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 29. Theoreme. The 39. Proposition. Equall triangles consisting vpon one and the same base, and on one and the same side: are also in the selfe same parallel lines.

SVppose that these two equall triangles ABC and DBC do consist vp∣pon one and the same base, namely, BC and on one and the same side. Thē I say that they are in the selfe same parallel lines. Drawe a right line from the point A to the point D. Now I say that AD is a parallel line to BC. For if not, then (by the 31. proposition) by the point A drawe vnto the right line BC a parallel line AE, and draw a right line from the point E to the

point C. VVherfore y^e triangle EBC is

[illustration]

(by y^e 37. propositiō) equal to the triangle ABC, for they consist vpon one and the selfe same base, namely, BC, and are in y^e selfe same parallels, that is, AE and BC. But the triangle DBC is (by supposition) equall to the triangle ABC•• VVherfore the triangle DBC is equal to the triangle EBC, the greater vnto the lesse: which is impossible. VVhere∣fore the line AE, is not a parallel to the line BC. And in like sorte may it be proued that no other line besides AD is a parallel line to BC, wherefore AD is a parallel line to BC. VVherfore equall triangles consisting vpon one and the same base, and on one and the same side, are also in the selfe same parallel lines: which was required to be proued.

This proposition is the conuerse of the 37. proposition.* 1.1 And here is to be noted that if by the point A, you draw vnto the line BC a parallel line, the same shal of necessitie either light vpō the point D, or vnder it, or aboue it. If it light vpō it, then is that manifest which is required: but if it light vnder it, then foloweth that absurditie which Euclide putteth, namely, that the greater triangle is equall to the lesse: which selfe same absurditie also will follow, if it fall aboue the point D. As for example.

Suppose that these equall triangles ABC and DBC do consist vppon one and the selfe same base BC, and on one and the same side. Then I

[illustration]

say, that they are in the selfe same parallel lines, and that a right line ioyning together their toppes is a parallel to the base BC. Draw a right line frō A to D. Now if this be not a parallel to the base BC, let AE be a parallel vnto it, and let AE fall without the line AD. And produce the line BD till it concurre with the line AE in the pointe E and draw a line from E to C. Wherfore the triangle ABC is equal to the triangle EBC: but the triangle ABC is equall to the triangle DBC: Wherfore the triangle EBC is equall to the triangle DBC. Namely, the whole to the part: which is impossible. Wherfore the parallel line fal∣leth not without the line AD. And Euclide hath proued that it falleth not within. Wherfore the line AD is a pa∣rallel vnto the line BC. Wherfore equall triangles which are on the selfe same side, and on one and the selfe same base, are also in the selfe same parallel lines: which was required to be proued.

An addition of Flussates.

The selfe same also followeth in parallelogrames.* 1.2 For if vpon the base AB be set on one & the same side these equal parallelogrames ABCD & ABGE, they shall of necessitie be in the selfe same parallel lines. For if not, but one of them is

set eyther within or without, let the parallelo∣grame

[illustration]

B•• being equall to the parallelograme ABCD be set within the same parallel lines: wherefore the same parallelograme BF beyng e∣quall to the parallelograme ABCD (by the 35. proposition) shall also be equall to the other pa∣rallelograme ABGE (by the first common sen∣tence) For the parallelograme ABGE is by sup∣position equall to the parallelogramme ABCD: whe••fore the parallelograme BF being equall to the parallelograme ABGE, the parte shall bee e∣qual to the whole, which is absurde. The same in∣conuenience also will followe, if it fall without. VVherefore it can neither fall within nor with∣out. VVherfore equall parallelogrames beyng vpon one and the selfe same base and on one and the same side, are also in the selfe same parallel lines.

An addition of Campanus.

* 1.3If a right line deuide two sides of a triangle into two equall partes: it shall be equidistant vnto the third side.

Suppose that there be a triangle ABC: and let there bee a right lyne DE, which let deuide the two sides AB and BC into two equall partes in the pointes D and E Then I say, that the line DE is a parallel to the line A

[illustration]

C. Drawe these two lines AE and DC. Now then imagining a line to be drawne by the point E parallel to the line AB, the triangle BDE shall (by the 38. proposition) bee equall to the triangle DAE (for their two bases AD and DB are put to be equall) And by the same reason the triangle BDE is equall to the triangle CED. VVherfore (by the first common sentēce) the triangles EAD and ECD are equall, and they are erected on one and the sel••e same base, namely, DE, and on one and the same side. VVherefore (by the 39. proposition) they are in the selfe same parallel lines, and the line which ioyneth together their toppes is a parallel to their base. VVherfore the lynes DE and AC are pa∣ralles: which was required to be proued.

Notes

• * 1.1

  This Theoreme the conuerse of the 37. proposi∣tion.
• * 1.2

  An addition of Fl••ssases.
• * 1.3

  An addition of Campanus.