  Feller's classic text 'An Introduction to Probability Theory and its
Applications' contains a derivation of the well known significant-digit law
called Benford's law. More specifically, Feller gives a sufficient condition
("large spread") for a random variable $X$ to be approximately Benford
distributed, that is, for $\log_{10}X$ to be approximately uniformly
distributed modulo one. This note shows that the large-spread derivation, which
continues to be widely cited and used, contains serious basic errors. Concrete
examples and a new inequality clearly demonstrate that large spread (or large
spread on a logarithmic scale) does not imply that a random variable is
approximately Benford distributed, for any reasonable definition of "spread" or
measure of dispersion
