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] [Illustration] THEOREM. _The sum of the sides of a spherical polygon is less than 360 deg.._ 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 deg., and its other two sides each 90 deg., the sum of the sides being 539 deg.. THEOREM. _The sum of the angles of a spherical triangle is greater than 180 deg. and less than 540 deg.._ 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 deg., but would exceed 180 deg. 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 deg.. 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._