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 Assumpt.

And now that in like and equall figures, being in like sort situate, the sides of like proportion are also equall (which thing was before in this proposition taken as graunted) may thus be proued.* 1.1 Suppose y^t the rectiline figures NH and SR be equall and like, and as HG is to GN, so let RQ be to QS, and let GH and QR be sides of like proportion. Then I say that the side RQ is equall vnto the side GH. For if they be vnequall, the one of them is greater then the other, let the side RQ be greater then the side HG. And for that as the line RQ is to the line QS, so is the line HG to the line GN, and alternately also (by the 16. of the fifth) as the line RQ is to the line HG, so is the line QS, to the lyne GN, but the line RQ is greater then the line HG. Wherfore also the line QS is grea¦ter then y^e line GN. Wherefore also y^e figure RS is greater then the figure HN but (by supposition) it is equall vnto it, which is impossible. Wherfore y^e line QR is not greater then y^e line GH. In like sorte also may we proue that it is not lesse then it, wherfore it is equall vnto it: which was required to be proued.

Flussates demonstrateth this second part more briefly,* 1.2 by the first corollary of the ••0. of this boke, thus. Forasmuch as the rectiline figures are by supposition in one and the same proportion, and the same pro∣portion is double to the proportion of the sides AB to CD, and EF to GH (by the foresaid corollary) the proportion also of the sides shall be one and the selfe same (by the 7. common sentence) namely, the line AB shall be vnto the line CD as the line EF is to the line GH.

Notes

• * 1.1

  An Assumpt.
• * 1.2

  An other demō∣stration of the second part af∣ter Flussates.