February 5, 1920] 



NATURE 



599 



and use r' instead of r as our radial co-ordinate. 

 Whether we use (A) or any other expression, we have 

 to find out from the expression itself the meaning of 

 the co-ordinates introduced. In the limiting case m = o, 

 the above expression agrees with the formula for polar 

 co-ordinates and time in a Euclidean world; hence it 

 is usual to call r the distance from the sun and t the 

 time. Hut there can be no exact identification of 

 variables in a non-Euclidean world with quantities 

 the definition of which presupposes a Euclidean world ; 

 and the only exact definition of r and t is that thev 

 are mathematical intermediary quantities which 

 satisfy equation (A). The variable t is In no sense an 

 absolute time ; it is specifically associated with the 

 sun, which In equation (A) is' regarded as the onlv 

 mass in the universe worth considering. 

 _ Without troubling about the approximate iden- 

 tification of t with our common notion of 

 time, our results may be stated in the following 

 form :— At a point in the laboratory (r = const.), rff, 

 for a light vibration from a solar atom differs from dt.. 

 for a terrestrial atom. It follows from the formula 

 (.\) that dx, and is, will differ in the same ratio, since 

 we are now concerned only with the relation of it 

 and ds on the earth. The intermediarv auantltv t is 

 thus_ eliminated; and the difference" in the ' light 

 received from solar and terrestrial sources is an abso- 

 lute one, which it Is hoped the spectroscope will detect. 



A. S. Eddinctox. 



The Straight Path. 



In m\- book, -'A Theory of Time and Space," 1 

 directed attention to the fact that in the simple four- 

 dimensional time-space theorv there are three types 

 of plane in addition to three" types of line. 



On p. 360 I stated the following results : 



"If .A, B, C be the corners of a general triangle 

 all whose sides are segments of one kind, then : 



"(i) If the triangle lies in a separation plane, the 

 sum of the lengths of anv two sides is greater than 

 that of the third side. 



"(2) If the triangle lies in an optical plane, the 

 sum of the lengths of a certain two sides is equal to 

 that of the third side. 



"(3) If the triangle lies In an acceleration plane, the 

 sum of the lengths of a certain two sides is less than 

 that of the third side." 



These results were published in 1914, and, in spite 

 of the fact that they were printed in Italics, so that 

 he who runs might read (that is to say, provided anvone 

 should run on the occasion of reading mv book); yet 

 I still find writers continually making statements to 

 the effect that the straight line in this geometry is the 

 shortest distance between Its extremities. 



.As a matter of fact, what I call a "separation 

 line " lies in all three types of plane, and is, con- 

 sequently, neither a minimum nor a maximum, wffile 

 an "inertia line" can only lie in acceleration planes, 

 and can easily be seen to be a maximum in the mathe- 

 matical sense. Further, a triangle cannot have all 

 Its sides formed of segments of "optical lines." 



I have long contended that the ' usual method of 

 approach to what is generally called the "theory of 

 relativity" is quite inadequate, and this is a further 

 illustration of my contention. 



_ Not only are our ordinary ideas as to space and 

 time disturbed, but also our ideas of simultaneous- 

 ness and our notions of "straight lines" in the 

 resulting four-dimensional geometry. 



From the midst of this wreckage a logical theory 

 has to be constructed, and the difficulty is to find any 

 firm basis at all. 



In the course of my own work T succeeded in finding 

 XO. 2623, VOL. 104] 



what appears to be such a basis in the relations of 

 before and after. 



On this basis I found it possible to construct a 

 theory of time and space (apart from gravitation) 

 which led to the same equations as those of Einstein, 

 but of such a nature as to be independent of the 

 I)artlcular observer, and therefore truly physical and 

 devoid of the subjectivity which seems to cling to 

 Einstein's theory. 



These relations are, in fact, what might be described 

 .IS physical invariants, and, with the help of certain 

 postulates concerning them, they serve as a basis for 

 a system of geometry. 



If this Investigation had been published in the 

 German language it would doubtless have attracted 

 more attention on the part of British physicists, who 

 might then have added the ideas of before and after 

 to their store of fundamental physical concepts. 

 Instead of this, however, I have seen no mention of 

 them at all in recent discussions on the so-called rela- 

 tivity theory. It is true, of course, that no analysis 

 of Einstein's recent work has as yet been made in 

 terms of the relations of before and after, but seeing 

 that these have proved a sufficient basis for the simple 

 theory corresponding to Euclidean space, and that 

 such relations do actually hold in our experience, it 

 does not seem unreasonable to suppose that with 

 modified postulates thev might serve as a basis for the 

 more general theory. 



With regard, however, to my statement that the 

 straight line in the simple theorv is not the shortest 

 distance between its extremities, I can imagine some 

 people casting doubts upon mv veracity. For the 

 benefit of those who do not believe me, I venture to 

 give some simple arithmetical examples. 



Taking v as unity, the length .t of the segment of 

 a separation line b^ween elements the co-ordinates cf 

 which are (x,„ y„, ?„, t„) and Cr,, v,, Sj, t,) Is given 

 by the equation : 



s' = {x-xJ + {y,-y„Y + {z,-z„y-{t-t,)'. 



Let A, B, C,, C,, C, be elements the co-ordinates 

 of which are as follows : 



On substituting these values we get ; 



.AB=io 



.\C, = i3 C,B = i3 



AC,= 5 C„B= s, 



-AC = 



<".,B= ?, 



Thus we have 



AC,+C,B>AB 

 .\C„ + C:.B = .\B 



.ac;+c;b<ab 



For the case of an inertia line the length s is given 

 by the equation : 



s'^{t,-t„r-(x-x„Y-(y,-yJ'-{s,-z„)'. 



.As before, let .A, B, C be elements the co-ordinates 

 of which are as follows : 



A 0000 

 B o o o 10 

 C 4005 



Here .AB=io, A€ = 3, CB=3. 

 Thus AC-I-CB<AB. 



These 'examples should be sufficient to give an air 

 of plausibility to my statements. A. A. Robb. 



Cambridge, January 23. 



