Page:The method of fluxions and infinite series.djvu/30

6 Reolution of affected Equations may be compendiouly perform’d in Numbers, and then I hall apply the ame to Species.

20. Let this Equation $$y^2 - 2y - 5 = 0$$ be propoed to be reolved, and let 2 be a Number (any how found) which differs from the true Root les than by a tenth part of itelf. Then I make $$2 + p = y$$, and ubtitute $$2 + p$$ for y in the given Equation, by which is produced a new Equation $$p^3 + 6p^2 + 10p - 1 = 0$$, whoe Root is to be ought for, that it may be added to the Quote. Thus rejecting $$p^3 + 6p^2$$ becaue of its mallnes, the remaining Equation $$10p - 1 = 0$$, or $$p = 0,1$$, will approach very near to the truth. Therefore I write this in the Quote, and uppoe $$0,1 + q = p$$ and ubtitute this fictitious Value of p as before, which produces $$q^3 + 6,39q^2 + 11,23q + 0,061 = 0$$. And ince $$11,23q + 0,061 = 0$$ is near the truth, or $$q = -0,0054$$ nearly, (that is, dividing 0,061 by 11,23, till o many Figures arie as there are places between the firt Figures of this, and of the principal Quote excluively, as here there are two places between 2 and 0,005) I write −0,0054 in the lower part of the Quote, as being negative; and uppoing $$ -0,0054 + r = q$$, I ubtitute this as before. And thus I continue the Operation as far as I pleae, in the manner of the following Diagram: