Article 3
Areas of Corresponding Sectors.

The areas of corresponding sectors have equal measures. For conceive the sectors $S_1,S_2$ divided up into infinitesimal corresponding sectors; then the respective infinitesimal corresponding triangles have equal measures (Art. 2); but the given sectors are the limits of the sums of these infinitesimal triangles, hence

In particular, the sectors $A_1{O}_1{P}_1,A_2{O}_2{P}_2$ have equal measures; for the initial points $A_1,A_2$ are corresponding points.

It may be proved conversely by an obvious reductio ad absurdum that if the initial points of two equal-measured sectors correspond, then their terminal points correspond.

Thus if any radii $O_1{A}_1,O_2{A}_2$ be the initial lines of two equal-measured sectors whose terminal radii are $O_1{P}_1,O_2{P}_2$, then $P_1,P_2$ are corresponding points referred respectively to the pairs of conjugate directions $O_1{A}_1,O_1{B}_1$, and $O_2{A}_2,O_2{A}_B$; that is,

Prob. 1.

Prove that the sector $P_1{O}_1{Q}_1$, is bisected by the line joining $O_1$, to the mid-point of $P_1{Q}_1$. (Refer the points $P_1,Q_1$, respectively, to the median as common axis of $x$, and to the two opposite conjugate directions as axis of $y$, and show that $P_1,Q_1$ are then corresponding points.)

Prob. 2.

Prove that the measure of a circular sector is equal to the radian measure of its angle.

Prob. 3.

Find the measure of an elliptic quadrant, and of the sector included by conjugate radii.