Chapter 1

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Experimental Determination of the Velocity of Light

Made at the U.S. Naval Academy, Annapolis.

By

Albert A. Michelson, Master U.S. Navy.

Note.

The probability that the most accurate method of determining the solar parallax now available is that resting on the measurement of the velocity of light, has led to the acceptance of the following paper as one of the series having in view the increase of our knowledge of the celestial motions. The researches described in it, having been made at the United States Naval Academy, though at private expense, were reported to the Honorable Secretary of the Navy, and referred by him to this Office. At the suggestion of the writer, the paper was reconstructed with a fuller general discussion of the processes, and with the omission of some of the details of individual experiments.

To prevent a possible confusion of this determination of the velocity of light with another now in progress under official auspices, it may be stated that the credit and responsibility for the present paper rests with Master Michelson.

Simon Newcomb, _Professor, U.S. Navy_, _Superintendent Nautical Almanac_.

Nautical Almanac Office, Bureau of Navigation, Navy Department, _Washington, February 20, 1880._

Table Of Contents.

Introduction Theory of the New Method Arrangement and Description of Apparatus Determination of the Constants The Formulæ Observations Separate results of Groups of Observations Discussion of Errors Objections Considered Postscript

Experimental Determination of the Velocity of Light.

By Albert A. Michelson, _Master, U.S.N._

Introduction.

In Cornu's elaborate memoir upon the determination of the velocity of light, several objections are made to the plan followed by Foucault, which will be considered in the latter part of this work. It may, however, be stated that the most important among these was that the deflection was too small to be measured with the required degree of accuracy. In order to employ this method, therefore, it was absolutely necessary that the deflection should be increased.

In November, 1877, a modification of Foucault's arrangement suggested itself, by which this result could be accomplished. Between this time and March of the following year a number of preliminary experiments were performed in order to familiarize myself with the optical arrangements. The first experiment tried with the revolving mirror produced a deflection considerably greater than that obtained by Foucault. Thus far the only apparatus used was such as could be adapted from the apparatus in the laboratory of the Naval Academy.

At the expense of $10 a revolving mirror was made, which could execute 128 turns per second. The apparatus was installed in May, 1878, at the laboratory. The distance used was 500 feet, and the deflection was about twenty times that obtained by Foucault.[1]

[Footnote 1: See Proc. Am. Assoc. Adv. Science, Saint Louis meeting.]

These experiments, made with very crude apparatus and under great difficulties, gave the following table of results for the velocity of light in miles per second:

186730 188820 186330 185330 187900 184500 186770 185000 185800 187940 ------ Mean 186500 ± 300 miles per second, or 300140 kilometers per second.

In the following July the sum of $2,000 was placed at my disposal by a private gentleman for carrying out these experiments on a large scale. Before ordering any of the instruments, however, it was necessary to find whether or not it was practicable to use a large distance. With a distance (between the revolving and the fixed mirror) of 500 feet, in the preliminary experiments, the field of light in the eye-piece was somewhat limited, and there was considerable indistinctness in the image, due to atmospheric disturbances.

Accordingly, the same lens (39 feet focus) was employed, being placed, together with the other pieces of apparatus, along the north sea-wall of the Academy grounds, the distance being about 2,000 feet. The image of the slit, at noon, was so confused as not to be recognizable, but toward sunset it became clear and steady, and measurements were made of its position, which agreed within one one-hundredth of a millimeter. It was thus demonstrated that with this distance and a deflection of 100 millimeters this measurement could be made within the ten-thousandth part.

In order to obtain this deflection, it was sufficient to make the mirror revolve 250 times per second and to use a "radius" of about 30 feet. In order to use this large radius (distance from slit to revolving mirror), it was necessary that the mirror should be large and optically true; also, that the lens should be large and of great focal length. Accordingly the mirror was made 1¼ inches in diameter, and a new lens, 8 inches in diameter, with a focal length of 150 feet was procured.

In January, 1879, an observation was taken, using the old lens, the mirror making 128 turns per second. The deflection was about 43 millimeters. The micrometer eye-piece used was substantially the same as Foucault's, except that part of the inclined plate of glass was silvered, thus securing a much greater quantity of light. The deflection having reached 43 millimeters, the inclined plate of glass could be dispensed with, the light going past the observer's head through the slit, and returning 43 millimeters to the left of the slit, where it could be easily observed.

Thus the micrometer eye-piece is much simplified, and many possible sources of error are removed.

The field was quite limited, the diameter being, in fact, but little greater than the width of the slit. This would have proved a most serious objection to the new arrangement. With the new lens, however, this difficulty disappeared, the field being about twenty times the width of the slit. It was expected that, with the new lens, the image would be less distinct; but the difference, if any, was small, and was fully compensated by the greater size of the field.

The first observation with the new lens was made January 30, 1879. The deflection was 70 millimeters. The image was sufficiently bright to be observed without the slightest effort. The first observation with the new micrometer eye-piece was made April 2, the deflection being 115 millimeters.

The first of the final series of observations was made on June 5. All the observations previous to this, thirty sets in all, were rejected. After this time, no set of observations nor any single observation was omitted.

Theory of New Method.

Let S, Fig. 1, be a slit, through which light passes, falling on R, a mirror free to rotate about an axis at right angles to the plane of the paper; L, a lens of great focal length, upon which the light falls which is reflected from R. Let M be a plane mirror whose surface is perpendicular to the line R, M, passing through the centers of R, L, and M, respectively. If L be so placed that an image of S is formed on the surface of M, then, this image acting as the object, its image will be formed at S, and will coincide, point for point, with S.

If, now, R be turned about the axis, so long as the light falls upon the lens, an image of the slit will still be formed on the surface of the mirror, though on a different part, and as long as the returning light falls on the lens an image of this image will be formed at S, notwithstanding the change of position of the first image at M. This result, namely, the production of a stationary image of an image in motion, is absolutely necessary in this method of experiment. It was first accomplished by Foucault, and in a manner differing apparently but little from the foregoing.

In his experiments L, Fig. 2, served simply to form the image of S at M, and M, the returning mirror, was spherical, the center coinciding with the axis of R. The lens L was placed as near as possible to R. The light forming the return image lasts, in this case, while the first image is sweeping over the face of the mirror, M. Hence, the greater the distance RM, the larger must be the mirror in order that the same amount of light may be preserved, and its dimensions would soon become inordinate. The difficulty was partly met by Foucault, by using five concave reflectors instead of one, but even then the greatest distance he found it practicable to use was only 20 meters.

Returning to Fig. 1, suppose that R is in the principal focus of the lens L; then, if the plane mirror M have the same diameter as the lens, the first, or moving image, will remain upon M as long as the axis of the pencil of light remains on the lens, and _this will be the case no matter what the distance may be_.

When the rotation of the mirror R becomes sufficiently rapid, then the flashes of light which produce the second or stationary image become blended, so that the image appears to be continuous. But now it no longer coincides with the slit, but is _deflected_ in the direction of rotation, and through twice the angular distance described by the mirror, during the time required for light to travel twice the distance between the mirrors. This displacement is measured by the tangent of the arc it subtends. To make this as large as possible, the distance between the mirrors, the radius, and the speed of rotation should be made as great as possible.

The second condition conflicts with the first, for the radius is the difference between the focal length for parallel rays, and that for rays at the distance of the fixed mirror. The greater the distance, therefore, the smaller will be the radius.

There are two ways of solving the difficulty: first, by using a lens of great focal length; and secondly, by placing the revolving mirror within the principal focus of the lens. Both means were employed. The focal length of the lens was 150 feet, and the mirror was placed about 15 feet within the principal focus. A limit is soon reached, however, for the quantity of light received diminishes very rapidly as the revolving mirror approaches the lens.

Arrangement and Description of Apparatus.

Site and Plan.

The site selected for the experiments was a clear, almost level, stretch along the north sea-wall of the Naval Academy. A frame building was erected at the western end of the line, a plan of which is represented in Fig. 3.

The building was 45 feet long and 14 feet wide, and raised so that the line along which the light traveled was about 11 feet above the ground. A heliostat at H reflected the sun's rays through the slit at S to the revolving mirror R, thence through a hole in the shutter, through the lens, and to the distant mirror.

The Heliostat.

The heliostat was one kindly furnished by Dr. Woodward, of the Army Medical Museum, and was a modification of Foucault's form, designed by Keith. It was found to be accurate and easy to adjust. The light was reflected from the heliostat to a plane mirror, M, Fig. 3, so that the former need not be disturbed after being once adjusted.

The Revolving Mirror.

The revolving mirror was made by Fauth & Co., of Washington. It consists of a cast-iron frame resting on three leveling screws, one of which was connected by cords to the table at S, Fig. 3, so that the mirror could be inclined forward or backward while making the observations.

Two binding screws, S, S, Fig. 4, terminating in hardened steel conical sockets, hold the revolving part. This consists of a steel axle, X, Y, Figs. 4 and 5, the pivots being conical and hardened. The axle expands into a ring at R, which holds the mirror M. The latter was a disc of plane glass, made by Alvan Clark & Sons, about 1¼ inch in diameter and 0.2 inch thick. It was silvered on one side only, the reflection taking place from the outer or front surface. A species of turbine wheel, T, is held on the axle by friction. This wheel has six openings for the escape of air; a section of one of them is represented in Fig 6.

Adjustment of the Revolving Mirror.

The air entering on one side at O, Fig. 5, acquires a rotary motion in the box B, B, carrying the wheel with it, and this motion is assisted by the reaction of the air in escaping. The disc C serves the purpose of bringing the center of gravity in the axis of rotation. This was done, following Foucault's plan, by allowing the pivots to rest on two inclined planes of glass, allowing the arrangement to come to rest, and filing away the lowest part of the disc; trying again, and so on, till it would rest in indifferent equilibrium. The part corresponding to C, in Foucault's apparatus, was furnished with three vertical screws, by moving which the axis of figure was brought into coincidence with the axis of rotation. This adjustment was very troublesome. Fortunately, in this apparatus it was found to be unnecessary.

When the adjustment is perfect the apparatus revolves without giving any sound, and when this is accomplished, the motion is regular and the speed great. A slight deviation causes a sound due to the rattling of the pivots in the sockets, the speed is very much diminished, and the pivots begin to wear. In Foucault's apparatus oil was furnished to the pivots, through small holes running through the screws, by pressure of a column of mercury. In this apparatus it was found sufficient to touch the pivots occasionally with a drop of oil.

Fig. 7 is a view of the turbine, box, and supply-tube, from above. The quantity of air entering could be regulated by a valve to which was attached a cord leading to the observer's table.

The instrument was mounted on a brick pier.

The Micrometer.

The apparatus for measuring the deflection was made by Grunow, of New York.

This instrument is shown in perspective in Fig. 8, and in plan by Fig. 9. The adjustable slit S is clamped to the frame F. A long millimeter-screw, not shown in Fig. 8, terminating in the divided head D, moves the carriage C, which supports the eye-piece E. The frame is furnished with a brass scale at F for counting revolutions, the head counting hundredths. The eye-piece consists of a single achromatic lens, whose focal length is about two inches. At its focus, in H, and in nearly the same plane as the face of the slit, is a single vertical silk fiber. The apparatus is furnished with a standard with rack and pinion, and the base furnished with leveling screws.

Manner of Using the Micrometer.

In measuring the deflection, the eye-piece is moved till the cross-hair bisects the slit, and the reading of the scale and divided head gives the position. This measurement need not be repeated unless the position or width of the slit is changed. Then the eye-piece is moved till the cross-hair bisects the deflected image of the slit; the reading of scale and head are again taken, and the difference in readings gives the deflection. The screw was found to have no lost motion, so that readings could be taken with the screw turned in either direction.

Measurement of Speed of Rotation.

To measure the speed of rotation, a tuning-fork, bearing on one prong a steel mirror, was used. This was kept in vibration by a current of electricity from five "gravity" cells. The fork was so placed that the light from the revolving mirror was reflected to a piece of plane glass, in front of the lens of the eye-piece of the micrometer, inclined at an angle of 45°, and thence to the eye. When fork and revolving mirror are both at rest, an image of the revolving mirror is seen. When the fork vibrates, this image is drawn out into a band of light.

When the mirror commences to revolve, this band breaks up into a number of moving images of the mirror; and when, finally, the mirror makes as many turns as the fork makes vibrations, these images are reduced to one, which is stationary. This is also the case when the number of turns is a submultiple. When it is a multiple or simple ratio, the only difference is that there are more images. Hence, to make the mirror execute a certain number of turns, it is simply necessary to pull the cord attached to the valve to the right or left till the images of the revolving mirror come to rest.

The electric fork made about 128 vibrations per second. No dependence was placed upon this rate, however, but at each set of observations it is compared with a standard Ut₃ fork, the temperature being noted at the same time. In making the comparison the sound-beats produced by the forks were counted for 60 seconds. It is interesting to note that the electric fork, as long as it remained untouched and at the same temperature, did not change its rate more than one or two hundredths vibrations per second.

The Observer's Table.

Fig. 9 Represents The Table At Which The Observer Sits. The Light From The Heliostat Passes Through The Slit At S, Goes To The Revolving Mirror, &c., And, On Its Return, Forms An Image Of The Slit At D, Which Is Observed Through The Eye-piece. E Represents The Electric Fork (the Prongs Being Vertical) Bearing The Steel Mirror M. K Is The Standard Fork On Its Resonator. C Is The Cord Attached To The Valve Supplying Air To The Turbine.

The Lens.

The lens was made by Alvan Clark & Sons. It was 8 inches in diameter; focal length, 150 feet; not achromatic. It was mounted in a wooden frame, which was placed on a support moving on a slide, about 16 feet long, placed about 80 feet from the building. As the diameter of the lens was so small in comparison with its focal length, its want of achromatism was inappreciable. For the same reason, the effect of "parallax" (due to want of coincidence in the plane of the image with that of the silk fiber in the eye-piece) was too small to be noticed.

The Fixed Mirror.

The fixed mirror was one of those used in taking photographs of the transit of Venus. It was about 7 inches in diameter, mounted in a brass frame capable of adjustment in a vertical and a horizontal plane by screw motion. Being wedge-shaped, it had to be silvered on the front surface. To facilitate adjustment, a small telescope furnished with cross-hairs was attached to the mirror by a universal joint. The heavy frame was mounted on a brick pier, and the whole surrounded by a wooden case to protect it from the sun.

Adjustment of the Fixed Mirror.

The adjustment was effected as follows: A theodolite was placed at about 100 feet in front of the mirror, and the latter was moved about by the screws till the observer at the theodolite saw the image of his telescope reflected in the center of the mirror. Then the telescope attached to the mirror was pointed (without moving the mirror itself) at a mark on a piece of card-board attached to the theodolite. Thus the line of collimation of the telescope was placed at right angles to the surface of the mirror. The theodolite was then moved to 1,000 feet, and, if found necessary, the adjustment was repeated. Then the mirror was moved by the screws till its telescope pointed at the hole in the shutter of the building. The adjustment was completed by moving the mirror, by signals, till the observer, looking through the hole in the shutter, through a good spy-glass, saw the image of the spy-glass reflected centrally in the mirror.

The whole operation was completed in a little over an hour.

Notwithstanding the wooden case about the pier, the mirror would change its position between morning and evening; so that the last adjustment had to be repeated before every series of experiments.

Apparatus for Supplying and Regulating the Blast of Air.

Fig. 10 represents a plan of the lower floor of the building. E is a three-horse power Lovegrove engine and boiler, resting on a stone foundation; B, a small Roots' blower; G, an automatic regulator. From this the air goes to a delivery-pipe, up through the floor, and to the turbine. The engine made about 4 turns per second and the blower about 15. At this speed the pressure of the air was about half a pound per square inch.

The regulator, Fig. 11, consists of a strong bellows supporting a weight of 370 pounds, partly counterpoised by 80 pounds in order to prevent the bellows from sagging. When the pressure of air from the blower exceeds the weight, the bellows commences to rise, and, in so doing, closes the valve V.

This arrangement was found in practice to be insufficient, and the following addition was made: A valve was placed at P, and the pipe was tapped a little farther on, and a rubber tube led to a water-gauge, Fig 12. The column of water in the smaller tube is depressed, and, when it reaches the horizontal part of the tube, the slightest variation of pressure sends the column from one end to the other. This is checked by an assistant at the valve; so that the column of water is kept at about the same place, and the pressure thus rendered very nearly constant. The result was satisfactory, though not in the degree anticipated. It was possible to keep the mirror at a constant speed for three or four seconds at a time, and this was sufficient for an observation. Still it would have been more convenient to keep it so for a longer time.

I am inclined to think that the variations were due to changes in the friction of the pivots rather than to changes of pressure of the blast of air.

It may be mentioned that the test of uniformity was very delicate, as a change of speed of one or two hundredths of a turn per second could easily be detected.

Method Followed in Experiment.

It was found that the only time during the day when the atmosphere was sufficiently quiet to get a distinct image was during the hour after sunrise, or during the hour before sunset. At other times the image was "boiling" so as not to be recognizable. In one experiment the electric light was used at night, but the image was no more distinct than at sunset, and the light was not steady.

The method followed in experiment was as follows: The fire was started half an hour before, and by the time everything was ready the gauge would show 40 or 50 pounds of steam. The mirror was adjusted by signals, as before described. The heliostat was placed and adjusted. The revolving mirror was inclined to the right or left, so that the _direct_ reflection of light from the slit, which otherwise would flash into the eye-piece at every revolution, fell either above or below the eye-piece.[2]

[Footnote 2: Otherwise this light would overpower that which forms the image to be observed. As far as I am aware, Foucault does not speak of this difficulty. If he allowed this light to interfere with the brightness of the image, he neglected a most obvious advantage. If he did incline the axis of the mirror to the right or left, he makes no allowance for the error thus introduced.]

The revolving mirror was then adjusted by being moved about, and inclined forward and backward, till the light was seen reflected back from the distant mirror. This light was easily seen through the coat of silver on the mirror.

The distance between the front face of the revolving mirror and the cross-hair of the eye-piece was then measured by stretching from the one to the other a steel tape, making the drop of the catenary about an inch, as then the error caused by the stretch of the tape and that due to the curve just counterbalance each other.

The position of the slit, if not determined before, was then found as before described. The electric fork was started, the temperature noted, and the sound-beats between it and the standard fork counted for 60 seconds. This was repeated two or three times before every set of observations.