Bowditch (compass) rule—departure corrections: For a closed traverse of perimeter L with closing error in departure ΔD, what is the correction to the departure of a side of length l according to Bowditch’s rule?

Introduction / Context:

Traverse adjustment distributes the misclosure among sides in proportion to their “weights.” Bowditch’s rule (the compass rule) assumes that linear and angular errors are proportional to traverse lengths, making corrections to departures and latitudes proportional to side lengths.

Given Data / Assumptions:

• Closed traverse with computed total departure misclosure ΔD (ΣD ≠ 0) and latitude misclosure ΔL (ΣL ≠ 0).
• Perimeter (total traverse length) = L; side length under correction = l.
• Ordinary plane surveying precision; no rigorous least-squares adjustment.

Concept / Approach:

Bowditch’s rule prescribes: Correction to departure of a side = −(l/L) × ΔD. Similarly, correction to latitude of a side = −(l/L) × ΔL. The negative sign indicates the correction is applied opposite to the sense of the misclosure so that the adjusted sums become zero.

Step-by-Step Solution:

Verification / Alternative check:

Check that ΣC_D = −ΔD and ΣC_L = −ΔL; if satisfied, the adjustment is internally consistent.

Why Other Options Are Wrong:

• (L/l)ΔD and −(l²/L²)ΔD: not part of Bowditch proportionality.
• Same-sign correction: would worsen the misclosure.
• Equal corrections: ignores length weighting.

Common Pitfalls:

• Forgetting the negative sign, leading to increased misclosure.
• Applying corrections to bearings instead of departures/latitudes.

Final Answer:

−(l/L) × ΔD