On the other hand Turner has recently made a very interesting point,[9] namely, that the fourth dimension as actually treated by the mathematicians is not time itself, but time multiplied by a constant—the velocity of light.[10] Without affecting the astronomical proofs of relativity at all, this simplifies our conceptions enormously. In ordinary everyday life time and space cannot be identical, any more than a yard can be identical with a quart. On what is known to physicists as the centimeter-gram-second system, distance is represented by l, mass by m, and time by t. Velocity is then distance divided by time,
, or as we say in English units, so many feet per second, and the fourth dimension may be expressed as time multiplied by velocity,
. That is to say, it is simply distance, just like the other three dimensions. To say that time is the fourth dimension from this point of view, appears to us just as ridiculous as it would be to attempt to measure the velocity of a train in quarts. It is quite correct, however, although unusual, to speak of a given train as moving at a speed of 10 quarts per square inch per second,
. This would be equivalent to a velocity of 33 miles per hour.
If I wish to give a complete dimensional description of myself in my four dimensions, I must give my length, my breadth, and my thickness, ever since I came into being, and also the course I have traversed through space since that time. This latter distance will be expressed in terms of a unit whose length is 186,000 miles, the distance traversed by light in one second. The distance which I travel through space annually is enormous, and very complex as to direction. It involves not merely my own motions as I cross the room, or take a train or steamer, but also those due to the rotation of the earth on its axis, its revolution round the sun, and the motion of the latter through the heavens. In general I travel, or in other words increase my length in the fourth dimension, by over 4,000 units a year. The fourth dimension accordingly, if this view is accepted, is simply a distance like the other three, and perfectly easy to understand.
We now come to the three actual tests by which the theory has been tried. The planets as is well known revolve about the sun in ellipses, with the sun in one of the foci. That is to say, the sun is not in the center, but a little on one side of it. The end of the ellipse where the planet comes nearest to the sun is called the perihelion, and here the planet is moving most rapidly. The other end is called the aphelion, and here the motion is slowest. According to Newton’s theory of gravitation, if a spherical sun possesses a single planet or companion, its orbit will be permanently fixed in space unless perturbed by some other body. If a second planet exist, it will cause the perihelion of the first slowly to advance. According to Einstein the mass of a planet depends in part on its velocity. It will therefore be less at aphelion where it is moving slowly than at perihelion where it is moving rapidly, consequently in addition to the Newtonian attraction we have another one which increases as we approach the sun. The effect of this will be to cause the perihelion of the orbit to advance, whether there is a second planet or not.
Among the larger planets Mercury has the most eccentric orbit, and it also moves most rapidly, so that it is particularly well adapted to test the relativity theory. The observed advance of its perihelion is 574″ per century, instead of the theoretical figure 532″, due to the other planets—a difference of 42″.[11] This has long been a puzzling discrepancy between observation and the law of gravitation. Prior to Einstein, attempts were made to eliminate it by assuming a certain oblateness of the solar disk. If the equatorial diameter exceeded the polar by only 0″.5 the whole advance would be accounted for, but not only has this ellipticity failed of detection, but if it existed, it should produce a very noticeable and inadmissible change in the inclination of Mercury’s orbit, amounting to about 3″ per century, as has been demonstrated by both Herzer and Newcomb.[12]