Copy letter from Newton to John Smith, dated 24 July 1675 Isaac Newton c. 425 words The Newton Project Falmer 2012 Newton Project, University of Sussex

1 f.

Published in H.W. Turnbull (ed), The Correspondence of Isaac Newton, vol. 1 (Cambridge: 1959), pp. 348-9

Letter from Newton [to John Smith?], dated 27 August 1675 [MS Add. 9597/2/18/44]
UKCambridgeCambridge University Library Macclesfield Collection MS Add. 9597/2/18/43
24 July 1675 England English Latin John Smith Unknown Cataloguer MathematicsCorrespondence Daniele Cassisa started tagged transcription Catalogue information compiled from CUL Janus Catalogue by Michael Hawkins Proofed by Robert Iliffe Code audit by Michael Hawkins
43.

SrSir,

I rec'dreceived yoryour former L'reLetter as well as yoryour later, and should have written to youu sooner, but that I stay'd to think of something ytthat might satisfy yoryour Desire; But though I can not hitherto doe it to my owne liking, yet that I may not wrack yoryour patience too much I have here wittwritt youu what occurrs to mee, wchwhich is only about facilitating yethe Extraccontion 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 Extraccontion of $℞$ to 14 or 15 places, besides a greater number of Addiconstions, Subducconstions & Divisions in those greater numbers: And therefore I have rather sent youu the following Notes about Extracting $℞$.

1.) When youu have extracted any $℞$ by common Arithmetick to 5 Decimal places, youu 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 wchwhich youu 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 wchwhich youu would finde the 6th decimal figure, if prosecuted, will give you all to the 11th decimal figure. 2) Youu may seek the $℞$ if youu will, to 5 Decimal places by the logarithm's, But then youu must finde the rest thus. Divide the propounded number $\begin{array}{c}\text{once}\\ \text{twice}\\ \text{thrice}\end{array}}$ by ytthat $℞$ 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 numbnumber, and $B$. its $\left\{\begin{array}{l}Q\\ C\\ QQ\end{array}\right\$ $℞$ extracted by Logarithms unto 5 decimal places: Note ytthat youu have according to my former Direccontion 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 youu will doe well to lett the Table of $QQ\phantom{\rule{0.5em}{0ex}}℞$ alone, til youu have done th'the other two, and then, if youu finde your time too short, print the $Q.$ & $C.$ $℞$ without troubling yoryour selfe any further. SrSir, I am, yoryour humble SrvantServant Is: Newton Cambridge July 24th 1675 Copia vera
For Mr Collins Copie Mr Newtons 2d L'reLetter conc:concerning extraccontion of $℞$