Manchester Memoirs, Vol. xli. (1897), No. 10. 9 



So far as regards the representation of functional 

 relations by Cartesian co-ordinates, this is merely a mode 

 of expression. We have not assumed any property of the 

 assemblage of points representing a continuous function 

 which is not contained in the formal definition. 



It is otherwise with the fundamental conception of 

 continuous magnitude from which we started in § 1. As 

 the method here followed runs counter to the arithmetical 

 tendency which is so marked a feature of the modern 

 Theory of Functions, some remarks are called for in 

 justification. In the writer's opinion, the question as to 

 whether a geometrical or an arithmetical basis is the 

 more appropriate foundation for the Calculus cannot be 

 answered absolutely. It is a question of point of view. 

 But in some of the most important applications of the 

 Calculus, it would appear that the geometrical basis is 

 not only legitimate, but is imposed on us by the nature 

 of the case. As regards Analytical Geometry this is a 

 truism ; but it is perhaps not so generally recognised 

 that the whole of Mathematical Physics is in the same 

 case. 



A few instances will suffice to make this clear. In 

 Dynamics, the ideal clock consists of a point describing 

 a straight line ; it follows that time in this subject has 

 exactly the same kind of continuity as length. A velocity 

 is a distance described in some standard time; a mass 

 is the ratio of two velocities acquired in the same time 

 under certain conditions ; and so on. 



It is when we pass from the ideal representations 

 of things which we construct in Theoretical Physics to 

 the question of concrete measurement that the arith- 

 metical view of the subject claims special attention. 

 This lies, however, beyond the scope of the present 

 paper. 



