6gS 



NA TURE 



[November 25, 1922 



word " geometry " is commonly understood, e.g. 

 by the Board of Education. 



One remark as to Newton and the apple, which 

 I intended to typify a supported observer and a 

 continuously falling observer, respectively. If, with 

 Mr. Cunningham, we take the apple to typify an 

 observer at first supported and afterwards free, the 

 apple's view of things is appallingly complicated — 

 compared even with Newton's. But that only the 

 more emphasises the point that the natural simplicity 

 of things may be distorted ad libitum by the process 

 of fitting into an unsuitable space-time frame. 



A. S. Eddington. 



Observatory, Cambridge, 

 November 3. 



I am obliged to the Editor for giving me an oppor- 

 tunity to add a few words in comment upon Prof. 

 Eddington's letter, and I do so in no captious spirit, 

 but because it seems to me that in these very funda- 

 mental discussions it is of the utmost importance 

 to clear away as many misunderstandings and 

 difficulties as possible ; to recognise that some 

 divergences are merely consequences of viewing the 

 same matter from different points of view, but that 

 others may be due to looseness of thought on one 

 side or the other ; and I am glad to be able to 

 recognise that most of the divergence of Prof. 

 Eddington's exposition of the meaning of Einstein's 

 theory from my own understanding of it is merely 

 part of the difference between our natural ways of 

 thinking. But two sentences in Prof. Eddington's 

 letter do sum up my difficulty in regard to his exposi- 

 tion so clearly that I would like to direct attention 

 to them. 



" He admits, however, that all measurements that 

 have ever been made are contained in the picture, 

 and, I might add, all measurements that ever will be 

 made. Thus we have a large number of measured 

 intervals available for discussion." 



In this sentence Prof. Eddington begs the whole 

 question with which I ventured to end my review 

 of his lecture. All measurements of length and all 

 measurements of time that were ever made are, 

 I agree, in the picture. But who ever measured 

 this physical "interval"? What is the absolute 

 scale of interval, and how is it applied ? Again in 

 Prof. Eddington's letter we read : ' ' Clearly if a wrong 

 geometrical system is used, the measured intervals 

 will expose it by their disagreement." Unfortunately 

 this is not at all clear to me, and I will try to explain 

 why. So far as I can see, all actual phvsical measure- 

 ments are records of observations of coincidences, 

 e.g. of marks on a scale with marks on another body. 

 That is to say, they correspond to intersections and 

 concurrences of world lines of distinct physical 

 elements. The significant feature of the four- 

 dimensional picture of the universe is therefore 

 merely the order of arrangement of such concurrences 

 along the world lines of these physical entities. All 

 else is of the nature of an arbitrarily adopted method 

 of description of these orders of arrangement and is 

 not contained in the picture itself. A geometrical 

 system is an analytical means of describing the 

 picture. The concurrences remain and their order 

 is unaltered, no matter how we change our geometrical 

 system. If I adopt a geometrical system other than 

 that of Einstein, I may find the mathematics more 

 complicated, but the actual observable facts recorded 

 are the same — just as the fact of the meeting of the 

 Great Northern, Great Eastern, Midland, and London 

 and North-Western Railways in Cambridge station 

 independent of any particular brand of map 



NO. 2769, VOL. I IO] 



or time-table. Of course a map which denied this 

 fact would be wrong — but the adoption of a different 

 geometrical system of attaching what I must not 

 call " interval " to the separateness of two events 

 does not break up a concurrence. It is just because 

 actual measurements will not be altered by any 

 change of the geometrical system that I cannot agree 

 with the sentence I have quoted. 



E. Cunningham. 

 St. John's College. Cambridge, 

 November n. 



The Time-Triangle and Time-Triad in 

 Special Relativity. 



Dr. Robb directs attention in Nature of October 

 28, p. 572, to the fact that there is much confusion 

 of thought with regard to the stationary value of 

 the integral fda in the special theory of relativity. 

 When the path is purely temporal, as Dr. Robb was 

 the first to point out, the integral is an absolute 

 maximum, not a minimum. Prof. Eddington has 

 also directed attention to this truth. The following 

 view may be of interest. I give mainly the results, 

 as the precise mathematical proof would occupy too 

 much both of space and time. 



Let A, B, C be the vertices (point-instants) of a 

 pure tune-triangle in the field of special relativity. 

 Suppose C precedes A, and A precedes B in proper 

 time ; then it may be proved that C precedes B, 

 i.e. proper time order is transitive. Then if cosh C 

 denotes the unit-scalar product of the vectors CA, CB, 

 and if a, p, 7 denote the real and positive intervals 

 BC, CA, AB, we have 



cosh C 



2a/3 



It may be proved that the expression on the right- 

 hand side is always positive and is greater than 

 unit}'. Thus C may be regarded as the real 

 invariant "hyperbolic angle" between the temporal 

 vectors CA and CB. This angle has the same metrical 

 value for all observers moving with uniform mutual 

 relative velocities. 



It can also be proved that a>/3. Hence, since 

 cosh C> 1, 



a>/3 + 7 . 



That is, the greatest side of a pure time-triangle is 

 greater than tin- sum of the other two sides. 



It follows at once that the stationary value of the 

 integral Ida, where the path is purely temporal, is 

 an absolute maximum. 



There is thus a real hyperbolic angle between any 

 two co-directional temporal vectors. The triangle 

 ABC has two real " internal " hyperbolic angles 

 (B and C), and one real " external " hyperbolic 

 angle A'. Besides the above formula we have 



cosh A'' 



cr-pJ 2 



cosh B 



y* + «*-fi* 



2.,-iy 27a 



Taking positive signs for intervals and angles, we 

 have 



sinh A' sinh B sinh C 



and cosh (B + C) = cosh A'. 



Thus the one real external angle of a time-triangle 

 is equal to the sum of the two real internal angles. 



The hyperbolic angle between two co-directional 

 temporal vectors has a perfectly definite physical 

 meaning, if the physics of special relativity is sound. 

 Let CA and CB be the time-axes used by two 



