---
title: "Closed/open traverse, Bowditch's rule"
subject: "Surveying II"
chapter: 1
key: "semester_4/surveying_ii:closed-open-traverse-bowditch-rule"
source: https://www.bctengineer.com/q/semester_4/surveying_ii/closed-open-traverse-bowditch-rule
site: BCT Engineering
---

# Closed/open traverse, Bowditch's rule

Surveying II · Chapter 1: Traversing

- Asked in: 2065 Chaitra (old course), 7 marks
- Video: [Balance the traverse by using Bowditch rule | Traverse adjustment | Latitude Departure Closing error](https://www.youtube.com/watch?v=Jkb8qO98QZA) — SBK Concept
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/surveying_ii/closed-open-traverse-bowditch-rule
- Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-1

## Question

What are closed and open traverses? Explain Bowditch's rule for balancing the traverse.

## Short answer

A closed traverse ends at the starting point or on known stations, providing a mathematical check; an open traverse ends at an unknown point. Bowditch's rule distributes the closing error in latitude and departure among lines in proportion to their lengths, assuming equal precision and accidental errors only.

## Answer

#### Closed and open traverses
- **Closed traverse:** a traverse that ends where it began (a *closed loop*) or that starts and ends on stations of known position (a *closed link/connecting traverse*). It gives a mathematical check on the field work.
- **Open traverse:** a traverse that starts at a known station and ends at a point of unknown position, so there is no closing check. It is used for roads, canals, pipelines and similar route surveys, and the angles and lengths must be checked by repeat measurement.

#### Bowditch's rule
**Bowditch (compass) rule:** the closing error in latitude and departure is distributed among the lines in proportion to their lengths.

$$
C_L=-\Sigma L_{at}\times\frac{l}{\Sigma l},\qquad C_D=-\Sigma D_{ep}\times\frac{l}{\Sigma l}
$$

where $l$ is the length of the line and $\Sigma l$ the perimeter (or total length of the traverse).

**Assumptions:**
- Angular and linear measurements are of equal precision, and the errors are accidental (random) only.
- The error in the linear measurement of a line is proportional to $\sqrt{l}$, and the error in angular measurement is inversely proportional to $\sqrt{l}$. On these assumptions, the correction to the latitude and departure of each line works out to be proportional to its length.
- The angular misclosure has already been distributed among the angles before the rule is applied.

It is simple, and it is the most commonly used rule in plane surveying.

#### Example
A four-sided closed traverse has ΣL = 494.58 m, with the computed latitudes and departures below. The errors are $e_N=\Sigma\text{Lat}=+0.26$ m and $e_E=\Sigma\text{Dep}=-0.13$ m, so $e=0.291$ m and the precision is $1:1701$.

| Line | L (m) | Lat | Dep | Corr. lat | Corr. dep | Adj. lat | Adj. dep |
|---|---|---|---|---|---|---|---|
| AB | 99.97 | +85.20 | +52.30 | -0.053 | +0.026 | +85.15 | +52.33 |
| BC | 149.85 | +30.10 | +146.80 | -0.079 | +0.039 | +30.02 | +146.84 |
| CD | 119.96 | -92.40 | -76.50 | -0.063 | +0.032 | -92.46 | -76.47 |
| DA | 124.80 | -22.64 | -122.73 | -0.066 | +0.033 | -22.71 | -122.70 |

Check: the corrections add up to $-e_N$ and $-e_E$, so the corrected latitudes and departures sum to zero. For line AB: $C_{lat}=-(+0.26)\times99.97/494.58=-0.053$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which is a closed traverse?
   A. One ending at an unknown point
   B. One that starts and ends at the same station, or at stations of known position
   C. One using only chain and tape
   D. One without any angle measurements
   Answer: B. A closed loop or a closed link gives a mathematical check on the field work. (Key point: Closed traverse)

2. Bowditch's rule distributes the latitude and departure errors in proportion to:
   A. Each line's latitude
   B. The angle at the station
   C. Each line's departure
   D. Each line's length
   Answer: D. C_L = -sum Lat x l / sum l; the correction is proportional to line length. (Key point: Bowditch rule)

3. e_N = +0.26 m, sum L = 494.58 m, AB = 99.97 m. Correction to AB latitude?
   A. -0.053 m
   B. +0.053 m
   C. -0.260 m
   D. -0.105 m
   Answer: A. C = -0.26 x 99.97/494.58 = -0.053 m. (Key point: Correction calculation)

4. After applying Bowditch corrections, the check is that:
   A. The corrections are all positive
   B. Corrected latitudes and departures sum to zero
   C. The perimeter is unchanged and equals zero
   D. All lines have equal correction
   Answer: B. The corrections sum to -e_N and -e_E, so the corrected lat and dep sums vanish. (Key point: Check after balancing)

## Close variants set in other exams

- [Closed/open traverse, transit rule](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aclosed-open-traverse-transit-rule) — Surveying II, chapter 1, asked 2×

## Related in Surveying II

- [Bowditch balancing of link traverse XABCY](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Abowditch-link-traverse-xabcy) — Surveying II, chapter 1, asked 2×
- [Balancing linked traverse, transit rule](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Alinked-traverse-misclosure-transit-rule) — Surveying II, chapter 1, asked 2×
- [Field measurements in traversing, Bowditch rule](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atraverse-field-measurements-bowditch-rule) — Surveying II, chapter 1
- [Need for traversing, types, Bowditch rule](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atraversing-necessity-types-bowditch) — Surveying II, chapter 1
- [Closed loop/link traverse, misclosure balancing](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aclosed-loop-link-traverse-angular-misclosure) — Surveying II, chapter 1
- [Field measurements in traversing, closed/open traverse](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atraverse-field-measurements-closed-open) — Surveying II, chapter 1

## Related in other subjects

- [Checking open traverse; Bowditch graphically](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aopen-traverse-check-bowditch) — Surveying I, chapter 4
- [Open traverse ABCDEFG, deflection angles](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aopen-traverse-deflection-angles-40-45) — Surveying I, chapter 4, asked 2×
- [Open traverse ABCDEFGH, deflection angles](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aopen-traverse-deflection-angles-81-45) — Surveying I, chapter 4, asked 2×
- [Open loop versus closed loop control](https://www.bctengineer.com/question.md?key=semester_6%2Fembedded_system%3Aopen-vs-closed-loop) — Embedded System, chapter 7, asked 6×