smc

If a man born in the 19th century was $x$ years of in the year $x^2$, it implies that the year the man was born was $x^2-x$. So, if the man was born in the first half of the 19th century, it means that $x^2-x<1850$. Noticing that $40^2-40=1560$ and $50^2-50=2450$, we see that $40<x<50$. We can guess values until we hit a solution. $43^2-43=1806$, so we see that the man had to have been $43$ years old in the year $1849=43^2$, so the answer is $\boxed{1806}$.