Gale's table, coordinates of all points

The bearings are given directly, so the Gale's table starts from the latitudes and departures. Bowditch's rule is used to balance the traverse, and the coordinates are then run from C (1000 N, 1500 E).

Step 1: Latitude and departure

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

Step 2: Closing error

$$\begin{aligned} e_N&=\Sigma\text{Lat}=-0.117\ \text{m},\qquad e_E=\Sigma\text{Dep}=+0.346\ \text{m}\\ e&=\sqrt{e_N^2+e_E^2}=0.3657\ \text{m}\\ \text{Relative error}&=\frac{e}{\Sigma L}=\frac{0.3657}{559.50}=\frac{1}{1530} \end{aligned}$$

Step 3: Bowditch correction

$$C_{lat}=-e_N\frac{L}{\Sigma L},\qquad C_{dep}=-e_E\frac{L}{\Sigma L}$$

The corrections are in the last two columns above (their sums cancel the errors). Adjusted values:

Step 4: Coordinates of all points

Starting from C = (1000.000 N, 1500.000 E), add the adjusted latitude and departure of CD, DE, EA and AB in turn:

Check: B plus the adjusted BC returns exactly to C (1000.000 N, 1500.000 E).

Answer: A (972.637 N, 1333.995 E), B (1030.036 N, 1367.756 E), C (1000.000 N, 1500.000 E), D (993.467 N, 1565.935 E), E (920.688 N, 1542.047 E); misclosure 0.3657 m, 1 in 1530.