Similarly in this proposition concerning spheres, if from any point in the plane of the circle formed by the intersection of the two spherical surfaces lines are drawn tangent to either sphere, these tangents are equal. For it is easily proved that all tangents to the same sphere from an external point are equal, and it can be proved as in plane geometry that two tangents to the two spheres are equal.
Among the interesting analogies between plane and solid geometry is the one relating to the four common tangents to two circles. If the figure be revolved about the line of centers, the circles generate spheres and the tangents generate conical surfaces. To study this case for various sizes and positions of the two spheres is one of the most interesting generalizations of solid geometry.
An application of the proposition is seen in the case of an eclipse, where the sphere O' represents the moon, O the earth, and S the sun. It is also seen in the case of the full moon, when S is on the other side of the earth. In this case the part MIN is fully illuminated by the moon, but the zone ABNM is only partly illuminated, as the figure shows.[92]
Theorem. The sum of the sides of a spherical polygon is less than 360°.
In all such cases the relation to the polyhedral angle should be made clear. This is done in the proofs usually given in the textbooks. It is easily seen that this is true only with the limitation set forth in most textbooks, that the spherical polygons considered are convex. Thus we might have a spherical triangle that is concave, with its base 359°, and its other two sides each 90°, the sum of the sides being 539°.
Theorem. The sum of the angles of a spherical triangle is greater than 180° and less than 540°.
It is for the purpose of proving this important fact that polar triangles are introduced. This proposition shows the relation of the spherical to the plane triangle. If our planes were in reality slightly curved, being small portions of enormous spherical surfaces, then the sum of the angles of a triangle would not be exactly 180°, but would exceed 180° by some amount depending on the curvature of the surface. Just as a being may be imagined as having only two dimensions, and living always on a plane surface (in a space of two dimensions), and having no conception of a space of three dimensions, so we may think of ourselves as living in a space of three dimensions but surrounded by a space of four dimensions. The flat being could not point to a third dimension because he could not get out of his plane, and we cannot point to the fourth dimension because we cannot get out of our space. Now what the flat being thinks is his plane may be the surface of an enormous sphere in our three dimensions; in other words, the space he lives in may curve through some higher space without his being conscious of it. So our space may also curve through some higher space without our being conscious of it. If our planes have really some curvature, then the sum of the angles of our triangles has a slight excess over 180°. All this is mere speculation, but it may interest some student to know that the idea of fourth and higher dimensions enters largely into mathematical investigation to-day.
Theorem. Two symmetric spherical triangles are equivalent.