Copy letter from Newton to John Smith, dated 24 July 1675

<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 1000^th, as well as by every 100^th ℞, 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 $\left\{\begin{array}{l}\text{double the Quotient}\\ \text{triple the}\mathrm{q}\text{of the Quotient}\\ \text{quadruple the}\mathrm{c}\text{of the Quotient}\phantom{}\end{array}\right\}\text{for the}\mathrm{℞}\left\{\begin{array}{l}\text{square}\phantom{\mathrm{Qu}}\\ \text{cube}\phantom{\mathrm{Qu}}\\ \text{square square}\phantom{\mathrm{Qu}}\end{array}\right.$
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 $2\mathrm{B}$ or $3\mathrm{B}\mathrm{B}$ or ${4\mathrm{B}}^{.c.}=\mathrm{D}$ & $\mathrm{B}+\frac{\mathrm{C}}{\mathrm{D}}$ shall be the ℞ desired. That is, the same Division, by which you would finde the 6^th decimal figure, if prosecuted, will give you all to the 11^th 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 $\left.\begin{array}{c}\text{once}\\ \text{twice}\\ \text{thrice}\end{array}\right\}$ by that ℞ prosecuting the Division alwayes to 11 Decimal places, and to the Quotient add ${\text{y}}^{\text{e}}\text{said}\mathrm{℞}\left\{\begin{array}{l}\text{once, \& halfe}\\ \text{twice, \& a third part}\\ \text{thrice, \& a quarter}\end{array}\right\}\begin{array}{c}\phantom{\text{o}}\\ \text{of the summ}\\ \text{shall be the}\end{array}\left\{\begin{array}{l}\text{square}\\ \text{Cube}\\ \text{square square}\phantom{\text{I}}\end{array}\right\}\mathrm{℞}\text{desired.}$

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:
$\left.\begin{array}{l}\phantom{\text{and}}\text{2)}\mathrm{B}+\frac{\mathrm{A}}{\mathrm{B}}\text{,}\phantom{\frac{\mathrm{A}}{{\mathrm{B}}^0}}\\ \text{and}\text{3)}2\mathrm{B}+\frac{\mathrm{A}}{{\mathrm{B}}^2}\text{, shall be the}\phantom{}\\ \phantom{\text{and}}\text{4)}3\mathrm{B}+\frac{\mathrm{A}}{{\mathrm{B}}^3}\text{,}\end{array}\right\}\begin{array}{l}\mathrm{Q}\phantom{\frac{\mathrm{A}}{{\mathrm{B}}^0}}\\ \mathrm{C}\phantom{\frac{\mathrm{A}}{{\mathrm{B}}^0}}\text{root desired}\\ \mathrm{QQ}\phantom{\frac{\mathrm{A}}{{\mathrm{B}}^0}}\end{array}$

Note that you have according to my former Direction but $76Q\mathrm{℞}$ & $88C\mathrm{℞}$ & $94QQ\mathrm{℞}$ to extract, whereof 10 are exact ℞. But I think you will doe well to lett the Table of $QQ\mathrm{℞}$ 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

< insertion from the left margin >

Cambridge July 24th 1675

Copia vera

< text from f 43r resumes >

<43v>

For M^r Collins Copie M^r Newtons 2^d Letter concerning extraction of ℞