Geometry, the One and Twentieth Book, Of Lines and Surfaces in Solids
1 A body or solid is a lineate broad and high 1 d xj.
For length onely is proper to a line: Length and breadth, to a surface: Length breath, and heighth joyntly, belong unto a body: This threefold perfection of a magnitude, is proper to a body: Whereby wee doe understand that are in a body, not onely lines of length, and surfaces of breadth, (for so a body should consist of lines and surfaces.) But we do conceive a solidity in length, breadth and heighth. For every part of a body is also a body. And therefore a solid we doe understand the body it selfe. As in the body aeio, the length is ae; the breadth, ai, And the heighth, ao.
2 The bound of a solid is a surface 2 d xj.
The bound of a line is a point: and yet neither is a point a line, or any part of a line. The bound of a surface is a line: And yet a line is not a surface, or any part of a surface. So now the bound of a body is a surface: And yet a surface is not a body, or any part of a body. A magnitude is one thing; a bound of a magnitude is another thing, as appeared at the 5 e j.
As they were called plaine lines, which are conceived to be in a plaine, so those are named solid both lines and surfaces which are considered in a solid; And their perpendicle and parallelisme are hither to be recalled from simple lines.
3 If a right line be unto right lines cut in a plaine underneath, perpendicular in the common intersection, it is perpendicular to the plaine beneath: And if it be perpendicular, it is unto right lines, cut in the same plaine, perpendicular in the common intersection è 3 d and 4 p xj.
Perpendicularity was in the former attributed to lines considered in a surface. Therefore from thence is repeated this consectary of the perpendicle of a line with the surface it selfe.
If thou shalt conceive the right lines, ae, io, uy, to cut one another in the plaine beneath, in the common intersections: And the line rs, falling from above, to be to every one of them perpendicular in the common point s, thou hast an example of this consectary.
4 If three right lines cutting one another, be unto the same right line perpendicular in the common section, they are in the same plaine 5. p xj.
For by the perpendicle and common section is understood an equall state on all parts, and therefore the same plaine: as in the former example, as, ys, os, suppose them to be to sr, the same loftie line, perpendicular, they shall be in the same nearer plaine aiueoy.
5 If two right lines be perpendicular to the under-plaine, they are parallells: And if the one two parallells be perpendicular to the under plaine, the other is also perpendicular to the same. 6. 8 p xj.
The cause is out of the first law or rule parallells. For if two right lines be perpendicular to the same under plaine, being joyned together by a right line, they shall make their inner corners equall to two right angles: And therefore they shall be parallells, by the 21. e v. And if in two parallells knit together with a right line, one of the inner angles, be a right angle: the other also shall be a right angle. Because they are divided by a common perpendicular; As in the example. If the angles at a, and e, be right angles, ai, and eo, are parallells, and contrariwise, if ai, and eo be parallells, and the angle at a, be a right angle, the angle at e, also shall be a right angle.
6 If right lines in diverse plaines be unto the same right line parallel, they are also parallell betweene themselves. 9 p xj.
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.
9. If a right line in one of the plaines cut, perpendicular to the common section, be perpendicular to the other, the plaines are perpendicular: And if the plaines be perpendicular, a right line in the one perpendicular to the common section is perpendicular to the other è 4 d, and 38 p xj.
The perpendicularity of plaines, is drawne out of the former condition of the perpendicle: And the state of plaines on each side equall betweene themselves, is fetch’d from a perpendicularity of a right line falling upon a plaine. Because from hence it is understood that the plaine it selfe doth lye indifferently betweene all parts signified by right lines: Which in a Booke with the pages each way opened, is perceived by the verses or lines of the pages, both to the section and plaine underneath, perpendicular as here thou seest.
10. If a right line be perpendicular to a plaine, all plaines by it, are perpendicular to the same: And if two plaines be unto any other plaine perpendiculars, the common section is perpendicular to the same. e 15, and 19 p. xj.
The first is a consectary drawne out of the 9 e. And the latter is from hence manifest, because that same common section is a right line, in any manner of lofty plaines intersected, perpendicular both to the common section and plaine underneath. For if the common section, were not perpendicular to the plaine underneath, neither should the plaines cutting one another be perpendicular to the plaine underneath, but some one should be oblique, against the grant, as here thou seest.
11. Plaines are parallell which doe leane no way. 8 d xj.
And
12. Those which divided by a common perpendicle. 14 p xj.
It is a consectary out of the 3, and 6 e. For if the middle right line be perpendicular to both the plaines, it is also to the right lines on either side cut, perpendicular in the common intersection: And the inner angles on each side, being right angles, will evince them to be parallels.
It is also out of the definition of parallels, at the 15 e ij.
And
13. If two paires of right in them be joyntly bounded, they are parallell. 15 p xj.
Such are the opposite walls in the toppe or ridge of houses. As let aei, and uoy, be plaine which have two payres of right lines, ea, and ia: Item uo, and yo, joyntly bounded in a, and o: And parallels, to wit ea, against uo: and ia, against yo. I say that the plaines themselves are parallels: For the right lines ue, and oa: item yi, and oa, doe knit together equall parallels, they shal by the 27 e v, be equall and parallels: And so they shall prove the equidistancie.
The same will fall out if thou shalt imagine the joyntly bounded to infinitely drawn out; for the plaines also infinitely extended shall be parallell.
14. If two parallell plaines are cut with another plaine, the common sections are parallels, 16 p xj.
As here thou seest the parallell plaines aeio, and uysr, cut by the plaine ljvf, the common sections lj, and fv, shall also be parallell: Otherwise they themselves, and therefore also the plaines in which they are, shall meete, as in the point t, which is against the grant.