1796.| 
vanced by your facetious correfpondent, No 
Conjurer, and as he has expreffed himfelf 
in fuch terms as renders it difficult to de 
termine what he would with us to infer 
from his remarks; I have, though without 
any pretenfions to an extenfive acquaint- 
ance with Analytics, thrown together a 
few fuch obfervations as to me appeared 
neceffary. 
To what Mr. S. has mentioned re{peét- 
ang the obfcurity pervading the analytics of 
Dr. Waring, 1 know not what to anfwer, 
becaufe J have never perufed the work: 
but if his objections to it have no greater 
weight than what he fays concerning the 
Formation of Equations, both the one and 
the other muft fall tothe ground. For 
my part, 1 am rather at a lofs to know 
what he refers to, when he is {peaking of 
s« ‘The old beaten track, which the ex- 
perience of twoages thows to end in mazes 
and quagmires.”’ Is he referring to the 
fame thing here, as he is when he talks 
of ‘“ fhowing the fallacy cf the modern 
mode of reafoning?”” J know not with 
what propriety that method of reafoning 
which has been followed in an “ old beaten 
track,” ever fince the year 1631, can be 
ctlled Modern reafoning.’’ But, without 
dwelling upon this point, I thall proceed 
to confider what he advances againft the 
forming of equations from the multiplica- 
tion of double terms: and here I can meet 
with only one remark that feemsto require 
& particular reply ; namely, where he can- 
not difcover more than one root to the 
equation 1?-+-#. a—b—ab==o0, he allows 
that this equation refults from the multi- 
plication of the binomials r-+aand .—s; 
and yet he fays it cannot relult from the 
multiplication of two fimple equations. 
Surely fuch an inference is clogged with 
inconfiftency : but he hasalfoadvanced a 
reafon for drawing fo ftrange a conclufion, 
for he informs us, that +-+-a can never be 
equal to nothing. If any perfon with to 
afk, why not? I hope he will not be told 
that his reafoning has been carried on to 
the Ne plus witra, and that he mutt reft {a- 
tisfied with Mr. S.’s tiple diait. The whole 
of that gentleman’s conclufions on this 
head are apparently drawn, for want of 
making a proper diftinétion between neg a- 
tive and imaginary quantities : one would 
judge from what he has faid, that negative 
Yeots cannot have place in Algebra; but if 
he confult, without prejudice, what has 
been advanced on the fubject by Lud/am, 
Maclourin, and Saunderfon (without men- 
tioning others) I think he will find abund- 
ant reafon to reverfe his opinion. Nay it 
would be no difficult matter to produce a jt 
/ 
Mathematical Corre{pondence. 
\ 
an 213 
variety of queftions (and fuch only as are” 
propofed toa {chool-boy when grounding 
him in the knowledge of quadratics) 
the {olution of which would evince that 
negative roots not only have place, but are 
of great utility in the algebraic branch, 
and may be fupported by fuch demonttra- 
tion as is not to be overthrown. 
I am inclined to think, that very few 
perfons will agree with Mr. §. when he 
afferts that, ‘¢ {he changes of figns in an 
equation have no reference at all ro the 
fuppofed nature of the roots, according to 
their quality of being pofitive or negative.’ 
I would therefore take the liberty of re- 
commend neg it to him. to perufe with at- 
tention fome elementary treatife, where 
the nature and formation of equations are 
difcuffed at length ; and if, after fo doing, 
he does not find occafion to retraét his af- 
{ertions, I am fure the analyfts will have 
reaton to wilh him again to take up his 
pen in order to help them out of the 
“ quogmires,’ and fet them upon ‘terre 
fama. As for myfelf, though I hope I 
fhall be always open to conviction, I can- 
not but obferve, that at prefent, I am fo 
far from coinciding with Mr. §. in his opi- 
nions, that I think nothing has tended {fo 
much to preduce precifion and expedition 
in the reduétion of the higher equations, 
as thofe rules which have been deduced 
from confidering their formation by the 
multiplication of equations of inferior 
degrees: 
With refpect to the obfervations of No 
Conjurer, but little feems neceffary to be 
faid; he has expatiated with fufficient 
drollery on the wonderful powers of xo. 
thing ; but I cruft he will not be difpleafed 
with me for mentioning one of the effeéts 
of this ‘¢ fhadow of a fhaic,”” which will 
prove, that it is in fome cafes worth while 
te underftand that zoihbg is convertible 
into fomething of confeguence. When 
Powell and Waring were competitors for 
the Lucafian profefforthip at Cambridge, 
in a little piece publithed on that occafion 
; Op pp 
by the latter, he faid that was equal 
WER 
to 4 when was =r. Powell thought 
this was abfurd, becaufe when p—1, then 


Pan Ii fe) : ‘ 
a ==. Waring replied to 
LP Ty 
this, that when the numerator of the frac- 
tion was actually divided by the denomi- 
nator, the quotient was /-L724- 3-74, and 
that the fumofthefe terms became 4 when 
p==t: although it does not always appear 
«‘ who cah decide when doctors difagree,”’ 
did in this inflance ; for it was decided 
fo 
