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But there is one outstanding objection to this theory that the stellar universe is infinite. There may be supposed to be no reason why the average brightness of stars should be any different in one part of space from what is in any other part; multiplying this average brightness by the average number of stars per unit volume (the average star-density that we suppose for the infinite universe), we will get the average amount of light issuing from a unit volume anywhere in space; let us call this product L. Now, as the apparent brightness of any source of light is inversely proportional to the square of the distance between that source and the observer, then, if we call that distance d, the average apparent brightness of a unit of volume at distance d from the observer could be represented as L/d². If we divide space into an infinite number of concentric spherical shells, with the observer at the center, each with equal thickness, let us say the unit distance divided by 4n, then, especially when the sphere is very large, the volume of each shell is approximately d². Multiplying the average apparent brightness of a unit volume at distance d by the volume of the shell of distance d, we find that the volume of each such shell is a constant, L. Since the stellar universe consists of an infinite number of such shells, each of which has the same apparent brightness, it follows that the brightness of the sky, or indeed of the smallest part of it, must be altogether infinite. The consequence of the theory of an infinite universe is obviously contradicted by facts.

Now, there is obviously some fallacy in this argument, but I can't find it. Any thoughts would be appreciated.