Page:Spherical Trigonometry (1914).djvu/21

§6] not equally distant from the plane of the circle; they may be called respectively the nearer and further pole; sometimes the nearer pole is for brevity called the pole.

6. A pole of a circle is equally distant from every point of the circumference of the circle.

Let $$O$$ be the centre of the sphere, $$AB$$ any circle of the sphere, $$C$$ the centre of the circle, $$P$$ and $$P'$$ the poles of the circle. Take any point $$D$$ in the circumference of the circle; join $$CD$$, $$OD$$, $$PD$$. Then $$PD=\surd(PC^2+CD^2)$$; and $$PC$$ and $$CD$$ are constant, therefore $$PD$$ is constant. Suppose a great circle to pass through the points $$P$$ and $$D$$; then the chord $$PD$$ is constant, and therefore the arc of a great circle intercepted between $$P$$ and $$D$$ is constant for all positions of $$D$$ on the circle $$AB$$.



Thus the distance of a pole of a circle from every point of the circumference of the circle is constant, whether that distance be measured by the straight line joining the points, or by the arc of a great circle intercepted between the points.

Definition. The length of the arc, measured along a great circle, from any point on a small circle to the nearer pole is called the spherical radius of the small circle.