# Draft Letter from Newton to John Wallis, c. July 1695

S^{r}

I am very much obliged to you for y^{e} pains in transcribing my \two/ Letters of 1676 & much more for your kind concern of {illeg} right being done me by publishing them. I have perused your transcripts {illeg}|o|f them & e{illeg}|x|amined y^{e} calculations & corrected some few places w^{ch} were amiss. The chief was in pag. 13 lin 29 where $\mathrm{\theta}+1=\mathrm{r}$ was written for $\frac{\mathrm{\theta}+1}{\mathrm{n}}=\mathrm{r}$. Which mistake made y^{e} examples in y^{e} next page seem faulty, tho they were not so. In y^{e} end of y^{e} 20^{{th}} page & beginning of y^{e} next, it may be convenient to print y^{e} words after this manner

Possum utiqꝫ cum conicis sectionibus conicis Geometrice comparare curvas omnes \(numero infinities infinitas)/ quare ordinatim applicatæ sunt

$\begin{array}{l}\frac{\mathrm{d}{\mathrm{z}}^{\mathrm{\eta}-1}}{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{\eta}}+\mathrm{g}{\mathrm{z}}^{2\mathrm{\eta}}}\phantom{\rule{1em}{0ex}}\text{vel}\phantom{\rule{1em}{0ex}}\frac{\mathrm{d}{\mathrm{z}}^{2\mathrm{\eta}-1}}{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{\eta}}+\mathrm{g}{\mathrm{z}}^{2\mathrm{\eta}}}\phantom{\rule{1em}{0ex}}\text{\&c}\\ \frac{\mathrm{d}{\mathrm{z}}^{\frac{1}{2}\mathrm{\eta}-1}}{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{\eta}}+\mathrm{g}{\mathrm{z}}^{2\mathrm{\eta}}}\phantom{\rule{1em}{0ex}}\text{vel}\phantom{\rule{1em}{0ex}}\frac{\mathrm{d}{\mathrm{z}}^{\frac{3}{2}\mathrm{\eta}-1}}{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{\eta}}+\mathrm{g}{\mathrm{z}}^{2\mathrm{\eta}}}\phantom{\rule{1em}{0ex}}\text{\&c}\\ \frac{\mathrm{d}}{\mathrm{z}}\sqrt{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{\eta}}+\mathrm{g}{\mathrm{z}}^{2\mathrm{\eta}}}\phantom{\rule{1em}{0ex}}\text{vel}\phantom{\rule{1em}{0ex}}\mathrm{d}{\mathrm{z}}^{\mathrm{\eta}-1}\times \sqrt{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{\eta}}+\mathrm{g}{\mathrm{z}}^{2\mathrm{\eta}}}\phantom{\rule{1em}{0ex}}\text{\&c}\\ \frac{\mathrm{d}{\mathrm{z}}^{\mathrm{\eta}-1}}{\sqrt{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{\eta}}+\mathrm{g}{\mathrm{z}}^{2\mathrm{\eta}}}}\phantom{\rule{1em}{0ex}}\text{vel}\phantom{\rule{1em}{0ex}}\frac{\mathrm{d}{\mathrm{z}}^{2\mathrm{\eta}-1}}{\sqrt{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{\eta}}+\mathrm{g}{\mathrm{z}}^{2\mathrm{\eta}}}}\phantom{\rule{1em}{0ex}}\text{\&c}\end{array}$

The explications of the two sentences w^{ch} w^{ch}ere concealed in letters set out of order \pag 13 & pag 24/ may|wil|l be best set in y^{e} margin. And in pag 13 over against the words [quam solertissimus Slusius ante annos duos tresve tecum coomunicavit, de qua tu (suggerente Collinio) {illeg} rescripsisti \eandem/ mihi etiam innotuisse] may be set this note in the margin this Note. Verba Collinij {illeg} Hoc intellexit Newtonus ex Epistola Collinij, \die 18/ Junij 18 1673, ad ipsum data, cujus \hæc sunt/ verba sunt, *As to Slusius metho{d} of Tangents it was by the|i|m well understood when pu|he| published h{is} boo{illeg}|k| De Mesolabio but \he/ did not then divulge it because he wou{ld} not prevent his friend Riccio who afterwards declining mathema{ti}cal studies desired Slusius to divulge it, who not obteining leisu{re} to write of it at large promised to send it to M ^{r} Oldenb{urgh} to publish in y^{e} Transactions. Before it arrived I writ to you {to} understand what you knew of it & having received y^{e{illeg}} An{swer,} imparted it to M^{r} Oldenburgh to send to Slusius to let him {know} that it was understood in England tho perchance not so lo{ng or} so soon as himself had attained it*.

As to the time of my finding the method of conv{ex} series, the exacte{illeg}|s|t account I can give of it is this, T{illeg} the year {illeg} 1664 not long before \michaelemas &/ Christmass I {illeg} read your {illeg}|y|our {sic} works & \& found y^{e} intercalation of your series \I think/ that winter. {illeg}/ in the notes I then look {on} {illeg} {illeg} infinitorum I {illeg}ala{illeg} \& {obser}{illeg}/ quanties into converging series by division & extraction of roots & thereby of squaring all curves. And then (that is in y^{e} beginning of the year 1666) I retired from the University into Lincolnshire to avoijd the plague danger of y^{e} plague.

|To D^{r} Wallis|