416 PROCEEDINGS OF THE AMERICAN ACADEMY. 



the angle i/' whose hyperbolic tangent is the relative velocity which is 

 the same by either of the assumptions. 



If now we have a third (5)-line t" making an angle with the first 

 (5)-line, and cf)' with the second, where 4>' = 4> — 'A? and if we call the 

 relative velocities corresponding to these angles 



V = tanh 4>, ^' = tanh 4>' , u = tanh \j/, 



then it is not true that v' = v — u, but since ({)' = 4> — \p, 



, V — u 



V = 



1 — vu 



by (6). This is the theorem regarding the addition of velocities ob- 

 tained by Einstein. ^° The true significance of this result cannot be 

 emphasized too strongly, namely, that the velocity as such can only 

 be determined after a set of axes have been arbitrarily chosen; 

 relative velocity, however, has a meaning independent of any co- 

 ordinate system. Furthermore it is not the relative velocities, but 

 the non-Euclidean angles, which are their hyperbolic anti-tangents, 

 which are simply additive. If we were constructing a new system 

 of kinematics uninfluenced by the historical development of the 

 science, it might be preferable to make these angles fundamental 

 rather than the velocities. 



Suppose that from a given (5) -line we lay off successively equal 

 angles, so that each line determines with the preceding line the same 

 relative velocitj^ then the angle measured from the given line increases 

 without limit, but its hyperbolic tangent, which is the velocity relative 

 to this line, approaches unity, that is, the velocity of light. The 

 relative velocity, therefore, determined by any two (5)-lines Avhatever, 

 is less than the velocity of light. The velocity of light itself appears 

 the same regardless of the choice of coordinate axes. This is the sec- 

 ond postulate of the principle of relativity. Indeed if angle, instead 

 of relative velocity, had been made fundamental, the motion of light, 

 as compared with all other motions, would have been characterized 

 by an infinite value of the angle. 



19. Let us return to our figure and consider once more the lines 

 that have been marked /, t', and .r, x'. If we take the /-line as the locus 

 of a stationary particle, then all points along the line x or along any 

 parallel line are said to be simultaneous, for along any line perpendicu- 

 lar to the ^axis the value of t is constant. In like manner if we con- 



20 Einstein, Jahrb. d. Radioak, 4, 423. 



