Every navigation problem splits into a direction and a distance. The compass handles the direction. The distance comes from one of two places — the paper, or your own legs — and a working field engineer checks each against the other.

From the paper. $d = m\,S$, read aloud d equals m times S. Here $m$ is the length you measure on the sheet with a ruler, $S$ is the scale denominator — the 50,000 in 1:50,000, a bare number with no unit at all — and $d$ is the real distance on the ground, in whatever unit $m$ was measured in. That last part is the trick: a representative fraction is a pure ratio, so one of ANYTHING on the sheet is $S$ of the same thing on the ground. Measure in centimetres, get centimetres. The only work left is the bridge to kilometres, and there are 100,000 cm in a kilometre — so on a 1:50,000 sheet one centimetre is half a kilometre, and on a 1:25,000 sheet it is a quarter. For a winding trail, lay a damp string along the route and straighten it against the ruler; the string is the poor navigator's measuring wheel, and it works.

From your legs. $d = n\,L$ — d equals n times L. $n$ is the count of paces, a bare integer, and $L$ is your calibrated pace length in metres. Calibrate it by walking a taped 100 m and dividing 100 by your count. Most navigators count only the left foot — a "double pace" of about 1.5 m — because sixty-odd is an easier running total to hold than a hundred and thirty.

The nugget that matters on a bad day: the count is honest and the length is what drifts. Your pace shortens uphill, in the dark, in snow, in bog, and under a heavy pack — by ten or fifteen percent, easily. Recalibrate for the terrain you are actually on, and treat somebody else's pace length as a rumour.

And the doctrine that runs through this whole chapter: units guide, they do not confess. Getting centimetres and kilometres to cancel proves you have not made THAT mistake. It cannot prove you read the right scale bar.