The fact that light is propagated with a definite speed was first
brought out by Ole Roemer at Paris, in 1676, through observations of the
eclipses of Jupiter's satellites, made in different relative positions
of the Earth and Jupiter in their respective orbits. It is possible in
this way to determine the time required for light to pass across the
orbit of the earth. The dimensions of this orbit, or the distance of the
sun, being taken as known, the actual speed of light could be computed.
Since this computation requires a knowledge of the sun's distance, which
has not yet been acquired with certainty, the actual speed is now
determined by experiments made on the earth's surface. Were it possible
by any system of signals to compare with absolute precision the times at
two different stations, the speed could be determined by finding how
long was required for light to pass from one station to another at the
greatest visible distance. But this is impracticable, because no natural
agent is under our control by which a signal could be communicated with
a greater velocity than that of light. It is therefore necessary to
reflect a ray back to the point of observation and to determine the time
which the light requires to go and come. Two systems have been devised
for this purpose. One is that of Fizeau, in which the vital appliance is
a rapidly revolving toothed wheel; the other is that of Foucault, in
which the corresponding appliance is a mirror revolving on an axis in,
or parallel to, its own plane.
[Illustration: FIG. 1.]
Fizeau.
The principle underlying Fizeau's method is shown in the accompanying
figs. 1 and 2. Fig. 1 shows the course of a ray of light which,
emanating from a luminous point L, strikes the plane surface of a
plate of glass M at an angle of about 45°. A fraction of the light is
reflected from the two surfaces of the glass to a distant reflector R,
the plane of which is at right angles to the course of the ray. The
latter is thus reflected back on its own course and, passing through
the glass M on its return, reaches a point E behind the glass. An
observer with his eye at E looking through the glass sees the return
ray as a distant luminous point in the reflector R, after the light
has passed over the course in both directions.
In actual practice it is necessary to interpose the object glass of a
telescope at a point O, at a distance from M nearly equal to its focal
length. The function of this appliance is to render the diverging
rays, shown by the dotted lines, nearly parallel, in order that more
light may reach R and be thrown back again. But the principle may be
conceived without respect to the telescope, all the rays being ignored
except the central one, which passes over the course we have
described.
[Illustration: FIG. 2.]
Conceiving the apparatus arranged in such a way that the observer sees
the light reflected from the distant mirror R, a fine toothed wheel WX
is placed immediately in front of the glass M, with its plane
perpendicular to the course of the ray, in such a way that the ray
goes out and returns through an opening between two adjacent teeth.
This wheel is represented in section by WX in fig. 1, and a part of
its circumference, with the teeth as viewed by the observer, is shown
in fig. 2. We conceive that the latter sees the luminous point between
two of the teeth at K. Now, conceive that the wheel is set in
revolution. The ray is then interrupted as every tooth passes, so that
what is sent out is a succession of flashes. Conceive that the speed
of the mirror is such that while the flash is going to the distant
mirror and returning again, each tooth of the wheel takes the place of
an opening between the teeth. Then each flash sent out will, on its
return, be intercepted by the adjacent tooth, and will therefore
become invisible. If the speed be now doubled, so that the teeth pass
at intervals equal to the time required for the light to go and come,
each flash sent through an opening will return through the adjacent
opening, and will therefore be seen with full brightness. If the speed
be continuously increased the result will be successive disappearances
and reappearances of the light, according as a tooth is or is not
interposed when the ray reaches the apparatus on its return. The
computation of the time of passage and return is then very simple. The
speed of the wheel being known, the number of teeth passing in one
second can be computed. The order of the disappearance, or the number
of teeth which have passed while the light is going and coming, being
also determined in each case, the interval of time is computed by a
simple formula.
Cornu.
The most elaborate determination yet made by Fizeau's method was that
of Cornu. The station of observation was at the Paris Observatory. The
distant reflector, a telescope with a reflector at its focus, was at
Montlhéry, distant 22,910 metres from the toothed wheel. Of the wheels
most used one had 150 teeth, and was 35 millimetres in diameter; the
other had 200 teeth, with a diameter of 45 mm. The highest speed
attained was about 900 revolutions per second. At this speed, 135,000
(or 180,000) teeth would pass per second, and about 20 (or 28) would
pass while the light was going and coming. But the actual speed
attained was generally less than this. The definitive result derived
by Cornu from the entire series of experiments was 300,400 kilometres
per second. Further details of this work need not be set forth because
the method is in several ways deficient in precision. The eclipses and
subsequent reappearances of the light taking place gradually, it is
impossible to fix with entire precision upon the moment of complete
eclipse. The speed of the wheel is continually varying, and it is
impossible to determine with precision what it was at the instant of
an eclipse.
The defect would be lessened were the speed of the toothed wheel
placed under control of the observer who, by action in one direction
or the other, could continually check or accelerate it, so as to keep
the return point of light at the required phase of brightness. If the
phase of complete extinction is chosen for this purpose a definite
result cannot be reached; but by choosing the moment when the light is
of a certain definite brightness, before or after an eclipse, the
observer will know at each instant whether the speed should be
accelerated or retarded, and can act accordingly. The nearly constant
speed through as long a period as is deemed necessary would then be
found by dividing the entire number of revolutions of the wheel by the
time through which the light was kept constant. But even with these
improvements, which were not actually tried by Cornu, the estimate of
the brightness on which the whole result depends would necessarily be
uncertain. The outcome is that, although Cornu's discussion of his
experiments is a model in the care taken to determine so far as
practicable every source of error, his definitive result is shown by
other determinations to have been too great by about {1/1000} part of
its whole amount.
Young and Forbes.
An important improvement on the Fizeau method was made in 1880 by
James Young and George Forbes at Glasgow. This consisted in using two
distant reflectors which were placed nearly in the same straight line,
and at unequal distances. The ratio of the distances was nearly 12:13.
The phase observed was not that of complete extinction of either
light, but that when the two lights appeared equal in intensity. But
it does not appear that the very necessary device of placing the speed
of the toothed wheel under control of the observer was adopted. The
accordance between the different measures was far from satisfactory,
and it will suffice to mention the result which was
_Velocity in vacuo_ = 301,382 km. per second.
These experimenters also found a difference of 2% between the speed of
red and blue light, a result which can only be attributed to some
unexplained source of error.
The Foucault system is much more precise, because it rests upon the
measurement of an angle, which can be made with great precision.
[Illustration: FIG. 3.]
Foucault.
The vital appliance is a rapidly revolving mirror. Let AB (fig. 3) be
a section of this mirror, which we shall first suppose at rest. A ray
of light LM emanating from a source at L, is reflected in the
direction MQR to a distant mirror R, from which it is perpendicularly
reflected back upon its original course. This mirror R should be
slightly concave, with the centre of curvature near M, so that the ray
shall always be reflected back to M on whatever point of R it may
fall. Conceiving the revolving mirror M as at rest, the return ray
will after three reflections, at M, R and M again, be returned along
its original course to the point L from which it emanated. An
important point is that the return ray will always follow the fixed
line ML no matter what the position of the movable mirror M, provided
there is a distant reflector to send the ray back. Now, suppose that,
while the ray is going and coming, the mirror M, being set in
revolution, has turned from the position in which the ray was
reflected to that shown by the dotted line. If [alpha] be the angle
through which the surface has turned, the course of the return ray,
after reflection, will then deviate from ML by the angle 2[alpha], and
so be thrown to a point E, such that the angle LME = 2[alpha]. If the
mirror is in rapid rotation the ray reflected from it will strike the
distant mirror as a series of flashes, each formed by the light
reflected when the mirror was in the position AB. If the speed of
rotation is uniform, the reflected rays from the successive flashes
while the mirror is in the dotted position will thus all follow the
same direction ME after their second reflection from the mirror. If
the motion is sufficiently rapid an eye observing the reflected ray
will see the flashes as an invariable point of light so long as the
speed of revolution remains constant. The time required for the light
to go and come is then equal to that required by the mirror to turn
through half the angle LME, which is therefore to be measured. In
practice it is necessary on this system, as well as on that of Fizeau,
to condense the light by means of a lens, Q, so placed that L and R
shall be at conjugate foci. The position of the lens may be either
between the luminous point L and the mirror M, or between M and R, the
latter being the only one shown in the figure. This position has the
advantage that more light can be concentrated, but it has the
disadvantage that, with a given magnifying power, the effect of
atmospheric undulation, when the concave reflector is situated at a
great distance, is increased in the ratio of the focal length of the
lens to the distance LM from the light to the mirror. To state the
fact in another form, the amplitude of the disturbances produced by
the air in linear measure are proportional to the focal distance of
the lens, while the magnification required increases in the inverse
ratio of the distance LM. Another difficulty associated with the
Foucault system in the form in which its originator used it is that if
the axis of the mirror is at right angles to the course of the ray,
the light from the source L will be flashed directly into the eye of
the observer, on every passage of the revolving mirror through the
position in which its normal bisects the two courses of the ray. This
may be avoided by inclining the axis of the mirror.
In Foucault's determination the measures were not made upon a luminous
point, but upon a reticule, the image of which could not be seen
unless the reflector was quite near the revolving mirror. Indeed the
whole apparatus was contained in his laboratory. The effective
distance was increased by using several reflectors; but the entire
course of the ray measured only 20 metres. The result reached by
Foucault for the velocity of light was 298,000 kilometres per second.
Michelson.
The first marked advance on Foucault's determination was made by
Albert A. Michelson, then a young officer on duty at the U.S. Naval
Academy, Annapolis. The improvement consisted in using the image of a
slit through which the rays of the sun passed after reflection from a
heliostat. In this way it was found possible to see the image of the
slit reflected from the distant mirror when the latter was nearly 600
metres from the station of observation. The essentials of the
arrangement are those we have used in fig. 3, L being the slit. It
will be seen that the revolving mirror is here interposed between the
lens and its focus. It was driven by an air turbine, the blast of
which was under the control of the observer, so that it could be kept
at any required speed. The speed was determined by the vibrations of
two tuning forks. One of these was an electric fork, making about 120
vibrations per second, with which the mirror was kept in unison by a
system of rays reflected from it and the fork. The speed of this fork
was determined by comparison with a freely vibrating fork from time to
time. The speed of the revolving mirror was generally about 275 turns
per second, and the deflection of the image of the slit about 112.5
mm. The mean result of nearly 100 fairly accordant determinations
was:--
Velocity of light in air 299,828 km. per sec.
Reduction to a vacuum +82
Velocity of light in a vacuum 299,910 ± 50
Newcomb.
While this work was in progress Simon Newcomb obtained the official
support necessary to make a determination on a yet larger scale. The
most important modifications made in the Foucault-Michelson system
were the following:--
1. Placing the reflector at the much greater distance of several
kilometres.
2. In order that the disturbances of the return image due to the
passage of the ray through more than 7 km. of air might be reduced to
a minimum, an ordinary telescope of the "broken back" form was used to
send the ray to the revolving mirror.
3. The speed of the mirror was, as in Michelson's experiments,
completely under control of the observer, so that by drawing one or
the other of two cords held in the hand the return image could be kept
in any required position. In making each measure the receiving
telescope hereafter described was placed in a fixed position and
during the "run" the image was kept as nearly as practicable upon a
vertical thread passing through its focus. A "run" generally lasted
about two minutes, during which time the mirror commonly made between
25,000 and 30,000 revolutions. The speed per second was found by
dividing the entire number of revolutions by the number of seconds in
the "run." The extreme deviations between the times of transmission of
the light, as derived from any two runs, never approached to the
thousandth part of its entire amount. The average deviation from the
mean was indeed less than {1/5000} part of the whole.
To avoid the injurious effect of the directly reflected flash, as well
as to render unnecessary a comparison between the directions of the
outgoing and the return ray, a second telescope, turning horizontally
on an axis coincident with that of the revolving mirror, was used to
receive the return ray after reflection. This required the use of an
elongated mirror of which the upper half of the surface reflected the
outgoing ray, and the lower other half received and reflected the ray
on its return. On this system it was not necessary to incline the
mirror in order to avoid the direct reflection of the return ray. The
greatest advantage of this system was that the revolving mirror could
be turned in either direction without break of continuity, so that
the angular measures were made between the directions of the return
ray after reflection when the mirror moved in opposite directions. In
this way the speed of the mirror was as good as doubled, and the
possible constant errors inherent in the reference to a fixed
direction for the sending telescope were eliminated. The essentials of
the apparatus are shown in fig. 4. The revolving mirror was a
rectangular prism M of steel, 3 in. high and 1½ in. on a side in cross
section, which was driven by a blast of air acting on two fan-wheels,
not shown in the fig., one at the top, the other at the bottom of the
mirror. NPO is the object-end of the fixed sending telescope the rays
passing through it being reflected to the mirror by a prism P. The
receiving telescope ABO is straight, and has its objective under O. It
was attached to a frame which could turn around the same axis as the
mirror. The angle through which it moved was measured by a divided arc
immediately below its eye-piece, which is not shown in the figure. The
position AB is that for receiving the ray during a rotation of the
mirror in the anti-clockwise direction; the position A´B´ that for a
clockwise rotation.
[Illustration: FIG. 4.]
In these measures the observing station was at Fort Myer, on a hill
above the west bank of the Potomac river. The distant reflector was
first placed in the grounds of the Naval Observatory, at a distance of
2551 metres. But the definitive measures were made with the reflector
at the base of the Washington monument, 3721 metres distant. The
revolving mirror was of nickel-plated steel, polished on all four
vertical sides. Thus four reflections of the ray were received during
each turn of the mirror, which would be coincident were the form of
the mirror invariable. During the preliminary series of measures it
was found that two images of the return ray were sometimes formed,
which would result in two different conclusions as to the velocity of
light, according as one or the other was observed. The only
explanation of this defect which presented itself was a tortional
vibration of the revolving mirror, coinciding in period with that of
revolution, but it was first thought that the effect was only
occasional.
In the summer of 1881 the distant reflector was removed from the
Observatory to the Monument station. Six measures made in August and
September showed a systematic deviation of +67 km. per second from the
result of the Observatory series. This difference led to measures for
eliminating the defect from which it was supposed to arise. The pivots
of the mirror were reground, and a change made in the arrangement,
which would permit of the effect of the vibration being determined and
eliminated. This consisted in making the relative position of the
sending and receiving telescopes interchangeable. In this way, if the
measured deflection was too great in one position of the telescopes,
it would be too small by an equal amount in the reverse position. As a
matter of fact, when the definitive measures were made, it was found
that with the improved pivots the mean result was the same in the two
positions. But the new result differed systematically from both the
former ones. Thirteen measures were made from the Monument in the
summer of 1882, the results of which will first be stated in the form
of the time required by the ray to go and come. Expressed in
millionths of a second this was:--
Least result of the 13 measures 24.819
Greatest result 24.831
Double distance between mirrors 7.44242 km.
Applying a correction of +12 km. for a slight convexity in the face of
the revolving mirror, this gives as the mean result for the speed of
light in air, 299,778 km. per second. The mean results for the three
series were:--
Observatory, 1880-1881 V in air = 299,627
Monument, 1881 V " = 299,694
Monument, 1882 V " = 299,778
The last result being the only one from which the effect of distortion
was completely eliminated, has been adopted as definitive. For
reduction to a vacuum it requires a correction of +82 km. Thus the
final result was concluded to be
_Velocity of light in vacuo_ = 299,860 km. per second.
This result being less by 50 km. than that of Michelson, the latter
made another determination with improved apparatus and arrangements at
the Case School of Applied Science in Cleveland. The result was
_Velocity in vacuo_ = 299,853 km. per second.
So far as could be determined from the discordance of the separate
measures, the mean error of Newcomb's result would be less than ±10
km. But making allowance for the various sources of systematic error
the actual probable error was estimated at ±30 km.
It seems remarkable that since these determinations were made, a period
during which great improvements have become possible in every part of
the apparatus, no complete redetermination of this fundamental physical
constant has been carried out.
The experimental measures thus far cited have been primarily those of
the velocity of light in air, the reduction to a vacuum being derived
from theory alone. The fundamental constant at the basis of the whole
theory is the speed of light in a vacuum, such as the celestial spaces.
The question of the relation between the velocity in vacuo, and in a
transparent medium of any sort, belongs to the domain of physical
optics. Referring to the preceding section for the principles at play we
shall in the present part of the article confine ourselves to the
experimental results. With the theory of the effect of a transparent
medium is associated that of the possible differences in the speed of
light of different colours.
Velocity and wave-length.
The question whether the speed of light in vacuo varies with its
wave-length seems to be settled with entire certainty by observations of
variable stars. These are situated at different distances, some being so
far that light must be several centuries in reaching us from them. Were
there any difference in the speed of light of various colours it would
be shown by a change in the colour of the star as its light waxed and
waned. The light of greatest speed preceding that of lesser speed would,
when emanated during the rising phase, impress its own colour on that
which it overtook. The slower light would predominate during the falling
phase. If there were a difference of 10 minutes in the time at which
light from the two ends of the visible spectrum arrived, it would be
shown by this test. As not the slightest effect of the kind has ever
been seen, it seems certain that the difference, if any, cannot
approximate to {1/1.000.000} part of the entire speed. The case is
different when light passes through a refracting medium. It is a
theoretical result of the undulatory theory of light that its velocity
in such a medium is inversely proportional to the refractive index of
the medium. This being different for different colours, we must expect a
corresponding difference in the velocity.
Foucault and Michelson have tested these results of the undulatory
theory by comparing the time required for a ray of light to pass through
a tube filled with a refracting medium, and through air. Foucault thus
found, in a general way, that there actually was a retardation; but his
observations took account only of the mean retardation of light of all
the wave-lengths, which he found to correspond with the undulatory
theory. Michelson went further by determining the retardation of light
of various wave-lengths in carbon bisulphide. He made two series of
experiments, one with light near the brightest part of the spectrum; the
other with red and blue light. Putting V for the speed in a vacuum and
V1 for that in the medium, his result was
Yellow light V : V1 = 1.758
Refractive index for yellow 1.64
Difference from theory +0.12
The estimated uncertainty was only 0.02, or 1/6 of the difference
between observation and theory.
The comparison of red and blue light was made differentially. The
colours selected were of wave-length about 0.62 for red and 0.49 for
blue. Putting V_r and V_b for the speeds of red and blue light
respectively in bisulphide of carbon, the mean result compares with
theory as follows:--
Observed value of the ratio V_r, V_b 1.0245
Theoretical value (Verdet) 1.025
This agreement may be regarded as perfect. It shows that the divergence
of the speed of yellow light in the medium from theory, as found above,
holds through the entire spectrum.
The excess of the retardation above that resulting from theory is
probably due to a difference between "wave-speed" and "group-speed"
pointed out by Rayleigh. Let fig. 5 represent a short series of
progressive undulations of constant period and wave-length. The
wave-speed is that required to carry a wave crest A to the position of
the crest B in the wave time. But when a flash of light like that
measured passes through a refracting medium, the front waves of the
flash are continually dying away, as shown at the end of the figure, and
the place of each is taken by the wave following. A familiar case of
this sort is seen when a stone is thrown into a pond. The front waves
die out one at a time, to be followed by others, each of which goes
further than its predecessor, while new waves are formed in the rear.
Hence the group, as represented in the figure by the larger waves in the
middle, moves as a whole more slowly than do the individual waves. When
the speed of light is measured the result is not the wave-speed as above
defined, but something less, because the result depends on the time of
the group passing through the medium. This lower speed is called the
group-velocity of light. In a vacuum there is no dying out of the waves,
so that the group-speed and the wave-speed are identical. From
Michelson's experiments it would follow that the retardation was about
{1/14} of the whole speed. This would indicate that in carbon bisulphide
each individual light wave forming the front of a moving ray dies out in
a space of about 15 wave-lengths.
[Illustration: FIG. 5.]
AUTHORITIES.--For Foucault's descriptions of his experiments see
_Comptes Rendus_ (September 22 and November 24, 1862), and _Recueil de
Travaux Scientifiques de Léon Foucault_ (2 vols., 4to, Paris, 1878).
Cornu's determination is found in _Annales de l'Observatoire de Paris,
Mémoires_, vol. xiii. The works of Michelson and Newcomb are published
_in extenso_ in the _Astronomical Papers of the American Ephemeris_,
vols. i. and ii. (S. N.)