Mathematician aged x in the year x squared

A famous logician who died in $1871$ liked to point out that in the calendar year $x^2$ his age was exactly $x$. If nobody lives longer than $120$ years, find $x$.

$x=43$. The year $x^2$ must lie between $1751$ and $1871$, so $x$ is $42$ or $43$. Being $42$ in $1764$ would mean a birth year of $1722$, too early to die in $1871$ within $120$ years, whereas $43$ in $1849$ means born in $1806$, which fits.