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

S^{r},

I rec'd yo^{r} former L're as well as yo^{r} later, and should have written to yo^{u} sooner, but that I stay'd to think of something y^{t} might satisfy yo^{r} Desire; But though I can not hitherto doe it to my owne liking, yet that I may not wrack yo^{r} patience too much I have here witt {sic} yo^{u} what occurrs to mee, w^{ch} is only about facilitating y^{e} Extraccon of ℞. The form̄ 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 Extraccon of ℞ to 14 or 15 places, besides a greater number of Addicons, Subduccons & Divisions in those greater numbers: And therefore I have rather sent yo^{u} the following Notes about Extracting ℞.

1.) When yo^{u} have extracted any ℞ by com̄on Arithmetick to 5 Decimal places, yo^{u} may get the figures of the other 6 places by Dividing only the Residuum by $\left\{\begin{array}{l}\text{double the Quotient}\\ \text{triple the}\phantom{\rule{0.5em}{0ex}}\mathrm{q}\phantom{\rule{0.5em}{0ex}}\text{of the Quotient}\\ \text{quadruple the}\phantom{\rule{0.5em}{0ex}}\mathrm{c}\phantom{\rule{0.5em}{0ex}}\text{of the Quotient}\phantom{\text{}}\end{array}\right\}\phantom{\rule{0.5em}{0ex}}\text{for the}\phantom{\rule{0.5em}{0ex}}\mathrm{\u211e}\phantom{\rule{0.5em}{0ex}}\left\{\begin{array}{l}\text{square}\phantom{\mathrm{Qu}}\\ \text{cube}\phantom{\mathrm{Qu}}\\ \text{square square}\phantom{\mathrm{Qu}}\end{array}\right.$

Suppose B. the Quotient or ℞ extracted to 5 Decimal places, and C. the last Residuum, by the Division of w^{ch} yo^{u} 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 w^{ch} yo^{u} would finde the 6^{th} decimal figure, if prosecuted, will give you all to the 11^{th} decimal figure.

2) Yo^{u} may seek the ℞ if yo^{u} will, to 5 Decimal places by the logarithm's, But then yo^{u} must finde the rest thus. Divide the propounded number $\left.\begin{array}{c}\text{once}\\ \text{twice}\\ \text{thrice}\end{array}\right\}$ by y^{t} ℞ prosecuting the Division alwayes to 11 Decimal places, and to the Quotient add ${\text{y}}^{\text{e}}\phantom{\rule{0.5em}{0ex}}\text{said}\phantom{\rule{0.5em}{0ex}}\mathrm{\u211e}\phantom{\rule{0.5em}{0ex}}\left\{\begin{array}{l}\text{once, \& halfe}\\ \text{twice, \& a third part}\\ \text{thrice, \& a quarter}\end{array}\right\}\phantom{\rule{0.5em}{0ex}}\begin{array}{c}\phantom{\text{o}}\\ \text{of the summ}\\ \text{shall be the}\end{array}\phantom{\rule{0.5em}{0ex}}\left\{\begin{array}{l}\text{square}\\ \text{Cube}\\ \text{square square}\phantom{\text{I}}\end{array}\right\}\phantom{\rule{0.5em}{0ex}}\mathrm{\u211e}\phantom{\rule{0.5em}{0ex}}\text{desired.}$

For instance

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

$\left.\begin{array}{l}\phantom{\text{and}}\phantom{\rule{3em}{0ex}}\text{2)}\phantom{\rule{0.5em}{0ex}}\mathrm{B}+\frac{\mathrm{A}}{\mathrm{B}}\text{,}\phantom{\frac{\mathrm{A}}{{\mathrm{B}}^{0}}}\\ \text{and}\phantom{\rule{3em}{0ex}}\text{3)}\phantom{\rule{0.5em}{0ex}}2\mathrm{B}+\frac{\mathrm{A}}{{\mathrm{B}}^{2}}\text{, shall be the}\phantom{\text{}}\phantom{\rule{0.5em}{0ex}}\\ \phantom{\text{and}}\phantom{\rule{3em}{0ex}}\text{4)}\phantom{\rule{0.5em}{0ex}}3\mathrm{B}+\frac{\mathrm{A}}{{\mathrm{B}}^{3}}\text{,}\end{array}\right\}\begin{array}{l}\mathrm{Q}\phantom{\frac{\mathrm{A}}{{\mathrm{B}}^{0}}}\\ \mathrm{C}\phantom{\frac{\mathrm{A}}{{\mathrm{B}}^{0}}}\text{root desired}\\ \mathrm{QQ}\phantom{\frac{\mathrm{A}}{{\mathrm{B}}^{0}}}\end{array}$

Note y^{t} yo^{u} have according to my former Direccon but $76\phantom{\rule{0.5em}{0ex}}Q\phantom{\rule{0.5em}{0ex}}\mathrm{\u211e}$ & $88\phantom{\rule{0.5em}{0ex}}C\phantom{\rule{0.5em}{0ex}}\mathrm{\u211e}$ & $94\phantom{\rule{0.5em}{0ex}}QQ\phantom{\rule{0.5em}{0ex}}\mathrm{\u211e}$ to extract, whereof 10 are exact ℞. But I think yo^{u} will doe well to lett the Table of $QQ\phantom{\rule{0.5em}{0ex}}\mathrm{\u211e}$ alone, til yo^{u} have done th' other two, and then, if yo^{u} finde your time too short, print the Q. & C. ℞ without troubling yo^{r} selfe any further.

S^{r}, I am, yo^{r} humble S^{r}vant

Is: Newton

< insertion from the left margin >Cambridge July 24th 1675

Copia vera

< text from f 43r resumes >