ALGORITHM OF IMAGINARY QUANTITIES. > lg$ 



being cubed, give unity for the result; we say, therefore, trated by an 

 that 1 has three roots; and, if we are simply requested to ^^^™P'«* 

 extract the cube root of 1, we may make it either 1 or 

 — i + i V^ — 3, or — ^ — 4- V — 3 ; for the cube of each ©f 

 these quantities is represented by 1. But if we are asked, 

 ■what is the cube root of ( — ^ -|- ^ V — 3)', we must not say, 

 that ( — 7 -|- -J- -v/ — 3)* nz 1, and therefore its root is also 

 equal to 1 ; for, as in this case we know, that 1 is the re- 

 presentative of the cube of a particular quantity, its cube 

 root must necessarily be that quantity, and no other. * 



Hence it appears, that there is no ambigjaity in extract- No ambiguity 

 ing the roots of Quantities, which are known to be gene- '"'^'^ we know 



, J „ . * ^ 1.- I- ^' J? • .-. howthequaa- 



rated trom the constant multiplication or a given quantity utywasgene- 



by itself, whether this quantity be real or imaginary. But ""^ted, 

 if, by the multiplication of two unequal factors, we »^'''ve "j "J'^^'V^^* 

 at any result, and have afterwards to extract the root of that unequal fac- 

 quantity, there is then nothing in the nature of the case ^°"' 

 to limit the root. If, for example, we find, that the pro- 

 duct ( — 1 4- i -/— 3) X (— T — T -/— 3) =: l; and ive have 

 afterwards to take the cube root of this product ; there 

 is nothing to indicate, that we ought to take one root in 

 preference to another : whence it follows, that a quantity 

 generated from the product of unequal factors has all the 

 generality of a quantity unconditionally assumed ; whereas 

 that which is generated from the product of equal factors ' 

 has not that generality,as it admits only of one root ; vrhile 

 the other quantity, made up of unequal factors, has as 

 many roots as there are" units in the index of the ppwer, 

 the root of which is to be extracted. 



Let us now see how far what has been said above will Application to 

 serve to explain the difficulty stated in my former letters, ^"^^ P?'"* ^'i 

 or rather in my last letter, to which I intend more parti- 

 cularly to direct the following remarks : 



The quantity which I proposed to square was the fol- 

 lowing, 



v^— ^ + 4-/— 3+ ^—k—i 



v/— 3, 



Now here the quantities under the cubic radicals are the 



two imaginary roots of the equation x' z= 1 ; which, for the 



O 2 sake 



