Letter from John Collins to Newton, dated 9 September 1676

<1r>

9^th September 1676

M^r Newton
Sir

I received yours with the returne of the Papers inclosed, it is likely being in hast I did not so warily word D^r Pells sense of the infinite Series, which whatsoever it be it matters not, for he is utterly a Stranger to that kind of doctrine, he doth indeed affirme he can gett the Sum of many Algebraick Nomes giving a roote in order to the raising of a Resolvend, by ayd of tables of Logarithmes or Sines, the great promises in his Idea being attended with few or noe performances or Communications suitable causeth his esteeme much to fall &c M^r Oldenburgh is gone into the Country for 10 dayes however hath sent you a Coppy of the Letters both of Leibnitz and Schurnhaus, who are persons of great worth and Candour, and though they pretend or assert that the doctrine of infinite Series may be enlarged or the Series themselves attained by more easy Principles than yours yet you having already taken the Paines in your owne method, and the Result though not the Journey the same in both, I thinke notwithstanding you would doe well to publish the same in Latin or permitt a Translation and the comming foorth thereof in English, you will at first perceive that Leibnitz out of these Data $\mathrm{AQ}=\mathrm{r}$ the radius and ${\mathrm{Q}}_1\mathrm{N}=\mathrm{z}$ hath a designe to find the Ordinate ${}_1\mathrm{B}_1\mathrm{D}$ by an Analyticall Calculation, by ayd of the Square of the Chord AD which will be found $=\frac{4{\mathrm{r}}^4}{\mathrm{r}\mathrm{r}+\mathrm{z}\mathrm{z}}$, for the finding whereof suppose the Chord of the Complement to the Semicircle to be likewise d{rawn}, and then there is given the ratio of those Chords such as r {to z an}d the Sum of their Squares $=4\mathrm{r}\mathrm{r}$ and out of such data by a{n ana}lyticall processe $\square \mathrm{AD}$ is found $=\frac{4{\mathrm{r}}^4}{\mathrm{r}\mathrm{r}+\mathrm{z}\mathrm{z}}$, And then it holds
$\square \mathrm{AN}:\square \mathrm{NQ}\colon\colon \square \mathrm{AD}:\square {\mathrm{D}}_1{\mathrm{B}}_1$ which he calls ${\mathrm{y}}^2$ ie $\mathrm{r}\mathrm{r}+\mathrm{z}\mathrm{z}:\mathrm{z}\mathrm{z}\colon\colon \frac{4{\mathrm{r}}^4}{\mathrm{r}\mathrm{r}+\mathrm{z}\mathrm{z}}:\frac{4{\mathrm{r}}^4\mathrm{z}\mathrm{z}}{\begin{array}{|c|}\hline 2\\ \hline\end{array}\begin{array}{c}\mathrm{r}\mathrm{r}\end{array}\begin{array}{c}+\mathrm{z}\mathrm{z}\end{array}}$ the roote whereof is $\frac{2\mathrm{z}{\mathrm{r}}^2}{\mathrm{r}\mathrm{r}+\mathrm{z}\mathrm{z}}=\mathrm{y}$ as he makes it
The reason of my Writing is to rectify a mistake viz that when he sayes habita ergo recta ${}_1\mathrm{B}_1\mathrm{D}\left(=\frac{2\mathrm{z}{\mathrm{r}}^2}{{\mathrm{r}}^2+{\mathrm{z}}^{\mathrm{r}}}\right)$ et recta ${}_1\mathrm{B}_2\mathrm{B}\left(=\frac{4{\mathrm{r}}^3\mathrm{z}\mathrm{\beta }}{\begin{array}{|c|}\hline 2\\ \hline\end{array}\begin{array}{c}{\mathrm{r}}^2+{\mathrm{z}}^2\end{array}}\right)$ habebitur valor rectanguli, ${}_1\mathrm{D}_1{\mathrm{B}}_2\mathrm{B}$ multiplicatis eorum Valoribus in se invicem habebitur inquam $\frac{8{\mathrm{r}}^5}{\begin{array}{|c|}\hline 3\\ \hline\end{array}\begin{array}{c}\overline{{\mathrm{r}}^2+{\mathrm{z}}^2}\end{array}}$ (this is not the Product but should be $\frac{8{\mathrm{r}}^5\mathrm{z}\mathrm{z}\mathrm{\beta }}{\begin{array}{|c|}\hline 3\\ \hline\end{array}\begin{array}{c}\overline{{\mathrm{r}}^2+{\mathrm{z}}^2}\end{array}}$, And whereas he sayes the Ordinate NP is $\frac{8{\mathrm{r}}^5{\mathrm{z}}^2}{\begin{array}{|c|}\hline 3\\ \hline\end{array}\begin{array}{c}\overline{{\mathrm{r}}^2+{\mathrm{z}}^2}\end{array}}$ this I thinke should also be $\frac{8{\mathrm{r}}^5\mathrm{z}\mathrm{z}\mathrm{\beta }}{\begin{array}{|c|}\hline 3\\ \hline\end{array}\begin{array}{c}\overline{{\mathrm{r}}^2+\mathrm{z}\mathrm{z}}{}_{\phantom{1}}{}^2\phantom{1}\end{array}}$
I hope you will be pleased to settle me as to this doubt, not else at present but that I am

Your much obliged
Servitor
John Collins

Viviani at Florence hath a treatise de
Loco Solido &c in the Presse, and here
father Bonds theory of the inclination of the
Magnet is almost finished

<1v>

To M^r Isaac Newton

fellow of Trinity Colledge
In

Cambridge