The correction in elevations due to the curvature and refraction is proportional to D2/ R

**Explanation:**

The correction in elevations due to the curvature and refraction is proportional to **D^2/R**.

**Curvature Correction:**
- When we consider a horizontal distance D on the Earth's surface, the Earth can be approximated as a sphere with radius R.
- Due to the curvature of the Earth, the line of sight between two points on the Earth's surface will not be a straight line, but rather a curved line.
- As a result, the elevation of a point will appear lower than it actually is when viewed from a distant point.
- This correction is proportional to the horizontal distance D and is given by D^2/R, where R is the radius of curvature of the Earth.

**Refraction Correction:**
- Refraction is the bending of light as it passes through different mediums with different refractive indices.
- The Earth's atmosphere has a varying refractive index due to changes in temperature, pressure, and humidity.
- This causes the path of light to deviate from a straight line and results in an apparent shift in the elevation of a point when viewed from a distant point.
- The refraction correction is also proportional to the horizontal distance D.

**Combined Correction:**
- The combined correction for curvature and refraction is proportional to D^2/R.
- This means that as the horizontal distance D increases, the correction increases, and as the radius of curvature R increases, the correction decreases.
- The correction is inversely proportional to the radius of curvature, indicating that the larger the radius of curvature, the smaller the correction will be for the same horizontal distance.

**Conclusion:**
- The correction in elevations due to the curvature and refraction is proportional to D^2/R, where D is the horizontal distance and R is the radius of curvature of the Earth.

We know
Cc =d²/2R
Cr =1/7 ×d²/2R
thus
Cc and Cr proportional to d²/R
where d is distance between levelling instrument and staff
R is radius of earth