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 fourtenth booke of Euclides Elementes after Flussas. (Book 14)

FOr that the fouretenth Booke, as it is set forth by Flus∣sas, containeth in it moe Propositions then are found in Hypsicles, & also some of those Propositions which Hypsicles hath, are by him somewhat otherwise de∣monstrated, I thought my labour well bestowed for the readers sake to turne it also all whole, notwithstan∣ding my trauaile before taken in turning the same booke after Hypsicles. Where note ye, that here in this 14. booke after Flussas, and in the other bookes follo∣wing, namely, the 15. and 16. I haue in alleadging of the Propositions of the same 14. booke, followed the order and number of the Propositions, as Flussas hath placed them.

¶The first Proposition. A perpendicular line drawen from the centre of a circle, to the side of a Pentagon inscribed in the same circle:* 1.1 is the halfe of these two lines taken together, namely, of the side of the hexagon, and of the side of the decagon inscribed in the same circle.

TAke a circle ABC, and inscribe in it the side of a pentagon, which let be BC, and take the centre of the circle, which let be the point D:* 1.2 and frō it draw vnto the side BC a perpendicular line DE: which produce to the point ••. And vnto the line E F put the line EG equall. And draw these right lines CG, CD, and CF. Then I say, that the right line DE (which is drawen from the centre to BC the side of the pentagon) is the halfe of ••he side•• of the decagon and hexagon, taken together.* 1.3 Forasmuch as the line DE is a perpendicular ••nto the line BC: therefore the sections BE and EC shall be equall (by the 3. of the third): and the line EF is common vnto them both; and the angles FEC and FEB, are right an∣gles, by supposition. Wherefore the bases BF and FC are equall (by the 4. of the first). But the line BC is the side of a pentagon, by constructi∣on. Wherefore FC which subtendeth the halfe of the side of the

[illustration]

pentagon, is the side of the decagon inscribed in the circle ABC. But vnto the line FC is, by the 4. of the first, equall the line CG, for they subtend right angles CEG, and CEF, which are con∣tained vnder equall sides. Wherefore also the angles CGE, and CFE, of the triangle CFG, are equall, by the 5. of the first. And forasmuch as the arke FC is subtended of the side of a decagon, the arke CA shall be quadruple to the arke CF: Wherefore also the angle CDA shall be quadruple to the angle CDF (by the last of the six••). And forasmuch as the same angle CDA, which is set at the center, is double to the angle CFA, which is set at the circumference, by the 20. of the third: therefore the angle CFA, or CFD, is double to the angle CD••, namely, the halfe of qua∣druple. But vnto the angle CFD or CFG, is proued equall the angle CGF: Wherefore the outward angle CGF, is double to the angle CDF. Wherefore the angles CDG and DCG, shall be equall. For vnto those two angles the angle CGF is equall, by the 32. of the first. Wherefore the sides GC and GD, are equall, by the 6. of the first. Wherefore also

the line GD is equall to the line FC, which is the side of the decagon. But vnto the right line FE is e∣quall the line EG, by construction. Wherefore the whole line DE is equall to the two lines C•• and FE. Wherefore those lines taken together (namely, the lines DF and FC) shall be double to the line DE. Wherefore the line DE (which is drawen from the centre perpendicularly to the side of the pen∣tagon) shal be the halfe of both these lines taken together, namely, of DF the side of the hexagon, and CF the side of the decagon. For the line DF which is drawen from the centre, is equall to the side of the hexagon, by the Corollary of the 15. of the fourth. Wherefore a perpendicular line drawen from the center of a circle, to the side of a pentagon inscribed in the same circle: is the halfe of these two lines taken together, namely, of the side of the hexagon, and of the side of the decagon inscribed in the same circle: which was required to be proued.

A Corollary.

If a right line drawen perpendicularly from the centre of a circle to the side of a pentagon, be diuided by an extreme and meane proportion: the greater seg∣ment shall be the line which is drawen from the same c••••tre to the side of an equilater triangle inscribed in the same circle. For, that li•••• (drawen to the side of the triangle) is (by the Corollary of the 12. of the thirtenth) the halfe of the line drawen from the centre to the circumference, that is, of the side of the hexagon: Wherefore the residue shall be the halfe of the side of the decagon. For the whole line is the halfe of the two sides, namely, of the side of the hexagon, and of the side of the decagon. But of the side of a decagon and of an hexagon taken together, the greater segment is the side of the hexagon (by the 9. of the thirtenth). Wherefore the greater segment of their halfes shall be the halfe of the hexagon, by the 15. of the fift: which halfe is the perpendicular line draw∣en from the centre to the side of the triangle, by the Corollary of the 12. of the thirtenth.

¶The second Proposition. If two right lines be diuided by an extreme and meane proportion:* 1.4 they shall be diuided into the selfe same proportions.

SVppose that these two right lines AB and DE be eche cut by an extreme and meane pro∣portion in the pointes F and Z. Then I say, that these two lines are diuided into the selfe same proportions, that is, that the line AB is in the point F diuided in like sort as the line DE is in the point Z. For if they be not in like sort cut, let one of them, namely, DE, be cut like vnto the line AB in the point C.* 1.5 So that let the line DE be to DC the greater part, as the greater part DC is to CE the lesse part, by the 3. definitiō of the sixt. But (by suppositi∣tion) the line DE is to the line DZ, as the

[illustration]

line DZ is to the line ZE. Wherefore the right line DE is diuided by an extreme and meane proportion in two pointes C and Z. But the proportion of DE to DC the lesse line, is greater then the propor∣tion of the same DE to DZ the greater line, by the 2. part of the 8. of the fift. But as DE is to DC, so is DC to CE: Wherefore the proportion of DC to CE, is greater then the proportion of DZ to ZE. And forasmuch as DZ is greater then DC, the proportion of DZ to CE shall be greater then the pro∣portion of DC to CE, by the 8. of the fift. Wherefore the proportion of DZ to CE, is much greater then the proportion of DZ to ZE. Wherefore one and the selfe same magnitude, namely, DZ, hath to CE the greater line, a greater proportion then it hath to ZE the lesse line, contrary to the second part of the 8. of the fift: which is impossible. Wherfore the right lines AB & DE, are not cut vnlike. Wher∣fore they are cut like, and into the selfe same proportions. And the same demonstration also will serue, if the point C fall in any other place. For alwaies some one of them shall be the greater. If therefore two right lines be cut by an extreme and meane proportion: they shall be cut into the selfe same propor∣tions: which was required to be proued.

¶The third Proposition. If in a circle be described an equilater Pentagon:* 1.6 the squares made of the side of the Pentagon and of the line which subtendeth two sides of the

Pentagon, these two squares (I say) taken together, are quintuple to the square of the line drawen from the centre of the circle to the circūference.

SVppose that in the circle BCG the side of a

[illustration]

Pentagon be BG: and let the line BC sub∣tend two sides thereof.* 1.7 And let the line BG be diuided into two equall partes by a right line drawen from the centre D: namely, by the diameter CDE produced to the point Z. And drawe the right line BZ. Then I say, that the right lines BC and BG, are in power quintuple to the right line DZ, which is drawen from the centre to the circumference. For forasmuch as (by the 47. of the first) the squares of the lines CB and BZ,* 1.8 are equall to the square of the diameter CZ: therefore they are quadruple to the square of the line DZ, by the 20. of the sixt (for the line CZ is double to the line DZ). Wherefore the right lines CB, BZ, and ZD, are in power quintuple to the line ZD. But the right line BG con∣taineth in power the two lines BZ and ZD, by the 10. of the thirtenth. For DZ is the side of an hexagon, & BZ the side of a decagon. Wherefore the lines BC and BG (whose powers are equall to the powers of the lines CB, BZ, ZD) are in power quintuple to the line DZ. If therefore in a circle be described an equilater Pentagon: the squares made of the side of the Pentagon and of the line which subtendeth two sides of the Pentagon, th••se two squares (I say) taken together, are quintuple to the square of the line drawen from the centre of the circle to the circumference.

¶A Corollary.

If a Cube and a Doderahedron be contained in one and the selfe same Sphere:* 1.9 the side of the Cube, and the side of the Dodecahedron, are in power quintuple to the line which is drawen from the centre of the circle which contai∣neth the Pentagon of the Dodecahedron. For it was proued in the 17. of the thirtenth, that the side of the Cube subtendeth two sides of the Pentagon of the Dodecahedron, where the sayd solides are contained in one and the selfe same Sphere. Wherfore the side of the Cube subtending two sides of the Pentagon, and the side of the same Pentagon, are contained in one and the selfe same circle. Wherefore, by this Proposition, they are in power quintuple to the line which is drawen from the cen∣tre of the same circle which containeth the Pentagon of the Dodecahedron.

The 4. Proposition. One and the selfe same circle containeth both the Pentagon of a Dodeca∣hedron,* 1.10 and the triangle of an Icosahedron described in one and the selfe same sphere.

LEt the diameter of the sphere geuen be AB, and let the bases of the Icosahedron and Do∣decahedron described in it,* 1.11 be the triangle MNR, and the pentagon FKH, and a∣bout them let there be described circles. by the 5. and 14. of the fourth. And let the lines drawne from the centres of those circles to the circumferences be LN and OK. Then I say that the lines LN and OK are equal, and therfore one and the selfe same circle contai∣neth both those figures. Let the right line AB, be in power quintuple to some one right line, as to the line CG (by the Corollary of the 6. of the tenth.) And making the cētre the poynt C, & the space CG, describe a circle DZG. And let the side of a pentagon inscribed in that circle (by the 11. of the fourth) be the line ZG. And let EG (subtending halfe of the arke ZG) be the side of a Decagon inscribed in that circle. And by the 30. of the sixt, diuide the line CG by an extreme & meane proportion in the poynt I.* 1.12 Now forasmuche as in the 16. of the thirtenth, it was proued, that this line CG (vnto whome the diameter AB of the sphere is in power quintuple) is the line which is drawne from the centre of the circle, which containeth fiue angles of the Icosahedron, and the side of the pentagon described

in that circle DZG, namely the line ZG

[illustration]

is side of the Icosahedron described in the Sphere, whose diameter is the line AB: therefore the right line ZG, is equal to the line MN, which was put to be the side of the Icosahedrō, or of his triāgular base. Moreouer, by the 17. of the thirtenth, it was manifest that the right line ••H (which subtendeth the angle of the pentagon of the Dodecahedron inscribed in the fore∣sayde sphere) is the side of the Cube, in∣scribed in the self same sphere. (For vpon the angles of the cube, were made the an∣gles of the Dodecahedron.) Wherefore the diameter AB is in power triple to F∣H, the side of the Cube (by the 15. of the thirtenth). But the same line AB is (by supposition) in power quintuple to the line CG. Wherefore fiue squares of the line CG, are equal to thre squares of the line FH: (for eche is equal to one and the self same square of the line AB). And forasmuche as EG the side of the Deca∣gon, cutteth the right line CG by an ex∣treme and meane proportion (by the co∣rollary of the 9. of the thirtenth): Like∣wise the line HK, cutteth the line FH, the side of the Cube by an extreeme and meane proportion (by the Co∣rollary of the 17. of the thirtenth): therfore the lines CG and FH, are deuided into the self same pro∣portions, by the second of this booke: and the right lines CI and EG, which are the greater segmentes of one and the selfe same line CG, are equal: And forasmuche as fiue squares of the line CG are equal to thre squares of the lines FH: therefore fiue squares of the line GE, are equal to thre squares of the line HK (for the lines GE and HK are the greater segmēts of the lines CG and FH). Wherefore fiue squ••re•• of the line•• CG & GE are equal to the squares of the 〈◊〉〈◊〉 ••H & HK, by the 1•• of the ••ift. But vnto the squares of the lines CG and GE•• is ••qual the squ••re of th•• ••ine ZG, by the 10. of the thirtēth: and vnto the line ZG the line MN was equal: wherfore fiue squares of the line MN, are equall to three squares of the lines FH, HK. But the squares of the lines •••• and HK, 〈◊〉〈◊〉 quintuple to the square of the line OK (which is drawne from the centre) by the third of this booke. Wherfore thre squares of the lines FH and HK make 15. squares of the line OK. And forasmuch as the square of the line MN is triple to the square of the line LN (which is drawne from the centre) by the 12. of the thirtenth, ther∣fore fiue squares of the line MN are equal to 15. squares of the line LN. But fiue squares of the line MN are equal vnto thre squares of the lines FH and HK. Wherefore one square of the line LN is equall to one square of the line OK (being eche the fiuetenth part of equal magnitudes) by the 15. of the fif••••. Wherfore the lines LN and OK, which are drawne from the centers, are equal. Wherefore also the circles NRM, and FKH which are described of those lines, are equal. And those circles contayne (by supposition) the b••ses of the Dodecahedron and of the Icosahedron described in one and the selfe same sphere. Wherfore one and the selfe same circle, &c. a•• in th•• pro••••sition: which was required to be proued.

The 5. Proposition. If in a circle be inscribed the pentagon of a Dodecahedron, and the trian∣gle of an Icosahedron,* 1.13 and from the centre to one of theyr sides, be drawne a perpendicular line: That which is contained 30. times vnder the side, & the perpendicular line falling vpon it, is equal to the superficies of that so∣lide, vpon whose side the perpendicular line falleth.

SVppose that in the circle AGE, be described the pentagon of a Dodecahedron, which let be ABGDE, and the triangle of an Icosahedron described in the same sphere, which let be AFH.* 1.14 And let the centre be the poynt C. ••••on which draw perpendicularly the line CI to the side of the Pentagon, and the line CL to the side of the triangle. Then I say that the rectangle figure contained vnder the lines CI and GD 30. times, is equal to

the superficies of the Dodecahedron: and that that

[illustration]

which is cōtained vnder the lines CL & AF 30. times is equal to the super••icies of the Icosahedrō described in the same sphere. Draw these right lines CA, CF, CG and CD.* 1.15 Now forasmuch as that which is cōtained vn¦der the base GD & the altitude IC, is double to the triangle GCD, by the 41. of the first: And fiue triangles like and equal to the triangle GCD do make the pen∣tagon ABGDE of the Dodecahedron: wherfore that which is contained vnder the lines GD and IC fiue times is equal to two pentagōs. Wherfore that which is contained vnder the lines GD and IC ••0. times is equal to the 12. pentagons, which containe the superfi∣cies of the Dodecahedron.

Againe that which is contained vnder the lynes CL and AF, is double to the triangle ACF: where∣fore that which is contained vnder the lines CL and AF three times is equal to two suche triangles as AF∣H is, which is one of the bases of the Icosahedron (for the triangle ACF, is the third part of the triangle AF∣H, as it is easie to proue, by the 8. & 4. of the first.) Wherfore that which is cōtained vnder the lines CL and AF. 30 times times, is equall to 10. such triangles as AFH i••, which containe the superficies of the Icosahedron. And forasmuch as one and the selfe same spher•• containeth the Dodecahedron of this pentagon, and the Icosahedron of this triangle (by the 4. of this booke ••) and the line CL falleth per∣pendicula••ly vpon the side of the Icosahedron, and the line CI vpon the side of the Dodecahedron: that which is 30. times contained vnder the side, and the perpendicular line falling vpon it, is equal to the ••uperficies of that solide, vpon whose side the perpendicul••r falleth. If therefore in a circle &c. as in the proposition: which was required to be demonstrated.

A Corollary.

* 1.16The superficieces of a Dodecahedron and of an Icosahedron described in one and the selfe same sphere are the one to the other, as that which is contained vn¦der the side of the one and the perpendicular line drawne vnto it from the cen∣tre of his base, to that which is contained vnder the side of the other, and the perpendicular line drawne to it from the centre of his base. For a•• thirty•• tim••s is to thirty times, so is once to once by the 15. of th•• fifth.

The 6. Proposition. The superficies of a Dodecahedron, is to the superficies of an Icosahe∣dron described in one and the selfe same sphere,* 1.17 in that proportion, that the side of the Cube is to the side of the Icosahedron contained in the self same sphere.

* 1.18SVppose that there be a circle ABG, & in it (by the 4. of this boke) let there be inscribed the side•• of a Dodecahedron and of an Icosahedron contained in on•• and the selfe same sphere. And let the side o•• the Dodecahedron be AG, and the side of the Icosahedron be DG. And let the centre be the poynt E: from which draw vnto those s••des, perpendicular lines EI and EZ. And produce the line EI to the poynt B, and draw the lin•• BG. And let the side of the cube con∣tained in the self same sphere be GC. Then I say that the superficies of the Dodecahedron i•• to the su∣perficies of the Icosahedron, as the line ••G, i•• to the li•••• GD. For forasmuche as the line EI bein•• diuided by an extreme and meane proportion, the greater segment th••rof shall be the lin•• EZ, by the corollary of the first of this booke:* 1.19 and the line CG being diuided by an extreme and meane pro∣portion, his greater segment is the line AG, by the corollary of the 17. of the thirtenth: Where∣fore the right lines EI and CG ••r•• cut proportionally by the second of this b••oke. Wh••r••fore as the line CG, is to the line AG, so is the line EI to the line EZ. Wher••fore that which it con∣tained vnder the extreames CG and EZ, is ••quall to that which i•• contayn••d vnder the meanes AG and EI. (by the 16. of the sixth.) But as that which i•• contained vnder the lin•••• CG and ••Z is to that which is contained vnder the lines DG and EZ, so (by the first of the sixth) i•• the lin•• CG to the

line DG, for both those parallelogrames haue o•••• and the

[illustration]

selfe same altitude, namely the line EZ. Wherfore as that which is contained vnder the lines EI and AG (which i•• proued equal to that which is contained vnder the line•• CG and EZ) is to that which is contained vnder the lines DG and EZ, so is the line CG to the li•••• DG. But as that which is contained vnder the lines EI and AG is to that which is contained vnder the lines DG and EZ, so (by the corollary of the former proposition) is the su∣perficies of the Dodecahedron, to the superficies of the I∣cosahedron. Wherfore as the superficies •••• the Dodecahe∣dron is to the superficies of the Icosahedron, so is CG the side of the cube, to GD the side of the Icosahedron. The superficies therefore of a Dodecahedron is to the superfi∣cies•• &c. as in the proposition, which was required to be proued.

An Assumpt.

The Pentagon of a Dodecahedron,* 1.20 is equall to that which is contained vn∣der the perpendicular line which falleth vpon the base of the triangle of the Icosahedron, and fiue sixth partes of the side of the cube, the sayd three so∣lides being described in one and the selfe same sphere.

Suppose that in the circle ABEG, the pentagon of a Dodecahedron be A••CIG, and let two sides thereof AB and AG be subtended of the right line BG. And let the triangle of the Icosahedron inscribed in the selfe same sphere,* 1.21 by the 4. of this booke, be AFH. And let the centre of the circle be the poynt D, and let the diameter be ADE, cutting FH, the side of the triangle in the poynt Z, and cut∣ting the line BG in the poynt K. And draw the right line BD. And from the right line KG cut of a third part TG, by the 9. of the sixth. Now then the line BG subtending two sides of the Dodecahedron, shalbe the side of the cube inscribed in the same sphere,* 1.22 by the 17. of the thirtenth: and the triangle of the Icosahedron of the same sphere shalbe A••H by the 4. of this booke. And the line AZ which pas∣seth by the centre D shall fall perpendicularly vpon the

[illustration]

side of the triangle. For forasmuch as the angles GAE & BAE are equall (by the 27. of the third•• for they are see vpon equall circumferences): therefore the ••ases BK and KG are (by the ••. of the first) equall. Wherefore the line BT contayneth 5. sixth partes of the line BG. Then I say that that which is contayned vnder the lines AZ and B∣T, is equall to the pentagon A••C••G. For forasmuch as the line ••Z is sesq••ialter to the line AD (for the line D∣•• is diuided into two equall partes in the poynt Z, by the corollary of the ••2•• of the thirtenth). Likewise by con∣struction, the line KG is sesquialter to the line KT: there∣fore as the line AZ is to the line AD, so is the line KG to the 〈◊〉〈◊〉 ••T. Wherefore that which is contayned vnde•• the 〈◊〉〈◊〉 AZ and KT, is equall to that which is con∣tayned vnder the meanes AD and KG, by the 16. of the sixth. But vnto the line KG is the line ••K ••roued equall. Wherefore that which is contayned vnder the lines AZ and KT is equall to that which is contayned vnder the lines AD and BK. But that which is contayned vnder the lines AD and BK is (by the 41. of the first) double to the triangle ABD. Wherefore that which is contayned vnder the lines AZ and KT is double to the same triangle ABD. And forasmuch as the pentagon ABCIG contayneth•• 〈…〉〈…〉 equall ••o the triangle ABD, and that which is contayned vnder the lines AZ and KT contayneth two such triangles: therefore the pentagon ABCIG is duple sesquialter to the rectangle parallelogramme contayned vnder the lines AZ and KT. And 〈…〉〈…〉 1. of the sixth, that which is cōteyned vnder the lines AZ and BT is to that which is contayned vnder the lines AZ and KT, as the base BT is to the base ••••T•• therefore that which is contayned vnder the lines AZ and BT is duple sesquialter to that which is contayned vn••••r the line AZ & KT. But vnto that which is contayned vnder the lines AZ and KT the pentagon ABCIG is proued duple sesquialter. Wher¦fore the pentagon ABCIG of the Dodecahedron is equall to that which is contayned vnder the per∣pendicular

line AZ, and vnder the line BT which is fiue sixe partes of the line BG.

¶ The 7. Proposition. * 1.23A right line diuided by an extreame and meane proportion: what propor∣tion the line contayning in power the whole line and the greater segment, hath to the line contayning in power the whole and the lesse segment: the same hath the side of the cube to the side of the Icosahedron contayned in one and the same sphere.

* 1.24TAke a circle ABE: and in it (by the 11. of the fourth) inscribe an equilater pentagon BZEC∣H: and (by the second of the same) an equilater triangle ABI. And let the centre thereof be the poynt G. And drawe a line from G to B. And diuide the line GB by an extreame and meane proportion in the poynt D (by the 30. of the sixth). And let the line ML contayne in power both the whole line GB and his lesse segment B∣D

[illustration]

(by the corollary of the 13. of the tenth). And draw the right line B•• sub••ēdi••g the angle of the pentagon, which shall be the side of the cube (by the corollary of the 17. of the thirtenth) •• and the line BI shall be the side of the Icosahedron, and the line ••Z the side of the Dodecahe∣dron by the 4. of this booke. Then I say that BE the side of the cube is to BI the side of the Icosahedron, as the line contayning in power the lines BG & GD is to the line contayning in power the lines GB and BD.* 1.25 For for∣asmuch as (by the 12. of the thirtenth) the line BI is in power triple to the line BG, and (by the 4. of the same) the squares of the line GB & BD are triple to the square of the line GD. Wherefore (by the 15. of the fifth) the square of the line BI is to the squares of the lines GB & BD (namely, triple to triple), as the square of the line B•• is to the square of the line GD (namely, as one is to one). But as the square of the line BG is to the square of the GD, so is the square of the line BE to the square of the line BZ. (For the lines BG, GD, and BE, BZ are in one and the same proportion, by the second of this booke. For BZ is the greater segment of the line BE, by the corollary of the 17. of the thirtenth). Wherefore the square of the line BE is to the square of the line BZ, as the square of the line BI is to the squares of the lines BG and BD. Wherefore alternately the square of the line BE is to the square of the line BI, as the square of the line BZ is to the squares of the lines GB and BD. But the square of the line BZ is equall to the squares of the lines BG and GD (by the 10. of the thirtenth). For the line BG is equall to the side of the hexagon, and the line GD to the side of the decagon, by the corollary of the 9. of the same. Wherefore the squares of the lines BG and GD, are to the squares of the lines G•• and BD, as the square of the line BE is to the square of the line BI. But the line ZB contayneth in power the lines BG and GD, and the line ML contayneth in power the lines GB and BD by construc∣tion. Wherefore as the line ZB (which contayneth in power the whole line BG and the greater seg∣ment GD) is to the line ML (which contayneth in in power the whole line GB and the lesse segment BD) so is BE the side of the cube to BI the side of the Icosahedron, by the 22. of the sixth. Wherefore a right line diuided by an extreame and meane proportion: what proportion the line contayning in power the whole line and the greater segment, hath to the line contayning in power the whole line and the lesse segment: the same hath the side of the cube to the side of the Icosahedron cōtayned in one and the same sphere which was required to be proued.

¶ The 8. Proposition. * 1.26The solide of a Dodecahedron is to the solide of an Icosahedron: as the side of a Cube is to the side of an Icosahedron, all those solides being described in one and the selfe same Sphere.

FOrasmuch as in the 4. of this booke, it hath bene proued, that one and the self same cir∣cle containeth both the triangle of an Icosahedron, and the pentagon of a Dodecahe∣dron described in one and the selfe same Sphere: Wherefore the circles, which cōtaine those bases, being equall, the perpendiculars also which are drawen from the centre of the Sphere to those circles, shall be equall (by the Corollary of the Assumpt of the 16 of the twelfth). And therefore the Pyramids set vpon the bases of those solides haue one and the selfe same altitude: For the altitudes of those Pyramids concurr•• in the centre. Wherefore they are in proportion as their bases are, by the 5. and 6. of the twelfth. And therefore the pyramids which compose the Dodecahedron, ar•• to the pyramids which compose the Icosahedron, as the bases are, which bases are the superficieces of those solides. Wherefore their solides are the one to the other, as their superficieces are. But the superficies of the Dodecahedron is to the superficies of the Icosahedron, as the side of the cube is to the side of the Icosahedron, by the 6. of this booke. Wher∣fore by the 11. of the fifth, as the solide of the Dodecahedron is to the solide of the Icosahedron, so is the side of the cube to the side of the Icosahedron, all the said solides being inscribed in one and the selfe same Sphere. Wherefore the solide of a Dodecah••dron is to the solide of an Icosahedron: as the side of a cube is to the side of an Icosahedron, all those solides being described in one and the self same Sphere: which was required to be proued.

A Corollary.

The solide of a Dodecahedron is to the solide of an Icosahedron,* 1.27 as the su∣perficieces of the one are to the superficieces of the other, being described in one and the selfe same Sphere: Namely, as the side of the cube is to the side of the Icosahedron, as was before manifest: for they are resolued into pyramids of one and the selfe same altitude.

¶ The 9. Proposition. If the side of an equilater triangle be rationall: the superficies shall be irra∣tionall,* 1.28 of that kinde which is called Mediall.

SVppose that ABG be an equilater triangle: and from the point A draw vnto the side BG a perpendicular line AD: and let the line AB be rationall. Then I say, that the superfi∣cies ABG is mediall.* 1.29 Forasmuch as the line AB is in power sesquitertia to the line AD (by the Corollary of the 12. of the thirtenth): of what partes the line AB containeth in power 12, of the same partes the line AD containeth in power 9:* 1.30 wherefore the resi∣due BD containeth in power of the same partes 3. (••or the line AB cōtaineth in power the lines AD and BD, by the 47. of the first). Wherfore the

[illustration]

lines AD and DB are rationall and commensurable to the rationall line set AB, by the 6. of the tenth. But forasmuch as the power of the line AD is to the power of the line DB in that proportion that 9. a square number is to 3. a number not square: therfore they are not in the proportion of square numbers, by the Corollary of the 25. of the eight. And ther∣fore they are not commensurable in length, by the 9. of the tenth. Wherefore that which is contained vnder the lines AD and DB, which are rationall lines commensurable in power onely, is mediall, by the 22. of the tenth. But that which is contained vnder the lines AD and DB, is double to the triangle ABD, by the 41. of the first. Wherefore that which is contained vnder the lines AD and DB, is equall to the whole triangle ABG (which is double to the triangle ABD, by the 1. of the sixt). Wherefore the triangle ABG is mediall. If therfore the side of an equilater triangle be ra∣tionall: the superficies shall be irrationall, of that kinde which is called Mediall: which was required to be proued.

A Corollary.

If an Octohedron and a Tetrahedron be inscribed in a Sphere whose dia∣meter is rationall:* 1.31 their superficieces shall be mediall. For those superficieces consiste of equilater triangles, whose sides are commensurable to the diameter which is rationall, by the 13. and

14. of the thirtenth, and therefore they are rationall. But they are commensurable in power onely to the perpendicular line, and therefore they containe a mediall triangle, as it was before manifest.

¶The 10. Proposition. * 1.32If a Tetrahedron and an Octohedron be inscribed in one and the self same Sphere: the base of the Tetrahedron shall be sesquitertia to the base of the Octohedron, and the supersicieces of the Octohedron shall be sesquialtera to the superficieces of the Tetrahedron.

* 1.33FOrasmuch as the diameter of the Sphere is in power sesquialtera to the side of the Tetrahe∣dron (by the 13. of the thirtenth) and the same diameter is in power duple to the side of the Octohedron (by the 14. of the same booke): therefore of what partes the diameter contai∣neth in power sixe, of the same the side of the Tetrahedron contayneth in power 4, and of the same the side of the Octohedron containeth in power 3. Wherefore the power of the side of the Tetrahedron, is to the power of the side of the Octohedron in the same proportion that 4. is to 3: which is sesquitertia. And like triangles (which are the bases of the solides) described of those sides, shall haue the one to the other the same proportion that the squares made of those sides haue. For both the triangles are the one to the other, and also the squares are the one to the other, in double proporti∣on of that in which the sides are, by the 20. of the sixth.* 1.34 Wherefore of what partes one base of the Te∣trahedron was 4: of the same are fower bases of the Tetrahedron 16: likewise of what partes of the same one base of the Octohedron was 3: of the same are 8. bases of the Octohedron 24. Wherfore the bases of the Octohedron are to the bases of the Tetrahedron, in that proportion that 24. is to 16: which is sesquialtera. If therefore a Tetrahedron and an Octohedron be inscribed in one and the selfe same Sphere: the base of the Tetrahedron shall be sesquitertia to the base of the Octohedron, and the super∣ficieces of the Octohedron shall be sesquialtera to the superficieces of the Tetrahedron: which was re∣quired to be proued.

¶The 11. Proposition. A Tetrahedron is to an Octohedron

[illustration]

inscribed in one and the selfe same Sphere,* 1.35 in proportion, as the rectan∣gle parallelogrāme contained vnder the line, which containeth in power 27. sixty fower partes of the side of the Tetrahedron, & vnder the line which is subsesquiocta••a to the same side of the Tetrahedron, is to the square of the diameter of the sphere.

* 1.36LEt vs suppose a Sphere, whose diame∣ter let be the line AB, and let the cen∣tre be the point H. And in it let there be inscribed a

[illustration]

Tetrahedron ADC, and an Octohedron AEKBG. And let the line NL containe in power ••7/64 of AC the side of the Tetrahedron. And let the line ML be in lēgth subsesquioctaua to the same side. Thē I say, that the Tetrahedron ACD is to the Octohedron AEB, as the rectangle parallelogramme contay∣ned vnder the lines NL and LM,* 1.37 is to the square of the line AB. Foras∣much as the line drawen frō the angle A by the centre H perpēdicular∣ly vpon the base of the Tetrahedron, falleth vpon the cēter T of the cir∣cle which containeth that base, and maketh the right line HT the sixth part of the diameter AB (by the Corollary of the 13. of the thirtenth): therefore the line HA (which is drawen from the centre to the circū∣ference) is triple to the line HT: and therefore the whole line AT is

to the line AH, 〈…〉〈…〉. Let the Tetrahedron ADC be cut by a plaine GHK passing by the centr•• H, and being parallel vnto the base DTC, by the Corollary of the 11. of the eleuenth. Now then the triangle ADC of the Tetrahedron, shall be cut by the right line KG, which is parallel to the line DC, by the 16. of the eleuenth. Wherfore as the line AT is to the line AH, so is the line AC to the line AG (by the 2. of the sixth). Wherefore the line AC is to the line AG sesquitertia, that is, as 4. to 3. And forasmuch as the triangles ADC, AKG, and the rest which are cut by the plaine KHG, are like the one to the other, by the 5•• of the sixth: the pyramids ADC and AKG, shall be like the one to the o∣ther, by the 7. definition of the ••leuenth. Wherefore they are in triple proportion of that in which the sides AC and AG are, by the 8. of the twelfth. But the proportion of the sides AC to AG is, as the proportion of 4. to 3. Now then, if, by the 2. of the eight, ye finde out 4. of the lest numbers in continu∣all proportion, and in that proportion that 4. is to 3: which shall be 64.48.36. and 27: it is manifest, by the 15 definition of the fifth, that the extremes ••4. to 27. are in triple proportion of that in which the proportion geuen 4. to 3. is: Or the quantitie of the proportion of 4. to 3. (which is 1. and 1/••) being twise multiplied into it selfe, there shall be produced the proportion of 64. to 27. Wherefore the Pyramis or Tetrahedron ADC is to the pyramis AKG, as 64. is to ••7: which is triple to the propor∣tion of 4. to 3. And forasmuch as the line AC is vnto the line AG in length sesquitertia: of what partes the line AC containeth in power 64: of the same partes doth the line AG containe in power 36. For (by the 2. of the sixth) the proportion of the powers or squares, is duple to the porportion of the sides which are as 64. is to 48.

Now then vpon the line RS which let be equall to the line AG, let there be an equilater triangle QRS described (by the first of the first).* 1.38 And from the angle Q, draw to the base RS a perpendicular line QT. And extend the line RS to the poynt X. And as 27. is to 64. (so by the corollary of the 6. of the tenth) let the line RS be to the line RX. And diuide ••he line RX into two equall partes in the poynt V, and draw the line QV. And forasmuch as the line RS is equall to the line AG, of what partes the line AC contayneth in power 64. of the same part the line RS contayneth in power 36. for it is proued that the line AG contayneth in power 36. of those partes:* 1.39 And of what partes the line RS contayneth in power 36 of the same partes the 〈◊〉〈◊〉 QT contayneth in power ••7. by the corollary of the 12. of the thi••tenth. Wherfore of what partes the line AZ contayneth in power 64. of the same parts the line Q∣T contayneth in power 27. Wherefore the right line QT shall be equ••ll to the right line LN by sup∣position. Agayne forasmuch as the line RS is put equall to the line AG: and of what partes the line R∣S contayneth in length 27. of the same parts is the line RX put to contayne in length 64. and of what partes the line RX contayneth in

[illustration]

length 64. of the same the line A∣C (which is in length sesquitertia to the line AG or RS) contay∣neth 36. Wherefore the line RV (which is the halfe of the line R∣X) containeth in lēgth of the same partes 32. of which the line AC contayned in length 36. Where∣fore the line RV is to the line A∣C subsesquioctaua: and therefore the line RV is equall to the line LM which is also subsesquioctaua to the same line AC. And forasmuch as the line NL is equall to the line QT, and the line LM to the line RV (as before hath bene proued) the rectangle parallelogramme contayned vnder the lines QT and RV, shall be equall to the rectangle parallelogramme, contayned vnder the line NL which is in power ••7/64 to the side AC, and vnder the line LM, which is in length subsesquioctaua to the same side A∣C. But that which is contayned vnder the lines QT and RV is double to the triangle QVR by the 41. of the first: and to the same triangle QVR is the triangle QXR duple by the first of the sixth. Where∣fore the whole triangle QXR is equall to that which is contayned vnder the lines QT and RV, and therefore is equall to the parallelogramme MN. And forasmuch as the line RX by supposition contay∣neth in length 64. of those partes of which the line RS contayneth 27: and the triangles QRX, and Q∣RS are, by the first of the sixth, in the proportion of their bases, that is, as 64. is to 27: but as 64. is to 27. so is the pyramis or tetrahedron ADC to the pyramis AKG: wherefore as the parallelogramme NM or the triangle QRX, is to the triangle QRS, so i•• the pyramis ADC to the pyramis AKG. And for∣asmuch as the semidiameter AH is the altitude of the pyramis AKG, and also of the two equall and like pyramid•• of the octohedron which haue their common base in the square of the octohedron (by the corollary of the 14. of the thirtenth): therefore as the base of the pyramis AKG (which is the tri∣angle QRS) is to two squares of the octoh••dron, that is, to the square of the diameter AB, which is e∣quall to those squares (by the 47. of the first), so is the pyramis AKG to the octohedron AEB, by the 6. of the twelfth. And forasmuch as the parallelogramme MN is to the base QRS, as the pyramis ADC is to the pyramis AKG, and the base QRS is to the square of the line BE, as the pyramis AKG is to the octohedron AEB: therefore by p••oportion of ••quality ta••ing away th•• meane•• (by the 22. of the fifth) as the parallelogramme NM is to the square of the line BE, so is the pyramis ADC to the octo∣hedron

AEB inscribed in one and the selfe same sphere. But the parallelogramme NM is contayned vnder the line NL which by supposition is in power ••7/•••• to AC the side of the tetrahedron ADC, and vnder the line LM which is also by supposition in length subsesquioctaua to the same line AC. Wher∣fore a tetrahedron & an octohedron inscribed in one and the selfe same sphere, are in proportiō, as the rectangle parallelogramme contayned vnder the line, which contayneth in power 27. sixty fower parts of the side of the Tetrahedron, and vnder the line which is subsesquioctaua to the same side of the Te∣trahedron, is to the square of the diameter of the sphere: which was required to be proued.

¶The 12. Proposition. * 1.40If a cube be contayned in a sphere: the square of the diameter doubled, is e∣quall to all the superficieces of the cube taken together. And a perpendicu∣lar line drawne from the centre of the sphere to any base of the cube, is e∣quall to halfe the side of the cube.

* 1.41FOr forasmuch as (by the 15. of the thirtenth) the diameter of the sphere is in power triple to the side of th•• cube: therefore the square of the diameter doubled is sex∣tuple to the base of the same cube. But the sextuple of the power of one of the sides contayneth the whole superficies of the cube 〈◊〉〈◊〉 or the cube is composed of sixe square superficieces (by the 2••. diffinition of th•• eleuenth) whose sides therefore are equall: wherefore the square of the di••meter ••oubled is equall to the whole super∣ficies of the cube. And forasmuch as the diameter of the cube, and the line which falleth perpendicularly vpō the opposite bases of the cube,* 1.42 do cut the one the other into two equall partes in the centre of the sphere which containeth the cube (by the 2. corollary of the 15. of the thirtenth) and the whole right line which coupleth the centres of the opposite bases, is equall to the side of the cube by the 33. of the first, for it coupleth the equall and parallel semidiameters of the bases: therefore the halfe thereof shall be equall to the halfe of the side of the cube by the 15. of the fifth. If therefore a cube be contayned in a sphere: the square of the diameter doubled is equall to all the superficieces of the cube taken together. And a perpendicular line drawne from the centre of the sphere to any base of the cube, is equall to halfe the side of the cube: which was required to be prou••d.

¶ A Corollary.

If two thirds of the power of the diameter of the sphere be multiplyed into the perpendicular line equall to halfe the side of the cube,* 1.43 there shall be produced a solide equall to the solide of the cube. For it is before manifest that two third partes of the power of the diameter of the sphere are equall to two bases of the cube. If therefore vnto eche of those two thirds be applyed halfe the altitude of the cube, they shall make eche of those solides equall to halfe of the cube, by the 31. of the eleuenth: for they haue equall bases. Wherefore two of those solides are equall to the whole cube.

You shall vnderstand (gentle reader) that Campane in his 14. booke of Euclides Ele∣mentes hath 18. propositiōs with diuers corollaries following of them. Some of which propositions and corollaries I haue before in the twelfth and thirtenth bookes added out of Flussas as corollaries (which thing also I haue noted on the side of those corol∣laries, namely, with what proposition or corollary of Campanes 14. booke they doo agree). The rest of his 18. propositions and corollaries are contained in the twelue for∣mer propositions and corollaries of this 14. booke after Flussas: where ye may see on the side of eche proposition and corollary with what proposition and corollary of Campanes they agree. But the eight propositions following together with their corol∣laries, Flussas hath added of him selfe, as he him selfe affirmeth.

The 13. Proposition. One and the self same circle containeth both the square of a cube, and the triangle of an Octohedron described in one and the selfe same sphere.

SVppose that there be a cube ABG, and an Octohedron DEF described in one and the selfe same sphere, whose diameter let be AB, or DH. And let the lines drawne from the cētres (that is the semidiameters of the circles which ctōaine the bases of those solides)•• be CA and ID. Then I say that the lines CA and ID are equal. Forasmuch as AB the diameter of the sphere which containeth the cube,* 1.44 is in power triple to BG the side of the cube (by the 15. of the thirtenth) vnto which side, AG the diameter of the base of the cube, is in power double (by the 47. of the first): which line AG is also the diameter of the circle, which cōtai∣neth the base (by the 9. of the fourth:) therfore AB the diameter of the sphere is in power sesquialter to the line AG: namely, of what partes the line AB, containeth in power 12. of the same the line AG, shal containe in pow∣er

[illustration]

8. And therfore the right line AC whiche is drawn from the cē∣tre of the circle to the circumference, contei∣neth in power of the same partes 2. Where∣fore the diameter of the sphere is in power sextuple to the lyne which is drawne from the centre to the cir∣cumference of the cir∣cle whiche containeth the square of the cube But the Diameter of the selfe same Sphere whych containeth the Octohedron, is one and the selfe same with the diameter of the cube, namely, DH, is equall to AB: and the same diameter is also the diameter of the square which is made of the sides of the Octohedron: wherefore the saide diameter is in power double to the side of the same Octohedron, by the 14. of the thirtenth. But the side DF is in power triple to the line drawne from the centre to the circumference of the cir∣cle which containeth the triangle of the octohedron (namely to the line ID) by the 12. of the thirtenth. Wherfore the selfe same diameter AB or DH, which was in power sextuple to the line drawne from the centre to the circumference of the circle which containeth the square of the cube, is also sextuple to the line ID drawne from the centre to the circumference of the circle, which containeth the trian∣gle of the Octohedron: Wherefore the lines drawne from the centres of the circles to the circumfe∣rences which containe the bases of the cube and of the octohedron are equal. And therfore the circles are equal, by the first diffinition of the third. Wherfore one and the selfe same circle containeth &c. as in the proposition: which was required to be proued.

A Corollary.

Hereby it is manifest, that perpendiculars coupling together in a sphere, the centres of the circles which containe the opposite bases of the cube and of the Octohedron, are equal. For the circles are equal, by the second corollary of the assumpt of the 16. of the twelfth: and the lines which passing by the centre of the sphere, couple together the centres of the bases, are also equal, by the first corollary of the same. Wherfore the perpendicular which coupleth together the opposite bases of the Octohedron, is equal to the side of the cube. For either of them is the altitude erected.

The 14. Proposition. An Octohedron is to the triple of a Tetrahedron contained in one and the

selfe same sphere, in that proportion that their sides are.

SVppose that there be an octohedron ABCD, and a Tetrahedron EFGH: vpon whose base FGH erect a Prisme,* 1.45 which is done by erecting from the angles of the base perpendicu∣lar lines equal to the altitude of the Tetrahedron: which prisme shalbe triple to the Tetra∣hedron EFGH, by the first corollary of the 7. of the twelfth. Then I say that the octohedron ABCD is to the prisme which is triple to the Tetrahedron, EFGH, as the side BC is to the side FG. For forasmuch as the sides of the opposite bases of the octohedron,* 1.46 are right lines touching the one the other, and are parellels to other right lines touching the one the other, for the sides of the squares which are cōposed of the sides of the octohedrō, are opposite: Wherfore the opposite plaine triangles, namely, ABC & KID, shalbe parallels, and so the rest

[illustration]

by the 15. of the eleuenth. Let the diameter of the Oc∣tohedron, be the line AD. Now then the whole Octo∣hedron is cut into foure equal and like pyramids set vp∣on the bases of the octohedron, and hauing the same al∣titude with it, & being about the Diameter AD: name∣ly the pyramis set vpon the base BID, and hauing his toppe the poynt A, and also the pyramis set vppon the base BCD, hauing his top the same poynt A. Likewise the pyramis set vpō the base IKD, & hauing his toppe the same poynt A: and moreouer the pyramis set vpon the base CKD, and hauing his toppe the former poynt A: which pyramids shalbe equal by the 8. diffinition of the eleuenth (for they eche consist of two bases of the octohedron, and of two triangles contained vnder the diameter AD and two sides of the octohedrō). Wher∣fore the prisme which is set vpon the base of the Octo∣hedron, and hauing the same altitude with it, namelye, the altitude of the parallel bases, as it is manifest by the former, is equal to thre of those pyramids of the Octo∣hedron, by the first corollary of the seuēth of the twelft. Wherefore that prisme shall haue to the other prisme vnder the same altitude, composed of the 4. pyramids of the whole octohedron, the proportion of the triangular bases, by the 3. corollary of the same. And forasmuch as 4. pyramids are vnto 3. pyramids in sesquitercia propor∣tion, therefore the trianguler base of the prisme which containeth 4. pyramids, is in sesquitertia proportion to the base of the prisme which containeth thre pyramids of the same octohedron, and are set vpon the base of the Octohedron and vnder the altitude thereof: that is, in sesquitercia proportion to the base of the Octohe∣dron. But the base of the same octohedron is in sesqui∣tertia proportion to the base of the pyramis, by the ••enth of this booke: Wherefore the triangular bases, namely, of the prisme which cōtaineth four pyramids of the octohedron, and is vnder the altitude thereof, are equal to the triangular bases of the prisme, which con∣taineth three pyramids vnder the altitude of the pyra∣mis EFGH. But the prisme of the octohedron is equal to the octohedron: and the prisme of the pyramis EF∣GH is proued triple to the same pyramis EFGH. Now then the prismes set vpō equal bases, are the one to the other as their altitudes are (by the corollary of the 25. of the eleuenth) namely, as are the parallelipidedons their doubles, by the corollary of the 31. of the eleuenth. But the altitude of the Octohedron is equal to the side of the cube contained in the same sphere, by the corollary of the 13. of this booke. And the side of the cube is in power to the altitude of the Tetrahedon in that proportion that 12. is to 16, by the 18. of the thirtenth: And the side of the octo∣hedron is to the side of the pyramis in that proportion that 18. is to 24. (by the same 18. of the thirtēth) which proportion is one & the self same with the proportiō of 12. to 16. Wherfore that prisme which is equal to the Octohedron, is to the prisme which is triple to the Tetrahedron, in that proportiō that the altitudes, or that the sides are. Wherfore an octohedrō is to the triple of a Tetrahedron cōtained in one and the selfe same sphere, in that proportion that their sides are: which was required to be de∣monstrated.

A Corollary.

The sides of a Tetrahedron & of an Octohedrō are proportionall with their altitudes. For the sides & altitudes were in power sesquitercia. Moreouer the diameter of the sphere is to the side of the Tetrahedron, as the side of the Octohedron is to the ••••de of the cube•• namely, the powers of eche is in sesquialter proportion, by the 18. of the thirtenth.

The 15. Proposition. If a rational line containing in power two lines, make the whole and the greater segment, and again containing in power two lines, make the whole and the lesse segment: the greater segment shalbe the side of the Icosahe∣dron, and the lesse segment shalbe the side of the Dodecahedron contayned in one and the selfe same sphere.

SVppose that AG be the diameter of the sphere which containeth the Icosahedron ABGC. And let BG subtend the sides of the pentagon described of the sides of the Icosahedron (by the 16. of the thirtēth.) Moreouer vpon the same diameter AG, or DF equal vnto it,* 1.47 let ther be described a dodecahedron DEFH, by the 1••. of the thirtenth, whose opposites sides ED and FH let be cut into two equal partes in the poynts I and K, and draw a line from I to K. And let the line EF couple two of the opposite angles of the bases which are ioyned together. Thē I say that AB the side of the Icosahedron is the greater segment which the diameter AG containeth in power together with the whole line, and line ED is the lesse segment, which the same diameter AG or DF containeth in power together with the whole.* 1.48 For forasmuche as the

[illustration]

opposite sides AB and GC of the Icosahedron being cou∣pled by the diameters AG and BC, are equal & parallels, by the 2. corollary of the 16. of the thirtēth: the right lines BG & AC which couple thē together are equal & paral∣lels by the 33. of the first. Moreouer the angles BAC & A∣BG being subtended of equal diam••ters, shall by the 8. of the first be equal, & by the 29 of the 〈◊〉〈◊〉, they shal be right angles. Wherfore the right line AG 〈◊〉〈◊〉 in power the ••wo lines AB and BG, by the 47. of 〈…〉〈…〉. And forasmuch as the line BG subtendeth the angle of the pentagon com∣posed of the sides of the Icosahedron, the greater segment of the right line BG, shalbe the right line AB, by the ••. of the thirtenth: which line AB, toge••her with the whole line BG, the line AG containeth in power. And forasmuch a•• the line IK coupling the opposite and parallel sides ED and FH of the Dodecahedron, maketh at those poyntes right angles, by the 3. corollary of the 17. of t••e thirtenth: the right line EF which coupleth together equal and pa∣rallel lines EI & FK, shalbe equal to the same line IK, by the 33. of the first. Wherfore the angle DEF shalbe •• right angle by the 29. of the first. Wherefore the diameter DF cōtaineth in power the two lines ED and EF. But the lesse segment of the line IK is ED the side of the Dodecahe∣dron, by the 4. corollary of the 17•• of the thirtenth. Wher∣fore the same line ED is also the lesse segment of the line EF (which is equal vnto the line IK): wherfore the diam••¦ter DF containing in power the two lines ED and EF (by the 47. of the first) containeth in pow••r•• ED the side of the dodecahedron, the lesse segment, together with the whole. If therfore a rational line AG or DF containing in power two lines AB and BG, doo make the whole line and the greater s••gment, and agai••e containing in power two lines EF and ED, do make the whole line and the lesse segment: the greater segment AB, shall be the side of the Icosahe∣dron, and the lesse segment ED shall be the side of the Dodecahedron contained in one and the selfe same sphere.

The 16. Proposition. If the power of the side of an Octohedron be expressed by two right line••

ioyned together by an extreme and me••ne proportion: the side of the Ico∣sahedron contained in the same sphere, shalbe duple to the lesse segment.

LEt AB the side of the Octohedron ABG containe in power the two lines C and H, which let haue that proportion that the whole hath to the greater segment (by the corollarye of the first proposition added by Flussas after the last propositiō of the sixth booke).* 1.49 And let the Icosahedron contained in the same sphere be DEF, whose side let be DE, and let the right line subtending the angle of the pentagon made of the sides of the Icosahedron be the line EF. Then I say that the side ED is in power double to the line H the lesse of those segmentes. Foras∣much as by that which was demonstrated in the 15. of this booke,* 1.50 it was manifest that ED the side of the Icosahedron is the great••r segment of the line EF•• and that the diameter DF containeth in power the two lines ED and EF, namely, the whole and the greater segment: but by suppo••ition the side AB cōtaineth in po∣wer

[illustration]

the two lines C & H ioined to∣gether in the self same proportiō. Wherefore the line EF is to the line ED, as the line C, is to the line H, by the ••. o•• this boke•• And alt••rna••••y by the 16. of the fiueth, the line EF is to the line C, as the line ED, is to the line H. And for∣asmuche as the line DF containeth in power the two lines ED and EF, and the line AB containeth in power the two lines C and H: therefore the squares of the lines EF and ED are to the square of the line DF, as the squares of the lines C and H to the square AB. And alternately, the squares of the lines EF and ••D, are to the squares of the lines C and H, as the square of the line DF is to the square of the line AB•• But DF the diameter is (by the 14. of the thirten••h) i•• power double to AB the side of the octo∣hedron inscribed (by supposition) in the same sphere. Wherefore the squares of the lines EF and ED, are double to the squares of the lines C and H. And therfore one square of the line ED is double to one square of the line H by the 12. of the fifth. Wherfore ED the side of the Icosahedron is in power duple to the line H, which is the lesse segment. If therfore the powe•• of the side of an octohedron be expres∣sed by two right lines ioyned together by an extreme and meane proportion: the side of the Icosahe∣dron contained in the same sphere, shalbe duple to the lesse segment.

The 17. Proposition. If the side of a dodecahedron, and the right line, of whome the said side is the lesse segment, be so set that they make a right angle: the right line which containeth in power halfe the line subtending the angle, is the side of an Octohedron contained in the selfe same sphere.

SVppose that AB be the side of a Dodecahedron, and let the

[illustration]

right line of which that side is the lesse segment be AG, name∣ly which coupleth the opposite sides of the Dodecahedron, by the 4. corollary of the 17. of the thirtenth:* 1.51 and let those lines be so set that they make a right angle at the point A. And draw the right line BG. And let the line D containe in power halfe the line BG (by the first proposition added by Flussas after the laste of the sixth). Then I say that the line D is the side of an Octohedron contayned in the same sphere. Forasmuche as the line AG maketh the greater segment GC the side of the cube contained in the same sphere (by the same 4. corol∣lary of the 17. of the thirtenth):* 1.52 and the squares of the whole line AG. and of the lesse segment AB are triple to the square of the greater segment GC, by the 4. of the thirtenth: Moreouer the diameter of the sphere, is in power triple to the same line GC the side of the cube (by the 15. of the thirtenth:

Wherfore the line BG is equal to the 〈◊〉〈◊〉. For it con••••ineth in power the two lines AB and AG (by the 47. of the first,) and therefore it containeth in power the triple of the line GC. But the side of the Octohedron contained in the same sphere, is in power triple to halfe the diameter of the sphere by the 14. of the thirtenth. And by suppo••••tion the line D contai•••••••• in pow•••• the halfe of the line BG. Wherefore the line D (containing in power the halfe of the same diameter) is the side of an octohe∣dron. If therfore the side of a Dodecah••dron and the right line of whome the said side is the lesse seg∣ment, be so set that they make a right angle: the right line which containeth in power halfe the line subtending the angle, is the side of an Oc••••••edron contained in the selfe same sphere: Which was re∣quired to be proued.

A Corollary.

Vnto what right line the side of the Octo••edron is in power sesquialter: vnto the same line the side of the Dodecahedron inscribed in the same sphere, is the greater segment. For the side of the Dodecahedron is the greater segment of the segment CG, vn∣to which D the side of the Octohedron is in power sesqui••lter, that is, is halfe of the power of the line BG, which was triple vnto the line CG.

¶The 18. Proposition. If the side of a Tetrahedron containe in power two right lines ioyned to∣gether by an extreme and meane proportion: the side of an Icosahedron described in the selfe same Sphere, is in power sesquialter to the lesse right line.

SVppose that ABC be a Tetrahedron, and let his side be AB,* 1.53 whose power let be diuided into the lines AG and GB, ioyned together by an extreme and meane proportion: name∣ly, let it be diuided into AG the whole line, and GB the greater se••ment (by the Corolla∣ry of the first Proposition added by Flussas after the last of the sixth). And let ED be the side of the Icosahedron EDF contained in the selfe same Sphere. And let the line which subtendeth the angle of the Pentagon described of the sides of the Icosahedron be EF. Then I say, that ED the side of

[illustration]

the Icosahedron is in power sesquialter to the lesse line GB.* 1.54 Forasmuch as (by that which was demonstrated in the 15. of this booke) the side ED is the gre••ter segment of the line EF which subtendeth the angle of the Pentagon. But as the whole line EF is to the greater seg∣ment ED, so is the same gr••••∣ter segment to the lesse (by the 30. of the sixth): and by suppo∣sition, AG, was the whole line, and G•• the greater segment: Wherefore as EF is to ED, so is AG to G••, by the second of the foure∣tenth. And alternately, the line EF is to the line AG, as the line ED is to the line GB. And forasmuch as (by supposition) the line AB containeth in power the two lines AG and GB: therefore (by the 4••. of the first) the angle AGB is a right angle. But the angle DEF is a right angle, by that which was de∣monstrated in the 15. of this booke. Wherefore the triangles AG•• and FED, are equiangle, by the ••. of the sixth. Wherefore their sides are proportionall: namely, as the line ED is to the line GB, so is the line FD to the line AB, by the 4. of the sixth. But by that which hath before bene demonstrated, FD is the diameter of the Sphere which containeth the Icosah••dron: which diameter is in power sesqui∣alter to AB the side of the Tetrahedron inscribed in the•• same Sphere, by the 13. of the thirtenth. Wher∣fore the line ED the side of the Icosahedron, is in power sesquialter to G•• the greater segment or lesse line. If therefore the side of a Tetrahedron containe in power two right lines ioyned together an ex∣treme and meane proportion: the side of an Icosahedron described in the selfe same Sphere, is in pow∣er sesquialter to the lesse right line.

¶ The 19. Proposition. The superficies of a Cube is to the superficies of an Octohedron inscribed in one and the selfe same Sphere, in that proportion that the solides are.

* 1.55SVppose that ABCDE be a Cube, whose fower diameters let be the lines AC, BC, DC, and EC produced on ech side. Let also the Octohedron inscribed in the selfe same Sphere be FGHK: whose three diameters let be FH, GK, and ON. Then I say, that the cube ABD is to the Octohedron FGH, as the superficies of the cube is to the super∣ficies of the Octohedron. Drawe from the centre of the cube to the base ABED, a per∣pendicular line CR. And from the centre of the Octohedron draw to the base GNH, a perpendicular line ••L.* 1.56 And forasmuch as the three diameters of the cube do passe by the 〈◊〉〈◊〉 C, therefore, by the 2. Corollary of the 15. of the thirtenth, ••here shall be made of the cube sixe pyramids, as thys pyramis ABDEC, equall to the whole cube. For there are in the cube ••ixe bases, vpon which fall equall perpendiculars from the cen•••••• by the Corollary of the Assump•• of the 16. of the twelfth, for the bases are contained in equall circl•• of the Sphere. But in the Octohedron the three diameters do make vpon the 8. bases, 8. pyramids, hauing their toppes in the centre, by the 3. Corollary of the 14•• of the thirtenth. Now the bases of the cube and of the Octohedron are contained in equall circles of the

[illustration]

Sphere, by the 13. of this booke. Wherefore they shall be equally distant from the centre, and the per∣pendicular lines CR and ••, shall be equall, by the Corollary of the Assumpt of the 16. of the twelfth. Wherefore the pyramids of the cube shall be vnder one and the selfe same altitude with the pyramids of the Octohedron, namely, vnder the perpendicular line drawen from the centre to the bases. Wher∣fore sixe pyramids of the cube, are to 8. pyramids of the Octohedron being vnder one and the same al∣titude, in that propo••tion that their bases are, by the 6. of the twelfth: that is, one pyramis set vpon sixe bases of the cube, and hauing to his altitude the perpendicular line, which pyramis is equall to the sixe pyramids, by the same 6. of the twelfth, is to one pyramis set vpon the 8. bases of the Octohedron, being equall to the Octohedron, and also vnder on•• and the selfe same altitude, in that proportion that sixe bases of the cube, which containe the whole superficies of the cube, are to 8. bases of the Octohedron•• which containe the whole superficies of the Octohedron. For the solides of those pyramids are in pro∣portion the one to the other, as their bases are, by the selfe same 6. of the twelfth. Wherefore ••he su∣perficies of the cube is to the superficies of the Octohedron inscribed in one and the selfe same Sphere, in that proportion, that the solides are: which was required to be proued.

¶ The 20. Proposition. If a Cube and an Octohedron be contained in one & the selfe same Sphere: they shall be in proportion the one to the other, as the side of the Cube is to

the semidiameter of the Sphere.

SVppose that the Octohedron AECDB be inscribed in the Sphere ABCD: and let the cube inscribed in the same Sphere be FGHIM: whose diameter let be HI,* 1.57 which is e∣quall to the diameter AC, by the 15. of the thirtenth: let the halfe of the diameter be AE. Then I say, that the cube FGHIM is to the Octohedron AECDB, as the side MG is to the semidiameter AE. Forasmuch as the diameter AC is in power double to BK the side of the Octohedron (by the 14. of the thirtenth) and is in power triple to MG the side of the cube (by the 15. of the same): therefore the square BKDL shall be sesquial••er to FM the square of the cube. From the line AE cut of a third part AN, and frō the line MG cut of like∣wise a third part GO, by the 9. of the sixth.* 1.58 Now then the line EN shall be two third partes of the line AE, and so also shall the line MO be of the line MG. Wherefore the parallelipipedon set vpon the base BKDL, and hauing his altitude the line EA, is triple to the parallelipipedon set vpon the same base, and hauing his altitude the line AN, by the Corollary of the 31. of the eleuenth: but it is also tri∣ple to the pyramis ABKDL which is set vpon the same base, and is vnder the same altitude (by the se∣cond Corollary of the 7. of the twelfth). Wherefore the pyramis ABKDL is equall to the parallelipi∣pedon, which is set vpon the base BKDL, and

[illustration]

hath to his altitude the line AN. But vnto that pa∣rallelipipedō, is double the parallelipipedon which is set vppon the same base BKDL, and hath to his altitude a line double to the line EN, by the Corollary of the 31. of the first and vnto the pyra∣mis is double the Octohedron ABKLDC, by the 2. Corollary of the 14. of the thirtenth. Where∣fore the Octohedron ABKDLC is equall to the parallelipipedon set vpon the base BKLD, & ha∣uing his altitude the line EN (by the 15. of the fifth). But the parallelipipedon set vpon the base BKDL, which is sesquialter to the base FM, and hauing to his altitude the line MO, which is two third partes of the side of the cube MG, is equall to the cube FG: by the 2. part of the 34. of the ele∣uenth. (For it was before proued that the base BKDL is sesquialter to the base FM). Now then these two parallelipipedons, namely, the paralleli∣pipedon which is set vpō the base BKDL (which is sesquialter to the base of the cube) and hath to his altitude the line MO (which is two third partes of MG the side of the cube) which paral∣lelipipedon is proued equall to the cube, and the parallelipipedon set vpon the same base BKDL, and hauing his altitude the line EN (which paral∣lelipipedon is proued equall to the Octohedron): these two parallelipipedons (I say) are the one to the other, as the altitude MO, is to the altitude EN (by the Corollary of the 31. of the eleuenth). Wherefore also as the altitude MO, is to the alti∣tude EN, so is the cube FGHIM, to the Octo∣hedron ABKDLC, by the 7. of the fifth. But as the line MO is to the line EN, so is the whole line MG to the whole line EA, by the 18. of the fifth. Wherefore as MG the side of the cube, is to EA the semidiameter, so is the line FGHIM to the Octohedron ABKDLC inscribed in one & the selfe same Sphere. If therefore a cube and an Octohedron be contained in one and the selfe same Sphere. they shall be in proportion the one to the other, as the side of the cube is to the semidiameter of the Sphere: which was required to be demonstrated.

A Corollary.

Distinctly to notefie the powers of the sides of the fiue solides by the power of the diameter of the sphere.

The sides of the tetrahedron and of the cube doo cut the power of the diameter of the sphere in∣to two squares which are in proportion double the one to the other. The octohedron cutteth the

power of the diameter into two equall squares. The Icosahedron into two squares, whose proportion is duple to the proportion of a line diuided by an extreame and meane proportion, whose lesse segmēt is the side of the Icosahedron. And the dodecahedron into two squares, whose proportion is quadru∣ple to the proportion of a line diuided

[illustration]

by an extreame and meane proportion, whose lesse segment is the side of the dodecahedron. For AD the diameter of the sphere, contayneth in power AB the side of the tetrahedron, and BD the side of the cube, which BD is in power halfe of the side AB. The diameter also of the sphere contayneth in power AC and C∣D two equall sides of the octohedron. But the diameter contayneth in power the whole line AE and the greater seg∣ment thereof ED, which is the side of the Icosahedron, by the 15. of this booke. Whe••fore their powers being in duple proportiō of that in which the sides are, by the first corollary of the 20. of the sixth, haue their proportion duple to the proportion of an extreame & meane proportiō. Farther the diameter cōtayneth in power the whole line AF, and his lesse segment FD, which is the side of the dodecahedron, by the same 15. of this booke. Wherefore the whole hauing to the lesse, •• double proportion of that which the extreame hath to the meane, namely, of the whole to the greater segment, by the 10. diffinition of the fifth, it followeth that the proportion of the power is double to the doubled proportion of the sides, by the same first corollary of the 20. of the sixth: that is, is quadru∣ple to the proportion of the extreame and of the meane, by the diffinition of the sixth.

An aduertisment added by Flussas.

By this meanes therefore, the diameter of a sphere being geuen, there shall be ge∣uen the side of euery one of the bodies inscribed. And forasmuch as three of those bo∣dies haue their sides commensurable in power onely, and not in length, vnto the dia∣meter geuen (for their powers are in the proportion of a square number to a number not square: wherefore they haue not the proportion of a square number to a square number, by the corollary of the 25. of the eight: wherefore also their sides are incom∣mensurabe in length by the 9. of the tenth): therefore it is sufficient to compare the powers and not the lengths of those sides the one to the other•• which powers are con∣tained in the power of the diameter: namely, from the power of the diameter, let there ble taken away the power of the cube, and there shall remayne the power of the Tetra∣hedron: and taking away the power of the Tetrahedron, there remayneth the power of the cube: and taking away from the power of the diameter halfe the power thereof, there shall be left the power of the side of the octohedron. But forasmuch as the sides of the dodecahedron and of the Icosahedron are proued to be irrationall (for the side of the Icosahedron is a lesse line, by the 16. of the thirtenth: and the side of the dedocahe∣dron is a residuall line, by the 17. of the same) therfore those sides are vnto the diame∣ter which is a rationall line set, incommensurable both in length and in power. Where∣fore their comparison can not be diffined or described by any proportion expressed by numbers, by the 8. of the tenth: neither can they be compared the one to the other: for irrational lines of diuers kindes are incōmēsurable the one to the other: for if they should be commensurable, they should be of one and the selfe same kinde, by the 103. and 105. of the tenth, which is impossible. Wherefore we seking to compare them to the power of the diameter, thought they could not be more aptly expressed, then by such proportions, which cutte that rationall power of the diameter according to their sides: namely, diuiding the power of the diameter by lines which haue that proportiō, that the greater segment hath to the lesse, to put the lesse segment to be the side of the Icosahedron: & deuiding the sayd power of the diameter by lines hauing the propor∣tion of the whole to the lesse segment, to expresse the side of the dodecahedron by the lesse segment: which thing may well be done betwene magnitudes incommensurable.

The ende of the fourtenth Booke of Euclides Elementes after Flussas.

Notes

• * 1.1

  The first pro∣position after Campane.
• * 1.2

  Construction.
• * 1.3

  Demonstra∣tion.
• * 1.4

  The 2. propo∣sition after Campane.
• * 1.5

  Demonstrati∣on leading to an impossibi∣litie.
• * 1.6

  The 4. propo∣sition after Campane.
• * 1.7

  Constru••t••on.
• * 1.8

  Demonstra∣tion.
• * 1.9

  This Corolla∣ry Campane also ••utteth after the 4. proposition in his order.
• * 1.10

  The 5. propo∣sition after Campane.
• * 1.11

  Construction.
• * 1.12

  Demonstra∣tion.
• * 1.13

  This is the 6. and 7. propo∣sitions after Campane.
• * 1.14

  Construction.
• * 1.15

  Demons••ra∣tion.
• * 1.16

  This Corolla∣ry Campane also addeth after the 7. proposition i•• his order.
• * 1.17

  The 5. propo∣sition a••ter Campane.
• * 1.18

  Construction.
• * 1.19

  Demonstra∣tion.
• * 1.20

  This Assumpt Campane also hath after the 8. proposition, in his order.
• * 1.21

  Construction.
• * 1.22

  Demonstra∣tion.
• * 1.23

  The 9. propo∣sition after Campane.
• * 1.24

  Construction.
• * 1.25

  Demonstra∣tion.
• * 1.26

  This Campane putteth a•• a Co∣rollary in the 9. proposition af∣ter his order.
• * 1.27

  This Corolla∣ry is the 9. proposition after Cam∣pane.
• * 1.28

  The 12. pro∣position after Campane.
• * 1.29

  Construction.
• * 1.30

  Demonstra∣tion.
• * 1.31

  The 13. propo∣sition after Campane.
• * 1.32

  The 14. pro∣position after Campane.
• * 1.33

  Demonstra∣tion of the first part.
• * 1.34

  Demonstrati∣on of the se∣cond part.
• * 1.35

  The 17. pro∣position after Campane.
• * 1.36

  Fir••t part of the construc∣tion.
• * 1.37

  First part of the Demon∣stration.
• * 1.38

  Second par•• of the c••n∣struction.
• * 1.39

  Second part of the D••∣monstration.
• * 1.40

  The 18. pro∣position after Campane.
• * 1.41

  Demonstra∣tion of the first part.
• * 1.42

  Demonstra∣tion of the second part.
• * 1.43

  The Corolla∣ry of the 8. proposition after Cam∣pane.
• * 1.44

  Dem••nstra∣tion.
• * 1.45

  Construction.
• * 1.46

  Demonstra∣tion.
• * 1.47

  Construction.
• * 1.48

  Demonstra∣tion.
• * 1.49

  Construction.
• * 1.50

  Demonstra∣tion.
• * 1.51

  Construction.
• * 1.52

  Demonstra∣tion.
• * 1.53

  Construction.
• * 1.54

  Demonstra∣tion.
• * 1.55

  Construction.
• * 1.56

  Demonstra∣tion.
• * 1.57

  Construction.
• * 1.58

  Demonstra∣tion.