ADAMS— THE EINSTEIN THEORY. 179 



the sun. In the same way, the two principles of physics that have 

 kept their vaHdity — the law of the conservation of energy and the 

 second law of thermodynamics — grew out of the failure to find any 

 violations of them. As is the case with these two laws, the prin- 

 ciple of relativity may be stated in a number of alternative ways. 

 From the gravitational point of view Jeans has stated this principle 

 " A planet cannot describe a perfect ellipse about the sun as focus," 

 and this statement expresses very distinctly the failure of the 

 Newtonian mechanics to account for all known physical phenomena. 



Now instead of trying to modify Newton's laws of motion, 

 Einstein goes back of them and uses views of space and time which 

 are different from those upon which the Newtonian mechanics is 

 founded. For the purpose of describing natural phenomena the 

 Euclidean space has almost universally been considered sufficient. 

 Whether or not Euclidean space represents anything which has 

 a real existence has been a doubtful question among mathematicians 

 from the earliest times. Other systems of geometry have been 

 developed, following closely the plan of Euclid, keeping some of his 

 axioms and rejecting others, and the consequences examined. 

 Riemann, however, in his essay on the " Hypotheses which are the 

 Foundation of Geometry " introduced a new system of geometry, and 

 the development of Riemann's geometry supplied the altered concep- 

 tion of space and time necessary for the Einstein theory. 



The Riemann geometry bears a relation to Euclidean geometry 

 somewhat analogous to the relation of direct action to action at a 

 distance in physics. According to Riemann, space is a three-dimen- 

 sional continuum, by which is meant that a point in space may be 

 represented continuously by three independent quantities, the co- 

 ordinates of the point. Riemann considered the more general prob- 

 lem of a continuum in which n independent coordinates are required 

 to specify a point, thus developing an w-dimensional geometry. In 

 order to define the metrical properties of space Riemann assumed 

 that the square of the distance between two infinitely near points 

 is a quadratic differential form of the relative cooordinates of the 

 points, with coefficients not constant, but functions of the coordi- 

 nates. In Euclidean space it is always possible to choose coordi- 

 nates — the usual rectangular coordinates — such that the square of 



