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 8. Theoreme. The 15. Proposition. If two right lines cut the one the other: the hed angles shal be equal the one to the other.

SVppose that these two right lines AB and CD, do cut the one the o∣ther in the point E. Then I say, that the angle AEC, is equall to the angle DEB.* 1.1 For forasmuch as the right line AE, standeth vpon the right line DC, making these angles CEA, and AED: therefore (by the 13. propositiō) the angles CEA, and AED, are equall to two right angles. Agayne forasmuch as the right line DE, standeth

[illustration]

vpon the right line AB, making these angles AED, and DEB: therfore (by the same propositi∣on) the angles AED, and DEB, are equall to two right angles: and it is proued, that the angles CEA, and AED, are also equall to two right an∣gles. VVherfore the angles CEA, and AED, are equall to the angles AED, and DEB. Take a∣way the angle AED, which is common to them both. VVherefore the angle remayning CEA, is equall to the angle remayning DEB. And in like sort may it be proued, that the angles CEB, and DEA, are equall the one to the other. If therefore two right lines cut the one the other, the hed angles shalbe equall the one to the o∣ther: which was required to be demonstrated.

* 1.2Thales Milesius the Philosopher was the first inuenter of this Proposition, as witnesseth Eudemius, but yet it was first demonstrated by Euclide. And in it there is no construction at all.* 1.3 For the exposition of the thing geuē, is sufficient inough for the demonstration.

Hed Angles,* 1.4 are apposite angles, caused of the intersection of two right lines: and are so called, because the heddes of the two angles are ioyned together in one pointe.

The conuerse of this proposition after Pelitarius.

* 1.5If fower right lines being drawen from one point, do make fower angles, of which the two oppo∣site angles are equall: the two opposite lines shalbe drawen directly, and make one right line.

Suppose that there be fower right lines AB, AC, AD, and AE, drawen from the poynt A, making fower angles at the point A: of which let the angle BAC be equall to the angle DAE, and the angle BAD to the angle CAE. Then I say, that BE and CD are onely two right lines: that is, the two right lines BA and AE are drawen directly,

and doo make one right line, and likewise the two right

[illustration]

lines CA and AD are drawen directly, and do make one right line. For otherwise if it be possible,* 1.6 let EF be one right line, and likewise let CG be one right line. And for∣asmuch as the right line EA standeth vpon the right line CG, therefore the two angles EAC and EAG, are (by the 13 proposition) equall to two right angles. And for∣asmuch as the right line B A standeth vpon the right line EF: therefore (by the selfe same) the two angles EAG and FAG are also equal to two right angles. Wherefore taking away the angle EAG, which is common to them both, the angle EAC, shall (by the thirde common sen∣tence) be equall to the angle FAG: but the angle EAC is supposed to be equall to the angle BAD. Wherefore the angle BAD is equall to the angle FAG, namely a part to the whole: which is impossible. And the sel••e same absurditie will follow, on what side soeuer the lines be drawen. Wherefore BE is one line, and CD also is oue line: which was required to be proued.

The same conuerse after Proclus.

If vnto a right line, and to a point thereof be drawen two right lines, not on one and the same side, in such sort that they make the angles at the toppe equall:* 1.7 those right lines shalbe drawen directly one to the other, and shal make one right line.

Suppose that there be a right line AB, and take a point in in C. And vnto the point in it C, draw these two right lines CD and CE vnto contrary sides, making the angles at the hed equal, namely, the angles ACD and BCE. Then I say, that the lines CD and CE are drawen directly, and do make one right line. For forasmuch as the right line CD standing vpō the right line AB, doth make the angles DCA and DCB equall to two right angles (by the 13 proposition:) and the angle DCA

[illustration]

is equall to the angle BCE: therefore the angles DCB and BCE are equal to two right angles. And forasmuch as vnto a certayne right line BC, and to a point thereof C, are drawen two right lines not both on one and the same side, making the side angles equall to two right angles, therefore (by the 14 proposition) the lines CD and CE are drawen directly, & do make one right line, which was required to be proued.

The same may also be demonstrated by an argument lea∣ding

[illustration]

to an absurditie.* 1.8 For if CE be not drawen directly to CD, so that they both make one right line, then (if it bee possible) let CF bee drawne directly vnto it. So that let DCF be one right line. And forasmuch as the tw•• right lines AB and DF do cutte the one the other, they make the hed angles equall (by the 15. proposition) Wher••fore the an∣gles ACD and BCF are equall: but (by supposition) the angles ACD and BCE are also equall. Wherefore (by the first common sentence) the angle BCE is equall to the angle BCF: namely, the greater to the lesse: which is im∣possible. Wherefore no other right line besides CE is dra∣wen directly to CD. Wherefore the lines, CD and CE are 〈…〉〈…〉, and make one right line: which was requi∣red to be proued.

* 1.9Of this fiuetenth Proposition followeth a Corrollary. VVhere note that a Corollary is a Proposition, whose demonstration dependeth of the demonstration of an other Proposition, and it appeareth sodenly, as it were by chance offering it selfe vnto vs: and therefore is reckoned as lucre or gayne. The Corollary which followeth of this proposition, is thus.

* 1.10If fower right lines cut the one the other: they make fower angles equall to fower right angles.

This Corollary gaue great occasion to finde out that wonderful proposition in∣uented of Pithagoras, which is thus.

* 1.11Only three kindes of figures of many angles, namely, an equilater triangle, a right angled figure of fower sides, and a figure of sixe sides, hauing equall sides and equal angles, can fill the whole space about a point, their angles touching the same point.

* 1.12Euery angle of an equilater triangle contay∣neth

[illustration]

two third partes of a right angle: sixe tymes two thirdes of a right angle make fower right an∣gles. VVherefore sixe equilater triangles fill the whole space about a point which is equal to fower right angles, as in the 1. figure. Also euery angle of a rectangle quadrilater figure is a right angle: wher¦fore fower of them fill the whole space as in the 2. figure. Euery angle of a sixe angled figure is equal to a right angle,* 1.13 and moreouer to a third part of a right angle. But a right angle, and a third part of a right angle, takē thre times, make 4. right angles: wherefore three equilater sixe angled figures fill the whole space about a point: which space (by this Corrollary) is equall to fower right angles: as in the third figure. Any other figure of many sid••, howsoeuer you ioyne thē together at the angles, shal either want of fower angles, or exceede them. By this Corrollary also it is mani••est that if mo then two lines, that is, three, or fower, or how many soeuer do cut the one the other in one point, all the an∣gles by them made at the point shalbe equall to fower right angles. For they fill the place of fower right angles. And it is also many sest, that the angles by those right lines made are double in number to the right lines which cutte the one the other. So that if there b•• two lines which cut the one the other, thē are there made fower angles equall to fower right angles: but if thre, then are there made sixe angles: if fower, eight angles, and so infinitly. Foreuer the mul••itude, or number of of the angles is dubled to the multitude of the ••igh•• lines which cut the one the other. And as the angles increase in multitude, so

diminish they in magnitude. For that that which is deuided is alwayes one and the selfe same thing, namely, fower right angles.

Notes

• * 1.1

  Demonstration.
• * 1.2

  Thales Milesius the ••irst inuen∣t••r of this pro∣position.
• * 1.3

  No construction in this proposi∣tion.
• * 1.4

  What hed an∣gles are.
• * 1.5

  The conuerse of this propositio af¦ter Pelitarius.
• * 1.6

  Demonstration leading to an absurditie.
• * 1.7

  The same con∣uerse after Peli∣tarius, which is demonstrated directly.
• * 1.8

  The same con∣uerse after Pro∣clus demonstra∣ted indirectly.
• * 1.9

  What a Corrol∣lary is.
• * 1.10

  A Corollary following of this proposition.
• * 1.11

  A wonderfull proposition in∣uented by P••∣••hagoras.
• * 1.12

  Euery angle o•• an equilater tri¦angle i•• equal ••o ••wo ••hird par•••••• of a right angle
• * 1.13

  Euery angle of a si••e angled fi∣gu••e i•• equall to a right angle, and to a third part of a right angle.