sumed that the difference of A and C is double of the difference of B and D. The former assumption is a particular case of V. 1, and the latter is a particular case of V. 5.
An important extension may be given to III. 20 by introducing angles greater than two right angles. For, in the first figure, suppose we draw the straight lines BF and CF. Then, the angle BEA is double of the angle BFA, and the angle CEA is double of the angle CFA; therefore the sum of the angles BEA and CEA is double of the angle BFC. The sum of the angles BEA and CEA is greater than two right angles; we will call the sum, the re-entrant angle BEC. Thus the re-entrant angle BEC is double of the angle BFC. (See note on I. 3-2). If this extension be used some of the demonstrations in the third book maybe abbreviated. Thus III. 21 maybe demonstrated without making two cases; III. 22 will follow immediately from the fact that the sum of the angles at the centre is equal to four right angles; and III. 31 will follow immediately from III. 20.
III. 21. In III. 21 Euclid himself has given only the first case; the second case has been added by Simson and others. In either of the figures of III. 2 1 if a point be taken on the same side of BD as A, the angle contained by the straight lines which join this point to the extremities of BD is greater or less than the angle BAD, according as the point is within or without the angle BAD; this follows from I. 21.
We shall have occasion to refer to IV. 5 in some of the remaining notes to the third Book; and the student is accordingly recommended to read that proposition at the present stage.
The following proposition is very important. If any number of triangles be constructed on the same base and on the same side of it, with equal vertical angles, the vertices will all lie on the circumference of a segment of a circle.
For take any one of these triangles, and describe a circle round it, by IV. 5; then the vertex of any other of the triangles must be on the circumference of the segment containing the assumed vertex, since, by the former part of this note, the vertex cannot be without the circle or within the circle.
III. 22. The converse of III. 22 is true and very important; namely, if two opposite angles of a quadrilateral be together equal to two right angles, a circle may be circumscribed about the quadrilaeral. For, let ABCD denote the quadrila-
