284 Mathematical Correfpondence. [O&. 
a with to fee your correfpondent bring forward, in fupport of what he advances, an example _ 
wherein we might inftance that +a will really hold for the product whofe factors are 4/—a and 
—a, as an inftance of the kind never occurred to me, nor perhaps to many more of your readers, 
although I have-often had occafion to obferve that —a would. 
If it was required to find two numbers, whofe fum fhall be 10, and produét 28, they would 
appear to be 5-+4/—3, and 5—,/—3. And now, if ./—3x4/—3=+33 5+4/—3X 
53 muft be —= 22, whereas their produét ought to be = 28, and confequently is 
abfurd. ; 
That imaginary quantities have no meaning, and are nothing at all, are new ideas. If ,/—a—=0, 
how is 4/—a>/4/—a==0 x o==a to be reconciled ? 
I fhail content myfelf with thefe few remarks, until I fee what anfwer they draw fromy 
EL o0-Cosa, and, in the mean time, am, fir, yours, &c. 
N, C. 

For the Montbly Magazine. 
4&4 NEW DEMONSTRATION OF THE RULE FOR FINDING THE SUM CF THE POWERS 
OF THE ROOTS OF ANY EQUATION, 
I, GIR Isaac NEwToNn is generally confidered as the firft who gave a rule for finding the furs 
of the powers of the roots of any equation ; but the merit of this difcovery is certainly due 
to the great, though negleéted mathematician, Albert Girard. Among his new Algebraic In- 
ventions, fome forrnulz are given for this purpofe, from which the Newtonian and others are 
eafily and direftly deducible. The fubje& itfelf is curious and important, not only becaufe the » 
conclufions are of ufe in finding the limits of equations, but becau‘e they ferve to demonftrate a 
number of fimple and general properties of curve lines. Various demonftrations of the common 
rulé have been exhibited by Maclaurin, and the other commentators upon Newton; and to thefe 
I fhall add the following, which may be thought to foflefs fome peculiar advantages. 
Il. Let x°—px2—3-pgx— 2x0 —3_L tan 4 1x95 4, &c. = 0 be the given equation, where 
site sc 2 : I = 
the coefficients p, 9, 7, 5, t, &c. are known quantities : and, putting x= —,; there refults — — 
; & z 
Tt I IE I I 
ee pee Ee a a .=0; 
p- serena! gn—2 Z 23 : 2n—4 ea ce 03 
or, I—/2-}-927—r23satmt2s, &C. =. 
Now, fuppofing 1—az%, I—fx, I—yz, I—22, Iz, &c. 

. to be the faGtors of this equation ; 
fo that a, B, y, & & &c. may be the roots of the given one, or the different values of a5 we 
have 1—/24927—r23 4 seta, &c. i 
== (i—az) (1—z) (1—y2) (t—dz) (1—22), &c.5 and taking the logarithms of both, Log. 
(1—pxz--7z?— 23 4 sa4—iz5-+, &c.)—=Log. (1—w2) + Log. (1—fx) + Log. (i--yz)-+ & ee 
pe A Ae AE AS : : 
But it is well known that Log. (1a) —a~— —{——--— &e. 
co DAS sual: AER 
rA7 Ad aA AS 
and Log. (7—a) —— ee ee &e. 
4 3 2 
Therefore) —2 (p—gz-trotaeszitizt— &c.) 
22 
ae (¢—q2rbrzraiz3-iz4— &c.)? 
3 
= = ( Paps 3 Li z4— &c.)% 
zt 
—— ( Payer ini Liste &c.)4 



4 
&e. 
as? o323 a4z4 @Sxd 
= —aAZs SSS eS = 
aE aie 
B Z ZL [95% 
fist catinthe Oh UE 
zm Z Ps 5s 
2 4 
6727 $323) 8424 «S25 
— 8S ae OO. 
2 3 4 5 
&c. 
Or, -& (pga ratensxitizim &e) 
