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II. The Elements of Geometry. 

 By Norman Campbell, Se.D* 



Summary. 



IT is maintained that the geometry of Euclid is best 

 interpreted as an attempt to deduce as many important 

 propositions as possible from the assumption that length, 

 angle, area (and perhaps volume) are magnitudes uni- 

 versally measurable by the methods that are actually 

 employed in experimental physics. All his chief pro- 

 positions (in so far as they are true) can be deduced from 

 that assumption without any other. 



This view is supported, not by a detailed analysis of 

 the Elements, but by a very summary sketch of the laws 

 that must be true if the assumption is to be acceptable. 

 In a sequel it is hoped to discuss similarly the foundations 

 of another branch of experimental geometry with which 

 Euclid is not directly concerned — namely the geometry of 

 position, which involves the concept of " space." 



1. There was formerly much discussion whether geometry 

 was an experimental or a mathematical science. It is now 

 generally agreed that there are two closely connected 

 sciences, one mathematical and one experimental. The 

 former, which has been defined as the study of multi- 

 dimensional series, consists of a logical development of 

 ideas which have no necessary dependence on the experience 

 of the senses. It does not consist of laws and cannot be 

 proved or disproved by experiment ; it can enter into 

 relation with experimental science only through theories 

 and by suggesting hypotheses which, interpreted suitably, 

 predict laws. The formulation of such theories, in which 

 Minkowski was the pioneer, is one of the most striking 

 features of modern mathematical physics. The experi- 

 mental science, on the other hand, is meaningless apart 

 from experience, and its propositions are true or false 

 according as they agree or disagree with experiment. 

 They are the very fundamental laws which involve only 

 the geometrical magnitudes such as length, angle, or area. 

 It may be noted in passing that the laws predicted by 

 geometrical theories are not in general geometrical laws, 

 but involve electrical, optical, or dynamical concepts. 



* Communicated by the Author. 



