<58r>

Sr

I doubt you think I have forgot to answer yor last letter, & to return you thanks for ye pains you took in copying out for me ye large letters of those two ingenious persons M. Leibnitz & M. Tschurnhause. As for what you propound about ye former's calculation, you have well corrected $\frac{8{r}^{5}}{\begin{array}{|c|}\hline 3\\ \hline\end{array}\begin{array}{c}\stackrel{‾}{rr+zz}\end{array}}$ by turning it to $\frac{8{r}^{5}zz\beta }{\begin{array}{|c|}\hline 3\\ \hline\end{array}\begin{array}{c}\stackrel{‾}{rr+zz}\end{array}}$ where it signifies an area, but ye ordinate NP is rightly $\frac{8{r}^{5}}{\begin{array}{|c|}\hline 3\\ \hline\end{array}\begin{array}{c}\stackrel{‾}{rr+zz}\end{array}}$, it being produced by dividing ye rectangle viz $\frac{8{r}^{5}zz\beta }{\begin{array}{|c|}\hline 3\\ \hline\end{array}\begin{array}{c}\stackrel{‾}{rr+zz}\end{array}}$ by its' base β.
You seem to desi{re} {yt} I {would} pu{illeg}|b|lish my method & I look upon yor advice as an act of singular friendship, being I beleive censured by divers for my scattered letters in ye Transactions about such things as no body els would have le{illeg}|t| come out wthout 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, wch is to let what I write ly by till I am out of ye way. As for ye apprehension yt 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 <58v> extracting affected roots is beyond it: but in order to series they seem to me laborious enough in comparison of ye ways I follow, \though for {other} ends they may be of excellent {use}/, As for ye 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 wthout it, by ye simple consideration of ye ordinatim applicatæ: not excepting ye method of reducing roots to fractions. The advantage of ye way I follow you may guess by the conclusions drawn frō it {illeg} wch 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 ye indices of their dignities be fractions or {illeg} surd quantities (suppose x{illeg}+{illeg} {$a{x}^{\lambda }+b{x}^{\mu }{y}^{\sigma }+c{y}^{\tau }=0$}, where x signifies ye base, y ye ordinate {illeg}, λ, μ, σ, τ, ye indices of ye 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 ye simplest figures it may be compared wth, be those figures Conic sections or others. And then by a direct & short way (I dare say ye shortest ye nature of ye 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 ye same rul{illeg}|e| compare them if they may be compared. This may seem a bold assertion because its' hard to <59r> say a figure mayor may not be squared or compa{red} wth another, but it's plain to me by ye 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 yt I am

Yors to serve you

Is. Newton

Cambridge.
Novemb. 8. 1676.

For Mr John Collins
at the Farthing Office in
Fanchurch Street

London
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