# Letter from Newton to John Collins, dated 8 November 1676

S^{r}

I doubt you think I have forgot to answer yo^{r} last letter, & to return you thanks for y^{e} pains you took in copying out for me y^{e} large letters of those two ingenious persons M. Leibnitz & M. Tschurnhause. As for what you propound about y^{e} former's calculation, you have well corrected $\frac{8{\mathrm{r}}^{5}}{\begin{array}{|c|}\hline 3\\ \hline\end{array}\begin{array}{c}\stackrel{\u203e}{\mathrm{r}\mathrm{r}+\mathrm{z}\mathrm{z}}\end{array}}$ by turning it to $\frac{8{\mathrm{r}}^{5}\mathrm{z}\mathrm{z}\mathrm{\beta}}{\begin{array}{|c|}\hline 3\\ \hline\end{array}\begin{array}{c}\stackrel{\u203e}{\mathrm{r}\mathrm{r}+\mathrm{z}\mathrm{z}}\end{array}}$ where it signifies an area, but y^{e} ordinate NP is rightly $\frac{8{\mathrm{r}}^{5}}{\begin{array}{|c|}\hline 3\\ \hline\end{array}\begin{array}{c}\stackrel{\u203e}{\mathrm{r}\mathrm{r}+\mathrm{z}\mathrm{z}}\end{array}}$, it being produced by dividing y^{e} rectangle ${\mathrm{}}_{1}{\mathrm{P}}_{1}{\mathrm{N}}_{1}{\mathrm{N}}_{2}{\mathrm{P}}_{3}$ viz $\frac{8{\mathrm{r}}^{5}\mathrm{z}\mathrm{z}\mathrm{\beta}}{\begin{array}{|c|}\hline 3\\ \hline\end{array}\begin{array}{c}\stackrel{\u203e}{\mathrm{r}\mathrm{r}+\mathrm{z}\mathrm{z}}\end{array}}$ by its' base β.

You seem to desi{re} {y^{t}} I {would} pu{illeg}|b|lish my method & I look upon yo^{r} advice as an act of singular friendship, being I beleive censured by divers for my scattered letters in y^{e} Transactions about such things as no body els would have le{illeg}|t| come out w^{th}out a subst{illeg}|a|ntial discours. I could wish I had {illeg}|c|ou{illeg}|ld| retract what has been done, but by that, I have learnt what's to my convenience, w^{ch} is to let what I write ly by till I am out of y^{e} way. As for y^{e} apprehension y^{t} M. Leibnitz's method may be more general or more easy then mine, you will not find any such thing. His observation about reducing all roots to fractions is a very ingenious one, & certainly his way of extracting affected roots is beyond it: but in order to series they seem to me laborious enough in comparison of y^{e} ways I follow, \though for {other} ends they may be of excellent {use}/, As for y^{e} method of Transmutations in general, I presume he has made further improvements then others have done, but I dare say all that can be done by it may be done better w^{th}out it, by y^{e} simple consideration of y^{e} ordinatim applicatæ: not excepting y^{e} method of reducing roots to fractions. The advantage of y^{e} way I follow you may guess by the conclusions drawn frō it {illeg} w^{ch} I have set down in my answer to M. Leibnitz: though I have not said all there. For there is no curve line exp{illeg}|r|est by any æquation of three terms, though{illeg} the unknown quantities affect one another in it, or y^{e} indices of their dignities be fractions or {illeg} surd quantities (suppose x{illeg}+{illeg} {$\mathrm{a}{\mathrm{x}}^{\mathrm{\lambda}}+\mathrm{b}{\mathrm{x}}^{\mathrm{\mu}}{\mathrm{y}}^{\mathrm{\sigma}}+\mathrm{c}{\mathrm{y}}^{\mathrm{\tau}}=0$}, where x signifies y^{e} base, y y^{e} ordinate {illeg}, λ, μ, σ, τ, y^{e} indices of y^{e} dignities of x & y, & a, b, c known quantities with their signes + or −) I say there is no such curve line but I can in less then half a quarter of an hower tell whether it may be squared or what are y^{e} simplest figures it may be compared w^{th}, be those figures Conic sections or others. And then by a direct & short way (I dare say y^{e} shortest y^{e} nature of y^{e} thing admits of for a general one) I can compare them. And so if any two figures exprest by such æquations be propounded I can by y^{e} same rul{illeg}|e| compare them if they may be compared. This may seem a bold assertion because its' hard to
say a figure mayor may not be squared or compa{red} w^{th} another, but it's plain to me by y^{e} fountain {I} draw it from, {though I} will not undertake {to} prove it to others. The same method extends to æquations of four terms {illeg} {illeg}|&| others also but not so generally. But I shall say no more at present but y^{t} I am

Yo^{rs} to serve you

Is. Newton