1815.] Scientijic InleUlge^ice. S9S 



really one of tlie two imaginary roots of 4, ■— (2 ± 2 v^ — 3) 

 resulting from the equation x- + 4 x + IG = 0. 



I am, Sir, your obedient servant, 



5, Datcman's Buildings, T r t* _ 



VII. Another Communication on the same adject, 



(To Dr Tliomson.l 

 SIR, 



My attention Tvas arrested by the seventh article of scientific 

 intelligence in your last number (p. .",15). Some mathematicians 

 have denied the universality of the doctrine of tliere being as many 

 roots to an equation as it has dimensions, but none have been able to 

 maintain that there are more. 1 therefore examined JMr. Lockhart's 

 proof with some attention : and 1 conceive tliat he cannot take it 

 ill if I endeavour to point out tiie source of his mistake. I think, 

 likewise, that you will be indebted to me for doing so; since the 

 " method for approximating towards the roots of cubic equations 

 belonging to the irreducible case," has justly given some weight to 

 the author's opinions ; and you must be desirous of not being the 

 means of propagating an error which can only be supported bv the 

 uutl'.ority of his name. 



I must begin by laying down that — ^ + ^ - -J _^ 



y/ — {'-^ — ^ V — S k not a cube root of 6-1, but of 8. To 

 show this in the simplest manner, we will substitute -'- + i \/^^3 

 in the place of V _ ('_a _ s. >^/ _ 3^ fo,- these two quantities 

 may easily be shown to lie equal by the rule for extracting the roots 



of binomial surds ; and then — 4..!.+ 3.^_3 _ 



-^^^ — i-j, =: — 1 + V — 3, which, when cubed, will be 



found to make up 8. But it may be asked how a mathematician so 

 well acquainted with algebraical processes, and esjiecially with culiie 

 equations, can have made such a mistake, and in wliat part of his 

 reasoning the fallacy lies? This question 1 think admits of a com- 

 plete answer ; for the error will be found in his manner of bringing^ 



out the value of (— 2 + Gj/ — .i) / — ( 1,3 — | v'~^3.' 

 It is [jcrfcctly clear that a a/ is equal to the square root of a* /' ♦ 

 but it escaped Mr. L. that by squaring his quantity he introduced an 

 ambiguity, since the square root of «* I' is ± n \/ !■ ; and in this 

 instance he ought to have taken the negative instead of the positive 

 root ; the value then would have been liG — 2S = 8, instead of 

 ;U> -t- 28 = M. To show that this is so, we have only to take the 



value above assigned toV'— {\^ — -} \^ — •^; and then we shall 

 find lhat(-i + G ^/ — A) • ( \ + a v/ — :i) = — 2K. 

 There are some particulars in your lust number which would not 



