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

An other demonstration to proue that as the superficies of the Dodecahe∣dron is to the superficies of the Icosahedron, so is the side of the cube to the side of the Icosahedron. † 1.1

LEt there be a circle ABC. And in it describe two sides of an equilater pen∣tagon (by the 11. of the fift) namely, AB and AC: and draw a right line from the point B to the point C. And (by the 1. of the third) take the centre of the circle, and let the same be D. And draw a right line from the point A to the point D, and extend it directly to the point E, and let it cut the line BC in the point G. And let the line DF be halfe to the line DA, and let

the line GC be treble to the line HC, by the 9. of the sixt.* 1.2 Now I say, that that which is contained vnder the lines AF and BH, is equall to the pentagon inscribed in the circle ABC. Draw a right line from the point B to the point D. Now forasmuch as the line AD is double to the line DF, there∣fore the line AF is sesquialter to the line AD.

[illustration]

Againe,* 1.3 forasmuch as the line GC is treble to the line CH, therefore the line GH is double to the line CH. Wherefore the line GC is sesquialter to the line HG. Wherefore as the line FA is to the line AD, so is the line GC to the line GH. Wherefore (by the 16. of the sixt) that which is contained vnder the lines AF & HG, is equall to that which is contained vnder the lines DA and GC. But the line GC is equall to the line BG (by the 3. of the third). Wherfore that which is contained vnder the lines AD and BG, is e∣quall to that which is contained vnder the lines AF and GH. But that which is contained vn∣der the lines AD and BG, is equall to two such triangles as the triangle ABD is (by the 41. of the first). Wherefore that which is contained vnder the lines AF and GH, is equall to two such triangles as the triangle ABD is. Wherefore that which is contained vnder the lines AF and GH ••iue times, is equall to ten triangles. But ten triangles are two pentagons. Wherefore that which is contained vnder the lines AF and GH fiue times, is equall to two pentagons. And forasmuch as the line GH is double to the line HC, therefore that which is contained vnder the lines AF and GH, is double to that which is contained vnder the lines AF and HC (by the 1. of the sixt). Wherefore that which is contained vnder the lines AF and CH twise, is equall to that which is contained vnder the lines AF and GH once. Take eche of those parallelogrammes fiue times. Wherefore that which is contained vnder the lines AF and HC ten times, is equall to that which is contained vnder the lines AF & GH fiue times, that is, to two pentagons. Wherefore that which is contained vnder the lines AF and HC fiue times, is equall to one pentagon. But that which is contained vnder the lines AF and HC fiue times, is equall (by the 1. of the sixt) to that which is contained vnder the lines AF and HB, for the line HB is quintuple to the line HC (as it is easie to see by the con∣struction) and they are both vnder one & the selfe same altitude, namely, vnder AF. Wher∣fore that which is contained vnder the lines AF and BH, is equall to one pentagon.

Notes

• † 1.1

  But first the Assumpt follow∣ing, the con∣struction wh••re••f here beginne••h, is to be proued.
• * 1.2

  The Assumpt, which also Flus∣sas putteth as an Assumpt a••ter the 6. propositiō.
• * 1.3

  Demonstration of the Assumpt.