PRINCIPLES OF NUMBER. 189 



may be discriminated from each other, and similarly with 

 the people, then there arises a resemblance between the 

 chairs and people, and this resemblance in number may be 

 the ground of inference. If on another occasion the chairs 

 are filled by people again, we may infer that these people 

 must resemble the others in number, though they need not 

 resemble them in any other points. 



Groups of units are what we really treat in arithmetic. 

 The number Jive is really i + i + i + i + i, but for the sake 

 of conciseness we substitute the more compact sign 5, or 

 the name five. These names being arbitrarily imposed in 

 any one manner, an indefinite variety of relations spring 

 up between them which are not in the least arbitrary. If 

 we define four asi + i+i + i, and five asi + i + i + i + i, 

 then of course it follows th&ijive - four + i ; but it would 

 be equally possible to take this latter equality as a defi 

 nition, in which case one of the former equalities would 

 become an inference. It is hardly requisite to decide how 

 we define the names of numbers, provided we remember 

 that out of the infinitely numerous relations of one number 

 to others, some one relation expressed in an equality 

 must be a definition of the number in question and the 

 other relations immediately become necessary inferences. 



In the science of number the variety of classes which 

 can be formed is altogether infinite, and statements of 

 perfect generality may be made subject only to difficulty 

 or exception at the lower end of the scale. Every existing 

 number for instance belongs to the class 



m + 7 ; 



that is, every number must be the sum of another number 

 and seven, except of course the first six or seven numbers, 

 negative quantities not being here taken into account. 

 Every number is the half of some other, and so on. The 

 subject of generalization, as exhibited in arithmetical or 

 mathematical truths, is an indefinitely wide one. In 



