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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