PROPOSITION XV.-THEOREM.

151. If four magnitudes are proportionals, their like powers and roots will also be proportional.

Let A B C D; then will

:

Raising both members of this equation to the nth power, we have

and extracting the nth root of each member, we have

Hence, by Prop. II., the last two equations give

and

A" : B" :: C" : D",

A: B:: C: Dà.

153. The CIRCUMFERENCE or PERIPHERY of a circle is its entire bounding line; or it is a curved line, all points of which are equally distant from a point within called the centre.

154. A RADIUS of a circle is any straight line drawn from the centre to the circumference; as the line CA, CD, or CB.

155. A DIAMETER of a circle is any straight line drawn through the centre, and terminating in both directions in the circumference; as the line AB.

All the radii of a circle are equal; all the diameters are also equal, and each is double the radius.

156. An ARC of a circle is any part of the circumference; as the part AD, AE, or EGF.

157. The CHORD of an arc is the straight line joining its extremities; thus EF is the chord of the arc EGF.

D

A

B

C

E

F

158. The SEGMENT of a circle is the part of a circle included between an arc and its chord; as the surface included between the arc A EGF and the chord EF.

159. The SECTOR of a circle is the

part of a circle included between an

D

C

E

F

G

arc, and the two radii drawn to the extremities of the arc; as the surface included between the arc AD, and the two radii CA, CD.

161. A TANGENT to a circle is a straight line which, how far so ever produced, meets the circumference in but one point; as the line CD. The point of meeting is called the POINT OF CONTACT; as the point M.

162. Two circumferences TOUCH each other, when they have a point of contact without cutting one another; thus two circumferences touch each other at the point A, and two at the point B.

163. A STRAIGHT LINE is INSCRIBED in a circle when its extremities are in the circumference;

C

A

B

as the line AB, or B C.

A

164. An INSCRIBED ANGLE is one which has its vertex in the circumference, and is formed by two chords; as the angle ABC.

165. An INSCRIBED POLYGON is one which has the vertices of all its angles

in the circumference of the circle;

as the triangle ABC.

C

B

166. The circle is then said to be CIRCUMSCRIBED about

168. The circle is then said to be INSCRIBED in the

polygon.

PROPOSITION I.-THEOREM.

169. Every diameter divides the circle and its circumference each into two equal parts.

Let AEBF be a circle, and AB a diameter; then the two parts AEB, AFB are equal.

A

F

E

B

For, if the figure A E B be applied to AFB, their common base A B retaining its position, the curve line AEB must fall exactly on the curve line AFB; otherwise there would be points in the one or the other unequally distant from the centre, which is contrary to the definition of the circle (Art. 152). Hence a diameter divides the circle and its circumference into two equal parts.

170. Cor. 1. Conversely, a straight line dividing the circle into two equal parts is a diameter.

For, let the line AB divide the circle AEBCF into two equal parts; then, if the centre is not in AB, let AC be drawn through it, which is therefore a diameter, and conse- A quently divides the circle into two equal parts; hence the surface AFC is equal to the surface A F C B, a part to the whole, which is impossible.

171. Cor. 2. The arc of a circle, whose chord is a diameter, is a semi-circumference, and the included segment is a semicircle.

172. A straight line cannot meet the circumference of a circle in more than two points.

For, if a straight line could meet the circumference ABD, in three points, A, B, D, join each of these points with the centre, C; then, since the straight lines CA, CB,

D

CD are radii, they are equal (Art.

155); hence, three equal straight

B

lines can be drawn from the same point to the same straight line, which is impossible (Prop. XIV. Cor. 2, Bk. I.).

PROPOSITION III. — THEOREM.

173. In the same circle, or in equal circles, equal arcs are subtended by equal chords; and, conversely, equal chords subtend equal arcs.

Let A DB and EGF be two equal circles, and let the arc AD be equal to EG; then will the chord AD be equal to the chord E G.