Dead reckoning is arithmetic with a plotting sheet: you know where you started, you know your course and speed, and you know how long. $\Delta\varphi = \dfrac{S\,t\cos C}{R}$ — delta phi equals S t cos C, over R.

$S$ is the speed over the ground in knots, $t$ the elapsed time in hours, $C$ the true course clockwise from north, and $\Delta\varphi$ the difference of latitude. $S\,t$ is the distance run; the cosine of the course splits it into the north-south component, and $S\,t\sin C$ is the east-west component, which navigators call the departure.

This chapter's marine and airborne lessons run in knots and nautical miles, and not out of nostalgia. One nautical mile is one minute of latitude, by definition. So a run of 24 nmi due north moves you 24 minutes of latitude, read straight off the side of the chart with dividers and no conversion whatsoever. Rewrite the plot in kilometres and you have thrown that away for nothing.

Two traps live here. Departure is not difference of longitude: to convert it you divide by the cosine of the middle latitude, and skipping that is what puts a high-latitude plot badly out. And a DR position is a reckoning, not a fix — it knows nothing about the current setting you sideways or the leeway the wind is making.

Which is where the next relation comes in. Take a real fix, compare it with the DR, and the gap is everything the reckoning missed: $D_r = \dfrac{\sqrt{\Delta n^{2} + \Delta e^{2}}}{t}$. $\Delta n$ and $\Delta e$ are the northing and easting from the DR position to the fix in nautical miles, $t$ is the hours the DR ran, and $D_r$ is the drift — the current's speed in knots. Its direction is the set, and note the naming convention: a current is named for where it flows TO, a wind for where it comes FROM. Two cautions: this vector is everything the DR did not account for, not the tidal stream alone, and it is an AVERAGE over the whole interval, so a stream that turned halfway through flatters the plot badly.

Finally, the number a plotter puts on its deviation bar: cross track error, $e_{xt} = R\arcsin\!\left(\sin\dfrac{d_{13}}{R}\sin\left(\theta_{13}-\theta_{12}\right)\right)$. $d_{13}$ is the distance run from the start of the track to where you actually are, $\theta_{12}$ is the bearing of the intended track, $\theta_{13}$ the bearing to your actual position, and $e_{xt}$ is the perpendicular offset — positive right of track, negative left. On any leg short of a few hundred miles it collapses to distance times the sine of the angle, which is the airman's 1-in-60 rule in disguise: one degree off course puts you one mile off after sixty.