174 SCIENCE PROGRESS 



The logical implications of such assertions are at least as 

 important to the pure mathematician as to the physicist, and 

 this may serve as our excuse for discussing a theory already 

 mentioned from other aspects, under other headings in Science 

 Progress. Prof. Eddington begins by the development of 

 a geometry of tensors of an extremely general kind, which 

 is entirely mathematical, and only proceeds later to the con- 

 struction of a physical theory by identifying the tensors with 

 quantities which we measure experimentally ; the natural 

 gauge-system provides the link which enables this to be ac- 

 complished. 



The properties of tensors in relation to any transformation 

 of co-ordinates are now generally understood. Some of those 

 now introduced are invariant for alterations of gauge-system, 

 and Prof. Eddington calls them in-tensors, and their simplest 

 illustration is a displacement, say dx^, whose components are 

 differences of co-ordinates (pure numbers) and not related to 

 any gauge. These in-tensors are essentially a new mathematical 

 development in the older Riemann geometry, with possible 

 consequences of great interest. 



The most immediate question raised by Prof. Eddington is 

 as follows : Given an in-vector representing a displacement 

 at any point P, can we find an exactly equivalent displace- 

 ment at a point P^ infinitely near to P ? Such an equivalence 

 must be assumed as existing, if a physical theory is ultimately 

 to be established, for otherwise the continuum is structureless, 

 and there are no resemblances in nature. 



The points of a continuum are enumerated, in order, by any 

 system of co-ordinates, and no change of co-ordinates changes 

 this order, and some preliminary statement of order must be 

 made in the very definition of a displacement, or it cannot be 

 a mathematical expression of any physical or structural relation. 

 Dr. Robb appears to have solved this problem of preliminary 

 order in a work recently published, which we notice later. 

 Prof. Eddington clearly holds the belief that the basis of truth 

 in the quantum theory will ultimately be elucidated by the 

 present method. Without expressing belief or disbelief in this 

 conclusion, we may at least agree with his statement that it is 

 probable, and that one or two sentences in what has already 

 been postulated generally might require modification if pheno- 

 mena were really discontinuous in the quantum sense. 



Naturally, in an article of this nature, we cannot give a 

 detailed consideration of the generalised geometry of tensors 

 which Prof. Eddington develops. We can only refer readers 

 to it, with a statement of the fact that it is quite clearly a 

 marked advance upon the geometry of Riemann, and yet, as 

 the author feels, not yet the final possible advance. We have 



