is indeed fetch'd from the angle in a semicircle, but by the equall or like angle of the tangent and end of the diameter.
As let the unequall sections be eio, and eao: the tangent let it be uey: And the angles in the opposite sections, eao, and eio. I say they are equall in the alterne angles of the secant and touch line oey, and oeu. First that which is at a, is equall to the alterne oey: Because also three angles oey, oea, and aeu, are equall to two right angles, by the [14 e v]. Unto which also are equall the three angles in the triangle aeo, by the [13 e vj]. From three equals take away the two right angles aue, and aoe: (For aoe, is a right angle, by the [21 e]; because it is in a semicircle:) Take away also the common angle aeo: And the remainders eao, and oey, alterne angles, shall be equall.
Secondarily, the angles at a, and i, are equall to two right angles, by the [14, e]: To these are equall both oey, and oeu. But eao, is equall to the alterne oey. Therefore that which is at i, is equall to, the other alterne oeu. Neither is it any matter, whether the angle at a, be at the diameter or not: For that is onely assumed for demonstrations sake: For wheresoever it is, it is equall, to wit, in the same section. And from hence is the making of a like section, by giving a right line to be subtended.
Therefore
28 If at the end of a right line given a right lined angle be made equall to an angle given, and from the
toppe of the angle now made, a perpendicular unto the other side do meete with a perpendicular drawn from the middest of the line given, the meeting shall be the center of the circle described by the equalled angle, in whose opposite section the angle upon the line given shall be made equall to the assigned è 33 p iij.
This you may make triall of in the three kindes of angles, all wayes by the same argument: as here the angle given is a: The right line given ei: at the end e, the equalled angle, ieo: The perpendicular to the side eo, let it be eu: But from the middest of the line given let it be yu. Here u, shall be the center desired. And from hence one may make a section upon a right line given, which shall receive a rectilineall angle equall to an angle assigned.
And