[a0 oe Cee. : 
MATHEMATICAL CORRESPONDENCE, 
iS aieninetenmeeenatl 
To the Editor of the Monthly Magazine. 
SIR, | ih 
HE Arithmetic of Imaginary Quantities, being a fubjeét on which mathematicians feem die 
vided, any attempts to fouad it fon true principles cannot be unacceptable. I have be-« 
low attempted to do this : how far I have fucceeced, I leave to an impartial. public to determine. 
Your early infertion of it will much oblige, Sir, your humble fervant, 
Newceafil.-upfion-Tyne, Fune 45, 4797. : J. Garnett. 

OBSERVATIONS ON THE MuLTIPticaTtion, Diviston, &c. OF IMPOSSIBLE OR 1IMA< 
GINARY QAANTITIES IN ALGEBRA. ‘ 
By J. Garnett, late Affiffant Aftrovomer at the Royal Obfervatory, Greenwich, xuw a Chemift 
ant Druggift, Newsa/ile-upjion-Tyze. 
DeriniT1oNn. The fquare root of +a is expreffed thus,/—aj of courfe, the fquare of this 
quantity (4/ —a) mutt be —a, being juft the reverfe. : 
This being premi‘ed, the do&trine retpediing the management of imaginary quantities, as they 
atually occur iin equations, is quite eafy ; for nothing more is required than a ftri& attention to 
the common rules ve!pecting the figns, viz. that “ like figns give plus, unlike minus.?’— ft may 
be thus exemplined : 
—) —— Comment bs #3 
Cafe 1. 4/—a fquared, or 4/—-aX4/—a== (by the definition) —a. 
The illuftration may be as follows: let x?--a==0, then by tranfpofitiony &c. x=4/—a; fube_ 
ftitute thig for x, and the equation will become 4/—aX4/-—u--2==0 ; hence, that the equa- 
tion may vanith, 4/—axX4/—a muft produce the produdt —-a, otherwife it will involve an 
abfurdity. " 
Cafe 2, fax Pb /—ox pba ty/ ab. 
x? Pe : 
It may be exemplified as follows: let’ -+--a==o, then by tranfpofition and evolution 
/ 
=— 
x2 a 
Pies ma; multiply now each fide by a/b, and it becomes x==4/—ab ; which refult is the 
fame, as if the operation for finding x had been performed by rea/ quantities only. 
Cafe 3. A aX if bf 4 X bf 10> 
a ; Sind Sa 
It may be thus illuftrated: let a— —==c, then by tranf{pofition and taking the root of both 
b chs 
pay pe eee eee ; ie 
fides V35 =4/----+:; multiply each fide by /—s and the’ equation will become 
5x? : ————et . . 
(/ += ==) xz=n/ab——bey agreeing with the conclufion deduced by folving the equation With 
real quantities only. 
Cafe 4. A a at This needs ..0 uluft:ation. 
/ re Tin 
Cafe 5. Jaan pia 2x +5 ty — > 
Se ai prensa ig. 
Jf a f/f —b=V/ 4 x= y- 
On infpeétion, the fecond terms appear to be particular cafes of cafe 2; therefore, the illu 
tration there given will ferve for this cafe, 
— — 4 x: I in ; 
Cafe 6. .f/—a+t 0 rs ao rte Santa RS Fp 
This cafe is only a particular one of cafe 3, and may be illuftrated in the fame way. 
Thefe fix cafes feem to comprehend all the cafes that can occur. 
Remark. It appears that (ér. Emerfon has inadvertently fallen into a miftake, by confider- 
ing abftractedly the imaginary quantitics in the Scholium, at p. 67, of his Algebra; thus _mak- 
Ang the product of 4/—a by 4/——b to be vA ~e ah, ox imaginary, and the a/ na by —i/—b to be 
MontTaty Mac, No, XXI. R ona 
