342 MEMOIRS OF THE AMERICAN ACADEMY. 



I am not aware that Mr. Jevons actually uses this latter process, but it is open to 

 him to do so. In this way, Mr. Jevons's algebra becomes decidedly simpler even than 

 Boole's. 



It is obvious that any algebra for the logic of relatives must be far more compli- 

 cated. In that which I propose, we labor under the disadvantages that the rnulti- 

 plication is not generally commutative, that the inverse operations are usually inde- 

 terminative, and that transcendental equations, and even equations like 



a b* — cde* _±_ fx _{_a; ? 



where the exponents are three or four deep, are exceedingly common. It is obvious, 

 therefore, that this algebra is much less manageable than ordinary arithmetical algebra. 

 We may make considerable use of the general formulas already given, especially of 

 (1)> (21)> and (27), and also of the following, which are derived from them : — 



(8S.) If a -< b then there is such a term x that a 4- x = b 



— 

 ( 



If a — <^ b then there is such a term x that b,x = a . 

 If b,x = a then a ^ b . 



If a -< b c-fca -< c-fcb 



(88 

 (89 



(90.) Ifa-<$ co-<cb. 



(91 



(92.) If a -< b <* -< ca , 

 (93.) Ifa-<b &-<:¥. 

 (94.) «,*-<>. 



tVfc 



If a — <T h ac -<" he . 





There are, however, very many cases in which the formula) thus far given are of 



little avail. 



^ Demonstration of the sort called mathematical is founded on suppositions of par- 

 ticular cases. The geometrician draws a figure ; the algebraist assumes a letter to 

 signify a single quantity fulfilling the required conditions. But while the mathe- 

 matician supposes an individual case, his hypothesis is yet perfectly general, because 

 he considers no characters of the individual case but those which must belong to 

 every such case. The advantage of his procedure lies in the fact that the logical 

 laws of individual terms are simpler than those which relate to general terms, because 

 individuals are either identical or mutually exclusive, and cannot intersect or be sub- 



ordinated to one another 



fore, more restricted to matters of 



Mathematical demonstration is not, there 

 n than any other kind "of reasoning. In 



