  Hilbert's Irreducibility Theorem is a cornerstone that joins areas of
analysis and number theory. Both the genesis and genius of its proof involved
combining real analysis and combinatorics. We try to expose the motivations
that led Hilbert to this synthesis. Hilbert's famous Cube Lemma supplied fuel
for the proof but without the analytical foundation and framework it would have
been heating empty air. The lemma is said to presage Ramsey Theory but we note
differences in motivation.

History and Overview