  We give a rigorous derivation of the free energy of (i) the classical Ising
model on the triangular lattice with translation-invariant coupling constants,
and (ii) the one-dimensional quantum Ising model. We use the method of Kac and
Ward. The novel aspect is that the coupling constants may have negative signs.
We describe the logarithmic singularity of the specific heat of the classical
model and the validity of the Cimasoni--Duminil-Copin-Li formula for the
critical temperature. We also discuss the quantum phase transition of the
quantum model.
