  The electromagnetic excitation of the $\Delta(1232)$-resonance plays an
appreciable role in the Lamb shift and hyperfine structure of muonic and
electronic hydrogen. Its effect appears at the subleading order
$\mathcal{O}(\alpha^5)$, together with other proton-polarizability
contributions from forward two-photon exchange. We use the large-$N_c$
relations for the nucleon-to-delta transition form factors to compute the
effect of the $\Delta(1232)$ in the hydrogen spectrum. We pay particular
attention to a subtile difference between predictions based on a direct
calculation of the two-photon exchange (or Compton scattering amplitudes) and
predictions based on the $\Delta(1232)$-production photoabsorption cross
sections. The mismatch is explained by studying the dispersion relations for
tree-level Compton scattering off the proton in more details.
