A traverse is a chain of measured courses round a parcel. Each course carries a length and a direction, and the whole method rests on turning that pair into two components that can simply be added up.

Directions are booked as azimuths: $\alpha$ — alpha — measured clockwise from north, from 0° to 360°. Its reverse is the back azimuth, $\alpha_b = \alpha \pm 180^{\circ}$, wrapped back into that range. Same line, looked along from the other end — and it is what the instrument is set to when the crew sights back to orient at a new station.

Each course is then split. The latitude is the north–south component: $\text{Lat} = L\cos\alpha$. The departure is the east–west component: $\text{Dep} = L\sin\alpha$. $L$ is the horizontal course length in metres and $\alpha$ its azimuth. Cosine for latitude, sine for departure — and the reason is that azimuths are measured from NORTH, so a course pointing due north is all latitude, and $\cos 0 = 1$. Swap the two and every plot closes beautifully in the wrong place. The signs are not decoration either: they carry south and west.

Round a closed loop the latitudes must sum to zero and so must the departures. They never quite do. The residuals $\Sigma\text{Lat}$ and $\Sigma\text{Dep}$ stand at right angles, so the gap between where the crew finished and the pin they started from is the hypotenuse: $E_c = \sqrt{(\Sigma\text{Lat})^{2} + (\Sigma\text{Dep})^{2}}$, the linear closure error in metres.

Then the number that actually gets reported. $N = \dfrac{P}{E_c}$, where $P$ is the total perimeter walked and $N$ the precision denominator, written 1:N. A 0.20 m miss round a 1,200 m loop is 1:6,000 — good boundary work. The same 0.20 m round a 200 m loop is 1:1,000, and would be rejected. That is the whole argument for a ratio: error only means something next to the distance it accumulated over.

And read the ratio the right way round. $N$ is a denominator. One part in eight thousand is finer than one part in three thousand, so a bigger N is a better traverse. It reads backwards to the eye, and mistaking it is how a rejected loop gets signed off.