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 after Pelitarius.

Suppose that the triangle geuen be ABC.* 1.1 Whose side AB let be produced vnto

the point D. Then I say, that the angle DBC is greater then either of the angles BAC and ACB. For forasmuch as the two lines AC and BC do concurre in the point C, and vpon them falleth the line AB: therefore (by the conuerse of the first peticion) the two inward angles on one and the selfe same side, are lesse

[illustration]

then two right angles. Wherefore the angles ABC and CAB are lesse then two right angles: but the angles ABC and DBC are (by the 13 proposition) equal to two right angles. Wherefore the two angles ABC and DBC are greater then the two angles ABC and BAC. Wherefore taking away the angle ABC, which is com∣mon to them both, there shall be left the angle DBC greater then the angle BAC. And by the same reason, forasmuch as the two lines BA and CA concurre in the point A, and vppon them falleth the right line CB, the two inward angles ABC and ACB are lesse then two right angles. But the angles ABC and DBC are equall to two right angles. Wherfore the two angles ABC and DBC, are greater then the two angles ABC & ACB. Wher∣fore taking away the angle ABC, which is common to them both, there shal remaine the angle DBC greater then the angle ACB: which was required to be proued.

Here is to be noted, that when the side of a triangle is drawen forth, the angle of the triangle which is next the outwa••d angle, is called an angle in order vnto it: and the other two angles of the triangle are called opposite angles vnto it.

* 1.2Of this Proposition followeth this Corrollary, that it is not possible that from one & the selfe same point should be drawen to one and the selfe same right line, three equall right lines. For from one point, namely, A, if it be

[illustration]

possible, let there be drawen vnto the right line BD, these three equall right lines AB, AC, & AD. And forasmuch as AB is equall to AC, the angles at the base are (by the fifth proposition) equall. Wherfore the angle ABC is equal to the angle ACB. Agayne forasmuch as AB is equall to AD, the angle ABD is (by the same) equall to the angle ADB: but the angle ABC was equall to the angle ACB. Where∣fore the angle ACB is equall to the angle ADB: namely, the outward angle to the inwarde & oppo∣site angle: which is impossible. Wherfore from one and the selfe same point, can not be drawn to one & the selfe same right line three equall right lynes: which was required to be proued.

* 1.3By this Proposition also may this be demonstrated, that if a right line falling vpon two right lines, do make the outward angle equall to the inward and oppo∣site angle, those right lines shall not make a triangle, neither shal they concurre. For otherwis•• one & the selfe same angle should be both greater, and also equal: which is impossible. As for example.

Suppose that there be two right lines AB and CD, and vpon them let the right line BE fall, making the angles ABD and CDE equall. Then I say, that the right lines AB and CD shall not concurre. For if they concurre, the foresaide angles abidyng equall, namely, the angles CDE and ABD: Then forasmuch as the angle CDE is the out∣ward angle it is of necessitie greater then the inward and opposite angle, & it is also e∣qual vnto it: which is impossible. Wherfore if the said lines cōcurre, thē shal not the an∣gles remayne equall, but the angle at the point D shall be encreased. For whether AB

abiding fixed you suppose the line CD to be moued

[illustration]

vnto it, so that they concurre, the space and distance in the angle will be greater: for how much more CD approcheth to AB, so much farther of goeth it from DE. Or whether CD abiding fixed, you ima∣gine the line AB to be moued vnto i••, so that they concurre, the angle ABD will be lesse, for there∣with all it comm••th nere vnto the lines CD & BD. Or whether you imagine either of them to be mo∣ued the one to the other, you shall finde that the line AB comming neere to CD, maketh the angle ABE lesse, and CD going farther from DE by reason of his motion to the line BD•• maketh the angle CDE to increase. Wherefore it followeth of necessitie, that if it be a triangle, and that the right lines AB and CD do conc••rre, the outward angle also shall be greater•• then the inward and opposite angle. For either the inward and opposite angle abiding fixed, the outward i•• increased: or the outwarde a∣biding fixed, the inward and opposi•••• is diminished•• or els both of them being moued till they concurre, the inwarde is diminished, and the outwarde is more increased. And the caus•• hereof is the motion of the right lines the one tending to that parte where it diminisheth the inwarde angle, the other tending to that part where it increaseth th•• outward angle.

Notes

• * 1.1

  An other De∣monstration af∣ter Pelitarius.
• * 1.2

  A Corrollary following of this proposition.
• * 1.3

  An other Cor∣rollary ••ollo∣wing also of the same.