Find the square root of 1849.

Step 1: Pair the digits of 1849 from the right.
The pairs are 18 and 49.

Step 2: Find the largest integer whose square is less than or equal to the first pair, 18.
$4^2 = 16$, which is less than 18.
$5^2 = 25$, which is greater than 18.
So, the first digit of the square root is 4.

$$\begin{array}{r} 4 \\[-3pt] \sqrt{18\,49} \\[-3pt] -16 \\[-3pt] \hline 2 \end{array}$$

Step 3: Bring down the next pair, 49, to form the new dividend, 249. Double the current quotient ($4 \times 2 = 8$) to get the new partial divisor.
Now, we need to find a digit $x$ such that $8x \times x \le 249$.
If $x=3$, then $83 \times 3 = 249$.

$$\begin{array}{r} 43 \\[-3pt] \sqrt{18\,49} \\[-3pt] -16 \downarrow \\[-3pt] \hline 249 \\[-3pt] -249 \\[-3pt] \hline 0 \end{array}$$

The square root of 1849 is 43.

The positive square root of 1849 is $\boxed{43}$.