Two things distort a long levelling sight, and conveniently they act in opposite directions. The earth curves away from the line of sight, making distant points read too low. The atmosphere bends the light downwards, partly cancelling that effect.
This post covers both corrections, their combination, and the distance to the visible horizon that follows from them.
Why the Corrections Are Needed
Recall from the introduction to levelling the distinction between two lines:
- A level line is at constant height above mean sea level, and is curved.
- A horizontal line is straight and tangential to it.
A levelling instrument gives you the horizontal line — but the surface you want to measure heights against is the level line. Over a long sight the two separate, and the correction accounts for that separation.
1. Correction for Curvature
Cc = d2 / (2R) (negative)
| Symbol | Meaning |
|---|---|
| d | Horizontal distance between A and B |
| R | Radius of the earth, 6370 km |
Cc = 0.07857 d2 metres (d in km)
Checking the Numerical Constant
It is worth verifying where 0.07857 comes from, since it makes the formula memorable rather than arbitrary.
Cc = d2/(2R) = d2/(2 × 6370) km = d2/12740 km
Converting to metres: (1000/12740) d2 = 0.0785 d2 m ✓
Why Curvature Correction Is Negative
The horizontal line of sight is a tangent to the level surface, so it rises above that surface as it goes.
A staff held at a distant point is therefore intersected higher up than it would be by a true level line. A bigger staff reading means the point appears lower than it really is — so the correction must be subtracted from the apparent difference, or equivalently the reading corrected downwards.
This is exactly the effect described in the introduction, where a point B genuinely at mean sea level would be misconstrued as being below it.
2. Correction for Refraction
Cr = (1/7) × d2/(2R) (positive)
= 0.01122 d2 metres (d in km)
Checking This Constant Too
0.07857 ÷ 7 = 0.01122 ✓
The refraction correction is exactly one seventh of the curvature correction, which makes it easy to remember and easy to check.
Why Refraction Is Positive
Light travelling through the atmosphere passes through air of decreasing density with height. This bends the ray downwards, curving it towards the earth’s surface.
Because the ray bends down, it strikes the distant staff lower than a straight ray would — which partly undoes the curvature effect. Refraction therefore works in the opposite direction to curvature, and its correction carries the opposite sign.
The One-Seventh Figure
The factor of 1/7 represents the average refraction of the atmosphere. That word matters — refraction depends on air temperature, pressure and humidity, all of which vary with weather and time of day.
So while the curvature correction is exact geometry, the refraction correction is only an average estimate. This is precisely why “variation in the average refraction” appears among the errors that reciprocal levelling eliminates.
3. Combined Correction
Since the two act oppositely, they partly cancel:
C = (6/7) × d2/(2R) (negative)
C = 0.06735 d2 metres (d in km)
Where 6/7 Comes From
Curvature (1 unit, negative) + Refraction (1/7 unit, positive) = 6/7 unit, negative
0.07857 × 6/7 = 0.06735 ✓
Refraction cancels one seventh of the curvature, leaving six sevenths. The combined correction remains negative, because curvature is the larger of the two by a factor of seven.
All Three Constants Together
| Correction | Fraction of d2/2R | Constant (d in km, C in m) | Sign |
|---|---|---|---|
| Curvature | 1 | 0.07857 | Negative |
| Refraction | 1/7 | 0.01122 | Positive |
| Combined | 6/7 | 0.06735 | Negative |
Note the arithmetic relationship running across the table: 0.07857 − 0.01122 = 0.06735. If you remember any two constants, the third follows.
Note the d² Dependence
All three corrections vary as the square of the distance. This has a strong practical consequence:
| Sight Distance | Combined Correction |
|---|---|
| 100 m (0.1 km) | 0.7 mm |
| 500 m (0.5 km) | 17 mm |
| 1 km | 67 mm |
| 2 km | 269 mm |
At ordinary sight lengths the correction is negligible — under a millimetre at 100 m. It only becomes significant on long sights, which is exactly why the introduction said the level and horizontal lines coincide over short distances but need correction over long ones.
Doubling the distance quadruples the correction, so it grows quickly once sights become long.
Distance to the Visible Horizon
Rearranging the combined correction formula gives the distance at which a point of given height disappears from view:
d = √(C / 0.06735)
C in metres, d in kilometres
Both curvature and refraction are taken into account in this expression.
Reading the Formula
Since C = 0.06735 d2, simply solving for d gives d = √(C/0.06735). It is the same relationship read backwards.
Its meaning is this: if an object stands C metres above the level surface, it will just be visible at a distance of d kilometres — beyond which the curvature of the earth hides it.
A Worked Example
For an observer whose eye is 1.5 m above the ground:
d = √(1.5 / 0.06735) = √22.27 = 4.72 km
So a person of average height standing on level ground sees the horizon roughly 4.7 km away. The result is intuitively reasonable, which is a useful check that the formula has been applied correctly.
Note the Square Root
Because the relationship is a square root, height buys distance slowly. Quadrupling your height only doubles how far you can see — which is why lighthouses and observation towers must be built very tall to gain a modest increase in range.
Formula Summary
| Quantity | Expression |
|---|---|
| Curvature correction | Cc = d2/2R = 0.07857 d2 m, negative |
| Refraction correction | Cr = (1/7)(d2/2R) = 0.01122 d2 m, positive |
| Combined correction | C = (6/7)(d2/2R) = 0.06735 d2 m, negative |
| Distance to visible horizon | d = √(C / 0.06735) |
| Radius of earth | R = 6370 km |
| Units | d in km, C in metres |
Quick Revision Notes
- Curvature correction is negative: Cc = d2/2R = 0.07857 d2 m.
- Refraction correction is positive and equals one seventh of the curvature correction: 0.01122 d2 m.
- Combined correction is negative and equals six sevenths of the curvature correction: 0.06735 d2 m.
- 0.07857 − 0.01122 = 0.06735.
- R = 6370 km; in all these formulas d is in km and C in metres.
- Curvature makes distant points appear lower; refraction bends light downwards and partly cancels it.
- All corrections vary as d2, so they are negligible on short sights and grow rapidly on long ones.
- Distance to the visible horizon: d = √(C / 0.06735), with both curvature and refraction included.
- The 1/7 factor is an average, since refraction varies with atmospheric conditions.
Mistakes Students Commonly Make
- Getting the signs backwards. Curvature is negative, refraction positive, combined negative.
- Adding the two corrections to get 8/7. They act oppositely, giving 6/7.
- Forgetting the square root in the visible horizon formula. Since C varies as d2, d must vary as √C.
- Mixing units. d must be in kilometres and C in metres for the constants to work.
- Using R = 6370 m instead of 6370 km.
- Treating the refraction correction as exact. The 1/7 is only an average and varies with atmospheric conditions.
- Applying these corrections to short sights where they are negligible — under a millimetre at 100 m.
Conclusion
Two effects distort long sights, and they oppose one another. The earth curves away from the straight line of sight, making distant points read low — a negative correction of 0.07857 d². The atmosphere bends light downwards and gives back one seventh of that, a positive 0.01122 d². What remains is six sevenths, the combined correction of 0.06735 d², still negative because curvature dominates. Reverse the same relationship and you get the distance to the visible horizon, which grows only as the square root of height.
Frequently Asked Questions
Why is a curvature correction needed in levelling?
Because the line of sight given by a level is straight and tangential, while the surface of constant height is curved. Over long sights the two diverge, making distant points appear lower than they are.
What is the curvature correction formula?
Cc = d²/2R, which equals 0.07857 d² metres when d is in kilometres and R is taken as 6370 km. It is negative.
Why is the refraction correction positive?
Because the atmosphere bends the ray of light downwards towards the earth’s surface, so it strikes the staff lower than a straight ray would, partly undoing the curvature effect.
What is the refraction correction?
One seventh of the curvature correction, that is 0.01122 d² metres, taken as positive.
What is the combined correction?
Six sevenths of the curvature correction, or 0.06735 d² metres, and it is negative because curvature is seven times larger than refraction.
How are the three constants related?
0.07857 minus 0.01122 equals 0.06735, so remembering any two gives the third.
What units are used in these formulas?
The distance d is in kilometres and the correction C is in metres.
Why are these corrections negligible on short sights?
Because they vary as the square of the distance. At 100 m the combined correction is under a millimetre, while at 2 km it is about 269 mm.
What is the formula for distance to the visible horizon?
d = √(C/0.06735), where C is the height in metres and d the distance in kilometres. Both curvature and refraction are taken into account.
Why is the one seventh factor only approximate?
Because atmospheric refraction depends on temperature, pressure and humidity, which vary with weather and time of day. The one seventh figure represents average conditions.