Imagine constructing two spheres around a given star, one ten times farther from the star than the other (if the radius of the inner sphere is R, the radius of the outer sphere is 10 R). Now let us subdivide each sphere into little squares, 1 square foot in area, and assume than on the inner sphere I could fit one million such squares. Since the area of a sphere increases as the square of the radius, the second sphere will accommodate 100 times the number of squares on the first sphere, that is, 100 million squares (all 1 square foot in area). Now, since all the light from the star goes through both spheres, the amount of light going through one little square in the inner sphere must be spread out among 100 similar squares on the outer sphere. This implies that the brightness of the star drops by a factor of 100, when we go from the distance R to the distance 10 R (see Fig. 8.1).
If we go to a distance of 20 R the brightness would drop by a factor of 400, which is the square of 20, for 30R there would be a decrease by a factor of 900 = (30)2, etc. Thus we conclude that