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SECTION XII. 
OF ТИК RESOLUTION OF EQUATIONS OF SEVERAL 
DIMENSIONS. 
B EFORE ive proceed to explain the methods of re 
solving cubic, biquadratic, and other higher equa 
tions, it will be requisite, in order to render that subject 
more clear and intelligible, to premise something con 
cerning the origin and composition of equations. 
Mr. Harriot has shewn how equations are derived by 
the continued multiplication of binomial factors into 
each other: according to which method, supposing эс — сг, 
x—by x—c, x—d, &c. to denote any number of such 
factors, the value of a, is to be so taken that some one of 
those factors may be equal to nothing: then, if they be 
multiplied continually together, their product must also 
be equal to nothing, that is, x — a x x—b x x—c x 
x — d &c. rr 0: in which equation x may, it is plain, be 
equal to anyone of the quantities a, b, c, d, &c. since any 
one of these being substituted instead of ,r, the whole ex 
pression vanishes. Пенсе it appears, that an equation 
may have as many roots as it has dimensions, or as are 
expressed by the nurnberof the factors, whereof it is sup 
posed to be produced. Thus the quadratic equation 
— 0 or х г ? £ x f ab — o, has 
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mi 
x — a x x ■ 
two roots, a and b\ the cubic equation x— a x x— b 
xx — c o, or 
— a I ab 
x 3 \ b>x z -\- ac L x —abc — o, has three roots, 
It 
fa* 
a, b, andc; and the biquadratic equation, x — a x 
x — b x x — c x x — d — 0, or 
/ i ■“ abc 
,■ -abdl 
ad f be > x -f- 
x Ar f- 
4 
i4 
^ bd f cd^ 
acd | 
—- bed 
x 4- abed r= o, 
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