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 5. Proposition. The side of a Pyramis diuided by an extreme and meane proportion, ma∣keth the lesse segment in power double to the side of the Icosahedron in∣scribed in it.

SVppose that ABG be the base of a pyramis:* 1.1 and let the base of the Icosahedron inscri∣bed in it, be CDE, described of three right lines, which being drawen from the angles of the base ABG cut the opposite sides by an extreme and meane proportion, by the for∣mer Proposition: namely, of these three lines AM, BI, and GI. Then I say, that AI the lesse segment of the side A••, is in power duple to CE the side of the Icosahedron. For, forasmuch as by the former Proposition, it was proued that the triangle CDE is inscri∣bed in an equilater triangle,* 1.2 whose angles cut the sides of

[illustration]

ABG the base of the pyramis by an extreme and meane proportion, let that triangle be FHK, cutting the line AB in the point F. Wherefore the lesse segment FA is equall to the segment AI, by the 2. of the fouretenth: (for the lines AB and AG are cut like). Moreouer the side FH of the triangle FHK is in the point D cut into two equall partes, as in the former Proposition it was proued, and FC∣ED also by the same is a parallelogramme: Wherefore the lines CE and FD are equall, by the 33, of the first. And for∣asmuch as the line FH subtendeth the angle BAG of an e∣quilater triangle, which angle is contained vnder the grea∣ter segment AH and the lesse segment AF•• therefore the line FH is in power double to the line AF or to the line AI the lesse segment, by the Corollary of the 16. of the fiue∣tenth. But the same line FH is in power quadruple to the line CE, by the 4. of the second: (for the line FH is double to the line CE). Wherefore the line AI being the halfe of the square of the line FH is in power duple to the line CE, to which the line FH was in power quadruple. Wherefore the side AG of the pyramis being diuided by an extreme and meane proportion, maketh th•• lesse segment AI in power duple to the side CE of the Icosahedron inscribed in it.

¶ A Corollary.

The side of an Icosahedron inscribed in a pyramis, is a residuall line. For the diameter of the Sphere which containeth the fiue regular bodies, being rationall, is in power ses∣quialtera to the side of the pyramis, by the 13. of the thirtenth: and therefore the side of the pyramis is rationall, by the definition: which side being diuided by an extreme and meane proportion, maketh

the lesse segment a residuall line, by the 6. of the thirtenth. Wherefore the side of the Icosahedron be∣ing commensurable to the same lesse segment (for the square of the side of the Icosahedron is the halfe of the square of the said lesse segment) is a residuall line, by that which was added after the 103. of the tenth booke.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.