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 21. Theoreme The 30. Proposition. Right lines which are parallels to one and the selfe same right line: are also parrallel lines the one to the other.

SVppose that these right lines

[illustration]

AB and CD, be parallel lines to the right line EF. Then I say, that the line AB is a parallel line to CD.* 1.1 Let there fall vpon these thre lines a right line GHK. And foras∣much as the right line GHK falleth vppon these parallell right lines AB and EF, therfore (by the proposition going before) the angle AGH is e∣quall to the angle GHF. Agayne for∣asmuch as the right line GK falleth vppon these parallell right lines EF and CD, therefore (by the same) the angle GHF is equall to the angle GKD. Now then it proued that the angle AGH is equall to y^e angle GHF, and y^t the angle GKD is equall to the angle GHF. VVherfore the angle AGK is equall to the angle GKD. And they are alternate angles: wherfore AB is a parallel line to CD. Right lines therfore which are parallels to one and the selfe same right line, are also parallel lines the one to the other: which was required to be proued.

Euclide in the demonstration of this proposition, setteth the two parallel lines which are compared to one, in the extremes, and the parallel to whome they are compared, he placeth in the middle, for the easier demonstration. It may also be proued euen by a principle onely. For if they shoulde concurre on any one side, they should concurre also with the middle line, and so should they not be parallels vnto it, which yet they are supposed to be.

* 1.2But if you will alter their position and placing, and set that line to which you will cōpare the other two lines, aboue, or beneath: you may vse the same demon¦stration which you had before. As for example.

Suppose that the lines AB and CD be

[illustration]

parallels to the line EF: and let both the lines AB and CD, be aboue, and let the line EF be beneath, and not in the middest. Vpon which let the right line GHK fall. And forasmuch as either of the angles KHD and HGB is equall to the angle HKE, (for they ar alternate angles) therfore they are (by the first common sentence) equall the one to the other. Wherefore (by the 28 proposition) the right lines AB and EF•• are paralles.

* 1.3But here if a man will obiect that the lines EK and KF, are parallels vnto the line CD,* 1.4 and therefore are parallels the one to the other. VVe will answere that the lines EK and KF are partes of one parallel line, and are not two parallel

lines.* 1.5 For parallel lines ar vnderstanded to be produced infinitly But EK being produced falleth vpon KF. Wherefor•• it is one and the selfe same with it, and not an other. Wherefore all the partes of a parrallel line are parallels, both to the right line vnto which the whole parallel line is a parallel, and also to al the parts of the same right line. As the line EK is a parallel vnto HD, and the line KE to the line CH. For if they be produced infinitly, they will neuer concurre.

Howbeit there are some which like not, that two distinct parellel lines, should be taken and counted for one parallel line: for that the continuall quan∣tity, namely, the line is cut asonder, and cesseth to be one. VVherefore they say, that there ought to be two distinct parallel lines compared to one. And therfore they adde to the proposition a correction, in this maner. Two lines being parallels to one line: are either parallels the one the other, or els the one is set directly againste the other, so that if they be produced they should make one right line. As for example.

Suppose that the lines CD and EF be parallels to one and the selfe same line AB,

[illustration]

and let them not be parallels the one to the other. Then I say, that the two lines CD & EF, are directly set the one to the other. For for as much as they are not parallel lines, let them concurre in the point G, And from the point G draw a line cutting the line AB in the point H. Now by the former proposition the angles AHG & HGC are equall to two right angles, but by the same propositiō, the angle AHG, is equall to the alternate angle HGE. Wherefore the angles HGC and HGF are equal to two right angles. Wherefore (by the 14 proposition) the lines CG and FG are drawen directly and make one right line. Wherfore al∣so the lines CD and EF are set directly the one to the other: and being produced they will make one right line.

Notes

• * 1.1

  Demonstration.
• * 1.2

  An other case in 〈◊〉〈◊〉 Problem••.
• * 1.3

  An obiection.
• * 1.4

  Answer.
• * 1.5

  Parallel lines are vnderstan∣ded to be produ∣ced infinitely.