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 22. Theoreme. The 32. Proposition. If two triangles be set together at one angle, hauing two sides of the one proportionall to two sides of the other, so that their sides of like proportion be also parallels: then the other sides remayning of those triangles shall be in one right line.

SVppose the two triangles to be ABC, and DCE, and let two of their sides AC & DC make an angle ACD, and let the said triangles haue two sides of the one, namely, BA and AC proportionall to two sides of the other,* 1.1 namely, to DC and DE, so y^t as AB is to AC, so let DC be to DE. And let AB be a parallell vnto DC, and AC a parallell vnto DE. Then I say, that the lines BC and CE are in one right line. For forasmuch as the line AB is a parallell vnto the line DC, and

[illustration]

vpon thē lighteth a right line AC:

[illustration]

therefore (by the 29. of the first) the alternate angles BAC and ACD are equall the one to the other. And by the same reason the angle CDE is equall vnto y^e same angle ACD. Wherefore the angle BAC is equall vnto the angle CDE. And foras∣much as there are two triangles ABC and DCE, hauing the angle A of the one equall to the angle D of the other, and the sides about the equall angles are (by supposition) proportionall, that is, as the line BA is to the line AC, so is the line CD to the line DE, ther∣fore the triangle ABC is (by the 6. of the sixt) equiangle vnto the triangl•• DCE. Wherefore the angle ABC is equall vnto the angle DCE. And it is proued, that the angle ACD is equall vnto the angle BAC. Wherefore the whole angle ACE is equall vnto the two angles ABC and BAC. Put the an∣gle ACB common to them both. Wherefore the angles ACE and ACB are e∣quall vnto the angles CAB, ACB, & CBA. But the angles CAB, ACB, and CBA, are (by the 32. of the first) equall vnto two right angles. Wherefore also the angles ACE and ACB are equall to two right angles. Now then vn∣to the right line AC, and vnto the point in it C, are drawen two right lines BC and CE, not on one and the same side, making the side angles ACE & ACB equall to two right angles. Wherefore the lines BC and CE (by the 14. of the first) are set directly and do make one right line. If therefore two triangles be set together at one angle hauing two sides of the one proportionall to two sides of y^e other, so y^t their sides of like proportion be also parallels: then y^e sides remay∣ning of those triangles shall be in one right line: which was required to be proued.

Although Euclide doth not distinctly set forth the maner of proportion of like rectiline figures, as he did of lines in the 10. Propositiō of this Booke, and in the 3. following it, yet as Flussates noteth, is that not hard to be done by the 22. of thys Booke. ••or two like rectiline figures being geuen to finde out a third proportio∣nall•• also betwene two rectiline superficieces geuen to finde out a meane propor∣tionall (which we before taught to do by Pelitarius after the 24. Proposition of this booke): and moreouer three like rectiline figures being geuen to finde out a fourth proportionall like and in like sort described, and such kinde of proportions, are easie to be found out by the proportions of lines. As thus. If vnto two sides of like proportion we should find out a third proportionall by the 11. of this boke•• the rectiline figure described vpon that line shall be the third rectiline figure pro∣portionall with the two first figures geuen by the 22. of thys booke. And if be∣twene two sides of like proportion be taken a meane proportionall by the 13. of thys Booke: the rectiline ••igure described vpon the sayd meane shall likewise be a meane proportionall betwene the two rectiline figures geuē by the same 22. of the sixt. And so if vnto three sides ge••en be found out the fourth side proportionall (by the 12. of this booke) the rectiline ••igure described vpon the sayd fourth line shall be the fourth rectiline figure proportionall. For if the right lines be proporti∣onall, the rectiline figures described vpon them shall also be proportionall, so that the said rectiline ••igures be like & in like sort described by the said 22. of the sixt.

Notes

• * 1.1

  Demonstra∣tion.