58
SrSir

I doubt you think I have forgot to answer yoryour last letter, & to return you thanks for yethe pains you took in copying out for me yethe large letters of those two ingenious persons M. Leibnitz & M. Tschurnhause. As for what you propound about yethe 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 yethe ordinate $\mathrm{NP}$ 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 yethe 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 $\mathrm{\beta}$. You seem to desire ytthat I would publish my method & I look upon yoryour advice as an act of singular friendship, being I beleive censured by divers for my scattered letters in yethe Transactions about such things as no body els would have let come out wthwithout a substantial discours. I could wish I had could retract what has been done, but by that, I have learnt what's to my convenience, wchwhich is to let what I write ly by till I am out of yethe way. As for yethe apprehension ytthat 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 yethe ways I follow, though for other ends they may be of excellent use, As for yethe 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 wthwithout it, by yethe simple consideration of yethe ordinatim applicatæ: not excepting yethe method of reducing roots to fractions. The advantage of yethe way I follow you may guess by the conclusions drawn frōom it wchwhich I have set down in my answer to M. Leibnitz: though I have not said all there. For there is no curve line exprest by any æquation of three terms, though the unknown quantities affect one another in it, or yethe indices of their dignities be fractions or surd quantities (suppose $\mathrm{x}$$+$ $\mathrm{a}{\mathrm{x}}^{\mathrm{\lambda}}+\mathrm{b}{\mathrm{x}}^{\mathrm{\mu}}{\mathrm{y}}^{\mathrm{\sigma}}+\mathrm{c}{\mathrm{y}}^{\mathrm{\tau}}=0$, where $\mathrm{x}$ signifies yethe base, $\mathrm{y}$ yethe ordinate , $\mathrm{\lambda}$, $\mathrm{\mu}$, $\mathrm{\sigma}$, $\mathrm{\tau}$, yethe indices of yethe dignities of $\mathrm{x}$ & $\mathrm{y}$, & $\mathrm{a}$, $\mathrm{b}$, $\mathrm{c}$ 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 yethe simplest figures it may be compared wthwith, be those figures Conic sections or others. And then by a direct & short way (I dare say yethe shortest yethe nature of yethe 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 yethe same rule compare them if they may be compared. This may seem a bold assertion because its' hard to
59 say a figure mayor may not be squared or compared wthwith another, but it's plain to me by yethe fountain I draw it from, though I will not undertake to prove it to others. The same method extends to æquations of four terms & others also but not so generally. But I shall say no more at present but ytthat I am

YorsYours to serve you Is. Newton
Cambridge. Novemb. 8. 1676.