MONISM IN ARITHMETIC. jg 



oo. We find, further, that in transferring the notions greater and 

 less to these symbols, -f oo is greater than any positive number, 

 however great, and oo is smaller than any negative number, how- 

 ever small. We read these new signs, accordingly, "plus infinitely 

 great" and "minus infinitely great." 



But even here arithmetic has not reached its completion, al- 

 though the combination of as many previously defined numbers as 

 we please by as many previously defined operations as we please 

 will stilllead necessarily to some previously defined number. Every 

 science must make every possible advance, and still one step in ad- 

 vance is possible in arithmetic. For in virtue of the laws of com- 

 mutation and association, which also fortunately obtain in multipli- 

 cation, just as we advance from addition to multiplication, so here 

 again we may ascend from multiplication to an operation of the third 

 degree. For, just as for a -j- a -(- a -f- a we read 4.0, so with the same 

 reason we may introduce some more abbreviated designation for 

 a. a. a. a. The introduction of this new operation is in itself simply 

 a matter of convenience and not an extension of the ideas of arith- 

 metic. But if after having introduced this operation we repeatedly 

 apply the monistic principle of arithmetic, the principle of no ex- 

 ception, we reach new means of computation which have led to un- 

 dreamt of advances not only in the hands of mathematicians but 

 also in the hands of natural scientists. The abbreviated designation 

 mentioned, which, fructified by the principle of no exception, can 

 render science such incalculable services, is simply that of writing 

 for a product of b factors of which each is called a, #*, which \vc 

 read a to the b th power. Here a new direct operation, that of / 

 hition, is defined, and from now on we are justified in distinguishing 

 operations which are not inverses of others, as addition, multiplica- 

 tion, and involution, by numbers of degree. Addition is the direct 

 operation of the first degree, multiplication that of the second de- 

 gree, and involution that of the third degree. In the expression 

 a b the passive number a is called the base, the aotive number b the 

 exponent, the result, the power. 



But whilst in the direct operations of the first and second de- 

 gree, the laws of commutation and association hold, here in involv 



