NOTION AND DEFINITION OF NUMBER. 



ji yf ANY essays have been written on the definition o number. 

 LV-L But most of them contain too many technical expressions, 

 both philosophical and mathematical, to suit the non-mathemati- 

 cian. The clearest idea of what counting and numbers mean may 

 be gained from the observation of children and of nations in the 

 childhood of civilisation. When children count or add, they use 

 either their fingers, or small sticks of wood, or pebbles, or similar 

 things, which they adjoin singly to the things to be counted or 

 otherwise ordinally associate with them. As we know from history, 

 the Romans and Greeks employed their fingers when they counted 

 or added. And even to-day we frequently meet with people to whom 

 the use of the fingers is absolutely indispensable for computation. 



Still better proof that the accurate association of such "other" 

 things with the things to be counted is the essential element of nu- 

 meration are the tales of travellers in Africa, telling us how African 

 tribes sometimes inform friendly nations of the number of the enemies 

 who have invaded their domain. The conveyance of the informa- 

 tion is effected not by messengers, but simply by placing at spots 

 selected for the purpose a number of stones exactly equal to the 

 number of the invaders. No one will deny that the number of the 

 tribe's foes is thus communicated, even though no name exists for 

 this number in the languages of the tribes. The reason why the 

 fingers are so universally employed as a means of numeration is, 

 that every one possesses a definite number of fingers, sufficiently 

 large for purposes of computation and that they are always at hand. 



Besides this first and chief element of numeration which, as we 





