lytica. 
4 
of thefe latter equations are frequently 
terms called negative, or impofhible; and 
the rélation of thei terms to the coefficients 
of tlre principal equation isya great ob- 
je of inquiry. In this art the profeffor 
was very iuccefsful, though little affiftance 
isto be derived from his writings, in look- 
ing for the real roots. We fiallnot, per- 
haps, Be deemed to depreciate his merits, 
if we place the feries for the fum of the 
powers of the roots of any equaticn, among 
the moft ingenious of his. ee yet 
we cannot ada, that it has very ute elully 
enlarged the bounds of {cience, or file 
the algebraift will ever find eecafion to in- 
troduce it into practice. We may fay the 
fame on many ingenious transformations 
of equations, on the difcovery of impoffi- 
ble roots, and fimilar exertions of undoubt- 
edly greattalents. ‘Th They have carried the 
aflumption to its utmoft limits; and the 
difficulty attending the fpeculation has 
rendered. 
tain iis real utility 5 oe 
may occafionaily ee 
the ‘Milcellan iea Analytica. 
they who reject it 
uleful hints from 
aphe as an nee was, we believe, 1 
the latter end of the year 1759, when he 
pebulhes the firit c chapter of the Mil 
nea Analytica, as a.fpecimen of his quali- 
fications for the profefiorthip ; ; and this, 
hapter he defended, in a reply to a pam- 
phet entitled, Obfervations cn the Firft 
Chapter of a book called Mifcellanea Ana- 
Here the profeflor was flrangely 
puzzled with the éommon paradox, that 
nothing divided by nothing may be equal 
to various finite quantities, and has re- 
courfe. to unqueftionable authorities in 
proof Fof this potion. The names of Mac- 
aurin, Saunderfon, De Moivre, Bernou- 
uli, Monmort, are rangedin favour of his 
opinion: but Dr. Powell was not fo eafily 
convinced, and returns to the charge, in the 
Defence of the Obfervations; to which the 
profefior replied in a Letier to the Rev. 
Dr. Powell, Fellow of St. John’s College, 
Cambr ide, in anfwer to his Obiervations, 
&c. Inthise OmtHOR EE it is certain that 
the profeflor gave evident prosfs of his 
abilities ; though itis equally certain that 
ke followed too implicitly the decifions 
oi his predeceilors. No apparent advan- 
tage, no authority, whatever, fhould in- 
duce mathematicians to {werve from the 
principles of right reafoning, on which 
their fcience is fuppofed to be peculiarly 
founded, _Accerding to Maclaurin, the 
Profeffcr, -a sama then, 
d others, If P= 
Qz—x2 
48. 3 Jdemoirs of Dr. Waring 
perfons more anxious to alcer- 
_ ‘ 
[ Feb. ie 
5 
es Bese og! 
when x = a, P is equal to on for, fay 


ices 
they, < : 1s equal to pee 
— x2 am x: ax 5 
- that is, when & is equal to a, P——7 
: : ~ a= #8 
I : - : 
or—— SButwhen x is equal to a, the nu- 
Tinea ele ‘ 
merator and denominator of the fraction 
Gi oS 
are both, in their langu 
To 
to nothing. ‘Therefore, nothing divided 
by nothing, is equal to. In the fame 


2a 
waa T a—x 4. 
manner a a ae :which 
a—xs a -axtxr™ a—x S 
when X is equakto @, becomes —-There- 
3 a? 
fore, nothing divided by nore is equal 

I I 
———— , that is, —_ 
a ghle Geagns Ra ieee. 
is abfurd. But we need only trace back. 
our eps to fee the fallacy in this mode of 
realoning. For P is ep to fome num- 
— 
> 
ber multiplied into — 
2 

af that is, when 
a— ce 
48 equal to a, P is equal-to fome num- 
ber multiplied into nothing, ana divided 
by pothing ; that is, P is, in that cafe, 
Ho number at all. Fer a—a cannot_be 
divided by a—x when x is equal to a, 
fince, In that cafe, @—x is no number 
at all, 
If, in the beginning of his career, the 
profeffor could admit iuch paralogifns i In- 
-tohis fpeculations ; and the writings ‘of 
the mathematicians, for nearly a century 
before him, may plead‘in his excafe; we 
are not to be furprifed that his difcoveries 
fliould be built rather en the affumptions - 
of others, than on any new principles of 
his own. Acquiefcing in the ftrange no- 
tion, that nothing could be divided by no- 
thine, and produce a a variety of numbers, 
he-as eafily adopted the pofition, <hat an 
equation-has as many roots‘as it has di- 
menfions.- Thus 2 and — 4 are {aid to be 
rcots of the equatien .° —ax=8, though 
4 can be the root only of the equation ; 
x?-—2 x8, which differs 1o materially from | 
- 
that in one cafe 2x is added, 
the preceding, 
in the other cafe it is fubtraGted from x2: 
Ailewances being made for this error in: 
the principles, 4 the dedudtions are, In pe- 
neral, ‘legitimately. made; and any one 
who can give himlelf the cable of demon- 
firating the propofitions, may find fuffi- 
cient empioyrment i the Pree feffor’s ana- 
lytics. Perhaps it will be {uficient for a 
ftudent to devote his time te the fimplett 
- 4 ae eats 
’ 
wage, equal | 
* which : 
