igS ALGORITHM OF IMAGINART QUANTITIES. 



Answer to the the reason of this restriction is obvious ; because though 

 question on ^j^ — ft . and consequently a* = ^b, which are both re- 

 quantities. presented by unity, or 1, yet the cube root of the formm* is 

 fl, and of the liitter 1 : as is evident from the preceding 

 part of this paper, the first (1,) being produced from the 

 cubing of a certain quantity a, and therefore having its 

 cube root — a ; and the other, being the product of two 

 unequal factors, possesses all the generality of 1 uncon- 

 ditionally assumed. 



Hence it appears, that there may be quantities equal to 

 each other, which, when placed under the same radicals, 

 lose their equality: the one o»" them being restricted to give 

 a paiticular root, and the other a different root; and there- 

 fore, in such cases, although the quantities and the radicals 

 are precisely the same, yet all equality between the two 

 ceases, and they must b6 considered as totally dissimilar 

 quantities. 



This is exactly the case in the examples proposed, as 

 may be made evident as follows: 



We have seen, that a* zr b, and b^ zz a; and if, in con- 

 sequence of this equality, we snake pur first result, ob«? 

 tainpd from squaring, viz. 



,* -i- 



Vb 



9 



+ 2 4- 



as we did in the nnmerical example, we should find, that; 

 in continuing the operation, we should not arrive at th^ 

 same result as in the former case; that is, by multiplying 

 again, in order to cube the expression, we should have 



A/a + Vb 



Va b + 2 V 



3 



4- Vb^ + 2 l^b + l/ab 



l + 3\/a+3^/b+l=: 



3 {\/a + Vb) + 2 

 which is different from the former result. 



The 



