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SECTION VII.
EVOLUTION,
E VOLUTION, or the Extraction of Roots, being di
rectly the contrary to Involution, or raising of powers,
is performed by converse operations, viz. by the division
of indices, as Involution was by their multiplication.
Thus the square root of t 6 , by dividing the exponent
by 2, is found to be T 5 ; and the cube root of x r ', by di
viding the exponent by 3, appears to be ar: moreover,
the biquadratic root of a -p x\ :i will be a T aj 2 ; and
the cube root of a a + avrP will be aa
In the same manner, if the quantity given be a frac
tion, or consists of several factors multiplied together,
its root will be extracted, by extracting the root ot
each particular factor.
Thus the square root of a l b z will be ah; that of
a will be — ; and that of
81 x a x caq-T.Tl
wi
16 x a
pa > fiTtfïL . Moreover, the square root of aa — xx\ z
4 x a — x
i
will be aa — xx | * ; its cube root aa — | ' ; and its
i
biquadratic root, a a — xxY ; and so of others. All
which being nothing more than the converse oi the
operations in the preceding section, requires no other
demonstration than what is there given.
Evolution of compound quantities is performed by
the following Rule,
First, place the several Terms, whereo f the given quan-
tit if is composed, in order, according to the dimensions of
some letter therein, as shall be judged most commodious ;
then let the root of the first term be found, and placed in
the quotient; which term being subtracted, lc\ thejirst tcr?n
of the remainder be brought doivn, and divided by twice
the first term of the quotient, or by three times its square,
or four times its cube, §c. according as the root to be ex
tracted is a square, cubic, or biquadratic one, 4'C. and let