  We report precision determinations of the beam normal single spin asymmetries
($A_n$) in the elastic scattering of 0.95 and 2.18~GeV electrons off $^{12}$C,
$^{40}$Ca, $^{48}$Ca, and $^{208}$Pb at very forward angles where the most
detailed theoretical calculations have been performed. The first measurements
of $A_n$ for $^{40}$Ca and $^{48}$Ca are found to be similar to that of
$^{12}$C, consistent with expectations thus demonstrating the validity of
theoretical calculations for nuclei with Z~$\leq20$. We also report $A_n$ for
$^{208}$Pb at two new momentum transfers (Q$^2$) extending the previous
measurement. Our new data confirm the surprising result previously reported,
with all three data points showing significant disagreement with the results
from the $Z\leq 20$ nuclei. These data confirm our basic understanding of the
underlying dynamics that govern $A_n$ for nuclei containing $\lesssim 50$
nucleons, but point to the need for further investigation to understand the
unusual $A_n$ behaviour discovered for scattering off $^{208}$Pb.
