TEE THEORY OF RELATIVITY 439 



have moved in consequence of the motion; second, since B's clock is 

 running slow, the time taken for light to traverse this too great dis- 

 tance is itself too great. Now if too great a distance is traversed in too 

 great a time, then the velocity will |-» „# "n T\' 



remain the same provided the factor j 1 j V 



which multiplies the distance is the fig. 7. 



same as that which multiplies the time. 



But unfortunately, or fortunately, a very little mathematics shows 

 that this multiplier is not the same. A sees too short a distance 

 being traversed by light in a second of time, and therefore 5's yardstick 

 is too short, and by an amount depending on the relative velocity of A 

 and B. Thus we are led to the astonishing general conclusion of the 

 relativity theory with reference to length: If two bodies are moving 

 relative to each other, then to an observer on the one, the unit of length 

 of the other, measured in the direction of this relative velocity, appears 

 to be shortened by an amount depending on this relative velocity. This 

 shortening must not be looked upon as due to the resistance of any 

 medium, but, as Minkowski puts it, must be regarded as purely a gift 

 of the gods, a necessary accompaniment of the condition of motion. The 

 same objection might be raised here as in the case of the time unit. 

 Perhaps the length of the yardstick appears to change, but does the real 

 length change ? But the answer is, there is no way of determining the 

 real length, or more exactly, the words real length have no meaning. 

 Neither A nor B can determine whether he is in motion or at rest abso- 

 lutely, and if B compares his measure with another one traveling with 

 him, he learns nothing, and if he compares it with one in motion rela- 

 tive to him, he finds the two of different length, just as A did. 



This startling fact, that a railway train as it whizzes past us is 

 shorter than the same train at rest, is at first a trifle disturbing, but 

 how much of our amazement is due to our experience, or lack of it. A 

 certain African king, on beholding white men for the first time, reasoned 

 that as all men were black, these beings, being white, could not be men. 

 Are we any more logical when we say that since in our experience no 

 yardsticks have varied appreciably on account of their velocity, hence it 

 is absurd to admit the possibility of such a thing. 



Perhaps it might be well at this point to give some idea of the size 

 of these apparent changes in the length of the time unit and the space 

 unit, although the magnitude is a matter of secondary importance. The 

 whole history of physics is a record of continual striving after more 

 exact measurements, and a fitting of theory to meet new corrections, 

 however small. So it need not occasion surprise to learn that these 

 differences are exceedingly minute ; the amazing thing, and the thing of 

 scientific interest, is that they exist at all. If we consider the velocity of 



