The Bimetallic Balance

John Harrison, the H4 Marine Watch, and the Longitude Prize

Through the autumn gales of November 1761, a silver watch measuring 5.2 inches in diameter ticked inside a four-locked mahogany box aboard the sixty-gun fourth-rate warship HMS Deptford as it beat southward toward Madeira. Four separate keys were held by Captain Dudley Digges, his first lieutenant, the governor-designate of Jamaica, and William Harrison, son of the self-taught Lincolnshire carpenter John Harrison who had spent thirty-one years in London building sea timekeepers. By the nineteenth day out from Spithead, dead-reckoning logs compiled from knotted lines, compasses, and hourglasses placed the warship at $13^\circ 50'$ West longitude. Harrison's watch, still keeping Portsmouth meridian time, indicated $15^\circ 19'$ West—nearly ninety nautical miles farther out to sea. Digges offered to buy the watch on the spot if Madeira appeared on schedule. At six o'clock the next morning, the lookout sighted Porto Santo dead ahead across the bow.

Determining latitude at sea requires only a vertical angle sighting of the noon Sun or the North Star, because latitude lines are fixed geometric parallels. Longitude has no natural celestial zero point because the Earth spins continuously at $360^\circ$ every twenty-four hours, or $15^\circ$ per hour. At the equator, one degree spans sixty nautical miles, meaning every four seconds of clock error translates to one nautical mile of positional drift. After Admiral Sir Cloudesley Shovell's flagship HMS Association and three warships struck the Western Rocks off the Isles of Scilly in October 1707, drowning roughly two thousand sailors in a single night, Parliament passed the Longitude Act of 1714. The statute offered £20,000 to anyone who could determine longitude across a six-week West Indies voyage to within half a degree (30 nautical miles), requiring a clock that drifted less than 2.8 seconds per day.

On dry land, pendulum clocks achieved that accuracy because their period depends on constant gravity $g$. Aboard a wooden ship pitching twenty degrees in ocean swells, effective gravity changes with every wave acceleration, rendering a pendulum useless. A marine clock had to rely instead on a spring-driven torsional oscillator—a circular balance wheel of moment of inertia $I$ coupled to a coiled steel hairspring of torsional stiffness $\kappa$, whose oscillation period $T$ obeys the harmonic relation:

In an ordinary eighteenth-century pocket watch, that relation broke down under sliding friction and temperature. Animal and vegetable oils thickened in North Atlantic spray and turned acidic in Caribbean heat, altering the impulse torque delivered to the balance wheel. Worse was thermal elasticity drift: when cabin temperature rose by $10^\circ\text{C}$, the brass balance wheel expanded outward, increasing $I$, while the steel hairspring lost elastic stiffness $\kappa$. Both effects acted together, slowing a watch by roughly ten seconds per day for every $10^\circ\text{C}$ rise.

After spending twenty-five years building three heavy brass sea clocks (H1, H2, and H3) weighing sixty to one hundred pounds, Harrison realized in 1753 that a small, high-frequency balance wheel recovers its phase stability under ship motion far better than a slow-moving beam. Built with components from watchmaker John Jeffries, H4 beat at five ticks per second ($2.5\text{ Hz}$ full cycles) with a three-inch steel balance wheel moving at high rim velocity.

To defeat friction without animal grease, Harrison crafted his bearings from lignum vitae, a dense tropical hardwood that exudes its own microscopic film of guaiacum resin, and shaped the verge escapement pallets from polished diamond blades. Because a diamond verge escapement is sensitive to variations in mainspring force, Harrison inserted a mechanical remontoire driven by a tiny auxiliary spring that rewound itself every 7.5 seconds, feeding constant torque to the escapement teeth. Finally, to cancel thermal drift in $\kappa$ and $I$, he riveted a thin strip of brass to a strip of steel to form a bimetallic curb finger. Because brass expands 1.6 times faster than steel when heated, rising temperatures bent the composite strip along the outer coil of the hairspring, shortening its active vibrating length to increase $\kappa$ in exact proportion to the thermal softening of the steel.

When HMS Deptford docked at Port Royal, Jamaica, on January 19, 1762, after eighty-one days at sea, H4 had lost just 5.1 seconds after applying its pre-calibrated rate, corresponding to 1.25 nautical miles of error. Despite years of obstruction by Astronomer Royal Nevil Maskelyne, who favored his own lunar-distance tables, King George III intervened in 1773 after testing Harrison's H5 watch and forced Parliament to pay the aged carpenter his remaining prize money. When Captain James Cook returned from his three-year Antarctic circumnavigation aboard HMS Resolution in July 1775 carrying Larcum Kendall's K1 exact copy of the H4 watch, his South Pacific islands were charted to within three nautical miles.