A.— MATHEMATICAL AND PHYSICAL SCIENCES. 21 



up Lis mind and renounced the whole movement. The present position, 

 then, is that the years 1918-1926 have been spent chiefly in researches 

 which, while they have contributed greatly to the progress of geometry, 

 have been on altogether wrong lines so far as physics is concerned, and we 

 have now to go back to the pre-1918 position and make a fresh start, with 

 the definite conviction that the geometry of space-time is Riemannian. 



Granting then this fundamental understanding, we have now to in- 

 quire into the axiomatics of the theory. This part of the subject has 

 received less attention in our country than elsewhere, perhaps because 

 of the more or less accidental circumstance that the most prominent and 

 distinguished exponents of relativity in England happened to be men 

 whose work lay in the field of physics and astronomy rather than in mathe- 

 matics, and who were not specially interested in questions of logic and 

 rigour. It is, however, evidently of the highest importance that we should 

 know exactly what assumptions must be made in order to deduce our 

 equations, especially since the subject is still in a rather fluid condition, 

 and there is a possibility of effecting some substantial improvement in it 

 by a partial reconstruction of the foundations. 



What we want to do, then, is to set forth the axiomatics of general 

 relativity in the same form as we have been accustomed to give to the 

 axiomatics of any other kind of geometry — that is, to enunciate the 

 primitive or undefined concepts, then the definitions, the axioms, and 

 the. existence-theorems, and lastly the deductions. In the course of the 

 work we must prove that the axioms are compatible with each other, and 

 that no one of them is superfluous. 



The usual way of introducing relativity is to talk about measuring- 

 rods and clocks. This is, I think, a very natural and proper way of 

 introducing the doctrine known as ' special relativity,' which grew out of 

 FitzGerald's hypothesis of the contraction of moving bodies, and was 

 first clearly stated by Poincare in 1904, and further developed by Einstein 

 in 1905. But general relativity, which came ten years later, is a very 

 different theory. In general relativity there are no such things as rigid 

 bodies — that is, bodies for which the mutual distance of every pair of 

 particles remains unaltered when the body moves in the gravitational 

 field. That being so, it seems desirable to avoid everything akin to a 

 rigid body — such, for example, as measuring-rods or clocks — when we 

 are laying down the axioms of the subject. The axioms should obviously 

 deal only with the simplest constituents of the universe. Now if one of 

 my clocks or watches goes wrong, I do not venture to try to mend it 

 myself, but take it to a professional clockmaker, and even he is not always 

 wholly successful, which seems to me to indicate that a clock is not one 

 of the simplest constituents of the universe. Some of the expounders of 

 relativity have recognised the existence of this difficulty, and have tried 

 to turn it by giving up the ordinary material clock with its elaborate 

 mechanism, and putting forward in its place what they call an atomic 

 clock ; by which they mean a single atom in a gas, emitting light of 

 definite frequency. Unfortunately the atom is apparently quite as compli- 

 cated in its working as a material clock, perhaps more so, and is less 

 understood ; and the statement that the frequency is the same under all 

 conditions, whatever is happening to the atom, is (whether true or not) 



