  We study the Reidemeister torsion and the analytic torsion of the $m$
dimensional disc in the Euclidean $m$ dimensional space, using the base for the
homology defined by Ray and Singer in \cite{RS}. We prove that the Reidemeister
torsion coincides with a power of the volume of the disc. We study the
additional terms arising in the analytic torsion due to the boundary, using
generalizations of the Cheeger-M\"{u}ller theorem. We use a formula proved by
Br\"uning and Ma \cite{BM}, that predicts a new anomaly boundary term beside
the known term proportional to the Euler characteristic of the boundary
\cite{Luc}. Some of our results extend to the case of the cone over a sphere,
in particular we evaluate directly the analytic torsion for a cone over the
circle and over the two sphere. We compare the results obtained in the low
dimensional cases. We also consider a different formula for the boundary term
given by Dai and Fang \cite{DF}, and we show that the result obtained using
this formula is inconsistent with the direct calculation of the analytic
torsion.

Differential Geometry