As here ae, and uy, right lines in diverse plaines suppose them to be parallell to io: I say, they are parallell one to another. For from the point i, let ia, and iu, be erected at right angles to io to cut the parallells, by the [17. e v]. Therefore, by the [3 e], oi, seeing that it is perpedicular to ia, and iu, two lines cutting one another, it is perpendicular to the plaine beneath. Therefore by the the [6 e], yu, and ea, are perpendicular to the same plaine: And therefore, by the same, they are parallell.
7 If two right lines be perpendiculars, the first from a point above, unto a right line underneath, the second
from the common section in the plaine underneath, a third, from the sayd point perpendicular to the second, shall be perpendicular to the plaine beneath. è 11 p xj.
It is a consectary out of the [3 e]. As for example, if from a lofty point a, ae, be by the [18 e v], perpendicular to e, a point of the right line io underneath: And from e the common section, by the [17 e v], there be eu, another perpendicular: Lastly ay, a lofty right line, be by the [18 e v], perpendicular unto eu, at the point y, ay shall be perpendicular unto the plaine underneath. For that ae is perpendicular to io, the same ae declineth neither to the right hand, nor to the left, by the [13 e ij]. And in that againe ay is perpendicular to eu, it leaneth neither forward nor backeward. Therefore it lyeth equally or indifferently, betweene the foure quarters of the world.
If the right line io, doe with equall angles agree to r, the third element.
8. If a right line from a point assigned of a plaine underneath, be parallell to a right line perpendicular to the same plaine, it shall also be perpendicular to the plaine underneath. ex 12 p xj.
As for example let the plaine be aeio: And the assigned point in it u: From this point a lofty perpendicular is to be erected. Let there be made from the point y, the perpendicular ys, unto the plaine underneath, by the [7 e]. And to it let ur, be made parallell by the [24 e v]. Now ur, seeing it is parallell to a perpendicular upon the plaine underneath, it shall be perpendicular to the same, by the [5 e].