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 way to do the same after Pelitarius, by parallel lines.

Suppose that the circle ge∣uen

[illustration]

be ABC,* 1.1 whose cētre let be the poynt F: and in it (by the former Proposition) in∣scribe an equilater and equi∣angle Pentagon ABCDE•• by whose fiue angles drawe from the centre beyond the circūference, fiue lynes, FG, FH, FK, FL, and FM. And it is m••nifest, that the fiue an∣gles at the cētre F, are equall, when as the fiue sides of the triangles within are equall, and also their bases. It is ma∣nifest also, that th•• fiue an∣gles of the Pentagon which are at the circumference, are deuided into ••••n equall ••n∣gles (by the 4. of the fir••••): Now then betwene the two lines FG and FH, draw the

line GH parallel to the side AB, and touching the circle ABC (which is done by a Pro∣position added by Pelitarius after the 17. of the third). And so likewyse draw these lines HK, KL, and LM, parallel to ech of these sides BC, CD, and DE, and touching the circle. And for asmuch as the lines FG and FH fall vpon the two parallel lines AB and GH,* 1.2 the two angles FGH, & FHG, are equall to the two angles FAB and FBA, the one to the other (by the 29. of the first). Wherefore (by the sixt of the same) the two lines FG and FH are equall. And by the same reason, the two angles FHK & FKH, are equall to the two angles FGH and FHG the one to the other: and the line FK is equall to the line FH, and therefore is equall to the line FG. And forasmuch as the angles at the poynt F are equall, therefore (by the 4. of the first) the base HK is equall to the base GH. In like sort may we proue, that the three lines FK, FL, and FM, are equall to the two lines FG and FH. And also that the two bases KL and LM, are e∣quall to the two bases GH and HK: and that the angles which they make with the lines FK, FL, and FM, are equall the one to the other. Now then draw the fift line MG: which shall be equall to

[illustration]

the foure former lines (by the 4. of the first) for that as we haue proued, the two lines FG & FM, are equall, & the angle GFM is equall to euery one of the angles at the poynt F. Thys line also MG toucheth the circle. For vnto the point where the line LM toucheth the circle whi∣che let be N, drawe the lyne FN. And it is manifest (by the 18. of the third) that either of the angle•• at the poynt N, is a right angle. Wher••fore for asmuch as the angle L of the triangle FLN, is equall to the angle M of the trian∣gle FMN, & the angle N of the one, is equall to the angl•• N of the other: and the lyne FN is cōmon to thē both, the line NL shall (by the 26. of the first) be equall to the line NM. And so is the line ML deuided equally in the poynt N. And forasmuch as the three sides of the triangle FGP are equall to the three sides of the triangle FMP, the angle P of the one shal be equall to the angle P of the other (by the 8. of the first). Wher∣fore either of thē is a right angle (by the 13. of the same). And forasmuch as the two an∣gles FMP and FPM of the triangle FMP, are equall to the two angles FMN & FNM of the triangle FMN, and the side FM is common to them both, therefore the line FP is equall to the line FN. But the line FN is drawen from the centre to the circumfe∣rence. Wherefore also the line FP is drawen from the centre to the circumference. And forasmuch as the line MG is perpendicular to the line FP, therefore (by the Corolla∣ry of the 16. of the third) it toucheth the circle. Wherfore the Pentagon GHKLM circumscribed about the circle is equilater: it is also equiangle, a•• it is easie to proue by the equalitie of the halfes: which was required to be done.

Notes

• * 1.1

  An other way to do the sam•• after Pelita∣rius.
• * 1.2

  Demonstra∣tion.