Page:The method of fluxions and infinite series.djvu/31

Rh 21. But the Work may be much abbreviated towards the end by this Method, especially in Equations of many Dimensions. Having first determin'd how far you intend to extract the Root, count so many places after the first Figure of the Coefficient of the last Term but one, of the Equations that result on the right side of the Diagram, as there remain places to be fill'd up in the Quote, and reject the Decimals that follow. But in the last Term the Decimals may be neglected, after so many more places as are the decimal places that are fill'd up in the Quote. And in the antepenultimate Term reject all that are after so many fewer places. And so on, by proceeding Arithmetically, according to that Interval of places: Or, which is the same thing, you may cut off every where so many Figures as in the penultimate Term, so that their lowest places may be in Arithmetical Progreffion, according to the Series of the Terms, or are to be suppos'd to be supply'd with Cyphers, when it happens otherwise. Thus in the present Example, if I desired to continue the Quote no farther than to the eighth place of Decimals, when I substituted $$0.0054 + r$$ for $$q$$, where four decimal places are compleated in the Quote, and as many remain to be compleated, I might have omitted the Figures in the five inferior places, which therefore I have mark'd or cancell'd by little Lines drawn through them; and indeed I might also have omitted the first Term $$r^3$$, although its Coefficient be $$0.99999$$. Those Figures therefore being expunged, for the following Operation there arises the Sum $$0.0005416 + 11.162r$$, which by Division, continued as far as the Term prescribed, gives $$- 0.00004852$$ for $$r$$, which compleats the Quote to the Period required. Then subtracting the negative part of the Quote from the affirmative part, there arises $$2.09455148$$ for the Root of the proposed Equation.

22. It may likewise be observed, that at the beginning of the Work, if I had doubted whether $$0.1 + p$$ was a sufficient Approximation to the Root, instead of of $$10p - 1 = 0$$, I might have suppos'd that $$6p^2 + 10p - 1 = 0$$, and so have wrote the first Figure of its Root in the Quote, as being nearer to nothing. And in this manner it may be convenient to find the second, or even the third Figure of the Quote, when in the secondary Equation, about which you are conversant, the Square of the Coefficient of the penultimate Term is not ten times greater than the Product of the last Term multiply'd into the Coefficient of the antepenultimate Term. And indeed you will often save some pains, especially in Equations of many Dimensions, if you seek for all the Figures Rh