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 33. Proposition. The solide of a Dodecahedron excedeth the solide of a Cube inscribed in

it, by a parallelipipedon, whose base wanteth of the base of the Cube by a third part of the lesse segment, and whose altitude wanteth of the altitude of the Cube, by the lesse segment of the lesse segment, of halfe the side of the Cube.

FOrasmuch as by the 17. of the thirtenth, and 8. of the fiuetenth, it w••s manifest, that the base of a cube inscribed in a dodecahedron, doth with his sides subtend t•••• angles of 4. pentagons cōcurring at one and the selfe same side of the dodecahedron:* 1.1 let that base of the cube be A∣BCD: and let the side wherat 4. bases of the dodecahedron circumscribed concurre, be EG: which shall contayne a solide AEBDGC set vpon the base ABCD. Diuide the sides AB and DC in∣to two equall partes in the poyntes L and N. And draw the line LN, which is a parallel to the side EG, as it was manifest by the 17. of the thirtenth. The perpendiculars also ER and GO which couple those parallels, are eche equall, to halfe of the side EG, and eche is the greater segment of halfe the side of the cube, and therefore the whole line EG is the greater segment of the whole line LN the side of the cube (by the foresayd 17. of the thirtenth). By the poyntes R and O, draw vnto the sides AB and CD parallel lines FH and IK. And draw these right lines EF, EH, GI and GK. Now forasmuch as the two lines FH & ER touching the one the other are parallels to the two lines IK and GO touching also the one the other,* 1.2 & not being in the selfe same playne with the two first lines: therfore the playne super••i∣cieces EFH and GIK passing by those lines are parallels, by the 15. of the eleuenth: which playnes do cutte the solide AEBDGC. Whererefore there are made fower quadrangled pyramids set vpon the rectangle parallelogrames LH, LF, NK, and NI, and hauing their toppes the poyntes E and G. And forasmuch as the triangles GOK and ERH are equall and like, by the 4. of the first, namely,* 1.3 they con∣tayne

[illustration]

equall angles comprehēded vnder equall sides, and they are parallels by construction, being set in the playnes GIK and EFH: the figures GKHE, OKHR, and GORE shall be parallelogrammes, by the di••••••nition of a parallelogramme, and therefore the solide GOKERH is a prisme, by the 11. diffini∣tion of the eleuenth. And by the same reason may the solide GOIERF be proued to be a prisme. And forasmuch as vpon equall bases NOKC, and RLBH, and vnder equall altitudes OG and RE are set pyramids: those pyramids shall be equall to ••hat pyramis which is set vpon the ••ase CKID (which is double to either of the bases NOKC, and RLBH) and vnder the same altitude OG, by the 6. of the twelfth. And forasmuch as the side GE is the greater segment of the line CB, the line KH, which by the 33. of the first, is equall to the line GE, shall be the greater segment of the same line CB, by the 2. of the fourtenth. Wherefore the residues CK and HB shall make the lesse segment of the whole line C∣B. But as the greater segment KH is to the two lines CK and HB the lesse segment, so is the rectangle parallelogramme OH to the two rectangle parallelogrammes OC and HL, by the 1. of the sixt. Wher∣fore the lesse segment of the parallelogramme NB, shall be the two parallelogrammes OC and HL. Put the line KM double to the line KC and draw the line MS parallel to the line CN. Wherefore the parallelogramme OKMS is equall to the parallelogrammes OC and HL, by the 1. of the sixth. Wher∣fore the pyramis set vpon the base OKMS contayneth two third partes of the prisme set vpon the selfe same base, by the 4. corollary of the 7. of the twelfth. Wherfore the prisme which is set vpon two third

partes of the base OKMS is equall to the two pyramids NOKCG and RLBHE. For the sections of a prisme are one to the other, as the sections of the base are, by the first corollary of the 25. of the ele∣uenth. But the sections of the base are as the sections of the line CB or KM, by the 1. of the sixt. Wher∣fore the two pyramids NOKCG and RLBHE, adde vnto the prisme GOKERH two third partes of the prisme set vpon the base OKMS. And forasmuch as the line KM is the lesse segmēt of the whole line BC (for it is equall to to the two lines CK and H••), and the prisme set vpon the base OKHR is cutte like vnto the line KM, namely, in eche are taken two thirdes, as hath before bene proued: the prisme equall to the two pyramids, shall adde vnto the prisme GOKERH, which is set vpon the grea∣ter segment KH, two th••••ds of the lesse segment. Wherefore in the line BC there shall remayne one third part of the lesse segment: and therefore in the rectangle parallelogramme NB which is halfe the base of the cube, there shall remayne the same third part of the lesse segment. And by the same reason may we proue that in the other pyramids ONDIG, and RLAFE, and in the prisme GOIERF is left the selfe same excesse of the base LAND, namely, the third part of the lesse segment. Wherefore the whole prisme contayned betwene the triangles IGK and FEH, and vnder the length of the greater * 1.4

[illustration]

segment and two third partes of the lesse segment of BC the side of the cube, is equall to the solide composed of 4. bases of the dodecahedron and set vpon the base of the cube. Wherfore the base of that prisme wanteth of the whole base of the cube onely a third part of the lesse segment: and the altitude of that prisme was the line GO, which is the greater segment of halfe the side of the cube. And foras∣much as vnto the triangle IGK, is double the rctangle parallelogramme set vpon the same base IK, (the side of the cube) and vnder the altitude GO, by the 41. of the first: it followeth that three rectan∣gle parallelogrammes set vpon the same base IK, the side of the cube, and vnder the altitude OG the greater segment of halfe the side of the cube, are sextuple to the triangle IGK. Wherefore those three rectangle parallelogrammes doo make one rectangle parallelogramme set vpon the base IK and vnder the altitude of the line GO tripled. But by the 7. diffinition of the eleuenth, there are sixe prismes equal and like vnto the foresayd prisme, being set vpon euery one of the sixe bases of the cube: which prismes are in proportion the one to the other as their bases are by the 3. corollary of the 7. of twelfth. Where∣fore the solide composed of these sixe prismes, shall want of the base ABCD the third part of the lesse segment, and taking his altitude of the foresayd rectangle parallelogramme, the sayd altitude shall be e∣quall to three greater segmentes (one of which is GO) of halfe the side of the cube.

Now resteth to proue that these three segmentes want of the side of the cube by the lesse segment of the lesse segment of halfe the sid•• of the cube. Suppose that AB the side of the cube be diuided into the greater segment AC, and into the lesse segment CB (by the 30. of the sixt). And diuide into two e∣quall partes the line AC in the poynt G, and the line CB in the poynt E. And vnto the line CG put the line CL equall. Now forasmuch as the lines AG and GC are the greater seg••••••tes of halfe the line AB, for ••che of

[illustration]

them is the halfe of the greater segment of the whole line A∣B: the lines EB and EC shall be the lesse segmentes of halfe the line AB. Wherefore the whole line C•• is the greater segment, and the line C∣E

is the lesse segment. But as the line CL is to the line CE, so is the line CE to the residue EL. Wher∣fore the line EL is the greater segment of the line CE, or of the line EB which is equall vnto it. Wher∣fore the residue LB is the lesse segment of the same EB (which is the les••e segment of half•• the side of the cube). But the lines AG, GC, and CL are three greater segmentes of the halfe of the whole line AB: which thre greater segmentes make the altitude of the foresayd solide: wherefore the altitude of the sayd solide wanteth of AB the side of the cube by the line LB, which is the lesse s••gment of the line BE. Which line BE agayne is the lesse segment of halfe the side AB of the cube. Wherefore the fore∣sayd solide consisting of the sixe solides, whereby the dodecahedron exceedeth the cube inscribed in it, is set vpon a base which wanteth of the base of the cube by a third part of the lesse segment, and is vnder an altitude wanting of the side of the cube by the lesse segment of the lesse segment of halfe the side of the cube. The solide therefore of a dodecahedron exceedeth the solide of a cube inscribed in it, by a parallelipipedon, whose base wanteth of the base of the cube by a third part of the lesse segment, and whose altitude wanteth of the altitude of the cube, by the lesse segment of the lesse segment of halfe the side of the cube.

¶A Corollary.

A Dodecahedron is double to a Cube inscribed in it, taking away the third part of the lesse segment of the cube, and moreouer the lesse segment of the lesse segment of halfe of that excesse. For if there be geuen a cube, from which is cut of a solide set vpon a third part of the lesse segment of the base and vnder one and the same altitude with the cube: that solide taken away hath to the whole solide the proportion of the section of the base to the base, by the 32. of the eleuenth. Wherefo••e from the cube is taken away a third ••art of the lesse segment. Far∣ther, forasmuch as the residue wanteth of the altitude of the cube, by the lesse segment of the lesse seg∣ment of halfe the altitude or side, and that residue is a parallelipipedon, if it be cut by a plaine superficies parallel to the opposite plaine superficieces, cutting the altitude of the cube by a point, it shall take away from that parallelipipedon a solide, hauing to the whole the proportion of the section to the altitude, by the 3. Corollary of the 25. of the eleuenth. Wherefore the excesse wanteth of the same cube by the thi••d part of the lesse segment, and moreouer by the lesse segment of the lesse segment of halfe of that excesse.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  Extend in the figure a line ••rom the point E to the point B.
• * 1.4

  Extend in the figure a line from the point E to the point B.