30 . Mathematical Corre[pandence. 
track, which the experience of two ages 
thows to end in mazes and quagmires, 
we fhould recal our wandering fteps, 
and endeavour to find out a better path 
im the recefles of fcience. 
That the pofition on which the mo- 
dern reafoning on the formation of 
equations is falfe, may, I think, be 
proved fatisfactorily to any one, who has 
been inftru¢ted in the firft rudiments of 
Algebra. For a quadratic equation, two 
fimple equations are multiplied together, 
generally denoted by #—a=o and «— 
=o; for a cubic equation, three fimple 
equations ; and for an equation of higher 
dimenfions, as many fimple equations as 
that equation has dimenfions. In the 
ficft inftance, x—u==o is multiplied into 
«—b—=o, and, confequently, the refult, x? 
—x.a-+b-Lab, is equal to nothing. Of the 
faft equation, there are evidently two 
roots, @ and 4, which may be afcertained 
without reference to the fuppofed mul- 
tiplication 5 and, in faét, this equation 
does not refult from the fuppofed multi- 
plication 5 for if x—a==o, the unknown 
quantity in the fecond equation ought 
not to be called x, but by fome other 
term, and then if the two equations are 
multiplied together, a—a=o, and y—s 
=o, the refult will be, xy—ay—dr 
-Lab=o; that is, the equation will be 
equal to nothing, when + is equal to a, 
or y is equal to &. 
I do not deny that an equation may be 
formed by the multiplication of double 
terms, and a fimple inftance will be the 
means of farther fhowing the fallacy of 
the modern mode of reafoning, and the 
falfehood of the affertion, that an equa- 
tion has as many roots as it has. dimen- 
fions. Let a and 6 be any determinate 
quantities, @ being greater than 4, and 
#, the unknown quantity, greater than a. 
By multiplying together a—a, and 
«—, we obtain the compound fum 
at—1. ati-ab. Wow, fince # is a 
variable quantity, I may fuppofe it to di- 
minifh, till it becomes equal to a, and, 
confequently, in that fituation, my com- 
pound form will become a quadratic 
equation, a?—pa-+yo. Let x be di- 
minifhed fill more, till it becomes equal 
to 4, and the compound form will again 
become a quadratic, whofe root is equal 
to 6, refulting not from the multiplica- 
tion *—a into w—*, but from that of 
a—x« into +—b. We have obtained, then, 
by this mode of framing a quadratic 
equation, the knowledge of the truth, 
that in equations of this form #2—px- 
[ Feb. 
g==o, there are two roots: and the fame 
truth is difcoverable in a much eafier 
manner, without this tedious procefs of 
multiplying, by a very flight infpeétion 
of the equations. 
But if fomething has thus ‘been done, 
though im a bad manner, by multiplying 
in one form of a quadratic, what are 
we to do in other cafes, when, for ex- 
ample, it is made to be 42-bpxr—g=0 ? 
---We are told that this will refult 
from the multiplication of #-a=o, 
into #—/=0, and, confequently, that 
the equation will full have two roots, a 
and 6. I allow, that it will refult from 
the multiplication of the double terms 
*e+a and w«—é, and that the refult 
may become, +?-+-4%. a—b—ab=0.--- 
But, whether I confider the formation of 
this equation, or inveftigate its peculiar 
nature, I cannot difcover more than 
one root, and it appears to me impof_fible, 
as it muft, I think, to every perfon, 
that it fhould have more than one root, 
which is 6. For aa can never become 
equal to nothing; and this equation can- 
not, therefore, refult from the multipli- 
cation together of two fimple equations. 
Again, from infpeéting the quadratic, 
it is difcovered at firft fight, that x can- 
not be equal toa. In this cafe, there- 
fore, it is not true, that an equation has 
as many roots as dimenfions; and J] 
might go on to prove the fame in equa- 
tions of ‘ higher dimenfions, fome of 
which will have as many roots as di- 
menfions, and others will not. The 
inveft:gating of the number of roots in 
an equation from the nature of its form, 
will lead to real fatisfaétory knowledge, 
of great ufe in the mixed mathematics, 
whilft the other mode of treating equa- 
tions, as produced from multiplying 
fimple equations together, or equations 
of lower dimenfions, has confounded a 
plain, fimple, and elegant {cience; ‘in- 
ftead of fharpening the faculties of the 
mind, has blunted its natural edge, and 
has made many a ftudent a mere tech- 
nical tranfpofer of figeres upon papery, ~ 
inftead of an accurate reafoner. 
The hmits of my paper do not per 
mit me to expatiate farther upon this fub- 
je&t ; and, indeed, it is unneceffary, till 
I hear with what reception my firft idea 
may meet among your fcientific corre- 
fpondents. They will fee clearly to what . 
extent my reafoning proceeds ; namely, 
that the changes of figns in an equation 
have no reference at all to the fuppofed 
nature of the roots, according to ‘their | 
hes | ‘quality 
