Theorem of the Addition of Velocities. The Experiment of Fizeau
 
Now in practice we can move clocks and measuring-rods only with velocities that are small compared with the velocity of light; hence we shall hardly be able to compare the results of the previous section directly with the reality. But, on the other hand, these results must strike you as being very singular, and for that reason I shall now draw another conclusion from the theory, one which can easily be derived from the foregoing considerations, and which has been most elegantly confirmed by experiment.
 
In Section VI we derived the theorem of the addition of velocities in one direction in the form which also results from the hypotheses of classical mechanics—This theorem can also be deduced readily horn the Galilei transformation (Section XI). In place of the man walking inside the carriage, we introduce a point moving relatively to the co-ordinate system K′ in accordance with the equation
x prime equals w t Superscript prime Baseline period
By means of the first and fourth equations of the Galilei transformation we can express x′ and t′ in terms of x and t, and we then obtain
x equals left-parenthesis v plus w right-parenthesis t period
This equation expresses nothing else than the law of motion of the point with reference to the system K (of the man with reference to the embankment). We denote this velocity by the symbol W, and we then obtain, as in Section VI,
upper W equals v plus w (A)
But we can carry out this consideration just as well on the basis of the theory of relativity. In the equation
x prime equals w t prime
we must then express x′ and t′ in terms of x and t, making use of the first and fourth equations of the Lorentz transformation. Instead of the equation (A) we then obtain the equation
upper W equals StartStartFraction v plus w OverOver 1 plus StartFraction v w Over c squared EndFraction EndEndFraction (B)
which corresponds to the theorem of addition for velocities in one direction according to the theory of relativity. The question now arises as to which of these two theorems is the better in accord with experience. On this point we axe enlightened by a most important experiment which the brilliant physicist Fizeau performed more than half a century ago, and which has been repeated since then by some of the best experimental physicists, so that there can be no doubt about its result. The experiment is concerned with the following question. Light travels in a motionless liquid with a particular velocity w. How quickly does it travel in the direction of the arrow in the tube T (see the accompanying diagram, Figure 3) when the liquid above mentioned is flowing through the tube with............