  We present extensive magnetic measurements of the
(Ca_xLa_{1-x})(Ba_{1.75-x}La_{0.25+x})Cu_{3}O_{y} (CLBLCO) system with its four
different families (x) having a Tc^max(x) variation of 28% and minimal
structural changes. For each family we measured the Neel temperature, the
anisotropies of the magnetic interactions, and the spin glass temperature. Our
results exhibit a universal relation Tc=c*J*n_s for all families, where c~1, J
is the in plane Heisenberg exchange, and n_s is the carrier density. This
relates cuprate superconductivity to magnetism in the same sense that phonon
mediated superconductivity is related to atomic mass.
