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 4. Probleme. The 16. Proposition. To make an Icosahedron, and to comprehend it in the Sphere geuen, wher∣in were contained the former solides, and to proue that the side of the Ico∣sahedron is an irrationall line of that kinde which is called a lesse line.

* 1.1TAke the diameter of the Sphere, namely, the line AB: and deuide it in the point C, so that let the line AC be quadruple to the line CB, by the 9. of the sixt. And describe vpō the line AB a semicircle ADB. And (by the 11. of the first) from the point Craise vp vnto the line AB a perpendicular line CD. And draw a right line from the point D to the point B. And describe a circle EFGHK whose line from the centre (which let be the point Z) to the circumference, let be equall to the line DB. And in the circle EFGHK describe (by the 11. of the fourth) an equilater and equiangle Pentagon ••igure EFGHK. And deuide the circumferences EF, FG, GH, HK, and KE, into two equall partes in the pointes L, M, N, X, O. Draw also these right lines LM, MN, NX, XO, and OL: and moreouer these lines OE, EL, LF, FM, MG, GN, NH, HX, XK, and FO, and they shall be the sides of an equilater decagon in∣scribed in the circle EFGHK, by the 29. of the third. Wherefore the figure LMNXO is an equilater pentagon, by the 29. of the third, and the right line EO is the side of a decagon ••r ten angled figure. Raise vp (by the 1••. of the eleuenth) from the pointes E, F, G, H, K, and the centre Z, vnto the plaine superficies of the circle, perpendicular lines EP, FR, GS, HT, KV, and ZW, and let ech of them be put equall to the line drawen from the centre of the circle EFGHK, to the circumference, namely, to the line ZE. Wherefore right lines drawen from W to P, from W to V, from W to T, from W to S, from W to R, shall be equall and parallels to right lines drawen from Z to E, from Z to K, from Z to H, from Z to G, and from Z to F, by the 6. and 7. of the eleuenth, and 33. of the first. Wherefore the plaine superficieces EFGHK, and PRSTV, which are extended by those parallel lines, are pa∣rallel

superficieces, by the 15. of the eleuenth. Wherefore making the centre the point W, and the space WP, or WV, describe a circle, and it shall passe, by the pointes T, S, F, and shall be equall to the circle EFGHK. For the semidiameters of eche are equall. And drawe these right lines PR, RS, ST, TV, VP, and they shall make a pentagon, whose sides shall be e∣quall to the sides of the Pentagon OLMNX, by the 29. of the first. For ech of them doth subtend two sides of the decagon, or

[illustration]

the fift part of equall circles. From the vpper pointes P, R, S, T, V, draw these lines PO, PL, RL, RM, SM, SN, TN, TX, VX, VO: which shall subtend right angles cōtained vnder the sides of the decagon EL∣FMGNHXKO, and the per∣pendicular lines PE, RF, SG, TH, VK.

Now forasmuch as the perpendicular lines PE, RF, SG, TH, and VK,* 1.2 are put e∣quall to the line ZE drawen from the centre, therefore they are equall to the side of an equi∣later hexagon inscribed in the same circle (by the Corollary of the 15. of the fourth). Where∣fore the right lines PO, PL, VO, and VX (which subtend the right angles contained vn∣der those perpendicular lines and the sides of the decagon) containe them in power, by the 47. of the first. But the side of a pentagon (namely, the side LO or PV) containeth in power the sides of an hexagon and of a decagon inscribed in one and the selfe same circle, by the 10. of

this booke. Wherefore the subtending lines PO, PL, VO, VX, TX, TN, SN, SM, RM, RL, containe in power the selfe same square that the sides of the pentagon OLMNX con∣taine, or that the sides of the pentagon PRSTV containe: and therefore those subtending lines are equall to the sides of the foresaid pentagons. Wherefore the triangles contained of those subtending lines and of the sides of the pentagons, and which are ten in number, name∣ly, PLO, OVP, VOX, VXT, TXN, TNS, SNM, SMR, RML, and RLP, are equilater. Againe produce the right line ZW on either side to the points Q & Y and vn∣to the side of the decagon, namely, to the line OE, put the lines ZY and WQ equal. And for∣asmuch as the right line QY is ere∣cted

[illustration]

perpendicularly to the plaine su∣perficies OLMNX, therefore it is also erected perpendicularly to the o∣ther plaine superficies PRSTV,* 1.3 by the Corollay of the 14. of the eleuēth. And drawe these right lines QP, QV, QT, QS, & QR: and these lines also YL, YM, YN, YX, and YO.

* 1.4Now forasmuch as the lines QP, QV, QT, QS, and QR do eche subtend right an∣gles contayned vnder the sides of an equilater hexagon & of an equilater decagon inscribed in the circle PRSTV or in the circle EFGHK (which two circles are equall) therfore the sayd lines are eche equal to the side of the pentagon inscribed in the foresayd circle by the 10.

of this booke, and are equall the one to the other, by the 4. of the first, (for all the angles at the poynt W which they subtend are right angles). Wherefore the fiue triangles QPV, QPR, QRS, QST, and QTV, which are contayned vnder the sayd lines QV, QP, QR, QS, QT, and vnder the sides of the pentagon VPRST, are equilater, and equal to the ten for∣mer triangles. And by the same reason the fiue triangles opposite vnto them, namely, the tri∣angles YML, YMN, YNX, YXO, and YOL, are equilater and equal to the said ten triangles. For the lines YL, YM, YN, YX, and YO do subtend right angles cōtayned vnder the sides. of an equilater hexagon and of an equilater decagō inscribed in the circle EFGHK, which is equall to the circle PRSTV. Wherefore there is described a solide contayned vnder 20. equilater triangles. Wherefore by the last diffinition of the eleuenth there is described an Icosahedron.

Now it is required to comprehend it in the sphere geuen, and to proue that the side of the Icosahedron is an irrationall line of that kinde which is called a lesse line. Forasmuch as the line ZW is the side of an hexagon, & the line WQ is the side of a decagon, therfore the line ZQ is diuided by an extreme and meane proportion in the point W, and his greater segmēt is ZW (by the 9. of the thirtēth). Wherfore as the line QZ is to the line ZW, so is the line Z∣W to the line WQ. But the ZW is equall to the line ZL by construction, and the line WQ to the line ZY by construction also: Wherefore as the line QZ is to the line ZL, so is the line ZL to the line ZY, and the angles QZL•• and LZY are right angles (by the 2. diffinition of the eleuenth): If therfore we draw a right line from the poynt L to the poynt Q, the angle YLQ shalbe a right angle, by reasō of the likenes of the triangles YLQ and ZLQ (by the 8. of the sixth). Wherfore a semicircle described vpō the line QY, shal passe also by the point L (by the assumpts added by Campane after the 13. of this booke). And by the same reasō al∣so, for that as the line QZ is the line ZW, so is the line ZW to the line WQ, * 1.5 but the line ZQ is equall to the line YW, and the line ZW to the line PW: wherefore as the line YW is to the line WP, so is the line PW to the line WQ. And therefore agayne if we draw a right line from the poynt P to the point Y, the angle YPQ shalbe a right angle. Wherfore a semi∣circle described vpon the line QY shal passe also by the point P, by the former assumpts: & if the diameter QY abiding fixed the semicircle be turned round about, vntil it come to the selfe same place from whence it began first to be moued, it shall passe both by the point P, and also by the rest of the pointes of the angles of the Icosahedron, and the Icosahedron shalbe comprehended in a sphere. I say also that it is contayned in the sphere geuen.

Diuide (by the 10. of the first) the line ZW into two equall parts in the point a. And for∣asmuch as the right line ZQ is diuided by an extreme and meane proportion in the point W, and his lesse segment is QW, therefore the segment QW hauing added vnto it the halfe of the greater segment, namely, the line Wa, is (by the 3. of this booke) in power quintuple to the square made of the halfe of the greater segment: wherefore the square of the line Qa is quintuple to the square of the line ••W. But vnto the square of the Qa, the square of the line QY is quadruple (by the corollary of the 20. of the sixth) for the line QY is double to the line Qa: and by the same reason vnto the square of the WA the square of the line ZW is quadruple: Wherefore the square of the line QY is quintuple to the square of the line ZW (by the 15. of the fiueth). And forasmuch as the line AC, is quadruple to the line CB, there∣fore the line AB is quintuple to the line CB. But as the line AB is to the line BC, so is the square of the line AB to the square of the line BD (by the 8 of the sixth, and corollary of the 20. of the same). Wherfore the square of the line AB is quintuple to the square of the line B∣D. And it is is proued that the square of the line QY is quintuple to the square of the line ZW. And the line BD is equall to the line ZW, for either of them is by position equall to the line which is drawen from the centre of the circle EFGHK to the circumference. Where∣fore the line AB is equall to the YQ. But the line AB is the diameter of the sphere geuen: Wherefore the line YQ, which is proued to be the diameter of the sphere contayning the Ico∣sahedron,

is equall to the diameter of the sphere geuen. Wherefore the Icosahedron is contay∣ned in the sphere geuē. Now I say that the side of the Icosahedron is an irrationall line of that kinde which is called a lesse line. For forasmuch as the diameter of the sphere is rational, and is in power quintuple to the square of the line drawen frō the centre of the circle OLMNX: wherefore also the line which is drawen from the centre of the circle OLMNX is rationall: wherefore the diameter also being cōmensurable to the same line (by the 6. of the tenth) is ra∣tionall. But if in a circle hauing a rationall line to his diameter be described an equilater pentagon, the side of the pentagon is (by the 11. of this booke) an irrationall line, of that kinde which is called a lesse line. But the side of the pentagon OLMNX is also the side of the Ico∣sahedron described, as hath before ben proued. Wherfore the side of the Icosahedrō is an irra∣tionall line of that kinde which is called a lesse line. Wherefore there is described an Icosahe∣dron and it is contayned in the sphere geuen, and it is proued that the side of the', Icosa∣hedron is an irrationall line of that kind which is called a lesse line. Which was required to be done, and to be proued.

A Corollary.

Hereby it is manifest that the diameter of the sphere, is in power quintuple to the line which is drawen from the centre of the circle to the circumference, on which the Icosahedron is described. And that the diameter of the sphere: is com∣posed of the side of an hexagon, and of two sides of a decagon described in one and the selfe same circle.

Flussas proueth the Icosahedron described, to be cōtayned in a sphere, by drawing right lines from the poynt, a, to the poyntes P and G after this maner.

Forasmuch as the lines ZW, WP are put equal to the line drawen from the centre to the circum∣ference•• and the line drawen from the centre to the circumference is double to the line aW, by con∣struction: therefore the line WP is also double to the same line aW. Wherefore the square of the line WP is quadruple to the square of the line aW (by the corollary of the 20. of the sixth). And those lines PW and Wa contayning a right angle PWa (as hath before bene proued) are subtended of the right line aP. Wherefore (by the 47. of the first) the line aP contayneth in power the lines PW, and Wa. Wherefore the right line aP is in power quintuple to the line Wa. Wherefore the right lines aP, and aQ being quintuple to one and the same line Wa, are (by the 9. of the flueth) equall. In like sorte also may we proue that vnto those lines aP and aQ, are equall the rest of the lines drawen from the poynt a to the rest of the angles R, S, T, V. For they subtend right angles contayned of the line Wa, and of the lines drawen from the centre to the circumference. And forasmuch as vnto the line Wa is equall the line Va, which is likewise erected perpendicularly vnto the other plaine superficies OLMNX: there∣fore lines drawen from the point a to the angles O, L, M, N, X, and subtending right angles at the point Z contayned vnder lines drawen frō the centre to the circumference, and vnder the line aZ, are equal not onely the one to the other, but also to the lines, drawen frō the sayde poynt a, to the former angles at the poynts, P, R, S, T, V••. For the lines drawen frō the centre to the circumference of ech circle are equall, & the line aW is equall to the line aZ. But the line aP is proued equal to the line aQ, which is the halfe of the whole QY. Therefore the residue aY is equall to the foresayd lines aP, aQ &c. Wherefore making the centre the poynt a, and the space one of those lines aQ, aP, &c. extende the su∣perficies of a sphere, & it shal touch the 12. angles of the Icosahedron, which are at the pointes O, L, M. N, X, P, K, S, T, V, Q, Y: which sphere is described, if vpō the diameter QY, be drawen a semicircle, and the sayd semicircle be moued about, till it returne vnto the same place from whense it began first to be moued.

¶ A Corollary added by Flussas.

The opposite sides of an Icosahedron are parallels. For the diameters of the sphere do fall vpon the opposite angles of the Icosahedron: as it was manifest by the right line QY. If therefore there be ima∣gined to be drawen the two diameters PN, and OM they shall concurre in the point F: wherefore the right lines which ioyne them together, PV, and LN, are in one and the selfe same playne superficies, by

the 2. of the eleuenth. And forasmuch as the alternate angles at the endes of the diameters are equall (by the 8. of the first): for the triangles contayned vnder equall semidiamete••s and the side of the Ico∣sahedron are equiangle: therefore (by the 28. of the first) the lines PV and LN are paralles.

Notes

• * 1.1

  First part of the construction.
• * 1.2

  First part of the demonstration.
• * 1.3

  Second part of the cōstruction.
• * 1.4

  Second part of the demōstratiō.
• * 1.5

  For the line Q∣W, is equall to the line IZ, & the line ZW, is commō to them both.

  This part is a∣gaine after∣ward demon∣strated by Flus∣sas.