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

* 1.125 An Icosahedron is a solide or bodily figure contained vnder twentie equall and equilater triangles.

As the solides before last mentioned are all

[illustration]

described by the number and forme of the su∣perficieces which containe them: so this body likewise is de••ined by that that it is contayned of twentie triangles equall, equilater, and e∣quiangle. And although this solide also be ve∣ry hard to conceaue, as it is commonly descri∣bed vpon a plaine (an example wherof you haue in the first figure here set): yet is it of necessitie that in that forme it be described, if we will vnderstand such descriptions as are set forth of Euclide touching that body in the fiue bookes following. Howbeit you may by it (although somewhat rudely) see the 20. triangles, which are imagined to be equall, equilater, and equian∣gle, if you consider ••iu: angles of fiue triangles to concurre together at a point. And forasmuch as there are in this solide 20. triangles, and euery t••iangle hath three angles, the concurse of the said triangles will be in twelue pointes. As in this example the pointes of the concurse are A, B, C, D, E, F, G, H, K, L, M, & N. Where note that in this plaine the two poyntes M and N are but one point, yet must ye imagine one of those pointes to be erected vpward, and the other down∣ward. Now the ••iue triangles which concurre in the point M, are these, BMD, DMF, FMH, HML, and LMB: the fiue triangles which concurre in the point N, and are imagined to be erected downward, are these, ANC, CNE, ENG, GNK, and KNA: the other ten triangles which include this body, are these, ABC, BCD, CDE, DEF, EFG, FGH, GHK, HKL, KLA, LAB. The second figure here appeareth more bodilike vnto the eye.

These ••iue solides now last defined, namely, a Cube, a Tetrahedrō, an Octohedron, a Dodecahedron and an Icosahedrō are called regular bodies.* 1.2 As in plaine superficieces, those are called regular figures, whose sides and angles are equal, as are equilater triangles, equilater pentagons, hexagons, & such lyke, so in solides such only are counted and called regular, which are cōprehēded vnder equal playne super∣ficieces, which haue equal sides and equal angles, as all these fiue foresayd haue, as manifestly appeareth by their definitions, which were all geuen by this proprietie of equalitie of their superficieces, which haue also their sides and angles equall. And in all the course of nature there are no other bodies of this condition and perfection, but onely these fiue. Wherfore they haue euer of the auncient Philosophers bene had in great estimation and admiration, and haue bene thought worthy of much contemplacion, about which they haue bestowed most diligent study and endeuour to searche out the natures & pro∣perties of them. They are as it were the ende and perfection of all Geometry, for whose sake is written whatsoeuer is written in Geometry. They were (as men say) first inuented by the most witty Pithago∣ras then afterward set forth by the diuine Plato, and last of all meruelously taught and declared by the most excellent Philosopher Euclide in these bookes following, and euer since wonderfully embraced of all learned Philosophers.* 1.3 The knowledge of them containeth infinite secretes of nature. Pithag••ras, Ti∣meus and Plato, by them searched out the cōposition of the world, with the harmony and preseruation therof, and applied these ••iue solides to the simple partes therof, the Pyramis, or Tetrahedrō they ascri∣bed to the ••ire,* 1.4 for that it ascendeth vpward according to the figure of the Pyramis. To the ayre they ascribed the Octohedron,* 1.5 for that through the subtle moisture which it hath, it extendeth it selfe euery way to the one side, and to the other, accordyng as that figure doth. Vnto the water they assigned the

Ikosahedron, for that it is continually flowing and mouing,* 1.6 and as it were makyng angle•• 〈…〉〈…〉 ••ide according to that figure. And to the earth they attributed a Cube,* 1.7 as to a thing stable•• 〈◊〉〈◊〉 and sure as the figure ••ignifieth. Last of all a Dodecahedron,* 1.8 for that it is made of P••ntago••, whose angles are more ample and large then the angles of the other bodies, and by that ••ea•••••• draw more •••• roun••nes, 〈◊〉〈◊〉 & to the forme and nature of a sphere, they assigned to a sphere, namely, 〈…〉〈…〉. Who so will 〈…〉〈…〉 in his Tineus, shall ••ead of these figures, and of their mutuall proportion•• ••••raunge ma••ter••, which h••re are not to be entreated of, this which is sayd, shall be sufficient for the 〈◊〉〈◊〉 of them and for th•• declaration of their diffinitions.

After all these diffinitions here set of Euclide, Flussas hath added an other diffinition, which 〈◊〉〈◊〉 of a Parallelipipedon, which bicause it hath not hitherto of Euclide in any place bene defined, and because it is very good and necessary to be had, I thought good not to omitte it, thus it is.

Notes

• * 1.1

  Twenty fiue diffinition.
• * 1.2

  Fiue regular bodies.
• * 1.3

  The dignity of these bodies.
• * 1.4

  A Tetrahe∣dron ascribed vnto the fire.
• * 1.5

  An octohe∣dron ascribed vnto the ayre.
• * 1.6

  An Ikosahedron assigned vnto the water.
• * 1.7

  A cube assigned vnto the earth.
• * 1.8

  A dodecahe∣dron assigned to heauen.