There are four ways to find the difference in height between two points, and they are strikingly different in principle. One uses a bubble, one uses trigonometry, one uses air pressure, and one uses the temperature at which water boils.
The choice between them comes down to a single question — can you actually reach the point?
The Four Methods
- Direct levelling
- Trigonometric levelling
- Barometric levelling
- Hypsometric levelling
The first is direct; the other three are all indirect methods, in which height is inferred from some other measured quantity.
1. Direct Levelling
This is the most common method of levelling. A spirit level fixed to the telescope of a levelling instrument is used to make the line of sight horizontal. All vertical distances are then measured with respect to this horizontal line of sight, and these vertical distances are used to determine the difference in elevation of various points.
Direct levelling is also called spirit levelling.
Why It Is Called Direct
The name is precise. The difference in height is obtained directly, by subtraction of two staff readings — no formula, no conversion, no assumed relationship.
That directness is exactly why it is the most common and the most accurate method. Every step in the chain is a physical measurement rather than an inference, so there is less to go wrong.
The alternative name comes from the spirit level — the bubble tube whose liquid gives the instrument its horizontal line of sight.
2. Trigonometric Levelling
The method of levelling in which the difference of elevations is determined indirectly from the horizontal distance and the vertical angle. Because trigonometric relations are used, the method is called trigonometric levelling.
The Formula
H = D tan θ
| Symbol | Meaning |
|---|---|
| H | Height of the top of the object above the horizontal line of sight |
| D | Horizontal distance to the object |
| θ | Vertical angle — the inclination of the line of sight to the horizontal |
Where the Formula Comes From
It is simple right-angled triangle trigonometry. The horizontal distance D is the base, the height H is the perpendicular, and the line of sight is the hypotenuse. Since tan θ = opposite over adjacent, H = D tan θ.
Note carefully that H is measured from the horizontal line of sight, not from the ground. To obtain the object’s actual elevation you must also add the height of the instrument above the station and the station’s own elevation.
When It Is Used
Trigonometric levelling is generally used when direct levelling becomes difficult. For example, the elevations of inaccessible points such as the peak of a mountain or the top of a tower can be determined by trigonometric levelling.
The Decisive Advantage
This is the point worth grasping. Direct levelling requires a staff held on the point — which means somebody has to physically stand there.
You cannot hold a levelling staff on a mountain peak, on the top of a transmission tower, or on the far bank of a river in flood. Direct levelling is simply unavailable.
Trigonometric levelling needs only that the point be visible. Measure a horizontal distance and read a vertical angle to the top, and the height follows — without anyone going near it.
Note the connection to the theodolite chapter: this is one of the extra uses listed there, and it works because a vertical angle is measured from the horizontal, which is exactly the θ in the formula.
3. Barometric Levelling
Another type of indirect levelling in which elevations are determined indirectly from changes in atmospheric pressure. The atmospheric pressure decreases with an increase in elevation.
The Instrument
- Generally, aneroid barometers are used for determining changes in atmospheric pressure.
- These barometers are known as altimeters.
- The aneroid barometer is not as accurate as the mercury barometer.
Its Character: Quick but Coarse
Barometric levelling is a quick method of levelling. Altimeters are commonly used to determine the altitude of aeroplanes.
The trade-off is clear. Pressure can be read instantly at any location, so heights over a wide area can be obtained very rapidly — but the accuracy is poor.
The reason for the poor accuracy is worth understanding. Atmospheric pressure does not depend on height alone; it also changes with weather. A passing depression can shift the reading by an amount equivalent to many metres of elevation, without anyone moving at all.
That is why barometric levelling suits reconnaissance over rough country and aircraft altitude, where speed matters more than precision — and why it is unsuitable for engineering work.
4. Hypsometric Levelling
Also a type of indirect levelling. The difference of elevations is determined by noting the temperature at which water starts boiling. As the altitude of the place increases, the boiling point of water decreases.
The Hypsometer
The working of a hypsometer to find the altitude of stations rests on the fact that the temperature at which water boils varies with atmospheric pressure.
The Chain of Reasoning
Hypsometric levelling is really barometric levelling measured a different way, and the logic runs in three links:
Altitude increases → atmospheric pressure decreases → boiling point decreases
Water boils when its vapour pressure equals the surrounding air pressure. Reduce the air pressure and less heat is needed to reach that condition — so the boiling point falls.
A hypsometer therefore measures temperature in order to infer pressure, in order to infer height. Two inferences stacked on one another, which is why it is no more accurate than the barometric method it depends on.
Altimeter versus Hypsometer
| Altimeter | Hypsometer | |
|---|---|---|
| Measures | Pressure directly | Boiling point temperature |
| Instrument | Aneroid barometer | Boiling water apparatus |
| Used for | Finding heights; aeroplane altitude | Finding altitude of stations |
All Four Methods Compared
| Method | Direct or Indirect | Quantity Measured | Typical Use |
|---|---|---|---|
| Direct (spirit) | Direct | Staff readings on a horizontal line of sight | Most common; general levelling work |
| Trigonometric | Indirect | Horizontal distance and vertical angle | Inaccessible points — mountain peaks, tower tops |
| Barometric | Indirect | Atmospheric pressure | Quick work; aeroplane altitude |
| Hypsometric | Indirect | Boiling point of water | Altitude of stations |
How to Choose
The sequence of questions is simple:
- Can you put a staff on the point? Use direct levelling — it is the most accurate.
- Can you see it but not reach it? Use trigonometric levelling.
- Do you need many rough heights quickly over rough country? Use barometric levelling.
Accuracy falls as you move down that list, which is why direct levelling remains the standard whenever it is possible at all.
Quick Revision Notes
- Four methods: direct, trigonometric, barometric, hypsometric. Only the first is direct.
- Direct levelling uses a spirit level fixed to the telescope to make the line of sight horizontal; also called spirit levelling; the most common method.
- Trigonometric levelling: difference of elevation from horizontal distance and vertical angle, using H = D tan θ.
- H is measured above the horizontal line of sight, not above the ground.
- Trigonometric levelling is used for inaccessible points such as mountain peaks and tower tops.
- Barometric levelling uses changes in atmospheric pressure, which decreases with increasing elevation.
- Aneroid barometers used for this are called altimeters; they are less accurate than mercury barometers.
- Barometric levelling is a quick method; altimeters determine aeroplane altitude.
- Hypsometric levelling uses the boiling point of water, which decreases as altitude increases.
- A hypsometer works because boiling temperature varies with atmospheric pressure.
Mistakes Students Commonly Make
- Saying barometric pressure increases with elevation. It decreases.
- Saying the boiling point rises with altitude. It falls, because pressure falls.
- Confusing altimeter and hypsometer. An altimeter measures pressure; a hypsometer measures boiling point.
- Treating H in H = D tan θ as the height above the ground. It is the height above the horizontal line of sight.
- Using the slope distance instead of the horizontal distance for D.
- Forgetting that direct levelling is also called spirit levelling.
- Saying the aneroid barometer is more accurate than the mercury barometer. It is less accurate.
- Classifying trigonometric levelling as a direct method. Only spirit levelling is direct.
Conclusion
Four methods, one deciding question — can you reach the point? Direct levelling puts a staff on it and subtracts two readings, which makes it both the simplest and the most accurate. Trigonometric levelling needs only a clear line of sight, so mountain peaks and tower tops come within reach through H = D tan θ. Barometric levelling trades accuracy for speed by reading falling air pressure, and hypsometric levelling reaches the same pressure indirectly through the temperature at which water boils.
Frequently Asked Questions
What are the four methods of levelling?
Direct levelling, trigonometric levelling, barometric levelling and hypsometric levelling. Only the first is a direct method.
What is direct levelling?
The most common method, in which a spirit level fixed to the telescope of a levelling instrument makes the line of sight horizontal, and all vertical distances are measured with respect to it. It is also called spirit levelling.
What is trigonometric levelling?
The method in which the difference of elevations is determined indirectly from the horizontal distance and the vertical angle, using trigonometric relations.
What is the trigonometric levelling formula?
H = D tan θ, where H is the height of the top of the object above the horizontal line of sight, D is the horizontal distance and θ is the vertical angle.
When is trigonometric levelling used?
When direct levelling becomes difficult, particularly for inaccessible points such as the peak of a mountain or the top of a tower, where a staff cannot be held.
What is barometric levelling?
Indirect levelling in which elevations are determined from changes in atmospheric pressure, which decreases as elevation increases.
What is an altimeter?
An aneroid barometer used for determining changes in atmospheric pressure for levelling purposes. Altimeters are commonly used to determine the altitude of aeroplanes.
Is an aneroid barometer as accurate as a mercury barometer?
No, the aneroid barometer is less accurate.
What is hypsometric levelling?
Indirect levelling in which the difference of elevations is determined by noting the temperature at which water starts boiling, since the boiling point decreases as altitude increases.
How does a hypsometer work?
It relies on the fact that the temperature at which water boils varies with atmospheric pressure. Since pressure falls with height, the boiling point can be used to infer altitude.