4o6 POPULAR SCIENCE MONTHLY 



required to know how to define it, because in going back from defini- 

 tion to definition a time must come when we must stop. But at what 

 moment should we stop ? 



We shall stop first when we reach an object which falls under our 

 senses or that we can represent to ourselves ; definition then will become 

 useless; we do not define the sheep to a child; we say to him: See 

 the sheep. 



So, then, we should ask ourselves if it is possible to represent to 

 ourselves a point of space. Those who answer yes do not reflect that 

 they represent to themselves in reality a white spot made with the 

 chalk on a blackboard or a black spot made with a pen on white paper, 

 and that they can represent to themselves only an object or rather the 

 impressions that this object made on their senses. 



When they try to represent to themselves a point, they represent 

 the impressions that very little objects made them feel. It is needless 

 to add that two different objects, though both very little, may produce 

 extremely different impressions, but I shall not dwell on this difficulty, 

 which would still require some discussion. 



But it is not a question of that; it does not suffice to represent one 

 point, it is necessary to represent a certain point and to have the means 

 of distinguishing it from an other point. And in fact, that we may be 

 able to apply to a continuum the rule I have above expounded and by 

 which one may recognize the number of its dimensions, we must rely 

 upon the fact that two elements of this continuum sometimes can and 

 sometimes can not be distinguished. It is necessary therefore that we 

 should in certain cases know how to represent to ourselves a specific 

 element and to distinguish it from an other element. 



The question is to know whether the point that I represented to 

 myself an hour ago is the same as this that I now represent to myself, 

 or whether it is a different point. In other words, how do we know 

 whether the point occupied by the object A at the instant a is the same 

 as the point occupied by the object B at the instant /?, or still better, 

 what this means? 



I am seated in my room; an object is placed on my table; during a 

 second I do not move, no one touches the object. I am tempted to 

 say that the point A which this object occupied at the beginning of 

 this second is identical with the point B which it occupies at its end. 

 Not at all; from the point A to the point B is 30 kilometers, because 

 the object has been carried along in the motion of the earth. We can 

 not know whether an object, be it large or small, has not changed its 

 absolute position in space, and not only can we not affirm it, but this 

 affirmation has no meaning and in any case can not correspond to any 

 representation. 



