Einfluss der Erdkrümmung und der atmosphärischen Brechung auf die Nivellierung

While leveling, due to Earth's curvature, the level’s line of sight is perpendicular to a plumb line only at one point and deviates further from the level line.

This deviation, C, is determined by considering a right-angled triangle with sides E, representing the Earth’s radius, D, representing a horizontal distance corresponding to a deviation C, and hypotenuse E + C. 

From the equation connecting the sides of the triangle and neglecting the value of C² as its magnitude is much less compared to the magnitude of E², an expression for the deviation is derived. 

This expression indicates that the Earth's curvature causes a horizontal line to deviate by 0.0785 meters over 1 kilometer, increasing with the square of the distance.

Also, during leveling, the atmospheric refraction causes the line of sight to bend downwards by about 0.011 meters in 1 kilometer, about one-seventh of the Earth’s curvature effect.

Combined, the effect of curvature and atmospheric refraction, cause the level's line of sight to deviate by approximately 0.0675 meters in 1 kilometer, varying with the square of the horizontal distance.