How to Adjust a Closed Traverse by the Bowditch Rule
Last updated 24 October 2026
A worked example of a four-station closed loop: balance the angles, work out bearings and coordinate changes, find the misclosure and spread it by the Bowditch (compass) rule.
What the Bowditch rule does
A closed traverse should end exactly where it started, but small errors in every angle and distance mean it never quite does. The Bowditch rule (also called the compass rule) spreads the leftover gap, the misclosure, over the legs in proportion to their length: a long leg takes a bigger share of the correction than a short one. It suits traverses where the angles and distances are of similar quality, which is most total station work.
The example: a four-station loop
Station A is known at E 1000.000, N 2000.000 and the bearing A → B is fixed at 060°00'00" from control. The loop runs clockwise A → B → C → D → A. The interior angle was observed at each station and every leg measured (horizontal distances, in metres):
| Station | Observed interior angle | Leg | Distance (m) |
|---|---|---|---|
| B | 092°45'18" | A → B | 120.454 |
| C | 088°30'37" | B → C | 98.314 |
| D | 091°45'32" | C → D | 119.875 |
| A | 086°58'13" | D → A | 101.000 |
| Sum | 359°59'40" | Perimeter | 439.643 |
Step 1: check and balance the angles
The interior angles of a closed figure with n sides add up to (n − 2) × 180°. Four sides make 360°, and the observed angles add up to 359°59'40": an angular misclosure of −20".
Compare it with what your specification allows before going on. A common limit is ±C√n seconds, with C set by the order of the work (for example 15", giving ±30" for four angles). This one passes, so the 20" is shared equally: +5" on each angle. The balanced angles are 092°45'23", 088°30'42", 091°45'37" and 086°58'18", which add up to exactly 360°.
Step 2: work out the bearings
Walking round clockwise, each new bearing is the previous bearing + 180° − the interior angle at the station you are on (take off 360° if the answer is more than 360°). From A → B = 060°00'00":
- B → C = 060°00'00" + 180° − 092°45'23" = 147°14'37"
- C → D = 147°14'37" + 180° − 088°30'42" = 238°43'55"
- D → A = 238°43'55" + 180° − 091°45'37" = 326°58'18"
- and back at A: 326°58'18" + 180° − 086°58'18" = 420°00'00" − 360° = 060°00'00", the bearing you started with.
That last line is the check that the angles were balanced and the bearings carried correctly.
Step 3: the coordinate changes and the misclosure
For each leg, ΔE = distance × sin(bearing) and ΔN = distance × cos(bearing). Round a closed loop they should add up to zero:
| Leg | Bearing | Distance | ΔE | ΔN |
|---|---|---|---|---|
| A → B | 060°00'00" | 120.454 | +104.3162 | +60.2270 |
| B → C | 147°14'37" | 98.314 | +53.1946 | −82.6800 |
| C → D | 238°43'55" | 119.875 | −102.4630 | −62.2202 |
| D → A | 326°58'18" | 101.000 | −55.0504 | +84.6785 |
| Sum | 439.643 | −0.0026 | +0.0053 |
The traverse misses its start by −2.6 mm in eastings and +5.3 mm in northings. The linear misclosure is √(2.6² + 5.3²) = 5.9 mm, and the precision is perimeter ÷ misclosure = 439.643 ÷ 0.0059, about 1 : 74,600. If the job asks for 1 : 10,000 or better, it passes. (See traverse misclosure and precision for what to do when it doesn't.)
Step 4: spread the misclosure by the Bowditch rule
The correction to each leg is the misclosure with its sign turned round, times that leg's share of the perimeter:
- correction to ΔE = −(misclosure in E) × leg length ÷ perimeter
- correction to ΔN = −(misclosure in N) × leg length ÷ perimeter
For A → B: +2.6 mm × 120.454 ÷ 439.643 = +0.7 mm in E, and −5.3 mm × 120.454 ÷ 439.643 = −1.5 mm in N. Doing every leg and adding the corrected changes on from A gives:
| Leg | Correction E (mm) | Correction N (mm) | To station | Adjusted E | Adjusted N |
|---|---|---|---|---|---|
| A → B | +0.7 | −1.5 | B | 1104.317 | 2060.226 |
| B → C | +0.6 | −1.2 | C | 1157.512 | 1977.544 |
| C → D | +0.7 | −1.4 | D | 1055.050 | 1915.323 |
| D → A | +0.6 | −1.2 | A | 1000.000 | 2000.000 |
| Sum | +2.6 | −5.3 |
Bowditch, Transit or least squares?
- Bowditch spreads the error by leg length. Use it when angles and distances are about equally good.
- Transit spreads it by the size of each ΔE and ΔN, so it changes bearings less. Use it when the angles are better than the distances.
- Least squares weighs every angle and distance by its standard error and gives residuals and a statistical test: the most rigorous choice for control.
Doing it without the arithmetic
Traverse & Level does every step above as you type: the angular check, the bearings, the misclosure, the precision against your requirement, and the adjusted coordinates by Bowditch, Transit, Equal or least squares, with Accepted or Rejected straight away. Save the job to your account, print the report with Surveyed, Checked and Approved boxes, and share it with your checker. For a quick one-off, the free traverse adjustment calculator works in the browser with nothing to sign in to.
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