36 SIGNS AND THEIR LAWS. [CHAP. if. 



changing its sign. This is in accordance with the algebraic rule 

 of transposition. 



But instead of dwelling upon particular cases, we may at once 

 affirm the general axioms : 



1st. If equal things are added to equal things, the wholes are 

 equal. 



2nd. If equal things are taken from equal things, the re- 

 mainders are equal. 



And it hence appears that we may add or subtract equations, 

 and employ the rule of transposition above given just as in com- 

 mon algebra. 



Again : If two classes of things, x and y, be identical, that is, 

 if all the members of the one are members of the other, then 

 those members of the one class which possess a given property z 

 will be identical with those members of the other which possess 

 the same property z. Hence if we have the equation 



then whatever class or property z may represent, we have also 



zas = zy. 



This is formally the same as the algebraic law : If both mem- 

 bers of an equation are multiplied by the same quantity, the 

 products are equal. 



In like manner it may be shown that if the corresponding 

 members of two equations are multiplied together, the resulting 

 equation is true. 



14. Here, however, the analogy of the present system with 

 that of algebra, as commonly stated, appears to stop. Suppose it 

 true that those members of a class x which possess a certain pro- 

 perty z are identical with those members of a class y which pos- 

 sess the same property z, it does not follow that the members of 

 the class x universally are identical with the members of the 

 class y. Hence it cannot be inferred from the equation 



zx = zy, 

 that the equation 



x = y 



is also true. In other words, the axiom of algebraists, that both 



