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 30*R* there would be a decrease by a factor of 900 = (30)^{2},
etc. Thus we conclude that