Once the closing error is known and found acceptable, it has to be distributed — spread across the traverse so that the survey closes perfectly and can be plotted.
Five methods exist, and choosing between them comes down to one question: which did you measure more precisely, the angles or the distances?
Why Adjustment Is Done
If the error of closure is within permissible limits, the traverse should be adjusted. The purpose is to reduce the error of closure to zero. The error is distributed among the various sides of the traverse such that the traverse geometrically closes.
The Condition That Comes First
Note the opening words — if the error of closure is within permissible limits. Adjustment is not a way of rescuing bad work.
If the closing error is too large, it signals a mistake rather than accumulated random error — a misread vernier, a miscounted tape length, a wrongly booked bearing. Distributing a blunder across all the sides would simply smear the error over the whole survey and hide it. In that case the fieldwork must be checked and repeated.
Adjustment is only legitimate when what remains is ordinary random error.
The Five Methods
- Arbitrary method
- Bowditch’s rule
- Graphical method
- Transit rule
- Axis method
1. Arbitrary Method
The linear misclosure is distributed arbitrarily, according to the discretion of the surveyor, based on field conditions. The surveyor may decide to make a larger correction to one or two sides where difficulties were experienced during the measurements.
When Judgement Beats Formula
This method sounds unscientific, but there is a real argument for it.
Every other method assumes error is spread evenly according to some mathematical law. But the surveyor was there. If one line crossed a swamp, or had to be measured in fading light, or was taped across broken ground while every other line ran over a level field, then it is simply true that most of the error came from that line.
Putting the correction where the error actually arose is more honest than pretending it was evenly distributed. The method’s weakness is equally clear — it is not reproducible, and two surveyors may adjust the same traverse differently.
2. Bowditch’s Rule
Bowditch’s rule, also called the compass rule, is generally used for adjusting a traverse in which the angles and distances are measured with the same precision.
Its Assumption
- Bowditch’s rule is based on the assumption that the errors introduced in the traverse are accidental (random) in nature.
- The probable error in a traverse line is assumed to be directly proportional to the square root of its length, that is e ∝ √L.
Where the Square Root Comes From
This should look familiar. It is exactly the law governing compensating errors in chaining — random errors accumulate as √L, not as L, because their signs are random and they partly cancel.
Bowditch’s rule is therefore not an arbitrary formula. It is the direct application of the statistics of random error to traverse adjustment, which is precisely why it assumes accidental errors and why it suits a traverse where angles and distances were measured with comparable care.
The Formulas
Error in latitude of any line = Total error in latitude × (Length of the line / Perimeter of the traverse)
Correction to latitude of any line = −ey × (Length of the line / Perimeter of the traverse)
Correction to departure of any line = −ex × (Length of the line / Perimeter of the traverse)
Reading the Formula
The controlling quantity is length of the line divided by the perimeter. So the correction each line receives is proportional to how long that line is.
This is intuitive — a longer line had more opportunity to accumulate error, so it takes a larger share of the correction. The shares automatically add up to the whole, since all the lengths together make the perimeter.
Effect on Lengths and Bearings
When a traverse is adjusted by Bowditch’s rule, both the lengths and bearings of the lines get changed. However, in comparison to the transit rule, the lengths are changed less and the angles are changed more.
3. Graphical Method
- Based on Bowditch’s rule.
- Generally used for a compass traverse.
- Sometimes also used for a theodolite traverse with low accuracy.
The graphical method achieves the same distribution as Bowditch’s rule but by construction on the drawing rather than by calculation — appropriate where the accuracy does not justify the arithmetic.
4. Transit Rule
The transit rule is used to balance a traverse in which the angular measurements are more precise than the linear measurements.
The Formulas
Error in latitude of any line = ey × (Numerical value of the latitude of the line / Arithmetic sum of all latitudes)
Correction to latitude of any line = −ey × (Numerical value of the latitude of the line / Arithmetic sum of all latitudes)
The Critical Difference From Bowditch
| Bowditch’s Rule | Transit Rule | |
|---|---|---|
| Correction proportional to | Length of the line | Latitude (or departure) of the line |
| Divided by | Perimeter | Arithmetic sum of all latitudes (or departures) |
Bowditch distributes according to total length; the transit rule distributes according to the component in that direction.
Note “Arithmetic Sum”
The denominator is the arithmetic sum, not the algebraic sum — that is, magnitudes added without regard to sign.
This matters enormously. The algebraic sum of latitudes is the closing error itself, which is nearly zero. Dividing by it would give absurd results. The arithmetic sum adds all the latitudes as positive numbers, giving a sensible total against which each line’s share is measured.
Effect on Lengths and Bearings
In the transit rule, the angles are changed less but the lengths are changed more.
Why This Makes It Right for Precise Angles
Here is the logic that ties the whole topic together.
If your angles are the more precise measurements, you want the adjustment to disturb them as little as possible — they are your good data. Let the corrections fall on the lengths instead, which are less reliable anyway.
That is exactly what the transit rule does. Conversely, Bowditch changes angles more and lengths less, which suits a traverse where both were measured with equal care and neither deserves protection.
An Important Consequence
If a line has zero latitude (or departure), there will be no correction to the latitude (or departure). In other words, if a line runs parallel to one of the coordinate axes, adjustment will not alter the bearing of that line. However, if a line is inclined to the coordinate axes, adjustment will affect the bearing.
This follows directly from the formula. Since the correction is proportional to the line’s latitude, a line with zero latitude gets zero correction — the proportion is zero.
Physically: a line running due east has no north–south component at all, so there is nothing in that direction to correct. Its bearing therefore survives adjustment untouched, which is a genuine advantage when a particular line’s direction matters.
5. Axis Method
In the axis method, corrections are applied only to lengths. This method is used to balance a traverse in which the angles are measured very precisely whereas the distances are not measured so precisely.
The Logical Extreme
The axis method takes the transit rule’s principle to its conclusion. If the angles are very precise, then do not touch them at all — apply every bit of correction to the lengths.
The three methods thus form a clear progression:
| Method | Angles Changed | Lengths Changed | Use When |
|---|---|---|---|
| Bowditch | More | Less | Angles and distances equally precise |
| Transit | Less | More | Angles more precise than distances |
| Axis | Not at all | Only lengths | Angles very precise, distances not |
Read the “angles changed” column downwards — more, less, not at all — as angular precision improves. The rule is simply: protect whatever you measured best.
All Five Methods Summarised
| Method | Basis | When Used |
|---|---|---|
| Arbitrary | Surveyor’s discretion from field conditions | Where specific lines were known to be difficult |
| Bowditch (compass rule) | Correction ∝ length / perimeter; assumes accidental errors, e ∝ √L | Angles and distances of equal precision |
| Graphical | Based on Bowditch’s rule | Compass traverse, or low-accuracy theodolite traverse |
| Transit | Correction ∝ latitude / arithmetic sum of latitudes | Angles more precise than distances |
| Axis | Corrections applied only to lengths | Angles very precise, distances not |
Quick Revision Notes
- Adjustment is done only if the closing error is within permissible limits; its purpose is to reduce the error of closure to zero.
- Five methods: arbitrary, Bowditch’s rule, graphical, transit rule, axis method.
- Bowditch’s rule is also called the compass rule and suits equal precision in angles and distances.
- Bowditch assumes errors are accidental (random), with e ∝ √L.
- Bowditch correction = −e × (length of line / perimeter).
- Bowditch changes lengths less and angles more.
- The graphical method is based on Bowditch’s rule and is used for compass traverses.
- Transit rule correction = −ey × (numerical latitude / arithmetic sum of all latitudes).
- The transit rule is used when angular measurements are more precise than linear, and changes angles less, lengths more.
- Under the transit rule, a line with zero latitude or departure receives no correction, so a line parallel to a coordinate axis keeps its bearing.
- The axis method applies corrections only to lengths, for traverses with very precise angles.
Mistakes Students Commonly Make
- Swapping Bowditch and transit. Bowditch uses length/perimeter; transit uses latitude/arithmetic sum of latitudes.
- Getting the “changes more” statements backwards. Bowditch changes angles more; transit changes lengths more.
- Using the algebraic sum in the transit rule denominator. It must be the arithmetic sum of magnitudes.
- Forgetting that Bowditch’s rule is also called the compass rule.
- Stating the Bowditch assumption as e ∝ L. It is e ∝ √L, the random error law.
- Adjusting a traverse whose closing error exceeds permissible limits. That indicates a mistake, which must be found rather than distributed.
- Saying the axis method corrects both lengths and angles. It corrects lengths only.
- Overlooking that under the transit rule a line parallel to a coordinate axis keeps its bearing unchanged.
Conclusion
Adjustment spreads the closing error until the traverse closes, but only when that error is small enough to be genuine random error rather than a blunder. Which method to use follows one principle — protect whatever you measured best. Bowditch’s rule distributes by length and assumes random errors growing as √L, suiting a traverse where angles and distances were equally careful. The transit rule distributes by latitude and departure, leaving precise angles largely alone. And the axis method, correcting lengths only, protects very precise angles completely.
Frequently Asked Questions
What is the purpose of adjusting a traverse?
To reduce the error of closure to zero by distributing it among the sides, so that the traverse closes geometrically. It is done only if the error of closure is within permissible limits.
What are the five methods of traverse adjustment?
The arbitrary method, Bowditch’s rule, the graphical method, the transit rule and the axis method.
What is Bowditch’s rule also called?
The compass rule.
When is Bowditch’s rule used?
For adjusting a traverse in which the angles and distances are measured with the same precision.
What assumption does Bowditch’s rule make?
That the errors introduced are accidental or random in nature, and that the probable error in a traverse line is directly proportional to the square root of its length.
What is the Bowditch correction formula?
Correction to latitude or departure of any line equals the total correction multiplied by the ratio of the length of that line to the perimeter of the traverse.
When is the transit rule used?
To balance a traverse in which the angular measurements are more precise than the linear measurements.
What is the transit rule formula?
Correction to the latitude of any line equals the total correction in latitude multiplied by the ratio of the numerical value of that line’s latitude to the arithmetic sum of all latitudes.
How do Bowditch and transit differ in their effect?
Bowditch’s rule changes the lengths less and the angles more, while the transit rule changes the angles less and the lengths more.
What happens to a line with zero latitude under the transit rule?
It receives no correction to its latitude, so a line running parallel to a coordinate axis keeps its bearing unchanged by the adjustment.
What is the axis method?
A method in which corrections are applied only to lengths, used to balance a traverse in which the angles are measured very precisely but the distances are not.