<43r>

Sir,

I received your former Letter as well as your later, and should have written to you sooner, but that I stay'd to think of something that might satisfy your Desire; But though I can not hitherto doe it to my owne liking, yet that I may not wrack your patience too much I have here writt you what occurrs to mee, which is only about facilitating the Extraction of ℞. The former Method might be applyed to determin all by every 1000th, as well as by every 100th ℞, but not with advantage, for it will require the Extraction of ℞ to 14 or 15 places, besides a greater number of Additions, Subductions & Divisions in those greater numbers: And therefore I have rather sent you the following Notes about Extracting ℞.

1.) When you have extracted any ℞ by common Arithmetick to 5 Decimal places, you may get the figures of the other 6 places by Dividing only the Residuum by
Suppose B. the Quotient or ℞ extracted to 5 Decimal places, and C. the last Residuum, by the Division of which you are to get the next figure of the Quotient, and D the Divisor (that is $2B$ or $3BB$ or ${4B}^{.c.}=D$ & $B+\frac{C}{D}$ shall be the ℞ desired. That is, the same Division, by which you would finde the 6th decimal figure, if prosecuted, will give you all to the 11th decimal figure.

2) You may seek the ℞ if you will, to 5 Decimal places by the logarithm's, But then you must finde the rest thus. Divide the propounded number $\begin{array}{c}\text{once}\\ \text{twice}\\ \text{thrice}\end{array}}$ by that ℞ prosecuting the Division alwayes to 11 Decimal places, and to the Quotient add ${\text{y}}^{\text{e}}\phantom{\rule{0.5em}{0ex}}\text{said}\phantom{\rule{0.5em}{0ex}}℞\phantom{\rule{0.5em}{0ex}}\left\{\begin{array}{l}\text{once, & halfe}\\ \text{twice, & a third part}\\ \text{thrice, & a quarter}\end{array}\right\}\phantom{\rule{0.5em}{0ex}}\begin{array}{c}\phantom{\text{o}}\\ \text{of the summ}\\ \text{shall be the}\end{array}\phantom{\rule{0.5em}{0ex}}\left\{\begin{array}{l}\text{square}\\ \text{Cube}\\ \text{square square}\phantom{\text{I}}\end{array}\right\}\phantom{\rule{0.5em}{0ex}}℞\phantom{\rule{0.5em}{0ex}}\text{desired.}$

For instance
let A be the number, and B. its $\left\{\begin{array}{l}Q\\ C\\ QQ\end{array}\right\$ ℞ extracted by Logarithms unto 5 decimal places:

Note that you have according to my former Direction but $76\phantom{\rule{0.5em}{0ex}}Q\phantom{\rule{0.5em}{0ex}}℞$ & $88\phantom{\rule{0.5em}{0ex}}C\phantom{\rule{0.5em}{0ex}}℞$ & $94\phantom{\rule{0.5em}{0ex}}QQ\phantom{\rule{0.5em}{0ex}}℞$ to extract, whereof 10 are exact ℞. But I think you will doe well to lett the Table of $QQ\phantom{\rule{0.5em}{0ex}}℞$ alone, til you have done the other two, and then, if you finde your time too short, print the Q. & C. ℞ without troubling your selfe any further.

Sir, I am,       your humble Servant

Is: Newton

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Cambridge July 24th 1675

Copia vera

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<43v>

For Mr Collins Copie Mr Newtons 2d Letter concerning extraction of ℞