March 12, 1920] 



SCIENCE 



259 



Granting, then, the principle of equivalence, 

 we can so choose axes at any point at any in- 

 stant that the gravitational field will disappear; 

 these axes are therefore of what Eddington 

 calls the " Galilean " type, the simplest pos- 

 sible. Consider, that is, an observer in a box, 

 or compartment, whidh is falling with the ac- 

 celeration of the gravitational field at that 

 point. He would not be conscious of the field. 

 If there were a projectile fired off in this com- 

 partment, the observer would describe its path 

 as being straight. In this space the infinitesi- 

 mal interval between two space-time points 

 would then be given by the formula 



dsr = dx\ -\- dx\ -|- dar^ -j- dx'i, 



where ds is the interval and x^, x^, x^, x^, are co- 

 ordinates. If we make a mathematical trans- 

 formation, i. e., use another set of axes, this 

 interval would obviously take the form 



ds' = gndx^i -f- g^a^xK -J- gssdx-, -\- g^dx^ 



-{- 2g^.,dXidx2 -{■ etc., 



where x^, x„, x^ and x, are now coordinates re- 

 ferring to the new axes. This relation involves 

 ten coefficients, the coefficients defining the 

 transformation. 



But of course a certain dynamical value is 

 also' attached to the g's, because by the transfer 

 of our axes from the Galilean type we have 

 made a change which is equivalent to the in- 

 troduction of a gravitational field; and the 

 £f's must specify the field. That is, these fir's 

 are the expressions of our experiences, and 

 hence their values can n!ot depend upon the 

 use of any special axes; the values must be the 

 same for all setections. In other words, what- 

 ever function of the coordinates any one g is 

 for one set of axes, if other axes are chosen, 

 this g must still be the same function of the 

 new coordinates. There are ten j/'s defined by 

 differential equations; so we have ten covariant 

 equations. Einstein showed how these g's 

 could be regarded as generalized potentials of 

 the field. Our own experiments and observa- 

 tions upon gravitation have given us a certain 

 knowledge concerning its potential ; that is, we 

 know a value for it which must be so near the 

 truth that we can properly call it at least a first 

 approximation. Or, stated differently, if Ein- 



stein succeeds in deducing the rigid value for 

 the gravitational potential in any field, it must 

 degenerate to the Newtonian value for the 

 great majority of cases with which we have 

 actual experience. Einstein's method, then, 

 was to investigate the functions (or equations) 

 which would satisfy the mathematical condi- 

 tions just described. A transformation from 

 the axes used by the observer in the following 

 box may be made so as to introduce into the 

 equations the gravitational field recognized by 

 an observer on the earth near the box ; but this, 

 obviously, would not be the general gravita- 

 tional field, because the field changes as one 

 moves over the surface of the earth. A solu- 

 tion found, therefore, as just indicated, would 

 not be the one sought for the general field ; and 

 another must be found which is less stringent 

 than the former but reduces to it as a special 

 case. He found himself at liberty to make a 

 selection from among several possibilities, and 

 for several reasons chose the simplest solution. 

 He then tested this decision by seeing if his 

 formulae would degenerate to Newton's law for 

 the limiting case of velocities small when com.- 

 pared with that of light, because this condi- 

 tion is satisfied in those cases to which New- 

 ton's law applies. His formulae satisfied this 

 test, and he therefore was able to announce a 

 " law of gravitation," of which Newton's was a 

 special form for a simple case. 



To the ordinary scholar the difficulties sur- 

 mounted by Einstein in his investigations ap- 

 pear stupendous. It is not improbable that 

 the statement which he is alleged to have 

 made to his editor, that only ten men in the 

 world could understand his treatment of the 

 subject, is true. I am fully prepared to be- 

 lieve it, and wish to add that I certainly am 

 not one of the ten. But I can also say that, 

 after a careful and serious study of his papers, 

 I feel confident that there is nothing in them 

 which I can not understand, given the time to 

 become familiar with the special mathematical 

 processes used. The more I work over Ein- 

 stein^s papers, the more impressed I am, not 

 simply by his genius in viewing the problem, 

 but also by his great technical skill. 



Following the path outlined, Einstein, as 



