71 G PROCEEDINGS OF THE AMERICAN ACADEMY. 



passes over the path mnm. If, however, the entire system is consid- 

 ered to be in absolute motion with a velocity v, the light must pass 

 over a different path mn'm' where nn' is the distance through which the 



m m' ga> > n if 



Figure 2. 



mirror moves before the light reaches it, and mm is the distance tra- 

 versed by the source before the light returns to it. 

 Obviously then, 



and 



Also from the figure, mn' = mn + nn', 



mn'm' = mnm + 2 nn' — mm'. 



Combining, we have 



mn'm' 1 1 



mnm >' 2 1 — /5 2 ' 



X — .^ 



c 

 Hence if we call the system in motion, instead of at rest, the calculated 

 path of the light is greater in the ratio _^ 2 . 



Now the velocity of light must seem the same to the observer, 

 whether he is at rest or in motion. His measurements of velocity de- 

 pend upon his units of length and time. We have already seen that a 



second on a moving clock is lengthened in the ratio , and 



therefore if the path of the beam of light were also greater in this same 

 ratio, we should expect that the moving observer would find no dis- 

 crepancy in his determination of the velocity of light. From the 

 point of view of a person considered at rest, however, we have j ust seen 



that the path is increased by the larger ratio -j 2 . In order to 



account for this larger difference, we must assume that the unit of 



length in the moving system has been shortened in the ratio * -. 



