8o6 



NATURE 



[February 17, 1921 



preted in terms of familiar ideas ; even to the most 

 revolutionary of mankind, familiarity is a source 

 of some satisfaction. The new theory is based on 

 ideas utterly unfamiliar, and it might be urged 

 that anything based on them must be the precise 

 contrary to explanation. But if we ask why we 

 are so ready to accept theories based on material 

 analogy, we shall find our reason in the fact that 

 such theories have actually turned out to possess 

 the amazing property of predicting unsuspected 

 laws. The theory of relativity also possesses that 

 property. Ought we not to extend, so as to in- 

 clude it, our notions of the proper limits of 

 physical theory, and to rid ourselves of the dis- 

 comfort of unfamiliarity by the simple process of 

 studying its ideas so closely that they become an 

 integral part of our mental equipment? 



It may be asked. Do theories, indeed, aim at 

 nothing but satisfactoriness and prediction? Is 



not their object rather to discover the true nature 

 of the real world? Such questions must be 

 answered by questions. Do physicists (1 say 

 nothing of mathematicians or philosophers) be- 

 lieve that anything is real for any reason except 

 that it is a conception of a true law or of a true 

 theory? Have we any reason to assert that 

 molecules are real except that the molecular 

 theory is true — true in the sense of predicting 

 rightly and interpreting its predictions in terms 

 of acceptable ideas? What reason have we ever 

 had for saying that thunder and lightning really 

 happen at the same time, except that the con- 

 ception of simultaneity which is such that this 

 statement is true makes it possible to measure 

 time-intervals? When these questions are 

 answered it will be time to discuss whether rela- 

 tivity tells us anything about real time and real 

 space. 



The Relation between Geometry and Einstein's Theory of Gravitation. 



By Dorothy Wrinch and Dr. Harold Jeffreys. 



nPHE term "geometry" has been used ever 

 -»- since the time of Euclid to denote two 

 completely distinct subjects ; but the formal simi- 

 larity of their propositions has been so close as 

 to obscure until recently the entire dissimilarity 

 of their status in scientific knowledge. The 

 Greek geometers seem to have been inspired 

 originally by the need for a satisfactory method of ' 

 surveying; at the same time, their logical turn of ^ 

 mind led them to present their results in the now 

 familiar form of a deductive science. The char- j 

 acteristics of such a science are that a certain 

 number of primitive propositions fj, now called ; 

 postulates, are stated at the beginning, and that i 

 from these, by a process of pure logic, further 

 propositions qj are one by one developed. But 

 this development is quite a separate process from 

 that of deciding whether the primitive propositions 

 are true or not, and if this is not done it is 

 impossible to assert that the deduced propositions 

 are true. 



Different sets of primitive propositions 

 po, p3, . . . would give different sets of deduced 

 propositions q^, q^, . . . and the complete working 

 out of these is a science in itself ; its results are 

 all, therefore, of the form " p^ implies q^," " p^ 

 implies qoi" ^"^^ so on. Euclid actually used in his 

 development several postulates which he never 

 explicitly stated, but which have been made ex- 

 plicit by modern writers ; our present object, how- 

 ever, is not to indicate these, but to consider his 

 geometry in the perfectly deductive form it would 

 have had if he had actually stated them. We 

 have noticed that in any other system in which 

 any one of Euclid's postulates is false, many of 

 his deduced propositions are also false. This, 

 however, does not affect his method in the least ; 

 all his arguments are independent of the truth 

 of the postulates, and in every case it is possible 

 NO. 2677, VOL. 106] 



to assert — and this is all the modern geometer 

 asserts — that if the postulates are true the pro- 

 positions are true. A system like Euclid's is, 

 therefore, a part of pure logic ; the large division 

 of pure logic that includes it as a very special 

 case is pure geometry. Of the many systems of 

 pure geometry now known, all are on just the 

 same footing, and there is no sense in which any 

 one of them is preferable to any other. 



Euclid's contemporaries, however, were not in- 

 terested merely in his logical method ; they wished 

 to identify the furrows in their fields with his 

 lines, and the fields themselves with his surfaces ; 

 and to have some justification for this it was 

 necessary to assume that his postulate's were true 

 of them. Only one example is needed to show 

 how formidable an assumption this was. In order 

 to prove one of his earliest propositions, Euclid 

 assumes that a triangle can be picked up, trans- 

 ported bodily, and deposited on top of another. 

 Imagine this process carried out when the tri- 

 angles are fields ! The impossibility of carrying 

 it out implies that a most important proposi'tion 

 was not proved for the very case to which they 

 contemplated applying his geometry, and hence 

 that, so far as the knowledge of that day went, 

 there was not the slightest reason for believing 

 that geometry was applicable for its original pur- 

 pose of earth-measurement. Yet its results, in 

 so far as they were capable of being applied in 

 actual surveying, seem to have been instantly 

 accepted. Why? It may have been due partly 

 to lack of disposition to criticise something that 

 the critics felt they could not have done better 

 themselves, a mental attitude that may perhaps 

 still occasionally exist ; but the chief reason 

 was probably that some of the deduced pro- 

 positions were directly verifiable, such as the 

 proposition that the equality of corresponding 



