MATHEMATICS 7 



things related, it follows that the science of mathematics 

 which employs them ultimately concerns real sub- 

 stantial things themselves, to the study of one aspect of 

 which it is above all devoted. 



Everything has number. Larger and smaller groups 

 of similar things differ in number, and we can readily 

 express these differences by spoken or graphic signs to a 

 certain extent. But the limitation of our faculties 

 makes it impossible for us to think or speak of a very 

 extensive series of numbers by entirely distinct symbols. 

 Merely spoken signs we may at once put on one side, as 

 they are entirely devoid of the permanence requisite to 

 enable them to serve for scientific purposes. The art 

 of expressing numbers by means of written signs is 

 called notation. 



Through the eye, many such numerical symbols 

 are very readily recognisable. Such is, for example, 

 the case with the numerical symbols depicted on dice 

 or cards, and it is conceivable that specially gifted 

 individuals might be able to distinguish and recollect 

 several hundreds of such absolutely different and dis- 

 tinct numerical symbols. That however would, after 

 all, be of but little utility for the study of very high 

 numbers, wherein the most gifted imaginations would 

 soon be reduced to adopt the method employed by 

 ordinary persons. . The method usually adopted consists 

 in dealing with numbers in groups, and groups of 

 groups, and groups of groups of groups, and so on, 

 according to some regular system, which for one reason 

 or another has come to be the one adopted. 



In the four fingers and thumb of each hand, and the 

 five toes of each foot, man possesses an easy and ready 

 means of incipient enumeration, and the words used by 

 various savage tribes to denote numbers, plainly show 



