The demonstration of this is very like unto the above mentioned, of five parts. And thus much of the secants, the Tangents doe follow.
19 If a right line be perpendicular unto the end of the diameter, it doth touch the periphery: And contrariwise è 16 p iij.
As for example, Let the circle given ae, be perpendicular to the end of the diameter, or the end of the ray, in the end a, as suppose the ray be ia: I say, that ea, doth touch, not cut the periphery in the common bound a.
This was to have beene made a postulatum out of the definition of a perpendicle: Because if this should leane never so little, it should cut the periphery, and should not be perpendicular: Notwithstanding Euclide doth force it thus: Otherwise let the right line ae, be perpendicular to the diameter ai. And a right line from o, with the center i, let it fall within the circle at o, and let oi, joyned together. Here in the triangle aoi, two angles, contrary to the [13 e vj], should be right angles at a, by the grant: And at o, by the [17 e vj].
The demonstration of the converse is like unto the former. For if the tangent, or touch-line ae, be not perpendicular to the diameter iou, let oe, from the center o, be drawne perpendicular; Then shall the angle oei, be right angle: And oie an acutangle: And therefore by the [22 e vj], oi, that is oy, shall be greater then oye, that is the part, then the whole.
Therefore
20 If a right line doe passe by the center and touch-point, it is perpendicular to the tangent or touch-line. 18 p iij.