Algebra

Since De Morgan spent his entire life in the 1800s and was $x$ years old in the year $x^2$ , then:

$\begin{aligned} \left \lceil \sqrt{1800} \right \rceil & \le x \le \left \lfloor \sqrt{1899} \right \rfloor & \small \color{#3D99F6} \text{where }\lceil \cdot \rceil \text{ and } \lfloor \cdot \rfloor \text{ denote the ceiling and floor functions respectively.} \\ \implies 43 & \le x \le 43 \\ \implies x & = 43 \\ x^2 & = 1849 \end{aligned}$

And he was born in the year $1849-43 = \boxed{1806}$ .

References:

• Ceiling function
• Floor function

He was born in $x^2 - x$ year.
We want $1800<x^2-x<1900$
The only positive integer solution of this inequality is $43$ .
Therefore he was born in:
$43^2-43=1806$