MATTER IN MOTION 105 



other existing paths. Now we may generalize 

 the law of Galileo, that two bodies starting 

 together always remain together as they fall, 

 by stating that if the paths of any two test- 

 bodies are tangent at one point they must coin- 

 cide throughout their whole extent. This ex- 

 tremely simple theorem, although it still gives 

 no quantitative law of gravitation, nevertheless 

 alread}^ contains a considerable part of the 

 remarkable gravitation theory of Einstein. 



The theorem which we have just announced 

 may be deduced from another which is even 

 more general, namely. If in a given field a path 

 passes through two neighboring points, no other 

 path can pass through those two points without 

 coinciding completely with the first. In other 

 words, this path is a line which is uniquely de- 

 termined by the two points. Now you will re- 

 member that in the second chapter we chose to 

 regard the line uniquely determined by two 

 points as the straight line, and we are immedi- 

 ately arrested by the thought that perhaps 

 these paths in a gravitational field may be the 

 straight lines of some new geometry, in the sense 

 that a great circle is the straight line in the 

 geometry of navigation. This thought has in 



