Check Gale's table ABCDEA, D at 2500 N

Method: check and adjust the angles, find the bearings from the known bearing, compute latitudes and departures, balance them by Bowditch's rule and compute the coordinates from the given station.

Step 1: Angular check and adjustment

For a closed traverse of $n=5$ sides, the sum of interior angles must be $(2n-4)\times 90^\circ=540^\circ$.

Observed sum = 540°00', so there is no angular error and no correction is needed.

Step 2: Bearings

The traverse is taken as clockwise (interior angles on the right), so

$$\text{Bearing of next line}=\text{Bearing of previous line}+180^\circ-\text{interior angle at the station}$$

Starting from AB = 40°00':

Check: carrying the bearing round the traverse brings back AB = 40°00', as given.

Step 3: Latitudes, departures and closing error

$$\text{Lat}=L\cos\theta,\qquad \text{Dep}=L\sin\theta$$

Closing error: $e_N=-0.054$ m, $e_E=-0.034$ m,

$$e=\sqrt{e_N^2+e_E^2}=0.0637\ \text{m},\qquad \text{relative error}=\frac{e}{\Sigma L}=\frac{0.0637}{250.37}=\frac{1}{3928}$$

This is well within the usual limit for theodolite traversing (about 1 in 3000 or better), so the work is acceptable.

Step 4: Bowditch correction and adjusted lines

Correction to a line $=-\text{(total error)}\times L/\Sigma L$ (tabulated above). The corrected latitudes and departures give the final length $=\sqrt{\text{Lat}^2+\text{Dep}^2}$ and bearing $=\tan^{-1}(\text{Dep}/\text{Lat})$:

Step 5: Coordinates (D = 2500.000 N, 4500.000 E)

Starting from D and adding the adjusted latitudes and departures in the order DE, EA, AB, BC, CD:

Check: C plus line CD returns to D (2500.000 N, 4500.000 E).

Answer: relative error 1 in 3928 (accurate work); coordinates A (2550.086 N, 4442.585 E), B (2586.276 N, 4472.950 E), C (2566.404 N, 4510.530 E), D (2500.000 N, 4500.000 E), E (2506.894 N, 4451.129 E).