442 On Imaghiary Cube Rooti. [June, 



1/ — 3 + V — '-"fj '^ - 3 - y — '-nV'j do not vanish. The 

 cube roots In respect of tlie root unity are 4. ~ ^/ - |s. and 

 _i_ + V ~ x^ ; l^ut if, under this conception, we should assimilate 

 the sum of these roots to the root of the equation 3 a; — a^ =*: 2, a 

 ercater mistake, in ray opinion, could not be made. In the same 

 manner may my quantity be divided by 4, and it will be a cube root 

 of unity, but never can it be conceived to be a root of the 

 equation 3 .r — a;* = 2 5 but if Dr. Tiarks's number be so 

 divided, it will be, together with — ^ + // — f, a root of the 

 equation a;^ — S x = 2. The equations 7 x — x' = . 6 and 

 ^ X — ar' = 2 have a similar root unity ; but it Is seen that all 

 equality is lost when they are converted into fractions, and this is 

 precisely the case of our two numbers. It is the province of the 

 lovers of the science to decide on the question. 



lam, Sir, your obedient servant, 

 jicy 9, 1S15. JajMes Lockhart. 



Another Communication at the sarne suljed. 

 (To Dr. Thomson ) 



SIR, ^oy 3. 1815. 



As the subject proposed by Mr. Lockhart on tlie algorithm of 

 imaginary quantities is one of considerable importance in a variety 

 of analytical investigations, you will be induced probably to admit a 

 few remarks on the two answers published in your last number. 



The first thing which appears singular is, that one of your corres- 

 pondents has shown Mr. Lockhart's expression to be the cube root 

 of 64, but under a ditferent form to that usually given ; and the 

 other, that it is not the square root of 64, but of 8. 



The fact is, that Mr. L.'s expression, ~ "^ ^ ~ + 



^~ — (13 _ ^ v^ _ 3), the same as all other quantities in which 

 the sign of the square root enters, admits of two values ; and as 

 there is no previous condhion, either of them may be employed j 

 aud the quantity will be accordingly either the (/ 64 or ^ 8. 

 R. N. D. is therefore too positive when he says, " It is not the cube 

 root of 64, but of 8." He Is also wrong In stating that by squaring 

 G a/ Z; an ambiguity Is Introduced ; for the ambiguity has place in 

 the *y b before the operation of squaring; in fact, the only case la 

 which there is no ambiguity Is when we know the origin of the 

 quantity whose root is to be extracted, as Is shown in one of the 

 latter numbers of Nicholson's Journal, where the object was to 



j 3 



explain why '/4--iV-S + -v^^ + i ^ -^» ^^ '"^^^^ « 

 known to be equal to 1-87938, or 2 sin. 70°, is not (when squared 

 hy the usual process) equal to the square of the same number. The 



