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

Reſolution of affected Equations may be compendiouſly perform’d in Numbers, and then I ſhall apply the ſame to Species.

20. Let this Equation $y^{2}-2y-5=0$ be propoſed to be reſolved, and let 2 be a Number (any how found) which differs from the true Root leſs than by a tenth part of itſelf. Then I make $2+p=y$, and ſubſtitute $2+p$ for y in the given Equation, by which is produced a new Equation $p^{3}+6p^{2}+10p-1=0$, whoſe Root is to be ſought for, that it may be added to the Quote. Thus rejecting $p^{3}+6p^{2}$ becauſe of its ſmallneſs, 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 ſuppoſe $0,1+q=p$ and ſubſtitute this fictitious Value of p as before, which produces $q^{3}+6,39q^{2}+11,23q+0,061=0$. And ſince $11,23q+0,061=0$ is near the truth, or $q=-0,0054$ nearly, (that is, dividing 0,061 by 11,23, till ſo many Figures ariſe as there are places between the firſt Figures of this, and of the principal Quote excluſively, 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 ſuppoſing $-0,0054+r=q$, I ſubſtitute this as before. And thus I continue the Operation as far as I pleaſe, in the manner of the following Diagram:

$y^{2}-2y-5=0$	${\begin{aligned}&+2,10000000\\&-0,00544852\\&{\overline {\,+\,2,09455148}},\&c.=y\end{aligned}}$
${\begin{aligned}2+p=y.\qquad &+y^{3}\\&-2y\\&-5\end{aligned}}$	${\begin{aligned}&+8+12p+6p^{2}+p^{3}\\&-4-2p\\&-5\end{aligned}}$
${\text{The sum}}$	$-1+10p+6p^{2}+p^{3}$
${\begin{aligned}0,1+q=p.\qquad &+p^{3}\\&+6p^{2}\\&+10p\\&-1\end{aligned}}$	${\begin{aligned}&+0,001+0,039+0,39^{2}+q^{3}\\&+0,06+1,2+6,\\&+1,+10,\\&-1,\end{aligned}}$
${\text{The sum}}$	$0,061+11,23q+6,39^{2}+q^{3}$
${\begin{aligned}-0,0054+r&=q\cdot q^{3}\\&+6,39^{2}\\&+11,239\\&+0,061\end{aligned}}$	${\begin{aligned}&-0,0000001\not 5\not 7\not 4\not 6\not 4+0,000\not 0\not 8\not 7\not 4\not 8r-\not 0,\not 0\not 1\not 6\not 2r^{2}+r^{3}\\&+0,0001837\not 0\not 8-0,068\not 0\not 4+\not 0,\not 3\\&-0,060642+11,23\\&+0,061\end{aligned}}$
${\text{The sum}}$	$+0,0005416\qquad +11,162r$
$-0,00004852+s=r.$

21.