lines EZ and ZT (by the 4. of the eleuēth) •• therefore the square of the line KI is equall to the squares of the lines EI and EK, but the square of the line EI is double to the square of the line EK (for it is double to the square of the line ZI, as hath bene proued, and the bases of the cube are equall squares). Wherefore the square of the line KI is triple to the square of the line KE, that is, to the square of the line ZI. But the right line ZI is equall to th•• right line DB, by position, vnto whose square the square of the di••meter AB is triple, by that which was demonstrated in the 13. Proposition of this booke. Wherefore the diameters KI & DB are equall. Wherefore there is described a cube KI, and it is comprehended in the Sphere geuen wherin the other

solides were contained, the diameter of which Sphere is the line AB. And the diameter KI or AB of the same Sphere, is proued to be in power triple to the side of the cube, namely, to the line DB, or ZI.

¶ Corollaryes added by Flussas.

First Corollary.

Hereby it is manifest, that the diameter of a Sphere containeth in power the sides both of a pyra∣mis and of a cube inscribed in it.

For the power of the side of the pyramis is two thirdes of the power of the diameter (by the 13. of this booke). And the power of the side of the cube is, by this Proposition, one third of the power of the sayd diameter. Wherefore the diameter of the Sphere contayneth in power the sides of the pyra∣mis and of the cube..

¶ Second Corollary.

All the diameters of a cube cut the one the other into two equall partes in the centre of the sphere which containeth the cube. And moreouer those diameters do in the selfe same point cut into two e∣quall partes the right lines which ioyne together the centres of the opposite bases.

As it is manifest to see by the right line LOF. For the angles LKO, and FIO, are equall, by the 29. of the first: and it is proued, that they are contained vnder equall sides: Wherefore (by the 4. of the first) the bases LO and FO are equall. In like sort may be proued, that the rest of the right lines which ioyne together the centres of the opposite bases do cut the one the other into two equall partes in the centre O.

Notes

• * 1.1

  First part of the demonstration.
• * 1.2

  Second part of the Constructiō.
• * 1.3

  Second part of the demonstra∣••ion.
• * 1.4

  By the 2. As∣sumpt of the 13. of this booke.
• * 1.5

  Third part of the demonstra∣tion.