May 20, 1922] 



NATURE 



635 



quantity-measures become hopelessly confused. Weyl 

 seems to have been the first to insist on keeping these 

 distinct. He brings a meaning into formulae which had 

 previously appeared to be artificial combinations. The 

 student who has with difficulty acquired some skill in 

 operating with tensors has to learn in addition how to 

 manipulate tensor-densities ; but the results repay the 

 extra labour. 



Einstein has amalgamated for us geometry and 

 mechanics. He has shown that if we have given an 

 exact specification of the geometry of a region of space 

 and time, that specification will also determine all the 

 mechanical properties existing in the region — gravita- 

 tional field, inertia, momentum, and stress. Einstein 

 accepted for this purpose the geometry of Riemann. 

 In 1918 Weyl showed that Riemannian geometry con- 

 tains a limitation which makes it appear inappropriate 

 to the description of a physical continuum from which 

 all action at a distance is excluded. He generalised 

 the geometry and so gave to the state of the world 

 additional degrees of freedom. Actually four addi- 

 tional quantities had to be fixed in this more general 

 specification of geometry ; and he identified these with 

 the four electromagnetic potentials. In this way the 

 whole electrodynamic scheme is associated with the 

 mechanical scheme, both being amalgamated with the 

 geometry of the world. There is, however, an element 

 of speculation in Weyl's unification which does not 

 appear in Einstein's ; the mechanical and geometrical 

 properties of the gravitational field are aspects of the 

 same phenomena ; the electrical and geometrical pro- 

 perties of the electromagnetic field are not shown to 

 be the same phenomena though they are supposed to 

 originate in the same source. 



Nearly half of Weyl's book is devoted to the de- 

 velopment on a logical basis of a system of geometry. 

 In this part we have to be content with laying a 

 foundation, with scarcely a hint of the well-known 

 phenomena of relativity which will follow. Knowing 

 Weyl's great reputation as a pure mathematician, we 

 felt some apprehension lest he should approach the 

 study of space as though it were a matter of pure 

 geometry. The fear was groundless. He recognises 

 fully that he is dealing with a physical subject ; and 

 in his geometry space is recognised at the outset as a 

 form of phenomena (p. 11), not a mere continuum of 

 n dimensions. Among the most novel investigations 

 is a justification of the Pythagorean metric (the quad- 

 ratic formula for the interval) by an argument involv- 

 ing the theory of groups. The reasoning is difficult to 

 follow. 



As the changes made in successive German editions 

 bear witness. Prof. Weyl is still developing his ideas. 

 We think that near the end of the present edition he 

 NO. 2742, VOL. 109] 



has reached conclusions which were not in his mind 

 at the beginning. Four pages from the end, after some 

 illuminating remarks on the two modes of transferring 

 a quantity from place to place by " persistence " and 

 by " adjustment " respectively, he decides that actual 

 transference by clocks and measuring-rods corresponds 

 to adjustment. Whilst this conclusion seems to be 

 undoubtedly correct, the reader has scarcely been pre- 

 pared for it, and indeed the existence of anything with 

 respect to which adjustment can be made has only been 

 demonstrated a few pages earlier. But the beginning 

 of the book needs reconsidering in the light of this 

 conclusion. How are we to reconcile the two following 

 statements ? 



(i) The same object, remaining what it is, could 

 equally well have been in some other place. The cor- 

 respondence between the portions of space occupied in 

 the two positions is called congruent transference (p. 11). 



(2) A measuring-rod even in a statical field does not 

 in general undergo a congruent transference (p. 308). 



. These are not the exact words, but I think that they 

 convey the sense intended. It would seem to follow 

 that a measuring-rod at another place and time is not 

 precisely the thing it was. But it must be remembered 

 that the statement (i) was enunciated as an axiom, 

 which we were expected to accept as a matter of 

 common experience ; it is no place for metaphysical 

 subtleties, which indeed Prof. Weyl is not hkely to 

 indulge in. There is, I believe, a direct contradiction 

 between the initial premises (i) and the final con- 

 clusion (2) which can only be removed by revising our 

 ideas as to the status of Weyl's ultra-Riemannian 

 geometry. In spite of its specialise'd character the 

 geometry of Riemann is the geometry of space and 

 time (" the form of phenomena "), as Einstein assumed. 

 Weyl's generalisation does not refer to actual space 

 and time ; but it gives us the needful mode of treat- 

 ment in graphical guise of those fundamental relations 

 which underlie the world of space and time and things. 



Not until the last third of the book do we enter on 

 the general theory of relativity. Then in a hundred 

 pages we hasten through all the main results, including 

 the re-formulation of mechanics to which Weyl has so 

 largely contributed. De Sitter's and Einstein's rival 

 views of a curved world are compared, and we gather 

 that the author is not so hostile as most continental 

 writers to the former. In either case, by noticing that 

 G2\/^ (not G^Jg) is the fundamental scalar-density of 

 zero dimensions, he is able to show that the cosmical 

 curvature-term appears naturally and inevitably. 

 Much of the more advanced theory depends on the 

 Hamiltonian method of stationary variation of a 

 volume-invariant. This is applied in two forms — 



(i) Variations arising from changes of the reference- 



