WORLDS OF PHYSICS AND OF SENSE 115 



and smaller, while of any two of the set there is always 

 one that encloses the other, then we begin to have the 

 kind of conditions which would enable us to treat them 

 as having a point for their limit. The hypotheses required 

 for the relation of enclosure are that (1) it must be transi- 

 tive ; (2) of two different spatial objects, it is impossible 

 for each to enclose the other, but a single spatial object 

 always encloses itself ; (3) any set of spatial objects such 

 that there is at least one spatial object enclosed by them 

 all has a lower limit or minimum, i.e. an object enclosed 

 by all of them and enclosing all objects which are 

 enclosed by all of them ; (4) to prevent trivial exceptions, 

 we must add that there are to be instances of enclosure, 

 i.e. there are really to be objects of which one encloses 

 the other. When an enclosure-relation has these pro- 

 perties, we will call it a " point-producer." Given any 

 relation of enclosure, we will call a set of objects an 

 " enclosure-series ' if, of any two of them, one is 

 contained in the other. We require a condition which 

 shall secure that an enclosure-series converges to a point, 

 and this is obtained as follows : Let our enclosure-series 

 be such that, given any other enclosure-series of which 

 there are members enclosed in any arbitrarily chosen 

 member of our first series, then there are members of 

 our first series enclosed in any arbitrarily chosen member 

 of our second series. In this case, our first enclosure- 

 series may be called a " punctual enclosure-series." Then 

 a "point" is all the objects which enclose members of a 

 given punctual enclosure-series. In order to ensure infinite 

 divisibility, we require one further property to be added 

 to those defining point-producers, namely that any object 

 which encloses itself also encloses an object other than itself. 

 The " points " generated by point-producers with this pro- 

 perty will be found to be such as geometry requires. 



