288 PROFESSOR DE MORGAN, ON THE 



2. The sign = is the only one of which the explanation is requisite 

 in the art of operation : it signifies an assertion of identity of operative 

 effect, and gives the right to substitute one side for the other, when 

 desired. Its use implies a postulate, the only one demanded : that a = b 

 gives A = B whenever A is derived from a by the same operations in 

 the same order, which produce B from b. 



3. The signs + and — are opposite in effect ; what one does the 

 other undoes : and is the symbol of a pair of such opposite operations 

 having been performed. Thus + a - a = 0. And such operations are 

 convertible in their order: thus + a — b + c = + c — b + a= —b + c + a, &c. 



4. The signs x and +- (or any substitutes for them) are opposite 

 in effect: and 1 is the symbol of a pair of such opposite operations 

 having been performed. Thus x a -*- « — 1. And these operations 

 are also convertible in their order : thus 



x a -r- b x c — xc-rixfl= -+ A x c x «, &c. 



5. The operations x and -J- are of a distributive character, when 

 performed upon the results of the operations + and — . Thus 



( + a) x ( + i — e) "m ( + a) x ( + b) + ( + a) x ( - c), &c. 



6. Like signs ( + and — ) produce + in all cases, and unlike signs 

 — . And like signs (x and ■*•) produce x in all cases, and unlike signs 

 ■*■ . And each pair of signs is, relatively to its own set, distributive. 



7. The signs and 1 may themselves be considered as subjects of 

 operation, and 1 + 1 is abbreviated into 2, 1 + 1 + 1 into 3, 1 + 1 + 1 + 1 

 into 4, and so on. 



8. The laws by which the symbol a b is used are a b x a c = a h+c and 

 (a b ) c = a hc . 



I believe the preceding rules to be neither insufficient nor redundant, 

 though I should be noways surprised to see them proved both the one 

 and the other; least of all if it were the latter. 



The most remarkable point in this separation is that the laws of 

 operation prescribe much less of connexion between the successive symbols 

 a + b, ab, and a h , than a person who has deduced these laws from an 



