Which measurement problem did Harrison's marine chronometers finally solve at sea?

Longitude. Latitude is readable from the Sun or the pole star; longitude requires knowing what time it is somewhere else at the moment you take your local noon, and every hour of difference is fifteen degrees. So the problem was a clock that kept time on a rolling, damp, temperature-swinging deck - and pendulums are useless at sea.

Parliament offered £20,000 in 1714 after a naval disaster off the Scilly Isles killed some 1,400 men. John Harrison, a self-taught Lincolnshire carpenter, spent decades on it: three large sea clocks with counter-oscillating balances and bimetallic strips to cancel temperature effects, then an abrupt change of direction to H4, a watch of 13 centimetres that lost about five seconds on a voyage to Jamaica.

The Board of Longitude, dominated by astronomers pursuing a rival lunar-distance method, kept moving the requirements - demanding repeat trials, the surrender of his machines, and the construction of duplicates by other hands. He was eventually paid most of the money after petitioning the king, at 79. The lunar method was not silly, incidentally, and remained in use as a cheaper backup for decades, because a chronometer that stopped left you with nothing.