Einstein from computations alone, without introducing any new constants or hypotheses whatever, showed, if the theory of relativity be accepted, that the sun should produce an acceleration of 43″ per century, thus entirely accounting for the observed discrepancy, far within the limits of accuracy of the observations. The only other planet whose orbit has a large eccentricity, and that is suitable for investigation, is the planet Mars. Here the discrepancy between observation and theory is very slight, only 4″, and a portion of that may be due to the attraction of the asteroids. This deviation is so slight that it may well be due entirely to accidental errors of observation, but however that may be, Einstein’s theory reduces it to 2″.7.
This all seems very satisfactory and complete, but the trouble with it is that the coincidence for Mercury is rather too good. It is based on the assumption that the sun is a perfect sphere, and that the density of its surface is uniform from the equator to the poles. This would doubtless be true if the sun did not revolve on its axis. In point of fact it does revolve, in a period in general of about 26 days. Consequently an object on its equator must experience a certain amount of centrifugal force. Therefore if its surface were of uniform density the shape of the sun would be an oblate spheroid.
It can be readily shown that the theoretical excess of the equatorial over the polar diameter, due to the centrifugal force, should amount to only 0″.04, an amount which could hardly be detected by observation, and might readily be concealed by a slight excess of equatorial over polar density. Any reasonable excess of density at the center would diminish this result but slightly. The molecular weight of the central material[13] is probably about 2. This computed equatorial excess is one-twelfth of the amount necessary to cause the observed advance, and should therefore cause an advance of the perihelion of about 3″.5 per century, reducing the difference between the observed advance and that caused by gravitation to 38″.5. According to Einstein the advance due to relativity should be, as we saw, 43″, a discrepancy of 4″.5 per century, or 10 per cent. Jeffreys has remarked that any discrepancy such as 10″ “would be fatal to a theory such as Einstein’s, which contains no arbitrary constituent capable of adjustment to suit empirical facts.”[14] It must be pointed out here however, that so far as known, this small correction to the motion of Mercury’s perihelion has not previously been suggested, so that there has been no opportunity hitherto for its criticism by others.
One of the eclipse photographs
The arrows pointing to the star-images have been inserted by hand; and the star-images themselves have had to be materially strengthened in order to make them show in the engraving at all.
Photograph submitted by Dr. Alexander McAdie, Harvard University, by courtesy of the Royal Observatory, Greenwich.
It was due largely to the success with Mercury that it was decided to put the relativity theory to another test. According to the Newtonian theory, as stated by Newton himself, corpuscles as well as planets have mass, and must therefore be attracted by the sun. According to Einstein, owing to their high velocity, this attraction must be twice as great as it would be according to the theory of gravitation. If the ray of light proceeding from a star were to pass nearly tangent to the sun’s limb it should be deflected 0″.87 according to Newton. According to the theory of relativity it should be deflected 1″.75. Stars of course cannot usually be observed near the sun. It is therefore necessary to take advantage of a total solar eclipse, when the sun is completely hidden by the moon, in order to secure these observations.
Two expeditions, one to Africa, and one to South America, observed successfully the total eclipse of May 29, 1919. The former was located on the Island of Principe in the Gulf of Guinea. The latter was located at Sobral, Brazil. Their equipment and results are shown in the following table, where the successive columns give the location, the aperture in inches of the telescopes employed, their focus in feet, the number of plates secured, the number of stars measured, their mean deduced deflection from their true positions by the attraction of the sun, and the deviations from the theoretical results.[15] In the first and last line of the table shown herewith, this
| Location | Aperture | Focus | Plates | Stars | Defl. | Dev. |
| Principe | 13 | 11 | 2 | 5 | 1″.60 | -0″.15 |
| Sobral | 13 | 11 | 19 | 12 | 0 .93 | (+0 .06) |
| Sobral,, | 4 | 19 | 8 | 7 | 1 .98 | +0 .23 |