The fundamental principle is simple to state: the positions of celestial bodies — the sun, moon, planets, and stars — in the sky are predictable. Given a nautical almanac (which tabulates the calculated positions of celestial bodies for each day and hour of the year), a sextant to measure the angle between a celestial body and the visible horizon, and an accurate clock to record the precise moment of the measurement, a navigator can calculate their position on Earth’s surface.
The practical execution of this principle involves more complexity. A sextant measurement must be made at a specific moment — typically local apparent noon for a sun sight, when the sun is at its highest point in the sky — and the measurement must be recorded along with the precise time to within a few seconds. Even small errors in the time record translate into substantial errors in the calculated longitude. The calculation itself requires a sequence of table lookups, arithmetic operations, and graphical plotting on a nautical chart.
Oldham had the sextant. She had a watch, found working in the bilge. She had nautical charts of the Pacific. She had the nautical almanac and the sight reduction tables that the Hazana had been equipped with as standard navigation tools.
What she reconstructed, working from the sun’s noon altitude each day to determine latitude, and from the difference between the sun’s calculated Greenwich Hour Angle and her watch-derived time to determine longitude, was a daily position. Each day at noon she took a sun sight and worked through the mathematics. Each position gave her a point on the chart. The sequence of points told her whether she was making progress toward Hawaii and whether she was on a track that would bring her close enough to find it.