  In this expository paper we transcribe two letters by Guido Castelnuovo, one
to Francesco Severi, the other to Beniamino Segre, and explain the contents of
both, which basically focus on the quest for an algebraic proof of the equality
between the analytic and the arithmetic irregularity and of the closedness of
regular 1-forms on a complex, projective, algebraic surface. Such an algebraic
proof has been found only in the 1980's by Deligne and Illusie.
