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. Proposition. One and the selfe same circle containeth both the Pentagon of a Dodeca∣hedron,* 1.1 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,* 1.2 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.* 1.3 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.

Notes

• * 1.1

  The 5. propo∣sition after Campane.
• * 1.2

  Construction.
• * 1.3

  Demonstra∣tion.