5_y_ - 20_y_³ + 16_y_⁵ = 6,
whence _n_ = 5, and _a_ = 6, and the Root will be =
½[⁵√](6 + √(35)) + ½/[⁵√](6 + √(35))
or because 6 + √35 = 11.916 whose Logarithm is 1.0761304, and its
Fifth part is 0.2152561, whose Arithmetical Complement is 9.7847439. The
Numbers belonging to these Logarithms are 1.6415 and 0.6091, whose half
Sum is 1.1253 = _y_.
But if it happen that _a_ is less than Unity then the Second Form, as
being more convenient, ought to be pitch'd on. So if the Equation had
been
5_y_ - 20_y_³ + 16_y_⁵ = 61/64,
then _y_ will be =
½[⁵√](61/64 + √(-375/4096)) +
½[⁵√](61/64 - √(-375/4096))
and if the Root of the Fifth Power can by any means be Extracted the
true and possible Root of the Equation, will thence Emerge, tho' the
Expression seems to insinuate an Impossibility. But the Root of the
Fifth Power of the Binomial
61/64 + √(-375/4096) is
¼ + ¼√(-15)
and so the same Root of the Binomial
61/64 + √(-375/4096) is
¼ - ¼√(-15)
the half Sum of which Roots is = ¼ = _y_.
But if that Extraction can not be perform'd, or may seem too difficult,
the thing may be solv'd by the help of a Table of Natural Sines, after
the following manner;
To the Radius 1 let _a_ = 61/64 = 0,95112 the Sine of some Arch which is
therefore 72° 23', whose Fifth part (because _n_ is equal to 5) is 14°
28' the Sine of it is 0.24981 = ¼ nearly.
The same is the Method of proceeding in Equations of higher Dimensions.
_A Discourse concerning the Action of the Sun and Moon on Animal
Bodies; and the Influence which This may have in many Diseases. By
_Richard Mead_, M. D. F. R. S._