Earth, Time, Direction and Magnetism
Earth shape and distance
The Earth is modelled as an oblate spheroid flattened at the poles; the geoid is the equipotential surface used for elevation. A nautical mile is one minute of arc of a great circle, 1852 m, so a degree of latitude is 60 nm. A great circle is the shortest path between two points and its direction changes continuously; a rhumb line keeps a constant track but is longer except on the equator or a meridian.
Departure and convergency
Departure (nm) = change of longitude in minutes × cos(latitude). Convergency, the angle between meridian directions, = change of longitude × sin(mean latitude). Both use latitude; the commonest error is to swap sine and cosine.
Time
The Earth turns 15° per hour, 1° in 4 minutes, 1 minute of arc in 4 seconds. UTC is the reference; local mean time differs by 4 minutes per degree of longitude, places to the east being later. Standard meridians every 15° define zone time, and the date changes when crossing the International Date Line, eastbound subtracting a day.
Direction and magnetism
True north is the chart datum. Variation is the angle between true and magnetic north; deviation is between magnetic and compass north. Isogonals join points of equal variation, and the agonic line joins zero variation. To go from true to magnetic, easterly variation is subtracted (variation east, magnetic least); westerly is added. Compass deviation is read from the calibration card for that heading.
Worked example: two points on 45°N separated by 4° of longitude have departure = 240 × cos 45°, about 170 nm, and convergency between their meridians = 4° × sin 45°, about 2.8°.
Traps: confusing variation with deviation, using the wrong sign, and forgetting that departure uses cos latitude while convergency uses sin latitude.