MATHEMATICS 



But perhaps an idea of the nature of the very 

 complex mathematics which underlies this theory 

 may be given. It is a question of what the mathe- 

 matician calls an invariant, and Einstein's general 

 position is that the group of interrelated phenomena 

 \ which constitute the universe must have some- 

 thing of an invariant nature about them and it 

 is not at all obvious that the laws we previously 

 built up for their action are in accord with such 

 a specification, though they may be nearly so in 

 accord. 



Now what is an invariant ? Everybody really 

 knows, though, like M. Jourdain, who talked prose 

 without knowing it, one may use an invariant in 

 ordinary reasoning without being aware of the 

 fact. The Principle of the Conservation of Energy 

 is perhaps the invariant most commonly used. 

 If we have any system of moving bodies, which is 

 self-contained, then, however the configuration of 

 the system may be changing in accordance with 

 dynamical laws, there is always a function, of the 

 positions and motions at any time, which is not 

 changing but preserving its value, and we call it 

 the total ILnergy. It expresses an intrinsic property 

 of the system, with a meaning which is in no way 

 dependent on the particular manner in which we 

 choose to measure the positions of the bodies at 

 any instant. As all will know, such measurements 

 depend on the use of co-ordinates, Cartesian, 



