946 



SCIENCE 



[N. S. Vol. XXXVII. No. 964 



symbol for hs^erbolic tangent, we may write 

 the equations of transformation in the form 



x^x' cosh + of sinh 0, 

 ct = x' sinh + ct' cosh 0. 



That is, if one end of the table, say B, is 

 apparently to the moving observer at a dis- 

 tance x' ahead of the other end, which is the 

 moving origin, A, then the stationary observer 

 knows that the real stationary distance is 

 x' cosh (}>. If the clock at B reads t' to the 

 moving observer, then the stationary observer 

 knows that the time which has elapsed from the 

 beginning of the motion is really t' cosh cj> ; and 

 at velocity v, this means that the origin A has 

 moved away from the stationary origin a real 

 distance vi' cosh <j) or ct' sinh (p. Hence the 

 real distance of B from the stationary origin 

 is 



x=^ x' cosh + ct' sinh 0. 



Also the stationary observer knows that the 

 clock at B is off from two causes, one its jKJsi- 

 tion, at a distance apparently x' from A, 

 which sets it back really by 



x'^' 



that is, 



vV{l-n 



x' sinh 



The other cause is that the time read on the 

 clock since the motion began is V , but the real 

 time as seen by the stationary observer, is 

 V cosh (^. Hence we have the equation 



ct = x' sinh + cV cosh 0. 



From these equations the stationary observer 

 could compute x' , which the moving observer 

 would think was the distance of B from his 

 moving origin, and the time t' on his moving 

 dock. We have 



xf ^^x cosh — ct sinh 0, 

 ct' = — X sinh + c< cosh 0. 



These equations evidently are much like the 

 first pair, and indeed we see that if we change 



' sinh = p/Vll ^^^), cosh0 = l/V(l— /S=)~, 

 Beoh0= V(l— /3^). 



the sign of (f>, that is, of /3, or finally of v — 

 which means that we imagine the moving ob- 

 server to be at rest and the stationary observer 

 to be relatively in motion — we have the second 

 set. We would therefore expect that if we 

 have two moving observers, with different ve- 

 locities V and v', we would find similar equa- 

 tions for their respective interpretations of 

 each other's data as to distance and time. 

 Thus indeed if 



X =: x" cosh \f/ -)- ct" sinh if/, 

 ct = — x" sinh i/' + ct" cosh ^, 



we find x' and ct' to be in terms of x" and ct" , 



a;' = re" cosh (0 — \p) — ct" sinh (0 — ^), 

 ct' ^= — x" sinh (0 — f) -\- ct" cosh (0 — ^t-). 



We see at once from this that the relative 

 velocity is not found by getting the difference 

 of the velocities v and v', but by getting the 

 difference of <p and i/', that is, the relative ve- 

 locity is 



t-anh (tanh"H' — tanh~*i''). 



After this long preliminary we come to the 

 paper before us which presents a full study 

 of the geometrical representation of these 

 facts, in a most elegant manner. The formulas 

 above are interpreted as representing a rota- 

 tion in a four-dimensional space, but not a 

 common space. The rotation in a common 

 space >vould involve the V — Ij and to pre- 

 serve the real numbers as reals, the space 

 chosen is a non-Euclidean space. After all, 

 the difference is really this, that certain terms 

 like rotation, perpendicular, etc., do not mean 

 what they ordinarily do, but have meanings 

 related to a given hyperbola, rather than to a 

 given circle. Thus really perpendicular lines 

 through the origin are conjugate diameters of 

 a circle whose center is the origin. In the 

 paper " perpendicular " still means conjugate, 

 but as to a hyperbola and not a circle. This 

 illustrates sufliciently the way in which the 

 terms appear. Only a careful study of the 

 paper itself can give a clear idea of the char- 

 acter of the presentation. The reader simply 

 needs to be on the alert as to the geometrical 

 meaning assigned here to familiar terms 

 whose meaning has been altered. 



