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
5_y_ + 20_y_³ + 16_y_⁵ = 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_.
But if in the aforegoing Equation the Signs are alternately Affirmative
and Negative; or which is the same thing if the Series be after this
manner,
_ny_ + ((1 - _nn_)/(2 × 3))_ny_³ + ((1 - _nn_)/(2 × 3)) ×
((9 - _nn_)/(4 × 5))_ny_⁵ + ((1 - _nn_)/(2 × 3)) ×
((9 - _nn_)/(4 × 5)) × ((25 - _nn_)/(6 × 7))_ny_⁷ + _&c._ = _a_.