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 2. Probleme. To describe an equilater & equiangle Icosidodecahedron, & to cōprehend it in a sphere geuen: and to proue that the diameter being diuided by an ex∣treame and meane proportion, maketh the greater segment double to the side of the Icosidodecahedron.

SVppose that the diameter of the sphere geuen be NL, and (by the 30. of the sixth) diuide the line NL by an extreame and meane proporti∣on in the poynt I: and the greater segment thereof let be NI.* 1.1 And vpō the line NI describe a cube by the 15. of the thirtenth: and about this cube let there be circumscri∣bed

[illustration]

a dode••ahedron, by the 17. of the thirtenth: & let the same be ABCDEFHKMO. And diuide euery one of the sides into two equall parts in the poynts Q, R, S, T, V, X, Y, Z, P, ••, ••, G. &c? and couple the sections with right lines, which shall subtend the an∣gles of the pentagons, as the lines PG, GV, VQ, QY, Y∣R, RQ, VT, TX, XV, and so the rest. Now forasmuch as these lines subtend equall an∣gles of the pentagons, and those equall angles are con∣tayned of equall sides (name∣ly of the halues of the sides of the pentagons: therefore those subtending lines are equall, by the 4. of the first. Wherefore the triangles G∣QV, YQR, VXT, and the rest which take away solide angles of the dodecahe∣dron, are equilater. Agayne forasmuch as in euery pentagon are described fiue e∣quall right lines, coupling the middle sections of the sides, as are the lines QV, V∣T, TS, SR, RQ: they describe a pentagon in the playne of the pentagon of the dodecahedron: and the sayd pentagon is contayned in a circle, namely, whose cen∣••re is the centre of a pentagon of the dodecahedron. For the lines drawne from that centre to the angles of this pentagon are equall, for that they are perpendicu∣lars vpon the bases cutte, by the 12. of the fourth. Wherefore the pentagon QRS∣TV is equiangle, by the 11. of the same. And by the same reason may the rest of the pentagons described in the bases of the dodecahedron be proued equall and like. Wherefore those pentagons are 12. in number: And forasmuch as the equall and like t••iangles, doo subtend and take away 20. solide angles of the dodecahedron, therefore the sayd triangles shall be 20. in nūber. Wherfore we haue described an Icosidodecahedrō by the diffinitiō, which Icosidodecahedrō is equilater, for that all the sides of the triangles are equal & cōmon with the pētagons: and it is also e∣quiangle. For euery one of the solide angles is made of two superficiall angles of an equilater pentagon, and of two superficiall angles of an equilater triangle.

Now let vs proue that it is contained in the Sphere geuen, whose diameter is NL.* 1.2 Forasmuch as perpendiculars drawen frō the centres of the Dodecahedron, to the midle sections of his sides, are the halfes of the lines, which couple the op∣posite midle sections of the sides of the Dodecahedron, by the 3. Corollary of the 17. of the thirtenth: which lines also, by the same Corollary, do in the centre diuide the one the other into two equall partes: therefore right lines drawen from that point to the angles of the Icosidodecahedron (which are set in those midle secti∣ons) are equall: which lines are 30. in number according to the number of the sides of the Dodecahedron: for euery one of the angles of the Icosidodecahedron are set in the midle sections of euery one of the sides of the Dodecahedron. Wher∣fore making the centre the centre of the Dodecahedron, and the space any one of the lines drawen from the centre to the midle sections, describe a Sphere, and it shall passe by all the angles of the Icosidodecahedron, and shall containe it.

And forasmuch as the diameter of this solide, is that right line, whose grea∣ter segment is the side of the cube inscribed in the Dodecahedron, by the 4. Co∣rollary of the 17. of the thirtenth, which side is NI, by supposition. Wherefore that solide is contayned in the Sphere geuen whose diameter is put to be the line NL.

* 1.3Now let vs proue that the

[illustration]

greater segment of the diame∣ter is duple to QV the side of this solide. Forasmuch as the sides of the triangle AEB are in the pointes Q and V diui∣ded into two equall partes, the lines QV and BE are pa∣rallels, by the Corollary of the 39. of the first. Wherefore as AE is to AV, so is EB to VQ, by the 2. of the sixth. But the line AE is double to the line AV. Wherefore the line BE is double to the line QV: by the 4. of the sixth. Now the line BE is equall to NI, or to the side of the cube, by the 2. Corollary of the 17. of the thirtenth, which line NI is the greater segment of the diameter NL. Wherefore the greater segment of the diameter geuen, is double to the side of the Icosidodecahedron inscribed in the Sphere geuen. Wherefore we haue described an equilater and equiangle Ico∣sidodecahedron, and contained it in a Sphere geuen, and haue proued that the di∣ameter thereof being diuided by an extreme and meane proportion, maketh hys greater segment double to the side of the Icosidodecahedron.

¶ An aduertisment of Flussas.

To the vnderstanding of the nature of this Icosidodecahedron, ye must well conceaue the passions and proprieties of both those solides, of whose bases i•• con∣sisteth, namely, of the Icosahedron and of the Dodecahedron. And although in it

the bases are placed oppositely, yet h••u•• they one to the other one & the s••me in∣clination. By reason wherof there he hidden in it the actions and p••••••••ons of the o∣ther regular solides. And I would haue thought i•• not impertinent to the purpose to haue set forth the inscriptions and circumscriptions of this solide, if w••nt of t••h•• had not hindred. But to the end the reader may the better a••••aine to the vnder∣standing therof, I haue here following briefly set forth, how it may in o•• about e∣uery one of the fiue regular solides be inscribed or circumscribed: by the helpe whereof ••e may, with small trauaile or rather none at all, so that he haue well p••y∣sed and considered the d••monstrations pertayning to the foresayd fi••e regular so∣lides, demonstrate both the inscription of the sayd solides in it, and the inscription of it in the sayd solides.

¶Of the inscriptions and circumscriptions of an Icosidodec••hedron.

An Icosidodecahedron may containe the other fiue regular bodyes. For it will receaue the angles of a Dodecahedron, in the centres of the triangles which subtend the solide angles of the Dodecahedron: which solide angles are 20. in nū∣ber, and are placed in the same order in which the solide angles of the Dodecahe∣dron taken away or subtended by them, are. And by that reason it shall receaue a Cube and a Pyramis contayned in the Dodecahedron: when as the angles of the one are set in the angles of the other.

An Icosidodecahedron receaueth an Octohedron, in the angles cutting the sixe opposite sections of the Dodecahedron, euen as if it were a simple Dode∣cahedron.

And it containeth an Icosahedron, placing the 12. angles of the Icosahedron in the selfe same centres of the 12. Pentagons.

It may also by the same reason be inscribed in euery one of the fiue regular bo∣dies: namely, in a Pyramis, if ye place 4. triangular bases concentricall with 4. bases of the Pyramis, after the same maner, that ye inscribed an Icosahedrō in a Pyrami••. So likewise may it be inscribed in an Octohedron, if ye make 8. bases thereof con∣centricall with the 8. bases of the Octohedron. It shall also be inscribed in a Cube, if ye place the angles which receaue the Octohedron inscribed in it, in the centres of the bases of the Cube. Moreouer, ye shall inscribe it in an Icosahedron, when the triangles compased in of the Pentagon bases, are concentricall with the trian∣gles, which make a solide angle of the Icosahedron. Finally, it shall be inscribed in a Dodecahedron, if ye place euery one of the angles thereof in the midle secti∣ons of the sides of the Dodecahedron, according to the order of the constructi∣on thereof.

The opposite plaine superficieces also of this solide are parallels. For the op∣posite solide angles are subtended of parallel plaine superficieces, as well in the an∣gles of the Dodecahedron subtended by ••ri••ngle••, a•• in the angles of the Icosahe∣dron subtended of Pentagons, which thing may easily be d••monstra••ed. More∣ouer in thi•• solide are infinit•• properti•••• & p••ssion••, springing of the solide•• wher∣of ••t is composed.

Wherefor•• it is manifest that a Dodecahedron & an Icosahedron, mixed, are

transformed into one & the selfe same solide of an Icosidodecahedron. A cube al∣so and an octohedrō are mixed and altered into an other solide, namely, into one and the same Exoctohedron. But a pyramis is transformed into a simple and per∣fect solide, namely into an Octohedron.

If we will frame these two solides ioyned together into one solide, this onely must we obserue.

In the pentagon of a dodecahedron inscribe a like pentagon, so that let the an∣gles of the pentagon inscribed be set in the midle sections of the sides of the pen∣tagon circumscribed, and then vpon the said pentagon inscribed, let there be set a solide angle of an Icosahedron, and so obserue the selfe same order in euery one of the bases of the Dodecahedron: and the solide angles of the Icosahedron set vpon these pentagons shall produce a solide consisting of the whole Dodecahe∣dron, and of the whole Icosahedron. In like sort, if in euery base of the Icosahedrō, the sides being diuided into two equall partes be inscribed an equilat•••• triangle, and vpon euery one of those equilater triangles be set a solide angle of a Dodeca∣hedron: there shall be produced the selfe same solide consisting of the whole Ico∣sahedron, & of the whole Dodecahedron.

And after the same order, if in the bases of a cube, be inscribed squares subten∣ding the solide angles of an Octohedron, or in the bases of an Octohedron, be in¦scribed equilater triangles subtēding the solide angles of a cube, there shall be pro∣duced a solide consisting of either of the whole solides, namely, of the whole cube and of the whole Octohedron.

But equilater triangles inscribed in the bases of a pyramis, hauing their angles set in the midle sections of the sides of the pyramis, and the solide angles of a pyra∣mis set vpon the sayd equilater triangles, there shall be produced a solide, consi∣sting of two equal and like pyramids.

And now if in these solides thus composed, ye take away the solide angles, there shalbe restored againe the first composed solides: namely, the solide angles taken away from a Dodecahedron and an Icosahedron composed into one, there shalbe left an Icosidodecahed••on: the solide angles takē away from a cube, and an octohedrō cōposed into one solide, there shalbe left an exocthedrō. Moreouer the solide angles taken away from two pyramids composed into one solide, there shal be left an Octohedron.

Flussas after this setteth forth certaine passions and properties of the fiue sim∣ple regular bodies: which although he demonstrateth not, yet are they not hard to be demonstrated, if we wel pease and conceiue that, which in the former bookes hath bene taught touching those solides.

Of the nature of a trilater and equilater Pyramis.

[ 1] A trilater equilater Pyramis, is deuided into two equal partes, by three equal squares, which in the centre of the pyramis cutte the one the other into two equal partes, and perpendicularly, and whose angles are set in the midle sections of the [ 2] sides of the pyramis. From a pyramis are taken away 4. pyramids like vnto the whole, which vtterly take away the sides of the pyramis, and that which is left

is an octohedrō inscribed in the pyramys in which all the solides inscribed in the pyramis are contained. A perpendicular drawne from the angle of the pyramis [ 3] to the base, is double to the diameter of the cube inscribed in it. And a right line [ 4] coupling the midle sections of the opposite sides of the pyramis, is triple to the side of the selfe same cube. The side also of the pyramis is triple to the diameter of the [ 5] base of the cube. Wherefore the same side of the pyramis is in power duple to [ 6] the right line which coupleth the midle sections of the opposite sides. And it is [ 7] in power sesquialter to the perpendicular which is drawne from the angle to the base. Wherefore the perpendicular is in power sesquitertia to the line which cou∣pleth [ 8] the midle sections of the opposite sides. A pyramis, and an Octohedron in∣scribed in it, also an Icosahedron inscribed in the same Octohedron, doo containe [ 9] one and the selfe same sphere.

Of the nature of an Octohedron.

Foure perpendiculars of an Octohedron, drawne in 4. bases therof from two [ 1] opposite angles of the said Octohedron, and coupled together by those 4. bases, describe a Rhombus, or diamond figure: one of whose diameters is in power du∣ple to the other diameter. For it hath the same proportiō that the diameter of the Octohedron, hath to the side of the Octohedron. An Octohedron & an Icosahe∣dron inscribed in it, do containe one and the selfe same sphere. The diameter of [ 2] the solide of the Octohedron, is in power sesquialter to the diameter of the circle [ 3] which containeth the base: and is in power triple to the right line which coupleth [ 4] the cētres of the opposite bases: and is in power * 1.4 duple superbipartiens tercias to the perpēdicular or side of the foresaid Rhombus: and moreouer is in lēgth triple to the line which coupleth the centres of the next bases. The angle of the incli∣nation [ 5] of the bases of the Octohedron, doth with the angle of the inclination of [ 6] the bases of the pyramis, make angles equal to two right angles.

Of the nature of a Cube.

The diameter of a cube, is in power sesquialter to the diameter of his base: [ 1] and is in power triple to his side: and vnto the line which coupleth the centres of [ 2] the next bases, it is in power sextuple. Moreouer the side of the cube is to the side [ 3] of the Icosahedron inscribed in it, as the whole is to the greater segment: vnto [ 4] the side of the Dodecahedron, it is as the whole is to the lesse segment: vnto the [ 5] side of the Octohedron, it is in power duple: and vnto the side of the pyramis, it is [ 6] in power subduple. Moreouer the cube is triple to the pyramis: but to the cube [ 7] the Dodecahedron is in a maner duple. Wherfore the same Dodecahedron is in a [ 8] maner sextuple to the sayd pyramis.

Of the nature of an Icosahedron.

Fiue triangles of an Icosahedron, do make a solide angle, the bases of which [ 1] triangles make a pentagon. If therfore from the opposite bases of the Icosahedron

be taken the other pentagon by them described, these pentagons shall in such sort cut the diameter of the Icosahedron which coupleth the forsaid opposite angles, that that part which is contained betweene the plaines of those two pentagons, shalbe the greater segment: and the residue which is drawne from the plaine to [ 2] the angle shall be the lesse segment. If the opposite angles of two bases ioyned to∣gether, be coupled by a right line, the greater segment of that right line is the side [ 3] of the Icosahedron. A line drawne from the centre of the Icosahedron to the an∣gles, is in power quintuple, to halfe that line which is takē betwene the pentagōs, or of the halfe of that line which is drawne from the centre of the circle which cō∣tayneth [ 4] the foresaid pentagon: which two lines are therefore equall. The side of the Icosahedron contayneth in power either of them, and also the lesse segment, [ 5] namely, the line which falleth from the solide angle to the pentagon. The diame∣ter of the Icosahedron contayneth in power the whole line, which coupleth the opposite angles of the bases ioyned together, and the greater segment thereof, [ 6] namely, the side of the Icosahedron. The diameter also is in power quintuple to the line which was taken betwene the pentagons, or to the line which is drawne from the centre to the circumference of the circle which containeth the pentagon [ 7] cōposed of the sides of the Icosahedron. The dimetient contayneth in power the right line which coupleth the centres of the opposite bases of the Icosahedron, [ 8] and the diameter of the circle which contayneth the base. Moreouer the sayd di∣metient contayneth in power the diameter of the circle, which contayneth the pentagon, and also the line which is drawne from the centre of the same circle to the circumference: That is, it is quintuple to the line drawne from the centre to [ 9] the circumference. The line which coupleth the centres of the opposite bases, con¦tayneth in power the line which coupleth the centres of the next bases, and also the rest of that line of which the side of the cube inscribed in the Icosahedron is [ 10] the greater segment. The line which coupleth the middle sections of the opposite sides, is triple to the side of the dodecahedron inscribed in it. Wherefore if the [ 11] side of the Icosahedron, and the greater segment thereof be made one line, the third part of the whole, is the side of the dodecahedron inscribed in the Icosahe∣dron.

Of the nature of a Dodecahedron.

[ 1] The diameter of a dodecahedron contayneth in power the side of the dode∣cahedron, and also that right line, vnto which the side of the dodecahedron is the lesse segment, and the side of the cube inscribed in it is the greater segmēt: which line is that which subtendeth the angle of the inclination of the bases, contayned [ 2] vnder two perpendiculars of the bases of the dodecahedron. If there be taken two bases of the dodecahedron distant the one from the other by the length of one of the sides, a right line coupling their centres, being diuided by an extreame and meane proportion, maketh the greater segment the right line which coupleth the [ 3] centres of the next bases. If by the centres of fiue bases set vppon one base, be drawne a playne superficies, and by the centres of the bases which are set vpon the opposite base be drawne also a playne superficies, and then be drawne a right line coupling the centres of the opposite bases, that right line is so cut, that eche of his partes set without the playne superficies, is the greater segment of that part which [ 4] is contayned betwene the playnes. The side of the dodecahedron is the greater

segment of the line which subtendeth the angle of the pentagon. A perpendicular [ 5] line drawne from the centre of the dodecahedron to one of the bases, is in power quintuple to half the line which is betwene the playnes: And therfore the whole [ 6] line which coupleth the centres of the opposite bases, is in power quintuple to the whole line which is betwene the sayd playnes. The line which subt••deth the [ 7] angle of the base of the dodecahedrō, together with the side of the base, are in power quintuple to the line which is drawne from the cētre of the circle, which contayneth the base, to the circumference. A secti∣on [ 8] of a sphere contayning three bases of the dodecahedron taketh a third part of the diameter of the sayd sphere. The side of the dodecahedron, and the line which [ 9] subtendeth the angle of the pentagon, are e∣quall to the right line which coupleth the middle sections of the opposite sides of the dodecahedron.

Notes

• * 1.1

  Construction of the Icosi∣dodecah••dron.
• * 1.2

  That the Icosi∣dodecahedron is contayned in the sphere geuen.
• * 1.3

  That the dia∣meter being deuided by an extreme and meane propor∣tion. &c.
• * 1.4

  That is as 8. 103••