3 If two right lines given making an angle, and knit together with a base, be continued, the first equally to the second, the second infinitly; a parallell to the base from the end of the first continuation unto the second, shall intercept the third proportionall. 11. p vj.
The Diagramme here also, and demonstration is in all
respects the same with our [30 e], or 2 c of Ramus.
Thus farre Ramus: And here by the judgement of the learned Finkius, two elements of Ptolomey are to be adjoyned.
32 If two right lines cutting one another, be againe cut with many parallels, the parallels are proportionall unto their next segments.
It is a consectary out of the [28 e]. For let the right lines ae. and ai, cut one another at a, and let two parallell lines uo, and ei, cut them; I say, as au, is to uo, so ae, is to ei. For from the end i, let is, be erected parallell to ae, and let uo, be drawne out untill it doe meete with it. Then from the end s, let sy, be made parallell to ai: and lastly, let ea, be drawne out, untill it doe meete with it. Here now ay, shall be equall to the right line is, that is, by the [26. e], to ue: and at length, by the [28. e], as ua, is to uo; so is ay, that is, ue, to os. Therefore, by composition or addition of proportions, as ua, is unto uo, so ua, and ue, shall be unto uo, and os, that is, ei, by the [27. e].
The same demonstration shall serve, if the lines do crosse one another, or doe vertically cut one another, as in the same diagramme appeareth. For if the assigned ai, and us, doe cut one another vertically in o, let them be cut with the parallels au, and si: the precedent fabricke or figure being made up, it shall be by [28. e.] as au, is unto ao, the segment next unto it: so ay, that is, is, shall be unto oi, his next segment.
The [28. e] teacheth how to finde out the third and fourth proportionall: This affordeth us a meanes how to find out