H. L. Fizeau
The results
Enter Cornu
Cornu’s Results
The Young/Forbes Result
Perrotin’s Procedures
Perrotin and Prim’s Results
Conclusions from Toothed-Wheel Experiments
Table 5: Toothed Wheel Experimental Values

 

H. L. Fizeau

The two methods of measuring the velocity of light, c, that have been considered to date have both been astronomical.  However, back in 1638, in his ‘Discorsi’, Galileo suggested the basis of a terrestrial experiment over a number of miles using lanterns, shutters and telescopes to timed flashes of light.  The Florentine Academy in 1667 tried the idea over a distance of one mile without any observable delay.  Just nine years later the reason became apparent:  Roemer’s value for c was so great in comparison to human reaction times in operating the lantern shutters that there was no hope of observing the finite travel time delay for c over one mile (or 1609.344 meters).

It was not until 1849 that the French physicist H. L. Fizeau overcame the problem in the following fashion.  In the first place it was desirable to have as large a distance as practical involved, instead of just one mile.  Fizeau used as his base-line the distance between two hills near Paris, Suresnes and Montmartre, measured as 8633 meters.1  As he had arranged to observe the returning beam of light, the total distance traveled was thus 17,266 meters.  Though this distance was large in comparison with the single mile used with lanterns, it was the second shortest base-line ever used in this type of experiment.

In place of the shutters on lanterns, Fizeau used a rotating wheel with 720 teeth and driven by clockwork made by Froment.2  Light from an intense source was focused on the rim of the wheel, then made into a parallel beam by a telescope, and traversed the 8633 meters.  There it was received by another telescope which focused the beam onto a concave mirror, sending the light back along the same path that it had just traveled.  The returned beam was viewed between the teeth in the wheel.  The system was focused with the wheel at rest and with the light shining between the gap in the teeth.  The wheel was then rotated, automatically chopping the beam into a series of flashes like the lantern shutters.

diagram courtesy of http://home.iprimus.com.au/longhair1/chap6.htm

 

The results

Initially, as the wheel’s rotation rate was increased from zero, the observed light intensity dropped, until the light flash that had passed between the teeth on the way out struck the next tooth on its return.  The observed light intensity was then at its minimum.  However, by increasing the wheel’s rotation rate, a situation arose where the returning flash went through the next gap to the one it passed through on the way out.  This gave an observed maximum intensity.  Fizeau found the first minimum occurred with the wheel rotating at 12.6 turns per second and the first maximum followed at 25.2 turns per second.3  The flash of  light had thus traveled the 17,266 Km distance in 1/(25.2 x 720) seconds.  Hence, in one second, the light traveled 25.2 x 720 x 17,266 Km.  This gives Fizeau’s value for c as 313,300 Km/s.  This is the usual figure given in textbooks.

Although this value for Fizeau’s result is often quoted,4 the technical literature prefers another value for the following reason.  Fizeau stated that his result was the mean of 28 measurements that gave c as 70,948 leagues of 25 to the degree per second (’70,948 lieues de 25 au degre’).5  As Dorsey points out, the meter was meant to be 1,10,000,000 part of the earth’s meridional quadrant and hence Fizeau’s league became equal to 40/9 Km.  Multiplication of 40/9 by 70,948 gives the technically accepted result of 315,300 Km/s as Fizeau’s value for c, the speed of light.6 

Fizeau’s experimental details were not given in his report, and one promised later after further work seems not to have been found.  However, in a quote from the French journal ‘L/Astronomie’, with no exact reference given, two further values appear against an 1855 date that are presumably Fizeau’s.7  They give c as 305,650 Km/s and 298,000 Km/s, thought this latter value may be a bad citation for Foucault’s initial work in 1862.

Enter Cornu

In 1872, A. Cornu used Fizeau’s method to determine c over a base-line of 10,310 meters from the École Polytechinque to Mont Valerien.8  This was in the nature of a preliminary experiment, and was refined later.  Of the 658 measurements made, 86 were determined using a weight driven motor and the remaining 572 using a clockwork spring drive.  The results obtained were later rejected by Cornu9 as being “affected by serious systematic errors”.10  Dorsey states that “The apparatus was crude; the precision was low”.11  This preliminary result that Cornu rejected was 298,400 Km/s in air or 298,500 ± 300 Km/s in a vacuum.12

In 1874, the Council of the Paris Observatoire, headed by LeVerrier, who was the Observatory Director, and Fizeau, decided to ask Cornu to obtain a definitive value for c.  The date was April 2, and the reason was that a transit of Venus was to occur on December 9th of that year.  A value for c accurate to one part in a thousand would be needed by astronomers observing the event.  Cornu complied with the request.

The sending telescope was mounted on the Paris Observatory and the light flashes sent to the tower of Montlhery where the collimator lens returned the chopped beam. The base-line was 22,910 meters.  Four smoked aluminum wheels of 1/10 to 1/15 mm thickness were used.  Three had pointed teeth numbering 144, 150, and 200 respectively.  The fourth wheel of 40 mm diameter has 180 square teeth.  The wheels could be rotated in either direction, which eliminated a number of errors.

The apparatus was powered by a weight-driven, friction-brake controlled device.  An electric circuit automatically left a record of wheel rotation rates on a chronograph sheet advancing 1.85 cm/s.  A 1/20 second oscillator was used to subdivide the one second intervals of the observatory clock.  Times were estimated to 0.001 second were claimed.  The main difficulty in observation was the determination of the exact moment of total eclipse of the returned flash as the background is always slightly luminous.  The speed of the wheel corresponding to the disappearance of the beam was noted, as was the speed for its re-appearance and a mathematical averaging procedure was adopted.13

Cornu’s Results

In his initial report of December 1874, Cornu announced that his preliminary reductions of the data resulted in a value for c of 300,330 Km/s in air.14  Multiplied by a refractive index for air of 1.0003 gave the vacuum velocity of 300,400 ± 300 Km/s.  After final reductions of the data gave a result of 300,170 Km/s for air, Cornu felt obliged to apply a correction for possible vibrations of the wheel which resulted in a value of 300,350 Km/s in air.15  The addition of the 82 Km/s to bring the value to that of a vacuum gave his final result again as 300,400 ± 300 Km/s to four figures.  In all, he claimed to make 630 measurements of c for this toothed wheel determination.  However, he records 624 sets of observations, of which he appears to have used 546 in the reduction process.16

Following the publication of the initial report, Helmert noticed that, for low speeds of the wheel, the c values were higher than the mean value, while for high wheel speeds, the values were lower than the mean.17

  This suggested that a systematic error was affecting the result such that c* = c (H/q) where c*  was the reported value of c, and H = 7.1.  The q term was the order of the eclipse so that for higher rotation speeds q was also higher.  Dorsey derived a similar function.18  Helmert’s correction was generally discussed, verified, and accepted, even as late as 1900, despite Cornu’s protestations at the treatment.19  The Cornu-Helmert value was 299,990 ± 200 Km/s.  Dorsey, after considerable analysis, decided that “The best value one can derive from the observations seems, from these data, to be 299.8 metametres/sec. in air with a possible range of ± 0.2”.20  After applying a refractive index correction, the final Cornu-Dorsey vacuum c value was 299,900 ± 200 Km/s.

The mean dates for Cornu’s experiments were 1872.0 for his preliminary work at the Ecole Polytechnique, and 1874.8 for those at the Paris Observatory.  It should be noted that Newcomb gives incorrect dates for these determinations and also attributes the value of 299,900 to a re-discussion of Cornu’s results by Listing.21  Though Listing had just published a paper (Astron. Nachr., Vol. 93, p. 369, 1878) on solar parallax, Cornu’s results were accepted without discussion.  This misinformation was transmitted by Michelson in his Tables22 and by Preston (‘The Theory of Light’, p. 511, 1901).  Michelson furthermore quotes the value as Listing’s even when referencing a work that correctly attributes it to Helmert (Phil. Mag., 6th series, Vol. 3, 1902, p. 334).  Later Michelson gave the result as 299,950 Km/s, different from Cornu, Helmert or Listing, leaving the origin of this figure lost in obscurity.23

The Young/Forbes Result

One of the main problems facing the toothed wheel method was the estimation of the exact moment of eclipse of the light beam.  Cornu overcame the problem by making pairs of observations on either side of the exact eclipse position and further pairing with reversed wheel rotation.  Young and Forbes in England in 1991 used a different technique.  From an observing station at Wemyss Bay, light was sent to two distant reflectors, instead of the normal one, in the hills behind Inellan.  The reflectors were in the same line but the nearer one was 16,835.0 feet from the observation post, and the other was at 18,212.2 feet distance.  The two images so formed were observed simultaneously.  The position of the eclipses or the maximum was not needed.  The speed of the cogwheel was measured instead at the time when both images appeared to be of equal intensity.

The advantage of this method is that the eye is extremely sensitive to slight differences in the intensity of adjacent images.  The extreme disadvantage of their arrangement consisted of the short base-line.  Even taking the most distant reflector, the base was only 5551.07 metres long.  This was by far the shortest for this type of experiment being not quite two-thirds of Fizeau’s base-length.  The problems of the short base asserted themselves, as did other experimental features that were not conducive to obtaining good results.  This determination was severely criticized by both Newcomb24 and Cornu.25  It is omitted in the definitive list of best c determinations treated by Birge.26  Dorsey comments, “it is generally admitted that their work is seriously in error, and is reported unsatisfactorily.”27  This Young/Forbes result was given as 301,382 Km/s28 “with an unwarranted accuracy but they give no probable error.”29

Perrotin’s Procedures

Joseph Perrotin was born in 1845 and was appointed as the Director of the Observatory at Nice in France during the late 1800’s.  His assistant at the Observatory was Prim, who had the details of their determination of c by Fizeau’s method published in 1908.30  Cornu was still alive at the inception of the project in 1898, and gave his counsel throughout its duration.  Cornu died in 1902, the year in which the experimental proceedings were finalized.  Perrotin died shortly after, in 1904, before completing the discussion, leaving Prim to release the final calculations in 1908.

In the initial work, centering about 1900.4, the base-line was from the Nice Observatory to the village of La Gaude, a distance of 11,862.2 metres.  In the more extensive series, that centered around 1902.4, a longer path of 45,950.7 metres to Mont Vinaigre was employed.  In the La Gaude series, a total of 1540 measurements of c were made, 607 by Perrotin and 933 by Prim.  The Mont Vinaigre experiment totaled 2465 measurements with Perrotin making 1452 and Prim 1013.  In all the complete exercise made 4005 measurements to come to the final value for the series.31

With Cornu as advisor, much of the equipment and procedure was the same.  The illuminator and the powering apparatus were those Cornu used, as were the chronograph, pendulums, and 1/20 second oscillator.  The microscope with variable magnification was also that used by Cornu in his determination.32  The only essentially different equipment items were the sending and collimator lenses and the toothed wheel itself.  This latter was 35.5 mm. diameter, and comprised 150 triangular teeth.  It differed from Cornu’s 150 toothed wheel only in the thickness of the aluminum, being 0.8 mm thick.  The stage was thus set for an interesting comparison between Cornu’s results and those of Perrotin, using virtually the same items of equipment, but separated in time by about 28 years.

Perrotin and Prim’s Results

As soon as the necessary reductions had been carried out by Perrotin, it was announced that the La Gaude series gave a value of 299,900 ± 80 Km/s.33 The mean date was 1900.4.  Under similar circumstances, Perrotin published the Mont Vinaigre result as 299,860 ± 80 Km/s for a mean date of 1902.4.34  On this occasion, he discussed both results and issued their average as 299,880 ± 50 Km/s, the mean date being 1901.4.35  This later figure was frequently quoted as Perrotin’s final definitive value.

Perrotin began writing up all the details of the two series, but had not completed the task at the time of his death in 1904.  Prim continued writing the discussion, issued in 1908, re-working the calculations in the process.  Prim’s discussion makes no mention whatever of the reports that were issued earlier.  Instead, he derived a value for the La Gaude series of 300,032 ± 215 Km/s.  However, he was unhappy with the method used to obtain the result.  The other option that he considered was a least-squares treatment, but he felt that the number of observations over the La Gaude path was too few to justify this approach.  Consequently, this La Gaude value was completely discarded as unsatisfactory.36 Prim then re-appraised the Mont Vinaigre experiments and treated them by the least-squares method to yield a final value of 299,901 ± 84 Km/s at the mean date of 1902.4.37 Although Dorsey criticized this value (he did not comment on the earlier reports), he did not derive a better one, despite extensive analysis.

The issuing of this final discussion resulted in some confusion, particularly since Prim’s final declared value of 299,901 Km/s for the Mont Vinaigre path was so close to the Perrotin value for the La Gaude path of 299,900 Km/s.  Many failed to realize that they were from a different series of experiments over a different base-line.  Even Michelson fell into this trap, quoting this final declared value as having been obtained over the shorter La Gaude path.38

Conclusions from Toothed-Wheel Experiments

Table 5 summarizes the above results by listing the fourteen values obtained by this method.  A least squares linear fit to all these data points gives a light speed decline of 164 Km/s per year, while a fit to the most reliable values, marked [*], gives a decline of 2.17 Km/s per year.  The confidence interval was 99.4% that c was not constant at its present value during the period covered by these experiments.  The Fizeau values have been described as pioneering experiments that were “admittedly but rough approximations…intended to ascertain the possibilities of the method.”39  The remaining  unstarred values were rejected by the experimenters themselves or have been severely criticized as outlined above.  Nevertheless, when all the data points are included, the decrease in the speed of light is even more evident.

The comparison between Cornu’s re-worked values and those obtained with substantially the same items of equipment by Perrotin is of great interest.  The Cornu mean is 299,945 Km/s, while the Perrotin mean is 299,887 Km/s.  The drop measured by this same equipment is thus 58 Km/s in 26.6 years.  It is fascinating to note that both Perrotin and Prim, despite problems with the analysis, independently obtained results in which the earliest determination resulted in the higher value for c.

If we ignore the bad citation of 1855, the only values that went against the decay trend in the entire of Table 5 were those rejected by the experiments themselves or those severely criticized by others.  When combined with the results of the two astronomical type determinations, the decay trend now appears in three different methods of measuring c.  These methods have involved about 30 different c measuring instruments, comprising at least six by the Roemer method, at least 21 for aberration, and at least three by the Fizeau method. A decay trend measured by 30 different instruments lengthens the odds against coincidence to roughly one in a billion.

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