WILSON AM) LEWIS. — RELATIX ITV. 391 



The further characteristics of this transformation must l)e deter- 

 mined Uy a study of the important physical invariants. Funchimental 

 amonfj tliese invariants is the velocity of lijrht, which by the second 

 postulate of the principle of relativity must he the same to all observ- 

 ers. Hence any line in our four dimensional manifold which repre- 

 sents motion with the velocity of light must hear the same relation 

 to every set of reference axes. This is a condition which certainly 

 cannot he fulfilled by any transformation of axes to which we are 

 accustomed in real Euclidean space. It is indeed a condition sufficient 

 to determine the properties of that non-Euclidean geometry which we 

 are to investigate. 



Minkowski, in his two papers on relativity,^ used two different 

 methods. In his first ami elaborate treatment of the subject he in- 

 troduced the imaginary unit V— 1 in such a way that the lines w^hich 

 represent motion with the velocity of light become the imaginary 

 invariant lines familiar to mathematicians who discuss the real and 

 imaginary geometry of Euclidean space. In this way, however, the 

 points of the manifold which represent a particle in position and time 

 become imaginary; the transformations are imaginary; the whole 

 method becomes chiefly analytical. In his second, a brief paper, 

 ^Minkowski makes use of certain geometrical constructions which 

 have their simplest interpretation only in a non-Euclidean geometry. 



4. It is the purpose of the present work to develop the four dimen- 

 sional non-Euclidean geometry which is demanded by the principle 

 of relativity, and to show that the laws of electromagnetics and 

 mechanics not only can be simply interpreted in this way but also are 

 for the most part mere theorems in this geometry. 



In the first sections we shall develop in some detail the non-Eucli- 

 dean geometry in two dimensions. For it is only by a thorough 

 comprehension of this simpler case that it is possible to proceed into 

 the more difficult domains involving three and four dimensions. This 

 part of the paper will be continued by a discussion of vectors and the 

 vector notation that will be employed. At this point it is possible 

 in a few simple cases to show the applications of the non-Euclidean 

 geometry to problems in kinematics and mechanics. 



The sections devoted to three dimensions will be occupied largely 

 with numerous analytical developments of the vector algebra, many of 

 which are directly applicable not only in space of higher dimensions 



^ Gesammelte Abhandlungen von Hermann Minkowski, Vol. 2, pp. 352- 

 404 and pp. 431-444. 



