(1) y = ½ n√ √ 1 + aa + a - ½ / n√ √ 1 + aa + a or,
(2) y = ½ n√ √ 1 + aa + a - ½ n√ √ 1 + aa - a or,
(3) y = ½ / n√ √ 1 + aa - a - ½ n√ √ 1 + aa - a or,
(4) y = ½ / n√ √ 1 + aa - a - ½ / n√ √ 1 + aa + a
For Example, Let the Root of this Equation of the Fifth Power be required 5y + 20y3 + 16y5 = 4 in which case n is = 5, and a = 4, and the Root, according to the first Form, is
y = ½ 5√ √ 17 + 4 - ½ / 5√ √ 17 + 4
which is Expeditiously resolved into Numbers after this manner.
√ 17 + 4 is equal to 8.1231, whose Logarithm is 0,9097164, and the fifth part of it is 0,1819433, the Number answering it 1.5203 = 5√ √ 17 + 4. But the Arithmetical Complement of 0.6577 is 9.8180567, the Number answering is 0.1819433 = 1 / 5√ √ 17 + 4 and the half difference of these Numbers is 0,4313 = y.
Here we may observe, that in the Room of the general Root, we may advantageously take
y = ½ √ 2a - ½ / n√ 2a if the quantity a be pretty large in respect of Unity. As if the Equation were 5y + 20y3 + 16y5 = 682, the Logarithm of 2a = 3.1348143 whose Fifth part is 0.6269628, the Number answering is 4.236, and the Number answering the Arithmetical Complement 9.3730372 is 0.236, the half difference of these Numbers is 2 = y.