Let the right line given bee ae. And let two equall peripheries from the ends a. and e. meete in i. and o. Then from those meetings let the right line io. be drawne. I say, That ae. is divided into two equall parts, by the said line thus
drawne. For by drawing the raies of the equall peripheries ia. and ie. the said io. doth cut the angle aie. into two equall parts, by the [11. e]. Therefore the angles aiu. and uie. being equall and equicrurall (seeing the shankes are the raies of equall peripheries, by the grant.) have equall bases au. and ue. by the [7. e iij]. Wherefore seeing the parts au. and ue. are equall, ae. the assigned right line is divided into two equall portions.
13. If a right line doe stand perpendicular upon another right line, it maketh on each side right angles: And contrary wise.
A right line standeth upon a right line, which cutteth, and is not cut againe. And the Angles on each side, are they which the falling line maketh with that underneath it, as is manifest out of Proclus, at the 15. pj. of Euclide; As here ae. the line cut: and io. the insisting line, let them be perpendicular; The angles on each side, to witt aio. and eio. shall bee right angles, by the [13. e iij].
The Rular, for the making of straight lines on a plaine, was the first Geometricall instrument: The Compasses, for the describing of a Circle, was the second: The Norma or Square for the true erecting of a right line in the same plaine upon another right line, and then of a surface and body, upon a surface or body, is the third. The figure therefore is thus.
Now Perpendiculū, an instrument with a line & a plummet of leade appendant upon it, used of Architects, Carpenters, and Masons, is meerely physicall: because heavie things
naturally by their weight are in straight lines carried perpendicularly downeward. This instrument is of two sorts: The first, which they call a Plumbe-rule, is for the trying of an erect perpendicular, as whether a columne, pillar, or any other kinde of building bee right, that is plumbe unto the plaine of the horizont & doth not leane or reele any way. The second is for the trying or examining of a plaine or floore, whether it doe lye parallell to the horizont or not. Therefore when the line from the right angle, doth fall upon the middle of the base; it shall shew that the length is equally poysed. The Latines call it Libra, or Libella, a ballance: of the Italians Livello, and vel Archipendolo, Achildulo: of the French, Nivelle, or Niueau: of us a Levill.
