THE THEORY OF RELATIVITY 441 



mathematics of the new subject. Some of the conclusions, however, can 

 be understood without much mathematics. For example, we can no 

 longer speak of a particle moving in space, nor can we speak of an 

 event as occurring at a certain time. Space and time are not inde- 

 pendent things, so that when the position of a point is mentioned, there 

 must also be given the instant at which it occupied this position. The 

 details of this idea, as first worked out by Minkowski, may be briefly 

 stated. With every point in space there is associated a certain instant of 

 time, or to drop into the language of mathematics for a moment, a point 

 is determined by four coordinates, three in space and one in time. We 

 still use the words space and time out of respect for the memory of these 

 departed ideas, but a new term including them both is actually in use. 

 Such a combination, i. e., a certain something with its four coordinates, 

 is called by Minkowski a world 'point. If this world point takes a new 

 position, it has four new coordinates, and as it moves it traces out in 

 what Minkowski calls the world, a world-line. Such a world-line gives 

 us then a sort of picture of the eternal life history of any point, and the 

 so-called laws of nature can be nothing else than statements of the rela- 

 tions between these world-lines. Some of the logical consequences of this 

 world-postulate of Minkowski appear to the untrained mind as bordering 

 on the fantastic. For example, the apparatus for measuring in the 

 Minkowskian world is an extraordinarily long rod carrying a length 

 scale and a time scale, with their zeros in coincidence, together with a 

 clock mechanism which moves a hand, not around a circle as in the 

 ordinary clock, but along the scale graduated in hours, minutes and 

 seconds. 



Some of the conclusions of the relativity mechanics with reference 

 to velocity are worth noting. In the classical mechanics we were accus- 

 tomed to reason in the following way: Consider a body with a certain 

 mass at rest. If it be given certain impulse, as we say, it takes on 

 a certain velocity. The same impulse again applied doubles this veloc- 

 ity, and so on, so that the velocity can be increased indefinitely, and can 

 be made greater than any assigned quantity. But in the relativity 

 mechanics, a certain impulse produces a certain velocity, to be sure; this 

 impulse applied again does not double the velocity ; a third equal impulse 

 increases the velocity but by a still less amount, and so on, the upper 

 limit of the velocity which can be given to a body being the velocity of 

 light itself. This statement is not without its parallel in another 

 branch of physics. There is in heat what we call the absolute zero, a 

 value of the temperature which according to the present theory is the 

 lower limit of the temperature as a body is indefinitely cooled. No 

 velocity then greater than the velocity of light is admitted in the rela- 

 tivity mechanics, which fact carries with it the necessity for a revision 

 of our notion of gravitational action, which has been looked upon as 

 instantaneous. 



