# Letter from Isaac Newton to Henry Oldenburg, dated 26 October 1676

Oct. 26. 1676

S^{r}.

Two dayes since I sent you an answere to M^{r}. Leibnitius excellent letter after it was gone running my eyes over a transcript that I had made to be taken of it I found some things w^{ch} I could wish altered and I since I cannot now do it my self I desire you would doe it for me (those things are amended in this transcript.)

I feare I have been something too severe in taking notice of some oversights in M^{r}. Leibnitius letter considering the goodness and ingenuity of the Author and that it might have beene my owne fate in writing hastily to have committed the like oversights, but |y|et they being I think real oversights I suppose he cannot be offended at it if you think any thing be expressed too severely pray give me notice of it and I will endeavour to mollify it unless you will doe for with a word or two of your owne I beleive M^{r}. Leibnitius will not dislike th{illeg}|e| Theoreme towards the beginning of my letter (pag 5) for squaring curve lines Geometrically Sometime when I have more leisure iti {sic} possible I may send him a fuller account of it explaining how it is to be ordered for comparing curvilinear lines figures with one another and how the simplest figure is to be found with which a compounded curve may be compared Some other things in M^{r}. Leibnitius letter I once thought to have to{illeg}|u|g|c|ht you upon as the resolution of affected æquations and the impossibility of a geometricall Quadrature of the circle in which M^{r} Gregory seemes to have tripped{illeg} but I shall add one thing here that the series of æquations for the section of any angle by whole numbers which M^{r}. Tscurnhans {sic} saith he can derive by any easy methode one from an other is conteined in that one æquation w^{ch} I put in the third section of the problems in {illeg}|m|y former letter for cutting an angle in a given ratio and in another æquation like that also the coefficients of those æquations may be all obteined by this progression

$1\times \frac{\stackrel{\u203e}{\mathrm{n}-0}\times \stackrel{\u203e}{\mathrm{n}-1}}{1\times \stackrel{\u203e}{\mathrm{n}-1}}\times \frac{\stackrel{\u203e}{\mathrm{n}-2}\times \stackrel{\u203e}{\mathrm{n}-3}}{2\times \stackrel{\u203e}{\mathrm{n}-2}}\times \frac{\stackrel{\u203e}{\mathrm{n}-4}\times \stackrel{\u203e}{\mathrm{n}-5}}{3\times \stackrel{\u203e}{\mathrm{n}-3}}\times \frac{\stackrel{\u203e}{\mathrm{n}-6}\times \stackrel{\u203e}{\mathrm{n}-7}}{4\times \stackrel{\u203e}{\mathrm{n}-4}}\times \phantom{\rule{0.5em}{0ex}}\text{\&c}$ The first coefficient being 1 the second $1\times \frac{\stackrel{\u203e}{\mathrm{n}-0}\times \stackrel{\u203e}{\mathrm{n}-1}}{1\times \stackrel{\u203e}{\mathrm{n}-1}}$, the third{illeg} $1\times \frac{\stackrel{\u203e}{\mathrm{n}-0}\times \stackrel{\u203e}{\mathrm{n}-1}}{1\times \stackrel{\u203e}{\mathrm{n}-1}}\times \frac{\stackrel{\u203e}{\mathrm{n}-2}\times \stackrel{\u203e}{\mathrm{n}-3}}{2\times \stackrel{\u203e}{\mathrm{n}-2}}$ &c and n being the number by which the Angle is to be cut. As if n be 5 then the series is $1\times \frac{5\times 4}{1\times 4}\times \frac{1\times 0}{3\times 2}$ that is $1\times 5\times 1\times 0$, and consequently the coefficients $1.5.5$ So if n be 6 the series is $1\times \frac{6\times 5}{1\times 5}\times \frac{4\times 3}{2\times 4}\times \frac{2\times 1}{3\times 3}\times 0$ that is $1\times 6\times \frac{3}{2}\times \frac{2}{9}\times 0$ & consequently y^{e} coefficients $1.6.9.2$. this Scribble is not fit to be seene by any body nor {scarc{illeg}|e|} my other letter in y^{t} blotted form I sent it unless it be by a freind{illeg}

|N^{o} 4|

Newton to Oldenburg

Camb

Drawer 5

3/K