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 sixtenth booke of the Elementes of Geometrie added by Flussas. (Book 16)

IN the former fiuetenth booke hath bene taught how to inscribe the fiue regular solides one with in an other. Now semeth to rest, to cōpare those solid•• so inscribed, one to an other, and to set forth their passion•• and pro∣prieties: which thing, Flussas considering, in this sixtēth booke added by him,* 1.1 hath excellently well and most conningly performed. For which vndoubtedly he hath of all them which haue a loue to the Mathematicals, de∣serued much prayse and commendacion: both for the great tra••ailes and payn••s (which it is most likely) he hath ta••••n in i••uenting such straunge and wonderfull propositions with their demonstrations, in this booke contayned, as also for participating and communica∣ting abrode the same to others. Which booke also, that the reader should want nothing conducing to the perfection of Euclides Elements: I haue with some trauaile translated, & for the worthines ••hereof haue added it, a•• a sixtenth booke to the 15. bookes of Euclide. Vouchsafe therefore gentle reader dili∣gently to read and peyse it, for in it shall you finde no•• onely matter strange and delec∣table, but also occasion of inuention of greater things pertayning to the natures of the fiue regular solid••s••

¶ The 1. Proposition. A Dodecahedron, and a cube inscribed in it, and a Pyramis inscribed in the same cube, are contained in one and the selfe same sphere.

FOr the angles of the pyrami•• are se•• in the ang••es of the cube wherein it is inscri∣bed (by the first of the fiuetenth•• and all the angles of the cube are set in the angles of the dodecahed•••••• circumscribed 〈…〉〈…〉 (〈◊〉〈◊〉 the 8. of the fiuetenth): And all the angles of the Dodecahedron, are set in the superficies of the sphere, by the 17. of the thirtenth. Wherefore those three solides inscribed one within an other, are contained in one and the selfe same sphere, by the first diffinition of the fiuetenth. A dodecahedron therfore and a cube inscribed in it, and a pyramis inscribed in the same cube, are contained 〈…〉〈…〉 ••••lfe same sphere.

〈…〉〈…〉

These three solides li〈…〉〈…〉elfe same Icosahedron, or Octohedron, or Pyramis. 〈…〉〈…〉me Icosahedron, by the, 5.11. & 12. of the fiuetenth: and they ar〈…〉〈…〉ctohedron, by the 4. 6. and 16. of the same: lastly they are inscribed in 〈…〉〈…〉 the first, 18. and 19. of the same. For the angles of all these solide〈…〉〈…〉 the circumscribed Icosahedron, or octohedron, or pyramis.

¶ The 〈…〉〈…〉 The proportion of a Dodecahedron circumscribed about a cube, to a Dodeca∣hedrō

inscribed in the same cube, is triple to an extreme & meane propartiō.

FOrasmuch as in the •••• corollary of the 13. of the fiu••••enth, it was proued, that the side of a Dodecahedron inscribed in a cube, is the lesse segment of the side of that cube de∣uided by an extreme and meane proportion: and the side of the dodecahedron ••ircum∣scribed about the same cube, is the greater segment of the side of the same cube (which thing also was taught in the 13. of the fiuetenth) the side of the Dodecahedron circum∣scribed, shalbe to the side of the Dodecahedron inscribed, as the greater segment of a right line deuided by an extreme and meane proportion, is to the lesse segment of the same, which proportion is called an extreme and meane proport•••••• by the diffinition, and by the 30. of sixth. But the proportion of like solide prolihedrons, is t••iple to the proportion of the side•• of like pro∣portion, by the corollary of the 17. of the twelueth. Wherefore the proportion of the Dodecahedron circumscribed about the cube, is to the dodecahedron inscribed in the same cube, in triple proportion of the sides ioyned together by an extreme and meane proportion: The proportion therefore of •• Do∣decahedron circumscribed about a cube to a dodecahedron inscribed in the same cube, is triple to an extreme and meane proportion.

The 3. Proposition. In euery equiangle, and equilater Pentagon, a perpendicular drawne from one of the angles to the base, is deuided by an extreme and meane proporti∣on by a right line subtending the same angle.

SVppose that ABCDF be a•• equiangle and ••quilater

[illustration]

pentagon:* 1.2 and from one of the angles namely, from A, let there be drawne to the base CD a perpendicu∣lar AG: and let the line BF subtend the angle BAP,* 1.3 which line BF let the line AD cut in the poynt I. Then I say that the line BF, cutteth the line AG by an extreme and meane proportion. For forasmuche as the angles GAF and GAB are equal by the 27. of the third, and the angles A••F and AFB, are equal by the 5. of the first•• therefore the ••••gles remaining at the poynt E, of the triangles AEB and AEF are equal: for that they are the residues of two right angles by the corollary of the 32. of the first. But the angle EGC, is by construction a right angle•• wherfore the lines BF & CD are parallels by the 28. of the first. Wherefore as the line DI is to the line IA, so i•• the line GE to the line EA, by the 2. of the sixth. But the line DA, is in the poin•• I deuided by an extreme and meane proportion, by the 8. of the thirtenth. Wherfore the line GA is in the poynt E, deuided by an extreme and meane proportion (by the ••. of the fourtenth). Wherfore in e∣uery equiangle and equilater pentagon, a perpendicular drawne from one of the angles to the base, is deuided by an ex••reme and meane proportion by a right line subte••ding the same angle••

¶ A Corollary.

The line which subtendeth the angle of a pentagon, is a parallel to the side opposite vnto the angle. As it was manifest in the lines ••F and CD.

The 4. Proposition. If frō the angles of the base of a * 1.4 Pyramis, be drawne to the opposite sides, right lines cutting the sayd sides by an extreme and meane proportion: they shall containe the bise of the Icosahedron inscribed in the Pyramis, which base shalbe inscribed in an equilater triangle, whose angles cut the sides of the base of the Pyramis by an extreme and meane proportion.

* 1.5SVppose that ABG be the base of a pyramis, in which let be inscribed an equilater triangle FKH, which is done by deuiding the sides into two equal partes. And in ••his triangle let there be inscribed the base of the Icosahedrō inscribed in the pyramis: which is described by deuiding the sides FK, KH, HF, by an extreme & meane proportiō in the poynts C, D, E, by the 19. of the fiuetēth. Againe let the sides of the pyramis, namely, AB, BG, and GA be deuided by an extreme and me••ne proportion in the poynts I, M, L, by the 30. of the sixth. And drawe these right lines AM, BL, GI.* 1.6 Then I say that those lines describe the triangle CDE of the Icosahedron. For forasmuch as the lines BG and FH are parallels, by the 2. of the sixth: by the point D let the line ODN be drawne parallel to either of the lines BG & FH. Wherfore the triangle HDN shalbe like to the triangle HKG, by the corollary of the 2. of the sixth. Wherfore either of these lines DN and NH shall be equal to the line DH, the greater segment of the line KH or FH. And forasmuch as the line FO is a parallel to the line HK, and the line OD to the line FH•• the line OD shall be equal to the whole line FH in the pa∣rallelogramme FODH, by the 34. of the

[illustration]

••irst. Wherefore as the whole line FH is to the grea••er segment FE, so shall the lines equal to them be, namely, the line OD to the line DN, by the 7. of the fifth. Wherfore the line ON is deuided by an extreme and meane proportion in the poynt D, by the 2. of the fourtenth. But the triangles AOD, AFE, and ABM, are like the one to the other, and so also are the triangles ADN, AEH, and AMG, by the corollary of the secōd of the sixth•• Wherefore as FE is to EH, so is OD to DN, and BM to MG. Whe••fore the line AM cutting the lines FH and ON, lyke vnto the line BG in the pointes E, D, M, describeth ED the side of the triangle of the Icosahedron ECD, which is descri∣bed in the sections E, C, D, by suppositiō. And by the same r••ason the lines BL and GI shall describe the other sides EC and CD of the same triangle. By the point E, let there be drawne to GI a parallel line PEQ. Now forasmuch as the lines BM and FE are parallels, the line AM is in the poynt E, cut like to the line AB in the poynt F, by the 2. of the sixth. Wherefore the line AE is equal to the line EM: and vnto the line EM also are equal either of the lines GD and DI: which ••re cut l••ke vnto the forsaid lines. Againe forasmuche as in the triangle ADI the lines DI and EP are parallels, as the line DI is to the line EP, so is the line AD to the line AE: but as the line AD is to the line AE, so is the line DG to the line EQ by the 2. of the sixth: wherefore as the line DI is to the line EP, so is the line DG to the line EQ: and alternately as the line DI is to the line DG, so is the line EP to the line EQ: but the lines DI and IG are equal: wherfore also the lines EP and EQ are equal. And forasmuch as the line AH is equal to the line FH, whose greater segmēt is the line HN•• therfore the whole line AN, is deuided by an extreme and meane proportion in the poynt H, by the ••. of the thirtenth. But as the line AN is to the line AH, so is the line AD to the line AE, by the 2. of sixth (for the line•• FH and ON are parallel••:) and againe as the line AD is to the line AE, so (by the same) is the line AG to the line AQ, and the line AI to the line AP: for the lines PQ, and GI are parallels: Wherefore the lines AG and AI are deuided by an extreme and meane proportion in the points Q & P: & the line AQ shalbe the greater segmēt of the line AG or AB. And forasmuch as the whol•• line AG is to the greater segment AQ, as the greater segment AI is to the residue AP: the line A•• shalbe the lesse segment of the whole line A•• or AG. Wherfore the li•••• PEQ (which by the poynt E passeth parallelwise to the line GI) cutteth the lines AG and BA by an extreme and meane proportion in the poynts Q and P. And by the same reason the line ••R (which by the poynt C, passeth parallelwise to the line AM) shall fall vpon the sections P and R: so also shal the line RQ (which by the poynt D passeth parallelwise to the line BL) fall vpo•• the sections RQ. Wherefore either of the lines PE and EQ shalbe equal to the line CD, in the parallelogrammes PD, and QC, by the 34. of the first. And forasmuch as the lines PE and EQ are equal, the lines PC, CR, RD and DQ shalbe likewise equal. Wh••rfore the triangle PRQ i•• ••quilater, and cutteth the sides of the base of the pyrami•• in the poyntes P, Q, R, by an extreme and meane proportion. And in it is inscribed the base ECD of the Ico∣sahedron contained in the for••sayd pyramis. If therefore from the angles of the base of a pyramis, be drawne to the opposite sid••s, right lines cutting the sayde sides by an extreme and meane proportion: they shall containe the base of the Icosahedron inscribed in the pyramis, which base shall be inscribed in an equilater triangle, whose angles cut the sides of the base of the pyramis by an extreme & meane propo••tion.

¶ A Corollary.

The side of an Icosahedron inscribed in an Octohedron, is the greater segment of the line, which being drawen from the angle of the base of the Octohedron cutteth the opposite side by an extreame and meane proportion. For, by the 16. of the fiuetenth, FKH is the base of the Octohedron, which containeth the base of the I∣cosahedron CDE: vnto which triangle FKH, the triangle HKG is equall, as hath bene proued. By the point H draw vnto the line ME a parallel line HT, cutting the line DN in the point S. Wherefore ES, DT, and ET, are parallelogrammes: and therefore the lines EH and MT are equall: and the lines EM and HT are like cut in the pointes D and S, by the 34. of the first. Wherefore the greater segment of the line HT is the line HS, which is equall to ED the side of the Icosahedron. But (by the 2. of the sixth) the line TK is cut like to the line HK by the parallel DM. And therefore (by the 2. of the fourtenth) it is di∣uided by an extreme and meane proportion. But the line TM is equall to the line EH. Wherefore also the line TK is equall to the line EF or DH. Wherefore the residues EH and TG are equall. For the whole lines FH and KG are equall. Wherefore KG the side of the triangle HKG is in the point T diui∣ded by an extreme and meane proportion in the point T, by the right line HT, and the greater segment thereof is the line ED the side of the Icosahedron inscribed in the Octohedron, whose base is the trian∣gle HKG (or the triangle FKH which is equall to the triangle HKG) by the 16. of the fiuetenth.

¶ The 5. Proposition. The side of a Pyramis diuided by an extreme and meane proportion, ma∣keth the lesse segment in power double to the side of the Icosahedron in∣scribed in it.

SVppose that ABG be the base of a pyramis:* 1.7 and let the base of the Icosahedron inscri∣bed in it, be CDE, described of three right lines, which being drawen from the angles of the base ABG cut the opposite sides by an extreme and meane proportion, by the for∣mer Proposition: namely, of these three lines AM, BI, and GI. Then I say, that AI the lesse segment of the side A••, is in power duple to CE the side of the Icosahedron. For, forasmuch as by the former Proposition, it was proued that the triangle CDE is inscri∣bed in an equilater triangle,* 1.8 whose angles cut the sides of

[illustration]

ABG the base of the pyramis by an extreme and meane proportion, let that triangle be FHK, cutting the line AB in the point F. Wherefore the lesse segment FA is equall to the segment AI, by the 2. of the fouretenth: (for the lines AB and AG are cut like). Moreouer the side FH of the triangle FHK is in the point D cut into two equall partes, as in the former Proposition it was proued, and FC∣ED also by the same is a parallelogramme: Wherefore the lines CE and FD are equall, by the 33, of the first. And for∣asmuch as the line FH subtendeth the angle BAG of an e∣quilater triangle, which angle is contained vnder the grea∣ter segment AH and the lesse segment AF•• therefore the line FH is in power double to the line AF or to the line AI the lesse segment, by the Corollary of the 16. of the fiue∣tenth. But the same line FH is in power quadruple to the line CE, by the 4. of the second: (for the line FH is double to the line CE). Wherefore the line AI being the halfe of the square of the line FH is in power duple to the line CE, to which the line FH was in power quadruple. Wherefore the side AG of the pyramis being diuided by an extreme and meane proportion, maketh th•• lesse segment AI in power duple to the side CE of the Icosahedron inscribed in it.

¶ A Corollary.

The side of an Icosahedron inscribed in a pyramis, is a residuall line. For the diameter of the Sphere which containeth the fiue regular bodies, being rationall, is in power ses∣quialtera to the side of the pyramis, by the 13. of the thirtenth: and therefore the side of the pyramis is rationall, by the definition: which side being diuided by an extreme and meane proportion, maketh

the lesse segment a residuall line, by the 6. of the thirtenth. Wherefore the side of the Icosahedron be∣ing commensurable to the same lesse segment (for the square of the side of the Icosahedron is the halfe of the square of the said lesse segment) is a residuall line, by that which was added after the 103. of the tenth booke.

¶The 6. Proposition. The side of a Cube containeth in power halfe the side of an equilater trian∣gular Pyramis inscribed in the said Cube.

FOr forasmuch as the side of the pyramis inscribed in the cube subtēdeth two sides of the cube which containe a right angle, by the 1. of the fiuetenth: it is manifest, by the 47. of the first, that the side of the pyramis subtēding the said sides, is in power duple to the side of the cube: Wherefore also the square of the side of the cube is the halfe of the square of the side of the pyramis. The side therefore of a cube containeth in power halfe the side of an equilater triangular pyra∣mis inscribed in the said cube.

¶ The 7. Proposition. The side of a Pyramis is duple to the side of an Octohedron inscri∣bed in it.

FOrasmuch as by the 2. of the fiuetenth it was proued, that the side of the Octohedron in∣scribed in a pyramis coupleth the midle sections of the sides of the pyramis. Wherefore the sides of the pyramis and of the Octohedron are parallels, by the Corollary of the 39. of the first: and therefore, by the Corollary of the 2. of the sixth, they subtend like trian∣gles. Wherfore (by the 4. of the sixth) the side of the pyramis is double to the side of the Octohedron, namely, in the proportion of the sides. The side therefore of a pyramis is duple to the side of an Octohedron inscribed in it.

¶ The 8. Proposition. The side of a Cube is in power duple to the side of an Octohedron inscri∣bed in it.

IT was proued in the 3. of the fiuetenth, that the diameter of the Octohedron inscribed in the cube, coupleth the centres of the opposite bases of the cube. Wherefore the said diameter is equall to the side of the cube. But the same is also the diameter of the square made of the sides of the Octohedron, namely, is the diameter of the Sphere which containeth it, by the 14. of the thirtenth. Wherefore that diameter being equall to the side of the cube, is in power double to the side of that square, or to the side of the Octohedron inscribed in it, by the 47. of the first. The side therefore of a Cube, is in power duple to the side of an Octohedron inscribed in it: which was requi∣red to be proued.

¶ The 9. Proposition. The side of a Dodecahedron, is the greater segment of the line which containeth in power halfe the side of the Pyramis inscribed in the sayd Dodecahedron.

SVppose that of the Dodecahedron ABGD the side be AB: and let the base of the cube inscribed in the Dodecahedron be ECFH, by the •••• of the fiuetenth. And let the side of the pyramis inscribed in the cube be CH, by the 1. of the fiuetenth.* 1.9 Wherefore the same pyramis is inscribed in the Dodecahedron, by the 10. of the fiuetenth. Then I say, that AB the side of the Dodecahedron is the greater segment of the line which contai∣neth in power halfe the line CH, which is the side of the pyramis inscribed in the

Dodecahedron.* 1.10 For forasmuch as EC the side of the cube be∣ing

[illustration]

diuided by an extreme and meane proportion maketh the greater segment the line AB, the side of the Dodecahedron, by the ••••rst Corollary of the 17. of the thirtenth: (For they are cont••ined in one and the selfe same Sphere (by the first of this booke): and the line EC the side of the cube contayneth in power the halfe of the side CH, by the 6. of this booke. Wherefore AB the side of the Dodecahedron, is the greater segment of the line EC, which containeth in power the halfe of the line CH, which is the side of the Dodecahedron inscribed in the pyramis. The side therefore of a Dodecahedron, is the greater segment of the line which con∣taineth in power halfe the side of the Pyramis inscribed in the said Dodecahedron.

¶The 10. Proposition. The side of an Icosahedron, is the meane proportionall betwene the side of the Cube circumscribed about the Icosahedron, and the side of the Dode∣cahedron inscribed in the same Cube.

SVppose that there be a cube ABFD, in which let there be inscribed an icosahedron CL∣IGOR, by the 14. of the fiuetenth.* 1.11 Let also the Dodecahedron inscribed in the same be EDMNPS, by the 13. of the same. Now forasmuch as CL the side of the Icosahedron is the greater segmēt of AB the side of the cube circumscribed about it, by the 3. Corolla∣ry of the 14. of the fiuetenth:* 1.12 and the side ED of the

[illustration]

Dodecahedrō inscribed in thesame cube is the lesse segmēt of the same side AB of the cube, by the 2. Corollary of the 13. of the fiuetenth: it followeth that AB the side of the cube be∣ing diuided by an extreme and meane proportion, maketh the greater segment CL the side of the Icosahedron inscribed in it, and the lesse segment ED the side of the Dodecahedron likewise inscrib••d in it. Wherefore as the whole line AB the side of the cube, is to the greater segment CL the side of the Icosahedron, so is the greater segment CL the side of the Icosahedron, to the lesse segment ED•• the side of the Dodecahedron, by the third definition of the sixth. Wherefore the side of an Icosahedron, is the meane proportionall betwene the side of the cube circum∣scribed about the Icosahedron, and the side of the Dodecahe∣dron inscribed in the same cube.

¶The 11. Proposition. The side of a Pyramis, is in power † 1.13 Octodecuple to the side of the cube in∣scribed in it.

FOr, by that which was demonstrated in the 18. of the fiuetenth, the side of the pyra∣mis is triple to the diameter of the base of the cube inscribed in it:* 1.14 and therefore it is in power nonecuple to the same diameter (by the 20. of the sixth). But the dia∣mer is in power double to the side of the cube, by the 47. of the first. And the dou∣ble of nonecuple maketh Octodecuple. Wherefore the side of the pyramis is in power Octodecuple to the side of the cube inscribed in it.

¶The 12. Proposition. The side of a Pyramis, is in power Octodecuple to that right line, whose

greater segment is the side of the Dodecahedron inscribed in the Pyramis.

FOrasmuch as the Dodecahedron and the cube inscribed in it, are set in one and the s••lf•• same pyramis, by the Corollary of the first of this booke: and the side of the pyramis cir∣cumscribed about the cube is in power octodecuple to the side of the cube inscribed, by the former Proposition: but the greater segment of the selfe same side of the cube, is the side of the Dodecahedron which containeth the cube, by the Corollary of the 17. of the thirtenth. Wherfore the side of the pyramis is in power octodecuple to that right line, namely, to the side of the cube, whose greater segment is the side of the Dodecahedron inscribed in the pyramis.

¶ The 13. Proposition. The side of an Icosahedron inscribed in an Octohedron, is in power duple to the lesse segment of the side of the same Octohedron.

FOrasmuch as in the 17. of the fiuetenth, it was proued, that the side of an Icosahedron inscri∣bed in a pyramis, coupleth together the two sections (which are produced by an extreme and meane proportion) of the side of the Octohedron which make a right angle: and that right angle is contained vnder the lesse segmentes of the sides of the Octohedron, and is subtended of the side of the Icosahedron inscribed: it followeth therefore, that the side of the Icosahedron which subtendeth the right angle, being in power equall to the two lines which containe the said angle, by the 47. of the first, is in power duple to euery one of the lesse segmētes of the side of the Octohedron which containe a right angle. Wherefore the side of an Icosahedron inscribed in an Octohedron, is in power duple to the lesse segment of the ••ide of the same Octohedron.

¶The 14. Proposition. The sides of the Octohedron, and of the Cube inscribed in it, are in power the one to the other † 1.15 in quadrupla sesquialter proportion.

SVppose that ABGDE be an Octohedron, and let the cube inscribed in it be FCHI. Then I say, that AB the side of the Octohedron, is in power quadruple sesquialter to FI the ••ide of the cube. Let there be drawen to BE the base of the triangle ABE a perpendicular AN: and againe let there be drawen to the same base in the triangle G••E the perpendicular GN: which AN & GN shall

[illustration]

passe by the centres F and I: and the line AF is duple to the line FN, by the Corollary of the 12. of the thirtenth. Wherfore the line AO is duple to the line OE, by the 2. of the sixth. For the lines FO and NE are parallels. And therefore the diameter AG is triple to the line FI. Wherfore the pow∣er of AG is * 1.16 noncuple to the power of FI. But the line AG is in power duple to the side AB, by the 14. of the thirtenth. Wherefore the square of the line AB, being ing the halfe of the square of the line AG, which is noncuple to the square of the line FI, i•• quadruple sesqui∣alter

to the square of the line FI. The sides therefore of the Octohed•••••• ••nd of the cube inscribed in it•• are in power the one to the other, in quadruple sesquialter proportion.

¶The 1••. Proposition. The side of the Octohedron, is in power quadruple sesquialter to that right line, whose greater segment is the side of the Dodecahedron inscribed in the same Octohedron.

FOrasmuch as in the 14. of this booke, it was proued, that the side of the Octohedron is in power quadruple sesquialter to the side of the cube inscribed in it: but the side of the cube being cut by an extreme and meane proportion, maketh the greater segment the side of the Dodecahedron circumscribed about it, by the 3. Corollary of the 13. of the fiuetenth: there∣fore the side of the Octohedron is in power quadruple sesquialter to that right line (namely, to the side of the cube) whose greater segment is the side of the Dodecahedron inscribed in the cube. But the Do∣decahedron and the cube inscribed one within an other, ar•• inscribed in one and the selfe same Octo∣hedron, by the Corollary of the first of this booke. The side therefore of the Octohedron, is in power quadruple sesquialter to that right line, whose greater segment is the side of the Dodecahedron inscri∣bed in the same Octohedron.

¶ The 16. Proposition. The side of an Icosahedron, is the greater segment of that right line, which is in power duple to the side of the Octohedron inscribed in the same Icosahedron.

SVppose that there be an Icosahedron ABGDFHEC: whose side let be BG or ••C•• and let the Octohedron ins••••ib••d in it be AKD••: and let the side therof be AL. Then I say, that the side ••C is the greater segment of that right line which is in power duple to the side AL. For forasmuch as figures inscribed and circumscribed haue o••e & the selfsame centre, by the Corollary of the ••1. of the fiuetenth, let the same be the point I. Now right line•• drawen by th•••• 〈◊〉〈◊〉 to the midle sections of the opposite sides, namely, the lines AID and KIL, do in the point I ••ut 〈…〉〈…〉 the other in∣••••

[illustration]

two ••quall 〈◊〉〈◊〉, and perpendicularly, by the Corolla∣ry of the 14. of the fiuetenth: and forasmuch as they couple the midle sections of the opposite lines BG and HF, ther∣fore they cut them perpendiularly:* 1.17 wherefore also the lines BG 〈…〉〈…〉, are parallels, by the 4. Corollary of the 14. of the 〈…〉〈…〉. Now then draw a line from B to H: and the sayd ••••ne BH shall be equall and parallel to the line KL, by the 33. of the first. But the line BH subtendeth ••w•• sides of the pentagon which is composed of the sides of the Icosahedron, namely, the sides BA and AH: Wherfore the line BH being cut by an extreme and meane proporti∣on maketh the greater segment the side of the pentagon, by the 8. of the thirtenth: which side is also the side of the Ico∣sahedron, namely, EC. And vnto the line BH the line KL•• is equall: and the line KL is in power duple to AL the side of the Octohedron, by the 47. of the first: for in the square AKDL the angle KAL is a right angle. Wherefore EC the side of the Icosahedron, is the greater seg∣ment of the line BH or KL, which is in power duple to AL ••he side of the Octohedron inscribed in the Icosahedron. Wherefore the side of an Icosahedron, is the greater segment of that right line, which is in power duple to the side of the Octohedron inscribed in the same Icosahedron.

¶The 17. Proposition. The side of a Cube is to the side of a Dodecahedron inscribed in it, in duple proportion of an extreame and meane proportion.

FOr it was manifes•• by the ••. corollary of the 13. of the fiuetenth, that the side of a cube diui∣ded by an extreame and meane pr••portion, maketh the lesse segment, the side of the dode∣cahedron inscribed in it: but the whole is to the lesse segment in duple proportion of that in which it is to the greater, by the 10. diffinitiō of the fifth. For the whole, the greater segmēt, and the lesse, are lines in continuall proportion, by the 3. diffinition of the sixth. Wherefore the whole namely the side of the cube, is to the side of the dodecahedron inscribed in it, namely, to his lesse seg∣ment, in duple propo••tion of an extreame and meane proportion', namely, * 1.18 of that which the whole hath ••o the greater segmen••, by the 2. of the fourtenth.

¶ The 18. Proposition. The side of a Dodecahedron is, to the side of a Cube inscribed in it, in con∣uerse proportion of an extreame and meane proportion.

IT was proued in the 3. corollary of the 13. of the fiuetenth, that the side of a Dodecahe∣d••on circumscribed about a Cube, is the greater segment of the side of the same Cube. Wherefore the whole side of the Cube inscribed is to the greater segment, namely, to the side of the dodecahedron circumscribed, in an extreame and meane proportion: wherefore by conuersion, the greater segment, that is, the side of the dodecahedron, is to the whole, namely, to the side of the Cube inscribed, in the conuerse proportion of an extreame and meane proportion, by the 13. diffinition of the fiueth.

¶ The 19. Proposition. The side of an Octohedron, is sesquialter to the side of a Pyramis inscri∣bed in it.

FOr (by the corollary of the 14. of the thirtenth) the Octohedron is cu••te into two quadrilater py••amids, one of which let be ABGDF:* 1.19 and let the centres of the cir∣cles which contayne the 4. bases of the Octohedron be K, E, I, C. And dr••w these right lines KE, ••I, IC, CK, and EC. Wherefor•• K••IC is a square, and one of the bases of the cube inscribed in the Octohedron, by the 4. of the fiuetenth. And foras∣much as the angles of a cube and of the pyramis in••cribed in it, are for in the centres of the bases of the Octohedron circumscribed about the cube, by the 6•• of the fiue∣tenth: and the side of the pyramis coupleth the opposite angle•• of the base of th•• cube, by the 1. of the fiuetenth, it is manifest that

[illustration]

the line EC is the side of the pyramis inscribed in the Octohedron ABGDF. Then I say that GD the side of the Octohedron, is sesquialter to EC the side of the pyramis inscribed in it. From the poynt A draw to the bases BG and FD perpendi∣culars AN and AM•• which (by the corollary of the 12. of the thirtenth) shall passe by the centres E and C.* 1.20 And draw the line NM. Now forasmuch, as B∣GDF is a square, by the 14. of the thirtenth, the lines NG and MD shall be parallels and equall. For the lines BG and FD are by the perpendicu∣lars cutte into two equall partes in the poyntes N and M (by the 3. of the third). Wherefore the lines NM and GD shall be parallels and equall, by the 33. of the first. And forasmuch as the lines AN and AM which are the perpendiculars of equall and like•• triangles are c••t a like in the poyntes •• and C, the lines EC and NM•• shall be parallels, by the 2. of the sixth: and therefore by the corollary of the same, the triangles AEC, and ANM shall be like. Wherefore as the line AN is to the line AE, so is the line NM to the line EC by the 4. of the sixth. But the line AN is sesquialter to the line AE, for the line AE is duple to the line EN, by the corollary of the 12•• of the thirtenth•• wherefore the line NM, or the line GD which is equall vnto it, is sesquialter to the line EC. Wherefore GD the side of the Octohedron, is sesquialter to EC the sid•• of the pyra∣mis inscribed in it.

¶ The ••0. Proposition. If from the power of the diameter of an Icosahedron, be taken away the power tripled of the side of the cube inscribed in the Icosahedron: the power remayning shall be sesquitertia to the power of the side of the I∣cosahedron.

LEt there be taken an Icosahedron ABGD: and l•••• two bases of the cube inscribed in it, ioyned together be EHKL and LKFC: and let the diameter of the cube be FH and the side be EH, and let the diameter of the Icos••h••dron be ••G, and the side be AB. Then I say, that if from the power of the diamet•••• GB, be taken away the power tripled of EH the side of the cube:* 1.21 the power remayning, shall be sesquetertia to the power of AB the side of the Icosahedron. For forasmuch as the centres of inscribed and circum∣scribed figures, are in one & the selfe same poynt, by the ••••rollary of the 21. of the 〈◊〉〈◊〉 the diame∣ters BG and FH shall in one and the selfe same poynt

[illustration]

cutte the one the other into two equall partes: for we haue before by the same corollary taught, that the toppes of equall and like pyramids doo in that poynt concurre, let that poynt be the centre I. Now the an∣gles of the cube, which are at the poyntes F and H are set at the centres of the bases of the Icosahedron by the 11. of the fiuetenth•• Wherefore the line FH shall be perpendicular to both the bases of the Icosahedrō, by the corollary of the assūpt of the 16. of the twelfth. Wherefore the line IB contayneth in power the two lines IH and HB, by the 47. of the first. But the line H∣B, is drawne from the centre of the circle which con∣tayneth the base of the Icosahedron namely, the angle B is placed in the circumference, and the poynt H is the centre. Wh••refore the whole line BG contayneth in power the whole lines FH and the diameter of the circle (namely, the double of the line BH) by the 15•• of the fiueth. But the diameter which is double to the line HB is in power sesquiterti•• to the side of the e∣quilater triangle inscribed in the same circle•• by the corollary of the ••••. of the thirtenth. For it is in pro∣portion to the side•• as the side is to the perpendicular, by the corollary of the 8. of th•• ••ixth. And FH the diameter of the cube, is in power triple to EH the side of the same cube, by the 15•• of the thirtenth. If therefore from the power of the diameter BG, be taken away the power tripled of EH the side of the cube inscribed•• that is•• the power of the line FH: the residue (namely, the power of the diameter of the circle which is duple to the line HB shall be sesquiterti•• to the side of the triangl•• inscribed in that circle: which selfe side is AB the side of the Icosahedron. If therfor•• from the power of the diameter of an Icosahedrō, be takē away the power tripled of the side of the cube inscribed in the Icosahedron, the power remayning shall be s••squitertia ••o the power of the side of the Icosahedron.

A Corollary.

The diameter of the Icosahedron, contayneth in power two lines, namely, the diameter of the cube inscribed, which coupleth the centres of the opposite ba∣ses, and the diameter of the circle which contayneth the base of the Icosahedron. For it was manifest, that BG the diameter contayneth to power the line FH which doupleth the centres, and the double of the line BH, that is, the diameter of the circle contayning the bas•• wherein i•• the centre H••

¶ The 21. Proposition. The side of a Dodeca••edron is the lesse segment of that right line, which is in power duple to the side of the Octohedron inscribed in the same Do∣decahedron.

LEt ther be taken a Dodecahedron AB••DCT, one of whose sides let be AB. And let the Octohedron inscribed in the Dodecahedron be EFLKI: one of whose sides let be EF. Then I say that AB the side of the Dodecahedron, is the lesse seg∣ment of a certayne right ••ine •• cut

[illustration]

by an extreame and meane pro∣portion) which is in power duple to EF the side of the Octohedrō inscribed in the Dodecahedron. Draw the diameters EL and FK of the Octohedron. Now they couple the midle sections of the opposite sides of the dode∣cahedron AB and GD, (by the 9. of the fiuetenth, & 3. corollary of the 17. of the thirtēth) & euery one of those diameters being diuided by an extreame and meane proportion, doo make the lesse segment, the side of the dodecahedron, by the 4. corollary of the same. Wherefore the side AB is the lesse segment of the line FK. But the line FK contayneth in power the two equall lines EF & EK, by the 47. of the first: for the angle FEK is a right angle of the square FEKL of the Octohedron. Wherfore the line FK is in pow∣er duple to the line EF. Wherefore the line AB (the side of the dodecahedron) is the lesse segment of the line FK, which is in power duple to EF the sid•• of the Octohedron. The side therefore of a Dodecahedron i•• the lesse segment of that right line, which is in power duple to the side of the Octohedron inscribed in the same Dod••cahedron.

¶ The 22. Proposition. The diameter of an Icosahedron is in power sesquitertia to the side of the same Icosahedron, and also is in power sesquialter to the side of the Pyra∣mis inscribed in the Icosahedron.

FOr forasmuch as it hath bene proued (by the 10. of this booke) that if frō the power of the diameter of the Icosahedrō be taken away the triple of the power of the side of the cube inscribed in it, there shalbe left a square sesquitertia to the square of the side of the Icosahedron: But the power of the side of the cube tripled, is the diameter of the same cube, by the 15. of the thirtēth: And the cube, & the pyramis inscribed in it are contai∣ned in one & the self same sphere, by the first of this booke, and in one & the self same Icosahedron by the corollary of the same. Wherfore one and the selfe same diameter of the cube, or of the sphere which cōtaineth the cube and the pyramis, is in power sesquialter to the side of the pyramis by the 13. of the thirtenth. Wherfore it followeth, that if from the diameter of the Icosahedron, be ta∣ken away the triple power of the side of the cube, or the sesquialter power of the side of the pyramis, which are the powers of one and the selfe same diameter, there shall be left the sesquitertia power of the side of the Icosahedron. The diameter therefore of an Icosahedron is in power sesquitertia to the side of the same Icosahedron, and also is in power sesquialter to the side of the Pyramis inscribed in the Icosahedron.

The 23. Proposition. The side of a Dodecahedron is to the side of an Icosahedron inscribed in it, as the lesse segment of the perpendicular of the Pentagō, is to that line which is drawne from the centre to the side of the same pentagon.

* 1.22LEt there be taken a Dodecahedron ABGDFSO. Whose side let be AS or SO: and let the Icosahedron inscribed in it be KLNMNE, whose side let be KL. From the two angles of the pentagon•• BAS and FAS of the Dodecahedron, namely, from the angle•• •• and F, let there be drawne to the common base AS perpendicular lines BC & FC: which shal passe by the centres K & L of the sayd pentagons, by the corollary of the 10. of the thirtēth. Draw

the lines BF and RO. Now forasmuche as the

[illustration]

line RO subtendeth the angle OFR of th•• pentagon of the dodecahedron, it shall cut the line FC by an extreme and meane proportion, by the 3. of this booke, let it cut it in the poynt I. And forasmuche as the line KL is the side of the Icosahedron inscribed in the Dodecahe∣dron, it coupleth the cētres of the bases of the dodecahedron: for the angles of the Icosahe∣dron are set in the centres of the bases of the dodecahedron, by the 7. of the fiuetenth. Now I say that SO, the side of the dodecahedron is to KL the side of the Icosahedron, as the lesse segment IF of the perpendicular line CF, is to the line LC which is drawne from the centre L to AS the side of the pentagon. For foras∣much as in the triangle BCF the two sides CB and CF are in the centres L and K cut like proportionally,* 1.23 the lines BF and KL shalbe pa∣rellels, by the 2. of the sixth. Wherefore the tri∣angles BCF, and KCL shall be equiangle, by the corollary of the same. Wherfore as the line CL is to the line KL•• so is the line CF to the line BF, by the 4. of the sixth. But CF maketh the lesse segment the line IF, by the 3. of this booke, and the lin•• BF maketh the lesse segment the line SO, namely, the side of the Dodecahedron, by the 2. corollary of the 13. of the fiuetenth. For the line BF which coupleth the angles B and F of the bases of the dodeca∣hedron, is equall to the side of the cube, which contayneth the dodecahedron, (by the .13. of the fiuetenth). Wherefore as the whole line C••, is to the whole line BF, so is the lesse segment IF to the lesse segment SO (by the 2. of the 14). But as the line CF is to the line BF, so is the line CL proued to be to the line KL. Wherefore as the line IF is to the line SO, so is the line CL to the line KL. Wherefore alternately by the 16. of the fiueth, as the line IF the lesse segment of the per∣pendicular of the pentagon FAS, is to the line LC which is drawne from the centre of the pentagon, to the base, so is the line SO the side of the Dodecahedron to th•• line KL the side of the Icosahe∣dron inscribed in it. The side therfore of a Dodecahedron is to the side of an Icosahedron inscribed in it, as the lesse segment of the perpendicular of the pentagon, is to that line which is drawne from the cen••re to the side of the same pentagon.

¶ The 24. Proposition. If halfe of the side of an Icosahedron be deuided by an extreme & meane proportion: and if the lesse segment thereof be taken away from the whole side, and againe from the residue be taken away the third part: that which remaineth shall be equal to the side of the Dodecahedron inscribed in the same Icosahedron.

SVppose that ABGDF be a

[illustration]

pentagon,* 1.24 containing fiue sides of the Icosahedron by the 16. of the thirtenth, and let it be inscribed in a circle, whose centre let be the point E. And vpon the sides of the pentagon, let there be rea∣red vp triangles, making a solide angle of the Icosahedron at the poynt I, by the 16. of the thirtenth. And in the circle ABD, inscribe an equilater triangle AHK. From the centre E drawe to HK the side of the triangle, and GD the side of the penta∣gon, a perpendicular line, which let be E∣CNM. And draw these right lines EG, ED, IG and ID. And deuide the line BG into two equal parts in the poynt T. And

drawe these lines IN, IT, TN, ET. And

[illustration]

forasmuche as in the perpendiculars IT & IN are the centres of the circles which containe the equilater triangles IBG, & IGD, by the corollarye of the first of the thirde. Let those centres be the points S and O. And draw the line SO. Deuide the line TB the half of BG the side of the Icosahedron by an extreme and meane proportion in the poynt R, by the 30. of the sixth, and let the lesse segment therof be RB. And forasmuch as the line SO cou¦pleth the centres of the triangles IBG, & IGD, it is by the 5. of the fiuetenth, the side of the Dodecahedrō inscribed in the Icosahedron, whose side is the line BG. From the side BG take away ••R the lesse segment of the halfe side. And from the residue GR take away the third part GV (by the 9. of the sixth.) Then I say that the residue RV is equal to SO the side of the Dodecahedron inscribed.* 1.25 For forasmuch as the perpendicular EN is in the poynt C deuided by an extreme and meane proportion, by the corollary of the first of the fourtenth, and the greater segment therof is the line EC, and vnto the line EC the line CM is equal, by the corollary of the 12. of the thirtenth: wherefore the line EC is to the line CN, as the line CM is to the same line CN, by the 7. of the fiueth. But as the line EC is to the line CN, so is the whole line ••N, to the greater segment E∣C, by the 3. diffinition of the sixth. Wherefore (by the 11. of the fiueth), as the whole line EN is to the greater segment EC, so is the line CM to the line CN. Wherefore the line CM, is deuided by an extreme and meane proportion in the poynt N, namely, is deuided like vnto the line EN, by the 2. of the fourtenth. Wherfore the line EM excedeth the line EN by the lesse segment of his halfe, namely, by MN. And forasmuche as EGD is the triangle of an equilater and equiangle pentagon ABGDF, and ETN is likewise the triangle of the like pentagon inscribed in the pentagon ABGDF: Therefore by the 20. of the sixth, the triangle ETN is like to the triangle EGD•• Wherefore as the line EG is to the line EN, so by the 4. of the sixth, is the line GD to the line NT. Wherefore the line GD (or BG which is equal vnto it) excedeth the line NT by the lesse segment of the halfe of BG. For the line EG did in like sort excede the line EN. But that lesse segment is the line BR. Wherefore the residue RG is equal to the line TN. And forasmuch as IBG is an equilater triangle: the perpendicular ST shalbe the halfe of the line SI which is drawne from the centre, by the corollary of the 12. of the thirtenth: wher∣fore the line IT excedeth the line IS by his third part. And forasmuche as the line SO which coupleth the sections, is a parallel to the line TN, by the 2. of the sixth. For the equal perpendiculars IT, and IN are cut like in the poynts S & O: therfore the triangles ITN & I••O, are like by the corollary of the second of the sixth. Wherfore as the line IT is to the line IS, so by the 4. of the sixth is the line TN to the line SO. But the line IT excedeth the line IS by a third part: wherfore the line TN, excedeth the line SO by a third part: but the line TN is proued equal to the line RG. Wherfore the line RG exce∣deth the line SO by a third part of himself, which is GV. Wherfore the residue RV, is equal to the line SO, which is the side of the dodecahedron inscribed in the Icosahedron, whose side is the line BG. If therfore halfe of the side of an Icosahedrō, be deuided by an extreme & meane proportion: and if the lesse segment therof be taken away from the whole side, and againe from the residue be takē away the third part: that which remaineth shall be equal to the side of the dodecahedron inscribed in the same Icosahedron.

The 25. Proposition. To proue that a cube geuen, is to a trilater equilater pyramis inscribed in it, triple.

SVppose that the cube geuen, be ABCH: and let the pyramis inscribed in it be AGDF. Then I say that the cube ABCH is triple to the pyramis AGDF. For forasmuche as the base AFD is common to the pyramis AFDB and AFDG, the pyramis AFDB shalbe set without the pyramis AFDG. Likewise the rest of the bases of the inscribed pyramis are common to the rest of the pyramids sorte without: which are these: the pyramis AGDC vppon

the base AGD: the pyramis AGF•• vpon the base AGF••

[illustration]

and the pyramis GDFH vpon the base GDF. Which py∣ramids taken without, are foure in number, equal and like the one to the other, by the ••. diffinition of the eleu••th. For euery one of them is contained vnder thr•• halfe squares of the cube, and one of the bases of the pyramis inscribed. Wherfore euery one of thē is cōtained vnder the halfe base of the cube, & the altitude of the cube. As the pyramis AL∣GF, hath to his base halfe of the square EH, namely, the tri∣angle EGF, & hath to his altitude, the altitude of the cube, namely, the line AE. Wherfore the sayd pyramis is the sixth part of the cube. For if the cube be deuided into two pris∣mes, by the plaine CBFG, the prisme ACBGEF, shalbe tri∣ple to the pyramis AEGF, hauing one & the selfe same base with it EGF, and one and the selfe same altitude EA, by the first corollary of the 7. of the twelueth. Wherefore the sayd outward pyramis AEGF is the sixth part of the whole cube. Wherfore also the same pyramis together wyth the other thre outwarde pyramids AFDB, AGDC, and GDFH, ••hal containe two third partes of the cube. Wherfore the resi∣due, namely, the pyramis inscribed AGDF, shal contain one third part of the cube. And therefore conuersedly the cube shall be triple to it: wherefore we haue proued ••hat a cube geuē triple to a trilater & equilater pyramis inscribed in it.

¶The 26. Proposition. To proue that a trilater equilater Pyramis is duple to an Octohedron in∣scribed in it.

LEt there be taken a trilater Pyramis ABCD: whose sixe sides let be cut into two e∣quall partes, in the pointes E, K, F, L, G, and H•• inscribing thereby an Octohedron in the pyramis, by the 2. of the fiuetenth. Wherefore the pyramids AEGH, BEFK, CFGL, & DKHL, fall without the Octohedron inscribed, by the same second of the fiuetenth. But the outward Pyramids (namely, AEGH, and the three other) are like vnto the whole pyramis, by the 7. definition of the eleuenth. For the bases of the whole pyramis are by parallel lines drawen in them cut into like triangles, by the Corollary of the 2. of the sixth, of which the foresayd pyra∣mids

[illustration]

are made. Wherefore the whole pyramis is to euery one of them in treble proportion of that in which the sides of like proportion are, by the 8. of the twelfth. But by construction, the proportion of the side A•• to the side A•• is duple. Wherefore the whole pyramis ABCD is octuple to the pyramis AEGH, and so is it to euery one of the pyramids which are equall to AEGH. For duple proportion multiplyed into it selfe twise maketh octuple. Wherefore it followeth that the 4. pyramids AEGH, ••EFK, CFGL, and DKHL, taken together, make the halfe of the whole pyramis ABCD. Where∣fore the residue, namely, the Octohedron EG∣LKHF, is the other half of the pyramis. Wher∣fore the pyramis is duple to the Octohedron. Wherefore we haue proued that a trilater e∣quilater pyramis is duple to an Octohedron in∣scribed in it.

¶ The 27. Proposition. To proue that a Cube is sextuple to an Octohedron inscribed in it.

LEt there be taken a cube ABCD, EFGH: whose 4. standing lines AE, BF, CH, & DG, let be cut into two equall partes in the pointes I, K, M, L: and•• by those pointes let there be extended a plaine KLMI: which shall be a square, and parallel to the squares BC & FH, by the 15. of the eleuenth. Wherefore in it shall be the base which is common to the two pyramids of the Octohedron inscribed in the cube, by that which was demonstrated in the third of the fiuetenth. Let that base be NPRQ, coupling

[illustration]

the centres of the bases of the cube: and vpon that base let be set the two pyramids of the Octohedron, which let be NPQRS, and NPQRT. And forasmuch as those two pyramids taken together, haue their altitude equall with the altitude of the whole cube,* 1.26 ech of them a part hath to his alt••••ude halfe the altitude of the cube, namely, halfe of the side of the cube, as the line KB. And forasmuch as the square KLMI is double to the square NRQP, by the 47. of the first: the other squares of the cube shall also be dou∣ble to the square NRQP. And forasmuch as the cube, as it was manifest by the last of the fiuetenth, is resolued into sixe pyramids, whose bases are the bases of the cube, & the altitudes the lines drawen frō the centre to the bases, which are equall to halfe the side of the cube•• it followeth that e∣uery one of the sixe pyramids of the cube, hauing his base double to the base of eche of the pyramids of the Octohe∣dron, and the selfe same altitude that the said pyramids of the Octohedrō haue, is double to either of the pyramids of the octohedrō, by the 6. of the twelfth. And forasmuch as euery one of the pyramids of the cube is equall to the two pyramids of the Octohedron, the sixe pyramids of the cube shall be sextuple to the whole Octohedron. Wherefore it is manifest, that a cube is sextuple to an Octohedron inscri∣bed in it.

¶The 28. Proposition. To proue that an Octohedron is quadruple sesquialter to a Cube inscri∣bed in it.

SVppose that the Octohedron geuen be ABCDEF: and let the cube inscribed in it be GHIK, VQRS. Then I say, that the Octohedron is quadruple sesquialter to the cube inscribed in it. Forasmuch as the lines drawen from the centre of the Octohedron, or of the Sphere which containeth it, vnto the centres of the bases of the Octohedron, are pro∣ued equall, by the 21. of the fiuetenth: and the angles of the cube are set in the centres of those bases, by the 4. of the fiuetenth: it followeth, that the selfe same right lines are drawen from one and the selfe same centre of the cube and of the Octohedron: for they haue eche one and the selfe same centre, by the Corollary of the 21. of the fiuetenth. Let that centre be the point T. Wherefore the base BDFC, which cutteth the Octohedron into two equall and quadrilater pyra∣mids, by the Corollary of the 14. of the thirtenth, shall also cut the cube into two equall partes, by the Corollary of the 39. of the eleuenth. For it passeth by the centre T, by that which was demonstra∣ted in the 14. of the thirtenth. And forasmuch as the base of the cube is in the 4. centres G, H, I, K, of the bases of the pyramis ABDFC, a plaine LNOM, extended by those pointes, shall be parallel to the plaine BDFC, by that which was demonstrated in the 4. of the fiuetenth, and shall cut the pyra∣mis in the pointes L, N, O, M: and the lines LN, BD, and NO, DF, shall be parallels, so also shall the lines OM, FC, and LM, BC: and the square GHIK of the cube shall be inscribed in the square LNOM, by the same. Wherefore the square LNOM is duple to the square GHIK, by the 47. of the first. From the solide angle A, let there be drawen to the plaine superficies BDFC, a perpen∣dicular, which let fall vppon it in the point T, and let the same perpendicular be AT, cutting the plaine LNOM in the point P. And it shall also be a perpendicular to the plaine LNOM, by the Corollary of the 14. of the eleuenth. Againe from the angle BAD of the triangle ADB, let there be drawen by the centre H of the triangle, to the base a line AHX. Wherefore the line AX is ses∣quialter to the line AH, by the Corollary of the 12. of the thirtenth. Wherefore the line AH is duple to the line HX. But the other lines AB, AD, AF, AC, and the perpendicular APT, are cut like vnto the line AHX, by the 17. of the eleuenth: Wherefore the line AP is double to the line PT. Wherefore the line AP is the altitude of the cube, for the line PT is the halfe thereof.

And forasmuch as vpon the base

[illustration]

GHIK of the cube, and vnder the altitude AP of the same cube, is set the pyramis AGHIK: the said pyramis is the third part of the cube, by the Corollary of the 7. of the twelfth. But vnto the pyramis AGHIK the pyramis ALNOM is duple, by the 6. of the twelfth, for the base of the one is double to the base of the other. Wherefore the pyramis ALNOM is two third partes of the cube. And forasmuch as the pyramids ALNOM, and ABDFC, are like, by the 7. defi∣nition of the eleuenth: therefore they are in triple proportion of that in which the sides of like pro∣portion AH to AX, or AL to AB, are, by the Corollary of the 8. of the twelfth. But the side AB is proued to be sesquialter to the side AL. Wherefore the pyramis A∣BCDF is to pyramis ALN∣OM, as 27. is to 8. (that is, in ses∣quialter proportion tripled: for the quantitie or denomination of sesquialter proportion, namely, 1 ½ multiplied into it selfe once maketh 2¼, which againe multiplyed by 1½ maketh 3 1/••, that is, 27. to 8.). But of what partes the pyramis ALNOM containeth 8, of the same the cube con∣taineth 12: namely, is sesquialter to the pyramis. Wherefore of what partes the cube containeth 12, of the same the whole Octohedron (which is double to the pyramis ABDFC) containeth 54. Which 54. hath to 12. quadruple sesquialter proportion. Wherefore the whole Octohedron is to the cube in∣scribed in it, in quadruple sesquialter proportion. Wherefore we haue proued that an Octohedron ge∣uen is quadruple sesquialter to a cube inscribed in it.

¶A Corollary.

An Octohedron is to a cube inscribed in it, in that proportion that the squares of their sides are. For by the 14. of this booke, the side of the Octohedron is in power quadruple sesquialter to the side of the cube inscribed in it.

¶The 29. Proposition. To proue that an octohedrō geuē, is * 1.27 tre∣decuple sesquialter to a trilater equila∣ter pyramis inscribed in it.

LEt the octohedron ge••en, be AB: in which let

[illustration]

there be inscribed a cube FCED, by the 4. of the fiuetenth, and in the cube let there be inscribed a pyramis FEGD, by the ••. of the fiuetenth. And forasmuche as the angles of the pyramis are (by the same first of the fiuetenth) set in the angles of the cube: and the angles of the cube are set in the centres of the bases of the Octohedron, namely, in the poyntes F, E, C, D, G by the 4. of the fiuetenth. Wherfore the angles of the pyramis, are set in the centres F, C, E, D of the octohedron. Where∣fore the pyramis FEDG is inscribed in the octohedron (by the 6. of the fiuetenth.) And forasmuche as the octohe∣dron

AB is to the cube FCED, inscribed in it quadruple

[illustration]

sesquialter (by the former propositiō): and the cube CDEF is to the pyramis FEDG inscribed in it triple, by the 25. of booke: wherefore three magnitudes being geuen, namely, the octohedron, the cube and the pyramis, the proportion of the extremes (namely, of the octohedron to the piramis) is made of the proportions of the meanes, (namely, of the octohedron to the cube, and of the cube to the pyramis,) as it is easie to see by the declaration vpon the 10. diffinition of the fiueth. Now then multiplying the quantities or denomi¦nations of the proportions (namely, of the octohedron to the cube which is 4 1/••, and of the cube to the pyramis, which is 3) as was taught in the diffinition of the sixth, there shalbe produced 13 1/••, namely, the proportion of the octo¦hedron to the pyramis inscribed in it. For 4 ½, multiplyed by 3. produce 13 ½. Wherefore the Octohedron is to the pyramis inscribed in it in tredecuple sesquialter proportion. Wherefore we haue proued that an Octohedron is to a tri∣later equilater pyramis inscribed in it, in tredecuple sesqui∣alter proportion.

¶ The 30. Proposition. To proue that a trilater equilater Pyramis, is noncuple to a cube inscribed in it.

SVppose that the pyramis geuen, be ABCD, whose two bases let be ABC, and DBC, and let their centres be the poynts G and I. And from the angle A, draw vnto the base B∣C a perpendicular AE: likewise from the angle D draw vnto the same base BC, a per∣pendicular DE: and they shal concurre in the section E by the 3. of the third and in them shalbe the cētres G and I, by the corollary of the first of the third. And forasmuch as the line AD is the side of the pyramis, the same AD shall be the diameter of the base of the cube which cōtaineth the pyramis, by the 1 of the fiuetēth.* 1.28

[illustration]

Draw the line GI. And forasmuch as the line GI coupleth the centre•• of the bases of the pyramis: the saide line GI shalbe the diameter of the base of the cube inscribed in the pyramis by the 18. of the fiuetenth. And forasmuche as the line AG is double to the line GE, by the corollarye of the twelueth of the thirtenth: the whole line AE shal be triple to the line GE: and so is also the line DE to the line IE. Wherefore the lines AD and GI are parallels, by the 2. of the sixth. And therefore the triangles AED, and GEI are like•• by the corollary of the same. And forasmuch as the tri∣angles AED, and GEI are like, the line AD•• shalbe triple to the line GI, by the 4. of the sixth. But the line AD is the dia∣meter of the base of the cube circumscribed about the py∣ramis ABCD, and the line GI is the diameter of the base of the cube inscribed in the pyramis ABCD: but the diameters of the bases are equemultiplices to the sides (namely, are in power duple). Wherfore the side of the cube circumscribed about the pyramis ABCD, is triple to the side of the cube, inscribed in the same piramis, by the 15. of the fiueth: but like cubes are in triple proportion the one to the other of that in which their sides are, by the 33. of the eleuenth: and the sides are in triple proportion the one to the other: Wherfore triple taken thre times bringeth forth twenty seuencuple, which is 27. to 1: for the 4. termes 27.9.3.1, being set in triple proportion: the proportion of the first to the fourth, namely, of 27. to 1. shalbe triple to the proportion of the first to the second, namely, of 27. to 9, by the 10. diffinition of the fiueth: which proportion of 27. to 1. is the proportiō of the sides tripled, which proportiō also is found in like solides. Wherefore of what partes the cube circumscribed containeth 27. of the same, the cube inscribed containeth one: but of what partes the cube circumscribed, containeth 27. of the same, the pyramis inscribed in it, containeth 9. by the 25. of this booke: wherfore of what partes the pyramis AB CD containeth 9. of the same, the cube inscribed in the pyramis, containeth one. Wherefore we haue proued that a trilater and equilater pyramis, is non••cuple to a cube inscribed in it.

¶ The 31. Proposition. An Octohedron hath to an Icosohedron inscribed in it, that proportion, which two bases of the Octohedron haue to fiue bases of the Icosahedron.

SVppose that the octohedron geuen be ABCD, and let the Icosahedron inscribed in it, be F∣GHMKLIO. Then I say that the octohedron is to the Icosahedron, as two bases of the octohedron, are to fiue bases of the Icosahedron. For forasmuche as the solide of the octohe∣dron consisteth of eight pyramids, set vpon the bases of the octohedron,* 1.29 and hauing to theyr altitude a perpendicular line drawne from the centre to the base: let that perpendicular be ER, or ES, being drawne from the centre E (which centre is common to either of the solides, by the corollary of the 21. of the fiuetenth) to the centres of the bases, namely, to the poyntes R and S. Wherefore for that thre pyramids are equal and like, they shalbe equal to a prisme set vpon the selfe same base, and vnder the selfe same altitude, by the corollary of the seuenth of the twelueth. But vnto this prisme is double that prisme which is set vpon the self same base, and hath his altitude duple, namely, the whole line RS by the corollary of the 25. of the eleuenth: for it is equal to the two equal and like prismes whereof it is composed. Wherfore the prisme set vpō the base of the octohedron, and hauing to his altitude the line RS is equal to six pyramids, set vpon six bases of the Octohedron, and hauing to their altitude the line ER. So there remaine two pyramids (for in the octohedron are 8. bases) which shall be equal to the prisme which is set vpon the third part of the base of the octohedron, and vnder the altitude RS. For prismes vnder one and the selfe same altitude, are in proportion the one to the other, as are their bases, by the corollary of the 7 of the

[illustration]

twelueth. Wherefore the two prismes which are set vppon the base of the octohedron, and vp∣on a third part therof, and vnder the altitude RS, are equal to the 8. pyramids of the Octohedron, or to the whole solide of the oc∣tohedron. And forasmuch as the Icosahedron inscribed in the oc∣tohedron, hathe his bases set in the bases of the Octohedron, by the 17. of the fiuetenth: it follow∣eth that the pyramids set vppon the bases of the Icosahedron, & hauing to their toppes one and the selfe same centre E, are con∣tained vnder the selfe same alti∣tude, that the pyramids of the octohedron are cōtained vnder. namely, vnder the line ER, or ES. And therefore a prisme, set vpon the base of the Icosahedron, and hauing his altitude double to the altitude of the pyramis, namely, the whole line RS, is equal to sixe pyramids set vpon the base of the Icosahedron, and vnder the altitude ER or ES, as we haue pro∣ued in the octohedron. Wherfore the 20. pyramids, set vpon the 20. bases of the Icosahedron, are equal to thre prismes set vpon the base of the Icosahedron, and vnder the altitude RS, and moreouer to an o∣ther prisme set vppon a thirde part of the base of the Icosahedron and vnder the same altitude RS, which prisme is a thirde part of the former prisme, by the corollarye of the 7. of the twelueth: for their proportion is as the proportion of the bases. Wherfore two prismes set vpon the base of the octo∣hedron, and a third part therof, and vnder the altitude RS, is to 4. prismes set vpon three bases of the I∣cosahedron, and a third part thereof, and vnder the same altitude RS, in the same proportion that the bases are, that is, as 4. third partes of the base of the Octodron (which are equal to one base, and 1/••;) to ten third partes of the base of the Icosahedron (which are equal to thre bases & 1/••;) or as two third partes of the base of the Octohedron, are to fiue thirde partes of the base of the Icosahedron. But two thirde partes of the base of the Octohedron, are to fiue thirde partes of the base of the Icosa∣hedron, as two bases are to fiue bases (by the 15. of the fifth, for they are partes of equemultiplices:) And two prismes of the Octohedron are to 4. prismes of the Icosahedron, as the solide of the Octohedron is to the solide of the Icosahedron, when as eche are equal to eche of the solides: Wherefore (by the ••••. of the fiueth) the solide of the Octohedron, is to the solide of the Icos••hedron inscribed in it, as two bases of the Octohedron, are to fiue bases of the Icosahedron. An Oc∣tohedron

therfore is to an Ico••ahedron inscribed in it, in that proportion, that two bases of the Octo∣hedron, are to fiue bases of the Icosahedron.

¶ The 32. Proposition. The proportiō of the solide of an Icosahedron to the solide of a Dodecahe∣dron inscribed in it, consisteth of the proportion of the side of the Icosahe∣dron to the side of the Cube contayned in the same sphere, and of the pro∣portion tripled of the diameter to the line which conpleth the centers of the opposite bases of the Icosahedron.

* 1.30SVppose that there be •• Dodecahedron, whose diameter let be HI, and let the Icosahe∣dron contained in the same sphere be ABGC, whose dimetient let be AC. And let the right line which coupleth the centres of the opposite base•• be BG. And let the dodeca∣hedron inscribed in the Icos••hedron be that which is set vpon the diameter BG, by the 5. of the fiuetenth. And let the side of the cube be DE, and let the side of the Icosahe∣dron be D••, both the sayd solides being described in one and the selfe same sphere. Thē I say that the proportion of the solide of the Icosahedron ABCG to the solide of the dodecahedron set vpon the diameter BG, inscribed in it, consisteth of the proportion of the line DF to the line DE, and of the proportion tripled of the line AC to the line BG.* 1.31 For forasmuch as the solide of the Icosa∣hedron ABGC is to the solide of the dodecahedron HI, being contayned in one and the selfe same sphere, as DF is to D••,

[illustration]

by the 8. of the fourtenth But the dodecahedron whose diameter is HI, is to the dodecahedron whose diamer is BG, in treble propo••tiō of that in which the diameter HI is to the diameter B∣G, by the corollary of the 17. of the twelfth: & the lines HI and AC are equall by supposition (namely, the diameters of one and the selfe same sphere). Wherefore as HI is to BG, so is AC to BG. Wherefore the pro∣portion of the extremes, namely, of the Icosahe∣dron ABGC to the Do∣decahedron set vppon the diameter BG which coupleth the c••ntres, cō∣sisteth (by the 5. diffiniti∣on of the sixt) of the pro∣portions of the me••nes, namely, of the proportiō of the Icos••hedron ABCG to the dodecahedron HI (which is one and the same with the proportio•• of DF to DE) and of the proportion of the same HI to the other dodecahedron set vpon the diamete•• BG, inscribed in the same Icosahedron ABGC, by the same 5. of the fiuetenth: which proportion is triple to the propo••tiō of the line HI (or the line AC) to GB which coupleth the centres of the oppo∣site bases of the Icosahedron. The proportion therefore of the solide of the Icosahedron to the solide of a D••••ecahedron inscribed in it, consisteth of the proportion of the side of the Icosahedron to the side of the Cube contayned in the same sphere, and of the proportion tripled of the diameter to the lin•• which coupleth the centres of the opposite bases of the Icosahedron.

¶ 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:* 1.32 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,* 1.33 & 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,* 1.34 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 * 1.35

[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.

¶The 34. Proposition. The proportion of the solide of a Dodecahedron to the solide of an Icosa∣hedron inscribed in it, consisteth of the proportion tripled of the diameter to that line which coupleth the opposite bases of the Dodecahedron, and of the proportion of the side of the Cube to the side of the Icosahedron inscri∣bed in one and the selfe same Sphere.

SVppose that

[illustration]

AHBCK be a Dode∣cahedron•• whose diamet••r let be AB: and let the line which coupleth the cē∣tres of the opposite ba∣ses be KH•• and let the Icosahedron inscribed in the Dodecahedron ABC, be DEE: whose diameter let be DE. Now forasmuch a•• o••e and the selfe same circle cōtaineth the pentagon of a Dodecahedron, & the triangle of an Ico∣sahedro•• described in one and the selfe same Sphere, by the 14. of the fourtenth: Let that cir∣cle

be IGO. Wherfore

[illustration]

IO is the side of the cube, and IG the side of the Icosahedron, by the same. Thē I say, that the proportion of the Dodecahedron AHB∣CK to the Icosahedron DEF inscribed in it, cō∣sisteth of the proportiō tripled of the line AB to the line KH, and of the proportion of the line IO to the line IG. For ••o••asmuch as the I∣cosahedron DEF is in∣scribed in the Dodeca∣hedrō ABC,* 1.36 by suppo∣sitiō, the diameter DE shalbe equal to the line KH, by the 7. of the fiue∣tenth. Wherefore the Dodecahedron set vpō the diameter KH shall be inscribed in the same Sphere, wherein the I∣cosahedron DEF is in∣scribed: but the Dodecahedron AHBCK is to the Dodecahedron vpon the diameter KH in tri∣ple proportion of that in which the diameter AB is to the diameter KH, by the Corollary of the 17. of the twelfth: and the same Dodecahedron which is set vpon the diameter KH, hath to the Icosahedron DEF (which is set vpon the same diameter, or vpon a diameter equall vnto it, namely, DE) that pro∣portion which IO the side of the cube hath to•• IG the side of the Icosahedron, inscribed in one & the selfe same Sphere, by the 8 of the fouretenth. Wherefore the proportion of the Dodecahedron AH∣BCK to the Icosahedron DEF inscribed in it, consisteth of the proportion tripled of the diameter AB to the line KH, which coupleth the centres of the opposite bases of the Dodecahedron (which propor∣tion is that which the Dodecahedron AHBCK hath to the Dodecahedron set vpon the diameter KH) and of the proportion of IO the side of the cube to IG the side of the Icosahedron (which is the pro∣portion of the Dodecahedron set vpon the diameter KH to the Icosahedron DEF described in one and the selfe same Sphere) by the 5. definition of the sixth. The proportion therefore of the solide of a Do∣decahedron to the solide of an Icosahedron inscribed in it, con••isteth of the proportion tripled of the diameter to that line which coupleth the opposite bases of the Dodecahedron, and of the prop••••tion of the side of the cube to the side of the Icosahedron inscribed in one and the selfe same Sphere.

The 35. Proposition. The solide of a Dodecahedron containeth of a Pyramis circumscribed a∣bout it two ninth partes, taking away a third part of one ninth part of the lesse segment (of a line diuided by an extreme and meane proportion) and moreouer the lesse segment of the lesse segment of halfe the residue.

IT hath bene proued that the Dodecahedron, together with the cube inscribed in it is contai∣ned in one and the selfe same pyramis, by the Corollary of the first of this booke. And by the Corollary of the 33. of this booke, it is manifest, that the Dodecahedron is double to the same cube, taking away the third part of the lesse segment, and moreouer the lesse segment of the lesse segment of halfe the residue, or of this excesse. But a pyramis is to the same cube inscribed in it nonecuple, by the 30. of this booke. Wherefore the Dodecahedron inscribed in the pyramis, and con∣taining the same cube twise, taking away the selfe same third of the lesse segment, and moreouer the lesse segment of the lesse segment of halfe the residue, shall containe two ninth partes of the solide of the pyramis (of which ninth partes eche is equall vnto the cube) taking away this selfe same excesse. The solide therefore of a Dodecahedron containeth of a Pyramis circumscribed about it two ninth partes, taking away a third part of one ninth part of the lesse segment (of a line diuided by an extmere and meane proportion) and moreouer the lesse segment of the lesse segment of halfe the residue.

¶The 36. Proposition. An Octohedron exceedeth an Icosahedron inscribed in it, by a parallelipi∣pedon set vpon the square of the side of the Icosahedron, and hauing to his altitude the line which is the greater segment of halfe the semidiame∣ter of the Octohedron.

SVppose that there be an Octohedron ABCFPL:* 1.37 in which let there be inscribed an Icosahedron HKEGMXNVDSQT•• by the ••6. of the fiuetenth. And draw the dia∣meters AZRCBROIF, and the perpendicular KO ••arallel to the line AZR. Then I say, that the Octohedron ABCFPL is greater th••n the Icosahedron inscribed in it, by a parallelipipedon set vpon the square of the side HK or GE, and hauing to his altitude the line KO or RZ: which is the greater segment of the semidiameter AR. Forasmuch as in the same 16. it hath bene proued, that the triangles KDG and KEQ are described in the bases APF and ALF of the Octohedron:* 1.38 therefore about the solide angle there remaine vppon the base FEG three triangles KEG, KFE, and KFG, which containe a pyramis KEFG. Vnto which pyramis shall be equall and like the opposite pyramis MEFG set vpon the same base FEG, by the 8. definition of the eleuenth. And by the ••ame reason shall there at euery solide angle of the Octohedron remayne two pyramids equall and like: namely, two vpon the base AHK, two vpon the base BNV, two vpon the base DPS, and

[illustration]

moreouer two vp∣on the base QLT. Now thē there shal be made twelue pyramids, set vpon a base contained of the side of the Ico∣sahedron, and vn∣der two le••••e seg∣mentes of the side of the Octohedron containing a right angle, as for example the base GEF, And forasmuch as the side GE subtē∣ding a right angle, is, by the 47. of the ••irst, in power du∣ple to either of the lines EF and FG, and so the ••••de•• KH is in power duple to either of the sides AH and AK: and either of the lines AH, AK, or EF, FG, is in pow∣er duple to eyther of the lines AZ or ZK which cōtayne a right angle, made in the triangle or base AHK by the perpendicular AZ. Wherfore it followeth that the side GE or HK, is in power quadruple to the triangle EFG or AHK. But the pyramis KEFG, ha∣uing his base EFG in the plaine FLBP of the Octohedron, shall haue to his altitude the perpendicu∣lar KO (by the 4. definition of the sixth) which is the greater segment of the semidiameter of the Octohedron, by the 16. of the fiuetenth. Wherfore three pyramids set vnder the same altitude and vpon equall bases, shall be equall to one prisme set vpon the same base, and vnder the same altitude, by the 1. Corollary of the 7. of the twelfth. Wherefore 4. prismes set vpon the base GEF quadrupled (which is equall to the square of the side GE) and vnder the altitude KO (or RZ the greater segment which is equall to KO) shall containe a solide equall to the twelue pyramids, which twelue pyramids make the excesse of the Octohedron aboue the Icosahedron inscribed in it. An Octohedron therefore excedeth an Icosahedron inscribed in it, by a parallelipipedon set vpon the square of the side of the Icosahedron, and hauing to his altitude the line which is the greater segment of halfe the semidiameter of the Octo∣hedron.

¶ A Corollary.

A Pyramis exceedeth the double of an Icosahedron inscribed in it, by a so∣lide, set vpon the square of the side of the Icosahedron inscribed in it, and hauing to his altitude that whole line of which the side of the Icosahedron is the greater segmēt. For it is manifest by the 19. of the fiuetēth, that an octohedrō & an Icosahedrō inscribed in it are inscribed in one & the self same pyramis. It hath moreouer bene proued in the 26. of this boke, that a pyramis is double to an octohedrō inscribed in it. Wherfore the two excesses of the two octohedrons (vnto which the pyramis is equal) aboue the two Icosahedrons (inscribed in the said two octohedrons) being brought into an solide, the said solide shalbe set vpon the selfe same square of the side of the Ico∣sahedron, and shall haue to his altitude the perpendicular KO doubled: whose double coupling the opposite sides HK and XM maketh the greater segment the same side of the Icosahedron, by the first and second corollary of the 14. of the fiu••••en••h.

The 37. Proposition. If in a triangle hauing to his base a rational line set, the sides be commen∣surable in power to the base, and from the toppe be drawn to the base a per∣pendicular line cutting the base: The sections of the base shall be commen∣surable in length to the whole base, and the perpendicular shall be commen∣surable in power to the said whole base.

SVppose that there be a triangle ABG, whose base BG let be a rational line set of purpose. And let the sides AB and AG be vnto the same BG commensurable at the least in power.* 1.39 And from the toppe A, draw vnto the base BG, a perpendicular, cutting the base in the point P. Then I say that the sections of the base, are commensurable in lengthe to the whole line BG, and that the perpendicular AP, is vnto the same base BG cōmensurable at the least in power. Pro∣duce on either side the line BG to the poyntes C and E. And vnto the line AG put the line GE equal, and vnto the line AB put the line BC equal. And vpon the lines CB, BG and GE describe squares B∣K, BD, and GL. And from the greater of the squares of the lines AB or AG, which let be GL cut of a parallelogramme EM equal to the lesse square BK (by the 45. of the first:) And (by the same) vnto the residue GM, let there be applied vpon the line GD an equall rectangle parallelogramme OD.* 1.40 Now for∣asmuch as the angles APB and APG are right angles, therfore (by the 47. of the first) the line AG con∣taineth in power the two lines AP and PG, and the line AB the two lines AP and PB. Wherfore how much the line AG containeth in power more then the line AB, so much also doth the line PG contain in power more then the line BP: namely, taking away the common square of AP, there is left the ex∣cesse of the square of PG aboue the square of BP. But the square of AG (which is GL) exceedeth the square of AB (namely, the square BK) by the rectangle parallelogramme GM or OD, by construction. Wherfore the square of PG exceedeth the square of BP, by the rectangle parallelogramme OD. And forasmuch as vnto the squares of AB and AG, are equal the squares of AP and PB, and of AP and P∣G: and their excesse is taken away, namely, the rectangle parallelogramme OD: there shallbe left the squares of AP and PO equal to the squares of AP and PB. And taking away the square of AP which is cōmon, the residues

[illustration]

namely, the squares of BP and PO shalbe e∣qual: and therefore their sides (namely, the lines BP and PO) are equal. And forasmuch as the squares GL and PK are (by suppositiō) rational, and therefore cōmēsurable their ex∣cesse OD, shalbe com∣mēsurable vnto thē by the 15. of the tēth. And therfore it is rationall by the 9. diffinition of the tēth. Wherfore the rational parallelogram OD, being applied vp∣on the rational line G∣D (or BG) maketh the

bredth OG rational and cōmensurable in lēgth to the whole line BG•• by the 20. of the tenth. But if the whole line BG be commensurable to one of the partes OG, the lines BO, OG, and BG sh•••••• be com∣mensurable, by the same 15. of the tenth. Wherfore also the line OG shalbe commensurable to the half of the line BO, namely, to the line PO, or PB, by the 12. of the tenth. And forasmuch as the two lines PO and OG are commensurable, the whole line PG shalbe commensurable to the line PO, or to the line PB, by the same 15. of the tenth. Wherfore either of the lines PG and PB shall be cōmensurable vnto the whole line PB, by the same. Wherefore the lines BG, PB and PG haue the one to the other that proportion which numbers haue, by the 5. of thirtenth. Wherfore the sections PB and PG of the base BG are commensurable in length ••o the same base, by the 6: of the tenth.

And now that the perpendicular AP is commensurable in power to the base BG,* 1.41 i•• thus pro∣ued. Forasmuch as the square of AB is by supposition, commensurable to the square of BG: and vnto the rational square of AB is commensurable the rational square of BP (by the 12. of the eleuenth) Wherfore the residue, namely, the square of PA is commensurable to the same square of BP, by the 2. part of the 15. of the eleuenth. Wherefore by the 12. of the tenth, the square of PA is commensurable to the whole square of BG. Wherefore the perpendicular AP is commensurable in power to the base BG, by the 3. diffinition of the tenth: which was required to be proued.

In demonstrating of this, we made no mention at all of the length of the sides AB and AG, but only of the length of the base BG: for that the line BG is the rational line first set: and the other lines AB and AG are supposed to be commensurable in power only to the line BG. Wherefore if that be plainely demonstrated, when the sides are commensurable in power only to the base, much more easily wil it follow, if the same sides be supposed to be commensurable both in length and in power to the base: that is, if their lengthes be expressed by the rootes of square nombers.

¶ A Corollary. 1.

By the former things demonstrated, it is manifest that if from the powers of the base, and of one of the sides, be taken away the power of the other side, and if the halfe of the power remaining, be applied vpon the whole base, it shall make the bredth that section of the base which is coupled to the first side. For from the po∣wers of the base BG, and of one of the sides AG, that is, from the squares BD and GL, the power of the other side AB, namely the square BK (or the parallelogramme EM) is taken away. And of the residue, (namely, of the square BD, and of the parallelogramme OD, or DR, which by supposition is equal vn∣to OD) the halfe (namely•• of the whole FR, which is PD, for the lines GR and PB are equal to the lines GO and PO) is applied to the whole line BG or GD: and maketh the bredthe the line PG the section of the base BG, which section is coupled to the first side AG. And by the same reason in the o∣ther side, if from the squares BD and BK be taken away the square GL, there shall remaine the rectan∣gle parallelogramme FO: For the parallalelogramme EM is equal to the square BK, and the parallelo∣gramme GM to the parallelogramme OD. Wherefore FP the halfe of the residue FO, maketh the bredth BP, which is coupled to the first side taken AB.

A Corollary. 2.

If a perpendicular drawne from an angle of a triangle do cut the base: the sections are to the other sides in power proportionall by an A∣rithmetical proportion. For it was proued that the excesse of the powers of the lines AG and AB is one and the same with the excesse of the powers of the lines PG and PB. If therfore the powers do equally excede the one the other, they shall by an Arithmet••cal proportion, be proportionall.

The ende of the sixtenth Booke of the Elementes of Geometrie added 〈…〉〈…〉.

Notes

• * 1.1

  The argu∣ment of the 16. booke.
• * 1.2

  Construction.
• * 1.3

  Demonstra∣tion.
• * 1.4

  By a Pyra∣mis vnder∣stand a Te∣trahedron throughout all this booke.
• * 1.5

  Construction.
• * 1.6

  Demonstra∣tion.
• * 1.7

  Construction.
• * 1.8

  Demonstra∣tion.
• * 1.9

  Construction.
• * 1.10

  Demonstra∣tion.
• * 1.11

  Construction.
• * 1.12

  Demonstra∣tion.
• † 1.13

  That is, a•• 18. to 1.
• * 1.14

  Demonstra∣tion.
• † 1.15

  That i••, as 9. to 2.
• * 1.16

  That is, as 18. to 2. or 9. to 1.
• * 1.17

  Draw in the figure a line from B to H.
• * 1.18

  What the duple of an extreme and meane pro∣portion is.
• * 1.19

  Construction.
• * 1.20

  Demonstra∣tion.
• * 1.21

  Demonstra∣tion.
• * 1.22

  Constrution.
• * 1.23

  Demonstra∣tion.
• * 1.24

  Construction.
• * 1.25

  Demonstra∣tion.
• * 1.26

  Demonstra∣tion.
• * 1.27

  That is at 13. 1/•• is to ••.
• * 1.28

  Demonstra∣tion.
• * 1.29

  Demonstra∣tion.
• * 1.30

  Construction.
• * 1.31

  Demonstra∣tion.
• * 1.32

  Construction.
• * 1.33

  Demonstra∣tion.
• * 1.34

  Extend in the figure a line ••rom the point E to the point B.
• * 1.35

  Extend in the figure a line from the point E to the point B.
• * 1.36

  Demonstra∣tion.
• * 1.37

  Construction.
• * 1.38

  Demonstra∣tion.
• * 1.39

  Construction.
• * 1.40

  First part of the demon∣stration.
• * 1.41

  Second part of the De∣monstration.