WILSON AND LKW IS. — RELATIVITY, 393 



fields of gravitational potential and gravitational force. Moreover, 

 it is shown that these fields are identical in mathematical form with 

 the electromagnetic fields, and that all the equations of the electro- 

 magnetic field must be directly applical)le to the gravitational. 



In an appendix a few rules for the use of Gibbs's dyadics, which have 

 occasionally been employed in the text, are stated. And a brief 

 discussion of some of the mathematical aspects of our plane non- 

 Euclidean geometry is given. 



The Non-Euclidean Geometry in Two Dimensions. 

 Translation or the Parallel Transformation. 



5. In discussing a non-Euclidean geometry various methods of 

 procedure are available; a set of postulates may be laid down, or 

 the difi'crential method of Ricmann may be followed, or the theory 

 of groups may be used as by Lie, or (if the geometry falls under the 

 general projective type, as is here the case) the projective measure 

 of length and angle may be made the basis. For our present purpose 

 we need not restrict ourselves to any one of these; but since the first 

 is familiar to all, we shall employ it as far as convenience permits. 

 Some of the other methods will, however, be briefly discussed in the 

 appendix, §§ 64, 60. 



With a \ie\v to simplicity we shall at first limit the discussion to the 

 case of a plane. Points and lines will be taken as undefined, and 

 most of the relations connecting them will be the same as in Euclidean 

 plane geometry. Thus : * 



1°. Through two points one and only one line can be drawn. 



2°. Two lines intersect in one and only one point, except that 



3°. Through any point not on a given line one and only one 

 parallel (non-intersecting) line can be drawn. 



4^. The line shall be regarded as a continuous array of points in 

 open order. 



6. In regard to congruence or "free mobility" it is important to 

 proceed more circumspectly than did Euclid. The transformations 

 of Euclidean geometry may be divided into translations and rotations, 

 of which the former alone are the same for our geometry. It seems 

 desirable, therefore, to discuss first and in some detail the postulates 



4 We make no claim of completeness or independence for these postulates, 

 wliich arc designed priiniirily to show tlio points of siniihirity or dissimilarity 

 between our geometrj- and the Euclidean. A like remark may be matlc with 

 re.spect to proofs of theorems. 



