Page:Spherical Trigonometry (1914).djvu/23

Draw a great circle through $A$ and $B$, meeting $CD$ and $CE$ at $M$ and $N$ respectively. Then $AO$ is perpendicular to $OC$, which is a straight line in the plane $OCD$; and $BO$ is perpendicular to $OC$, which is a straight line in the plane $OCE$; therefore $OC$ is perpendicular to the plane $AOB$ (Euclid, XI. 4); and therefore $OC$ is perpendicular to the straight lines $OM$ and $ON$, which are in the plane $AOB$. Hence $M{\widehat {O}}N$ is the angle of inclination of the planes $OCD$ and $OCE$. And the angle $$A{\widehat {O}}B=A{\widehat {O}}M-B{\widehat {O}}M=B{\widehat {O}}N-B{\widehat {O}}M=M{\widehat {O}}N.$$

9. Definition. When two circles intersect, the angle between the tangents at either of their points of intersection is called the angle between the circles.

The angle of intersection of two great circles is equal to the enclination of their planes.

For, in the figure of the preceding Article, the tangents at $C$ to the circles $CD$ and $CE$, lying in the planes of these circles respectively, are perpendicular to their common radius $OC$, which is the line of intersection of the planes. Hence the angle between the tangents is the angle of inclination of the planes.

In the figure to Art. 6, since $PO$ is perpendicular to the plane $ACB$, every plane which contains $PO$ is at right angles to the plane $ACB$. Hence the angle between the plane of any circle and the plane of a great circle which passes through its poles is a right angle.

10. Two great circles bisect each other.

For since the plane of each great circle passes through the centre of the sphere, the line of intersection of these planes is a diameter of the sphere, and therefore also a diameter of each great circle; therefore the great circles are bisected at the points where they meet.