  We give a Pfaffian formula to compute the partition function of the Ising
model on any graph $G$ embedded in a closed, possibly non-orientable surface.
This formula, which is suitable for computational purposes, is based on the
relation between the Ising model on $G$ and the dimer model on its terminal
graph $G^T$. By combining the ideas of Loebl-Masbaum \cite{Loeb11}, Tesler
\cite{Tes2000}, Cimasoni \cite{Cim09, Cim10} and Chelkak-Cimasoni-Kassel
\cite{Chel15}, we give an elementary proof for the formula.
