responding angles at the centre, as may be shown by chang ing the order of the couplets in the preceding proportion.

Cor. 2. In equal circles, angles at the centre are proportional to their intercepted arcs; and the reverse, whether they are commensurable or incommensurable.

Cor 3. In equal circles, sectors are proportional to their angles, and also to their arcs.

Scholium. Since the intercepted arcs are proportional to the corresponding angles at the centre, the arcs may be taken as the measures of the angles. That is, if a circumference be described from the vertex of any angle, as a centre, and with a fixed radius, the arc intercepted between the sides of the angle may be taken as the measure of the angle. In Geometry, the right angle which is measured by a quarter of a circumference, or a quadrant, is taken as a unit. If, therefore, any angle be measured by one-half or two-thirds of a quadrant, it will be equal to one-half or two-thirds of a right angle.

An inscribed angle is measured by half of the arc included between its sides.

There may be three cases: the centre of the circle may

lie on one of the sides of the angle; it

may lie within the angle; or, it may lie without the angle.

1o. Let EAD be an inscribed angle, one of whose sides AE passes through the centre: then will it be measured by half of the arc DE

E

For, draw the radius CD. The external angle DCE, of the triangle DCA, is equal to the sum of the opposite interior angles CAD and CDA (B L. P. XXV., C. 6). But, the triangle DCA being isosceles,

the angles D and A are equal; therefore, the angle DCE is double the angle DAE Because DCE is at the centre, it is measured by the arc DE (P. XVII., S.): hence, the, angle DAE is measured by half of the arc DE; which was to be proved.

B

E

2o. Let DAB be an inscribed angle, and let the centre lie within it then will the angle be measured by half of the arc BED.

For, draw the diameter AE. Then, from what has just been proved, the angle DAE is measured by half of DE, and the angle EAB by half of EB: hence, BAD, which is the sum of EAB and DAE, is measured by half of the sum of DE and EB, or by half of BED; which was to be proved.

3o. Let BAD be an inscribed angle, and let the centre lie without it: then will it be measured by half of the arc arc BD.

For, draw the diameter AE. Then, from what precedes, the angle DAE is measured by half of DE, and the angle BAE by half of BE: hence, BAD, which is the difference of BAE and DAE, is measured by half of the

difference of BE and DE, or by

B

half of the arc BD; which was to be proved.

obtuse; for it is measured by half the arc BAC, greater than a semi-circumference.

Cor. 4. The opposite angles A and C, of an inscribed quadrilateral ABCD, are together equal to two right angles; for the angle DAB is measured by half the arc DCB, the angle DCB by half the arc

DAB hence, the two angles, taken together, are measured by half the circumference: hence, their sum is equal to two right angles.

Any angle formed by two chords, which intersect, is mea sured by half the sum of the included arcs.

Let DEB be an angle formed by the intersection of the chords AB and CD: then will it be measured by

half the sum of the arcs AC and DB.

by half of FDB; that is, by half the sum of FD and DB, or by half the sum of AC and DB; which was to be proved.

The angle formed by two secants, intersecting without the circumference, is measured by half the difference of the included arcs.

Let AB, AC, be two secants: then will the angle BAC be measured by half the difference of the arcs BC and DF

Draw DE parallel to AC: the arc EC will be equal to DF (P. X.), and the angle BDE equal to the angle BAC (B. I., P. XX., C. 3.). But BDE is measured by half the arc BE (P. XVIII.): hence, BAC is also measured by half the arc BE;

that is, by half the difference of BC

B

and EC, or by half the difference of BC and DF; which was to be proved.

PROPOSITION XXI. THEOREM.

An angle formed by a tangent and a chord meeting it at the point of contact, is measured by half the included

arc.

Let BE be tangent to the circle AMC, and let AC be a chord drawn from the point of contact A: then will the angle BAC be measured.

by half of the arc AMC.

D

AD.

For, draw the diameter The angle BAD is a right angle (P. IX.), and is measured by half the semi-circumference AMD (P. XVI., S.); the angle DAC is measured by half of the arc DC (P. XVIII.): hence, the angle BAC, which is equal to the sum of the angles is measured by half the sum of the arcs AMD or by half of the arc AMC; which was to be proved.

B

BAD and DAC, and DC,

The angle CAE, which is the difference of DAE and DAC is measured by half the difference of the arcs DCA and DC, or by half the arc CA.

6