SPACE AND TIME 177 



for neither at present has perceptual validity — that is, 

 exists in the world of real things. As conceptions both 

 are equally valid ; both are equally ideals, not involved 

 in our sense - impressions themselves, but which the 

 reasoning faculty has discovered and developed as a 

 means of classifying different types of sense-impressions 

 and of resuming in brief formulae their relationships and 

 sequences. 



Thus geometrical truths apply with absolute accuracy 

 to no group whatever of our sense-impressions ; but they 

 enable us to classify very wide ranges of phenomena by 

 aid of the notions of position, size, and shape. Geometry 

 enables us to predict with absolute certainty a variety of 

 relations between sense-impressions, when these impres- 

 sions do not involve more than a certain keenness in our 

 senses, more than a certain degree of exactness in our 

 measuring instruments. The absolute sameness and con- 

 tinuity demanded by geometrical conceptions do not exist 

 as limits in the world of perceptual experience, but only 

 as approximations or averages.^ In precisely the same 

 way the theory of atoms treats of ideal conceptions ; it 

 enables us to classify another and different range of sense- 

 impressions, and to formulate their mutual relations to 

 a certain degree of keenness again in our senses, or of 

 exactness in our scientific apparatus. Should the atom 

 become a perception as well as a conception, this would 

 not invalidate the usefulness of geometry. Very probably, 

 however, if we could magnify a football up to the size of 

 the earth, so that the perceptual atom, if it existed, would 

 have a size between small shot and a football, we should 

 find that the sense-impressions which the atom was con- 

 ceived to distinguish and resume, had themselves dis- 

 appeared under the new conditions.''^' In other words, our 

 scientific conceptions are valid for the world as we know 



^ Geometry might almost be termed a branch of statistics, and the defini- 

 tion of the circle has much the same character as that of Quetelet's Vhomme 

 nioyen. 



^ The visibility and tangibility of bodies may possibly be described by the 

 motion of atoms, but we cannot predict that a single atom would be either 

 visible or tangible, still less " bounded by a surface." 



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