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 13. Theoreme. The 19. Proposition.

Like triangles are one to the other in double proportion that the sides of lyke proportion are.

SVppose the triangles like to be ABC and DEF, hauing the angle B of the one triangle, equal vnto the angle E of the other triangle, & as AB is to BC, so let DE be to EF, so that let BC & EF be sides of like proportion. Then I say that the proportion of the triangle ABC vnto the triangle DEF is double to the proportion of the side BC to the side EF. Vnto the two lines BC and EF (by the 10. of the sixth) make a third lyne in proportion BG, so that as BC is to EF, so let EF be to BG, and draw a lyne from A to G. Now forasmuch as AB is to BC, as DE is to EF, therfore alter∣nately (by the 16. of the fifth) as AB is to DE, so is BC to EF.* 1.1 But as BC is to EF, so is EF to BG, wherfore also (by the

[illustration]

11. of the fifth) as AB is to DE, so is EF to BG. Wherfore the sides of the triangles ABG & DEF, which include y^e equal angles are reciprokally proportionall. But if in triangles hauing one angle of the one equall to one angle of y^e other, the sides which include y^e equall an∣gles, be reciprokal, the triangles also (by y^e 15. the sixth) shal be equall. Wherfore the triangle ABG is equall vnto y^e triangle DEF. And for y^t as y^• line BC is to y^e line EF, so is the line EF to y^e line BG: but if there be three lines in proportion, the first shall haue to the third double proportion that it hath to the second (by the 10. de∣finition of the fifth) therfore the line BC hath vnto the line BG double propor∣tion that it hath to the line EF. But as BC is to BG, so (by the 1. of the sixth) is the triangle ABC to the triangle ABG. Wherfore the tiangle ABC is vnto the triangle ABG in double proportion that the side BC is to the side EF. But the triangle ABG is equall to the triangle DEF. Wherfore also the triangle ABC is vnto the triangle DEF in double proportion that the side BC is to the side EF. Wherfore lyke triangles are one to the other in double proportion that the sides of like proportion are: which was required to be proued.

Corollary.

* 1.2Hereby it is manifest that if there be three right lines in pro∣portion, as the first is to the third, so is the triangle described vpon the first, vnto the triangle described vpon the second, so that the sayd triangles be like, and in lyke ••ort described, for it hath bene proued that as the lyne CB is to the line BG, so

is the triangle ABC to the triangle DEF: which was re∣quired to be demonstrated.

Notes

• * 1.1

  Demonstra∣tion.
• * 1.2

  A Corollary.