NUMBER AND MAGNITUDE 645 



ception of the mathematical continuum of n dimensions may be led 

 up to quite naturally by a process similar to that which we discussed 

 at the beginning of this chapter. A point of such a continuum is 

 denned by a system of n distinct magnitudes which we call its co- 

 ordinates. 



The magnitudes need not always be measurable; there is, for in- 

 stance, one branch of geometry independent of the measure of mag- 

 nitudes, in which we are only concerned with knowing, for example, 

 if, on a curve ABC, the point B is between the points A and C, and 

 in which it is immaterial whether the arc A B is equal to or twice 

 the arc B C. This branch is called Analysis Situs. It contains 

 quite a large body of doctrine which has attracted the attention of 

 the greatest geometers, and from which are derived, one from another, 

 a whole series of remarkable theorems. What distinguishes these 

 theorems from those of ordinary geometry is that they are purely 

 qualitative. They are still true if the figures are copied by an un- 

 skilful draughtsman, with the result that the proportions are distorted 

 and the straight lines replaced by lines which are more or less curved. 



As soon as measurement is introduced into the continuum we have 

 just defined, the continuum becomes space, and geometry is born. But 

 the discussion of this is reserved for Part II. 



