An experiment to look at twice
A beam crosses the room and returns before a mirror has turned a hair’s breadth. That tiny turn is enough to measure the speed of light.
Light is so quick that a human reaction is hopelessly slow. Astronomers found a way around that problem: make the journey enormous.
Jupiter’s moon Io disappears into Jupiter’s shadow on a regular schedule. Ole Rømer found that the observed timings shift as Earth’s distance from Jupiter changes. The extra journey takes extra time: light cannot travel instantaneously. This was a measurement of a light-travel delay on the scale of Earth’s orbit. [2]
James Bradley explained the annual aberration of starlight: Earth’s motion changes a star’s apparent direction, rather as moving through falling rain changes the angle at which it meets you. The small angle, together with Earth’s orbital speed, gives the speed of light. A second astronomical route—but still dependent on the scale of the solar system. [3]
Hippolyte Fizeau sent light between Suresnes and Montmartre, about 8.6 km apart. A turning toothed wheel let the beam out through a gap; at the right speed, the next tooth blocked its return. His result was about 315,000 km/s. Now the distance and the timing device were both on Earth. [4]
With N equally spaced teeth and equally wide gaps, a gap gives way to the next tooth in half a tooth pitch. If the wheel turns at f revolutions per second, this takes 1/(2Nf) seconds. At the first extinction, that is the light’s round-trip time.
2L / c = 1 / (2Nf) ⇒ c = 4LNf
The disappearance supplies the timing mark; no one has to react to the light’s arrival. This is the ideal equal-tooth/equal-gap model. In practice, observers compared repeated extinctions and measured the wheel’s speed.
Foucault’s next move was to turn an almost impossibly short time into a small but observable angle.
A narrow illuminated mark is imaged through the optics. A small rotating mirror sends its light toward a fixed concave mirror, which returns it. By the time the beam comes back, the rotating mirror has changed orientation.
These are ray-path schematics, not a scale reconstruction. The source and viewing optics are omitted from the return diagram. Historical lenses and a partially reflecting plate separated the illumination from the observed image.
For a one-way path L, light travels 2L between its two encounters with R. The delay is t = 2L/c.
At f turns per second, the mirror’s angular speed is ω = 2πf. During the delay it turns by θ = ωt.
Reflection doubles the mirror’s rotation: α = 2θ. A measured displacement, with calibrated viewing optics, gives this angle.
Angles in radians · f in revolutions per second · L is one way
A mirror’s normal bisects the angle between incident and reflected ray lines. Hold the incident ray fixed and rotate the normal by θ: the reflected ray must rotate through 2θ to keep those angles equal. There are two different factors of two here: one for the outward-and-back journey, one for reflection.
A flat distant mirror would return the right ray only for a very narrow alignment. A concave reflector centred on the rotating mirror sends a range of outward directions back toward it. With imaging optics, the brief returns form a repeatedly renewed image that can be examined with a microscope. The experiment uses a continuing beam; drawing one light packet is just a way to follow the delay. See the rotating-mirror apparatus manual [5] for a modern optical arrangement.
More than a race
Foucault first used this approach to show that light travels more slowly in water than in air. That contradicted the faster-in-water prediction of the emission theory then under debate. In 1862 he made an absolute measurement of about 298,000 km/s, roughly 0.6% below the modern vacuum value. [6] A room-sized experiment could now challenge an astronomical measurement.
Fizeau watched light fade as his wheel blocked its return. Foucault instead examined a magnified image of a ruled glass target. The museum describes vertical marks spaced 0.1 mm apart. That is their spacing, not their thickness or the diameter of a dot. [1]
Source size, diffraction, imperfect focus, and vibration give every real image some width. The useful feature here was a line’s position against a reference. The sources cited here do not establish a reliable original line width, so there is no defensible single “dot diameter” to quote.
These figures come from Alfred Cornu’s 1900 review. They describe a sub-millimetre displacement inspected with magnification. Cornu stressed its limitations as well as its ingenuity. [11]
The long light path ran from the rotating mirror to the return optics and back. Lenses brought the returning image to a local measuring plane, inspected through an eyepiece near the source. Extra mirrors could fold a long path into a room. The observer did not stand 20 metres away and look for a dot moving on a wall.
Keep three scales separate: the light-travel path, the lens geometry converting angle into displacement, and the eyepiece magnification making the image easier to inspect. Our calculator’s bare-screen distance D substitutes for the lens geometry; it is not the eye’s distance from the apparatus.
An ENS teaching problem uses a simplified Foucault arrangement: a 20 m mirror-to-reflector path, 800 rev/s, a source 0.546 m from the lens, and its conjugate image 21 m from the lens. These are the teaching model’s parameters, not a complete survey of the surviving apparatus. [12]
Using modern c, the round trip takes 133.4 ns and the beam turns through 1.341 mrad (0.07685°). The lens gives an effective lever arm of 20 × 0.546 / 21 = 0.520 m:
s ≈ 0.520 × 0.001341 m
≈ 0.000697 m = 0.697 mm
At 400 rev/s, this geometry gives half as much. The calculator’s default 0.671 mm uses 400 rev/s and a bare screen at D = 1 m: similar numbers, different arrangements.
In this unfolded thin-lens model, let a be source-to-lens distance and b be lens-to-rotating-mirror distance. The far conjugate is b + L from the lens. A small angular change α produces s = [aL/(b + L)]α; here b = 1 m. This optical scale factor must be known before the image shift can determine c.
Locating a feature and resolving two separate features are different tasks. A line’s centre or edge can be aligned with a fine reference even if the line is wider than the smallest detectable displacement. Think of aligning two ruler marks. Good contrast, focus, and magnification help; the position still comes from a calibrated scale.
A 0.7 mm shift spans seven 0.1 mm divisions. A 0.005 mm increment is one twentieth of a division. For intuition, 25× angular magnification makes the shift subtend roughly the angle of a 17.5 mm displacement seen at the conventional 25 cm viewing distance. It does not turn the physical shift into 17.5 mm or multiply the information in the image.
If a 0.7 mm shift has uncertainty 0.005 mm, that contribution alone is about 0.7%. If 0.005 mm instead characterizes each of two independent readings, their difference has about √2 times that uncertainty: approximately 1% of the shift. These are illustrative calculations, not a reanalysis of Foucault’s observation book.
With optical scale factor K, c = 8πfLK/s. A 1% error in s produces about a 1% error in c, in the opposite direction. Path length, rotation rate, optical calibration, blur, and drift also matter. Repetition reduces random scatter; it cannot remove a miscalibrated scale or systematic optical bias. This was an impressive measurement with real limits.
Keeping a speed steady and knowing its value are two separate jobs.
Froment built the turbine carrying Foucault’s mirror. For the 1862 experiment, the organ builder Aristide Cavaillé-Coll provided the air supply and pressure regulation. An organ needs a steady wind supply; so does a mirror whose speed must remain steady. Cavaillé-Coll later recalled watching the experiment and comparing the turbine’s note with a tuning fork. [7]
The mechanical principle is a balance: air supplies a driving torque, while drag and friction oppose it. The speed settles where these balance. A valve changes the air flow and therefore the operating speed. Stable pressure reduces fluctuations; it does not, by itself, tell you the number of turns per second.
Diapason is French for tuning fork. Its two elastic prongs vibrate at a repeatable natural frequency. Compare a nearly matching rotating-machine tone with the fork: the combined sound swells and fades in beats. The beat rate is the difference between the frequencies.
LISTENING EXPERIMENT / FOUCAULT’S MIRROR
Play a fixed 400 Hz reference and an adjustable “mirror” tone together. Turn the air-flow dial until the swelling and fading become as slow as possible. The mirror starts off pitch; its frequency stays hidden until you check.
Drag to search; use the arrow keys or Fine buttons near the match. Listen for several seconds: a very slow beat can sound steady at first. Matching frequency does not require matching phase.
Two pure tones are mixed into the same output so you hear their interference. This models the once-per-revolution shaft tone; it is not a recording of the historical turbine. A new attempt shifts the dial’s hidden calibration, so its centre is not an answer key. Sound stops when you leave the tab.
Adjust the air flow and the beats slow as the tones approach unison. The fork is a reference; it neither powers the mirror nor automatically locks its speed. This is an operator making a comparison and correcting the flow.
There is a catch: you must know what makes the sound. A once-per-revolution tone has frequency f, but k blade passages per revolution produce kf, and harmonics can also be audible. Matching any whistle to “400 Hz” does not prove 400 revolutions per second. In his account, Cavaillé-Coll specifically identified the turbine’s shaft tone and checked it against the optical speed reference. [7]
The 1862 precision check
A clockwork mechanism turned a wheel with 400 teeth at 1 revolution per second. The returning light provided one flash per mirror revolution. At 400 mirror revolutions per second, a flash came each time the wheel advanced by one tooth, so the repeating tooth pattern appeared stationary. The observer adjusted the air flow to stop the pattern drifting. [7]
The wheel itself ran at just 60 rpm; the mirror ran at 24,000 rpm. As with any stroboscope, other frequency ratios can give apparently stationary patterns. A coarse speed estimate and identifying the intended synchronization are necessary.
This is a different role for a toothed wheel from Fizeau’s: here it checks the mirror’s rotation, rather than blocking a beam after a city-wide round trip.
A stamped musical note is not a calibration. You need to relate the fork’s oscillations to a measured interval of time. You can do that without counting hundreds of vibrations with your eyes.
Sound adjacent forks in a closely spaced series. Their beats may be only a few per second, slow enough to count against a pendulum clock. Count over a longer interval to reduce the effect of starting and stopping a little early.
Scheibler’s tonometer used a series spanning an octave. If the lowest fork is F and the highest is exactly 2F, the sum of all the adjacent frequency differences is 2F − F = F. The measured differences therefore reveal the absolute frequency of the first fork—and then every fork in the series. [8]
F = b₁ + b₂ + … + bₙ
For example, 50 intervals of 4 beats/s would establish endpoints of 200 and 400 Hz. These are illustrative numbers, not the inventory of Scheibler’s actual instrument. Establishing the exact octave also needs a careful harmonic comparison; any octave error carries into the result.
The beat counts still need a trustworthy clock. A pendulum clock’s rate can be checked over long intervals against astronomical time observations. The chain ends in a time standard; it never depends on knowing the speed of light. Fork temperature, damage, mounting, and the meaning of “vibration” matter too: older texts may count half-swings, whereas one hertz means one complete cycle per second.
A vibrating stylus can trace a fork’s motion on a moving smoked surface while a clock supplies time marks. Count complete waves between two time marks: F = N/Δt. The recording converts a rapid motion into a stationary pattern that can be inspected at leisure. This explains a general historical calibration method, rather than claiming that it was the calibration procedure for Foucault’s particular fork.
Later laboratory standards compared a fork directly with a standard pendulum. Today a microphone or contact sensor and a calibrated electronic frequency counter do the counting, with a timebase traceable to a time standard. The principle is still cycles divided by elapsed time. [9]
The cleverness is in the chain: a slow clock checks a fast rotation; that rotation reveals a much shorter light-travel time.
04 / YOUR OPTICAL BENCH
Change the journey, rotation rate, and screen distance. The outputs update as you type.
Illustration only; the beam angle is exaggerated.
t = 2L / c
θ = 2πft
α = 2θ
x = D tan(α)
Here c = 299,792,458 m/s, the exact vacuum value. [10] At small angles, x ≈ Dα. The calculation uses the tangent.
Double L: light is away twice as long, so the deflection doubles. Double f: the mirror turns twice as far. Double D: the spot shift doubles, but the angle is unchanged. Set the rotation to zero, then reverse it.
With the example values, the light’s entire outward-and-back journey lasts about 133 nanoseconds. The mirror turns only 0.0192°, yet a screen one metre away shows a shift of about 0.67 mm.
Path: L is the one-way vacuum-equivalent optical distance, not the source-to-screen distance. For a uniform medium of index n and physical length ℓ, use L = nℓ (more precisely, group index for a pulse). In air, taking n ≈ 1 is a useful introductory approximation.
Readout: x is the shift on a bare, unfolded screen. Foucault measured a magnified image through lenses and a microscope; converting that shift to α requires the actual optical geometry. These example values illustrate his method and are not a prediction of the shift at the museum instrument’s eyepiece.
Geometry: the model assumes a steady angular speed, an ideal retracing return, and a small rotation during transit. It ignores finite apertures, image blur, lens magnification, and vibration. Inputs giving |α| greater than 0.1 rad are rejected because this simple small-deflection layout would no longer be a reliable physical model. Zero path or observation distance is allowed as a mathematical limit.
Predicting versus measuring: this calculator assumes c to predict a shift. In the experiment, one instead measured the shift and inferred c. For this ideal screen geometry, α = atan(x/D), then c = 8πfL/α, using consistent signs and nonzero values.
Historical apparatus, modern ray optics, and illustrative calculations are distinguished above. These links let you follow the trail.
Collection identity and inventory 10177. Its online caption gives 400 turns/minute; contemporary evidence below gives 400 turns/second. This page uses the latter.
A modern implementation, useful for the ray optics and the distinction between beam deflection and an imaged displacement.
First-person recollection of the 400-tooth clockwork, 400 rev/s mirror, and agreement with a Scheibler tuning fork. The comparison is evidence of a cross-check; it should not be retold as a tuning fork mechanically governing the 1862 apparatus.
Beat counting, the octave chain, and establishing frequencies. The 50-interval numerical example on this page is illustrative.
A review by a light-speed experimenter: approximately 0.7 mm displacement, 25× eyepiece magnification, and position readings around 0.005 mm. Separate scan, p. 9 (PDF).
A simplified Foucault optical arrangement used for the worked calculation. Its 800 rev/s example is distinguished from the 400 Hz listening exercise.