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 third Proposition. If 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, 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. 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, 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: 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.

Notes

• The 4. propo∣sition after Campane.
• Constru••t••on.
• Demonstra∣tion.
• This Corolla∣ry Campane also ••utteth after the 4. proposition in his order.