815.] On Imaginary Cube Roots. 441 



It would be strange indeed to call the first binomial ^ 8, and the 

 latter ^ 64. 



The binomial on the left hand being, then, by common consent 



and usage, equal to v' ^^^ + -S or ^ 64, it follows that my 

 number is a true cube root of 64, and not of 8, which your cor- 

 respondent N. R. D. affirms it to be, and that I have properly, and 

 in conformity with the practice of algebraists, taken the jwsitive 

 square root of 7^4. 



I conceive, therefore, that I have now only to show that the 

 quantity is different from the known forms of the cube roots of 64. 

 Dr. Tiarks has divided a:3 — 64 by a; — 4, and by means,of the 

 quotient he obtains — 2 ± V — 12, which are the cube roots 

 connected with the equation x3 — 48 x = 128, where by Cardan's 



3 - Z 



rule the roots are represented by i/ 64 + V 4096 — 4096 + 



3.- ^ ■ -_ 



V 64 — a/ 4096 — 4096, and where, by the roots previously 

 exhibited depending on i and v, the cube roots become — 2 ± >y 

 — 12; but these are the cube roots of binomials in their vanishing 

 state, in which state they have functions and connexions widely 

 different from those deduced from binomials which are not eva- 

 nescent. 



The means taken by Dr. Tiarks to prove my quantity to be equal 

 to — 2 — 2 v' — 3 is by no means sufficient. 



This, as well as the correctness of my assertion, may be suffi- 

 ciently evidenced by the nature of vanishing fractions ; and on this 

 evidence, and not on any ambiguity of expression, I entirely rest 

 my opinion. 



If binomials are not in a vanishing state, one of the roots of the 

 equation from which the binomials are deduced will, by a simple 

 operation, become extinct ; but all the roots will be preserved if the 

 binomials are evanescent. 



Thus let 3 j; - x' = 2 



OTx — x' = 2xl— X 



X — x^ 



.'. -rrr = X' + X = 2 



Here the roots are preserved, because the binomials connected 

 with the given equation vanish, liut 



Let 7 X — x* = 6 

 X — x' = 6 — 6x 



orx— x' — 6x1 — X 



• • 1 - jr 



Here the value of unity is extinct, because tlie binomials 



