Page:Spherical Trigonometry (1914).djvu/20

2 to the plane; take any point $$D$$ in the section and join $$OD$$, $$CD$$. Since $$OC$$ is perpendicular to the plane, the angle $$OCD$$ is a right angle; therefore $$CD=\surd(OD^2-OC^2)$$. Now $$O$$ and $$C$$ are fixed points, so that $$OC$$ is constant; and $$OD$$ is constant, being the radius of the sphere; hence $$CD$$ is constant. Thus all points in the plane section are equally distant from the fixed point $$C$$; therefore the section is a circle of which $$C$$ is the centre.

3. Definitions. The section of the surface of a sphere by a plane is called a great circle if the plane passes through the centre of the sphere, and a small circle if the plane does not pass through the centre of the sphere. Thus the radius of a great circle is equal to the radius of the sphere.

4. Through the centre of a sphere and any two points on the surface a plane can be drawn; and only one plane can be drawn, except when the two points are the extremities of a diameter of the sphere, and then an infinite number of such planes can be drawn. Hence only one great circle can be drawn through two given points on the surface of a sphere, except when the points are the extremities of a diameter of the sphere. When only one great circle can be drawn through two given points, the great circle is unequally divided at the two points; we shall for brevity speak of the shorter of the two arcs as the arc of a great circle joining the two points.

5. Definitions. The axis of any circle of a sphere is that diameter of the sphere which is perpendicular to the plane of the circle; the extremities of the axis are called the poles of the circle. The poles of a great circle are equally distant from the plane of the circle. The poles of a small circle are