THE FOURTH DIMENSION. 



73 



other homogeneous things. These other things are even now, among 

 uncivilised peoples, the ten fingers of the two hands. They may, 

 however, be simple strokes, or, as in the case of dice and dominoes, 

 black points on a white background. Third, we substitute for the 

 result of this association some concise symbol or word ; for example, 

 the Romans substituted for three things counted, three strokes placed 

 side by side, namely: III ; but for greater numbers of things they 

 employed abbreviated signs. The Aztecs, the original inhabitants 

 of Mexico, had time enough, it seems, to express all the numbers 

 up to nineteen by equal circles placed side by side. They had ab- 

 breviated signs only for the numbers 20, 400, 8000, and so forth. 

 In speaking, some one same sound might be associated with the 

 things counted ; but this method of counting is nowadays employed 

 only by clocks : the languages of men since prehistoric times have 

 fashioned concise words for the results of the association in question. 

 From the notion of number, thus fixed as the result of counting, man 

 reached the notion of the addition of two numbers, and thence the 

 notion that is the inverse of the last process, the notion of subtrac- 

 tion. But at this point it clearly appears that not every problerr 

 which may be propounded is soluble ; for there is no number which 

 can express the result of the subtraction of a number from one 

 which is equally large or from one which is smaller than itself. The 

 primary school pupil who says that 8 from 5 "won't go" is per- 

 fectly right from his point of view. For there really does not exist 

 any result of counting which added to eight will give five. 



If humanity had abided by this point of view and had rested 

 content with the opinion that the problem "5 minus 8" is not solv- 

 able, the science of arithmetic would never have received its full 

 development, and humanity would not have advanced as far in civ- 

 ilisation as it has. Fortunately, men said to themselves at this 

 crisis: "If 5 minus 8 won't go, we'll make it go; if 5 minus 8 does 

 not possess an intelligible meaning, we will simply give it one." As 

 a fact, things which have not a meaning always afford men a pleas- 

 ing opportunity of investing them with one. The question is, then, 

 what significance is the problem "5 minus 8" to be invested with? 



