Gale's table for traverse ABCDA

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 $n=4$ sides, the sum of interior angles must be $(2n-4)\times 90^\circ=360^\circ$.

Observed sum = 360°00'05", so the angular error is 0°00'05" (excess). The correction is distributed equally: 1.25" (subtracted) per angle.

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 CD = 314°58'04":

Check: carrying the bearing round the traverse brings back CD = 314°58'04", as given.

Step 3: Latitudes, departures and closing error

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

Closing error: $e_N=-0.090$ m, $e_E=+0.068$ m,

$$e=\sqrt{e_N^2+e_E^2}=0.1127\ \text{m},\qquad \text{relative error}=\frac{e}{\Sigma L}=\frac{0.1127}{239.50}=\frac{1}{2125}$$

Step 4: Bowditch correction and adjusted lines

Correction to a line $=-\text{(total error)}\times L/\Sigma L$. 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 (C = 500.000 N, 500.000 E)

Answer: the final adjusted lengths and bearings are as tabulated above, and the traverse closes with a relative error of 1 in 2125.