  We present the first lattice-QCD result for the Zemach and Friar radii of the
proton and neutron. Our calculation includes both quark-connected and
-disconnected diagrams and assesses all sources of systematic uncertainties
arising from excited-state contributions, finite-volume effects and the
continuum extrapolation. At the physical point, we obtain for the proton $r_Z^p
= ( 1.013 \pm 0.010\ (\mathrm{stat}) \pm 0.012\ (\mathrm{syst}) )~\mathrm{fm}$
and $r_F^p = ( 1.301 \pm 0.012\ (\mathrm{stat}) \pm 0.014\ (\mathrm{syst})
)~\mathrm{fm}$. These numbers suggest small values of the Zemach and Friar
radii of the proton, but are compatible with most of the experimental studies.
