If the horizontal distance between the staff point and the point of observation is d, then the error due to curvature of earth is proportional to A. $${\text{d}}$$ B. $$\frac{1}{{\text{d}}}$$ C. $${{\text{d}}^2}$$ D. $$\frac{1}{{{{\text{d}}^2}}}$$

The correct answer is: C. $d^2$

The error due to curvature of earth is proportional to the square of the horizontal distance between the staff point and the point of observation. This is because the curvature of the earth is a function of the distance from the observer. The further away the object is, the more curved the earth appears.

Option A is incorrect because the error due to curvature of earth is not proportional to the horizontal distance.

Option B is incorrect because the error due to curvature of earth is not inversely proportional to the horizontal distance.

Option D is incorrect because the error due to curvature of earth is not inversely proportional to the square of the horizontal distance.