770 



SCIENCE 



[N. S. Vol. XXXI. No. 803 



make such effects sensible. The velocities 

 which occur in some of the newly investi- 

 gated domains of physics are just as new 

 and outside our former experience as the 

 fifth dimension. 



Returning now to the magnitude of this 

 difference of opinion as to the distance be- 

 tween the clocks, it is easy to show that, 

 from our point of view, the moving ob- 

 server overestimates the distance in the 

 ratio 



l/(l-)3=). 



So that it may be said in general that 

 lengths in the direction of motion, which 

 lie says are equal, we say are unequal in 

 this same ratio. 



On lengths perpendicular to the direction 

 of motion our estimates agree. 



Now let us ask ourselves : What are cor- 

 responding lengths in the two systems? 

 Corresponding lengths may with propriety 

 be given the same name, "meter" for in- 

 stance. The condition that two lengths 

 should be "corresponding" is simply that 

 each observer comes to the same conclusion 

 with respect to the other length. 



The lengths AB and MN are not "cor- 

 responding, ' ' for the moving observer says 

 that MN is equal to AB, while we say AB 

 is less than MN, in the ratio (1 — ^S^). If, 

 however, we mark off on our platform a 

 length which is a mean proportion between 

 our estimate of the length AB and the 

 length MN, this length, say ME, will "cor- 

 respond" to the length AB, for we shall 

 then say, that AB is less than ME in the 

 ratio VI — /8^, while the moving observer 

 will say that ME is less than AB in the 

 same ratio. 



Thus any length, in the direction of mo- 

 tion, on a moving system is estimated less 

 in the ratio VI — P'^ by a "stationary" 

 observer. 



Or, put in a better way, an observer 

 viewing a system which is moving with re- 



spect to him, sees all lengths, in the direc- 

 tion of motion, shrunken in the proportion 

 y/1 — P^, where /3 is velocity with ivhich 

 the system is passing him in terms of the 

 velocity of light. 



We have now reached two results, which 

 we may summarize thus ; first, clocks which 

 a moving observer calls in unison do not 

 appear in unison to a "stationary" ob- 

 server, the clock in advance as regards 

 motion appeai'ing behind the other in time, 

 and second, distances in the moving sys- 

 tem appear shortened in the direction of 

 motion in the ratio VI — P'- In the above 

 we can, of course, interchange the words 

 "moving" and "stationary." 



Next let us turn our attention to the 

 unit of time in each system. It is not hard 

 to show that the unit of time in the moving 

 system will appear to it^ greater than ours 

 in the ratio 1/ VI — P'- This is due to the 

 fact that in the moving system forward 

 clocks are behind in time. 



In the measurement of time we assume 

 a certain standard motion to be taking 

 place at a constant rate and then take as a 

 measure of time the total displacement 

 which this motion has caused. Time 

 measurement with an ordinary clock is ob- 

 viously a special case of this general rule. 



The moving observer can adopt as his 

 unit of time the time it takes light, moving 

 with the characteristic^ space velocity V, 

 to travel a certain distance d and return 

 to him. 



Suppose d is in the direction of motion, 

 and the light after traveling a certain dis- 

 tance in the direction of motion is reflected 

 back to the observer. He will then write 

 t = d/V. 



We, however, "know" that he is overesti- 

 mating the distance d in the ratio 1/Vl— /^^ 



^That the moving observer's estimate of V can 

 not change with his velocity follows of course 

 from the first postulate. 



