The Rotating Cogwheel

Hippolyte Fizeau, Léon Foucault, and the Terrestrial Speed of Light

From the bell tower of his father's house in the western Paris suburb of Suresnes in July 1849, the French physicist Hippolyte Fizeau ignited an oxyhydrogen limelight lamp and focused its brilliant white beam through a telescope objective pointed east across the dark Seine valley. Five miles away, high on the hill of Montmartre at a measured distance of 8,633 meters, a second telescope caught the incoming light and bounced it straight back along the identical atmospheric path using a spherical return mirror at its focal plane. Between the lamp and the eyepiece in Suresnes, Fizeau spun a small brass disk four centimeters in diameter whose outer rim had been milled by the master Parisian horologist Paul Gustave Froment into 720 square-cut teeth separated by 720 identical gaps.

For two centuries before Fizeau's experiment, measuring the speed of light inside a terrestrial setup had seemed physically impossible. Galileo Galilei had tried in 1638 by stationing two observers with shuttered lanterns on Tuscan hilltops less than a mile apart, discovering only that human nerve-and-muscle reaction time—roughly two-tenths of a second—was tens of thousands of times slower than the transit of the light beam. The only successful measurements had relied on the solar system itself as a laboratory: in 1676, Ole Rømer timed the sixteen-minute cumulative lag in the eclipses of Jupiter's moon Io as Earth orbited toward and away from Jupiter, and in 1728 James Bradley measured the twenty-arcsecond stellar aberration angle caused by Earth's orbital velocity. Both astronomical methods required knowing the exact radius of Earth's orbit, which was itself uncertain by several percent.

Across Fizeau's 8,633-meter baseline between Suresnes and Montmartre, the round-trip flight path $2D$ totaled 17,266 meters. Travelling at roughly 300,000 kilometers per second, a pulse of light completed that journey in just 57.5 microseconds. No mechanical clock or electrical chronograph in 1849 could start and stop across fifty-seven millionths of a second.

Fizeau bypassed single-shot timing by converting a microsecond time interval into a continuous mechanical frequency. As an outgoing beam of light passed through one of the 720 open slots around the rim of his brass wheel, it raced to Montmartre, reflected off the distant mirror, and arrived back at the wheel 57.5 microseconds later. When the wheel sat stationary or spun slowly, the returning light passed cleanly back through the same open slot it had exited, appearing in Fizeau's eyepiece as a bright starlike point alongside the shadowy silhouettes of the gear teeth.

Driven by a falling weight and a train of clockwork step-up gears calibrated by an acoustic siren and revolution counter, Fizeau gradually accelerated the brass cogwheel. As rotational frequency $f$ climbed to exactly 12.6 revolutions per second, the wheel rotated by $1/1,440$th of a full circle—half the angular pitch of one tooth-and-gap pair—during the 57.5 microseconds the light spent in the air over Paris. Every pulse of light that escaped through a gap arrived back at Suresnes to find a solid brass tooth sitting squarely in its path, and the star in the eyepiece vanished into darkness. Doubling the wheel's speed to 25.2 revolutions per second brought the next open gap into position, restoring the bright star; tripling it to 37.8 revolutions per second produced a second eclipse. From the distance $D$, the number of teeth $N = 720$, and the first-eclipse rotational frequency $f$, the speed of light $c$ followed directly from the kinematic relation:

Plugging in $N = 720$, $D = 8,633\text{ m}$, and $f = 12.6\text{ Hz}$, Fizeau calculated a terrestrial light speed of 315,300 kilometers per second—roughly five percent too high because judging the exact center of minimum brightness across a flickering five-mile air path was difficult.

His former collaborator and rival, Jean Bernard Léon Foucault, immediately saw how to fold Fizeau's ten-mile outdoor flight path into a single laboratory room. Instead of chopping the beam with a toothed wheel, Foucault bounced a narrow pencil of sunlight off a tiny flat glass mirror driven by a high-pressure steam-and-air turbine devised by Charles Cagniard de la Tour. In April 1850, spinning his mirror at up to 800 revolutions per second, Foucault sent light through a three-meter water tube alongside an open air path. During the twenty nanoseconds the light took to traverse the room and bounce back, the turbine mirror rotated by a small angle $\theta$, deflecting the returning image sideways on a micrometer eyepiece scale. The water beam deflected further than the air beam, proving definitively that light travels slower in water than in air and refuting Newton's corpuscular emission theory, which had predicted the exact opposite. In 1862, using a folded twenty-meter optical path inside a single Paris room and an air-driven mirror spinning at four hundred revolutions per second, Foucault measured the speed of light at 298,000 kilometers per second—within 0.6 percent of the modern constant.