The Surveyor in Foure Bookes

and D-A-B in the greater Segment are angles of the same Segment, for the same reason. And all angles of this kind are called mixed angles, because they are contained under a right line and a curved one. Of which two Segments, the lesser has always the lesser angle; and the greater, the greater angle.

A geometric diagram of a circle with a horizontal chord connecting points A on the left and B on the right. Point C is at the top of the circle and point D is at the bottom.

DEFIN. XVII.

An angle in a Section, or Segment, is when two right lines are drawn from any point in the arch line, to the ends or extremes of the chord line; the angle in that point of the arch line is called an angle in a Section or Segment.

Euclid, Book 3, Def. 7.

A geometric diagram of a circle with a horizontal chord AC. Point B is on the upper arc, and point D is on the lower arc. Lines connect B to A and C, and D to A and C, forming two inscribed triangles sharing the same base chord.
As the angle A-B-C in the lesser Segment is an angle in a Section, or Segment, by reason that the two right lines B-A and B-C are drawn from the point B in the arch line to the ends or extremes of the chord line A-C. And also the angle A-D-C in the greater Segment is an angle in a Section, or Segment, because the two right lines D-A and D-C are drawn from the point D in the arch line to the ends or extremes of the chord line A-C. And here note, the greater Section has in it the lesser angle, and the lesser Section the greater angle, contrary to the mixed angles in the precedent DEFINITION mentioned. And here also is to be noted, by the declaration of this and the former DEFINITION, the difference between an angle of a Segment, and an angle in a Segment; the first being called a mixed angle, and this a right-lined angle.

DEFIN. XVIII.

If two right lines be drawn from any one point in the circumference of a Circle, and receive any part of the same circumference, the angle contained under those two lines is said to belong and to be correspondent to that part of the circumference so received.

Euclid, Book 3, Def. 8.

As the angle B-A-C contained under the right lines A-B and A-C, drawn from the point A and receiving the circumference B-D-C, by this DEFINITION is said to belong, subtend, and pertain unto the circumference B-D-C. And if right lines be drawn from the centre