Answer: E

Small Hint:

His birth year is $x^2-x$

Big Hint:

Use the requirement that $x^2$ is a nineteenth-century year and test the nearby integer $x$

Solution:

The year $x^2$ must lie in the nineteenth century, so $x=43$ because $43^2=1849$ while the neighboring squares fall outside the relevant range. His birth year is therefore $$x^2-x=1849-43=1806,$$ which is in the first half of the century.

Thus, the correct answer is E.