318 



MEMOIRS OF THE AMERICAN ACADEMY. 



wider definitions of these symbols. As we are to employ the usual algebraic signs 



as far as possible 



proper to begin by lay 



down defin 



of the various 



algebraic relations and operati 



The following will, perhaps, not be objected to. 



General Definitions of the Algebraic Signs. 



Inclusion in or being as small 



The 



quence holds 



that * 



If 



and 



then 



x 



y 



x 



y 



z 



z . 



Equality is the conjunction of being 



and 



converse. To say. that 



x 



y is to say that x 



y an ^ y 



x. 



Being less than is being as small as with the exclusion of its converse. 



To 



say 



that x < y is to say that x 



y, and that it is not true that y 



x. 



Being greater than is the converse of being less than, 

 say that y <C x. 



Addition is an associative operation. That is to say,f 



To say that x ]> y is to 



-t y) -f? * 



X 



(y-fc«) 



Addition is a commutative operation. That is, 



-ty 



y + 



Invertible addition is addition the corresponding inverse of which is determ 

 The last two formulas hold good for it, and also the consequence that 



If 



and 

 then 



x + y 

 x-\-y' 



y 



z 



z 



y 



* I use the sign 



-< in place of < My reasons for not liking the latter sign are that it cannot be written rapidly 

 enough, and that it seems to represent the relation it expresses as being compounded of two others which in reality 

 are complications of this. It is universally admitted that a higher conception is logically more simple than a lower 



one under it. Wh 



a broader concept is more simple than a narrower one included under it. Now all equality is inclusion in, but the 

 converse is not true ; hence inclusion in is a wider concept than equality, and therefore logically a simpler one. 

 On the same principle, inclusion is also simpler than being less than. The sign < seems to involve a definition by 



the 



t I write a comma below the sign of addition, except when (as -is the case in ordinary algebra) the corresponding 



determinat 





