# Unarranged fragments, mostly relating to the dispute with Leibniz

## Out of M^{r} Gregories Letter of the 17^{th} of May 1671

This method of infinite Series hath no good successe in the second Segments of round Solids, at least so farre as I can improoue them yet such as it is you shall haue it

Sit igitur Ellipsis ADCE, cuius Axis transversus AC. Coniugatus DE; Centrum B, supponatur nunc Sphærois generari ex revolutione Semiellipseos ADC, circa Axem AC, secetur Sphærois a duobus Planis parallelis, per Puncta K, B, Axi DB normalibus, et a duobus Planis Parallelis per Puncta B, F, axi BC normalibus:

Sit $\mathrm{BF}=\mathrm{a}$, $\mathrm{BK}=\mathrm{b}$; erit pars Sphæroidis KIFB a quatuor dictis Planis comprehensa = $=\frac{1680{\mathrm{r}}^{6}\mathrm{b}\mathrm{a}-280{\mathrm{r}}^{4}{\mathrm{b}}^{3}\mathrm{a}-42{\mathrm{r}}^{2}{\mathrm{b}}^{5}\mathrm{a}-15{\mathrm{b}}^{7}\mathrm{a}-\mathrm{\&c}}{840{\mathrm{r}}^{5}}-\frac{1680{\mathrm{r}}^{6}\mathrm{b}{\mathrm{a}}^{3}+280{\mathrm{r}}^{4}\mathrm{b}{\mathrm{a}}^{3}+280{\mathrm{r}}^{4}{\mathrm{b}}^{3}{\mathrm{a}}^{3}+126{\mathrm{r}}^{2}{\mathrm{b}}^{5}{\mathrm{a}}^{3}+75{\mathrm{b}}^{7}{\mathrm{a}}^{3}+\mathrm{\&c}}{5040{\mathrm{r}}^{5}\mathrm{c}2}-\frac{16{\mathrm{r}}^{6}\mathrm{b}{\mathrm{a}}^{5}+8{\mathrm{r}}^{4}{\mathrm{b}}^{3}{\mathrm{a}}^{5}+6{\mathrm{r}}^{2}{\mathrm{b}}^{5}{\mathrm{a}}^{5}+5{\mathrm{b}}^{7}{\mathrm{a}}^{5}+\mathrm{\&c}}{320{\mathrm{r}}^{5}{\mathrm{c}}^{4}}-\frac{48{\mathrm{r}}^{6}\mathrm{b}{\mathrm{a}}^{7}+40{\mathrm{r}}^{4}{\mathrm{b}}^{3}{\mathrm{a}}^{7}+42{\mathrm{r}}^{2}{\mathrm{b}}^{5}{\mathrm{a}}^{7}+45{\mathrm{b}}^{7}{\mathrm{a}}^{7}+\mathrm{\&c}}{2688{\mathrm{r}}^{5}{\mathrm{c}}^{6}}-\mathrm{\&c}$ in infinitum. This Series is nothing but a Congeries of other Series's all of them being infinite, yet is the best I can haue to this Purpose, I hope M^{r} Newton hath better, and that with his Leaue yee will informe mee, If you would haue it agree to the Sphære, yee shall only put c in place of r, which will render it more Simple, I can giue such a Series as this for any \the second/ Segment of a{illeg}|ny| round Solid, and if you like this, I shall giue a Series for the second Segments of an Hyperbl|o|lick Spindle, which I imagine is of greater Consequence than any thing else for Guaging –

$\mathrm{AB}=\mathrm{x}$. $\mathrm{BC}=\mathrm{o}=\mathrm{CD}$. $\mathrm{BE}=\mathrm{y}$. $\mathrm{HF}=\stackrel{.}{\mathrm{y}}\mathrm{o}=\mathrm{IK}$. $\mathrm{KG}=\stackrel{..}{\mathrm{y}}\mathrm{o}\mathrm{o}$. $\mathrm{GI}=\stackrel{.}{\mathrm{y}}\mathrm{o}+\stackrel{..}{\mathrm{y}}\mathrm{o}\mathrm{o}$. ${\mathrm{EF}}^{\mathrm{q}}=\mathrm{o}\mathrm{o}+\stackrel{.}{\mathrm{y}}\stackrel{.}{\mathrm{y}}\mathrm{o}\mathrm{o}$. $2\mathrm{EF},\overline{)\mathrm{EF}}={}^{2}\stackrel{.}{\mathrm{y}}\stackrel{..}{\mathrm{y}}\mathrm{o}\mathrm{o}$. $\overline{)\mathrm{EF}}=\frac{\stackrel{.}{\mathrm{y}}\stackrel{..}{\mathrm{y}}\mathrm{o}\mathrm{o}}{\mathrm{EF}}.\mathrm{G}$ ${\mathrm{GF}}^{\mathrm{q}}=\mathrm{o}\mathrm{o}+\stackrel{.}{\mathrm{y}}\stackrel{.}{\mathrm{y}}\mathrm{o}\mathrm{o}+2\stackrel{.}{\mathrm{y}}\stackrel{..}{\mathrm{y}}{\mathrm{o}}^{3}+\stackrel{..}{\mathrm{y}}\stackrel{..}{\mathrm{y}}{\mathrm{o}}^{4}$. ${}^{2}\mathrm{GF},\stackrel{.}{\overline{)\mathrm{GF}}}=2\stackrel{.}{\mathrm{y}}\stackrel{..}{\mathrm{y}}\mathrm{o}\mathrm{o}+2\stackrel{..}{\mathrm{y}}\stackrel{..}{\mathrm{y}}{\mathrm{o}}^{3}+2\stackrel{.}{\mathrm{y}}\stackrel{.}{\stackrel{..}{\mathrm{y}}}{\mathrm{o}}^{3}+\mathrm{\&c}$ $\stackrel{.}{\overline{)\mathrm{GF}}}=\frac{\stackrel{.}{\mathrm{y}}\stackrel{..}{\mathrm{y}}\mathrm{o}\mathrm{o}+\stackrel{..}{\mathrm{y}}\stackrel{..}{\mathrm{y}}{\mathrm{o}}^{3}+\stackrel{.}{\mathrm{y}}\stackrel{.}{\stackrel{..}{\mathrm{y}}}{\mathrm{o}}^{3}}{\mathrm{GF}}$. $\frac{\stackrel{.}{\mathrm{y}}\stackrel{..}{\mathrm{y}}\mathrm{o}\mathrm{o}}{\mathrm{EF}}\mathrm{GL}=2\mathrm{o}${illeg} $\mathrm{GL}={}^{2}\mathrm{p}=2\stackrel{.}{\mathrm{y}}\mathrm{o}+\stackrel{..}{\mathrm{y}}\mathrm{o}\mathrm{o}$o $\mathrm{EL}=2\times \mathrm{o}$. $\mathrm{FN}=\mathrm{q}.\frac{\sqrt{\mathrm{o}\mathrm{o}+\mathrm{p}\mathrm{p}-2\mathrm{p}\mathrm{q}+\mathrm{q}\mathrm{q}}}{\sqrt{\mathrm{x}}}+\frac{\sqrt{\mathrm{o}\mathrm{o}+\mathrm{p}\mathrm{p}+2\mathrm{p}\mathrm{q}+\mathrm{q}\mathrm{q}}}{\sqrt{\mathrm{x}}+\mathrm{o}}=\mathrm{Min}=\mathrm{R}+\mathrm{S}$. ${\mathrm{R}}^{2}=\frac{\mathrm{o}\mathrm{o}+\mathrm{p}\mathrm{p}-2\mathrm{p}\mathrm{q}+\mathrm{q}\mathrm{q}}{\mathrm{x}}$. $\mathrm{S}\mathrm{S}=\frac{\mathrm{o}\mathrm{o}+\mathrm{p}\mathrm{p}+2\mathrm{p}\mathrm{q}+\mathrm{q}\mathrm{q}}{\mathrm{x}+\mathrm{o}}$. $2\mathrm{R}\stackrel{.}{\mathrm{R}}=\frac{\mathrm{-2}\mathrm{p}\mathrm{q}+2\mathrm{q}\stackrel{.}{\mathrm{q}}}{\mathrm{x}}$. $2\mathrm{S}\stackrel{.}{\mathrm{S}}=\frac{\mathrm{+2}\mathrm{p}\stackrel{.}{\mathrm{q}}+2\mathrm{q}\stackrel{.}{\mathrm{q}}}{\mathrm{x}+\mathrm{o}}$ $\frac{-\mathrm{p}\stackrel{.}{\mathrm{q}}+\mathrm{q}\stackrel{.}{\mathrm{q}}}{\mathrm{R}\mathrm{x}}+\frac{\mathrm{p}\stackrel{.}{\mathrm{q}}+\mathrm{q}\stackrel{.}{\mathrm{q}}}{\mathrm{S}\mathrm{x}+\mathrm{S}\mathrm{o}}=0$. $-\mathrm{S}\mathrm{p}\stackrel{.}{\mathrm{q}}\mathrm{o}+\mathrm{S}\mathrm{q}\stackrel{.}{\mathrm{q}}\mathrm{o}-\mathrm{S}\mathrm{p}\stackrel{.}{\mathrm{q}}\mathrm{x}+\mathrm{S}\mathrm{q}\stackrel{.}{\mathrm{q}}\mathrm{x}+\mathrm{R}\mathrm{x}\mathrm{p}\stackrel{.}{\mathrm{q}}+\mathrm{R}\mathrm{x}\mathrm{q}\stackrel{.}{\mathrm{q}}=0$ $\frac{\sqrt{\mathrm{o}\mathrm{o}+\mathrm{p}\mathrm{p}+{}^{2}\mathrm{p}\mathrm{q}+\mathrm{q}\mathrm{q}}}{\sqrt{\mathrm{x}+\mathrm{o}}}$ in $-\overline{)\mathrm{p}\mathrm{o}+\mathrm{q}\mathrm{o}-\mathrm{p}\mathrm{x}+\mathrm{q}\mathrm{x}}=\frac{\sqrt{\mathrm{o}\mathrm{o}+\mathrm{p}\mathrm{p}-2\mathrm{p}\mathrm{q}+\mathrm{q}\mathrm{q}}}{\sqrt{\mathrm{x}}}\times \overline{)\mathrm{x}\mathrm{p}+\mathrm{x}\mathrm{q}}$. $\sqrt{\mathrm{o}\mathrm{o}+\mathrm{p}\mathrm{p}+2\mathrm{p}\mathrm{q}+\mathrm{q}\mathrm{q}}$ in $-\overline{)\mathrm{p}+\mathrm{q}}\times \overline{)\mathrm{x}+\mathrm{o}}\times \sqrt{\mathrm{x}+\mathrm{o}}=\sqrt{\mathrm{o}\mathrm{o}+\mathrm{p}\mathrm{p}-2\mathrm{p}\mathrm{q}+\mathrm{q}\mathrm{q}}$ in $\overline{)\mathrm{p}+\mathrm{q}}\times \mathrm{x}\times \sqrt{\mathrm{x}}$ $\sqrt{\mathrm{o}\mathrm{o}+\mathrm{p}\mathrm{p}+2\mathrm{p}\mathrm{q}+\mathrm{q}\mathrm{q}}$ in $\sqrt{\mathrm{x}+\mathrm{o}}=\sqrt{\mathrm{o}\mathrm{o}+\mathrm{p}\mathrm{p}-2\mathrm{p}\mathrm{q}+\mathrm{q}\mathrm{q}}$in $\sqrt{\mathrm{x}}$. $\mathrm{o}\mathrm{o}\underset{.}{\mathrm{x}}+\mathrm{p}\mathrm{p}\underset{.}{\mathrm{x}}+2\mathrm{p}\mathrm{q}\mathrm{x}+\mathrm{q}\mathrm{q}\underset{.}{\mathrm{x}}+{\mathrm{o}}^{3}\mathrm{x}+\mathrm{o}\mathrm{p}\mathrm{p}+2\mathrm{p}\mathrm{q}\mathrm{o}+\mathrm{q}\mathrm{q}\mathrm{o}=\mathrm{o}\mathrm{o}\underset{.}{\mathrm{x}}+\mathrm{p}\mathrm{p}\underset{.}{\mathrm{x}}-2\mathrm{p}\mathrm{q}\mathrm{x}+\mathrm{q}\mathrm{q}\underset{.}{\mathrm{x}}$ 2pq{illeg} $4\mathrm{p}\mathrm{q}\mathrm{x}+{\mathrm{o}}^{3}\mathrm{x}+\mathrm{o}\mathrm{p}\mathrm{p}-2\mathrm{p}\mathrm{q}\mathrm{o}+\mathrm{q}\mathrm{q}\mathrm{o}=0$. $\mathrm{o}\mathrm{o}+\mathrm{p}\mathrm{p}+2\mathrm{p}\mathrm{q}+\mathrm{q}\mathrm{q}$ in $\mathrm{q}\mathrm{q}-2\mathrm{p}\mathrm{q}+\mathrm{p}\mathrm{p}$ in ${\overline{)\mathrm{x}+\mathrm{o}}}^{3}=\mathrm{o}\mathrm{o}+\mathrm{p}\mathrm{p}-2\mathrm{p}\mathrm{q}+\mathrm{q}\mathrm{q}$ in $\mathrm{p}\mathrm{p}+2\mathrm{p}\mathrm{q}+\mathrm{q}\mathrm{q}$ in \${\mathrm{x}}^{3}$/ $\mathrm{o}\mathrm{o}\mathrm{p}\mathrm{p}-2\mathrm{o}\mathrm{o}\mathrm{p}\mathrm{q}+\mathrm{o}\mathrm{o}\mathrm{q}\mathrm{q}$ in ${\mathrm{x}}^{3}+3\mathrm{x}\mathrm{x}\mathrm{o}+\overline{)3\mathrm{x}\mathrm{o}\mathrm{o}}+\mathrm{p}\mathrm{p}\mathrm{q}\mathrm{q}-2{\mathrm{p}}^{3}\mathrm{q}+{\mathrm{p}}^{4}$. $\begin{array}{r}+\mathrm{o}\mathrm{o}\mathrm{q}\mathrm{q}-2\mathrm{o}\mathrm{o}\mathrm{p}\mathrm{q}\mathrm{o}\mathrm{o}\mathrm{p}\mathrm{p}\\ \phantom{\mathrm{oooo}}-2-\phantom{\mathrm{oooo}}\end{array}$ in $\overline{){\mathrm{x}}^{3}}\begin{array}{r}+\mathrm{o}\mathrm{o}\mathrm{p}\mathrm{p},+{\mathrm{p}}^{4}\\ -2\mathrm{o}\mathrm{o}\mathrm{p}\mathrm{q}\end{array}$

$\frac{1}{16}{\mathrm{b}}^{4}-\frac{4}{8}{\stackrel{.}{\mathrm{b}}}^{3}\mathrm{v}-\frac{1}{4}{\stackrel{.}{\mathrm{b}}}^{4}+\frac{3}{2}{\stackrel{.}{\mathrm{b}}}^{3}\mathrm{v}+\frac{1}{4}{\stackrel{.}{\mathrm{b}}}^{4}-\stackrel{.}{\mathrm{b}}3\mathrm{v}+\frac{1}{4}\mathrm{a}\mathrm{a}\mathrm{b}\mathrm{b}-\mathrm{a}\mathrm{a}\mathrm{b}\mathrm{v}+\mathrm{a}\mathrm{b}\mathrm{b}\mathrm{h}-2\mathrm{a}\mathrm{b}\mathrm{h}\mathrm{v}=\mathrm{h}\mathrm{a}\mathrm{b}\mathrm{b}$ $\frac{1}{16}{\mathrm{b}}^{4}+\frac{1}{4}\mathrm{a}\mathrm{a}\mathrm{b}\mathrm{b}-\mathrm{a}\mathrm{a}\mathrm{b}\mathrm{v}-2\mathrm{a}\mathrm{b}\mathrm{h}\mathrm{v}=0$. $\mathrm{v}=\frac{4{\mathrm{a}}^{2}{\mathrm{b}}^{2}+{\mathrm{b}}^{4}}{16\mathrm{a}\mathrm{a}\mathrm{b}+32\mathrm{a}\mathrm{h}\mathrm{b}}=\frac{{\mathrm{a}}^{2}\mathrm{b}+\frac{1}{4}{\mathrm{b}}^{3}}{4\mathrm{a}\mathrm{a}+8\mathrm{a}\mathrm{h}}=\frac{\mathrm{a}\mathrm{a}\mathrm{b}+\frac{1}{4}{\mathrm{b}}^{3}}{8\mathrm{a}\mathrm{h}}$. $\frac{\mathrm{a}\mathrm{a}+\frac{1}{4}\mathrm{b}\mathrm{b}}{\mathrm{v}}=\frac{8\mathrm{a}\mathrm{h}}{\mathrm{b}}=\frac{8\mathrm{a}\mathrm{h}}{\mathrm{b}}=\mathrm{FH}$. $\mathrm{BC}=\mathrm{a}$. $\mathrm{CG}=\frac{1}{2}\mathrm{b}\colon\colon \frac{8\mathrm{a}\mathrm{h}}{\mathrm{b}}$. $4\mathrm{h}=\mathrm{HL}$

$\sqrt{\mathrm{o}\mathrm{o}+\mathrm{p}\mathrm{p}+2\mathrm{p}\mathrm{q}}$ in $\mathrm{q}-\mathrm{p}$ in $\overline{)\frac{\mathrm{x}+\mathrm{o}}{\mathrm{x}}}\frac{3}{2}=\sqrt{\mathrm{o}\mathrm{o}+\mathrm{p}\mathrm{p}-2\mathrm{p}\mathrm{q}}$ in $\mathrm{q}+\mathrm{p}$. $\mathrm{o}\mathrm{o}+\mathrm{p}\mathrm{p}+2\mathrm{p}\mathrm{q}$ in $\overline{)\mathrm{p}\mathrm{p}-2\mathrm{p}\mathrm{q}}$ in x^{{illeg}} $1+\frac{3\times \mathrm{o}}{\mathrm{x}}=\mathrm{o}\mathrm{o}+\mathrm{p}\mathrm{p}-2\mathrm{p}\mathrm{q}$ in $\mathrm{p}\mathrm{p}+2\mathrm{p}\mathrm{q}$ $\mathrm{o}\mathrm{o}\mathrm{p}\mathrm{p}-2,\mathrm{o}\mathrm{o}\mathrm{p}\mathrm{q}+{\mathrm{p}}^{4}\mp 2{\mathrm{p}}^{3}\mathrm{q}$ in $1+3\frac{\mathrm{o}}{\mathrm{x}}=\mathrm{o}\mathrm{o}\mathrm{p}\mathrm{p}+2,\mathrm{o}\mathrm{o}\mathrm{p}\mathrm{q}+{\mathrm{p}}^{4}\pm 2{\mathrm{p}}^{3}\mathrm{q}$ $\frac{3,{\mathrm{o}}^{3}\mathrm{p}\mathrm{p}-6{\mathrm{o}}^{3}\mathrm{p}\mathrm{q}+3{\mathrm{p}}^{4}\mathrm{o}}{\mathrm{x}}=4,\mathrm{o}\mathrm{o}\mathrm{p}\mathrm{q}$. $3\mathrm{p}\mathrm{p}\mathrm{o}\mathrm{o}+3{\mathrm{p}}^{4}=4\mathrm{p}\mathrm{q}\mathrm{x}\mathrm{o}+6\mathrm{p}\mathrm{q}\mathrm{o}\mathrm{o}$. $\mathrm{q}=\frac{3\mathrm{p}\times \mathrm{EF}\mathrm{q}}{4\mathrm{x}\mathrm{o}}$ $\frac{4\mathrm{x}\mathrm{o}}{3\mathrm{p}}=\mathrm{GR}=\frac{\mathrm{EF}\mathrm{q}}{\mathrm{q}}$

p. 11. l. 4.

3^{dly} We do not dispute about the antiquity symbols of the symbols used by Newton for \of/ fluents fluxions & moments \Summs & Differences. And yet/ tho the symbol $\overline{)\frac{\mathrm{a}\mathrm{a}}{4\mathrm{x}}}$ be used by M^{r} Newton \for fluents or {illeg}s is summs is/ be older then the symbol $\int \frac{\mathrm{a}\mathrm{a}}{4\mathrm{x}}$ used \in the same sense/ by M^{r} Leibnitz, in the same sense & all the symbols of fluxions used by M^{r} Newton be older then any symbols of fluxions used by M^{r} Leibnitz \& M^{r} Newtons had symbols from fluxions {like} before he wrote his Letter dated 24 Octob 17|6|76, but M^{r} Leibnitz has none to this day/ & the rectangles under the fluents \fluxions/ & the mom letter o be older s used by M^{r} Newton for moments be \are/ older then the symbols dx & dy \|used in y^{e} same sense d| by M^{r}Leibnitz|.| in the same sense/. |But| These are only ways of Notation & signify nothing {th} to the method it self w^{ch} may be without them. [All the use that can be made of them is to shew that M^{r} Newtons \Leibnitz/ method was as old as the use of the symbols dx & dy & Th M^{r} Newtons as old as his use of any symbols for the same things.]

— as in the Principia. In the first Proposition of his book De quadratura Curvarum he used prickt Letters but did not make them necessary to his method. for in the Introduction to that book he described that|is| method \at large/ & illustrated it with various examples without making any use of prickt letters. That book was printe first printed in the year 1714 but the \a copy of the/ first Proposition was copied out of it & \thereof was copied &/ \at y^{e} request of D^{r} Wallis was copied &/ sent to |y^{e}| D^{r} Wallis by M^{r} Newton in his Letter of 27 August 1692 & printed the next year in the second Volume of the Doctors works. \It was made use of upon all occasions, by M^{r} Newton when he wrote his Principia Philosophiæ/ And so many things are mentioned /cited\ out of this book{illeg} |it| in M^{r} L Newtons Letter of 24 Octob 1676 that one may reasonably conclude that the book was writ before that time. In his Analysis per æquationes &c — — — — — trifling. And if it were not, yet M^{r} Newton has the advantage \as/ was mentioned above. For prickt letters are put for fluxions or veloci w^{ch} are motions \& finite quantities & {sic} a{illeg}/ & never signify moments {w} {illeg} or different|c|es which are parte infinitely small parts generated by fluxion|.| unless w|W|here the \symbols of fluxions (whether prickt letters or others/ are multiplied by the letter o either to expres or understood \the rectang make them infinitely little the rectangles are put for moments,/ or where fluxions are considered as the exponents of m{illeg}|o|ments, & accordingly the factor o in the factor o being understood, the symbols of fluxions are put being \are/ put for the exponents of moments, but in such cases the factor o is always understood. {illeg} This symbol \factor/ M^{r} Newton always expresses when he would is demonstrating a Proposition but when he is only investigating a Propop|s|ition he usually neglects to write it down Now M^{r} Leibnitz has no symbols of fluxions in his method & theref. all M^{r} Newtons symbols of fluxions are the oldest in the kind. And as for the dx & dy of M^{r} Leibnitz they are {illeg} not to be compared w^{th} the symbols of fluxions but w^{th} the symbols of Moments that is with the rectangles under the symbols of fluxions & the {illeg} letter o used by M^{r} Newton And those rectangles are the oldest being used by M^{r} Newton in his Analysis A. O w^{ch} \sent by/ D^{r} Barrow sent to M^{r} Collins in Iuly 1669. But whereas And tho our great Mathematician tells us that M^{r} Newton by putting o for the increment of x loses all the advantage of the differential method: the contrary is true. M^{r} Leibnits by putting dx for the letter o has lost a considerable part of the advantange of M^{r} Newtons method without adding any thing new to it besides h{illeg}|i|s symbols. For the Method by M^{r} Leibnitz's way of Notation is not Geometrical nor demonstrative,|.| Nor y It is only fitted for investigation, & is not so expedite \for that {illeg}poss/ as M^{r} Newtons method when he \who in such cases/ neglects to write down the letter o, & \so/ uses but one letter to represent moments the other \other letter {illeg}/ being understood. And {illeg} These advantages it has lost, & had|s| added no new ones. For there is nothing that can be done by it but what may be done by M^{r} Newton's method & that w^{th} as much or more dispatch.

But \whereas/ M^{r} Leibnitz has told us in the Acta Eruditorum \had/ changed the Letters a & e used by D^{r} Barrow into d dx & dy he tells us in the Acta Eruditorum mensis Iunij 1686 pag 297 |that this change did not create a new method was not necessary to y^{e} method| Malo autem dx & simila adhibere \saith he/ quam literas pro illis quia istud dx est modificatio quædam ipsius x, ut sola quando id fieri opus est litera x cum suis scilicet potestatibus & differentialibus calc\ul/um ingredieatur, & relationes trancendentes {sic} inter x et aliud exprimantur: Qua ratione etiam lineas transcendentis inter x et aliud exprim æquatione {illeg}|E|xplicare licet. So then by the confession of M^{r} Leibnitz the by his own confession might have used letters as D^{r} Barrow did & the method would have been the same, but he chose rather to use the letters dx symbols dx & dy for as more convenient. And yet he from these symbols he gave the method the name of the differential method as if it was \had been/ a new method. At w^{ch} rate a man might put any letter for y^{e} Abscissa of a curve & modify the letter variously to represent the Ordinate, the {le} area, the length, the subtangent, & the radius of curvity & call this a new methodu, & give it a new name to extinguish the memory of former authors & make himself famous for inventing a new method of solving \all sorts of/ Problemes relating to curve lines.

— as in the Principles. M^{r} Newton seldome uses prickt letters till he has brought Problems to equations & by those equations is to deduce fluxions from fluents or fluents from fluxions. How he deduces fluxions from fluents he has told is|u|s in the first Proposition of his book the Quadratura Curvarum & there he uses prickt letters & by them shews how from æquations involving fluents to deduce the first econd & third fluxions of the fluents & so on in infinitum & this he does by one very short rule comprehended in five lines How Also in explaining how he extracts a fluent out of an equation involving the |its| fluxions he {illeg} uses prickt{illeg} letters & extends the Proposition to y^{e} 2^{d} 3^{d} & fl{illeg} following fluxions. In his Letter dated 242 Octob 1676 he set down this his method comprehended his method in these two sentences Data æquatione fluentes quotcun quantitates involvente fluxiones invenire & vice versa. And Vna methodus consistit in extractione fluentis ex æquatione fluxionem ejus involvente, altera &c. When D^{r} Wallis was printing this|e| second Volume of his works he descr wrote to M^{r} Newton to explain those sentences to him & M^{r} Newton {illeg} & M^{r} Newton did it by prickt letters coppying sending him a copy of his book the first Proposition of his book the Quadratura Curvarum & of another Paper This was in y^{e} year 1692. In y^{e} year 1686 M^{r} Newton wrote in the Scholium upon the second Lemma of \the second book of/ his Principia Philosophiæ {illeg} that th ten years before upon his writing to M^{r} Leibnitz about his Method of fluxions M^{r} comprised in this sentence Data æquatione quotcun fluentes quantitates involvente Fluxiones invenire & vice versa. M^{r} Leibnitz wrote back that he had also fallen into such a method & communicated his method \scarce/ differing only in from M^{r} Newtons except in the forms of words \names/ & symbols. And M^{r} Newton's symbols here related unto were those of prickt letters with pricks. And indeed in his Letter of 24 Octob 1676 so many things are quoted out of book of Quadratures that one may thence conclude \y^{t}/ this book was writ before that time \letter./ But in the Principia Philosophiæ & \in/ the Letters of M^{r} Newton \to M^{r} Collins & M^{r} Oldenburg/ & th|h|is Analysis published in the Commercium Epistolicum there was & in his Analysis he had no where occasion to treat of Equations involving fluents {illeg}|&| fluxions. In y^{e} Introduction to the book of Quadratures M^{r} M^{r} Newton explained the method of fluxions at large & illustrated it with various examples & yet made no \without making any/ use of prickt letters w^{th} pricks. And will o^{r} great Mathematician say that M^{r} {illeg} M^{r} Newton did not then understand the Method of fluxions because there are no such Letters in that Introduction

$\mathrm{r}\mathrm{x}-\frac{\mathrm{r}}{\mathrm{q}}\mathrm{x}\mathrm{x}=\mathrm{y}\mathrm{y}$. $\mathrm{r}\stackrel{.}{\mathrm{x}}-\frac{2\mathrm{r}}{\mathrm{q}}\mathrm{x}\stackrel{.}{\mathrm{x}}={}^{2}\mathrm{y}\stackrel{.}{\mathrm{y}}$. $\mathrm{r}\stackrel{..}{\mathrm{x}}-\frac{2\mathrm{r}}{\mathrm{q}}\stackrel{.}{\mathrm{x}}\stackrel{.}{\mathrm{x}}=2\mathrm{y}\stackrel{..}{\mathrm{y}}+2\stackrel{.}{\mathrm{y}}\stackrel{.}{\mathrm{y}}$. $\mathrm{r}-\frac{2\mathrm{r}}{\mathrm{q}}\mathrm{x}.2\mathrm{y}\colon\colon \stackrel{.}{\mathrm{y}}.\stackrel{.}{\mathrm{x}}\colon\colon \mathrm{PN}.\mathrm{MN}$. $\stackrel{.}{\mathrm{x}}=1$. $\stackrel{.}{\mathrm{x}}\stackrel{.}{\mathrm{x}}=0=\stackrel{..}{\mathrm{x}}$. $\frac{\mathrm{r}-\frac{2\mathrm{r}}{\mathrm{q}}\mathrm{x},\stackrel{.}{\mathrm{x}}}{2\sqrt{\mathrm{r}\mathrm{x}-\frac{\mathrm{r}}{\mathrm{q}}\mathrm{x}\mathrm{x}}}=\stackrel{.}{\mathrm{y}}=\frac{\mathrm{r}\frac{1}{2},\overline{)\mathrm{q}-2\mathrm{x}},\stackrel{.}{\mathrm{x}}}{2\sqrt{\mathrm{q}\mathrm{q}\mathrm{x}-\mathrm{q}\mathrm{x}\mathrm{x}}}-\frac{2\mathrm{r}}{\mathrm{q}}=2\stackrel{..}{\mathrm{y}}\sqrt{\mathrm{r}\mathrm{x}-\frac{\mathrm{r}}{\mathrm{q}}\mathrm{x}\mathrm{x}}+\frac{\mathrm{r}\mathrm{r}-\frac{4\mathrm{r}\mathrm{r}}{\mathrm{q}}\mathrm{x}+\frac{4\mathrm{r}\mathrm{r}}{\mathrm{q}\mathrm{q}}\mathrm{x}\mathrm{x}}{2\mathrm{r}\mathrm{x}-\frac{2\mathrm{r}}{\mathrm{q}}\mathrm{x}\mathrm{x}}$.

$\stackrel{..}{\mathrm{y}}=\frac{-\frac{2\mathrm{r}}{\mathrm{q}}-\frac{\frac{1}{2}\mathrm{r}\mathrm{r}}{\mathrm{r}\mathrm{x}-\frac{\mathrm{r}}{\mathrm{q}}\mathrm{x}\mathrm{x}}+\frac{2\mathrm{r}}{\mathrm{q}}}{2\sqrt{\mathrm{r}\mathrm{x}-\frac{\mathrm{r}}{\mathrm{q}}\mathrm{x}\mathrm{x}}}$. $\stackrel{..}{\mathrm{y}}=-\frac{\mathrm{r}\stackrel{.}{\mathrm{x}}\stackrel{.}{\mathrm{x}}}{\mathrm{x}-\frac{\mathrm{x}\mathrm{x}}{\mathrm{q}}\sqrt{\mathrm{r}\mathrm{x}-\frac{\mathrm{r}}{\mathrm{q}}\mathrm{x}\mathrm{x}}}$. the {sic}

$\mathrm{PS}\mathrm{q}-\mathrm{PH}\mathrm{q}=2\mathrm{SHN}$. $\frac{\mathrm{q}\mathrm{q}-2\mathrm{q}\mathrm{t}}{2\mathrm{SH}}=\mathrm{HN}$. $\mathrm{SH}=\sqrt{\mathrm{q}\mathrm{q}-\mathrm{q}\mathrm{r}}$. $\mathrm{HN}=\frac{{\mathrm{q}}^{\frac{1}{2}}in\mathrm{q}-2\mathrm{r}}{\sqrt{\mathrm{q}-\mathrm{r}}}=\frac{\mathrm{q}\mathrm{q}-{}^{2}\mathrm{q}\mathrm{t}}{2\sqrt{\mathrm{q}\mathrm{q}-\mathrm{q}\mathrm{r}}}$. $\mathrm{CN}=\frac{{}^{2}\mathrm{q}\mathrm{q}-2\mathrm{q}\mathrm{t}-\mathrm{q}\mathrm{r}}{2\sqrt{\mathrm{q}\mathrm{q}-\mathrm{q}\mathrm{r}}}$. CA{illeg} $\mathrm{AN}=\frac{\mathrm{-2}\mathrm{q}\mathrm{q}+2\mathrm{q}\mathrm{t}+\mathrm{q}\mathrm{r}+\mathrm{q}\sqrt{\mathrm{q}\mathrm{q}-\mathrm{q}\mathrm{r}}}{2\sqrt{\mathrm{q}\mathrm{q}-\mathrm{q}\mathrm{r}}}$. $\frac{2\mathrm{d}\mathrm{y}}{\mathrm{MN}}=\frac{2\mathrm{d}\sqrt{\mathrm{r}\mathrm{x}-\mathrm{x}\mathrm{x}}\times \mathrm{d}-2\mathrm{x}}{2\mathrm{d}\mathrm{x}-2\mathrm{x}\mathrm{x}}$. $\frac{2\mathrm{y}\mathrm{y}}{\mathrm{r}-\frac{2\mathrm{r}}{\mathrm{q}}\mathrm{x}}=\mathrm{MN}=\frac{2\mathrm{r}\mathrm{x}-2\frac{\mathrm{r}}{\mathrm{q}}\mathrm{x}\mathrm{x}}{\mathrm{r}-\frac{2\mathrm{r}\mathrm{x}}{\mathrm{q}}}=\frac{2\mathrm{q}\mathrm{x}-2\mathrm{x}\mathrm{x}}{\mathrm{q}-2\mathrm{x}}$ $\frac{2\mathrm{d}\mathrm{y}}{\mathrm{MN}}=\frac{\mathrm{d}\times \mathrm{d}-2\mathrm{x}\times {\mathrm{r}}^{\frac{1}{2}}}{{\mathrm{d}}^{\frac{1}{2}}\sqrt{\mathrm{d}\mathrm{x}-\mathrm{x}\mathrm{x}}}$. d{illeg}$\mathrm{d}\frac{1}{2}\times \mathrm{p}\frac{1}{2}$ ${\mathrm{d}}^{\frac{1}{2}}{\mathrm{r}}^{\frac{1}{2}}=\frac{{\mathrm{d}}^{\frac{1}{2}}}{{\mathrm{d}}^{\frac{1}{2}}}.\frac{{\mathrm{p}}^{\frac{1}{2}},\stackrel{.}{\mathrm{x}},{\mathrm{d}}^{\frac{1}{2}}}{2\sqrt{\mathrm{d}\mathrm{x}-\mathrm{x}\mathrm{x}}}$ $\begin{array}{l}\mathrm{3477\u221f142857}\\ \mathrm{1042\u221f5}(0299815\\ \overline{)\phantom{0}6954285714}\\ \phantom{0}3470714286\\ \phantom{0}\overline{)3929428571}\\ \phantom{00}341285715\end{array}$ $\begin{array}{r}341285715\\ \overline{)312942857}\\ 28342858\\ \overline{)27817143}\\ 525715\\ 347714\end{array}$ $\begin{array}{r}178001\\ \overline{)173857}\\ 144\end{array}$ $\begin{array}{r}\mathrm{19\u221f63}\\ \overline{)\phantom{00}126}\\ \phantom{00}\mathrm{7\u221f56}\end{array}$

If the {illeg} net d value of y^{e} goods were 100^{li} the Queens duty would be 52.2.6 as above, and the summ of the value & duty augmented by the 7^{th} part thereof would be namely $\left[\begin{array}{rrr}152.& 2.& 6\\ 21.& 14.& \frac{5}{7}\end{array}\right]\stackrel{\text{li}}{173}.17.\frac{5}{7}$ would be the gross pr value by the candle For the allowance of 6 & $6\u2064\frac{1}{2}$ per cent in the eighth part of y^{e} whole. Say therefore: As \in this case/ the gross value by the candle 173.17.0|1|$\frac{5}{7}$ is to the Queens duty 52^{li}.2.6^{d} {illeg} \in this case, so is/ that is, as 1 to 0∟299815, & so is all other cases the gross value to y^{e} Queens duty in all other cases.

Wherefore if you multiply the gross value by the ratio 0∟299815 you will have the Queens duty.

As if the Q gross value be 100^{li} this Queens duty w sum multiplied by the Queens duty \said ratio/ will {illeg} produce the Queens duty {illeg} 29^{li}∟9815 or 29^{l}.19.{0}\7/$\frac{1}{2}$ The truth of which \method/ will thus appear

## China ware sold for 100^{li}

$\begin{array}{rrr}3477143)& \overline{)\mathrm{1042\u221f5}\phantom{00000}}& (0.2998151\\ 0.3010299& 6954286\phantom{00}& \phantom{0.0000000}\\ 0.0752575& 31294287\phantom{0}& \phantom{0.0000000}\\ \mathrm{b\; 1.\; 1\u221f1892}\phantom{10}\mathrm{a}\phantom{-10\mathrm{b}\u2064\frac{1}{2}}& 31294287& \phantom{0.0000000}\\ 0.1892\phantom{00}\mathrm{a}-\mathrm{b}\phantom{10\u2064\frac{1}{2}}& & \phantom{0.0000000}\\ 0.0946\phantom{10}5\mathrm{a}-5\mathrm{b}\phantom{1\u2064\frac{1}{2}}& 27817144& \phantom{0.0000000}\\ 0.054\phantom{01}6\mathrm{b}-5\mathrm{a}\phantom{1\u2064\frac{1}{2}}& 34771& \phantom{0.0000000}\\ 0.162\phantom{0}18\mathrm{b}-15\mathrm{a}\phantom{\u2064\frac{1}{2}}& \overline{)\phantom{00000}17386}& \phantom{0.0000000}\\ 189\phantom{0}21\mathrm{b}-17\u2064\frac{1}{2}\mathrm{a}& 1042499628& \phantom{0.0000000}\\ 18\u2064\frac{1}{2}\mathrm{a}-22\mathrm{b}\phantom{021\mathrm{b}-17\u2064\frac{1}{2}\mathrm{a}}& \overline{)348}& \phantom{0.0000000}\\ 37.44.//5\u2064\frac{2}{7}.6\u2064\frac{2}{7}.\phantom{15\mathrm{a}}& 499976& \phantom{0.0000000}\end{array}$ The right method of Computation as is conceived by the Auditors is to find the ratio of the Queens duty to the gross price in any one case & then in all cases to multiply the gross price by that ratio.

Suppose \Let/ the net value of the goods be 100^{li} & the in this case the Queens duty will be 52^{l}.2.6 as above; & the summ of this value & duty augmented by the 7^{th} part thereof \(we{illeg} |(the| the allowed|an||ce| for prompt payment & warehouse room)/, namely 173^{li}.17^{s}.1^{d}$\frac{5}{7}$ will be the gross value by the candle. For the allowance of 6 & $6\u2064\frac{1}{2}$ per cent eighth part of the whole is allowed for 7^{th} part in And as this value is to the Queens duty in this case so is 1 to ,299815, & so is the gross vales to the Queens duty in any other case.

As if the gross value be \were/ 100^{li}, this multiplied by ,299815 will produce the Queens duty 29,9815 that is 29^{li}.19.7$\frac{1}{2}$. The truth of w^{ch} method will thus appear.

And if the gross value were 3748^{li}.10^{s}, this multiplied by ,299815 will produce the Queens duty 1123^{li}∟8565, that is 1123^{li}.17^{s}.1^{d}$\frac{1}{2}$.

Now the truth of this method may be thus proved.

## China ware sold for 100^{li}

<4v>
Now that 29^{li}.19.7$\frac{1}{2}$ is the Queens just duty will further appear by the following computation.

The western Churhes in their external profession of faith were now macedonians, except that they still used the language of one hypostasis as equipollent to that of \one usia &/ substance. There might be many who believed the holy Ghost to be God but they did not yet begin to speak out. I The easter Chur / & began to by|e| shy of one anothers communion. For the little Council of Alexandria had agreed that the bishops {illeg} who had abolished t consented to the abolishing \of/ the use of the word usia with its compounds, {illeg}ed (except the ringleaders) should be looked upon as p{illeg} bishops who had been only deceived & should not lose their bishopricks if they would b in time they would retract & come over to the \Nice/ Council of Nice. But now the Latines finding themselves strong enough resolved \{illeg}/ to deprive \them/ all {bisho} the bishops who would not {illeg} retract so soon as they could get an Emperor for their purpose. And began \also/ to break \break friendship w^{th} the Macedonians &/ declare for the party of Athanasius against them.

The friendship between the Macedonians & \the/ Churches of Italy {illeg} lasted about seven \five or six/ years. F{illeg} [And all this time the bishop of Rome & {taat}{illeg} {torton} Auxentius governed the diocess of Millain. For when Athanasius & the bishop{illeg} & Lybia \wrote to the Africans/ (w^{ch} was in the year 371) Auxentius was not excommunicated by the b^{p} of {illeg} those of his party in Italy. nor Valent V V{r}g{an}t{ius} by those of P{illeg} but] For Basil wa{s} made bishop of Cæsarea in Iune A.C. 361 & soon after the Coun dieity of the holy ghost was published by the Councile of Rome & Illyricum w^{ch} met at \one &/ the same time. afterwards wrote his epistole to 73^{d} epistole in w^{ch} he saith saith {sic} that when he met with Sabinus the deacon he wrote by him to the whom y^{e} western bishops had sent with letters into y^{e} east, he wrote [by him] to y^{e} bishops of Illyricum & to those of Italy & Gallia. The Concils therefore of I|R|ome & Illyricum met in y^{e} year 371 or 372 (for they met before the death of Athanasius. And in these Councils the deity of the holy Ghost was proclaimed, & Auxentius condemned. The Council of Rome consisted of 73 bps &

The friendship \& communion/ between the Macedonians & the Churches of Italy & the west lasted, I think, till Pope Damasus called \at Rome/ a Council of 93 bishops at Rome whic|o|h sent Elpidius & {illeg} w^{th} a letter to a Council convened at the same time in Illyricum, in w^{ch} letter they thus describe their faith. When the evil . . . . . . erring bishops. By these words they signify that the communion between the Greek & Latin Churches was not yet broke off. The Churches of the whole Empire whether for or against the consubstantiality of the Son, for or against the Deity of the Holy Ghost, were hitherto of one communion composed hitherto one \visible/ catholick Church, notwithstanding \the differences of opinions/ that many of its members were corrupt in their opinions \☉/ & the Latines {of the} were meditating {illeg} to \excommunicate &/ throw out of their \communion/ all those bishops. And {C} to shew that they were in earnest they excommunicated Auxentius bishop of Millain but had yet done it, except in excommunicating {illeg}t{ed}

The Council of Illyricum having recieved this letter — — — . . . others used that of three. They \Basil/ wrote also at the same time \by Sabinus/ to the bishops of Illyricum. For Basil T|in|hiis|his| 73^{d} epistle writes in his 23|7|3^{d} Epistle \he writes thus/: Ego enim cum in Sabinum inciderim in Sabinum diaconum quo nuncio huc ad nos illi [occidentales] usi sunt, ad Illyricos scripsi, ad Italos præterea & Gallos episcopos & nonnullos privatim. These Councils therefore of Rome & Illyricum \& Gallia/ were convened after Basil was made bishop, that is, after Iune 371, & so may be placed in the year 372. For Athanasius heard of them before his death and died in May 373. The controversy therefore w^{ch} about the deity of the holy Ghost w^{ch} Athanasius had been hitherto labouring to kindle brake out into a flame in the year 372

In another Epistle written this year to the Churches of Italy & Gallia Basil exprest himself in this manner. We are seized w^{th} a persecution . . . . . . . . . . . {illeg}t / during the reign of Valens the Christians of the Greek Empire brake into several parties. The churche of w^{ch} were most numerous \generality were {sic}/ were those who \for the sake of peace/ contended for the language of the scriptures & in conformity to the Apostles rule of holding fast the form of sound words & to y^{e} decree of the \Church catholick in y^{e}/ Council of Antioch \& Church catholick/ against Paul of Samosat rejected the use of the word usia with its compounds \as novel & tending to faction & {illeg} sabellianes Paulinianism/ & forbore \the/ curious inquiries \of the Gnostick/ {illeg}{o} \Nicolaitans into/ the metaphysical nature of the deity God & Christ & the holy Ghost as novel & tending to Montanism Paulinianism & Sabellianism, & forbore the curious \inquiries/ of the Nicolaitans or men of science falsly so called, into the metaphysical nature of the supreme Beings: co contenting themselves with the declaration that the Son was like the father according to the scriptures & for doing so were called {illeg} falsly called Arians by the Saintworshippers \For they anathematized as well the novel language of Arius on y^{e} one hand as that of the Gnosticks on the other/. Next to them \in number/ were tha|e| {Camaus}iæ he{m} homousiams who owned not the deity of the holy Ghost & were falsly called Macedonians by the Saint worshippers, as if Macedonius was the author of that opinion.

${\overline{)\mathrm{a}+\mathrm{x}}}^{\mathrm{m}}\times {\overline{)\mathrm{a}-\mathrm{x}}}^{\mathrm{n}}={\mathrm{y}}^{\mathrm{p}}$. $\mathrm{a}+\mathrm{x}=\mathrm{z}.\stackrel{.}{\mathrm{x}}=\stackrel{.}{\mathrm{z}}.\mathrm{a}-\mathrm{x}=\mathrm{v}.-\stackrel{.}{\mathrm{x}}=\stackrel{.}{\mathrm{v}}$ ${\mathrm{z}}^{\mathrm{m}}{\mathrm{v}}^{\mathrm{n}}={\mathrm{y}}^{\mathrm{p}}.\mathrm{m}\stackrel{.}{\mathrm{z}}{\mathrm{z}}^{\mathrm{m}-1}{\mathrm{v}}^{\mathrm{n}}+\mathrm{n}{\mathrm{z}}^{\mathrm{m}}\stackrel{.}{\mathrm{v}}{\mathrm{v}}^{\mathrm{n}-1}={}^{\mathrm{p}}\mathrm{y}^{\mathrm{p}-1}\stackrel{.}{\mathrm{y}}$ $\mathrm{m}{\mathrm{z}}^{\mathrm{m}-1}{\mathrm{v}}^{\mathrm{n}}-\mathrm{n}{\mathrm{z}}^{\mathrm{m}}{\mathrm{v}}^{\mathrm{n}-1}.\mathrm{p}{\mathrm{y}}^{\mathrm{p}-1}\colon\colon \stackrel{.}{\mathrm{y}}.\stackrel{.}{\mathrm{x}}\colon\colon \mathrm{y}.\mathrm{subtang.}=\mathrm{subp}$. $\frac{\mathrm{m}\mathrm{y}}{\mathrm{z}}-\frac{\mathrm{n}\mathrm{y}}{\mathrm{v}}.\mathrm{p}\colon\colon \mathrm{y}.\mathrm{subtang}=\frac{\mathrm{p}\mathrm{m}}{\mathrm{z}}-\frac{\mathrm{p}\mathrm{n}}{\mathrm{v}}=\mathrm{subperpendic}$ p $\mathrm{p}\mathrm{m}\mathrm{a}+\mathrm{x}-\frac{\mathrm{p}\mathrm{n}}{\mathrm{a}-\mathrm{x}}.\mathrm{y}\colon\colon \stackrel{.}{\mathrm{y}}.\stackrel{.}{\mathrm{x}}\colon\colon \stackrel{.}{\mathrm{y}}.1.\frac{\mathrm{p}\mathrm{m}}{\mathrm{a}+\mathrm{x}}-\frac{\mathrm{p}\mathrm{n}}{\mathrm{a}-\mathrm{x}}=\mathrm{y}\stackrel{.}{\mathrm{y}}$ $\overline{)\frac{\mathrm{p}\mathrm{m}}{\mathrm{a}+\mathrm{x}}}-\overline{)\frac{\mathrm{p}\mathrm{n}}{\mathrm{a}-\mathrm{x}}}=\frac{1}{2}\mathrm{y}\mathrm{y}$.

Pag. 12. l 6. + The Postscript not being sent to M^{r} Newton he did not think himself concerned to meddle with it till at length M^{r} l'Abbé Conti pressed him to write an Answer that the Pos the Post{illeg} Postscript & the Answer might both be shewed to the King & continued to press him after the King had seen the Poscript \& the Answer was not writ till after the King had seen the Postscript. This Answer was a follows./

Bet{illeg}t|w|een Letter 6 & 7 v|in|sert this Paragraph Introduction to the Observations.

The Answer of M^{r} Leibnitz being sent open to M^{r} Remond at Paris to be sent from thence to M^{r} l'Abbe Conti, & copies of M^{r} Newton's & M^{r} l@Abbe Conti's Letters being sent also thither. M^{r} Newton declined to return \an/ answer in writing & only drew up the following Observations to satisfy his frends privately that he could easily \it was easy to/ have returned an Answer had he thought \it been/ fit to let M^{r} Leibnitz go on with his politiques.

At the end of all the Letters add the Paragraph in y^{e} German Elogium of M^{r} Leibnitz.

$\frac{1}{30{\mathrm{a}}^{3}}+\frac{1}{112{\mathrm{a}}^{3}}+\frac{1}{18{\mathrm{a}}^{3}}\times \frac{{\mathrm{z}}^{7}}{{\mathrm{a}}^{10}}=\frac{4}{45}+\frac{1}{112}=\frac{493}{448}$

$\frac{560}{5040}$

Tempus autem \quæ quantitas est hic fluens hic/ expono per lineam rectā uniformiter fluentem crescentem vel ac temporis momentum per particulam lineæ eodem temporis momento genitam, et inde lineam voco quantitatem uniformiter fluentem, et particulam ejus voco momentum ejus, et quantitates alias omnes simul augescentes voco fluentes et earum partic{illeg}|u|las eodem temporis momento genitas voco fluentium momenta et velocitates augescendi voco quantitatū fluxiones] et particulas singulis temporis momentis genitas nominando momenta {illeg} — et has motuum vel incrementorum velocitates nom (ex fluxione temporis) nominando fluxiones

× (i.e. fluxiones pro differentijs)

✝ ubi Leibnitius Inventor nominatur,

pag. 44. lin 17 — de qua sequentia \paulatim e{illeg}/ anno 1676 ex chartis antiquioribus extraxi. \Anno vero 1676 priores decem/ olim excogitavi et anno 1676 \Propositiones/ in formam sequentem redegi, undecimam vel eodem anno vel paulo post adjuxi, et duodecimam ex Epistolis anno 1676 scriptis jam addidi.

ut videre licet in Lib. 11, Prop. XIV cas. 3.

Quoniam methodus meus generalis ex methodo fluxionem et methodo Serierum convergentium convergentium {sic} componitur, id{illeg} et \quantitatum in series resolu{t}arum/ momenta prima secunda, tertia cætera sunt \{illeg}t/ terminis serierum \correspondentibus serierum terminibus/ momento temporis genitis proportionalis|a| ideo terminis s in resolutione Problematum nonnunquam usus sum terminis serierum. Sit A quantitas fluens, \&/ $\mathrm{B}={\mathrm{A}}^{\frac{\mathrm{m}}{\mathrm{n}}}$ quantitas alia fluens. Augeatur quantitas A momento O, et quantitas B evadet ${\overline{)\mathrm{A}+\mathrm{O}}}^{\frac{\mathrm{m}}{\mathrm{n}}}$. quæ in seriem resoluta fit ${\mathrm{A}}^{\frac{\mathrm{m}}{\mathrm{n}}}+\frac{\mathrm{m}}{\mathrm{n}}\mathrm{O}{\mathrm{A}}^{\frac{\mathrm{m}-\mathrm{n}}{\mathrm{n}}}+\frac{\mathrm{m}\mathrm{m}-\mathrm{m}\mathrm{n}}{2\mathrm{n}\mathrm{n}}\mathrm{O}\mathrm{O}{\mathrm{A}}^{\frac{\mathrm{m}-2\mathrm{n}}{\mathrm{n}}}+\frac{{\mathrm{m}}^{3}-3\mathrm{m}\mathrm{m}\mathrm{n}+2\mathrm{n}\mathrm{n}}{6{\mathrm{n}}^{3}}{\mathrm{O}}^{3}{\mathrm{A}}^{\frac{\mathrm{m}-3\mathrm{n}}{\mathrm{n}}}+\mathrm{\&c}$. Et hujus terminorum fluxione{illeg} primæ sunt $\frac{\mathrm{m}}{\mathrm{n}}{\mathrm{A}}^{\frac{\mathrm{m}-\mathrm{n}}{\mathrm{n}}}$.

Et ipsius ${\mathrm{A}}^{\frac{\mathrm{m}}{\mathrm{n}}}$ fluxio prima est $\frac{\mathrm{m}}{\mathrm{n}}{\mathrm{A}}^{\frac{\mathrm{m}-\mathrm{n}}{\mathrm{n}}}$ & hujus fluxio est {illeg} $\frac{\mathrm{m}\mathrm{m}-\mathrm{m}\mathrm{n}}{\mathrm{n}\mathrm{n}}{\mathrm{A}}^{\frac{\mathrm{m}-2\mathrm{n}}{\mathrm{n}}}$, Et hujus fluxio {illeg} ${\mathrm{m}}^{3}-3\mathrm{m}\mathrm{m}\mathrm{n}+2\mathrm{m}\mathrm{n}\mathrm{n}$

To

S^{r} Isaac Newton at the Lower Side of Leicester fields

Westminster.

Office {C}

suposd Golden square.

<7v>For S^{r} Isaac Newton at his house in S^{t} Martins lan\e/ \street/ nigh Leîsterffields

These

<6bis(r)>Honored Sir

In obedience to my fathers Commands I am oblig'd to continue here at london for sometime longer S^{r} W^{m} Quintin haueing promis'd my friends in the Countrey that I shall be preferd in the Customes, now seeing that I haue Qualified my self for that purpose, & that Iam in expectation dayly to come into M^{r} Iohn Selbys place at Whitby, he being to be promoted to a Collection in the South, & that now Sir the Welfare of our Family seems much to depend on my good success. Y^{r} kindnes has been extraordinary to me, for otherwise Imight haue suffer'd very much. I apply dayly to the Treasury & hope that M^{r} Iohn Selbys Warrant & mine will be granted us & that how glad I shall be to be in a Condition to support my Dear Father, that has liu'd well in \y^{e}/ world S^{r} you must not think much with me for being so free with you Imust Confess Iue been a great trespasser upon you but Ihope y^{r} Goodness will pardon me, & that my Dear father prays dayly you may continue my friend, & not let us sink now, when there is so fair a probability of my getting now into business, Iue' taken abundance of pains & the great fatigue Iue' undergone & hardships here as {illeg}|a|^{r} {illeg}|al|most inexpressible. I hope I shall demonstrate, when please god I am in business my gratitude to you & how much I am

Honored Sir

His patet me anno 1676 & annis minimùm quin \vel septem/ prioribus methodum & Analysin \methodum/ generalem habuisse reducendi Problemata ad æquationes fluxionales. Et ubi ex hujusmodi æquationibus tam affectis quam simplicibus eliciendi quantitates fluentes, et {illeg}qua{n}p{illeg}perin et in hac methodo quando assuma{illeg} series pro quantitatibus ignotis \assumere/ et assumptas terminorum coefficientes & indices dignitatum ex conditionibus Problematis determinare, Et inter computandum fluxiones et momenta ex fluentibus & fluentes \vicessim/ ex fluxionibus & momentis deducim & in huic finem me Theoremata plura pro Quadratura figurarum & inventione fluentium \per hanc methodum/ concinnasse, & nomen Analyseos huic methodo impossuissem methodos augentium {illeg} General{illeg} et Barrovij methodo meæ me nomen Analyseos hui methodo olim \ab anno 1669/ imposuisse \propter usum ejus in Solvendis problematis/ \propter \generalem/ usum hujus methodi \in {illeg} solutionionis problematibus/ me nomen Analyseos eidem ab anno 1669 imposuisse/ & huic. affines fuisse methodos Tangentium Gregorij, et Barrovij, propterea quod Methodum Slusij similiter producerent, sed [eos \{ubro}/ de me nunquam conquestos fuisse licut {illeg} sed a] & Barrovium Analysin meam ut methodum novam cum Co ad Collinium mississe misisse.

Brevitate verborum effectum est ut Scholium præcedens male intellectū fuerit, et propterea rem totam fusius enarrabo.

Cum vero D. Leibnitius post annos septem \anno 1684/ elementa hujus Analyseos in lucem emitteret & silentio præteriret literas meas p{illeg}æ {o} ea omnia quæ vead hanc methodum spectantia quæ vel ab Oldenburgo accept|e|rat vel in many|u| Collinij viderat: posui Scholium superius ut inde constaret me primum de hac methodo scripsisse et [elementa ejus in Lemmate præcedente posita synthetice demonstrata non habuisse aliunde] & Lemma superius ab editis Leibnitianis *non fuisse desumptum.*

Terminos serirum pro momentis nonnunquam usus sum{illeg} propter analogiam. Nam hi termini per terminos {illeg} 1. 1× \correspondentes/ seriei numeralis $1.1\times 2.1\times 2\times 3.1\times 2\times 3\times 4$ {illeg}|m|ultiplicati convertuntur in momenta.

<7v>Et his admitus D. Leibnitius {illeg} qu{illeg}{sam} Collinium consulebat de compendium in man hocce in manu Collinij videre potuit. Is enim hoc tempore Collinium de commercio Gregorij et meo consuluit et partem litererum nostrarum in ejus manu vidit & ab Oldenburgo paulo ante postulaverat ut demonstrationem mearum serierum a Collinio procuraret: quæ Demonstratio extabat in hoc Compendio

Analysis vero de qua hic locutus sum eadem est cum Analysi \per series/ cujus {illeg}|c|ompendium Barrovius \anno 1669/ ad Collinium misit

Ad hæc /His\ respondit D. Leibn. 27 Aug 1676 in hæc verba.

Hoc compendium est Analysis illa per series \fluxiones et mom{enta}/ quam Ionesius edidit.

Hæc est Analysis illa per series, fluxiones et momenta cujus specimen Barrovius noster anno 1669 cum Collinio communicavit ut supra.

<8r>Vbi symbolum o{illeg} ut quantitates fracta

In Epistolis meis 10 Decem 1672 & 24 Octob 1676 \datis/ dixi quantitates surdas methodum meam non morari, Et hæc{illeg} \hanc/ res|m| exempl{illeg}|o|{illeg} habere{illeg} \explic{illeg}ui|o|/ in Analysis mea a Ionesio edita, pag. 14. Substituatur uti \in æquatione/ pro quantitate radicale symbolum ubi quodvis; Tractetur symbolum illud ut quantitas fluens; Et completo opere pro symbolo et ejus fluxione scribatur quantitas radicalis et ejus fluxio.

Newtonus quantitatem aliquam ut uniformiter fluentem spectat Newtonus [fluxionem \exponentis/ temporis exponit per unitatem fluxiones aliarum quantitatum per alia symbola,] momentum temporis per exponit per Newtonus {illeg} \designat/ Tempus \uti/ per quantatem {sic} quamcun uniformiter fluentem {illeg} momentum {illeg} \fluxionum/ ejus designat per literam {a} Leibnit unitatem & momentum per literam o \Newtonus designat/: Aliarum quantitatum fluentes N designato per alia symbola et momenta \earum/ per symbola illa ducta in sym momentum o, e et{illeg}|st| a|A|reas \vero/ curvarum \designat/ per ordinatas ductas in quadrato inclusas. Leibnitius pro fluxionibus nulla habet symbola, pro momentis præfigit literam {illeg}t symbolis fluentium \literam d/, pro et pro Areis præ figit Literam s symbolis {O}{illeg} Ordinatarum \literam/. Newtonus caculo suo usus est in Analysi quam Barrovius cum Collinio mense Iulio anni 1669 communicavit; \et/ Leibnitius hanc methodum tum in Analysi {a}|i|lla tum alias in Epistolis 10 Decem. 1672, 13 Iun 1676 & 24 Octob 1676 valde generalim esse \& \ad/ {illeg}endas curvarum areas longitudines curvitat{e}s (sc. per fluxiones secundas descripsit,/ designavit /{illeg} significavit\ & verbis partim apertis partim fig descripsit partim figuratis celavit, [eam ad Curvarum areas longitudines, centra gravitatis \soliditates/ & curvaturas (sc. per fluxiones secundas) sese extendere, ut et \ad/ inversa Tangentium Problemata aliq|a| difficiliora{illeg}, sino vero ad omnia pene dixerat problemat si forte numeralia quædam Diophantæis similia excipiantur.] Leibnitius suam communicare cœpit anno 1677 \mense Iunio/ Et ut suam faceret scripsit {Et} in hæc verba. Newtono assentior *Clarissimi Slusij methodum \tangentium/ nondum esse absolutam \celeberrimo/ Newtono assentior: Et jam a multo tempore rem tangentium longe generalius tractavi scilicet per differentias Ordinatarum*. Sed anno tamen superiore methodum differentialem minime invenerat. Scripsit enim 27 Aug. 1676 *multa esse adeo mira et implexa ut ne ab æquationibus pendeant ne a Quadraturis: qualia sunt (ea multis alijs) Problemata methodi Tangentium inversæ; quæ etiam Cartesius in potestate non esse fassus est*. [Annis 1675 & 1676 D. Leibnitius quadraturam \quandam/ circuli componebat {illeg} limabat et poliebat vulgari more: sed postquam\ea/ Analysin gra{illeg}isse novam invenerat prolixius exponere vulgari more quæ Analysis sua nova paucis exhibet non satis operæ pretium videbatur. Leibn. in Actis Eruditorum Anno 1691 p. 178.]

sed acceptis Newtono Literis rescripsit {illeg} anno sequente mense Iunio methodum differentialem \ut ut o olim a sc inventam/ communicare cœpit & ut his verbis. *Clarissimi Slusij methodum tangentium nondum esse absolutam \celeberrimo/ Newtono assentior: et jam a multo tempore rem tangentium longe generalius tractavi scilicet per differentias Ordinatarum* &c. Quod methodum hanc a multo tempore invenerat probandum est. Nam D. Leibnitius pro se testis esse {illeg} non potest. Iniqus esset Iudex qui in rebus controversis hominē quemvis pro seipso testem admitteret.

Et vires more considerat \hase considerat, non/ ut qualitates primaries, \primas causis destitutas aut rebus \corporibus// essentiales, {illeg}d et causes destitutas \considerat/ sed ut qualitates \{lats} patent{illeg}/ quaru{m} causes ignoramus. Et vires considerat ut qualitates primas causis destitutas aut corporibus essentiales sed quarum causes ignoramus.

*Sir Isaac Newton*

*These*

Hon'd Sir,

I take leave to put You in mind of bringing with You to the Royal Society Your Key of the Iron Chest which contains the Common Seal, the Lease being ingross'd and ready for the Seal to be affixt to it.

Crane-Court, Nov. 9^{th} 1713.

I am

Your most Obliged and Obedient Serv^{t}.

Io: Thorpe

Galileo argued that uniform gravity by acting equally in equal times upon a falling body would produce equal velocities \of descent/ in those times, or that the whole force imprest, the whole time of descent & the whole velocity acquired \in falling/ would be proportional to one another; but the whole descent or space described would be \arise from the time & velocity together & there be in a compound ratio of them both, or as the/ as the {sic} \time of descent & the velocity of descending together that is as the/ squares of the times of \{in ther of there}/ descent. And hereby he demonstrated that the a Projectile in a space void of resistance describes a Parabola. And Mathematicians unanimously agree that he was in the right, And M^{r} Leibnitz himself being one of them. And yet he measures the force imprest, not by the velocity acquired to w^{ch} {it} is proportional, but by the space of descent to w^{ch} it is not proportional. And if a body ascend, grav the gravity of the body by acting \upon it/ equally in equal times will take of equal velocities of ascent in those equal times, or that |②| the whole force imprest, |①| the whole time of ascent to any height |③| & the whole velocity taken of in that as{illeg} time are proportional to one another \but not to the space of ascent./ But The whole space of ascent will be as the velocity & time together \arise from the time & velocity together & be in compound ratio of them both/, that is, as the square of either of them. And upon these rules of desce ascending & descending, Galileo demonstrated that projectiles would, in spaces void of resistance, {d}escribe Parabolas. And all Mathematicians acquiesce (not excepting M^{r} Leibnitz himself) unanimously agree that he was in the right. And it is proportional but by the space of ascent to which it is not proportionall.

et in epistolis supra impressis significavit {illeg} methodum suam ad tangentes directe & inverse et {illeg}l{illeg}qu{illeg}d \ad/ alia \etiam/ problemata \genera{illeg}/ extendere, & method{illeg} Inter Leibnitius et e{illeg}lis [et methodum suam in tangentibus directe et inverse per exempla exposuit.] Leibnitius negaverat Tangentes per inverse per æquationes trac vel \&/ quadraturi|a|s tractavi posse sed Lecta Newtoni literis animum advertit ad methodos tangentium & consideratio diffentiarum {sic} in methodis illis primam lucem ipsi affundit.

Leibnitius hic fat{a}tur methodus determinandi Tangentes per differentiam|s| \linearum/ primam ipsi lucem affunde{ri}t|isse| \Leibnitio/ id est methodum|s| Archimedis a Fermatio, Gregorio, Barrowo restitut{u}m|a| & promotam, Scripserat Newtonus methodum suam ad tangentes de{termi}nisse in omni problemat{illeg} genere. \a Newtono ad æquationes {illeg}is \quasvis in indeterminatas/ & motuum velocitates applicatam./

Vidit hoc Fermati{o}|u|s antea, determinando punctum flexus contrarij.

Nulla est hæc analogia. Differentiæ non sunt summarum differentiæ, {illeg} nec relationem habent \habent/ ad summas \habent {illeg} nisi quatenus sunt/ nisi quantitatum|es| infinite parvorum D Leibnitius methodum suam \minores/ aliunde habe ad \{P}{illeg}am/ D. Leibnitius aliunde accepit.

Methodus \igitur/ determinandi Tangentes per differentias line{illeg}|á|rum primam lucem affuderat Leibnitio \(p / id est methodus Archimedis a Fermatio Gregorio Barrowo restituta & promota, a Newtono ad quantitatum {illeg} \a{illeg}/ augmenta seu momenta generaliter applicata. Hujus methodi exempl{illeg}|a|{illeg} in problemate tand|g|entium directo (p ) & inverso (p ) & in Quadratura curvarum p \et {illeg} applicatione \methodo/ serierum p / Newtonus dederat. [Et Lebnitius {illeg} subinde \postea/ de methodo tangentiū amplianda cogitare cœpit (p ) & cum \ipsius/ Hudeni{illeg}|o| de eadem locutus est (p) & methodum tangentium per differentias ut omnium optimam in animo revolvens (p \p /) invenit methodū \incidit in/ differentialem \(p )/ et eandem cum Newtoniana statim conferens similitu similitudinem de contulit {illeg} & per omnia similem deprehendit p cum {illeg}t{illeg} ta primam lucem int|d|e habuit p & methodum {illeg} differentialem \differentialē sic reperiam/ cum Newtoniana statim contulit \id/ in problemate tangentium directa p & inversa|o| p et in Quadraturis p & similem deprehendit \animadvertit/ p sed in Actis Lipsicis prætendit se hoc non prius animadvertisse quam opera Newtoni et Wallisij prodiere. Sic etiam oblitus est literarum Oldenburgi per quas s] Leibnitius ex his deprehendit similitudinem in methodorum p sed] Et ex his D. Leibnitius statim deprehendit similitudinem methodorum ab initio p

Sic {et} {sic}

|Et| His admonitus D. Leib D. Leibnitius in methodum {illeg} tangentium \per differentias/ animum advenit \intendit/ (p \46, 47/ 87, 88) & methodum differentialem Newtonianæ similem \esse/ statim ab initio deprehendit p. 90, 91, 93.

$\frac{\mathrm{NH}}{\mathrm{CN}+\mathrm{FG}}=\frac{\mathrm{e}\mathrm{e}\mathrm{n}\mathrm{o}-\mathrm{a}\mathrm{n}\mathrm{o}\mathrm{o}\times {\mathrm{e}}^{5}}{{\mathrm{e}}^{3},\mathrm{e}\mathrm{e}\mathrm{n}\mathrm{n}\mathrm{o}\mathrm{o}-\frac{\mathrm{a}\mathrm{n}\mathrm{n}{\mathrm{o}}^{3}}{2}}=\frac{{\mathrm{e}}^{4}-\mathrm{a}{\mathrm{e}}^{2}\mathrm{o}}{\mathrm{e}\mathrm{e}\mathrm{n}\mathrm{o}-\frac{1}{2}\mathrm{a}\mathrm{n}\mathrm{o}\mathrm{o}}=\frac{\mathrm{e}\mathrm{e}}{\mathrm{n}\mathrm{o}}-\frac{\mathrm{a}}{2\mathrm{n}}$

$\frac{\mathrm{CF}}{2\mathrm{FG}}=\frac{\mathrm{n}\mathrm{o}}{\mathrm{e}}\times \frac{{\mathrm{e}}^{5}}{\mathrm{e}\mathrm{e}\mathrm{n}\mathrm{n}\mathrm{o}\mathrm{o}+\mathrm{a}\mathrm{n}\mathrm{n}{\mathrm{o}}^{3}}=\frac{{\mathrm{e}}^{4}}{\mathrm{e}\mathrm{e}\mathrm{n}\mathrm{o}+\mathrm{a}\mathrm{n}\mathrm{o}\mathrm{o}}=\frac{\mathrm{e}\mathrm{e}}{\mathrm{n}\mathrm{n}}-\frac{\mathrm{a}}{\mathrm{n}}$

$\mathrm{CP}.\mathrm{CG}\colon\colon \sqrt{\mathrm{CN}}.\sqrt{\mathrm{GF}}$ & arcus HC, CP erunt synchroni, et $\mathrm{HC}-\mathrm{CP}$ erit decrementum momentaneum ex resist & grav. Est $\mathrm{HC}-\mathrm{HN}$ {illeg} increm momentaneaum ex grav. Ergo $2\mathrm{HC}-\mathrm{CP}-\mathrm{HN}$ decrem. moment. ex grav. resist. $=\mathrm{HC}-\mathrm{CQ}$

*a* D Leibnitius hic fatetur se methodum differentialem a relatione tangent Differentiarum ad Tangentes \didicisse/ habuisse, id est a method{illeg}|o| determinandi Tangentes per Differentias \didicisse So{thian}t/ Archimedes, Fermatius, Gregorius, \&/ Barrowu{s} methodus\um/ \{illeg}s/ Differentiarum ad Tangentes applicuerant\e/: {illeg} Newtonus {illeg} se Fermatio {illeg} se a Fer se primam \hic Lucem primam/ a Fermatio \hic accepisse {illeg}t fatetur/ {illeg}pe{illeg} esse a{illeg}{tos} et \hanc/ methodum \a Fermatij s{illeg} acceptam/ G{illeg} {ne}|a|d æquationes abstractas \& proportiones motuum/ applicando, et per æquationes \& proportiones motuū/ ad op{illeg} \a{illeg}i{a}{illeg} quævis/ problemata \quævis/ quæ \per/ æquationes tractari possunt generalem re{illeg}d{illeg}tus Et {l}{illeg} s{illeg}g{illeg}t per proportiones motuum et hoc {cum} method \& proportiones motuum & series tractari possunt applicu{illeg} {s}{illeg}{ic} reddidit valde generalem/ & se talis methodi compotem \applicando {illeg}a{u}{illeg}m esse & methodum Slusij ejusdem corollariū/ esse significavit in epistolis supra impressis pag & De{inde} d{illeg} Leibnitius Deinde cum scriberet \& {illeg} {quibusdam} methodū illustravit p{illeg} pag &/ \Et cum/ D. Leibnitius \scripsissit/ methodum inversam tangentium impotestate esse \et methodum Slusij esse cor{ollarium} ejusdem/ ab æquationibus \& quadraturis/ non pendere, Newtonus respondit e{illeg}|s{illeg}|dem \hanc etiam/ in potestate esse Et tum dem{illeg} et exemplum \generale/ dedit solutionis per quadraturam|s| Curvarum ubi datur relatio inter latera duo quævis trianguli quod ab ordinata, tangente & subtangente constituitur. Et his ominibus {sic} Et tum demum D. Leibnitius his omnibus admo{t}eitus {illeg} methodum differentialem {illeg} \{illeg} methodum generalem quæsivit per/ relationem differentiarum ad Tangentes. habuit quæsivit & |in| Newtonianam invenit incidit quæ methodo Newtoni responderet, & se in ejusmodi methodum incidisse mox significavit scripsit ad Oldenburgum significavit in ult{illeg} s{illeg}{d} per Epistolā pag {de} quam videas pag supra impressam pag Methodus tangentium a Slusio publicata a Dixerat Newtonus methodum tangentium a Slusio {illeg} ascriptam, a principijs suis \methodo sua generali/ facillime fluere. Slusius Facillime fluit hæc methodus a Differentijs. Et Leibnitius hic fatetur se methodum differentialem a relatione Differentiarum ad Tangentes habuisse. Et tum demum co{m} D. Leibnitius animum ad\vertit ad/ methodos tangentium, & consideratio Differentiarum in methodis illis primam lucem ipsi affundit Et met problemata tangentium inversa {a}li{illeg} alia similia a tang ab æquationibus & quadraturis pendere se primam negasse & {illeg} \subinde/ a Newtono didicisse {illeg} statim oblitus est Vide p. 65, {illeg}|85|, 86, 93.

\Vidi ^{b} mox{illeg}/ b Hoc ante{illeg}|a| Fermatius qui, per \differentiam primam evanescentem id est per/ osculum determinavit punctum flexus contrarij.

c Et notavi ^{c} mirabilem &c c Nulla est hujusmodi {illeg}|a|nalogia Differentiæ \non/ sunt summarum differentiæ; non sunt ejusdem generis quantitates cum summis; [Leibnitius primam lucem aliunde habuit.] sunt ad summas ut momentum ad tempus, vel punctum ad lineam, non ut radix ad potentiam. Leibnitius primam lucem aliunde habuit.

Sic etiam oblitus est liter\ar/um Oldenburgi per quas series suas omnes accepit p.

he saith that when he was in London the second time he saw some of my Letters in the hands of M^{r} Collins \cheifly those w^{ch} related to series/ & observed that {illeg} {sic} in one of them I acknowledged my ignorance in the dimensions of the vulgar figures except the Sphæroid \Cissoid. &/ & \that the Committee had omitted this. But/ being told y^{t} this was in my Letter of 24 Octob. 1676 & was published {in} the Commerc. p 74 he acknowledges that it was, & adde|s|d {sic} that \he would cite another insta{nce.}/ i|I|n another of m{y} Letters w^{ch} he then saw (meaning a Letter dated 1672 & not yet publish{ed} \said that/ I acknowleded {sic} my ignorance that I could not find the second sections (or segm{ents} of Sphæroids, & that the Committee acknowledged had omitted this.

{illeg} were as much as was proper in tha{t} {illeg}ere I it being {illeg} of tha{illeg}|t| book to enter into disputes about {illeg} {M}^{r} Pell at that {illeg} notice of Mercators series for the H{illeg}d M^{r} Collins had some {illeg}communited mine & Gregories series to the Math{ematicia}ns at home & abroad, {illeg} might meet with some of them either at London or Paris without being {illeg} with M^{r} Collins.] & might be able to give him notice of

{illeg} ${\mathrm{x}}^{\mathrm{n}-1}\stackrel{.}{\mathrm{x}}=\stackrel{.}{\mathrm{y}}.{}^{\mathrm{n}}\mathrm{x}^{\mathrm{n}}\stackrel{.}{\mathrm{x}}=\stackrel{.}{\mathrm{y}}\mathrm{x}$.

Provided nevertheless that over & above the summs above mentioned there may be issued yearely out of the coinage monnys of England, the summ of 300^{li} per an' for the salary of the General of the Mint in Scotland, 50^{li} per an' for {t}he salary of the Clerk of the Bullion & 50^{li} per an' in part of the salary {of} the Warden of the said Mint untill the next voidance of their places.

{illeg} in the place \Scholium Passage Paraph/ there referred unto I do not find one word to this purpose. On the con{trary} {I} there represent that I sent notice to M^{r} Leibnitz of my Letter bef Method {illeg} sent notice to me of his method & left him to make it appear that he had {illeg} method before the date of my Letter, I that is, eight months \at least/ before the date {illeg} also And by referring to the Letters w^{ch} passed between M^{r} Leibnitz & {illeg} before, I left the Reader to consult those Letters & interpret the {illeg} Paragraph thereby. For by those Letter{s} {illeg}ld understand \see/ that I {illeg} {o}f that Method & the Method of Ser{ies} {illeg} five years before {illeg} {t}hose Letters that is, in the year 16{illeg} {M}^{r} Leibnitz knew nothing

quæ beneficio serierum ad omnia pene problemata se extenderet, etiam ad inversa tangentium problemata {ab} aliq|d| difficiliora \se extenderet/ & methodum exemplis illustrarem literis vero transpositis hanc sententiam involventibus [Data æquatione fluentes quotcun quantitates involvente fluxiones invenire, et vice versa] fundamentum ejus celare, at exemplar epis Epistolæ cujusdam {ad} Anno 1672 add Collinium dat{illeg} accepisset etiam exemplar Epistolæ a me anno 1672 ad Collinium scriptæ \in/ qua in dixeram methodum Tangentium Slusij Corollarium esse hujus methodi generalis, Cum methodus Tangentium Slusij fundaretur in di in \proportione/ differentij|æ| Ord Ordinatarum duarū sibi proximarum ad differentiam Abscissarum et anno 1672 ad Collinium scripsissem hanc methodum esse Corollarium methodi generalis quæ citra molestum ullum calculum ad resolvent|d|um abstrusiora problematum genera de curvitatibus, Areis, longitudinibus, centris gravitat{illeg}|is,| curvarum &c & ad quantitates surdas minime hæret, deinde exemplar autem hujus Epistolæ ad D Leibnitium, mittere anno 1676 ad D. Leibnitium mitteretur; et eodem anno literis alijs ad ipsum missis significarem me compotem esse methodi ducendi tangentes determinandi maximas & minimas ducendi tangentes quadrandi curvilineas et similia peragendi quæ in terminis surdis æque ac in rationalibus procederet, et cujus benefici{o} qu{a} beneficio serierum ad omnia pene problemata {illeg} [etiam ad inversa tangentium alia difficili ora] se extenderet, [et methodum exemplis illustrarem] respondet Le D. Leibnitius id sibi non videri; esse {illeg}|e|nim multa us ad eo mira et implexa ut ne ab æquationibus pendeant ne ex quadraturs, qualia sunt ex multis alijs problemata methodi tangentius inversæ. Cum autem rescripsissem \inversa de Tangentibus problemata esse in potestate, alia illis difficiliora &/ methodum tangentium Slusij ex meis Principijs statim consequi, quætiones {sic} sol & similiter quæstiones de maximis & minimis {e}t |re|solvi & quadraturas reddi fal|c|iliores & alia similia peragi, et Analysin meam as|d| surdas quantitates \aut ad Curvas suas vocant Mechanicas/ non hærere, eandem vero exemplis illustrassem sed fundamentum ejus \literis transpositis/ hanc sententiam involventibus [Data æquatione quotcun fluentes quantitates involvente fluxione invenire et vice versa] fundamentum ejus celassem{illeg}: Respondit D. Leibnitius in hæc verba. Clarissimi Slusij Methodum Tangentium [id est methodum per differentias Ordinatarum] nondum esse absolutam Newtono assentior Celeberrimo Newtono assentior &c et jam a multo tempore rem Tangentium longe generalius tractavi, scilicet per differentias Ordinatarum. Deinde methodum suam communicavit a mea vix abludentem præterquam in verborum et notarum formulis. Vtrius fundamentum continetur in hoc Lemmate.

Hoc fecerunt Gregorius in ejus \Prop 7/ Geometriæ universalis anno 1668 impressæ \Prop. 7/ et Barrovius in ejus Lect 10 anno 1669 impressa, id methodo consimili. Idem D. Leibnitius facere potuisset jam a multo tempore. Sed inversa tangentium Problemata differentiales & quadraturas reducere jam anno superiore minime noverat. Vide pag 65 l. 14, 15. Cum vero a Newtono didicerat Clarissimi Slusij methodum Tangentium nondum esse absolutam, sed Corollarium esse methodi generalis quæ extenderet se citra molestum ullum calculum ad abstrusiora problematum genera, etiam ad inversa tangentium problemata alia diffil|c|iliora, & quæ ad quantitates surdas & Curvas Mechanicas minime hæreret, & cujus ope Quadraturæ redderentur faciliores: cœpit is methodum Tangentium per differentias Ordinatarum ab alijs traditam jam longe generalius tractare quam antea, & novo nomine novis symbolis ut suam ornare. Quo effectum est

Id est, $\mathrm{y}=\frac{\mathrm{x}\stackrel{.}{\mathrm{x}}}{\sqrt{\mathrm{a}\mathrm{x}\prime -\mathrm{x}\mathrm{x}}}$ vel $\stackrel{.}{\mathrm{y}}=\frac{\mathrm{x}\stackrel{.}{\mathrm{x}}}{\sqrt{\mathrm{a}\mathrm{x}-\mathrm{x}\mathrm{x}}}$. Et nota quod Diffe\re/ntiæ rectius decerentur partes. Sunt enim partes summarum {illeg} non et \{&} non autem/ differentiæ, {illeg}|n|e aliaquam habent relationem ad summas nisi quatenus sunt earum partes.

id {illeg} & n{illeg} {illeg} id generaliter per \s/ quadraturas sine ejus methodo ge id in t semper per quadraturas solas abs methodo generaliore quando datur relatio inter latere trianguli {illeg} TBC.

Hæc Isagoge et Corollarium Propositionis ultimæ scripta sunt ubi liber prodijt: reliquus in MS reliqua ex MS antiquo impressa sant.

id nonnunquam per quadraturas nonnunquam geralius {sic} method{o}|i|s generaliore|i|bus.

alia difficiliora. Vide etiam

Simplicior est expressio $\stackrel{.}{\mathrm{y}}\sqrt{\mathrm{a}\mathrm{x}-\mathrm{x}\mathrm{x}}=\mathrm{x}\stackrel{.}{\mathrm{x}}$. maximas et minima ded|t|erminari quæstiones de maximis et minimis alias quasdem determinari, & Problem

Vide etiam pag 30 & pag 47 lin 4

Gregorius methodum serierum ex unit|c|a tantum serie Newtoniana|m| \methodo|u|m/ invenit & Newtonum tamen inventorē primum \libere/ agnovit: Leibnitius pluribus edoctus, fuit. se nihil ab alijs Vide pag 30 & pag 47 lin 4, 8. & 71, 72.

LM $\frac{2\mathrm{a}\mathrm{n}\mathrm{n}{\mathrm{o}}^{3}}{{\mathrm{e}}^{5}}\times \frac{2}{8\mathrm{n}\mathrm{o}}\times \frac{2{\mathrm{e}}^{3}}{\mathrm{n}\mathrm{n}\mathrm{o}\mathrm{o}}$ $\frac{4\mathrm{a}\mathrm{n}\mathrm{n}}{8\mathrm{e}{\mathrm{n}}^{3}}=\frac{\mathrm{a}}{2\mathrm{e}\mathrm{n}}$

$\frac{2\mathrm{a}\mathrm{n}\mathrm{n}{\mathrm{o}}^{3}}{{\mathrm{e}}^{5}}\times \frac{\mathrm{n}\mathrm{o}}{\mathrm{e}}$ ad $\frac{8{\mathrm{n}}^{4}{\mathrm{o}}^{4}}{4{\mathrm{e}}^{6}}$ a. n

Hic est tractatus de Analysi per æquationes numero terminorum infinitas supra impressus.

Hinc liquet methodum fluxionum non tantum Newtono ante annū 1669 Newtono involuisse sed etiam ad magnum perfectionis gradum eo tempore provectam fuisse.

|②| In the year 1671 I wrote a Tract concerning the method|s| of converging series & the method of fluxions jointly and|but| did not finish it, that part of it being wanting which related to the solution of the inverse meth Problems w^{ch} could not be reduced to quadratures, as I mentioned in my letter to M^{r} Oldenburg dated 24. Octob 1676 &|w|hich |was| published by D^{r} Wallis in the third Volume of his works. From this Tract \I extracted/ in the year 1676 I extracted the following book of Quadratures; & therein I copied from the former Tract without any alteration the Tables set down in \the/ tenth Proposition for squaring the simpler \some/ Curves, or red{ucin} \&/ comparing them with the Conic Sections. In my Letter of \to M^{r} Collens to M^{r} Collins/ dated 8 Novem. 1676 \found by M^{r}Iones a mongst his Papers & published by his|m| & dated 8 Novem. 1676/ I had relation to this Book in saying Nulla extat Curva — — — adeo generaliter. This mentioned in set down in \relates to/ Coroll. 2. Prop. 10 of this Book.

And in the second Lemma of the 2^{d} Book of Principles I demonstrated the Elements of the method of Fluxions.

|③| ∥ This Book was in the hands of M^{r} Ralpson & D^{r} Halley in the year 1691 as the former has bef{illeg}t attested in print before his death & the latter still attests. And \in the year 1692/ at the request of D^{r} Wallis I sent hi to him the first Proposition of this Book with examples in first & second fluxions & he published \before the end of the year he printed the/ the same in the second Volume of {his} Works w^{ch} \Volume/ came abroad in April 1693. And this is \was/ the first time that any Rule was published for finding second third fourth & other differences. But the Book continued in MS till the year 1704. And then in publishing it I wrote a Preface in w^{ch} I affirmed that I found the method of fluxions gradually in the years 1665 & 1666. For I thought then that I might safely write this because D^{r} Wallis in the Preface to the thir first Volume of his works published \in spring./

④ In the abovementioned Letter of {the} 24 Octob 1676 at the request of M^{r} Leibnitz I described how before the Plague w^{ch} raged in London in the years 1695 & 1696 by considering how to interpole the series of D^{r} Wallis I found the method of converging series \{illeg} together w^{th}/ the Rule for {illeg} converting the powers & dignities of Binomials into such series, & that D^{r} Barrow about the time \that/ the Logarithmotechnia of M^{r} Mercator came abroad sent to M^{r} Collins a compendium of thi|e|se series. This compendium was {illeg}|A| copy of this Compendium was in the hand\writing/ of M^{r} Collins was found by M^{r} Iones in the among the papers of M^{r} Collins & published after it had been collated {illeg} w^{th} y^{e} original w^{ch} M^{r} Iones borrowed of me for that end. The title thereof was *Analysis per series numero terminorum infinitas*. And in this Tract there are instances of calculating by the method of fluxions, \this method being interwoven with that of Series./ & it is therin affirmed that by this method of the method of series extends to all Problems & that by the help thereof ejus beneficio curvarum area & longitudinis &c (*id modo fiat) exacte & Geometrice* determinantur |And therefore I then understood the method of Fluxions so far as it is conteined in the first five or six Propositions of the book of Quadratures. And by the Testimony of D^{r} Barrow & M^{r} Collins I understood it thus far some years before that time For M^{r} Collins in a Letter to M^{r} Strode| And how this is done is explained in my said Letter of 24 Octob 1676. & illustrated with examples. And this is not to be done without the method of fluxions so far as it is conteined in y^{e} first five or six Propositions of the book of Quadratures. And therefore the method of fluxions so far as it is conteined in those Propositions was known to me when I wrote the said Letter of 24 Octob 1676 & before that even when I wrote the Analysis per Æquationes numero terminorū infinitas w^{ch} was in the yeare 1669 \it was known to me/ & {choun} |by the Testimony of D^{r} Barrow & M^{r} Collins it was known to me| some years before that. For M^{r} Collins in a letter to M^{r} Strode dated 26 Iuly 1672 & published by Order of the R. S. in the Commercium Epistolicum wrote thus. Mense Septembri 1668, Mercator Logarithmotechniam suam edidid|t| suam, quæ specimen hujus methodi (i.e. serierum infinitarum) in unica tantum figura, nempe {illeg}|Q|uadraturam Hyperbolæ continet. Haud multo post quam in publicum prodierat Liber, exemplar ejus Cl. Wallisio Oxonium misi qui suam de eo judicium in Actis Philosophicis statim fecit; alium Barro{v}i{o} Cantabrigiam qui quasdam Newtoni chartas (qui jam extemplo remisit: e quibus et ALIIS, quæ OLIM ab Auctore cum Barrovio communicata fuerant, patet illam Methodum a dicto Newtono aliquod|t| annis antea \ALIQVOT ANNIS ANTEA/ excogitatam & modo universali applicatam fuisse: ita ut ejus ope in quavis Figura Curvilinea proposita quæ una vel pluribus {illeg}|p|roprietatibus definitur Quadratura vel Area dictæ Figuræ, accurata si possibile sit \ACCVRATA SI POSSIBILE SIT/, sin minus infinitè vero propinqua, {Et} Evolutio vel longitudo lineæ curvæ, Centrum gravitatis Figuræ; solida ejus rotatione genita, & eorum superficies; sine ulla radicum extractione obtineri queant. {illeg}o So then by the testimony of D^{r} Barrow founded upon papers communicated by me to D^{r} Barrow \him/ before the Logarithmotechnia came abroad \from time to time/, I had brought the method to the perfection here described, some years before the Logarithmotechnia came abroad. In another Letter, M^{r} Collins saith, above two years before. So then \By these testimonies it appears therefore that/ I had the method therefore in the perfection here described before September 1666, & by consequence then understood the method of fluxions \that is/ so far as it is described in the first five or six Propositions of this Book of Quadratures to square curves exactly when it may be done, or at least by continual approximation, & by consequence that I then understood the method of fluxions so far \at least/ as it is conteined in the first five or six Propositions of the book of Quadratures. Which \together w^{th} the testimony of D^{r} Wallis/ I recon sufficient to justify me in {illeg} what I said in the Introduction to this Book.

|①| In December 1672 upon notice from M^{r} Collins that M^{r} Iames Gregory had a method of drawing Tang improved the methods of tangents of D^{r} Barrow so as to draw tangents without calcuclation & that M^{r} Slusius had such another Method w^{ch} he intended to communicate to M^{r} Oldenburg, I wrote the following Letter to M^{r} Oldenburg \Collins/ dated 10 Decem 1672. Ex animo gaudeo D. Barrovij — — — — reducendo eas ad series infinitas. These last words {illeg} you have \here/ a general description of the |great| extent of the method of fluxions, with an example of it in drawing of Tangents. The wor{illeg} the {sic} method here sp{illeg}t described was interwoven with the \another/ method of resolving converging Series in w^{ch} I reduce equations to converging series.

|②| For in the year 1671 I wrote a Tract concerning the method of

NB. Hunc Librum {illeg} de Quadraturis \MS/ Halleius & Ralphsonus Anno 1691 manibus suis tractarunt uti posterior publice testatus est et posterior adhuc testatur. Propositionem primam cum exemplis in differentijs primi fluxionibus primis & secundis \inveniendis/ D. Wallisius anno 1672 a me accepit & eadem in secundo ejus Volumine operum ejus impressa fu{illeg} anno proximo lucem vidit, & prima fuit hujus generis Regula quæ lucem vidit; es\t/qus \Regula illa/ verissima et maxime {illeg} universalis. {illeg} In Lemmate secudo {sic} libri |scdi| Principiarum mathematicorum Philosophiæ, Schol Regulam illam synthetice demonstra & in Scholio quod Lemmati subjunxi Propositionem dixi me Propositionem eandem \totidem syllabis/ posui dixi me eandem totidem syllabis prius in Epistola ad 24 Octob. 1676 ad Oldenburgum data posuisse. Hanc Epistolam D. Wallisius in tertio operum suorum edidit volumine edidit, Et ibi habetur hæc Propositio totidem literis tanquam fundamentum methodi cujusdam ducendi tangentes, determinandi maxima & minima, quadrandi figuras & similia peragendi, In qua metho & in hac {m} et hic \in qua/ non hæreti|{ur}| ad Æquationes Radicalibus unam vel utram Indefinitam Quantitatem involventibus utcun affectas. Et eodem fundamento dixi me ad Theoremata quædam generaliora quadrandi curvas pervenisse. Et Theorema primum \(sc. pro/ ib\ub/i {sic} posui et exemplis illustravi addidi me pro Trinomijs etiam et alijs quibusdam Regulas \quasdam/ concinasse. Est autem \hoc/ Theorema primum idem cum Propositione quinta libri de Quadraturis, Et Theo\re/ma primum pro Trinomijs idem cum Propositione sexta ejusdem libri. Et hæ duæ Propositiones pendent a Propositionibus quatuor primis ejusdem libri. Et propterea methodus fluxionum quatenus continetur in Propositionibus sex primis Libri de Quadraturis mihi innotuit anno 1676. Sed et Propositio septima et octava, ejusdem {illeg} \sunt/ generis. Nona autem ac decima requirentur ad solutionem Problematis quod posui in Epistola ad D. Collinium Novem. 8 1676 data \& ad Ioneso edita/ ve\r/bis sic Latine redditis Nulla extat Figura curvilinea cujus Æquatio — — — haud tamen adeo generaliter. ☉ ☉ Et in eadem Epistola Ordinatæ Cuvilinearum {sic} quæ cum Ellipsi et Hyperbola comparari possunt eædem ponuntur | habentur ac in Tabula secunda Prop. X libri de Quadraturis. Ideo Tabula illa (et propterea methodus fluxionum) mihi innotuit cujus ope \Tabula/ constructa fuit) mihi ante {illeg} innotuit anno 1676: immò et annis aliquot antea. Nam dixi in Epistola illa me Theoremata pro Comparatione Curvarum cum Conicis Sectionibus in Catalogum dudum retulisse.

Anno igitur 1676 Methodum fluxionum intellexeram quatenus in \hoc/ Libro |hocce| de Quadrature|is| Curvarum exponitur. Sed et anno 1671 eandem intellexerā Nam in Epistola mea \prædicta/ ad D. Oldenburgum 1|2|4 Iunij \Oct. 1676/ scriptam, dixi me in Tractatu quem tunc ante quinquennium scripseram de Seriebus conscripseram me \etia/ alia præter methodum serierum haud pauca congessisse inter quæ erat methodus Tangentium ducendi Tangentes quam solertissimus Slusius ante annos duos tresve tecum communicavit; de qua tu (suggerente Collinsio) rescripsisti eandem mihi etiam innotuissi. Dein Deinde Diversa ratione in eam incidimus. Et subinde dico me in hanc methodum tangentium incidisse per methodum quandam |quæ ad quæ ad quæstiones de maximis et minimis, de que quadraturis & alijs se extendit & quantitates surdas non moratur, &| cujus fundamentum continetur in hac sententia, Data æquatione fluentes quotcun quantitates involvente, Fluxiones invenire; et vice versa. De his omnibus \hic/ loquor tant|q|uam mihi cognitis ante quinquennius|m| \cognitis/. Sed et in Epistola {illeg} [ad D. Collinium 10 Decem. 1672 scripta data hanc methodum sic descripsi. Ex animo gaudeo — — — ne grave ducas. Methodus fluxionum hic perspicue describitur. Et præterea, in Epis] eadem

— nec prius cum amicis communicare cœpit quam ab Anglia discesserat, {illeg} |& postqꝫ|u|am ab Olden|burgo acceperat, remisit \{illeg} acce{illeg}/ ut Anglis ignotam.

Methodum exhibendi Arcum cujus sinus datus Leibnitius ab Oldenburgo postea quæsivit, Maij 12 1612 {sic}, ideo {illeg} nondum intellexit

\p. 38./ *a* Collinius jam ante quadrennium series Newtonianas ante triennium Gregorianas cum amicis communicare cœpit. Leibnitius in Anglia diversabatur A|a|nno superiore & hujusmodi series nondum communicaverat, nec prius cum amicis communicare cœpit quam ab Anglia discesserat, Et \subinde/ cum series aliquas ab Oldenburgo acceperat, quas a suis diversas esse \tum/ fatebatur, \tandem/ unam {illeg}bum \acceptarum/ remisit \Oldenburgo/ ut Anglis ignotam. Nullas autem communicavit nisi quas ab Oldenburgo acceperat.

*b* Methodum exhibendi arcum cujus sinus datur Leibnitius ab Oldenburgo postea quæsivit, Maij 12 1676, ideo nondum intellexit \habuit./ Vnde nec methodum jam habuit perveniendi ad seriem alteram \numerorum rationalium/ quam se primum invenisse hic jactabat \quæ|ā| uti eandem esse dicit/ Sola methodus transmut{illeg}|a|toria quam postea communicavit non dat arcum ex Sinu, ideo non est methodus de qua hic agitur, sed post inventa fui fuisse \videtur/. Series \forsan duas/ habere potuit sed methodum qua inventæ sunt nondum habuit.

p. 41. Hanc seriem D. Collins initio anni 1671 a Gregorio acceperat ut supra; D. Leibnitius cum amicis in Gallia opusculum de eadem hoc anno communicare cœpit, celata hac Epistola.

p. 42. |*a*| His verbis Leibnitius series acceptas a suis diversas esse testatur \confessus est/ /testatur\ et earum tamen unam pro sua \unam tanen {sic} acceptarum/ quasi Anglis ignotam præ s{illeg}a anno proximo \pro sua/ remisit. Miror quænam fuerint \ipsius/ series quibuscum non potuit series ab Oldenburgo missas \jam/ comparare non potuit, Si aliquas ab communicatas diversas habuit certe b eas nunquam communicavit mirum est quod eas nunquam communicavit in lucem produ{c}it protulit \et quare eas ab acceptas/ lucem nunquam viderunt. siquis habuit ab acceptis diversas

p. 42. *b* Hoc nunquam fecit Leibnitius . . . . . . vindicandi, cum incidisset tandem in transmutatione figurarum cujus beneficio series illa inveniri potuit \prætendens hanc/ quas longe diversam ab ijs quas per Mohrum acceperat, Anno & quasi alios multos [Anne Leibnitius ignorabat se hanc seriem a] quasi \a nemine acceptam quia/ longe diversam ab ijs quas per Mohr Georgium Mohr acceperat & ab alio nemine acceptam. Certe series hæc Leibnitio ipso teste non est Leibnitiana quia non diversa ab ijs quas ab Oldenburgo acceperat. Certe Leibnitiana non est quia seriem meam In Epistola 26 Octob. 1674 {illeg} data dicebat \seriem {illeg} seriem suam/ suam inventam esse eadem methodo \inventam esse/ cum serie \pro {illeg}endo arcu ex \dato/ sinu (Hæc series/ prima...earum quas per Mohrum \mox/ accepitat jam methodum hanc ignorat \{Ex} Quomodo invenietur jam ignorat/ \methodum inveniendi/ sibi mitti \suar/ postulat et \ut eandem obtineret/ sua {illeg} ab his longe diversa circa hanc rem meditata vicissim promittit seriem scilicet quam ab Oldenburgo acceperat|,| Vide ejus Epistolam Maij 12, 1676 non eadem methodo (ne sua non esset) sed alia {illeg} nova aliqua{illeg} inventam demonstratam.] Series hasce mirabatur ut valde ingeniosas sed demonstrare nondum potuit Postulat igitur demonstrationem sibi mitti, et ut eandem obtineat, sua ab his longe diversa circa hanc rem meditata promittit.

*** ad verba [Via quadam sic satis singulari] notetur. Leibnitius in Epistola 26 Octob 1674 data dicebat series suas \una et/ eadem methodo inventas esse. \& hanc methodum jam vocat viam quandam satis singularum./ Series pro arcu ex dato sinu prima est {illeg}arum quas per Mohrum mox accepit. Series hisce {illeg} \per Mohrum/ acceptas ut valde ingeniosas laudabat sed demonstrre nondum pot{uit}|era|t. Postulabat igitur demonstrationem sibi mitti, et ut eandem obtine\re/at sua ab his longe diversa circa hanc rem meditata promittebat.] Sed {n}ecrium suo{illeg} series suas ab Oldenburgianis diversas, nec \et/ viam singularem qua ejusmodi series prodirent, nunquam misit communicavit.

$\mathrm{DG}=\mathrm{e}$. $\begin{array}{r}\mathrm{e}\mathrm{e}-2\mathrm{a}\mathrm{o}-\mathrm{o}\mathrm{o}(\mathrm{e}-\frac{\mathrm{a}\mathrm{o}}{\mathrm{e}}-\frac{\mathrm{n}\mathrm{n}\mathrm{o}\mathrm{o}}{2{\mathrm{e}}^{3}}\\ \phantom{\mathrm{ee}}-2\mathrm{a}\mathrm{o}-\mathrm{o}\mathrm{o}\\ \phantom{\mathrm{ee}}-\frac{2\mathrm{a}\mathrm{o}+\frac{\mathrm{a}\mathrm{a}\mathrm{o}\mathrm{o}}{\mathrm{e}\mathrm{e}}}{\mathrm{o}-\frac{\mathrm{n}\mathrm{n}\mathrm{o}\mathrm{o}}{\mathrm{e}\mathrm{e}}}\end{array}$ {☾} $\frac{\frac{\mathrm{a}\mathrm{a}\mathrm{o}\mathrm{o}}{\mathrm{e}\mathrm{e}}+\mathrm{o}\mathrm{o}}{2\mathrm{e}-\frac{2\mathrm{a}\mathrm{o}}{\mathrm{e}}}=\frac{\mathrm{n}\mathrm{n}\mathrm{o}\mathrm{o}}{2{\mathrm{e}}^{3}}$. $22)\mathrm{o}\mathrm{o}+\frac{\mathrm{a}\mathrm{a}\mathrm{o}\mathrm{o}}{\mathrm{e}\mathrm{e}}=\mathrm{FG}=\frac{\mathrm{n}\mathrm{n}\mathrm{o}\mathrm{o}}{2{\mathrm{e}}^{3}}$

E{illeg}

Quasi

Seriem Gregorianam ab Oldenburgo acceptam Mense Maio a suis distinuerat. Opusculum de hac serie compositum hoc anno cum amicis in Gallia communicare cœpit celata Oldenburgi {illeg} communicatione. Seriem eandem \quasi/ Anglis ignotam jam promittit se missur{illeg} cum Oldenburgo communicaturum, & proximo anno communicavit. Et prætendit se cum amicis in Gallia ante biennium \id est anno 1672/ communicasse. Prætendit|eb||at| etiam ante menses quatuordecim

Series quas mense Aprili O suas in

Series mense Aprili ab Oldenburgo missas, Leibnitius a suis diversas esse dist{illeg}it \diversas esse agnovit/ Inter {illeg} Missarum una erat series Gregorij pro arcu \circuli/ ex tangente altera alia erat series Newtoni pro tangente ex arcu ex sinu. Has pro suis Leibnitius \{illeg}/ {illeg}e agnovit \Opusculū/ De serie priore Leibnitius ho compositum Leibnitius hoc anno cum amicis in Gallia communicavit|re| cœpit, {illeg} Methodus inveniendi seriem posteriorem Leibnitius \anno proximo/ postulavit ab Oldenburgo \Celata Oldenburgi epistola./ Et jam promittit se hanc seriem \quasi Anglis ignotam/ cum Oldenburgo communicaturum, quasi et eandem \simili/ fide dicit se eandem jam plusquam biennio abhinc cum Geometris in Gallia communicasse. Seriem alteram easdem laudabat ut \novas &/ valde ingeniosit|a|s & sibi tum primum cognita{illeg}|s|{illeg} postuland{illeg}|bit|\{a}t/ \et {illeg}|in|de occasionem accipuit exempl{illeg}/ /& ideo postulabat.\ earum demonstrationem |{illeg}| ad se mitti, quasi series nullas ab Oldenburgo prius accepisset. Et hoc parto . . . . . acceperat.

*a* Quasi Leibnitius nesciret hanc s{illeg} /hanc Quadraturam {illeg}\ Anglis \Gregorianam esse &c/ innotuisse|.| vel ab Ol

*b* Anno 1673 Leibnitius in Anglia commorans de hujusmodi serie{m}|bus| aliquid audire potuit. Annū|o| proximi|o| sscripsit \jactabat/ se hujusmodi seriem unam at {illeg}|a|lteram habere \sed nullam cum Anglis communicavit/, {illeg} anno tertio mense Aprili vel Mai{illeg}|o| accepit \ab Oldenburgo/ ejusmodi series \aliquot/ quas a suis diversa, esse agnovit. Eodem anno opusculum de serierum acceptarum una cum amicis in Gallia communicare cœpit, celata Oldenburgi e{illeg}|p|istola. {illeg} Et jam promittit se hanc seriem quasi Anglis ignotam Oldenburgo remissurum.

*b* Quasi ante Annum easdem non accepisset ab Oldenburgo.

*c* Opusculum prædictum de Quadratura arithmetica D. Leibnitius polire perrexit.

When M^{r} Leibnitz published the series o Gregory at his ow & such other things as he had notice of from England \for preventing disputes & doing justice to every body/: he should have acknowledged who |the| correspondence he had kept w^{th} M^{r} Oldenburg & what \he/ had received from England or seen in the hands of M^{r} Collins relating to the things w^{ch} he published.

*a* He omits the greatest part of my method of series & its connexion with the method of Fluxions.

*b* He gives M^{r} Fatioo the lye who had seen my ancient papers, & pretends that when I represented that I wro in wh{illeg} in the year 1671 I wrote a tract on the method of series & another. y^{t} method founded on this Proposition *Data æquatione fluentes quantitates involvente \invenire fluxiones/* & I did not dream of the calculus of fluents & fluxions

*c* He affirms falsly that I use prict letters for the differential characters For fluxions & differences are quantities of a different kind.

*d* He places the invention of the methods in the invention of the \the/ symbols as as if used in them as if new names & new symbols ought to pass for new inventions.

*e* He affirms falsly that in my Principles I had frequent occasion to use my calcus {sic} of fluxions for a after I had invented the Propositions by Analysis I demonstrated them by composition.

*f* He affirms falsly that in this book there is no footstep of this Calculus for y^{e} 2^{d} Lemma of the second book conteins the elements of this calculus And the Scholium upon this Lemma asserts this Calculus to my self.

*g* Prickt letters appeared in the second volume of the Works of D^{r} Wallis w^{ch} was printed in the year 16693 {sic} & mentions that he had the what he there printed of this matter he had from me in the year 1692. My Tract of Quadratures was handed about among in London in 1691. I made much use of it in writing my book of Principles & composed it many years before but never placed the method in the use of prickt letters: for the Introduction to this book was writ in y^{e} year 1704 without p{illeg} & conteins a description of y^{e} method without y^{e} use of prickt letters.

*h* Incrementum constans ipsius x nunquam notabam per x punctatam notatu uno puncto|.| Sed p Illud noto per {illeg} $\stackrel{.}{\mathrm{x}}\mathrm{o}$; & ubi $\stackrel{.}{\mathrm{x}}$ est unitas, per o

*i* Notatio mea per o sub falso dis|c|itur calculi differentialis commoda destruere. Notatio est elegantior et utilior.

*k* Regulam circa gradus utiliores falsam non dedi Mathematicus ille eminens erravit. Annon hic est Bernoullius.

*l* Apparet me ab Autore nostro falso accusatum Leibnitius|m| \Anno 1689/ ubi scripsit Tentamen De motuum Cœlestium causis, rectam methodum differentiandi differentiali non intellexisse. Rectam methodum Wallisius noster e literis nostris anno 16{illeg}|9|2 ad eum missis descripsit \anno/ in secundo Operum volumine {illeg} anno 1693 in lucem emisso, Eandem \recte/ descripsi in Propositione prima libri de Quadraturis; et hic liber manibus amicorum Londini terebatur anno 1691, & subsidio mihi fuit \annis 168{illeg}9, 1{84}|68|4 1683, 1684, 1685 & 1686/ ubi scribebam Principia Philosophiæ et multa ante conscriptus fuit. Et recte capiendo fluxiones fluxionum et momenta momentorum inveni Demonstrationem Theorematis Kepleriani anno 1677 & Curvaturam Curvarum ante annum 1673 ut ex epistola mea 10 Decem 1672 ad Collinium data manifestum est.

*a* literas punctatas uno duobus tribus &c punctis superpositis non adhibit pro dx, d{illeg}|{d}|x, d^{3}x; dy, ddy &c nunquam adhibuit. Hæ sunt quantitates diversi generis. Ex verbis ejus *nunc adhibet* incidit suspicio quod is fuit author Epitomis libri \De/ Quadratura Curvarum in Actis Erudit. anno 1705 ips impr{æ}|e|ssæ unde nata est hæc controversia.

*b* In Principijs Naturæ Mathematicis, calculo fluxionum utendi nulla erat occasio. Invent{illeg}|{a}| sunt Propositiones per hanc Analysin, demonstratæ vero per synthesin.

*c*

1 Methodum Serierum a me inventam minuit.

2 Somnia narrat. Ex Epistola 24. Octob. 1676 data certissimum est me fluxionales æquationes per ea tempora habuisse.

3 fallitur. Pro {illeg} symbolis differentij|a|rum non utor symbolis fluxionum. Hæ sunt quantitates diversi generis.

4 Fallitur. Methodi non consistunt in formulis verborum et symbolorum. In tractatu de quadraturis olim scripto literis punctatis utor, in ejus Præfatione nuper scripta literis punctatis utor et tamen methodum fluxionum li|a|bsque literis punctatis describo.

5 Falitur. {illeg} In libro Principiorum nullam habui occasionem utendi calculo fluxionum Propositiones inveni per Analysin demostravi {sic} per synthesin. Analysis al{illeg} tamen ita \per Demonstrationes syntheticas ita/ elucet ut Marchio Hospitalius scripserit {illeg} librum pene totum ex hac Analysi constare et ipse Leibnitius agnoverit me omnium primum specimine publice dato ostendisse quod partem hujus methodi nobilissimam et latissime patentem haberem: eam utique qua solidum minimæ resistentiæ, Curva celerrimi descensus, Catenaria et Vellaria inventa fuerunt.

6 Fallitur. Nam Lemma secundum libri secundi & Scholium ejus continet elementa hujus calculi.

7 Fallitur. Nam Volumen tertium seri editum fuit anno 1699. Extant literæ punctatæ in volumine secundo quod prodijt \impressum fuit/ anno 1693. Liber autem de Quadraturis in manibus amicorum terebatur anno 1691 et deinceps & in meis anno dum Principia Philosophiæ scripsi|ber||em| & multo ante sc{ri}ptus \compositus/ fuit|.| ut ex Epist

8. Fallitur. Incrementum constans ipsius x nunquam notabam per $\stackrel{.}{\mathrm{x}}$ punctatum uno puncto. Illud noto per $\stackrel{.}{\mathrm{x}}\mathrm{o}$: et u{illeg}|b|i $\stackrel{.}{\mathrm{x}}$ est unitas, per o.

9. Fallitur. Notatio mea per o non destruit commodo calculi differentialis se e contra, commodior est.

10 Fallitur. Regulam circa gradus ulteriores veram dedi.

11 Annon Ber Mathematicus ille \eminens/ Bernoullius est? Anne author Epistola seipsum hic citavit.

12 Accusationem probare deb{it}|ui|t. Non probata pro calumnia haberi debet.

13 Rectam methodum Wallisius noster e literis nostris anno 1692 ad eum \missis/ in secundo Operum volumine anno 1693 impresso descripsit. Eandem in libro de Qua Proportione prima Libri de Qua olim compositis de Quadraturis habetur.

13 Methodum veram Leibnitius ipse non habuit \Anno 1689/ ubi scripsit Tentamen suum de motuu motuum cœlestium causis. Post annum 1690 methodus illa {illeg} cœpit \a Leibnitianis/ intelligi. Anno 1696 Dn. Marchio Hospitalius eandem edidit. Methodum veram Wallisius a me accepit anno 1692 & \anno proximo/ in secundo Operum Volumine an edidit. Eadem describitur in Propositione prima libri de Quadraturis olim scripti. A fluxionibus secundis {illeg} A momentis secundis multæ Propositiones in Libris Principiorum pendent. & Hujus generis est {illeg} determinatio Problematis Kepleri Propositionis Keplerianæ quam anno 1677 inveni. Dixi etiam in Epistola 10 Decem 1672 me per \ad Collin{um} data/ methodum de qua ibi locutus sum, id est per methodum fluxionem curva \ad/ |curva|turas c|C|urvarum inveni{s} se extendere{,}. Hæ autem pendent a fluxionibus secundij|s|. {illeg} etiam \Regulam utiq {sic}/ Fluxion{is}|ū| uti primas|r|ū intelligenti Regula secundarum latere noti potest cum sit eadem cum Regula primarum ut Propositionem primam Libri de Quaduris {sic} consulenti statim patebit

complaint that Newton in his Principles & book of Quadratures had used fluxions for the differences of M^{r} Leibnitz \the inventor of {ean} {illeg}{thor}/ as Honoratus Faber substituted progressions of Motion {as} for the method of Cavellerius.

There have hitherto been \printed/ only a few copies of the Commercium Epipstolicum relating to the infinitesimal method

The Commercium Epistolicum relating to the Infinitesimal method

|1.| It was hoped that a few copies of the \following/ Commercium Epistolium {sic} printed off & sent {sen} printed off & sent to Mathematicians who were able to judge of these things, But might {h}{illeg}|it| have silenced the dispute about \about/ the dispute about the Infinitesimal Method \the {sic} complaint that M^{r} Newtons book of Quadratures was a peice of Plagiary/: but M^{r} Leibnitz & his friends have {illeg}d{illeg} declined answering it, & endeavoured to run the dispute into a wrangle & squabble about other matters \\such {as} {illeg}/ occult qualities, miracles, gravity, \attraction/ sensoriums, the perfection of the world, a vacuum atoms the solving of problems & the like: all w^{ch} are nothing to the purpose/ which are nothing to the purpose. & some of them still continue to squabble And tho M^{r} Leibnitz is dead, yet this squabbling humour is still continued by some of his friends. And therefore it has been thought fit to publish this Commercium Epistolicum together w^{th} the given \Account given/ thereof in the Philosophical Transactions for Ianuary 171$\frac{4}{5}$ & so leave it to posterity to judge of this matter by the ancient Records.

|3| Against the credit of these Records it has been objected that the Committee of the R. Society published every thing that made against M^{r} Leibnitz & omitted every thing that made against M^{r} Newton. In my s{illeg} But This|e| objection was only a pretence {illeg} made by M^{r} Leibnitz but had nothing more in it then clamour. For in his a Postscript of his first Letter to Abbe Conti, \for proving this/ he wrote that in his second voyage into England *M ^{r} Collins shewed him part of his correspondence & he observed \there/ that M^{r} Newton avowed \acknowledged/ his ignorance in many things & said (among other things) that he had found nothing about the dimension of the celebrated curves|il|inears besides the dimension of the Cissoid But* \saith he/

*they have supprest all this*. And M

^{r}Newton in his Letter of \to Abbe Conti dated/ 26 Feb. 171$\frac{5}{6}$ replied that this passage was \not suppressed but/ published in the Commercium pag 74 lin. 10 & 11. M

^{r}Leibnitz in his Answer written to Abbé Conti Apr. 9 1716 acknowledged that he had been mistaken, but saith he, I'le cite another instance. M

^{r}Newton affirmed \acknowledged owned/ in one of his Letters to M

^{r}Collins that he could not find the content of second sections (or second segments) of Spheroids or such like bodies: but they have not inserted this passage or Letter in the Commercium Epistolicum \& that the Commmitte had omitt{illeg}ed that/. And M

^{r}Newton in his Remarks upon the Letter of M

^{r}Leibnitz replyed that whether \he/ (M

^{r}Newton) could solve that Problem or not was nothing to the purpose point in question, & that {illeg} if the Committee had omitted it they would have done right, it being nothing to the point in question, but on the contrary, M

^{r}Collins in a Letter to M

^{r}Bertet Iames Gregory y

^{e}24

^{th}of December 1670, & in an other to M

^{r}Bertet the 21

^{th}of Feb. 1671, both printed in the Commercium Epistolicum pag. 25|4|, 26, wrote that his metho his (M

^{r}Newtons) method extended to second segments of round solids. And that M

^{r}Oldenburg wrote the same thing to M

^{r}Leibnitz himself the 8

^{th}of of {sic} December 1674. See the Commercium Epistolicum pag 39. Another thing objected against{e} the credit of those Records ha{illeg} was this \So that the accusation [amo{ounts} to nothing more th{a}n a Cavil] was groundless & amounts to nothing more then a piece of railery./

In a flying paper dated 29 Iuly 1713 M^{r} Leibnitz called in question the letter of M^{r} Oldenburg sent to him with 8 or 9 series & the Answer to it in the han \date/ by w^{ch} he [M^{r} Leibnitz acknowledged the receipt. of that Letter.

|2| The Letters of M^{r} Leibnitz to M^{r} Oldenburg are w^{th} M^{r}|a|re dated 3 Feb. 20 Feb. 30. Mar. 26 Apr. 24 May. 8 Iune, 1673; 15 Iuly, 26 Octob. 1674; 30 March, 15 Apr. 20 May, 12 Iul{illeg}|y|, 28 Decemb. 1675; & 12 May, 27 {O}Aug. \18 Novem/ 1676 & \21 Iune &/ 12 Iuly 1677. And these are \were/ /are\ all \except the 3^{d} & the the {sic} f{illeg}er|ive| last entered {sic}/ entered in in {sic} the Letters books of the R. S. N^{o} 6 pag 35, 34, * 101, 115, 137; & N^{o} 7 pag. 93, 110, 213, 235, 149, 189, And the Originals of all these letters {illeg} except that of 27 Aug. 1676 w^{ch} was published by D^{r} Wallis, \& that of 18 Novem following/ are stilll extant in the hand writing of M^{r} Leibnitz. several of these letters were also published by D^{r} Wallis {as} The Letters of 15 Iuly & 26 Octob. 1674; 12 Iuly & {illeg} 28 Decem. 1675 & 21 Iune \& 12 Iuly/ 1677 were also published by {M}|D|^{r} Wallis. And \all/ these Letters together w^{th} the Letters of M^{r} Oldenburg to M^{r} Leibnitz w^{ch} are copies of w^{ch} are still preserved make up {illeg}|th|e continued correspondence between M^{r} Leibnitz & M^{r} Oldenburge from Feb 3 167$\frac{2}{3}$ to the death of M^{r} Oldenburg w^{ch} in excepting that the Letter by w^{ch} M^{r} Leibnitz desired the conception out \a collection/ of Gregories Letters & that of M^{r} Oldenburg to M^{r} Leibnitz w^{ch} accompanied those Excerpta by w^{ch} he sent \to M^{r} Leibnitz/ that collection together w^{th} Newtons Letter of 13 Iune 1676 & that of M^{r} Collins dated 14 Iune 1676. The Letters of M^{r} Leibnitz ran upon numbers till Iune 8 1673. Then \M^{r} Leibnitz began to study the higher Geom &/ the correspondence ceased {both} for a time. And the next year in Iuly 15 M^{r} Leibnitz renewed the correspondence beginning his Letter of Iuly 15^{th} with these words *Diu est quod nullas a me habuisti litteras*. And from thence forward the correspondence continues without intermission to the death of M^{r} Oldenburg except that the aforesaid two Letters are wanting [the latter of w{illeg}|h|ich is extant but was not published?|]| having nothing in it material to the controversy.] So {illeg} {sic} then the Commercium Epistolicum of M^{r} Oldenburg w^{th} M^{r} Leibnitz \the English & M^{r} Collins M^{r} Oldenburg M^{r} Collins & {illeg}l{illeg} M^{r} Newton/ is here printed entire except those two Letters. And For M^{r} Collins \& M^{r} Newton/ had no correspondence with him except \only/ by means of M^{r} Oldenburg Leibnitz .

|4| When the Commercium Epistolicum came abroad \w^{ch} was in the end of the year 1712./ M^{r} Leibnitz to avoid answering it pretended \d{illeg}ing all/ the two next years that he had not seen it, but had desired an Eminent Mathematician to examin it & nor was at leasure to examin it, & therefore but had referred it the judgment of an able & impartial Mathematician. And the judgment of this Mathematician dated 7 Iune 1713 was inserted into a scurrillous paper dated 29 Iuly 1713 & de|i|spersed over all Europe, without telling the name of the Mathematician or \that/ of the Printer or City where it was printed. [But in the end of the year 1715 & beginng {sic} of the year following M^{r} Leibnitz to \give/ the per judgment of the Mathematician more credit began to tell the world that it was M^{r} Leibnitz Iohn Bernoulli & to make this probable he {illeg} translated this|e| Letter of 13 Iune 1713 into French & sent the transla in the translation omitted the sentence quemadmodum ab eminente quodam Mathematico dudum notatum est, by w^{ch} words the {illeg} author of the Letter cited M^{r} Iohn Bernoulli as a person different from himself.] This And this translati [And this translation he sen inserted into his letter of to Madam Pelniz & Madam Kilmanseg dated Apr 18 Apr. 1716 & caused it it to be printed in Holland Decem. 1715]

|6| In this flying paper it is said that prickt letters did not appeared first in the third Volume of D^{r} Wallis his works. {H} And that M^{r} Newton did not understand how to find second differences till it was commonly understood by others. But this is a mistake misrepresentation. The third Volum e came abroad in 1699. But prickt letters appeared in the second Volume \w^{ch} came abroad/ in Spring 169|8|3. In this Volume D^{r} Wallis published the first Proposition of |M^{r}| Newtons book of Quadratures illustrated with examples in prickt letter{s}|f|irst & second differences. And this is the first line that any Rule came abroad for finding 2^{d}, 3^{d}, {illeg} \&/ fourth differences & {illeg} & {at} & fift differences & so on. He \D^{r} Wallis/ published also at the same time the fift Proposition of the Book of Quadratures & reprented {sic} that he had received these things from M^{r} Newton by Letters dated Aug 27 & Sept 17 1702. This Proposition depends upon the four preceding Propositions & therefore the first five Propositions of that Book were in being in the year 16 1792 {sic}. This Book \in MS/ was in the hands of M|D|^{r} Halley & M^{r} Ralpson in the year 1691 as M^{r} Ralpson has publickly attested & D^{r} Halley still attests. And this \was before the differential method came abroad began to make a noise abroad &/ may suffice to clear this Book from the accusation \of plagiary/ w^{ch} gave a beginning to these disputes. In the XIV^{th} Proposition of the second Book of Principles you have an instance of solving Problemes by second Differences. And this \was the {illeg}fore {illeg} the differential {illeg}/ is the first instance of that kind made publick.

It is true that M^{r} Leibnitz was the first who published a Rule for first differences. But he should then have acknowledged {illeg} (as he did in his Letter of 21 Iune 1677 written to M^{r} Oldenburg) that he knew by Letter M^{r} Newtons Letters that M^{r} Newton had the a Method w^{ch} did the same things. Or rather, he should have acknowledged that M^{r} Newton gave him light into the method. For D^{r} Wallis in the Preface to the two first volumes of his works published in spring 1695 wrote that the method of fluxions & differential Calculus is one and y^{e} same {d} method differing only in some modes of expression & that M^{r} Newton in his Letters of 13 Iune & 24^{th} Octob 1676 expained {sic} to M^{r} Leibnitz this Method found by him ten years before that time or above; that is, in the year 1666 or before. The editors of the Acta Eruditorum the next year in giving an account of the|se| two Volumes cited a sentence out of it this Paragraph & therefore knew of it. And D^{r} Wallis gave notice of it to M^{r} Leibnitz himself {b} {sic} in a Letter to him dated & printed in the third Volume of his works

The Original Letters & {illeg} & B {sic} Letter Books wh from whence the Commercium Epistolicum was printed are still kept in the \Archives of the/ R. S. & [have been sometimes shewed to strangers & some of them compared with the Comercium Epistolicum. [And particularly the Letter{illeg} of M^{r} Oldenburg dated Apr 15 167{illeg}|5|, & the Answer to it in the handwriting of M^{r} Leibnitz dated 20 May have been have been {illeg} viewed \& collated with the print/ by many strangers some of w^{ch} & coll{illeg} \altogether/ some of whom knew the hand of M^{r} Oldenburg.] & collated by them collated with the print. {illeg} & no question is made in England of their sincerity M{^{r}} They have also upon occasions been shewed to strangers [& some \by/ of them been collated with \some of/ the printed Letters before the{illeg} \by/ who have \viewed them &/ collated some of them with the printed Letters to their satisfaction.

It has been Objcted {sic} with clamour that M^{r} when M^{r} Newton wrote the Scholium at the end of the Book of Quadratures he did not understand second differences: but the Objector knew that by the first Proposition of the Book y^{t} he did understand second differences.

## An Account of what has been done

since the publishing of the

Commercium.

The Commercium Epistolicum was published before the end of the year 1712 & h|n|o answer has hitherto been given to it: for it is unanswerable. M^{r} Leibnits to avoid answering it p{illeg} p{illeg} pretended the M^{r} Leibnitz pretended the two first years that he had not seen the Book being then at Vienna he had not seen the Book, but had written to an able & impartial Mathematician to give his judment {sic} upon it. And the {illeg} Answer or pretended answer of the Mathematician dated 7 Iune 16713 was inserted into a f{illeg}l scurrilous paper dated 29 Iuly following & printed without the name of the author or printer or Mathematician or place where it was printed & \the paper was/ dispersed over Europe: & it was two years & an half before M^{r} Leibnitz began to tell the word that the Mathematician was Iohn Bernoulli. The Letter {T} ascribed to him was in these words. Videtur N...sis occasionem nactus serierum opus multum promovisse per extractiones radicum, quas primus in usum adhibuit et quidem in ijs excolendis ut verisimile est ab initio omne suum studium posuit; nec credo — — — alijs fuisset familiaris &c Thus far the Mathematician. His first argument against M^{r} Newton is founded upon a \ridiculous/ supposition that prickt letters are essential to the method of fluxions, w^{ch} is not so. In the Introduction to the Book of Quadratures he|M^{r}| \Newton/ teaches the method of fluxions & yet there is illustrates it with examples & yet there is not a prickt letter in all that Introduction. In the second Lemma of the second Book of Principles M^{r} Newton demonstrated the Elements of the Method of fluxions: {illeg}|&| yet there is not a prickt letter in all that Lemma. In his Letters of Octob. 24. 1676 {illeg} 10 Decem 1673 \& 24 Octob 1676/ he d{illeg} wrote that he had a general Method w^{ch} readily gave the method of \of/ solving Problems of w^{ch} the method of Slu Tangents of Slusius was a Corollary & w^{ch} stuck not at surds & was founded in the latter of these two letters he added that this foundation of this Method was founded in the sentence Data æquatione fluentes quotcun æquat{a} quantitates involvente fluxiones invenire & vice versa. And will any impartial man affirm that when M^{r} Newton wrote those \two/ Letters, he did not so much as dream of the method of Fluxions because there deducing fluents from fluxions & fluxions from fluents because there are no prickt letters in those two Letters. In his Book of Quadratures he used prickt Lett letters, in his o{illeg} In h Sometimes he used prickt letters \& sometimes letters/ without pricks {illeg} or other symbols & sometimes he wrote of it w^{th}out any symbols. What symbols he used in his Analysis per series numero terminorum infinitas has been shewed above. There he used the symbol $\overline{)\frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}}$ in the very same The Mathematian {sic} affirms same sence in w^{ch} M^{r} Leibnitz uses the symbol $\int \frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}$, & therefore \in those days he/ dreamt of the summatory method before M^{r} Leibnitz \w^{ch} is the inverse of b{illeg}ing the differential/. And when M^{r} Leibnitz was in England the second time & saw in the hands of M^{r} Collins many he saw my Lette M^{r} Newtons Letter of 24 Octob. 1676 as he has confest in his Letter of {illeg} in w^{ch} this Analysis is mentioned by the name of Compendium serierum, & in the hands of M^{r} Collins he saw many of the Letters of D M^{r} Newton & M^{r} Gregory & had especially that w^{ch} ran principially upon series & had \then/ an opportunity to ask for this Compendium serierum & see this symbo.

The Mathematician says further that M^{r} Newton in his Principles of Philosophy had frequent occasion to use prict letters & yet used them not: but the truth is, he had no occasion to use them in that book. For that book is written by composition after the manner of the Ancients: but \yet/ the Analysis by w^{ch} it was invented shines through the composition.|;| |& this was acknowleged formerly both by the marq. de l' Hospital & by M^{r} Leibnitz himself as has been mentioned above|

The second argument of the Mathematician against M^{r} Newton to prove that the calculus of fluxions was not older then the differential Calculus is The Mathematician saith further that prickt letters appeared first in the third volume of the works of D^{r} Wallis many years after the differential calculus had obteined every where. But he hath misrepresented the matter. Prickt letters came abroad The third Volume of D^{r} Wallis \Works/ came abroad in y^{e} year 1699 & prickt letters appeared in his second volume w^{ch} came abroad in spring 1693, w^{ch} was two years before D^{r} Wallis heard that the Differential Method began to be celebrated abroad

The second argument of the Mathematician to prove that the calculus of fluxions was not older then the differential calculus is that M^{r} Newton did not understand how to find the fluxion of fluxions or difference of differences. And yet if you {be}{illeg} the Mathematician knew by the first {illeg} Proposition of the Book of Principles \Quadratures/ but Newton did know understand how to find the difference of differences \or second difference/ & the difference of second differences or third difference & so on in infinitum. And if you look into the Book of Principles lib. 2 Prop. 4 you will see that he the there knew how to find the difference of moments or second difference. And I do not meet with any ancienter instance made publick of working in second differences.

Pag. 1. lin. 20. — prima (p. 37, 38.) Add: Et symbola Leibnitij nondū obtinuerunt in Anglia.

Newtonus Tractatum sua|u|m de Quadraturis edidi die ante {illeg}quam ederetur \dudum/ scrip t{u}|sit|s erant{illeg} quippe \Nam/ plurima ex eo citata sunt in Epistolis 24 Octob. & 8 Novem 1676 (p. 34) In Epistola priore ponuntur \recitantur/ Ordinatis Curvarum quarum collationem cum Sectionibus Conicis Newtonus tum ante annos quin \(seu |(|seu anno 1671)/ in Tabulam retulerat \(pag. 178./ In Epistola posteriore \anno 1711)/ a Ionesio edita Corollarium secundum Propositionis decimæ verbis fusius enarratur, his verbis. *Nulla extat Curva cujus Æquatio ex tribus constat terminis — haud tamen adeo generaliter*. Hæc sunt omnium quæ in Libro de Quadraturis habentur difficilima & ab {sic} methodo fluxionum \aut simili/ inveniri non potuerunt & abunde satis demonstrant, methodum Fluxionum & Momentorum quatenus in \illam quatenus in in Propositionibus decem primis/ Libra ill{illeg}|i||us| habetur, Newtono innotuisse anno 1676 & \aut/ /&\ antea.

Object. 1. Newtonus introducis|t| attractiones & vult grav \i.e. qualitates occultas/ & vult attractionem gravitatem est attractionem id est qualitatem occultam \ejusmodi esse/.

Resp. Quo sensu Newtonus voce attractionis utitur, ostenditur \ex ejus/ supra (pag. 35, 36): \Princ libro Principiorum:/ & plenius patet ex sequentibus. In Introductione ad Sect. XI Lib. 1 Newtonus hæc habet verba {illeg}|sic| scribit. Qua de causa jam pergo motum exponere corporum se mutuo trahentium, considerando vires centripetas tanquam a|A|ttractiones, quamvis fortasse si physice loquamur, verius dicantur impulsus. In Mathematicis jam versamu{r}|s|, & propterea missis disputationibus Physicis, familiari utimur sermon{illeg}|e|, quo possimus a Lectoribus Mathematicis facilius intelligi. Et \in {sic}/ petulo post in Scholio S{illeg} sub finem hujus Sectionis: Vocem Attractionis hic generaliter usurpo pro corporum conatu quocun accedendi ad invicem sive conatus i{illeg}|l|le fiat per ab actione corporum vel se mutuo petentium, vel per spiritus emissos se agitantium, sive is ab actione Ætheris, aut Aeris Medijve cujuscun seu corporei seu incop|r|porporei {sic} oriatur corpora in{illeg}|no|tantia in se invicem utcun impellentis. Eodem sensu generali usurpo vocem Impulsus, non species virium & qualitates Physicas sed quantitates & proportiones Mathematicas in hoc Tractatu expendaris, ut in Definitionibus explicui In Mathesi investigandæ sunt virium quantitates & rationes illæ quæ ex conditionibus quibuscun positis consequuntur: deinde ubi in Physicam descenditur, conferendæ sunt hæ rationes cum Phænomenis ut innotescat quænam virium \conditiones/ singulis corporum attractivorum \viribus/ competant

Object. 2. Newtonus asserit Deum habere sensorium

Resp. Newtonus per Sensorium \nihil aliud/ intelligit \quam/ spatium infinitum tanquā locum sensetis in quo Deus omnia sentit. Sic enim liquitur de Deo: Is totus est sui sui {sic} similis, totus oculus, totus auris, totus cerebrum, totus brachum, totus vis sentiendi intelligendi et agendi: sed more minime humano, more minime corporeo, more nobis prorsus {illeg} incognito. Vt cæcus non habet idæam colorum, sic nos ideam non habemus modorum quibus Deus sentit & intelligit omnia. Corpore omni & figura corporeo destituitur: ideo videri non potest, nec sub specie rei alicujus corporei coli debet.

Object. 3 Newtonus introducit miracula in Philosophiam

Resp. Leibnitius vocat miracula, quæ quotidie eveniunt nemine admirante.

And whereas he claimed one of the methods of regression w^{ch} M^{r} Newton sent him \M^{r} Newton sent to him/ at his own request \M^{r} Newton sent to him/ & w^{ch} upon the first reading he did not know to be his own \nor understand/ he claimed afterwards claimed \as his own invention/ by pretending that he had \had {sic} forgot it till he/ found it in one of his old Papers: it lies upon \him/ in point of cadour {sic} & justice {illeg} either to prove the {sic} he had found it before M^{r} Newton, Or, \& forgot it before he wrote to M^{r} Newton for it or else/ publickly & in express words to renounce {illeg} his claim to it for preven for preventing future disputes about it.

It is therefore expected that M^{r} Leibnitz de renounce all right to this method as first Inventor, as well as to the Differential method of Mouton as second Inventor. For second Inventors have no right. The sole right is in the first Inventor untill another finds out the same thing apart: in w^{ch} {illeg} \in which case/ & then to take away the right of the first Inventor & divide it between him & that other, would be an Act of Injustice.

Endeavours have been used to make the learned world beleive that M^{r} Newton has been introducin|es|g occult qualities into Philosophy, & the Editors of the Acta Leipsiensia have published that M^{r} Newton as|f|firms that gravity cannot be explained mechanically. And if this be done to prejudice the Cartesians & \such/ other Philosophers as place all natural Philosophy in Mechanical \hypothetical/ explications of Phænomena by mechanism, at, against M^{r} Newton, it is an indirect way of proceeding & shews the w

M^{r} Leibnitz in one & the same Letter (his Letter of 1711 I has opposed his own candor to \the arguments of/ D^{r} Keill as if it were injustice to question it, \that is to refuse his \own/ testimony for himself,/ & t{illeg}d questioned \in the same Letter \he/ has \in an arbitrary manner/ without who much as one argument cal{illeg} \to justifie/ himself called/ the candor of M^{r} Newton for publishing that he found the diff method of fluxions \graduall{illeg}|y|/ in the years 1665 & 1666; & now it mutually comes upon him to satisfy the w{or}l satisfy the world about his own candor & integrity in the ten or twelve particulars \last/ above mentioned. *Nec lex est justior ulla.*

And whereas he has {illeg} in his Theodicee accused M^{r} Newton of making gravity an occult quality & a miracle \& thereby introducing |& of \thereby/ introducing occult qualities &| miracles into Philosophy thereby such opinions/ & yet M^{r} Newton has no where declared his \any/ opinion about gravit the cause of gravity|:| or it lies upon him in point of candor & justice to beg M^{r} Newtons pardon in {illeg} publickly for {illeg} endeavouring by the|i|ss indirect \such indirect & unfair/ practises to prejudice the world against him. The Philosophy w^{ch} M^{r} Newton in his Principles & Opticks has pursued is experimentall & meddles not w^{th} |therefore it & experimental philosophy teaches not it is not the business of Experimental Philosophy to teach| the causes of things any further then they can be proved by experiments. And if M^{r} Leibnitz would forbear to trouble the world with his opinions \whereas M^{r} Leibnitz is of opinion that nothing \is done/ by election without a reason, or in other words that {sic}/ that all things are governed by fate & necessity & that {mak} nothing more then matter & motion is requisite \to/ produce all the Phænomena in nature & that man himself is a meer machine; {illeg} untill he can prove these things by experiments, his Philosophy would be of better credit. \God is a supramundane \intelligence &/ the souls of men do not act upon their bodies & that but all/ \animal motion is performed without any other cause than mechanism {illeg} {illeg}|i|f opinion is {illeg} & man/ \himself is a meer machine & God is a supramundane intelligence:/ if he would forbear to trouble the world with such \irreligious/ opinions \& teach nothing more then/ untill he can prove them /conjectures & teach nothing more then \he/ can be\ Proved \by experiments/ his Philosophy would be of better credit.

And whereas M^{r} Newton sent him at his own request a method of regression w^{ch} upon the first reading he did not know to be his own nor understand, but so soon as he understood it he claimed \it/ as invented his own by pretending that he found it had forgot it untill he found it amongst his in {on} in his old papers: it lies upon him in point of candor & justice to prove either to prove that he was the first inventor of this method, or to renounce his claim to it for preventing future dispute about it.

None fuller of Miracles & occult qualities than that.

None more occult mysterious and miraculous \& trifling/ than that w^{ch} [destroys all the arguments for a Deity taken from Phænomena &] \& {illeg}/ supposes every thing, proves nothing, & excludes the proof of a Deity from {illeg} phænomena.

M^{r} Leibnitz in his Theodicee has accused M^{r} Newton of introducing occult qualities & miracles into philosophy {illeg} upon a supposition that he make gravity an occult quality miraculously seated in the will of God. And yet M^{r} Newton has writte in his Optiqus (Quest 22 has written expresly, to the contrary. Quam ego attractionem appello, saith he Qua causa efficiente hæ attractiones [gravitatis, virtutis magneticæ et Electricæ {illeg} \vel alterius cujuscun/ attractiones], {illeg} h{æ} peragantur, saith he, in id vero hic non inquiro. Quam ego attractionem appello, fieri sane potest ut ea efficiatur impulsu vel alio aliquo in universum solummodo vim aliquam significare intelligatur qua sit illa vis. Nam ex phænomenis Naturæ illud{illeg} nos prius edoctos leges & proprietates ill|st|ius attractionis; quam Et in id inquirere par sit quanam efficiente causa peragatur attractio. And a little after he mentions the same forces \attractions/ as forces w^{ch} by phænomena appear to have a being in nature & tho their causes be not yet known & distinguishes them from occult qualities w^{ch} flow from are supposed to flow from the specific forms of things And in the beginning of his Principia Philosophiæ speaking of the centripetal forces & attractions he saith: Virium causas et sedes physicas his \jam/ non expendo. And a little after: Voces autem attractionis impulsus vel propensionis in centrum cujus \cujuscun/ in centrum, indifferenter & pro se mutuo promiscue usurpo, has vires non Physice sed Mathematice tantum considerando. Vnde cæveat Lector ne per hujusmodi voces cogitet me speciem vel modum actionis causamve aut rationem physicam alicubi definire, vel centris (quæ sunt puncta physica mathematica) vires vere et physice tribuere, si forte aut centra trahere aut vires centrorum esse dixero.

And whilst M^{r} Leibnitz has taught that God is intelligentia supramundana & that all animal motion (even that of man) is purely mechanical, & that there no election of any thing without a reason, that is, none all at all, but all things are by fate & necessity: it lies upon him to satisfy the world that |it| \may be proved by phænomena that/ there is a God in whom we live & move & have our being.

And whereas M^{r} Newton sent him at his own request a method of Regression, w^{ch} upon the first reading he did not know to be his own, nor understood it, but so soon as he understood it he claimed as his own by pretending that he had found it long before, & \had/ forgot it, as he peceived {sic} by his old papers: it lies upon him in point of candor & justice, {la}|ei|ther to prove that he was the first inventor of this method, or to renounce his claim to it for preventing future disputes.

It is therefore expected that M^{r} Leibnitz do renounce all right to this method as first inventor, as well as to the Differential Method of Mouton as second Inventor. For second Inventors have no right. The sole right is in the first Inventor untill another finds out the same thing apart: in which case to take away y^{e} right of the first Inventor & divide it between him & that other would be an Act of injustice.

And whereas he has in his Theodiceus accused M^{r} Newton of making gravity an occult quality, & \& {sic} an immediate Act of the Deity w^{ch} he calls/ a miracle \immediately seated in the will of God/ & thereby of introducing occult qualities & miracles into Philosophy; & yet M^{r} Newton has no where declared \any such opinion {illeg}r|o|r/ any opinion about the cause of gravity |or about the cause{illeg} of any other sort of attractions: but speaks of them as forces whose causes are not yet known|: it lies upon M^{r} Leibnitz in point of candor & justice, to beg M^{r} Newton's pardon publickly for endeavouring by such indirect & unfair practises to prejudice the learned part of Europe against him. The Philosophy w^{ch} M^{r} Newton in his Principles & Opticks has pursued is experimental, & it is not the business of experimental Philosophy to teach the causes of things any further then they can be proved by experiments. And the Philosophy of M^{r} Leibnitz would be of better credit \freer from miracles & of better credit/ if he would prove his Propositions by experiments & be silent where experiments fail him. No Philosophy can be freer from occult qualities \mysteries/ & miracles then that in w^{ch} Propositions are proved by experiments, & w^{ch} is silent \which/ where experiments are wanting, is silent, or at the most speaks doubtfully, & by way of inquiry leaves the matter to further inquiry. None more trifling then that w^{ch} supposes every thing & proves nothing.

✝ but on the contrary declared his opinion very expresly about it in these words. For this r For this reason M^{r} Newton is silent about the causes of gravity, & has there occurring no experiments or phænomena by which he can discover it prove \it appe{r}ed it could be proved/ what it is \was the causes thereof/. And this he has abundantly declared In his Principles, {illeg}{(}{nd} neare the beginning, in these words: Virium causas et sedes Physicas jam non expendo. And a little after: Voces attractionis impulsus vel propensionis cujuscun in centrum, indifferenter & pro se mutuo promiscue usurpo, has vires non physice sed Mathematice tantum considerando. Vnde caveat Lector ne per hujusmodi voces cogitet me speciem vel modum {illeg}|a|ctionis causamve aut rationem physicam alicubi definir{e} vel centris (quæ sunt puncta mathematica) vires vere {illeg} & physice tribuere si forte aut centra trahere aut vires centrarum esse dixero. And in the end of his Opticks,^{[1]}: Qua causa efficiente hæ [gravitatis, \attractiones/ |[|sc. gravitas vis magnetica et electrica aliæ] attractiones peragantur, in id vero hic non inquiro. Quare ego attractionem appella, fieri sane potest ut ea impulsu efficiatur impulsu vel alio aliquo modo nobis incognito. Hanc vocem attractionis ita hic accipi velim ut in universum solummodo vim aliquam significare intelligatur qua corpora, ad se mutuo tendu|{e}|nt, cuicun demum causæ attribuenda sit illa vis. Nam ex phænomenis naturæ illud nos prius edoctos oportet quænam corpora se invicem attrahant & quænam sint leges & proprietates istius attractionis; quam in id inquirere par sit quanam efficiente causa peragutur attractio. And a little afft{illeg}|e|r, he mentions the same attractions as forces w^{ch} by phænomena appear to have a being in nature \& may be made usefull in explaining Phænomena tho their Nature/ tho their causes be not yet known & which may produce \by the causes of/ various effects {illeg} & distinguishes them from occult qualities w^{ch} are supposed to flow from the specific forms of things. But \And in the Scholium all the end of his Principles, after he has mentioned the properties But of gravity he adds: Rationem vero harum gravitatis proporietatum ex Phæmenis {sic} &c . . . . non habent./ Notwithstanding all this M^{r} Leibnitz has accused M^{r} Newton of making gravity an occult quality & \a miracle/ an immediate act of the Deity \God/ w^{ch} \act/ he calls a miracle. \And therefore/ It lies upon him \therefore/ in point of candor & justice to beg M^{r} Newton's pardon publickly for endeavouring by such indirect & unfair practises to prejudice the learned part of Europe against him.

And whereas the|is| this Philosophy tends to destroy \of M^{r} Liebnitz is thought by some to weaken destroy \weaken// all the arguments for a Deity taken from Phænomena, it lies upon him in point of religion to beg pardon of all mankind who beleive a Deity God either to show that his Philosophy is not of such a nature \& to declar{illeg}|e| that he is not offended w^{th} |M^{r}| Newton for insisting upon such arguments/, or to beg pardon of all mankind who beleive a God |And yet y^{e} Editors of the Acta Eruditorum have accused him of denying w|t|hat y^{e} cause of gravity is mechanical tho he is silent about that cause| & have compared an Agent w^{ch} he calls a \subtile/ Spirit to D^{r} |H| Mores hylarc{h}ic Principle. tho he is silent about the nature & cause of that Agent, & M^{r} Leibnitz has accused him of making Gravity an occult quality & a miracle.

The Philosophy which M^{r} Newton has published in his Principles & Opticks has pursued is experimental, & it is not the business of experimental Philosophy to teach the causes of things any further then they can be proved by experiments. \We are not to fill this Philosophy w^{th} opinions w^{ch} cannot be proved by Phænomena./ C I In this Philosophy Hypotheses have no place unless by way Quære or as as conjectures \or Questions proposed/ which may deserve \to be/ examinati|ed|on. for this r by experiments. \We are not to fill this Philosophy with opinions w^{ch} cannot be proved by experiments./ For this reason M^{r} Newton in his Optiques distinguished those things w^{ch} were made certain by experiments from those things w^{ch} remained uncertain & w^{ch} he therefore proposed in the end of his Opticks in the form of Queres. For this reason is|n| the Preface to his Principles, when he had mentioned the Motions of the Planets Comets Moon & Sea as deduced in this book from gravity, he added: Vtinam cætera Naturæ Phænomena ex Principijs Mechanicis e{illeg}|o|dem argumentandi genere derivare liceret. Nam multa me movent ut nonnihil suspicer ea omnia ex viribus naturæ quibusdam pendere posse quibus corporum particulæ per causas nondum cognitas vel in se mutuo impelluntur & secundum figuras regulares cohærent, vel ab invicem fugantur & recedunt: quibus viribus ignotis Philosophi hactenus Naturam frustra tentarunt And in the end of this book in the second Edition, he said that he forbore to describe the effects of this attraction for want of a sufficient number of experiment to determin the laws \of its acting/. And for the same reason he is silent about the cause of gravity, there occurring no experim^{ts} or phænomena by which he might prove what was the cause thereof. And this he hath abundantly declared in his Principles, neare the beginning thereof, in these words: Virium causas et sedes Physicas jam non expendo. And a little after: Voces attractionis, impulsus, vel propensionis cujuscun in centrum, indifferenter & promiscue se mutuo promiscue usurpo, has vires non physice sed Mathematice tantum considerando. Vnde caveat Lector ne per hujusmodi voces cogitet me speciem vel modum actionis causamve aut rationem phyiscam alicubi definire vel centris (quæ sunt puncta mathematica) vires vere et physice tribuere, si forte aut centra trahere aut vires centrorum esse dixero. And in the end of his Opticks: Qua causa efficiente hæ attractiones, [sc. gravitas vis magnetica et electrica aliæ] peragantur, hic non inquiro. Quam Ego attractionem appello, fieri sane potest ut ea efficiatur impulsu vel alio aliquo modo nobis incognito. Hanc vocem attractionis ita hic accipi \velim/ ut in universum solummodo vim aliquam significare intelligatur qua corpora ad se mutuo tendant cuicun demum causæ attribuenda sit illa vis. Nam ex phænomenis naturæ illud nos prius e doctos oportet quænam corpora seinvicem attrahant, & quænam sint leges & proprietates istius attractionis; quam in id inquirere par sit quanam efficiente causa peragatur attractio. And a little after he mentions the same attractions as forces by w^{ch} by Phænomena appear to have a being in nature tho their causes be not yet known, & distinguishes them from occult qualities w^{ch} are supposed to flow from the specific forms of things. And in the Scholium at the end of his Principles after he had mentioned the properties of gravity, he added: Rationem vero harum gravitatis proprietatum ex Phænomenis nondum potui deducere, & hypotheses non fingo. Quicquid enim ex phænomenis non deducitur Hypothesis vocanda est; & Hypotheses seu Metaphysicæ seu Physicæ seu Qualitatum occultarum seu Mechanicæ in Philosophia experimentali locum non habent. — Satis est quod Gravitas revera existat et agat secundum leges a nobis expositas, et ad corporum cœlestium & maris nostri motus omnes sufficiat. And after all this one would wonder that M^{r} Newton should be reflected upon for not explaining the cause of gravity by an Hypothesis, as if it were a crime to content himself with certainties established by experiments & phænomena & let uncertainties alone. And yet \the Editors of the Acta Eruditorum have accused \him/ of denying that the cause of Gravity is mechanical tho he is/ M^{r} Leibnitz \silent about it & of introducing a spirit that cause, & M^{r} Leibnitz/ has accused him ^{✝}^{[2]} of making gravity an occult quality & a miracle It lies upon him \M^{r} Leibnitz/ therefore in point of candor & justice to beg M^{r} Newtons pardon publickly for endeavouring by such indirect & |&| unfair \& tricking/ practises to \defame him &/ prejudice the learned part of Europe against him.

In the Acta Erudorum {sic}Erudorum for March 1714, the pag. 140|2| & t{illeg} its said that M^{r} Newton denies that the cause of gravity is mechanical; & yet he neither affirms nor denys nor affirms it, \in any of his writings,/ nor gives any opinion about it. Its said also that he lays down a certain new Hypothesis concerning a subtile spirit perrading the pores of bodies, perhaps the same with the Hylarchic principle of D^{r} More Henry More, or some other \spirit/ of less value then Hypotheses unless it be the Æther or subtile matter of the Cartesians. It appears by certain experiments tryed by M^{r} Hawksby before the Royal Society that electric bodies attract constantly at small distances even without rubbing. The Agent by which this attraction is performed M^{r} Newton calls a subtile spirit,|.| And because he does not explain it by an exp{illeg} Hypothesis, the but leaves its nature \cause/ & properties to be discovered by experiments; they turn his silence to ridicule & make Hypotheses for him. And by such like representations they endeavour to persuade the world that in point of Philosophy he is much inferior to M^{r} Leibnitz & in point of Mathematicks not only to M^{r} Leibnitz but also to M^{r} Bernoulli & M^{r} Tschurnhause, & therefore not likely to be the first inventor of the method of Fluxions.

And whereas M^{r} Leibnitz calls God \tells us that \God (the/ God in w{ic}|ho|m we live & move & have o^{r} being) is/ *Intelligentia supramundana*, & under the applys the name of miracles not only to extraordinary & supernatural events but even to the laws of nature if imprest upon her by the will of God, & under the \name of miracles excludes all this|e| actions from being ca of the first cause from being/ considered in Nature \experimental/ Philosophy, & teaches that nothing happens without a reason or in other words that there is nothing done by election \choise/ but all things are governed by fate & \in his Hypothesis of an Harmonia præstabilita asserts that the soul acts not upon the body, but/ that all animal motion, even in man himself, is performed by meere mechanism it lies upon him in point of religion \either/ to satisfy the world that he has not been at work to \his philosophy doth not tend to enervate &/ \undermine & enervate &/ explode all the arguments for a supreme governour of the Vniverse taken from Phænomena; or else to satisfy beg pardon \for this/ of all makind mankind who beleive a Deity God [& that he beleives that Man is some animal motion in Man is not merely mechanical.|]| Extraordinary events w^{ch} cannot be referred to the natural causes are not within the limits of Natural Philosophy: but] & that it \{illeg}/ not in pursuit of this designe that he \make him cry out {illeg} against M^{r} Newton/ cries out against M^{r} Newton for introducing occult qualities & miracles into Philosophy, & & applys the name of miracles not only to extraordinary \or supernatural/ events but even to the laws of Nature if imprest upon her by the will of God.

Anno Christi finiente
Apogæum Solis
Motus medius Longitudo Solis ab Ap {illeg}{lel}|ogæ|o
Motus medius Solis ab Apogæo.
Motus medius ☉^{is} ab Æquinoxio.
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Here he represents that he knew nothing by M^{r} Newton's Letters but that he had a certain method of Tangents, b{u}t but when his Principles came abroad he und{o} abundantly understood that he had gone much further in these matter methods: now he \But he is {since} gone back from what he acknowledged here &/ tells us that the Principles are written in the manner of the ancients & have nothing of the new Analysis in them, nor make it appear that M^{r} Newton knew any thing of these methods when he wrote that book. It lies upon therefore in candor & justice to acknowledge ther|at| when M^{r} Newton wrote his Principles he had a method of resolving such difficult & valuable difficillima et pulcherrima quæ etiam mistæ Matheseos Problemata quæ sine sine sine calculo differentiali AVT SIMILI non temere quisquam pari facilitate tractabit. For that \book is/ full of such Problemes & therefore could not be invented in the j{a}|u|dgment of M^{r} Leibnitz himself could not be written \have been invented/ sine c|C|alcula|o| Differentiali AVT SIMILI It lies upon him also {illeg} in candor & justice to acknowledge that the|a|t {sic} before he wrote his Letter of 21 Iune 1677 in w^{ch} he began first to mention \describe propose/ the Differential Method, he did understand by M^{r} Newtons three Letters above mentioned, that when he M^{r} Newton in those days had a method of drawing tangents to Curves determining maxima & minima, drawing tangents to Curves analytical or mechanical, determining the \& finding & the tangents/ lengths areas, lengths sol curvities, solid contents, centers of gravity of curves & curvilinear figures analytical or mechanical & this without taking away surds, & that this method in conjunction w{ith} |gave the Areas {illeg}|o|f Curvilinear fugures {sic} in Series which in certain cases brake off & became finite, & {illeg} compared figures w^{th} y^{e} Conic sections & |readily gave the Method of Tangents of Slusius as a branch or Corollary thereof, & gave also the Areas of Curvilinear figures|| by the help of finite & infinite equations together extended to inverse Problemes of Tangents & others more difficult & \even/ to almost all Problems except perhaps some numeral ones like those of Diophantus. He is also to acknowledge that when M^{r} Newton told him how general his methods were he could not beleive it no|bu|t forbear to \in his Letter of 27 Aug. 1676/ represented it improbable. Quod dicere vi [saying Quod dicere videmini pleras difficultates (exceptis Problematibus Diophantæis) ad series infinitas reduci; id mihi non videtur. Sunt enim] And then it lies upon him to make the world understand why he did not upon publishing his own method he concealed all this & did not candidly tell his Readers what he knew of the MET Methodus SIMILIS & whose it was. & that it was invented before his own & gave in conjunction with D^{r} Barrows method of Tangents gave him light into his \own/, [And he is further to acknowledge that the Principia Philosophiæ is a Book full of such Problemes as were not to be invented witho sine calculo differentiali AVT SIMILI.] & seemed incredible \to him/ when he was first told of it, & put him upon considering how to improve the methods of tangents \then extant/ & particularly that|os|e of Slusius & Barrow w^{ch} gave him light into it. For he tells us upon another occasion \has told D^{r} Wallis in his Letter of 28 May 1697)/ that he found this method by observing that Differences answered to Tangents & summs to Quadratures. Epist. ad

And whereas M^{r} Leibnitz tells us that God (the God in whom we live & move & have o^{r} being) is INTELLIGENTIA supra SVPRAMVNDANA, & applys the name of miracles not only to extraordinary & supernatural events but even to the laws of Nature if imprest upon her by the will of God, so that for atoms to be hard \or heavy/ by the will of God without a mechanical cause would be a miracle & {illeg} for atoms by consequence for atoms or any thing else to exist by the standing will of God would be a standing miracle & thus by the name of mirales {sic} excludes all the actions of the first cause from being considered in experimental Philosophy, & teaches that nothing happens without a reason, or in other words that there is nothing done by choise, but all things are governed by fate, & in his Hypothesis of an Harmonia præstabilita asserts that the soul of acts not upon the body but all animal motion, even in man himself, is performed by mere mechanism: it lies upon him in point of religion to satisfy mankind that his philosophy doth not tend to enervate & explode all the arguments for a supreme governor of y^{e} Vniverse taken from the Phænomena \of nature/; & that while he asserts that nothing happens without a reason, or in other words that there is nothing done by choise but \all things/ are governed by fate, he doth not mean mechanically, & that his complaining of accusing M^{r} Newton for introducing occult qualities & miracles into Philosophy proceeds not from notions of this kind.

It is true that M^{r} the Philosophy & M of these two \Gentlemen/ differ very much. The one beleives \teaches/ that God, (the God in whom we live & move & have our being) is every where Omnipresent: the other that he is INTELLIGENTIA supramundana SVPRAMVNDANA a God that's no where i{illeg}|n| the world. \Whence it follows that he cannot act upon the world without a miracle/ The {one} bele{illeg}|i|ve|s| that animal motion in man is not purely mechanical; the other that it is.|,| The one that the soul \or mind (according to/ the Hypothesis of an Harmonia præstabilita) not \never/ acting upon the body. The one \teaches/ that a Philosophers are to prov argue from Phænomena to the{\re/} first \immediate la{illeg}\a/{illeg}est next/ causes \thereof/ & from those \causes/ to the next & so on till we come to the first causs|e|: the other that all the actions of God are miracles & miracl & all the laws imprest upon Nature by the will of G{illeg}|o|d are both \perpetual/ miracles & occult qualities & therefore not to be considered by a Philosopher|s|. But why must \it go for a miracle or wonder if God has any thing to do with the world & why must/ all the arguments for a Deity taken from phænomena be exploded by such \new hard-/names? {D} {sic} Is it philosophical to rail & scold instead of arguing {illeg} \For certainly/ Philosophers are to reason without railing, \&/ not to rail without reasoning.

The one beleives that \sees no reason why/ God may \not/ chuse by the power of his will in things indifferent: the other teaches that nothing is done without a reason, meaning (I think) that all things are by fate. The one teaches that God (the God in whom we live & move & have o^{r} being) is omnipresent: the other &c

The one \for want of experiments to decide the question/ doth not affirm whether the cause of gravity be mechanical or not mechanical the other that it is a miracle if it be not mechanical. The one by way of inquiry attributes it to the will of the creator that the \least/ particles of matter are hard: the other attributes the hardness of matter to conspiring motions & calls it a miracle if the cause of this hardness be other then mechanical. The one doth not affirm that animal motion in man is purely mechanical: the other that it is \purely mechanical/; the soul or mind (according to the Hypothesis of an Harmonia præstabilita) never acting upon the body; \I think he means or never/ without a miracle.

<27v>The one proceeds upon the evidence of Experiment & Phænomena & stops where such evidence is wanting: the other propounds \is taken up with/ Hypotheses & propounds them not to be examined by Experim^{ts} but to be received w^{th}out examination The one doth not presume to say that God cannot chuse to act by the power of his will in matters indifferent: the other affirms that nothing himself is done without a reason

And why must the laws of Nature be call'd miracles & occult qualities (that is to say wonderfull improbabilities if not impossibilities absurdities) if derived from the will of God

M^{r} Leibnitz tells us in his Tentamen de motuum Cœlestium causis \after he had mentioned D^{r} Barrows indefinitely little triangle for drawing of tangents & said \\(according to the Scholium upon M^{r} Newtons X^{th} Lemma)/ that there were infinite degrees of infinites & infinitely littles he// tells us: Si motus exponatur per lineam communem quam dato tempore mobile absolvit, impetus seu velocitas exponetur per lineam infinite parvam, & ipsum elementum velocitatis, qualis est gravitatis sollicitatio, vel conatus centrifugus per lineam infinities infinite parvam. At hæc Lemmatū loco annotanda duxi pro methodo nostra quantitatum incomparabilium & Analysi infinitorum, tanquam doctrinæ hujus novæ Elementa. And this is all one as to say that if the space described be considered as the fluent, the velocity must be considered as the first fuxion {sic} & the variation of the velocity as the second fluxion, \& so on/ The|i|s was the foundation of M^{r} Newton's arguing in his Principles & this (w^{th} a little variation of the Language) M^{r} Newton makes the foundat{i} one of the a \Leibnits/ recc{e}|o|ns among the Elements of his Analysis. And while M^{r} Newton|'s| in his \book of/ Principles is founded upon these Elements & {illeg} in that book he argues right. about the{illeg}: it is certain that when he wrote that book, he had not y^{n} forgotten the method of second fluxions. {sic} \understood the elements of |of| that Analysis & upon w^{ch} M^{r} Leibnits founded the cal that Analysis & of the/ infinitesimal Analysis & had not then forgotten the method of second fluxions.

And M^{r} Leibnitz himself in his Letter of 21 Iune 1677 in explaining \how/ his Method gave \gave {sic}/ \shewed how it gave shewed how it gave/ the method of Tangents of Slusius & proceeded without striking at taking away surds & extended to Quadratures, \& then |& then|/ dela|cl|ared himself of opinion that \since/ M^{r} Newtons method \did these things it/ was of the same kind. Arbitror \saith he,/ qua celare voluit Newtonus saith he de Tangentibus ducendis, saith he, ab his non abludere. Quod addit, ex hoc eodem fundamento Quadraturas quo reddi faciliores me in sententia hac confirmat, nimirum semper figuræ illæ sunt quadrabiles quæ sunt ad æquationem differentialem. And before this M^{r} Newton \had/ told him further in his three Letters above mentioned, that his method determined Maxima & Minima, gav gave the lengths, \&/ curv{a}|i|ties &|o|f centers of gra \{Q} {sic}/ Curves, {illeg} & centers of gravity of curvilinear figures, & compared such figures \curves/ with the Conic Sections & squared them by \infinite/ series \series or æquations/ w^{ch} in certain cases brake off & became finite, & by the help of finite & infinite series \æquations/ together re{illeg}ded to \resolved/ inverse problems of tangent & others more difficult, & even to almost all Problems except perhaps some numeral ones like those of Diophantus. \And/ When M^{r} Leibnitz was \first/ told all this the great extent of the Method, he could not beleive it, but represented it improbable saying Quod dicere videmini pleras difficultates {ad} (exceptis Problematibus Diophantæis) ad Series infinitas reduci; id mihi non videtur. Sunt enim multa us mira & implexa ut ne ad|b| æquationibꝰ pendeant ne ex Quadraturis: [Qualia sunt (ex multis alijs) Problemata methodi tangentium inversæ. And M^{r} Newton replied that his Analysis extended to such inverse Problemes & others more difficult] It lies upon him therefore \in candor & justice/ to acknowledge that before he found the Differential method \or at least before he wrote his Letter of 21 Iune 1677,/ he knew all this & wondred at it: And to give an Account why he concealed it \his knowledge thereof/ when he published the Differential method \as his own/ & |why he| did not \then/ let the Germans know what he meant by the Differential methodus SIMILIS, {illeg}t \or/ whose it was, {illeg}|o|r what he had learnt \of it/ by his correspondence w^{th} M^{r} Oldenb. concerning it.

And — Analysis extended to almost all sorts of Problemes & {illeg} \represented it/ so general that M^{r} Leibnitz himself could not beleive it in answer exprest his disbeleif of it. untill \before/ he found the differential method. It lies upon him therefore in candour & justice to aknowledge this & to give an Account why he did not acknowledge any thing thereof when he first published the differential method. For it was not enough to mention a methodus SIMILIS without saying whose it was & of what antiquity & extent according to the notice he had from England.

— I do not know. But this I knot|w| that \[after M^{r} Leibnitz had appealed to y^{e} R S./ M^{r}O there lies no appeal from the Committee of the R. Society to M^{r} Bernoulli. M^{r} Berno M^{r} Leibnitz] M^{r} Bernoulli{illeg} is a party man & can be no judge. And M^{r} Leibnitz objected against can be on Iudge in this matter. {illeg}d He is M^{r} Leibnitz He is too much of a party, & what M^{r} Leibnitz objected against M^{r} D^{r} Keill namely that he is *Homo novus & rerum anteactarum parum peritus*, & the same objection lies against M^{r} Bernoul{li} M^{r} Leibnitz \allows/ that in the differential method letters may be used instead of the diff

But this I know that M^{r} Bernoulli had the \differential/ Method from M^{r} Leibnitz & is the chief of his disciples, \& his writ |& gave his opinion in the| Acta Leipsica before he saw the Commercium Epistolicum/ & that {illeg} what M^{r} Leibnitz objected against D^{r} Keill may {illeg}b he is \at w^{ch} time he was/ homo novus & rerum anteactarum parum peritus, as M^{r} Leibnitz objected against D^{r} Keil, & that what he wrote afterwards \he saw the Commercium/ was in his own defense. [And for M^{r} Leibnitz to appeal|ed| from the Committee of the Royall Society to his confident whose opinion he knew the chief \man/ of his party who h{ad]} & his skill in Mathematicks will not mend the matter. He h {sic} is skilful enough to know that in taxing me with ignorance of \in/ second differences he was mistaken & that the {illeg} some errors in M^{r} Leibnitz his Tentamen de motuum cœlestium causis proceeded from his want of skill in those differences but he has \not/ hitherto been too partial {to} acknowledged {at} what he knows.

— & that before the end of the year 1666 I [sometimes used letters with one or two pricks for quantities involving first or second fluxions &] wrote a small Tract on this subject w^{ch} was the grownd of that larger Tract w^{ch} I wrote in the year 1671 both w^{ch} are still in my custody & are founded upon the fist|rs|t Proposition of the Book of Quadratures explained & demonstrated by letters without pricks tho at the same time I sometimes used & that when I wrote in a Geometrical manner without prickt letters or other symbols; & that when I wrote the larger of those two Tracts I had made my Analysis composed of the methods of Series & Fluxions together so universal —

And therefore he is the Agressor. In both \both/ his \two/ Letters to D^{r} Sloan that p{illeg}{i}{illeg} of \(that/ dated 4 Mart. 1711 & \1711 & that dated/ 29 Decem. 1711, he pressed the R. Socity to condemn D^{r} Keil & \& before I medled in this matter/ challenged me to declare my opinion|.| in this matter. His words in his last \second/ letter are: *Ita vestræ æquitati committo, annon coercendæ sint variæ & injustæ \[Keilij]/ vociferationes, quas ipsi Newtono, Viro insigni et gestorum optime conscio, improbari arbitror; ejus sententiæ suæ libenter daturum Iudicia mihi persuadeo*. And therefore he is the aggressor|.| and ought to prove should have proved

## P. S.

He \M^{r} Leibnitz/ seems to say that what he formerly allowed concerning my having found a method like his own proceeded from his beleiving me, but since what I have retracted what I allowed \acknowledged/ formerly he may be allowed to do the like. Here especially since the probabilities rem{illeg} observed by Bernoulli, Here he accuses Bernoulli of calling my candor in question {illeg} which is the {v}er, viz^{t} in the Libel above mentioned. M^{r} Bernoulli may pretend |its probable that no Probleme can be solved by the method of fluxions without the use of prict letters &| the|a|t in the Introduction to the Book of Quadratures the Method of fluxions is not explained because there are no prickt letters {illeg} in it \that Introduction/, & that in the second Lemma of the second book of the Principia Philosophiæ the Elements of the method of fluxions are not demonstrated because there are no prick letters in it, & {illeg} that from the sentences [ set down in my letter of 24 October 1676] viz Data æquatione fluentes involvente fluxiones invenire & vice versa, et \and/ Vna methodus consit|s|it {sic} in extractione fluentis ex æquatione fluxionem involvente it cannot be inferred that I {oh}ad then knew how \then knew the first Proposition of the Book of Quadratures/ had at that time a method of reducing Problems to fluxional equations \or of resolving such æquation/ because there are no prickt letters in those sentences. And that in the Analysis per æquationes numero terminorum infinitas there are no specimens of the {illeg} Method of fluxions because there are no prickt letters. And that to the Schediasma the Proposition in the Schediasma hereunto annexed is no part of the Method of fluxions because it is explained without prickt letters tho it be the very same with the first Proposition in the book of Quadratures upon w^{ch} the Method of fluxions is founded. And that its improbable that any {illeg} cal|n| solve \no/ Problemes can be solved by the method of fluxions without the use of prickt letters. But if these things are only probable then its probable \possible/ that they \it/ may be otherwise & that I might have the Method of fluxions above 50 years ago notwithstand any thing w^{ch} hath been said to the contrary. {illeg} But other men will rather inferr from these things that I had the method of fluxions when fifty years ago {illeg} putting such symbols for fluxions as I thought fit, & M^{r} Leibnitz was of the same mind before he knew what symbols I used, as in his Letter of 21 Iune 1677 & where he allowed me a Methodus similis & in the Acta Eruditorum where he {illeg} spake again of for October 1684 p 473 where he {obtiner} again mentioned a Methodus si{l} similis extending to the higher sort of Problemes & in his Letter to me dated {illeg} 17 Mar 1673 st. n. hereunto annexed wherein he acknowledged of his own accord that by the Principia Philosophiæ it appeare I had shewed that I had \such/ a methodus similis & added that he also had endeavoured to reduce s|t|he difficulter Problems to such an Analysis. And in the Acta Eruditorum for Iune 1686 he left me at liberty to use \pag 297 he/ allowed that \{illeg} |in| the Differential method/ one might use letters instead of dx & such like character\symbol/s, & thereby left me at liberty to use what symbols I thought fit. And further, I do not put prickt letters for differences or moments as M^{r} Bernoulli supposes as M^{r} Bernoulli supposes, but for fluxions w^{ch} are finite quantities of another kind, the first \Differences l the one/ being infinitely little{illeg} the last finite & fluxions quan {ma} magnitudes \parts/ & fluxions \the other/ finite motions velocities of motion. And whereas M^{r} Leibnitz pretends that \since/ I have retracted what I foll|r|merly alleged him, {illeg} therefore he may be allowed to do the like; he should have proved that had retracted, & if I had done an ill thing yet that would not authoriz him to do the like. And whereas he pr{illeg} questions my credit in what I said of D^{r} Wallis, he will find the truth of what I said |in| the Paragraph of P|th|is Preface hereunto annexed

Leibnitius eadem fide chartam totam volantem scribere potuit qua citationem prædictam delevit, & \verisimile est quod/ Menkenius eandem imprimere prælo committere commisit. Nam Leibnitius chartas imprimendas \ad/ ipsū mittere solebat. Et Bernoullius Epistolam sine nomine imprimendā ad Leibnitium non misit

In prima sua ad Abbatem de Comitibus Epistola Leibnitius scripsit — — —

{Et} hæc. Quæstio dirimatur notandum est \{clari}us {illeg} alligatur \De Ad hanc Questionē spectat// quod \D./ Wallisius noster {[}|C|eleberrimus Professor Oxoniensis, Propositionem primam Libri de Quadraturis \exemplis in fluxionibus prim{illeg} & secundis in veniendis illustratam./ {illeg} edidit \anno 1693/ in Volumine secundo Operum suorum pag 392. Et hæc fuit Regula omnium prima \antiquissima/ quæ lucem vidit pro differentijs differend{illeg} {ut} fluxionibus secundis tertijs quartis cæteris in infinitum inveniendis Eandem Newtonus demonstravit synthetice in Lemmate secundo Libri secundi Principiorum: cum Propositionem sine Demonstratione prius posuisset in Epistola ad Oldenburgum 24 Octob. 1676 ut ad Leibnitium mitteretur & ibi significasset eandem esse fundamentum methodi generalis de qua scripserat tum ante decem annos quin, id est anno 1671. In hujus Propositionis solutione habetur Algorithmus Methodi fluxionum.

Gregorius scripsit ad Collinium 2{illeg} 5 Sept. 1671, se ex Barrovij methodis tangentes ducendi methodum generalem ducendi Tangentes & Geometricam ducendi tangentes ad omnes Curvas sine calculo: & Slusius se similem methodum habere mense Novembri 1672 scripsit ad Oldenburgum. Et Newtonus ad Collinium 10 Decem 1672 scripsit in hæc verba: Ex animo gaudeo D. Barrovij nostri reverendi Lectiones Mathematicas \exteris/ adeo placuisse, neque parum me juvat intelligere eos [Slusium et Gregorium] in eandem mecum incidisse ducendi Tangentes methodum & And {illeg}. Et subinde methodum suam tangentium descripsit, & addidit hanc methodum esse corollarium Methodi generalis solvendi abstrusiora Problemata & non hærere ad quantitates surdas. Epistolas totas Gregorij & Newtoni habes infra in Commercio, & earum Exemplaria Oldenbus|r||g|\ꝰ/ misit ad Leibnitium inter Excerpta ex Gregorij Epistolis 26 Iunij 1676; et Leibnitius Prælectiones Barrovij mense Novembri proximo secum tulit in Germaniam.

Sunto jam ipsarum $\mathrm{AB}=\mathrm{x}$ & $\mathrm{BC}=\mathrm{y}$ \utim Epistola Newtoni seu Abissa {sic} $\mathrm{AB}=\mathrm{x}$ & Ordinata $\mathrm{BC}=\mathrm{y}$. Et sint earum/ fluxiones p et q & momenta op et oq, et in AB producta capiatur $\mathrm{BE}=\mathrm{o}\mathrm{p}$, erigatur Ordinata EF parallela BC & occurrens Curvæ in F et compleatur parallelogrammum BCGE et erit $\mathrm{FG}=\mathrm{o}\mathrm{q}$. Proponatur æquatio quævis {illeg}|qu|antitates duas fluentes x et y involvens, puta ${\mathrm{x}}^{3}-2\mathrm{x}\mathrm{x}\mathrm{y}+\mathrm{b}\mathrm{x}\mathrm{x}-\mathrm{b}\mathrm{b}\mathrm{x}+\mathrm{b}\mathrm{y}\mathrm{y}-{\mathrm{y}}^{3}=0$ \ut in Epistola illa/. Multiplicetur omnis æquationis terminus per indicem dignitatis x et productum divisum per x (vide{illeg}|l|icet $3{\mathrm{x}}^{2}-4\mathrm{x}\mathrm{y}+2\mathrm{b}\mathrm{x}-\mathrm{b}\mathrm{b}$,) vocetur R. Multiplicetur omnis æquationis terminus per indicem dignitatis y & productum divisum per y ,|(|videlicet $\mathrm{-2}\mathrm{x}\mathrm{x}+2\mathrm{b}\mathrm{y}-3\mathrm{y}\mathrm{y}$) vocetur S. Et per Regulam in Epistola illa Newtoni traditam erit subtangens $\mathrm{BD}=\frac{\mathrm{S}\mathrm{y}}{\mathrm{R}}$, vel potius $=\frac{\mathrm{-5}\mathrm{y}}{\mathrm{R}}$ propterea quod AB et BD ducantur in partes contrarias. Sed ex methodo tangentium Barrovij est $\mathrm{FG}.\mathrm{GC}\colon\colon \mathrm{BC}=\mathrm{y}.\mathrm{BD}$. adeo $\mathrm{BD}=\frac{\mathrm{o}\mathrm{p}\times \mathrm{y}}{\mathrm{o}\mathrm{q}}=\frac{\mathrm{p}\mathrm{y}}{\mathrm{q}}$. Ergo $\frac{\mathrm{p}\mathrm{y}}{\mathrm{q}}=-\frac{\mathrm{S}\mathrm{y}}{\mathrm{R}}$ et facta reductione $\mathrm{R}\mathrm{p}+\mathrm{S}\mathrm{q}=0$ \Hæc est æquation involvens fluxiones/. Et si plures sint quantitates fluentes eadem operatio instit{illeg}|u|ta in{illeg} singulis dabit{illeg} æquationem involventem fluxiones \exhibat solutionem/ \omnium/. Hæ Et hæc Regula illud omne comprehendit quod Leib{nitius} ad Newtonum rescripsit in Epistola sua anno 1677 ad{illeg} rescripsit, ut et illud omne quod in Actis Eruditorum anno 1684 in lucem edidit. Exhibet enim hæc Regula solutionem Propositionis, [Data æquatione fluentes quotcun quantitates involventes invenire fluxiones; quam posui in Epistola 24 Octob 1676 ad O] primæ Libri de Quadraturis.

Propositionem quintam libri de Quadraturis Wallisius edidit — — — — ab ipso excogitatam, i.e. anno 1666 aut antea; nec non teste N. Fatio de Duillier Et his præmissis qui chartas antiquas Newtoni viderat, & contra seipsum testimonium perhibuit. Et his præmissis legatur jam Recensio Commercij Epistolici & Consulatur Commercium ipsum ubi de factis dubitatur.

Hæc Regula Hoc Theorema {illeg}tur \Sic inventio fluxionum ex æquatione fluentis involvente/ ex Epistola Newtoni {cf} method{o}|i|s Tangentium Barrovij & Gregorij a Newtono citati facillime fluit. [Idem vero exhibet solutionem Propositionis primæ Libri de Quadraturis.] \deducitur/ Et {illeg}s hujus Theorematis \sic ex Newtoni Epistola derivatum/ invent{um}|io| illud omne comprehendit \illud omne/ quod Leibnitius ad Newtonum rescripsit anno 1677 rescripsit, ut et illud omne quod in Actis Eruditorum anno 1684 in de hac methodo in lucem edidit. Exhibet enim hoc Theorema solutionem. Propositionis primæ Libri \Newtoni/ de Quadraturis Newtono \Idem/ vêro \Newtono/ innotuit|s||se| anno 1686 \quemad ut/ manifestum est ex Lem. 2 Lib. 2 Princip. {u}t \ut/ et ann{is}|o| 1676, manifestum est 1672 & 1671 \quemadmodum/ manifestum est ex Epistolis Newtoni 13 Iunij & 24 Octob. anni {i}|a|nni illius, ex quibus uti Wallisius scripsit Newtonum Leibniti methodum exposuisse, et \Leibnitio. Idem Newtono innotuisse etiam/ anno 1672 {illeg} manifestum est ex Epistola ejus Epistola jam explianta ijs quæ \jam dict{a sunt}/ de Epistola ejus hoc anno scripta, sunt jam dicta sunt jam dicta fuerunt.

Iudex sive celando et nomen suum & nome|i|na eorum quos judicat, libellum potius quam f{illeg}d{illeg} sententium judicis dictitat.

*a* Hæc Epistola, cum celetur et Iudicis et Iudicatī nomen, Libellum defamatorum magis sapit quam æqui \probi/ judicis sententiam.

*b* Algorithmus habetur &

*c* In Analysi per series Newtonus utitur

*d* Nullam habuit occasionem. Propositiones \non invenit per synthesin sed/ per Analysin inventas demonstravit synthetice ut in Geometriam admitterentur.

*e* Volumen tertium.

*f* Eodem sensu

*g* In libro

*h* Eminens ille

*i* Recta methodus

ubi Ordinatim Applicata \(ceu momentum areæ) fluentis)/ deducitur ex æquatione Ab|sc|issam et Aream involvente deducitur.

Recueil Tom. 2. p. 4, 5, 36, 52, 53.

In Epistola ad Cometissam de Kilmanseg

Agnovit postea Oldenburgium ad se mississe des Essays

Ad hanc Quæstionem spectat quod Leibnitius differentias \& methodum differentialem/ vocat quæ Newtonus momenta, Et Methodus differentialis eadem est cu{illeg} momentorum: Et \quod/ methodus fluxionū Newtoni non magis differt a methodo differt a methodo deffentialij {sic} quam differt a methodo momentorum. [Fluxiones sunt velocitates quibus m{illeg}nd {mome} \Vna et eadem est methodus nisi quatenus Newtoni methodus sit amplior/ Momenta vel differentiæ sunt partes \genitæ/ quantitatem \genitarum/, fluxiones sunt velocitates quibus partes illæ generantur. Leibnitius \& Newtonus/ considerat|n|t solas partes \genitas/, Newtonus considerat etiam partes \velocitates/ quibus \partes/ generantur. Newtoni methodus complectitur methodum Leibnitij.]

Vna et eadem est methodus quoad partes differentius vel momenta. Hæc sunt partes quantitatum genitarum. Vter considerat has partes Fluxiones sunt velocites {sic} quibus partes generantur. Vter considerat partes. Newtonus considerat etiam partes quibus velocitates quibus partes generantur. Newtoni methodus complectitum methodum Leibnitij.

Leibnitius ita series a se missas Epistolam Oldenburgij 15 Apr 1675 ad se missam accepit seriebus refertam, \seriebus refertam accepit/ Sed Series nunquam si fas est credere nunquam contulit cum suis.

Ad hanc Quæstionem spectat quod Leibnitius differentias & methodum differentialem vocat quam|s| Newtonus momenta & methodum momentorum quod methodus momentorum & methodus fluxionum eadem sit. Momenta sunt partes quas Leibnitius differentias vocat, fluxiones sunt velocitates quibus partes generantur. Leibnitius considerat partes, Newtonus considerat etiam velocitates. & Newtoni methodus est amplior & Leibnitij methodum complectitur.

Et hæc est Regula ducendi tangentes quam Newtonus posuit in Epistola illa posuit ut partem aliquam \vel specimen/ vell Corollarium vel Specimen {mea} Methodi suæ generalis:

Ducatur jam secundum methodum Barrovij a Gregorio promotam Ordinata nova EF priori BC proxima & compleatur parallelogrammum BCGE et pro \momentis/ BE vel CG & GF scribamus p et q et erit $\mathrm{q}.\mathrm{p}\colon\colon \mathrm{y}.\mathrm{BD}$. seu $\frac{\mathrm{p}\mathrm{y}}{\mathrm{q}}=\mathrm{BD}=-\frac{\mathrm{S}\mathrm{y}}{\mathrm{R}}$ et facta reductione \prodit/ $\mathrm{R}\mathrm{p}+\mathrm{S}\mathrm{q}=0$. Hæc æquatio ubi duæ tantum sunt fluentes involvit earum momenta. Et ubi plures momenta. {illeg} vel ut Leibnitius loquitur, omnium differente|i|as. Et Theorema hocce quod sic ex \specimine in/ Newtoni Epistola \posito/ facillime deducitur, totum illud \omne/ comprehendit quod Leibnitius anno 1677 ad Newtonum rescripsit, ut et illud omne quod in Actis Eruditorū anno 1684 in lucem edidit. {illeg} [et illud omne quod Newtonus in {illeg} Lem 2 Lib. 2 Princip. et Prop. 1 Lib. de Quadraturis edidit.] Hoc Theorema exhibet solutionem Propositionis primæ Libri de Quadraturis, ideo {illeg} solutio Propositionis \illius/ anno 1672 Newtono innotuit.

In libro de Analysi per Series Fluxiones ac Differentias \anno 1711/ a Ionesio edito extat Fragmentum Epistolæ D. Newtoni ad D. Collinium Nove. 8 1676 data ubi Newtonus scribit in hæc verba. Nulla extat Curva cujus æquatio — possint comparo. — Eadem methodus Æquationes quatuor terminorum alias complectitur, haud tamen adeo generaliter. Hactenus Newtonus. Hæc autem abs Methodo fluxionum fieri non possunt. Indicant vero methodum quadrandi Curvlineas in libro de Quadraturis expositam, eous productam\motam/ fuisse ante 8 Novem 1676.

In Epistola Newtoni ad Oldenburgum 24 Octob. 1676 ha {b}|{d}| citantur \ponuntur/ Ordinatæ Curvilinearum in Tabulam tum olim Catalogum tunc olim re{illeg} qu{s}|{a}|rum collationes cum Conicis sectionibus Newtonus in c|C|atalogum tunc olim {in} {illeg} retulerat. Earundem Curvarum \et/ {illeg}{E}odem ordine et ijs literis literis c|C|ollationes \cum Con. Sect./ describuntur in Tabula posteriore in libro de Quadraturis: ideo \Tabula illa composita fuit est/ methodus quadrandi Curvas eous producta, fuit annis aliquod|t| ante annum 1676. Id quod abs methodo fluxionum fieri non potuit.

In \prædicta/ Analysi prædicta per series \prædicta/ quam Barrovius anno 1669 ad Collinium misit Newtonus exponit \tempus per abscissam Curvæ/ fluentem per aream Curvæ ejus fluxionem \ejus fluentis/ per Ordinatam & momentum ejus per fluxionem ductam in temporis momentum o. Et sub finem Tractatus illus dat specimen calculi. Et his præmissis \Leibnitius/ methodum Newtoni serierum Newtoni in Tractatu illo descriptam postulabat a Collinio per literas ad Oldenburgum 12 Maij 1676, potuit videre in manibus Collinij proximo mense Octobri ubi in Angliam venit & in manibus Collinij vidit epistolas plures Newtoni Gregorij & aliorum, eas præsertim quæ circa series versabantur ut ipse in epistolis ad D. Abbatem de Comitibus & Cometissam de Kilmansegger agnovit.

Et his præmissis legatur jam Recensio Commercij Epistolici & consulatur Commercium ipsum ubi de factis dubitatur.

✝ Tandem agnovit iterum in Epistola sua ad {t}|C|ometissam de Kilmansegger, ut supra in Præfatione ostenditur

— Et Mathematicum esse Bernoullium \ipsum/ scripsit, & charta{s} illa{s} volunte{m} denuo dispersæ & Gallice in Hollandia imprima auctoritate Bernoullij \imprimi curavit/ \&/ Problemata Bernoullij Analystis Anglis solvendam proposuit Et auctorite Bernoulij \Bernullij Indivisa N constituti/ amicos suos rerum mathematicarum inscios passim contra Newtonum passim ciere conatus est, cum tamen Bernoullius.

Commercium Epist. p. 32, 5{illeg}|3{0}|, 3{illeg}, 39, 42, 43, 44, 45, 49 58, 87, 88, 96.

Charta volans p in Commer p.

<31v>In Epistola Charta \illa/ 29 Iulij 1713 data Leibnitius Epistolam Oldenburgi qua 15 April {sic} 1675 datam qua Oldenburgus series aliquot & inter alias seriem Gregorij quam L ad Leibnitium misit et inter alias seriem Gregorij quam Leibnitius postea ut suam edidit in dubium vocare conatus est, dicendo: Tale quiddam Gregorium habuisse ipsi Angli & Scot. Wallisius, Hookius Newtonus & junior \Gregorius/ ultra triginta sex annos ignoraverunt & Leibnitij esse inventum crediderunt. At hæc Epistola ut et E{illeg} in Libro Epistolico Regiæ Societatis asservata, ut et Epistola autographa Leibnitij {illeg}q se series \missas/ recepisse agnoscentis, cum ijsdem epistolis in Commerio {sic} editis, coram exteris plu\ri/mus collata sunt in qu{illeg}d Comite de Kilmansegger, Abbate de Comitibus, {illeg} ministris aliquot aut publicis exterorum {quot} Principum & et {sic} alijs exteirs non paucis \Anno 1715/ collatæ sunt et \ab omnibus/ agnitæ; & Leibnitius ipse a {an}no proximo in Epistola sua ad D. Cometissam de Kilmansegger 18 Apr. 1716 \data &/ a D. {illeg} Deso Maiseaux edita, idem agnovit narrando historiam epistolis in Commercio editis conformem. Narrat enim quomodo Londinum venit initio anni 1673, cum Oldenburgo amicitiam contraxit, a Pellio nostro apud D. Boyle audivit \quod/ Mercatorum seriem invenisset pro Hypola, librum comparavit & secum asportavit in Galliam, ibi sub Huygenio cœpit meditationes Geometricas {illeg} gustare, parvo tempore multum profecit, & invenit seriem \suam/ pro circulo, dein sic pergit Nous crûmes que j'etois le primier, qui avois fait quelque chose de tel sur le circle; & j'en écrivis sur ce ton-la a [15 Iulij & 26 Octob] a M. Oldenbourg en 1674 avec qui auparavant je ne telles choses, quoique nous eussiouns échangé deja pleusieurs [Feb. 20, Mart. 30, Apr 26 Maij 24 & Iunij 8] plusieurs Lettres. M. Oldenburg m'ecrit [15 Apr 1675] m'envoya des & essays. Cependant le mieu [Londinum missa 27 Aug. 1676] fut asser applaudi par M. Newton même [24 Octob. 1676] per M^{r} Newton même. Il est trouvé par apres [anno 1712] qu'une nomme M. Gregory avoit trove just\e/ment la même series que moi. Mais c'est ce qu' j'appris tard. Hic Leibnitius agnoscit se recepisse ab Oldenburgo *de essays* exempla serierum Newtoni, et in eadem Olde inter hæc exempla erat series Gregorij ut in Commercio videre licet. At Newtonus Wallisius & Gregorius junior et Hugenius hanc seriem a Gregorio ad Collinium ab Collinio Oldenburgo ad Leibnitium missam fuisse, per ea tempora ignorarunt.

Contra fidem epistolarum in Commercio editarum scripsit insuper Leibnitius — videntur

Altamen ut accusationem probaret — — — in Commercio pag. 39.

Cæterum Leibnitius in prima sua ad Abbatem — — — . ostendit

Subinde in prima sua ad

D. Leibnitius autem in proxima — — — — Leibnitius.

Altamen post ejus mortem — — — — referri debet. Et hæc Quæstio est utrum Leibnitius sit inventor Methodi & pro differentijs igitur Leibnitianis Newtonus adhibet semper [ex quo usus est hac methodo] adhibuit fluxiones, quemadmodum Honoratus Fabrius motuum progressus Cavallerianæ methodo substituit.

Ad hanc Quæstionem spectat \quod Algorithmus methodi habiatur in Propositione prima Libri de quadraturis &/ quod D. Wallisius Propositionem primam Libri de Quadraturis illustratam \illam cum/ exemplis inveniendi fluxiones primas et secundas edidit annno 1693 in Volumine secundo operum suorum pag. 393. — id est anno 1671.

Eodem spectat quod Iacobus Gregorius scripsit ad Collinium 5 Sept 1671 — — — — — Propositionis primæ libri de Quadraturis.

Propositionem quintam Libri de Quadraturis — de factis dubitatur.

— si fieri potest. Hoc artificium \anno 1669/ Newtono innotuit|s||se| patet per anno 1669 \uti/ patet ex Analysi per series,^{[3]} immo et annis aliquot antequam Mercatoris Logarithmotechnia prodiret (id est anno 1666 aut antea) \teste Barrouꝰ uti/ p{illeg}|a|tet ex E{p}{is} per testimonium ex Epistola ^{✝}^{[4]} Collini {illeg}|i| ad D. Strode. per testimonium Barrovij Propositio \illa/ quinta pendet a quatuor prioribus; ideo Methodus fluxionus|m| quatenus continetur in Propo{illeg}|s|itionibus quin primis libri Principiorum de Quadraturis Newtono innotuit anno 1666 aut antea.

Newtonus incidit in Theorem\emata/ suam de refractionibus, & coloribus lucis ineunte anno 1666. Sub idem tempus invenit etiam methodum|os| serierum et fluxionum. De his omnibus scripsit \Tractatus/ anno 1671 {illeg}|ut| in lucem im mitterentur. Sed anno proximo subortæ per d cum nonnulla communicasset de natura lucis, subortæ statim per diversorum Epistolas objectionibus alijs refertas, crebræ interpellationes, ipsum prorsus a consilio deterruerunt, et effecerunt ut seipsum argeret imprudentiæ \(ut ipsa olim locutus est)/ quod umbram captando, eatenus perdiderat quietem suam, rem prorsus substantialem. Ab his \omnibus igitur/ edendis abstinuit igitur us ad annum 1704 anno 32, & anno 1704 \us ad annum 1704/ et \interea/ aliqua tantum communicavit privatim cum amicis. Librum vero MS de Quadraturis \literis punctatis refertum/ Halleius noster & Ralphsonus anno 1691 manibus suis tractarunt ut alter testatum reliquit & alter adhuc testatur. Eodem libro Newtonus \annis 1685 & 1686/ co{illeg} inveniendis Principijs Philosophiæ se plurimum usum fuisse profitetur. Et quæ in Epistolis \tribus/ Newtoni anno 1676 scriptis \ex eodem/ citantur, indicant ipsum \eundem/ jdem tum in MS extitisse.

Et his præmissis —

^{r} Isaac Newton

Bar.

^{t} Mart ins Street

{illeg}

Kill

$-\frac{15{\mathrm{x}}^{4}}{4096\mathrm{a}}$

$\begin{array}{r}524\\ \overline{)-15}\\ 509\end{array}$

$+\frac{131{\mathrm{x}}^{4}}{1024\mathrm{a}}$

Dominus Brunkerus qu quadravit Hyperbolam per hasce duas series $\frac{1}{1\times 2}+\frac{1}{3\times 4}+\frac{1}{5\times 6}+\frac{1}{7\times 8}+\frac{1}{9\times 10}$ &c et $\frac{1}{2\times 3}+\frac{1}{4\times 5}+\frac{1}{6\times 7}+\frac{1}{8\times 9}$ &c. H|E|t hæ series cædam sunt cum seriebus $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\frac{1}{7}+\frac{1}{8}$ &c et $\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}$ &c ut colligendo summas binorum terminorum patebit. Impressa est autem Brunkeri Quadrad|t|ura Anno 1668 mense Aprili \id/ in Actis Philosophicis Num. 34. Et paulo post impressa est Mercatoris Logarithmotechnia cum eadem Quad\rat/ura promota. Et mox Gregorius Mercatoris Quadraturam eandem Geometrice demonstravit post{illeg}|it|a hac prima Propositione. Si *fuerint quantitates continue proportionales A, B, C D, E, F, &c numero terminorum infinitæ, quarum prima et maxima A; erit A–B ad A ut A ad summam omnium; hoc {s}|e|nim passu demonstratur apud Geometras*. Hæc Gregorius. {Et} Per hanc Propositionem Geometricam prodit quadratura Hyperbolæ abs Divisione Arithmetica. Mercator vero Propositionem probavit per Divisionem \Arithmeticam/. Sed methodum generalem. Quadrandi Curvas per hujusmodi series nec D. Brunker nec Mercator nec Gregorius invener|it|unt. {illeg}utere{illeg} |Gregorius hujusmodi methodum diu quæsivit & vix tandem sub finem anni 1670 invenire potuit. Newtonus hujusmodi methodum aliquam primus i{illeg}is omnium prod|t|ulit. &{c}|

Proposuerat Wallisius in Arithmetica sua Q seriem Curvarum cujus ordinatæ sunt $1,\sqrt{\mathrm{a}\mathrm{x}-\mathrm{x}\mathrm{x}},\mathrm{a}\mathrm{x}-\mathrm{x}\mathrm{x},\overline{)\mathrm{a}\mathrm{x}-\mathrm{x}\mathrm{x}}\sqrt{\mathrm{a}\mathrm{x}-\mathrm{x}\mathrm{x}},\mathrm{a}\mathrm{a}\mathrm{x}\mathrm{x}-{}^{2}\mathrm{a}{\mathrm{x}}^{3}+{\mathrm{x}}^{4}$ &c et areæ $\mathrm{x},\mathrm{*},\frac{\mathrm{a}\mathrm{x}\mathrm{x}}{2}-\frac{{\mathrm{x}}^{3}}{3},\mathrm{*},\frac{\mathrm{a}\mathrm{\alpha}{\mathrm{x}}^{3}}{3}-\frac{2\mathrm{a}{\mathrm{x}}^{4}}{4}+\frac{{\mathrm{x}}^{5}}{5}$, {illeg}|&|c, \existente Abscissa x/ Et quòd si hæc series \posterior/ in locis alternis qua{pravi} \interpolari/ possit habebitur area circuli: Invenit Newtonus methodum interpolandi hanc seriem hanc seriem. Et eadem methodo interpolavit seriem potestatum binomij $1.\mathrm{a}+\mathrm{x}.\mathrm{a}\mathrm{a}+2\mathrm{a}\mathrm{x}+\mathrm{x}\mathrm{x}.{\mathrm{a}}^{3}+3{\mathrm{a}}^{2}\mathrm{x}+3\mathrm{a}{\mathrm{x}}^{2}+{\mathrm{x}}^{3}$ &c. Et ponendo \Ponatur uti/ literam aliquam \puta n/ pro indice binomij, \et per Methodum Newtoni/ {illeg}t /prodibit\ {illeg}x{+} \hujusmodi seriem|s|/ ${\overline{)\mathrm{a}+\mathrm{x}}}^{\mathrm{n}}={\mathrm{a}}^{\mathrm{n}}+{}^{\mathrm{n}}\mathrm{a}^{\overline{)\mathrm{n}-1}}\mathrm{x}+\mathrm{n}\times \overline{)\mathrm{n}-1}\times {\mathrm{a}}^{\overline{)\mathrm{n}-2}}{\mathrm{x}}^{2}+\mathrm{n}\times \overline{)\mathrm{n}-1}\times \overline{)\mathrm{n}-2}\times \mathrm{a}\overline{)\mathrm{n}-3}\times {\mathrm{x}}^{3}+\mathrm{\&c}$. Et per {illeg} \{hujusmodi}/ seriem invenit Regula \inde/ pervenit ad methodum generalem |ubi n potest esse namerus integer, vel fractus vel surdus affirmativus vel negativus, et a et x nomina quæcun designant. Et hæc series eadem est cum Regula generali quam Newtonus in Principio Epistolæ suæ primæ cum Leibnitio communicavit. Sic Newtonus pervenit ad methodum generalem| quadrandi Curvas quarum ordit{illeg}|n|atæ sunt binomiorum potestate dignitates quæcun, sive nomina qu sint quantitates simplices vel \sive/ compositæ, sive et indices dignitatum sint numeri inti{illeg}|eg|ri vel {illeg}|f|racti vel surdi et affirmativi vel negativi, ut ex epistola|i|s ejus ad {D} Oldenburgum cum {illeg} Leibnitio communicandis intelligo. Deinde animadvertit idem per divisiones et extractiones radicum fieri & methodum promovit ad radices æquationum affectarum sive æquationes illæ sint affectæ solas quantitates \fluentes/ involvant sive etiam fluxiones, |Et methodum etiam \primus invenit/ assumendi terminos serierum et eosdem per collationem terminorum homogeniorum determinandi. {illeg} Hæc {illeg} invenerat antequam| ut ex ijsdem epistolis et alijs ejus scriptis patet.

Slusius {illeg} methodum ducendi tangentes ab his \ex tribus/ Lemmatibus conse/qu{i}\\qui/ derixesse dixit, & Lemmata impressa sunt in Actis Philosophicis. At quomodo methodus ab istis Lemmatibus derivetur \consequatur/ nondum s{illeg}d{illeg} \constat./ Annon Slusius Anglis i{illeg}iserit methodum illam aliunde habuerit hau{ser} habuerit.

Leibnitius vero ne unam quidem seriem earum quas sibi arrogorare conatus est primus invenit: {ala} Newtonus Methodum perveniendi ad has Series anno 1676 a Newtono accepit, acceptæ nihil unquam addit|d|it. Newtonus methodum et primus invenit et perfec|i|tam reddidit.

Quinetiam Regula jam descripta methodum fluxionum quam Leibnitius differentialem vocat in se involvit. |[|Si \terminus primus binomij sit quantitas fluens &/ terminus secundus Binomij sit mo|mo|mentum termini primi, terminus \primus/ seriei erit momentum dignitas {illeg}{um} primi seu quantitas \quantitatis/ fluens|t|is, terminus secundus erit momentum \primum/ dignitatis, tertius quartus quintus & reliqui {illeg} datas habebunt rationes ad ter momentorum secundum tertium quartum & reliqua in infinitum.|]| Si momentum dignitatis quantitatis fluentis desideretur s] Siquidem Seriei cujuscun terminus secundus sit momentum termini primi et tertius quartus quintus et reliqui sint ad momenta termini primi in datis rationibus, si modo Binomij nomen secundum sit momentum primi nominis primi. Verbi gratia sit o momentum fluentis x et quæratur momentum dignitatis ${\mathrm{x}}^{\mathrm{n}}$, id est quæratur differentia inter ${\mathrm{x}}^{\mathrm{n}}$ & ${\overline{)\mathrm{x}+\mathrm{o}}}^{\mathrm{n}}$ et resolvendo ${\overline{)\mathrm{x}+\mathrm{o}}}^{\mathrm{n}}$ in seriem ${\mathrm{x}}^{\mathrm{n}}+\mathrm{n}\mathrm{o}{\mathrm{x}}^{\mathrm{n}-1}+\mathrm{\&c}$, secundus seriei terminus |$\mathrm{n}\mathrm{o}{\mathrm{x}}^{\mathrm{n}-1}$| erit differentia illa quæsita. {Pr} Sit jam p momentum quantitatis fluentis y et quæratur mom{illeg}|e|ntum quantitatis fluentis ${\mathrm{x}}^{\mathrm{m}}{\mathrm{y}}^{\mathrm{n}}$ {illeg} et resolve id est quæratur differentia inter ${\mathrm{x}}^{\mathrm{m}}{\mathrm{y}}^{\mathrm{n}}$ et ${\overline{)\mathrm{x}+\mathrm{o}}}^{\mathrm{m}}{\overline{)\mathrm{y}+\mathrm{p}}}^{\mathrm{n}}$ et resolvendo binomia in series et multiplicando series in se mutuo prodibit series cujus secundus terminus est {illeg} diff $\mathrm{m}\mathrm{o}{\mathrm{x}}^{\mathrm{m}-1}\times \mathrm{n}\mathrm{p}{\mathrm{y}}^{1-1}$ est differentia quæsita. Et hæc est methodi momentorum et fluxionum quam Leibnitius differentialem vocat Demonstratio omnium brevissima & maxime naturalis. Communicavit igitur Newtonus cum Leibnitio \per Epistolam suam primam/ fundamentum \verissimum \verum// methodi differentialis a quo methodus illa sponte fluit et \per quod/ brevissime et optime demonstratur. [Sed et in Epistola|i|s su{a} secunda \alijs/ methodum illam tantum non demonstra \se habre {sic} dixit &/ verbis disertis tantum non descripsit: Leibnitius \autem/ methodum illam nondum demonstravit neq \nunquam/ melius demonstrare potest. Sed et Newtonus methodum {intram} serierum & methodum fluxionum s|o|b maximam affirmitatem \semper/ inter se s|c|onjuxit et simul tractavit. Et Leibnitius affirmitatem illam agnovit \in Act Lips. Anno 1693 mense Apr./ ubi series infinitas a methodo differentiali derivate cot|n|atus est. Scribit ibi Leibnitius in hæc verba. *Cum antea series infinitæ fuerint quæsitæ cum primo inventore Nicolao Mercatore Holsato per divisiones et cum summo Geometra Isaaco Newtono per extractiones; visum mihi fuit posse easad eas perveniri per commodius et universalius per suppositionem ipsius seriei quæsitæ tanquam inventæ ita ut terminorum coefficientes ex successu definirentur*. Rectius dixisset seriem infinitam continue proportionalium ejus summam veteribus innotuisse, D. Brunkerū p esse primum inventorem Quadraturæ per seriem infinitam. Mercatorem propositionem veterum per sum de summa seriei infinitæ continue proportionalium demonstrasse per divisionem et inde derivasse & ampliasse Quadraturam Hyperbolæ inventam a{illeg} Brun per seriem infinitam a Brunkero inventam. Newtonum regulam invenisse generalem reducendi binomia & binomialia omnia ad series infinitas & per hanc Regulam & extractiones radicum affectarum method{illeg}|u|m generalem tractandi curvas trancendentes Analytice & quadrandi curv{illeg}s|ilin||eas| omnes & similia peragendi primum \omnium/ invenisse. Rectius \Nullibi/ agnovis|t|se Leibnit{u}s|iu|s \{illeg}/ in Actis Lipsiensibus se series qua series \aliquot/ infinitas ab Oldenburgo primum accepi Anno 1675|3| accepisse & Regulam Nuetoni reducendi binomia in series anno proximo a Newtow {acce}pisse didicisse eas præsertim quæ ad circulum spectant. Nullibi agnovit se Regulam Newtoni accepisse reducendi binomia in series infinitas quæ prim{illeg} fuit methodum generalem aperiat quadrandi curvilineas & similia peragendi. Nullibi agnovit Se{illeg} se vidisse quasdam Newtoni epistolas ex quibus dided|ix|it Newtonum vel Methodum differentialem vel huic methodo simillimam primum invenisse, Nullibi agn et Sententia aliqua literis transpositis designata celasse. Nullibi agnovit Newtonum in Epistola 24 Octob. 1676 data signifi{illeg}|c|asse Inversa Problemat{illeg} de Tangentibus {Pro} Problemata esse in potestate alia illis difficiliora: ad quæ solvenda se usum esse duplici methodo una concinniori alteræ universaliori. et utram sententia literis transpositis designata celasse. Sed his silentio traditis Leibnitius L|C|e jam visum esse jam scribit \quod/ *visum sibi esse fuit posse series Newtonianas perveniri commodius et universalius per suppositionem ipsius seriei quæsitæ tanquam inventæ ita ut terminorum cofficientes {sic} ex successu definiantur*.

Quinetiam sub finem epistolæ secundæ sub finem 24 Octob 1676 datæ Newtonus {Le} inversa de tangentibus Problemat{illeg}|a| esse in potestate alia illis difficiliora ad quæ solvenda duplici usus est methodo et {illeg}|u|na concininniore altera generalione {sic} & methodum utram tentatijs deprimit hisce designavit sententijs. Vna methodus consistit in extractione fluentis quantitatis ex æquatione simul in{sic} fluxionem ejus. Altera tantuum in assumptione seriei pro quanti vente tate qualibet incognita ex qua cætera commode derivari possunt; et in collatione terminorum homologorum æquationis resultantis ad eruendos terminos assumptæ seriei.

Hæc Regula verbis alijs ita sonat. Ex Binomij F dignitate Fluentem e{illeg}re

Hanc seriem anno 1669 Newtono innotuisse patet ex Analysi supra impressa pag 19 lin 19, 20

1 Methodus tam fluxionum quam serierum {hanc} infirmitarum fundatur in operationibus quatuor vel quin \generab|l|libus ex Analysi supra impressa pag 19 lin 19, 20/ quarum hæc est prima. Et per hanc operationem Fluens ex æquatione quacun non affecta extrahitur s eruitur.

2 Operatio secunda genere|a|lis qua fluens ex æquatione non affecta extrahitur.

|p. 636. l. ult.| Id est, Data æquatione fluent{illeg} quo{illeg}|t|cun fluentes \quantitates/ involvente invenire fluxiones et e{illeg}t{illeg} vice versa. Solvitur vero sumendo secundos ter{illeg} terminos serierum \pro momentis primorum./ B Sit æquatio {illeg} ${\mathrm{x}}^{\mathrm{n}}=\mathrm{y}$. [Fluat x uniformiter & sit ejus momentum $\stackrel{.}{\mathrm{x}}\mathrm{o}$ \fluat etiam y et sit ejus momentum $\stackrel{.}{\mathrm{y}}\mathrm{o}$/. Resolvatur binomium ${\overline{)\mathrm{a}\mathrm{x}+\mathrm{a}\stackrel{.}{\mathrm{x}}\mathrm{o}}}^{\mathrm{n}}$^{{illeg}} in seriem et series $\mathrm{a}{\mathrm{x}}^{\mathrm{n}}+{}^{\mathrm{n}}\mathrm{a}\stackrel{.}{\mathrm{x}}\mathrm{o}{\mathrm{x}}^{\mathrm{n}-1}+\mathrm{\&c}$. Et secundas seriei t{illeg} $=\mathrm{y}+\stackrel{.}{\mathrm{y}}\mathrm{o}$ et {illeg}.] Solvitur vero per operationem primam in Epistola superiore descriptam \sumendo secundos serie{illeg}rum terminos pro flu{illeg} momentis primorum/. Sit æquatio o{illeg} ${\mathrm{x}}^{\mathrm{n}}=\mathrm{y}$. Flua\n/t x \et y/ uniformiter & sit ejus fluxio \earum/ momentum|ans| $\mathrm{o}\stackrel{.}{\mathrm{x}}$ & ipsius y momentum $\mathrm{o}\stackrel{.}{\mathrm{y}}$. Res{illeg}lvatur binomium in serie{illeg} \et fluendi velocitates. $\stackrel{.}{\mathrm{x}}$ et $\stackrel{.}{\mathrm{y}}$/ Et erit ${\overline{)\mathrm{x}+\mathrm{o}\stackrel{.}{\mathrm{x}}}}^{\mathrm{n}}=\mathrm{y}+\mathrm{o}\stackrel{.}{\mathrm{y}}$. Resolvatur binomium in seriem et ${\mathrm{x}}^{\mathrm{n}}+\mathrm{n}\mathrm{o}\stackrel{.}{\mathrm{x}}{\mathrm{x}}^{\mathrm{n}-1}+\mathrm{\&c}=\mathrm{y}+\mathrm{o}\stackrel{.}{\mathrm{y}}$ et ablatis æqualibus ${\mathrm{x}}^{\mathrm{n}}$ et y manebunt æqualia $\mathrm{n}\mathrm{o}\stackrel{.}{\mathrm{x}}{\mathrm{x}}^{\mathrm{n}-1}+\mathrm{\&c}=\mathrm{y}+\mathrm{o}\stackrel{.}{\mathrm{y}}$ et ablatis æqualibus ${\mathrm{x}}^{\mathrm{n}}$ et y manebunt æqualia $\mathrm{n}\mathrm{o}\stackrel{.}{\mathrm{x}}{\mathrm{x}}^{\mathrm{n}-1}$ et{illeg}{ij} et $\mathrm{o}\stackrel{.}{\mathrm{y}}$, quibus per o divisis fit $\mathrm{n}\stackrel{.}{\mathrm{x}}{\mathrm{x}}^{\mathrm{n}-1}=\stackrel{.}{\mathrm{y}}$. Est igitur $\mathrm{n}\stackrel{.}{\mathrm{x}}{\mathrm{x}}^{\mathrm{n}-1}$ Et similiter Sit æquatio ${\mathrm{x}}^{\mathrm{m}}{\mathrm{z}}^{\mathrm{n}}=\stackrel{.}{\mathrm{y}}$ et simili operatione prodibit $\mathrm{m}\mathrm{x}{\mathrm{x}}^{\mathrm{m}-1}\stackrel{.}{\mathrm{z}}{\mathrm{z}}^{\mathrm{n}-1}=\stackrel{.}{\mathrm{y}}$. Et idem fit in singulis terminu|is|{illeg} æquationum compositarum. Et hoc et|s|t fundamentum verum methodi fluxionum, Newtonus inven{illeg} et {d}{illeg} fundamentum verum et demonstratio optima brevissima et maxime generalis et Newtono ab initio inotuit ut ex ejus Analysi per æquationes infinitas (pag 3 Reg. 1. & pag 19) manifestum est. Newtonus \communicavit/ invenit methodum serier|s|um \infinita|s|{illeg}/ Leibnitius invenit secundos terminos serierum Newtonus communicavit cum Leibnitio methodum suam perveniendi ad series infinitas, Leibnitius secundos serierum terminos invenit.

p {illeg}|6|34 *a* Tempus hy{illeg}|b|erno inter annos 1664 et 1665

p. 634 \l. 11/ *a* Regula prima.

l. 30. *b* Regula secunda.

p. 636 l. 19 Analysis per æquationes infinitas supra impressa

l 45. Ad verba: Div{illeg}|e|rsa ratione in eam incidimus, nota: [Imo Slusius per Lemmata suæ quæ in Epistola quæ in Actis Philosophicis ut fundamentum methodi suæ \hujus/ impressa sunt in eam incidere non potuit, sed methodum aliunde habuit.] Slusius i{illeg} pr{illeg}bus methodum Riccio sibi minime arrogavit sed Riccio cu{e}|j|dam tr{illeg}|{i}|buit & \{illeg} methodum a se abjudicaret/ Lemmata posuit \ponendo/ a quibus methodus derivari non potest. Qu{illeg} Extant Lemmata in Actis Philosophicis /Newtonus methodum primus communicavit & verum ejus fundamentum solus posuit. Qui aliter sentit doceat quomodo Leibnitius methodus a Lemmatibus illis derivari possit.\

p. 635 l. 1. Exemplum \generale/ quadraturæ curvarum per methodum fluxionum. {illeg}

p 635 l. 1 Exemplum generale quadrandi curvas per methodum fluxionem & series infinitas conjunctim. Vid. pag. 18. lin 31.

p. 639. Pag 639 l. 1. Ex his patit{illeg} Series Newtoni{illeg}|a||na| exeorum quantitatem indefinitam (id est fluentem) unam vel plures in se semper involv{illeg}|unt|, a|i|deo ad methodum fluxionum perinent.

Ib. lin. 20. Ex his p{illeg}|a|tet \Propositiones de Newtoni de/ Quadraturam Curvarum ante annum 1676 inventas fuisse.

P. 640. l. 7. \NB/ D. Brunker Hyperbolam per hanc seriem minus {illeg} \$\frac{1}{2}+\frac{1}{12}+\frac{1}{30}+\frac{1}{56}+\mathrm{\&c}$ id est \seu/ per hanc/ $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}$ (conjunctis binis terminis) primus quadravit. \Mercator quadraturam hanc ampliavit./ Gregorius circulum per hanc seriem $1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}$ &c quadravit. Leibnitius hanc Quadraturā sibi ab Oldenburgo accepit|.| & sibi arrogavit. Newtonus quadravit Circulum per hanc seriem $1+\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\frac{1}{9}+\frac{1}{11}-\mathrm{\&c}$ quadravit seu $1+\frac{1}{15}-\frac{1}{63}+\frac{1}{143}$ &c quadravit

P. 644. l. 17. Id est, Vna methodus ...... assumptæ seriei. Prior methodus est Operatio tertia vel Regula tertia generalis qua methodus \Analysis per/ serier|s|um \infinitas/ et fluxionē|e|s s{illeg} perficitur. Posteri{illeg}|o|r est Regula quarta et ultima. \Duæ/ Primæ Regulæ in principia \epistola/ priora epistol{illeg} {f}{illeg}itæ f{illeg} \{illeg} traduntur/ et \prima quidem/ dat quantitatem fluentes|m| ex equationibus non affectis, secunda dat Fluentes|m| ex æquationibus affectis \fluxionem ejus non involventibus/, tertia dat fluentes|m| ex \æqu./ affectis fluxionem ejus involventibus. Quarta dat fluentes ex omnibus. Hi{illeg}|a|sce quatuor operationi|e|bus|s| pervenitur|d|i ad fluentes Newtonus ante annum 1676 invenerat. Ab eo tempore his nihil additum est. Leibnitius ex tempore de his cogitare cœpit Et Newtonum esse harum omnium \Regularum/ inventorem primum nemo dubitat. Et in Regulis hisce quatuor methodus tota comprehenditur \moment{illeg} fluentium et momentorum/ tam in Æquationibus infinitis quam in finitis comprehenditur, cum inventio fluxibus ex momen per Reg. 1 \cum momenta fluentium/ facile inveniantur \per {R}eg 1/ ut Supra expositum est.

## In Leibnitij Epistolam secundam.

Pag. 648. lin 20. E{illeg} {illeg}|H|is verbis patet Leibnitium jam primum calculum differentialem cum amicus \jam primum/ communicare cæpisse et methodum de d differentias didicisse ex secundis termi potes ponendo secund fluentem cum ejus momento pro binomio habuisse secundos terminos dignitatum binomij pro momentis dig{illeg} (vel (ut ipse loquitur) differentijs) dignitatum, perinde ut Newtonus antea fecerat. Vide pag 19.

Inveniri possunt plures hujusmodi series assumendo series|m| arearum pro lubitu, {illeg} \et/ inv|d|eniendo computando series|m| Ordinatarum & inter methodo Newtoniana sub initio hujus Epistolæ exposita interpolando seriem utram; vel assumendo seriem quamlibet ordinatarum ad Curvas quæ per finitas \æquationes/ quadrari possunt et interpolando seriem arearum. Et

Methodus prior est \tertia in his Epistolis tradita posita/ Regula tertia generalis \(in his Epistolis tradita)/ extrahendi quantitates \indeterminatas &/ fluentes & et posterior est Regula quarta. \Quantitas fluens {illeg}{st}{illeg}bilis/ Per Regulam primam fluent|s|es \et equs{illeg} momentum/ ex æquationibus non affectis, per secundam ex æquationibus affectis \fluxionem non involventibus/, per quartam ex tertiam ex æquationibus \affectis/ {illeg} fluxionem simul involventibus, per quartam ex conditionibus Problematis eructur et extrahitur. Et momentum fluentis \invenitur {illeg}/ per Regulam primam invenitur ut supra, [{illeg} vel etiam dicendo quod si fluens sit ut area curva momentum ejus et fluxu erit ut Ordinata] \vel etiam per Ordinatam Curvæ cujus area cujus Abscissa uniformiter {illeg} est ut fluens?/ Et |Et| sic Newtonus \{illeg}/ Regulis hisce quatuor totum flu Analysi \suam/ totam per fluentes & earum momenta \id/ {illeg} æquatio{n} et \Analysin id/ in æquationibus tam infinitis quam finitis ha{illeg}t in Epistolis suis Epistolis suis complexus Est cum Leibnitio communicandis \jam/ complexus \fuit/ est. & methodum fluxionum tantum Newtonum \Et eum/ esse harum quatuor Regularum inventorem primum nemo dubitat. Et [Analysin \eandem/ his Regulis complexam nemo ampliorem {illeg}d{illeg}d reddidit alij aliter exponsu explicuerunt, nemo ampliorem reddidit, Leibnitius ex literis jam primum \communicabat/ ded{illeg}t{illeg} anno sequente \suis verbis/ cum amicis ca{illeg} Leibnitius anno sequente \1677 sequente/ proferre cœpit \&/ nondum totam protulit, Newtonus Anno 1669 cum Collin Collinio \Collinius deinceps cum amicis/ communicabat ut supra.

p. 64{illeg}|5| lin. 25|9|. Ad verba ^{a} *aliter se habere solet*. *a* Id est, ad s{illeg}|o|lutionem Problematis methodus generalis jam descriptæ|a| sæpe requiritur.

p. 648 lin 19 Ad verba ^{a} in posterum. *a* Cœpit agitur Leibnitius methodum suam differentialem jam primum cum amicis communicare

p 649 lin 24. Ad verba celare voluit Pluribus argumentis Leibnitius se jam methodum differentialem Methodum

p 648 lin 24|8|. I|A|d verba. *Idem est de cæteris potentijs*. Posita serie potentiarum binomij Leib Ex secundis terminis potentiarum binomij Leibnitius didicit hic colligit differentias. suas. Newtonus {illeg}|qu|ando ex secundis terminis potentiarum quand{illeg}|| ex secundo termino seriei \{g}{illeg}/ potentias omnes involventis momenta sua colligeb|r|at. Eadem est utrius computandi ratio ut conferenti computationem Leibnitianā hic positam cum computatione Newtonim|a|na s|i|n Analysi sua pag. superius impressa pag 19, statim patebit.

p 649 \lin 24/ Ad verba *celare voluit* notetur. Leibnitius Multis argument{illeg}s \docet Leibnitius/ se in methodum incidisse quæ cum Newtoniana con per omnia congruit.

Analysin per fluentes et earum momenta Newtonus in æquationibus tam infinitis quam finitis Newtonus in his Epistolis ad Regulas quatuor reduxit. \P{illeg}r/ prima|m| est trac|h|tio\hitur/ fluent|s|is \& ejus momenta {illeg}/ ex Binomi{o}|j||s|ū {sic} at adeo ex æquationibum|s| quibuscun non affectis \et momentum fluentis. simul prodit/. Per secundam extrahitur fluens ex æquationibus affectis fluxionem non in{illeg}|v|olventibus, Per tertiam extrahitur fluens ex æquationibus affectis fluxionem simul involventibus. Per quartam extrahit fluens ex conditionibus Problematis. Regulæ duæ primæ in principo|i|o Epistolæ superioris {illeg} duæ {illeg} \anno 1665 inventæ sunt p{illeg}s{tis} {illeg}gruentis/ apertæ|è| tradantur \tradunt{illeg}|u|r & anno 1665 inventæ sunt (pag/: duæ ultimæ in fine hujus celabantur \hic/ occult{illeg}|a||ntur| tradebantur. Harum Regularum Newtonum esse primum inventorem nemo dubitat. Analysin eandem alij aliter explicuerunt, nemo ampliorem reddidit. Collinius anno \1669/ a D Barro accepit

In the same Letter of 27 Aug. 1676 after M^{r} Leibnitz had done with had described his Quadrature of the circle & Hyperbola equilateral Hyperbola, he added: Vicissim ex seriebus Regressuum pro Hyperba {sic} hanc inveni — — — — — quæ in Newtoni Epistola expressa est; scilicet erit $\mathrm{n}=\frac{\mathrm{l}}{1}+\frac{{\mathrm{l}}^{2}}{1\times 2}+\frac{{\mathrm{l}}^{3}}{1\times 2\times 3}+\frac{{\mathrm{l}}^{4}}{1\times 2\times 3\times 4}+\mathrm{\&c}$. .... Quod regressum ex arcubus attinet incideram ego directe in Regulam, quæ ex dato arcu{illeg} Sinum complementi exhibet. Nempe sinus Complementi $=1-\frac{{\mathrm{a}}^{2}}{1\times 3}+\frac{{\mathrm{a}}^{4}}{1\times 2\times 3\times 4}$ &c. Sed postea quo deprehendi ex ea illam nobis communicatam pro inveniendo Sinu recto qui est $\frac{\mathrm{a}}{1}-\frac{{\mathrm{a}}^{3}}{1\times 2\times 3}+\frac{{\mathrm{a}}^{5}}{1\times 2\times 3\times 4\times 5}$ &c posse demonstrari. |And yet in the same Letter he desired M^{r} Newton to explain further to him the extraction of affected roots & the method of regressiō by w^{ch} alone these series were to be found.| [By this one would think that M^{r} Leibnitz had found these four series or at least the three first of them by the inverse method of series or method of Regressions, \sometime/ before he received that method in M^{r} Newtons Letter, \& that he now understood that method very well, & yet/ [& yet he wanted both the invers & direct method of series when he wrote his Letter of 12 May 1676 desiring M^{r} Oldenburg to procure him the Demonstration of the direct & invers series expressing the relation between the Arc & the Sine, & by the Demonstration meaning the direct & inverse methods of finding those series And \when at his request/ M^{r} Newton in his Letter of 13 Iune 1676 had sent him those methos|d|s, he did not understand those methods but w{illeg} in his one would think that in point of candour he should not have been so forward in turning them against the Author, as if the Inventor of the method did not know how to make use of it. And much more would one think that he understood the method of regressions by which{illeg} he pretended to have found the inverse method of series the three first of those four series. For when had \And yet/ in the same Letter of 27 Aug 1676 aft{e} [when he had put in his claim to that Method the three first of those series as found by that method, he forgot himself & after a few lines] he desired M^{r} Newton to explain it \further/ to him.] His words are. *Sed desderaverim {sic} ut {illeg} Newtonus Clarissimus Newtonus nonnulla quo amplius explicet: ut Originem Theorematis quod initio posuit,|:| Item modum quo quantitates p, q, r in suis operationibus invenit: Ac deni quomodo in Methodo Regressuum se gerat ut cum ex Logarithmo quærit numerum* \Ne enim explicat quomodo id ex methodo sua derivetur. He pretended to two series for the/ |Two of the four series w^{ch} he pretended to were for finding the Nūber from y^{e} Log. & yet in the same Letter he desired M^{r} N. to tell him y^{e} method of finding them.|

When M^{r} Newton \had/ received this Letter, he wrote back that all the said four Series had been communicated to M^{r} by him to M^{r} Leibnitz, the two first being one & the same series in different circumstance, & the third being the excess above \of/ the Radius above the versed sine \which was communicated/. Whereupon M^{r} Leibnitz \has/ desisted from his claim.

And|He| w|W|hen M^{r} Newton \in the same Letter/ had explained what M^{r} Leibnitz desired he added two series for faciliating the method of Regressions And M^{r} Leibnitz in his Answer dated 21 Iunij 1677 desired that M^{r} Newton would send him in the continuation of those two series against him {illeg} But a month \three weeks/ after in a Letter dated 12^{th} Iuly 1677 he added: Relectis Newtoni literis video id facile non tantum ex ejus extractionibus derivari sed et altera illa methodo sub finem literarum ejus exposita inveniri qua me quo aliquando usum in veteribus meis schedis reperi{illeg}o. Sed cum in exemplo quod forte in manus meas inciderat sumpseram, nihil prodijsset elegans solita impatientia eam porro adhibere neglexisse. If That is, had he had found the method of Series so long ago, that before the year 1676 he had forgot the inverse method before the year 1676 when the two series shewed him by Mohr would have made him understand the use of it it had then remembred it & before the year 1675 when the several \eight/ series sent him by M^{r} conteined many instances of the use of it if he had then remembred it & before the year 1674 when the series \Theorem/ for finding the arc by the sin whose sine given would give afforded him an instance of the usefulness of it if he had then remembred it. And before the year 1673 when the series for finding the Arc whose tangent is given would have afforded him an instance of the usefulness of it if he had then remembred it. [He had \therefore/ invented the method of series so early that he had forgot one half of it before he left L (the method of Regression) before he left London.] He could not find the inverse method before he had the direct method. M^{r} Newton {illeg}|a|t his request had sent him both methods. As soon as he had them & understood them, he replied that he had found them so long ago as to have forgot that he had \ever/ found the inverse method. In the year 1668 M^{r} O In the year 1676 M^{r} Oldenburg \Mohr/ had given him an elegant Instance of this method of Regressions \as above/. In the year before that M^{r} Oldenburg had sent him three of or elegant instances of it. In the year before that he \pretended to have/ had a Theoreme for finding the Arc whose sine was given & wanted a Theoreme for finding the sine whose Arc was given w^{ch} would have been \an/ elegant instance. In y^{e} year 1668 Vicount Brounker & the M^{r} Mercator published a Theoreme for finding the Logarithm whose Number was given, M^{r} M{illeg} Leibnitz & the \inverse/ Theorem for finding the Number from the Logarithm given would have been an elegant instance. \This Theoreme he wanted & yet he threw away the method method of finding it/ He wanted it his Theorem so much as to write three lines to M^{r} Oldenburg for the method of finding such Theoremes \the method of finding this & such like Theorem/, & the last time & the last desired expresly that M^{r} Newton would explain to him quomodo in methodo Regressuum se gerat ut cum ex Logarithmo quærit Numerum. But as soon as he understood it \the method/, it was his own: for he {illeg} used it long ago \then/ found in his old papers that he had used it long ago, but had neglected it for want of an elegant instance. And if he had found the inverse method of series so long ago as to have forgot that he had found it, much more had he found the direct method long ago, A man may forget things & find them again in his old papers, but he {illeg}t to in cases of right such pretences are not to be regarded w^{th}out proof. M^{r} Newton deserved to have been treated w^{th} more candour & gratitude.

In the Acta Eruditorum Mensis Ianuarij 1689 pag 37 {illeg} M^{r} Leibnitz making mention of the method of Series, w^{ch} Mercator represents that Mercator found them by division & Newton enlarged the method by extractions of roots both pure & affected & then adds. A me ut obiter hic dicam, methodo serierum promovendæ, præter transformationem figurarum irrationalium linearum in rations|a|les symmetras (voco autem rationales quarum Ordinatæ semper ex abscissis haberi possunt in numeris rationalibus) excogitata est ratio pro curvis transcendenter datis, ubi ne extractio quidem locum habet. Assumo enim seriem arbitrariam, eam ex legibus problematis tractando obtineo ejus coefficientes. This Transformation of Figures is no part of the method of series. It's only a Lemma for \towards/ doing that \sometimes/ by the division of Wallis, (not always but in a very few cases,) w^{ch} might may be always \always/ done more readily by the extraction of roots \without it/ The \other/ method of assuming an arbitrary series is M^{r} Newton's. In his Letter of 1676 he set it down in this sentence. A{tl} Altera [methodus consistit] tantum in assumptione seriei pro quantitate qualibet incognita ex qua cætera commode derivari possunt et in collatione terminorum seriei resultantis homologorum æquationis resultantis ad eruendos terminos assumptæ seriei. So then M^{r} Leibnits has no right to the invention of any part of the method of Series.

Yet there is a sort of series the invention of w^{ch} is due to him, & those he mentions in y^{e} end of his Letter dated 3 Feb. 1673. Modum habeo, saith he, summam inveniendi seriei fractionum in infinitum decrescentium, quarum Numerator unitas, nominatores vero numeri Triangulares, aut Pyramidales, aut Triangulo-triangulares &c. These are found in the following manner {illeg} $\begin{array}{r}\phantom{+}1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\\ \overline{)-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}-\frac{1}{5}-\frac{1}{6}}\\ \phantom{+}\frac{1}{2}+\frac{1}{6}-\frac{1}{12}+\frac{1}{20}+\frac{1}{30}+\mathrm{\&c}=1.\end{array}$ $\begin{array}{r}\phantom{+}\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\\ -\frac{2}{2}-\frac{2}{3}-\frac{2}{4}-\frac{5}{5}-\frac{2}{6}\\ \overline{)+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}}\\ \phantom{+}\frac{1}{3}+\frac{1}{12}-\frac{1}{30}+\frac{1}{60}+\mathrm{\&c}=\frac{1}{2}\end{array}$ And these are found in the following manner.

De serie \From the series/ $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}$ &c aufer \{os}/ terminos omnes post primum et manebit \take all the terms but the first & there will remain/ $1=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}+\frac{1}{42}+\mathrm{\&c}$. De hac serie aufer omnes terminos post primum & manebit \From this series take all the terms but the first & there will remain/ $\frac{1}{2}=$$\frac{4}{12}+\frac{6}{72}+\frac{10}{600}+\frac{12}{1280}+\mathrm{\&c}$ seu =$\frac{1}{3}+\frac{1}{12}+\frac{1}{30}+\frac{1}{60}+\frac{1}{105}$ & sic deinceps \so on/. And in generall, from even \any/ regular series by subducting all the terms but the first or two first or three or four first will give a new series equal to y^{e} first or two first or three or four first terms.

This method seems to consist in subducting all the terms of \from/ any regular series exce all the terms except the first or two first, or three or four first. From the series $\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}$, &c subduct all the terms but the first & there will remain $1=\frac{1}{1,2}+\frac{1}{2,3}+\frac{1}{3,4}+\frac{1}{4,5},+\mathrm{\&c}$ \$=\frac{1}{1}-\frac{1}{2}+\frac{1}{3}-$/ And from this series take all the terms but the first & there will remain $\frac{1}{2}=\frac{2}{1,2,3}+\frac{2}{2,3,4}+\frac{2}{3,4,5}+\frac{2}{4,5,6}$, &c. And from the first series take all y^{e} terms but the two first & there will remain $\frac{3}{2}=\frac{2}{1,3}+\frac{2}{2,4}+\frac{2}{3,5}+\frac{2}{4,6}$ &c And from this series $\frac{1}{1}+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\frac{1}{9}$, &c take all the terms but the first & there will remain $1=\frac{2}{1,3}+\frac{2}{3,5}+\frac{2}{5,7}+\frac{2}{7,9}$, &c.

He pretended to two series for finding the number whose Logarithm was given, & yet wāted the method of finding those series. And when \M^{r}/ Newton {illeg}|h|ad further explained it {illeg} he still desired him t|a|gain to explain it further & t|a|t length when he understood it he wrote back that he had it long ago as he found in his old Papers, but had neglected it for want of an elegant example.

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By his \In the year 168|7|5 when he wrote the letter {sic}/ letter to M^{r} Collins printed by D^{r} Wallis {illeg} I gather that he had \did/ not then \use/ the differential method as he would have done had he then known it.

It does not appear that he used {illeg} discovered his knowleding any thing of it before the receipt of S^{r} Is. Newtons two Letters from M^{r} Oldenburg A.C. 1676

S^{r} Isaac then

In those letters letters S^{r} Isaac discovered that he knew it & had written a treatise of it five years before. He represented that it was a method w^{ch} did not stick at fractions & surds, w^{ch} extended to the Problemes of Quadratures direct & inverse t{illeg}|a|ngēts & others more difficult & gave an instance of its great extent in the problemes of Quadratures setting \down/ a general series which ran in infinitum & gave the quadrature in an infinite series when it was not expressible by a finite æquation but when it was expressible by a finite equation, brake off & gave that equation & w{illeg} wrote in the same letter. that th{illeg} he other ser that this was the first of Theoreme of those w^{ch} he there spake of \& therefore he had at that time other Theoremes of a higher kind derived from that method./ This Theoreme is the fift in his book of Quadratures & thence it is certain that the is for squaring the sixt is of the same kind for & |d|oth{er} the same thing in trinonomials {sic} w^{ch} the fift doth in binomials & the third & fourth are introductory to the fift & sixt & therefore these four Propositions were then known to him as the result of the method of fluxions: & so were the s{illeg}d ninth & tenth with the Tables of quadratures set down in a|t|he Scholium of the 10^{th}. For he tells in the second of the two letters above mentioned he \did/ set{illeg} down the Ordinates of these curved|s| & t{illeg}s gave \saith said/ that he had \tables of all/ their Quadratures: at w^{ch} Tables could not have been computed w^{th}out the knowledg of the 9^{th} & 10^{th} Propositions. In the same letters he sets down ænigmatically this Probleme Data æquatione fluentes quotcun quantitates in\vol/vente invenire fluxiones & contra vice versa. And the first part of this Probleme is the first Problem in the book of quadratures & the second Probleme is a very easy \& obvious/ corollary of the first Probleme or rather it is nothing else then the first pro Probleme applied to the case of Quadratures & therefor it must be allowed that S^{r} I. N. had before {illeg} the Propositions in the book of Quadratures were known to S^{r} Isaac before the writing of the said two letters to M^{r} Oldenburg & by consequence that he had \then/ not only invented the method of fluxions but also carried it to a higher pitch then M^{r} Leibnitz was able to carry \it under the name of/ the differential method during the 30 years w^{ch} followed before y^{e} publishing of the book of Quadratures notwithstanding the light w^{ch} M^{r} Newton gave him into it \this method/ |it| by those two letters. And by all these considerations it is manifest that M^{r} Newton was the first inventor of the method. And the same thing is further manifest by his little Tract de Analysi per series numero terminorum infinitas \written in the year 1669 \✝/ ✝ & mentioned in the \M^{r} Newtons/ second Letter of M^{r} L to M^{r} Oldenburg & lately published by M^{r} Iones. & by the letters of D^{r} Barrow & others/ found amongs the papers of M^{r} Colling|s| & p{illeg} published w^{th} that book \Tract/ by M^{r} Iones.

S^{r} Isaac was therefore the first inventor of this method, & the next question is whether M^{r} Leibnit & how far M^{r} Leibnit{illeg}|s| was a collateral inventor, or had it from S^{r} Isaac. And that he might have it from S^{r} Isaac is {illeg} [And if he had it not from S^{r} Isaac by means of the lett correspondence w^{ch} he kept with M^{r} Collins yet he before the writing of the \M^{r} Newtons/ two letters above mentioned to M^{r} C receipt of M^{r} Newton's two letters above mentioned yet a man of his parts might easily have it from those two letters.] For the infinite series described in those letters are nothin{illeg}|g| else then the method {illeg} the summ of the indefinite or fluent quantity & all its {illeg}|d|ifferences. The first term or {illeg}g is the fluent quantity, the second is the first difference, the {f}|t|hird is the second difference, & so on in infinitum. [Let x {illeg} be a fluent quantity of {illeg} & dx {illeg} & let o be its difference & the binomium quantity ${\overline{)\mathrm{x}+\mathrm{o}}}^{\frac{\mathrm{m}}{\mathrm{n}}}$ being by M^{r} Newton's rule \set down in these let first of/ resolved into \those two letters resolved into/ an infinite series, becomes ${\mathrm{x}}^{\frac{\mathrm{m}}{\mathrm{n}}}+\frac{\mathrm{m}}{\mathrm{n}}\mathrm{o}{\mathrm{x}}^{\frac{\mathrm{m}-\mathrm{n}}{\mathrm{n}}}+$ *m−n* &c, where ${\mathrm{x}}^{\frac{\mathrm{m}}{\mathrm{n}}}$ is the first term ${\mathrm{x}}^{\frac{\mathrm{m}}{\mathrm{n}}}$ is a the fluent quantity & $\frac{\mathrm{m}}{\mathrm{n}}\mathrm{o}{\mathrm{x}}^{\frac{\mathrm{m}-\mathrm{n}}{\mathrm{n}}}$ the second term $\frac{\mathrm{m}}{\mathrm{n}}\mathrm{o}{\mathrm{x}}^{\frac{\mathrm{m}-\mathrm{n}}{\mathrm{n}}}$ is the \its/ first difference. Now instead of o if w^{th} M^{r} Leibnits you put the symbol dx the series becomes ${\mathrm{x}}^{\frac{\mathrm{m}}{\mathrm{n}}}+\frac{\mathrm{m}}{\mathrm{n}}d\mathrm{x}{\mathrm{x}}^{\frac{\mathrm{m}-\mathrm{n}}{\mathrm{n}}}+\mathrm{\&c}$ w^{ch} $\frac{\mathrm{m}}{\mathrm{n}}\mathrm{dx}{\mathrm{x}}^{\frac{\mathrm{m}-\mathrm{n}}{\mathrm{n}}}$ is the in the language of M^{r} Leibnitz is the difference of the quantity ${\mathrm{x}}^{\frac{\mathrm{m}}{\mathrm{n}}}$. A] Let the binom{illeg} |For| By the rule set down in M^{r} Newtons first Letter Letter the binomium ${\overline{)\mathrm{x}+d\mathrm{x}}}^{\frac{\mathrm{m}}{\mathrm{n}}}$ is resolved into this infinite series ${\mathrm{x}}^{\frac{\mathrm{m}}{\mathrm{n}}}+\frac{\mathrm{m}}{\mathrm{n}}{\mathrm{x}}^{\frac{\mathrm{m}-\mathrm{n}}{\mathrm{n}}}d\mathrm{x}+\frac{\mathrm{m}\mathrm{m}-2\mathrm{m}\mathrm{n}}{\mathrm{n}\mathrm{n}}{\mathrm{x}}^{\frac{\mathrm{m}-2\mathrm{n}}{\mathrm{n}}}dd\mathrm{x}+\mathrm{\&c}$ where the first term ${\mathrm{x}}^{\frac{\mathrm{m}}{\mathrm{n}}}$ being the fluent the second term $\frac{\mathrm{m}}{\mathrm{n}}{\mathrm{x}}^{\frac{\mathrm{m}-\mathrm{n}}{\mathrm{n}}}\mathrm{dx}$ is the \its/ first difference & the third term $\frac{\mathrm{m}\mathrm{m}-2\mathrm{m}\mathrm{n}}{\mathrm{n}\mathrm{n}}{\mathrm{x}}^{\frac{\mathrm{m}-2\mathrm{n}}{\mathrm{n}}}\mathrm{ddx}$ is its second difference & so on in infinitum. And wherein I pray does M^{r} Leibnitz his differential method consist but in the invention of these differences

And in maintainging it declined to give any reasons for his claim, insisted upon his own candor as if it were jnjust to question it, pressed that M^{r} Newton should declare his opinion as the only Man now alive who sufficiently understood this matter, refused to contend with any man but M^{r} Newton as if {illeg} younger men were novices & uncapable of understanding the ancient Letters & Papers w^{ch} remained upon this subject & under the person of a nameless Mathematician & two other pretended nameless correspondents, set on foot a triple Libel against M^{r} Newton full of \indecent/ railing & injurious {illeg}|a|ccusations without any proof. Whether that Libel was writ by himself or his correspondents is not material. If by his correspondents, they were imployed by him & he is answerable for what they have writ untill he produces their names. And if he refuses to produce their names he will deserve to be reputed the Author.

As for the symbols used in his method

4 As for the symbols used by him \M^{r} Leibnitz/ they are of a later date \then M^{r} Newtons/. He has no symbols for fluxions: M^{r} Newtons are the oldest \& the only symbols/ of that kind. His \The differ/ symbols \of differences/ ax & dy are later by eight or ten years then the rectangles under the fluxions & the moment{illeg} o used by M^{r} Newton in the same sense. And thes symbols \of summs/ $\int \mathrm{x}$, {f} $\int \mathrm{y}$ are later by 16 or 20 years then M^{r}Newtons of including the Ordinate or Fluxion in a square to signify the Area or fluent.

5 And if the Methods be compared, M^{r} Newtons is \the/ more eleg{illeg}|a|nt because he uses but one infinitely or indefinitely small quantity signified by the letter o. It is \the/ more Natural & Geometrical because founded upon the rationes primæ quantitatum nascentium w^{ch} have a being in Geometry, whilst indivisibles \infini/ & quantitates \minimæ or/ primæ nascentes upon w^{ch} the method differential method is founded have no being. \There are rationes primæ \quantitatum/ nascentium but not \differentq|ia|uantitates/ prima nascentes./ Nature generates quantities by continual flux or increase & the ancient Geometers admitted such a generation of areas & solids where they drew one line into another by motion to generate an area & the area into a line to generate a solid. But the summing up of indivisibles to generate an area or solid was never yet admitted into Geometry. M^{r} Newton's method is \also of greater extent being/ adapted either for finding out a Proposition or for demonstrating it: M^{r} Leip|b|nitzes is only for finding it out. When the work succeeds not in finite equations M^{r} Newtons method has recours to converging series & thereby his method becomes incomparably more universal then that of M^{r} Leibnitz w^{ch} is confined to finite equations. And where the law of the fluxions is not known F but the fluxions are had only in a few particular cases M^{r} Newton finds that Law quam proxime by drawing a Curve line through any number of given points & thence deduces the fluents & solves \solution of/ the Probleme; & on this Account also his method is more universal then that of M^{r} Leibnitz. [So then the Differential method is only a part \branch/ of M^{r} Newtons general method, & if M^{r} Leibnitz would have that part to be his own, it lies upon him to prove that he had it before the year 1677. All other arguments are in vain.|]| M^{r} Leibnitz indeed pretends to a share in the method of series The method of M. Fermat de maximis & minimis M^{r} Newton \in M. yo/ retained & made universal without usining {sic} any more symbols then one for infini indefinitely small quantities. D^{r} Barrow began to use more symbols then one for infinitely small quantities \drawing of Tangents/ & {illeg} M^{r} Leibnitz changed the Symbols of D^{r} Barrow into dx & into dx & dy & {illeg} from this Notation gave the method a new name without ever acknowledging \himself/ obliged to the Doctor. And M^{r} Newton made the method general in the years 1665 & 1666, M^{r} Leibnitz in th{e y}ear 1677. And if he \M^{r} Leinits/ pretends to have done \found/ it early|i|er, it lies upon him to prov{e} {illeg} & to give the world a{n} \fair/ account when & by what means he found it.

M^{r} Newton {illeg} publishin|ed|g his Treatise of Quadratures in the year 167\0/4. This Treatise had lain by him many years & that it might not be taken for a new piece \was written long before/ many things being cited out of it in his Letter of 24 Octob 171 1676. That it might not be taken for a new piece M^{r} Newton repeated what D^{r} Wallis had published nine years before without being \then/ contradicted \when/ by the Editors of the Acta Lipsiensia or by M^{r} Leibnitz himself, M^{r} namely that he found \the found/ the method of fluxions \was invengr|te|aduallyd by degrees/ {illeg}p {in} in the years 1665 & 1666. \Herupon/ the Editors of those Acta who composed {(} \in Ianuary 1705/ in the style of M^{r} Leibnitz (who in those days made extracts of books for the) represēd that M^{r} Leibnitz was the first inventor of the method & y^{t} M^{r} Newton |had| substituted fluxions for differences \which was all one as to tax him with falshod & plagiary./. this was in Ianuary 16|7|05 And M^{r} Keil in an Epistle published in the Philosophical Transactions for May & Iune 1708 replied: \retorted the accusation, saying:/ {illeg} Fluxionum Arithmeticam sine omni dubio primus invenit New D. Newtonus ut cuilibet ejus Epistolas a Wallisio editas legenti facile constabit. Eadem tamen Arithmetica postea mutatis Nomine & Notationis modo, a Domino Leibnitio in Actis Eruditorum edita est. And this was the beginning of the p^{r}sent controversy.

M^{r} Leibnitz in a Letter to D^{r} Sloan dated 4 Martij st. n. 1711 appeald to M^{r} Newton & \gave his reasons ag^{t} M^{r}Keill/ \&/ desired that the R. Society would make \cause/ M^{r} Keil to make a publick recantation. M^{r} Keill \shewed |shewed the Acta Lipsiensia to M^{r}| M^{r} Newton t|w|he|o| Acta Lipsiensia {illeg} had not seen then & {sic}/ chose rather to explain & defend what M^{r}Ke he had affirmed \then to rit{tr}act/. And M^{r} Leibnitz in a second Letter to D^{r} Sloan dated 29 Decem 1711 insisted upon his own candor as if it would be injustice to exprest that he should defend it \instead of making good his accusation insisted only upon his own candor, as if it would be unjust to question it &/ \against M^{r}Keil;/, [justified what had been published in the Acta Lipsiensia, & yet \[& yet/ allowed that M^{r} Newton had found the method of fluxions by himself, \apart though though not so early;/] called M^{r} Keil a novice unacquanted with things past & \one that/ acted g without h{i} author{y} {illeg}frō M^{r} Newton, & a clamorous man, \& s^{d} y^{t} y^{e} Acta Lips. had given every mā his due/ & appealed to the judgment of M^{r} Newton himself as the only man (now Barrow, Gregory, Oldenburg, Collins & Wallis were dead) who understood those matters rembred \who last/ knew what had been done formerly. But M^{r} Newton had given his opinion before in the Introduction to his Book of Quadratures & M^{r} Leibnitz had a mind that he should retract it \desired that M^{r} Newton himself would give his opinion/ in this matter. He knew y^{t} M^{r} Newton had already given his opinion in the Introduction to his Analysis \book of Quadratures/: but M^{r} Newton must retract that opinion \retract that opinion/ & allow that he had substituted fluxions for y^{e} differences of M^{r} Leibnitz \was/ the first Inventor, & then M^{r} Leibnitz would allow him to be \that M^{r} Newton was/ the second Inventor & sufer{illeg} him to be quiet. \or not be quiet./

The R. Society therefore {illeg}|h|aving as much authority over M^{r} Leibnitz as over M^{r} Keil & being now twice pressed by M^{r} Leibnitz to interpose & seeing no reason to condemn M^{r} Keill w^{th}out inquiring into y^{e} matter \& that M^{r} Leibnitz declined to make good his accusation/ {illeg}dered \appointed/ Committee & that neither M^{r} Newton nor M^{r} Lebnitz \(the only persons alive who remeembred what had passed in these matters 40 years ago)/ could be witnesses for or against M^{r} Keil, \& that M^{r} Leibnits instead of making good his accusation declined to make good his accusation/ appointe{illeg} & that there were no others alive who remembred what had passed in these matters 40 years ago, appointed a Committee to search old Letters & Papers & report their opinion thereupon, And ordered the Letters & Papers to be published together with the opinion of their Committee, by w^{ch} opinion {illeg}|M|^{r} Newton was the oldest inventor had the method [above five years before] the year the writing of his Letter dated 13 Iune 1676 & sometime] before the writing of his Letters dated 10 Decem 1672 & th{illeg} above five years before the writing of his Letter dated 13 Iune 1676 & \even/ before the {illeg} writing of his Analysis communicated by D^{r} Barrow to M^{r} Collins in Iuly 1669; & so was the first inventor in the year 1669 or before & it appeared not that M^{r} Leibnitz had it before the year 1677.

Its objected that M^{r} {illeg} the R. Society heard have not heard M^{r} Leibnitz reasons, & therefore their judgment is void. But this is a sophistical Objection this is a sophistical objection. They

If it be pretended that M^{r} Leibnitz has not yet produced his reasons for himself: \I answer that/ in all the Letters w^{ch} passed between him & M^{r} D^{r} Wallis about these matters he was no [wherein he defended himself {sic} \wherein he defended himself as well as/ as well as he was able], he was not a produced not \an{e}|y|/ one reason to prove \either/ that he had the differential method before the year 1677 or that M^{r} Newton had it not be above ten years before, as D^{r} Wallis |had| affirmed. \that time./ He produced not \any/ one argument against M^{r} Fatio to prove that M^{r} Newton was the oldest inventor by many years. \He did not then deny that M^{r} Newton was the first inventor/ In his first Letter ag^{t} M^{r} Keill he produced what \his/ reasons against what M^{r} Keill had said. But|And| after M^{r} Keil had explained himself & answered those reasons: he produced no more reasons but in his next Letters for want of reason{s} cried out. Quæ D. Ioannes Keillius nuper ad Te scripsit, candorem meum apertius quam ante oppugnant: quem ut ego hac ætate, post tot documenta vitæ, Apologia defendam, & cum homine docto sed novo, & parum perito rerum anteactarum cognitore, nec mandatum habente ab eo cujus inter{illeg}|s|t, tanquam {p}ro Tribunali litigam nemo prudens æquus probabit. That is, he told the R. S{ociety} that they would be unjust unless they would allow him to be judge \a witness/ in his own {cas}e.

And am not I as good a witness that I invented the methods of series & fluxions in the year 1665 & improved them in the year 167|6|6, & that I still have in my custody several mathematical papers some of written in the years 167|6|4, 1665 & 1666 some of w^{ch} happen to have been dated & that in one of them dated 13 Novem 1665 the direct method of fluxions is \illustrated with Examples & demonstrated, & that down in this it is/ set down in these Problem|posit|e{illeg} /words\: *An Equation being given expressing the Relation of two or more lines x, y, & z & described in the same time by two or more moving bodies A, B, C &c, to find the relation of their velocities* p, q, r &c. And \that/ the {illeg} resolution of this Problem is there set down in these words *Set all the terms — — — — — relation of p, q, r &c* \& that this Resolution is there illustrated with examples & demonstrate{d}/ And that in a\no/ther Paper dated 16 May 1666 a general method of resolving Equations is Problems by motion is set down in seven Propositions the last of which — — — — quantitatum inter se. And that in confirmation of all this I am ready to produce the original papers at this time I had made my Analysis composed of the methods of series & fluxions together so universal as to reach to all almost all sorts of Problemes as I mentioned in my Letter dated 13 Iune 1676. These \things/ {t}|d|o not rely meerely upon my own testimony, but \the/ ancient papers \Manuscripts/ themselves are ready to be produced. {illeg}

— And that \in this Tract/ when any of the \the/ area {illeg}|a|rising from any of the terms {illeg}|i|n the Valor of the Ordinate cannot be expressed by Vulgar Analysis I represent it by prefixing the symbol ▯ to y^{e} term. As if the Abscissa be x & the Ordinate $\mathrm{a}\mathrm{x}-\mathrm{b}+\frac{\mathrm{b}\mathrm{b}}{\mathrm{b}+\mathrm{x}}$, the area will bee $\frac{1}{2}\mathrm{a}\mathrm{x}\mathrm{x}-\mathrm{b}\mathrm{x}+\u25af\frac{\mathrm{b}\mathrm{b}}{\mathrm{b}+\mathrm{x}}$. And that in this This \the same/ Tract I sometimes used a Letter with one prick for f quantities involving first fluxions & the same Letter with two pricks for quantities involving second fluxions. And that {illeg} the larger Tract w^{ch} I wrote in the year 1671 was founded upon this smaller & Tract & began with the Reduction of finite quantities to converging Series & with these \the solution of/ two Prop|b|lems \& the s set down in these words/. 1. *Relatione quantitum fluentium inter se data fluxionum relationem determinare.* 2. *Exposita Æquatione fluxiones quantitatum involvente invenire relationem quantitatum inter se*. The original Manuscripts \are still extant to/ {illeg}|h|ave {sic} \justify what is here represented & have been seen/ been seen {sic} by Mathematicians many years ago. {illeg} M^{r} Fatio mentions|e||d| them in|a|t his the end of his book of \abo entituled/ Fruit walls improved, & \which book was/ published in the year 1699. And in \my/ Letter wh{illeg} of 24 Octob 1676 I mentioned the last of these Tracts s|a|s written five years before that{illeg} Letter & about the method of series & another method which was founded in this sentence Data æquatione fluentes quotcun quantitates involvente invenire fluxiones & vice versa. And in the end of that Letter I added that I had a method of extracting fluents out of equations involving their fluxions: which implies that I then knew how to reduce Problems to fluxional equations. And by my Letters of 10 Decem. 1672 & 13 Iune 1676 it appears that I had then made my Analysis composed of the methods of Series & fluxions together so universal as to reach to all|m|ost all sorts of Problems. But M^{r} Leibnitz has {no} acknowledged that in Feb 1673 he knew & for some months after, he knew little or nothing of the higher Geometry but \at length/ learnt it afterwards of M^{r} Hu{illeg}|y|gens at Paris, [& pretends only \not/ that he found the Differential Method in \before/ the year 1676 but \nor/ brings not so much as one argum^{t} to prove that he found it before the year 1677, except that w^{ch} I am \still/ going to answer at the w^{ch} is drawn from his solving the Problem of Beaune the pretence that he solved the Probleme of Beaune by his method had s \unles/ *certa Anslysis* in his Letter of 27 Aug. 1676 {illeg}is \{illeg}y \{must}/ signifies/ the differential Analysis] & that he had not the differential method till after the year 1675

The first of these \shews what was the opinion of/ was writ before M^{r} Leibnitz \before he/ knew any thing more of my method then what he had from my Letters \& Pap/ writ in or before my the year 1676 & from the Principia use y^{t} {illeg} use that I made of this Principia Philosophiæ the second s mathematica,|.| & the f{illeg} The second shews was the notion of the English Mathematicians \tradition in England/ before they \we/ heard that this Method began to be celebrated in Holland as invented by M^{r} Leibnitz. And|m|ong the said Letters & Papers were also several papers written by my self in the years 1664, 165|6|5 & 1666 some of which happen to be dated

Et {illeg} \Me/ Commerci{illeg}\um/ Epistolic{illeg} oblivione minime respondet\um olivioni {sic} tradere conæretur tanquam nimis longo/ prolixo cui responderet, nec t{e}|a|men satis prolixūm {sic} ob Epistolas qui ob {illeg}b epistolas t{illeg}.

Commercio Epistolico tanquam {illeg} respondere recusaret, & convitijs uteretur

^{1}Cum D. Leibnitius ^{5}Epistolas præcedentes prius mitteret in Galliam quam earum tertia in Angliam veniret, ^{2}Commercio Epistolico {illeg} respondere \aperte/ recusaret nulla pro se argumenta \licet provocatur|s|/ afferret, & convitijs utere verbose ser & convitijs uteretur jus s{illeg} ^{3}causam suam probare licet provocatur|s| minime aggrederetur, & a^{4}verbose scriberet, et ^{4}convitijs uteretur \&/: ^{6}Newtonus minime rescripsit, sed Observationes quasdam sequentes in tertiam illam scriptas cum amicis solummodo communica{illeg}

Cum D. Leibnitius \convitijs uteretur et/ add{u}\e/ non posset ut vel Commercio Epistolico responderet vel argumenta pro se afferret; {illeg}s{illeg} a convitijs \nam/ abstineret, et \cum nec cum præcedentes/ Epistolas præcedentes \is in Galliam prius/ {illeg}|p|rius {sic} mitteret in Galliam quam earum tertia in Angliam veniret & prætenderet se hoc facere ut testes haberet \et alias etiam pro & alijs utetur convitijs argumentis contumelas adhiberet/: Newtonus minime rescripsit sed Observationes sequentes in tertiam illam \Epistolam/ scriptas cum amicis solum{illeg}|m|odo communicavit.

## P. S.

When the Committee of the R. S. published the Commercium Epistolicum the Papers in my custody were not produced, \& it will be more usefull to read the progress by which the method was invented./ Among my papers written in the years 1664, 1665 & 1666 some happen to be dated: amongst w^{ch} is that which follows

## 13 Novem. 1665

Probleme.

An Equation being given &c

In a little Tract written in the

In a paper dated 16 May 1666 my method of resolving Problems by my motion is set down in 7 Propositions the last of which is the same with that conteined in the Paper of 13 Novem 1666|5| tho exprest in other words.

In a little Tract written in October 1666 the same method is set downin the same seven Propositions \but the seventh is enlarged/ & an eighth is added to them. That which is added to the seventh is in these words

Note that if there happen to be in any equation either a fract or surd quantity or a mechanical one (i.e. w^{ch} cannot be Geometrically computed but is expressed either by the area or lengh {sic} or gravity or content of some curve line or solid &c) to find in what proportion the unknown quantities increase or decrease do thus. Take two ........ required

Exempl. 1. To find p & q the motions of x & y whose relation is $\mathrm{y}\mathrm{y}=\mathrm{x}\sqrt{\mathrm{a}\mathrm{a}-\mathrm{x}\mathrm{x}}$ ........ w^{ch} was required

Examle {sic} 2. If ${\mathrm{x}}^{3}-\mathrm{a}\mathrm{y}\mathrm{y}+\frac{\mathrm{b}{\mathrm{b}}^{3}}{\mathrm{a}+\mathrm{y}}-\mathrm{x}\mathrm{x}\sqrt{\mathrm{a}\mathrm{y}+\mathrm{x}\mathrm{x}}=0$ is the relation is the relation sought.

Example 3. If x=AB upon the {illeg} $\stackrel{.}{\mathrm{x}}=\mathrm{AB}+\mathrm{BC}=\sqrt{\mathrm{a}\mathrm{x}-\mathrm{x}\mathrm{x}}$, [that is if $\mathrm{BC}=\sqrt{\mathrm{a}\mathrm{x}}$ to the line $\mathrm{AB}=\mathrm{x}$ the line BC\$=\sqrt{\mathrm{a}\mathrm{x}-\mathrm{x}\mathrm{x}}$/ be ordinately applied.] \& the superficies be called z/ Suppose that $\mathrm{z}\mathrm{z}+\mathrm{a}\mathrm{x}\mathrm{z}-{\mathrm{y}}^{4}=0$ is the relati{illeg}|o|n between x y z

Example 3. If to y^{e} line $\mathrm{AB}=\mathrm{x}$, the Ordinate $\mathrm{BC}=\sqrt{\mathrm{a}\mathrm{x}-\mathrm{x}\mathrm{x}}$ be appl ordinately applied at right angles, & y be put for any indeterminate line BE & the area superficies ABC be called z: Suppose that $\mathrm{z}\mathrm{z}+\mathrm{a}\mathrm{x}\mathrm{z}-{\mathrm{y}}^{4}=0$ is the {op} relation between the lines x y & z whose motions are p, q, & r, & that p & q are desired. And The Equation $\mathrm{z}\mathrm{z}+\mathrm{a}\mathrm{x}\mathrm{z}-{\mathrm{y}}^{4}=0$ gives (by Prop. 7) $2\mathrm{r}\mathrm{z}+\mathrm{r}\mathrm{a}\mathrm{x}+\mathrm{p}\mathrm{a}\mathrm{z}-4\mathrm{q}{\mathrm{y}}^{3}=0$. Now erecting the perpendicular $\mathrm{AD}=1$ & completing the parallelogram {illeg} ABHD I consider that the superficies $\mathrm{ABHD}=\mathrm{AB}\times \mathrm{BH}=\mathrm{x}\times 1=\mathrm{x}$, that is, 1. $\sqrt{\mathrm{a}\mathrm{x}}$ 1. $\sqrt{\mathrm{a}\mathrm{x}-\mathrm{x}\mathrm{x}}\colon\colon \mathrm{p}.\mathrm{r}$. Or $\mathrm{r}=\sqrt{\mathrm{a}\mathrm{x}-\mathrm{x}\mathrm{x}}$. Which valor of r being substituted into the equation $2\mathrm{r}\mathrm{x}+\mathrm{r}\mathrm{a}\mathrm{x}+\mathrm{p}\mathrm{a}\mathrm{z}-4\mathrm{q}{\mathrm{y}}^{3}=0$, gives $\overline{)2\mathrm{p}\mathrm{z}+2\mathrm{p}\mathrm{x}}\times \sqrt{\mathrm{a}\mathrm{x}-\mathrm{x}\mathrm{x}}+\mathrm{p}\mathrm{a}\mathrm{z}-4\mathrm{q}{\mathrm{y}}^{3}=0$, w^{ch} was required.

The eighth Proposition was this. If two bodies A & B with their velocities p & q describe the lines x & y, & an equation be given expressing the relation between one of the lines x & the ratio $\frac{\mathrm{p}}{\mathrm{q}}$ $\frac{\mathrm{q}}{\mathrm{p}}$ w^{ch} or the ratio of their motions q & p: to find the other line y. And the resolution of this Proposition is here set down by \in/ these steps Rules.

1 If the valor of $\frac{\mathrm{q}}{\mathrm{p}}$

1. Get the valor of $\frac{\mathrm{q}}{\mathrm{p}}$ & if it be rational & its denominator consists but of one term, multiply that valor by x & divide each term of it by the logarithm of x in that term [that is by the index of its dignity] the Quote shall be the valor of y. As if $\mathrm{a}{\mathrm{x}}^{\frac{\mathrm{m}}{\mathrm{n}}}=\frac{\mathrm{q}}{\mathrm{p}}$ then is $\frac{\mathrm{n}\mathrm{a}}{\mathrm{n}+\mathrm{m}}{\mathrm{x}}^{\frac{\mathrm{n}+\mathrm{m}}{\mathrm{n}}}=\mathrm{y}$.

2 But if the denominator of the valor of $\frac{\mathrm{q}}{\mathrm{p}}$ consist of more terms then one, it may \[usually]/ be reduced to such a form that the denominator of each part of it shall have but one term so that y may be then found by the precedent Rule Which Reduction is thus performed. 1. If \If all the terms of the Denominator be not multiplied by x or xx or x^{3} &c/ Increase or diminish x untill the last term of the Denominator vanish. 2. And when all the terms of the Denominator are multiplied by x, xx or x^{3} &c divide the numerator by the Denominator (as in decimal numbers) untill the Quotient untill the Quotient {sic} consist of such parts none of whose Denominators are so multiplied by x, xx x^{3} &c, & begin the division in those terms in w^{ch} x is of the \its/ fewest dimensions: unless the If then the terms in the valor of $\frac{\mathrm{q}}{\mathrm{p}}$ be such as was before required the valor of y may be found by the first Rule only it must be so much incre diminished or increased as it was before diminished or increased by increasing or diminishing x. But if the denominator of any term of more terms then one, find those \the/ parts of y's valor w^{ch} correspond to severally.

Example 1. If $\frac{\mathrm{x}\mathrm{x}}{\mathrm{a}\mathrm{x}+\mathrm{b}}=\frac{\mathrm{q}}{\mathrm{p}}$, {illeg} then by division is $\frac{\mathrm{x}}{\mathrm{a}}-\frac{\mathrm{b}}{\mathrm{a}\mathrm{a}}+\frac{\mathrm{b}\mathrm{b}}{{\mathrm{a}}^{3}\mathrm{x}+\mathrm{a}\mathrm{a}\mathrm{b}}=\frac{\mathrm{x}\mathrm{x}}{\mathrm{a}\mathrm{x}+\mathrm{b}}=$$\frac{\mathrm{q}}{\mathrm{p}}$ (as may appear by multiplication) Therefore (by the first Rule) $\frac{\mathrm{x}\mathrm{x}}{2\mathrm{a}}-\frac{\mathrm{b}\mathrm{x}}{\mathrm{a}\mathrm{a}}+\u25af\frac{\mathrm{b}\mathrm{b}}{{\mathrm{a}}^{3}\mathrm{x}+\mathrm{a}\mathrm{a}\mathrm{b}}=\mathrm{y}$.

Examp. 2. If $\frac{{\mathrm{x}}^{3}}{\mathrm{a}\mathrm{a}-\mathrm{x}\mathrm{x}}=\frac{\mathrm{q}}{\mathrm{p}}$. I suppose $\mathrm{x}=\mathrm{z}-\mathrm{a}$, or $\frac{{\mathrm{z}}^{3}-3\mathrm{a}\mathrm{z}\mathrm{z}+3\mathrm{a}\mathrm{a}\mathrm{z}-{\mathrm{a}}^{3}}{2\mathrm{a}\mathrm{z}-\mathrm{z}\mathrm{z}}=$$\frac{\mathrm{q}}{\mathrm{p}}$

<39v>And by division $-\frac{\mathrm{a}\mathrm{a}}{\mathrm{z}\mathrm{z}}-\mathrm{z}+\mathrm{a}+\frac{\mathrm{a}\mathrm{a}}{4\mathrm{a}-{}^{2}\mathrm{z}}=\frac{\mathrm{q}}{\mathrm{p}}$, as may appear by multiplication.

And substituting $\mathrm{x}+\mathrm{a}$ into the place of z, you have $\mathrm{x}-\frac{\mathrm{a}\mathrm{a}}{2\mathrm{x}+2\mathrm{a}}+\frac{\mathrm{a}\mathrm{a}}{2\mathrm{a}-2\mathrm{x}}=\frac{\mathrm{q}}{\mathrm{p}}=$$\frac{{\mathrm{x}}^{3}}{\mathrm{x}\mathrm{a}-\mathrm{x}\mathrm{x}}$

<39v>And therefore by the first Rule, $\frac{\mathrm{x}\mathrm{x}}{2}+\u25af-\frac{\mathrm{a}\mathrm{a}}{2\mathrm{x}+2\mathrm{a}}+\u25af\frac{\mathrm{a}\mathrm{a}}{2\mathrm{a}-2\mathrm{x}}=\mathrm{y}$.

He saith that he formerly beleived me & on that account acknowledged that I had found the {illeg} method of fluxions apart, but now \{illeg}/ \now M./ Bernoulli has given him reason to suspect {illeg} me, he is at liberty to retract. But his Letter of to M^{r} N. dated $\frac{7}{17}$ Mar. 1693 \& herunto annexed/ was written before he had any other notice of my method then from M^{r} Oldenburgh M^{r}. Collins & the Principia Philosophiæ, & therefore he is not at liberty to retract.

He questions my credit in what I say about D^{r} Wallis & therefore I have subjoyned the Paragraph in the \his/ Preface of D^{r} Wallis together with the|i|s Doctors Letters relating to it. The D^{r} there plainly tells him that he had the Meth that M^{r} N. in the year 1676 \I/ explained to him the Method of fluxions invented \by me/ ten years before \those days/ or above. D^{r} Keil has not affirmed so much M^{r} L. beleiv did not think fit to contradict M D^{r} Wallis in those days but is now very angry angry at D^{r} Keill.

## P. S.

When the Committee of the Royal Society published the Commercium Epistolicum, the {illeg} Letters & Papers in my c|C|ustody were not produced. Among them the were \was/ the following Letter of M^{r} Leibnitz, \& a{illeg} Letter of D^{r} Wallis, dated/ \both/ s{aid} which upon {sic} \both w^{ch} upon/ a fresh occasion have been produced & in more \left/ in the Archives of the R. Society,|.| Among them were also several papers written by my self in the years 1664, 1665 & 1666 some of w^{ch} happen to be dated. And one of them writ in the year \november/ 16{illeg}|6|5 {v} {sic} hereunto subjoyned.

## Illustri v|V|iro

ISAAC NEWTON

Goth{illeg} Godefridus Gulielmus Leibnitius S.P.D.

Quantum tibi . . . . . . Vale. Dabam Hanoveræ $\frac{7}{17}$ Martij 1693.

In another Paper dated 16 May 1666 my method of resolving Problems by motion is set down in seven Propositions the last of w^{ch} is the same with that conteined in this paper of Nov 13 Nov. 166{6}|5| {illeg} And in a third paper da \small treatise written/ in November 1666, the same seven Propositions are set down again & the seventh is improved by shewing how to proceed without sticking at fractions or surds or such quantities as are now called transcendent. And an eighth Proposition is added conteining the inverse method of fluxions so far as I had then attained it, namely \by the methods of Quadratures & particularly/ by the three Rules upon w^{ch} the Analysis per æquationes numero terminorum infinitas is founded & by \most of/ the Propositions \Theorems/ set down in the Scholum {sic} to the tenth Proposition of the Book of Quadratures, [most of which are {illeg} \here/ set down in this Paper.] {I} And in the Tract which I wrote in the year 1671 the two first Propositions are {sic} And \that/ in this Paper I sometimes used a letter with one prick for quantities involving {f}{illeg} first fluxions & the same letter with two pricks for quantities involving fi{illeg} second fluxions. And \that/ in the Tract which I wrote in the year 1671 the two first Propositions were these. {illeg} \1/ Relatione quantitatum fluentium inter se data fluxionum relationem determinare. 2 Exposita æquatione fluxiones quantitatum involventes, invenire relationem quantitatum inter se. |All w^{ch} are to be compared with my Letters of 10. Decem. 1672, 13 Iune 1676 & 24 Octob. 1676. published in the Commercium Epistolicum & with the Scholium to the second Lemma of the second Book of Mathematical Priciples {sic} of Philosophy.| And in my Letter of 24 Octob. 1676 I represented that I had written a Tra D^{r} sent to M^{r} Collins a Tra compendium of the method of series about

that time that Mercator published his Logarithmotechnia & this Compendium is the Analysis per Series now extant. I mentioned also that t{ok}en five years before the writing that Letter that I had is in the year 1671 I had written a Tract concerning the method of series & another method together, & that the other method was founded in this sentence Data æquationes flentes {sic} quotcun quantitates involvente fluxiones invere {sic} & vice versa.

I suppose he means be{illeg} that \because/ he finds no prickt letters there. And by the same way of arguing he may say \& M^{r} Bernoulli may pretend/ that they find nothing of that Analysis in the method of fluxions in the Introduction to the Book of Quadratures, tho nothing else be there described that whole Introduction be {illeg} \entirely/ taken up in describing that \this/ Method & illustrating it with Examples [& my saying there that I found this Method in the years 1665 & 1666 gave occasion to this controversy]. I there \In that Introduction I/ say that I invented (not prickt letters, but) the method of fluxions in those years \the years 1665 & 1666/, & describe it in that Introduction \there/ without {illeg}|t|he use of prickt letters, as {illeg}|I| did long \about thirty years \twe &// /long\ before in my Analysis above mentioned. Let the world judge whether the method described in this Introduction & said here to be invented by me in the years 1665 & 1666 \gradually in those years/ be not one & the same method with \that/ described in the said Analysis \w^{ch} was \&/ communicated by D^{r} Barrow to M^{r} Collins/ 47 years ago & in my Letters of 10 Dec 1672 & 24 Octob 1676 & by me{illeg} concerning which I say in my Letter of 13 Iune 1676 that \Analysis by/ my method of series \thereby/ extends to almost all sorts of Problems (except perhaps some numeral one{illeg}|s| like those of Diophantus) but becomes not \altogether/ universal without the assistance of this other method |it| the help of some further methods, w^{ch} in next Letter \dated 24 Oc{illeg}|t|ob 1676 I describe to be the direct & invers methods of fluxions the method/ I explain to be the direct & invers method describe by these sentences Data æquatione fluentes quotcun quantitates involvente fluxiones invenire, et \vice/ versa of extracting fluents out of equations involving their fluxions, & the method of assum \{illeg}|a|ssuming/ arbitrary series. Although I said \in that Letter/ that the foundation of the method of w^{ch} I wrote a Tract in the year 1671 was founded in this sentence Data æquatione fluentes quotcun quantitates inolvente fluxiones invenire & vice versa

— & D^{r} Wallis (Vol 2 Opper. pag 393 lin. 32) {will} has told him that th{illeg}|a|t this method needs no further explication then that w^{ch} I gave of it in my Letter of 24 Oct. 1676.

That of D|M|^{r} Leibnitz shew that he understood that I had such a gener{a} *Methodus similis* as he {illeg} men before he knew any thing more of it then what he had from my Letters in the year 1676 & from the Principia Philosophiæ: that of D^{r} Wallis that the tradition in England \news tradition/ that I invented the method describ [mentioned there by the D^{r}] was \much/ older \in England/ then the tradidtion \newes/ that M^{r} Leibnitz invented it the first beginning in the year 1676 as old \in england/ as the year 1676, at w^{ch} time copies of my Letters were sent to him by M^{r} Oldenburg.

Cum D. Leibnitius Epistolas præcedentes in Galliam prius mitteret quam Responsem ejus in Angliam veniret et Responsum illud scurriliter scriptum esset |convilijs sentiret l{illeg} abundaret & A Autho|; Newtonus minime rescripsit, sed Observationes sequentes in Responsum illud \{illeg}|{a}| se scripsit|ta|s {et}/ cum amicis \tantum/ communicavit.

& {illeg} convilijs abundaret, et {illeg} Author Commer antiq Commercio Epistolico minime responderetur; Newtonus non rescripsit, ne monumenta antiqua sed seq. Observationes se in Responsum illud a{illeg} a se compositas cum amicis tantum communicavit.

Cum D. Leibnitias {sic} Epistolas præcedentes in Galliam prius misisset \mitteret/ quam Re{s c} earum tertia in Angliam veniret, & Newtonus minime rescripsit sed sed {sic} Observationes sequ suas \sequentes/ in tertiam illam scriptas cum amicis tantum \tantum/ communicavit.

## Observationes in Epistolam præcedentem.

And that in \or before/ Se 1665 I invented the first Proposi

That in the year 1665 I invented the firs method|s| of \series &/ fluxions & still have in my custody a paper dated 13 Novem. 1665 in which the first Proposition of the Book of Quadratures is propounded & Resolved illustr\a/ted with examples & Demonstrated Geometrically, the letters p, q, r being here put for the fluxions of the fluents x, y, z. \And/ That in the year 1666 I improved this method \(extended it to second fluxions & |& extended it to second fluxions|/ & before the end of the year \{&}/ sometimes \& {sic}/ used \sometimes letters without pricks & sometimes letter without pricks/ letters with one or two pricks for for firs quantities involving first or second fluxions & \before the end of the year/ wrote a small Tract on this subject which was the ground of that larger Tract which I wrote in the year {illeg}|1|67{illeg}|1|, both which

to be added \the/ to complete the series (for he has no where said t said in all that book that he uses the word \phrases *second & third*/ *differences* in the sence of M^{r} Leibnitz) the Scholium will be true w^{th}\out/ any correction.] & the Objection will vanish.

This account of the Com

{C}{illeg}

The credit \force & evidence/ of the Commercium Epistolicum {illeg} an & of the account here given of that matter stands upon the authority of the ancient Letters & papers \there partly by D^{r} Wallis & partly/ by order of the R. Society \the originals of those published by order of the Society are still preserved./ If M^{r} Leibnits or his those \ancient/ Letters & Papers are \altogether/ against M^{r} Leibnitz it is his own fault. If the Notes upon them for enabling the reader to compare them & understand them at one reading, have in any thing misrepresented them, M^{r} & h Leibnitz & his friends are at liberty to set shew wherein they are misrepresented & to set the matter right set right the representation. [If they only tell a contrary story \If in any thing they make for him they are at liberty to explain it. But/ M^{r} without proving what they affirm, M^{r} Leibnitz can be no witness in his own cause & his friends at in Germany know nothing \of the matter/ but what they have from M^{r} Leibnitz or from the Records themselves.] The friends of M^{r} Leibnitz in Germany know nothing \more/ of what passed between him & M^{r} Oldenburg when he was in England or France {illeg}d then what he tells them or what they find in the Letters w^{ch} \then/ passed between them. He cannot be a witness in his own cause, & therefore what \stories/ they affirm \tell/ in his favour concerning those things, {illeg}|i|s not to be regarded unless they prove their affirmations by t any further then the truth thereof appears out of ancient records] But they are not at liberty to tell stories without at discretion without proving what they say. M^{r} Leibnitz |c|is|a||n| not to be ad \be/ an evidence in his own cause & his friends know nothing of the matter \his correspondence with the English 40 years ago in his/ but what they \travells 40 years ago but what they have {sic}/ have from him or from the ancient {illeg} Letters & Papers records. the Letters & papers then written & still preserved.

It has been said that M^{r} Newton in the Scholium at the end of his Book of Quadratures has put the third fourth & fift terms of a converging Series respectively equall to the 2^{d} 3^{d} {illeg}|&| 4^{th} differences of y^{e} first Term, & therefore did not understand the method of second third & fourth differences. But in the first Proposition of that Book he shewed how to find the first second third & following fluxions & therefore did in infinitum & therefore when he wrote that Book, w^{ch} was before the year 1676, he did understand the method of all the fluxions \& by consequence of all the Differences/. And if he did not understand it when he added that Scholium to the \end of the/ book, w^{ch} was in the year 1704, it was \must have been/ because he had \then/ forgot it. And so the Question is only whether M^{r} Newton had forgot the method of second differences before the year 1704.

In the tenth Proposition of the second Book of the \his/ Principia Philosophiæ in describing \some of/ the uses of the terms of a converging series for the solving of Problemes |he| tells us that if the first term of {illeg}|th|e series represent the Ordinate BC of any Curve \line/ ACG, & DG be another Ordinate infinitely neare to the former. CBDI be a parallelogram infinitely narrow whose side DI cuts the Curve in G & the its tangent CF in F: the second term of the series will represent the line IF, & the third term the line FG. Now the line FG is but half the second difference of the Ordinate And therefore M^{r} Newton when he wrote his Principia, did put the third term of the series equall to half the second difference of the first term & by consequence had not then forgotten the method of second differences. ^{‡}^{[5]}

Nor is it likely that when he added the Scholium to then end of his book of quadratures he had the meaning of the fi \forgot the/ first Proposition of the Book. \*/^{[6]} If the word [ut,] which has been {illeg} in that Scholium has been accidentally omitted between the words [erit] and [ejus] be restored; that Scholium will agree with the rest of his writings, & the objection will vanish.

[He that knows how to find the first fluxion of any line & to expose that fluxion by a \another/ line, knows how to find the fluxion of that line & to expose it by a third line & \to/ find the fluxion of this line & expose it by a fourth line & so on perpetually, that is, he knows how to find the second third & fourth fluxions of the first line The method is one & the same in all the fluxions. The method is y^{e} And there is the same reason of the method of moments or differentias: so that \And/ he who understands it in the first differences understands it in all y^{e} rest.]

In the year 1692 when at the request of D^{r} Wallis, he sent to him |a copy of| the solution of /explication /explication\ of\ the Proposition, *Data æquatione flu{illeg}entes quotcun quantitates invenire*, he extended it to \gave examples af thereof in first/ second & third fluxions as you may see in the second Volume of the Doctors works pag 391, 392, & 393 & 396. And therefore he had not then forgotten the method of second fluxions. & {illeg} differences.

Nor is it likely that \in the year 1704/ when he added the aforesaid Scholium to the end of the book of Quadratures, he had forgotten \both/ the first Proposition of y^{e} book \& the last upon w^{ch} that Scholium was written./ If y^{e} word [ut] w^{ch} in that Scholium has bee may have been accidentally omitted between the words [erit] & [ejus] be restored; that scholium will agree with the rest of his writings, [supposing that he used the language of second third & fourth differences in the language \sence/ of M^{r} Leibnitz. But if he used that \language/ in some other sense of his own, suppose to signify the \second third & fourth/ differences w^{ch} rem

*S ^{r}*

I beg you will not think mee impertinent in troubling \you/ with this. it is upon y^{e} account of y^{e} Medall

Proceribus populis concentientibus

I think it will be better thus

Senatu Populo sancientibus

Proceres is but a Metaphor {&} is properly Off y^{e} Off Beams in buildings, and expresses rather cheif Officers than Noblemen

Populos is {rea}d in y^{e} plurall number but I think ofter in y^{e} singular sancire leges in Tully's word

I am yo^{r} most obedient Serv

S: Garth

pag. 4. l. 7, add.

It has been said that M^{r} Newton in the Scholium upon {illeg} y^{e} Proposition of y^{e} 2 book of his Principles put the third term of a converging series equal to the secon the second difference whereas it is equa of the first term whereas it{illeg} is but half that difference, & therefore M^{r} Newton did not then understand the method of second differences. But the Objector is mistaken himself {illeg} it may \with as much reason be retorted/ be retorted that the Objector is in an error \{an} charging M^{r} Newton w^{th} an error is in an error himself/ & therefore did not understand the method of second differences himself: \And M^{r} Newton was/ For M^{r} Newton in that Scholium puts the third term of the series equall to but half the second difference of the first term|.| thereof.

It has been said that M^{r} Newton in the Scholium at the end of {illeg}|t|his book of Quadratures did has put the terms {a} second third fourth & fift terms of a series converging series \respectively/ equall to the 2^{d} 3^{d} & 4^{th} differences of the first term, respectively, & therefore when he wrote that Scholium that is, in the year 16 1704 \he/ did not understand the method of second third & fourth differences. |That Scholium was added to y^{e} book in y^{e} year 1704 but the book \of Quad/ except that Scholium & the Introduction was {illeg}| But M^{r} Newton in the first Propositiō of the Book w^{ch} \it self/ was writt before the year 1676 explained how to find & in the first Proposition thereof \he has/ shewed b{e}|y| an \a/ generall Rule how to find the first second third & following fluxions in infinitum & therefore did then understand the method of all the fluxions. He that knows how to find the first fluxion of a \any/ line & to expose that fluxion by another line knows how to find that that other line the fluxion of that other line {illeg} w^{ch} is the second fluxion of that other \the first/ line. {O}{illeg} on perpetually the second fluxion of the second line is the third fluxion of the first line, & so on perpetually And if this fluxion be exposed by a third line the fluxion of this line will b{e} the third fluxion of the first line & so on perpetually. And therefore he that understands the method of first fluxions understands the method of all the rest. And there is the same reason of differences.

M^{r} Newton in the tenth Proposition of the second book of his Principia Philosophiæ, in describing the use that may be made of the severall terms of a converging series tells us that if the first term of the Series represent the Ordinate of a Curve BC the second will represent the line {illeg} IF, \&/ the third the line FG. & {illeg} Now the line FG is but half the diffe the second difference of the A Ordinate, & therefore M^{r} Newton \in applying |in applying these| series to the solution of Problems did then/ puts the third term equal to half the second difference, [& by consequence t to reconcile this with the scholium at the end of the Book of Quadratures the word [ut] w^{ch} has been \twice/ accidentally omitted is to be restored in the line 11 & between the words *erit* and *ejus* is to be restored :|[|unless you say that M^{r} Newton when he wrote his Principia understood the method of second differences, but did not u did not understand it when he wrote the Scholium.] \as it really is./

M^{r} Newton therefore when he wrote thi|e|s Principia Philosophiæ & {illeg}s book \& {sic} long before that, when he wrote/ the first Proposition of his book of Quadratures understood the method of second differences. And if he did not understand it \ten years ago/ when he wrote the Scholium, it was \which was but tenn years ago, it must be/ because he has forgot it. But there is no need of saying that he had forgot it: for \in that Scholium/ if the word [ut] w^{ch} has been \twice/ accidentally omitted between the words [erit] & [ejus] \in that scholium/ be restored, that Scholium will agree with the rest of his writings.

It has been represented that the R. Society have passed sentence without hearing both parties & therefore their sentence is voyd. B{u} And indeed M^{r} Leibnitz pressed them to condemn M^{r} Keil without hearing both parties; But a|A|nd told them that his own candour could not be questioned without injustice: Which is the same thing as to tell them that they would be unjust if they did not \allow M^{r} Leibnitz to be a witness in his own cause &/ condemn M^{r} Keil without hearing both parties. But the R. Society do not take upon th act as a Court of Iustice Iudicature, nor have they been desired to give their opinion in this matter unless by M^{r} Leibnitz against M^{r} Keill.

\And |Some months|/ After the Commercium Epistolicum was published M^{r} Leibnitz pretendin|ed|g that he had was not at \had/ not seen it nor was at leasure to answer \consider/ it, & that he had \therefore/ desired an able & impartial Mathematian {sic} to examin the matter, & had received his answer dated 7 Iune 1713. {illeg} And this answer was published in Germany with additions. \It is full of affirmations without any proof./ The able Mathematician & the author of the additions are not named & so the whole stands upon the credit of M^{r} Leibnitz, & even the style is re{uo}{illeg}ed to be his reputed his We h As his Letters were full of inflexions against M^{r} Keill so this paper is full of reflexions against M^{r} Newton. And M That you may compare it with the Acct above written we have here subjoyned it

Several accounts of this Commercium having been published abroad all of them very imperfect: its has been thought fit to publish the Account w^{ch} follows.

It has been represented \in Germany/ that M^{r} Leibnitz first found out the differential Calculus in numbers, & then by inventing the Analysis of infinitesimals translated this Calculus to Geometry. [And indeed M^{r} Leibnitz \in the year 1673/ pretended to the calculus differentialis of Mouton in numbers & was reprehended for {illeg} by D^{r} Pell for intruding into other mens inventions & wrote an Apology for himself w^{ch} is printed in the Commercium Epistolicum. And In this Apology he represented that he had found out several things apart without knowing what Mouton had done before, & had added some things of his own, but unluckily instanced in a property of numbers [natural triangular pyramidal tria\n/gulo, triangular] observed before by Mons^{r} Paschal whose book he had seen, He re & yet represented that Mons^{r} Paschal had not observed it. He tells us also that from given numbers multiplied after a certain manner he could produce very {illeg}y the numbers of very many series proceeding in infinitum, & that he could solve many Problemes in Progressions by adding subducting multiplying & \or/ dividing the Progressions, as by dividing an unit by a series of natural, triangular, Pyramidal, or triangulo-triangular numbers; & \principally/ that he could find the summ of a series of fractions so produced. See the mystery. From the series $\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}$ &c subduct all terms but the first ter & the first term $\frac{1}{1}$ will remain $\frac{1}{2}=\frac{2}{1\times 2\times 3}+\frac{2}{2\times 3\times 4}+\frac{2}{3\times 4\times 5}+\frac{2}{4\times 5\times 6}+\mathrm{\&c}$ And from the first series take all the terms but the two first & there will remain $\frac{3}{2}=\frac{2}{1\times 3}+\frac{2}{2\times 4}+\frac{2}{3\times 5}+\frac{2}{4\times 6}+\mathrm{\&c}$ {illeg} When one of M^{r} Newtons se From the series $\frac{1}{1}+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\mathrm{\&c}$ subduct all the terms but the first & there will remain $1=\frac{2}{1\times 3}+\frac{2}{3\times 5}+\frac{2}{5\times 7}+\mathrm{\&c}$. And by subducting this series from the last but one there will remain $\frac{1}{2}=\frac{2}{2\times 4}+\frac{2}{4\times 6}+\frac{2}{6\times 8}+\mathrm{\&c}$ that is $1=\frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}+\mathrm{\&c}$ as above. And these were the \great/ inventions of M^{r} Leibnitz about Numbers in those days w^{ch} M^{r} Leibnitz pretended to in those days.] But this is to tell us that he first found out the differential calculus of Mouton & then by inventing the Analysis of infinitess|i|mals translated the \Moutons/ calculus to Geometry. The invent{i} His pretending to The invention of Moutons method is nothing to the purpose. It lies upon him to prove that he invented the Analysis of infinitesimals before the year 1677. In a letter to D^{r} Wallis dated 28 May 1697 he tells us that the consideration of differences & summs in series of numbers gave him the first light by observing that Differences answered to tangents & summs to quadratures: that is, it put him upon considering D^{r} Wallis's summatory method of Tangents Quadratures & D^{r} Barrow's differential method of Tangents|.| & beginning where D^{r} Barrow left off, as the Marquess de L'Hospital has observed. But this consideration alone could not give him light into the method of drawing Tangents without sticking at surds & into the universal extent of this method for solving all sorts of Problems. That light he had by his correspondence w^{th} M^{r} Oldenburg as is in the years 1676 & 1 as is manifest by what has been said above.

It has been pretended \suggested/ by M^{r} Leibnitz \himself/, that the Committee of R. Society published only those papers w^{ch} made against him & suppressed those that made for him. B

the symbol o, t|T|he proportion of the fluxions (w^{ch} are finite quantities) giving \all {sic}/ all the other moments. It is more natural & geometrical

And whereas M^{r} Leibnitz has represented that the use of the letter o is vulgar & destroys the advantages of the Differential Method

The reason of his inconsistency in this matter was his computing by a wonderfull unphilosophical errour, the quantity of impulsive force from the quantity of matter acquired by a falling body from the quantity of its matter & \of/ the space de{l}|s|cribed by it in falling; recconing the force acquired to be in a compound ratio of the matter & \the/ space \together/ described \by it/. Now \matter is as the weight thereof &/ the space described is as the square of the line of \its/ falling, & therefore according to M^{r} Leibnitz the force acquired in falling is as the in a compound ratio of the matter falling & in a falling body weight of the falling body & the square of the time of its falling. And by consequence, where the weight remains the same, the force acquired in falling will be as the square of the time So that if the time \of falling/ be divided into equal parts, & in the first part of y^{e} time four degrees of force will be acquired, in the three first parts of time nine degrees of force will be acquired, & so on perpe in the four first parts of time sixteen degrees of force will be acquired, & so on b{illeg} if in th And therefore if in the first part of the time one degree of force impulsive force be acquired in the second part of the time three degrees of force will in the third will be acquired, in the third part of the time five degrees of force will be acquired in the fourth part of time seven degrees of force will be acquired, & so on. And if \so/ the \weight or/ gravity of the body which is supposed to be uniform & by wh by its action impresses these impulsive forces upon the body, acts with three times more force in the second parts of \the/ time then in the first & with five times more force in the third part of the time then in the first & with seven times more force in the fourth part of the time then in the first, & so on. Which is as much as to say that the th falling falling body grows heavier & heavier as it falls, & becomes three times heave|i|er in the \midle of the/ second part of the time then in the \middle of the/ first & five times heavier in the \middle of the/ third part of the time then in the first middle of the first, & so on. Or that the weight of the body is proportional to the time of its falling: And by consequence that in the beginning of the first part of the time the body would hath|ve| no weight at all. Which is contrary to the Hypothesis of uniform gravity & to experience it self.

The Theory of Projectiles invented by Galilæo is founded upon the Hypothesis of uniform gravity, {illeg} & is generally approved by Mathematicians. Now uniform gravity is that which acts with an uniform force & in equal times {illeg} equally \{illeg}/ upon it & by acting {in}{illeg} by acting equall \by acting with equal forces upon {illeg}|th|e body/ communicates equal forces to it. If in the first part of time it communicates one degree of force & gives one degree of velocity \to the falling body/; in the second part of time by acting as much as in the first, it will communicate another degree of force & give another degree of velocity to the falling body; & in the third part of time it will generate a third part of force & a third degree of velocity & so on perpetually: & therefore the time of falling \the time of falling/, the force imprest, & the velocity of descent will be always proportional to one \another/ one another into the time of falling & to one another. But the space described by the falling body arises partly from the time of descent & partly from the velocity of the falling body & therefore will be in a compound ratio of them both or as the s{illeg}|q|uare of either of them, & by consequence as the square of the force.

And so if two equal bodies be thrown directly upwards tho one with double \a/ velocity to {illeg} double to that of y^{e} other, the swifter body will {illeg}rise four times higher then the other in four \three/ time twice the time & in that time the action of gravity upon it \for taking away the force by w^{ch} it ascends/ will be double to the action of gravity upon the other body, & therefore the force by w^{ch} it ascended & w^{ch} is taken away by that action is only double to the force by w^{ch} the other body ascended.

A body therefore of one pound weight is not (as M^{r} Leibnitz supposes in the Acta Eruditorum ad Annum 1686 pag. 162) throw in Vacuo four {f}{illeg}t times as high but sixteen times as high by the same quantity of impulsive force wherewith a body of four pound weight is thrown one foot high. The ground of his error is that he confounds [Acta Erudit: ad Ann. 1686 pag. 162; & ad Ann. 1690 pag. 234; & ad Ann. 1691 pag. 439; & an|d| Ann. 1695 pag. 155] the spaces described \in unequal times/ by bodies falling downwards or thrown upwards with the spaces described in equal times by bodies rising & falling in equal times at the ends of the unequal arms of a ballance. Whereas in the first case the spaces \described/ are as the time & the velocity together, that is, as the square of the velocity: in the second case the times being equal the spaces \described/ are only as the velocity|i|es.

And because tis true that in an horizontal place, where gravity neither adds to nor takes from the Velocity, a body of four pound weight will at the same be carried one foot by the same force \imprest at once/ whereby a body of one pound weight will at \in/ y^{e} same time be carried on four feet: therefore he supposes [A{n}|c|ta Erudit: ad Ann. 1686 pag 162,] that in a perpendicular plane wherein the force is not imprest at once but Gravity continually augments or diminishes the force impr velocity in proportion to the time of its acting; a body of four pounds weight will be thrown upwards one foot by the same force imprest at once whereby a body of four \one/ pound weight will be thrown upwards \carried up/ four foot; or that a body of One pound weight will in falling four foot, will acquire the same impulsive velocity force as a body i{illeg}|of| four pounds weight in falling one foot. Then w^{ch} nothing can be more contrary to experience both to reason & experience.

$\mathrm{BD}=\mathrm{AB}=\mathrm{a}$. $\mathrm{BC}=\mathrm{x}$. $\mathrm{BDEC}=\mathrm{y}$. $\mathrm{Cc}=\mathrm{o}$. $\mathrm{Bc}=\mathrm{x}+\mathrm{o}$. $\mathrm{CEec}=\mathrm{o}\mathrm{x}\frac{\mathrm{a}\mathrm{a}}{\mathrm{a}+\mathrm{x}+\mathrm{o}}=\stackrel{.}{\mathrm{y}}\mathrm{o}$. $\frac{\mathrm{e}+\mathrm{f}\mathrm{x}+\mathrm{g}\mathrm{x}\mathrm{x}+\mathrm{h}\mathrm{y}}{\mathrm{h}\mathrm{o}+\mathrm{m}\mathrm{x}+\mathrm{n}\mathrm{y}}=\mathrm{z}$. Or $\mathrm{z}=\mathrm{e}+\mathrm{f}{\mathrm{x}}^{\mathrm{m}}+\mathrm{h}{\mathrm{x}}^{\mathrm{p}}{\mathrm{y}}^{\mathrm{q}}$. 109^{{d}{illeg}t}.17^{gr}+41.8=151.33

p. 38 l. 14. The Principle upon w^{ch} he founded his Dynami is erroneous The forces of equal bodies are not as the spaces by them in unequal times but as the spaces described by them in equal times

P. 44. l. 1. For fluxions write moments. For fluxions are not differences.

P. 46. l. 15 3. M^{r} Newton in 1704 in the Introduction to his book of Quadratures published that he found the Method of Fluxions gradually in the years 1665 & 1666, the next year in giving an account of this book in the Acta Lips. for this was called in question & M^{r} Newton was accused of Plagiary. M^{r} Keill in 1711 defended him.

p. 49. l 10. In the Acta Eruditorum. for

P 49. l. 10 This was \written by M^{r} Leibnitz himself/ in pursuit of the charge of Plagiary published in the Acta Lips 1705 as above & conteined a letter of M^{r} Bernoulli \to M^{r} Leibnitz/ dated 7 Iulij 171 Iune 173|1|3 written to the same purpose. M^{r} Leibnitz was then at Vienna & the Copies of the Commercium were sent to him thither by several hands \from England/. And{illeg} thereupon he wrote to M^{r} Bernoulli to examin the Commercius|m| {illeg} \{to}/ \appealed from the judgm^{t} of the Committee/ \to/ that of M^{r} Bernoulli, desiring him to examin the Book Book, & inserted M^{r} Bernoulli's Answer into & pretending that he had not yet seen it himself; & M^{r} Bernoulli's answer dated 7 Iune 1713 he inserted into anoth a Paper of his own dated 29 Iulij 1713 & caused them to be published without their names, as if written by other persons unconcerned. And D^{r} Keill answered this Paper in the Iournal Literair.

P. 49. l. 19. The pretence that M^{r} Leibnitz wo intended to write a Commercium is a sham. The

P. 52. l. 7. Hoc factum est anno 1673 ad finem vergente.

P. 53. l. 6. Hæc \Newtoniana Leibnitiu{ss}/ non didit|c|it \vidit/ ante annum 1676.

P. 53. l penult. M^{r} Newton {illeg} scripsit Lemma II Lib. II Princip. & Scholium subjunctum, non ut methodum Leibnitio concederet sed ut eandem sibi vindicaret.

before his Lectures came abroad, {illeg} — — — — — because {they} did many more things of this kind. [He has not produced the least proof that he knew the method before the year 1677, but on the contrary in his Letter of 27 Aug. 1676 he questi declared his disbeleif that my methods were so general t|a|s I had described & said that many Problems, & among others the inverse Problems of Tangents, could not be reduced to æquations or quadratures; & placed the perfection of Analysis in Analytical Tables of Tangents & the Combinatory Art, saying of the first, *Nihil est quod novim in tota|TOTA| Analysi momenti majoris*; & of the second, *Ea vero nihil differt ab ANALYSI ILLA SVPREMA ad cujus intima Cartesius non pervenit: est enim ad eam cont|s|tituendam opus Alphabeto cogitationum humanarum*. And in a Letter from Amsterdam to M^{r} Oldenburg 2{illeg}|8| Novem. 1676 he was thinking to perfect the improve the Method of Tangents of Slusius by such a Table of Tangents.] In that Letter he placed the perfection of Analysis in other methods then the differential, & represented it improbable that my methods should be so general as I had affirmed

6. Post verba [Metaphysica in Idæis, Idææ et Philosophia omnis vera in Phænomenis fundantur, & incepit Newtonus a Phænomenis: Momortus ab idæis (clausis oculis) incipiendum esse & {illeg}e \clausis oculis & ab hujusmodi somnijs/ ad phænomena per{illeg}|g|endum \esse/ contendit.] adde: Sed cum audiamus quid Leibnitius ipse quam cui Author noster cum Bernoullijs \cum Bernoullijs/ duobus tantopere favet, he de de Metaphysicæ sanæ notionibus olim scripsit|.| in In charta enim quam [in Actis Leipsicis anno 1694 mense {sic} Martio {illeg}] *De Primæ Philosophiæ Emendatione & de Notione Substantiæ* scripsit quæ in Actis Eruditorum Lipsicis anno 1694 mense Martio impressa fuit hæc habet. Video pleros qui Mathematicis doctrinis delectantur — — — — hoc commodo caremus. Hactenus Leibnitius.

11. Post verba [rimari philosorum {sic} est.] adde. In Principiorum editione nova Newtonus \secunda sub finem,/ Newtonus scripsit se rationem proprietatum gravitatis ex phænomenis deducere nondum potuisse & Hypotheses se non fingere. Et postquam descriptionem Systematis manda{ni} motus Planetarum Cometarum \Æquinoxionem/ & maris nostri a vi gravitatis per phænomena et mathematicas demonstrationes derivasset subjungit motus particularum corporum a spiritu quadam s electri{illeg}|c|a \quodam quo attractio electrica perficetur/ quam maxime pendere, sed hæc paucis exponi non posse. Et Editores Actorum Lipsiensium (Anno 16 1714 mense

6. {A} \In/ Metaphysicæ sanæ notionibus.] Metaphysicam sanam intelligit Cartesianam \qua uti ab ideis inmatis ad existentiam veru{g}|m| pergitur/. Audiamus igitur \autem quid/ Cartesius Leibnitius de h{æ}c Metaphysicæ d{illeg} sanæ notionibus olim scripsit. Is enim in t|d|isseratione {sic} de Metaphysica sanæ notionibus De Primæ Philosophiæ emendatione & de Notione substantiæ in Actis Lipsicis anno 1694 mense Martio impressa, hæc habet. Video, ait, pleros qui Mathematicis doctrinis delectantur .......... hoc commodo caremus. Hactenus Leibnitius. Metaphysica uti in Idæis, Ideæ & Philosophia omnis vera in Phænomenis fundat|n|tur, et incipit Newtonus a Phænomenis: Momortus ab Idæis \Idæas/ [ Idæas non a Phænomenis deducendas \esse/ sed clausis oculis formandas, esse [& ab h{illeg} \hujusmodi/ somnijs] ab Idæis ad phænome pergendum esse contenditur deinde oculos \deinceps/ apperiendos esse ut phænome|na| beneficio Idæarum {illeg}l{illeg} videamus idæarum clare & distincte videamus.

qu{a} uti Ideæ non deducuntur a phænomenis sed innatæ esse finguntur, non \pergitur \disputatur// a Phænomenis a de existentiam r{eu}m \sed/ sed ab innatis Idæis existentia Dei res {illeg}ato{ri} spatium \extensionem/ esse corpus & res non extensis existere probatur ab Idæarum in{na}tarū figmentis.

— materiam in forma quacun positam, ope legum naturæ formas omnes quarem est capax successive induere assumere tandem ad illam quæ est hujus mundi devenire; philosophiam omnem naturalem &c

Vires attrahentes, tanquam qualitates occultas a Newtono introductas, Leibnitius in Tractatu de Bonitate Dei explod alibi \hostilite {illeg} animo/ explodit. Idem hic facit Momortus {illeg} [quasi hæ vires in Gallia damnat{æ|a|}{illeg} essent {illeg} Newtonus induceret a] id fraudulanter \quasi vires \attrahentes/ in Gallia damnatas: attrahentes a Newtono essent quas Newtonus induceret/ At \Q{ua}tates occultas &/ Vires attrahentes in Gallia damnatus Newtonus non int{illeg} inducit — — — per fractonem excitatis.

2B. Demonstratum Vacuum se dedisse.] Media|o||num| maxime fluid{a}|o|ꝝ qualia sunt Aer Aqua et Argentum vivum, resistentiam prope omnem a vi intertiæ partium fluidi oriri {&} & hujusmodi resis vim inertiæ proportionalem esse densitati fluidi Newto adeo resistentiam fluidorum Med{illeg}|i|orum summe fluidorum proportionalem esse densitati \eorundem/ Mediorum quamproxime Newtonus {et} ratione {illeg} experimentis ostendit, et legem resistentiæ \in talibus fluidis/ exposuit multis experimentis confirmavit. Spatia autem cœlestia cum \In/ Vacuo Boyliano confe{illeg}{illeg}|ri|t, in quo resistentia {illeg} corporum minime sentitur \cadentium/ longe minor epe \reperitur/ quam in aere & {illeg} si forte aliqua sit, sentiri tamen non potest In spatijs cœlestibus quæ supra Atmosphæram \sunt/ & aere omni magis \sunt/ vacua, sunt resistentiam min{illeg} adhuc minorem esse statuit, Et \et ad Philosophiam cœlorum sufficere/ quod spatia illa medijs resistentibus vacua sint|.|, Newtonum ad Philosophiam cœlorum sufficiter credit, ad {illeg} Quod \Cœlo/ Med{illeg}|i|o prope omni corporeo vacua sint \esse/ Newtonus credit sed credit sed {illeg} Philosophiam cœlorum in hac sententia non fundat {} {sic} Sufficit quod m|M|edio omni resistente prope destituuntur. Ad Medio corporeo non resistente plena sint, Quæstio est ad Philosophiam cœlorum minime spectan{t}|s|{illeg} s|S|ed ejusmodi \tamen/, medium corporeum dari Newtonus tamen non credit nisi forte duo sint corporum genera alterum cujus partes vi|m| inertio habent inertiæ, alterum cujus partes vim talem non habent.

5B. A In experientia universali nixa] Quanam experientia memb constat motus omnes animalium et hominis ipsius mere mechanicas esse {illeg} & ab actione voluntatis minime oriri nisi quatenus omnes cogitandi et volendi actione|i|s a \quibus oriuntur{s} motus isti oriuntur, in/ motu corporum consistet|ar|e?

6. In metaphysicæ san{illeg}|æ| notionibus] Metaphysicam sanam intelligit Cartesianam: Qua uti . . . . . . devenire statuit. Sed \{illeg} nec/ Ideas substantiarum nos habere aut \Sed neq|c|ꝫ Cartesit|u|s alicubi probavit/, facultatem cogitandi rem cogitantem esse |at corpus omne quiescere quod in corpore ambiente quiescit aut vim et {sic}| aut \ne aut/ rem omnes extensam extensionem esse, aut \nec{illeg}/ motum corporū in sola translatione relativa causis tere sine vi inertia consistere, aut \ne/ \aut/ rem|s| cogitantes null{illeg}spatio præse{illeg} adesse, aut Deus non esse omnipræsentem per substantiam suam Cartesius alicubi probab|v|it. Hæc omnia sunt meræ Hypotheses. \spatio non adesse, seu nullibi esse, ne|aut| Ideas \nobis/ innatas non posse \esse aut nos/ aut \nos/ ideas substantiarum nos habere./ Hæc omnia sunt Hypotheses. Metaphysica in Ideis abstractis fundatur, Philosophia vera in Phænomenis. In Philosophia vera oculis apertis disputamus, In Metaphysica clausis oculis disputamus, in Philosophia vera videmus et a rebus visis phænomenis veritatis deducimus. Metaphysica a Th Theogonia Gentium antiquarum originem habui habuit qua uti Mundum totum animas mortuorum aut partes esse Dei aut ejus potentias esse disputarunt \somniando/ statuerunt seu \finxerunt id est/ Deum esse rerum \omnium/ naturam: Physica de rebus cognitis et earum causis tantum tracta [causas rerum visarum et cognitarum a{illeg} per argumenta certa \missis religionibus/ deducit] \missis genti{l}|u|m religionibus/ incipit a rebus visis et cognitis et earum \rationes &/ causas investigat, id per argumenta si fieri potest mathematica.

be found by the same method, & then subjoyns: Nec quic{k}|q|uam hujusmodi scio ad n|q|uod hæc methodus id varijs modis sese non extendit. Imo tangentes ad Curvas Mechanicas (siquando id non alias fiat) hujus ope d{illeg}|u|cuntur. Et quicquid vulgaris Analysis per æquationes ex finito terminorum numero constantes (quando id sit possibile) perficit, hæc per æquationes infinitas semper perficit: Ut nihil dubitaverim etiam nomen Analysis etiam huic tribuere.

Tempus uti per quantitatem quamcun uniformiter fluentem, fluxionem ejus per unitatem, et momentum per literam o Newtonus designat. Aliarum quantitatum fluentes \fluxiones/ designat is per alia symbola, et momenta earum per symbola illa ducta in momentum o, et areas curvarum per ordinatas quadrato inclusas. D. Leibnitius pro fluxionibus nulla habet symbola, pro momentis literam d præfigit symbolis fluentium, et pro areis præ fig literam \∫/ præfigit symbolis Ordinatarum. Newtonus calculo suo usus est in Analysi quam Barrovius cum Collinio mense Iulio anni 1669 communicavit; et hanc methodum tum in Octob. 1676 valde generalem esse significavit, & verbis partim apertis differentialem vocat, anno 1676 minime invenerat. Scripsit enim ad Oldenburgum eo anno, Augusti 27, in hæc verba *multa esse adeo mira et implexa ut ne ab æquationibus pendeant ne a Quadraturis: qualia sunt (ex multis alijs) problemata methodi Tangentium inversæ, quæ etiam Cartesius in potestate \non/ esse fassus est*. At acceptis Newtoni literis, anno sequente in Literis ad Oldenburgum 21 Iunij datis, methodum differentialem ut olim a se inventam communicare cœpit his verbis. *Clarissimi Slusij methodum tangentium nondum esse absolutam celeberrimo Newtono assentior: et jam a multo tempore rem tangentium longe generalius tractavi scilicet per differentias Ordinatarum. — Hinc in posterum nominando dy differentiam duarum proimarum y &c*. Quod methodum hanc a multo tempore invenerat probandum est. Nam D. Leibnitius pro se testis esse non potest. Iniquissi\mus/ esset Iudex qui \in rebus controversis/ hominem quemvis pro seipso testem admitteret.

## GEORGI\VS/ LVD. D. G. MA{illeg}. BRIT. FRAN. ET HIB. REX. FIDEI DEFENSOR.

## Ecclesia tuta.

And such Letters were writ in Council the name of the Elders & people; \together/ the people being present at the writing of them, & giving their assent \(Act xv. 12 6, 12, 22, 23:)/ for the Elders were \did/ not to \act as/ Lords {in} /{did} \were not to act as take the oversight of the flock \of God/ as Lords |over Gods heritage but, as ensamples to the flock.| The Presiden{t}/\ was to act {illeg} by the advice & with the consent of the Council & the {illeg} President & Council were to ad{illeg}vise w^{th} the peo{p} acquainte the people therewith & to advise the & teach them & procure their assent for avoiding ill will discord \faction/ & schism.

Fluxionum ei cognov{illeg} serierum infinitarum incidisse.

In the winter between the years 1664 & 1665 upon reading D^{r} Wallis's Arithmetica Infinitorum & trying to interpole his series \progressions/ for squaring the circle, I found out first an infinite series for squaring the circle & then another infinite series for squaring the Hyperbola \& soon after/. This latter series was the same with that published by M^{r} Mercator between \about/ 3 or 4 years after .|[|We found out this series by different methods & \Who found out this series first I do not {em} know But And/ two years before he published it I \had/ found out the \general/ method of finding squaring curves by such series by the help of division & extraction of roots.] & two years before he published it I had a a {sic} general method of squaring curves by such series by the help of division & extraction of roots.

pag tertia, et {illeg} specimen ejus methodi cum Gre Specimen Sed et specimen ejus methodi Collinius cum Gregorio communicavit. {illeg} Et Anno 1676 \Newtonus/ eandem in Epistola quadam quoad series infinitas in epistola|i|s quadam ad celeberrimum virum G. G. L D. Leibnitium literis \duabus/ duabus ad D. Oldenburgum datis et ad D. Leibnitium missis, quoad series infinitas fuse explicuit, quoad Fluxiones vero {f}{illeg} subobscuri|e|{illeg} insinuavit. Scripsit enim se tunc compotem fuisse in epistolarum secunda Die 24 Octobris data, se tunc compotem fuisse — Et ibidem subnectit Theorema subnectit pro Binomijs quod quadraturam ad finita exhibet æquatione ubi fieri potest, ad infinita verò ubi qua{illeg}|d|ratura per finitam æquationem exhiberi \exhiberi/ nequit. Quo exemplo satis ostendit methodum utram et fluxionum, et \illar/ serierum infinitarū una et e{illeg}|a|ndem esse qu{illeg}t{illeg}d quod|a|d fundamentum, sed illam fluxionum generaliorem esse & {illeg}t{illeg} {illeg}lla ad illam {illeg} serierum infinitarum se habere ut genus ad speciem vel totum ad partem |fluxionum abs methodo serierum infinitarum perfici non posse| Hoc Theorema primum vocat hoc est primum Seriei Theorematum — patebit. Scripsit vero Newtonus methodum ex dictis haud difficile colligi, se vero \impræsentia/ ne fuse exponere cogeretur, eandem his literis transpositis celare (6accdæ13eff7i3l9n404qrr4st12vx. Quæ quidem ordinatæ hanc conficiunt sententiam Data æquatione Fluentes, quotcun quantitates involventes invenire Fluxiones et vice versa. Et hisce autem \Literis/ anno 1677 acceptis Leibnitius respondit se in parilem methodum incidisse uti Princip Phil. Natur. Math. pag 253 &c videre licet: cujus tamen explicationem (in cæteris suis inventis publicandis non raro nimis præ properus, uti in Act. Erudit. plus semel conspici queat) ad Annum us 1684 distulerit.

D. Iaco{illeg}|b|us Gregorius ex serie aliqua \unica/ Newtoni \serie/ quam a Collinio acceperit, {illeg} methodum ipsam serierum infinitarum collegit. D. Leibnitius circa annum 1675 Londini fuit, & postea cum Collinio commercium Epistolicum habuit ac|et| t{um} dem \subinde/ epistolas Newtoni ab Oldenburgo accepit; & tum demum rescripsit se methodūm {sic} Newtonianæ non dissimilem habere. Qua ratione \{illeg}/ in hanc methodum inciderit, ipse {illeg}p{illeg} novit: nobis sufficit quod Newtonus sit primus ejus inventor. Ipse Leibni \Is/ inventionem terminorum serierum infinitarum primus docuit, et secundus seriei cujus terminus est quantitas illa quam Leibnitius differentiam vocat primi termini & cujus \adeo/ inventionem \ipse/ sibi arrogat

Sed et Newtonus hanc methodum longius provexit quam alius {illeg} Ipsam enim ad quadraturam Curvarum per series infinitas quæ in {illeg} æquationes finitas vertuntur \ante {annum} 1676/ applicuit. I{illeg}

Ipsam etiam ad æquationes fluxionales — nequeunt. Et methodum generalem tradi{illeg}t \jam invenerat/ extrahendi fluxiones ex æquationibus fluxionalibus{illeg}.

Methodos insuper universaliores ad comparationes Curvarum quæ geometrice quadrari non possunt eodem tempore tradiderat — subministraverit.

Et has methodos \insuper/ ad enodanda naturæ phænomena in Principijs suis mathematicis p{illeg} adeo feliciter applicuit ut ejus inventis cultores methodi differentialis vix aliquid addiderint. Motus autem Planetarum in Ellipsibus \Tandem verò/ per gravitatem versus {illeg} umbilicum Planetas autem in Ellipsibus per gravitatem versus Solem revolvi & ar Planetas autem in se mutuo graves esse & maxime in solem, & gravitate ab \sua/ versus \circum/ solem re{illeg} in Ellipsibus revolvi ac radijs ad solem ductis areas describere temporibus proportionales, Newtonus ante annum 1677 demonstravit\re didicit/, & demonstrationem cum Societate Regia anno 1684 communicavit. Produe|ijt|runt hæc \demonstratio/ in lucem anno 1687, et Leibnitius {illeg} anno 1689 Theorema a Newtonianum, \sed/ \in Actis etiam seipsicis descripti{illeg} /differentiali \sed/ errante,\/ calculo suo differentiali ad incudem revocavit semel \aliqua{illeg}{ts} {illeg}/ at iterum {e}rravit nec tamen \non/ dubitavit asserere sit \quo/ propro mart{a} i{illeg} idem Theorema incidisse Theorema idem invenisse ut suum faceret. Asseruit enim se Theorema illud invenisse, sed c{illeg} in calculo aliquoties erravit, Nondum enim ideo nihil invenit. Sed nec methodum jam di{c}ice satis didicerat qua Newtonus Principia sua sua mathematica inven|st|it|g||avit|, Calcul Operari quidem in differentijs primis jam ante didicerat sed mentem Newtoni {q} modum operandi in differentijs secunds {sic} nondum ut ex erroribus ullos \ejus/ manifestum est. At ex eo tempore Leibnitius et alij hanc methodum ad Problemata difficiliora {illeg} applicare ceperunt et methodum differentialem|is| in quæstionibus difficilioribus a Leibnitio et Bernoulli enodandis e{nc}ol{eri} cœpit, facto \e{illeg}t/ initia a |re|solutione Problematum de resistenti{illeg}|j|{illeg}s mediorum & motibus Planetarum corporum cœlestium, quæ Newtonus antea \in Princip/ tractaverat per methodos \{illeg}/ fluxionum antea tractaverat enodat{illeg} antea confecerat & tractaverat, {illeg} in lucem ed ut ex ejus Principijs Mathematicis manifestum est Methodus igitur fluxionum quam Leibnitius differentialem vocat, non prius apud exteros excoli et celebrari cœpit, quàm Newtonus eandem {illeg} apud nos ad summum fastigium ex|v|exerat. {et}

Iustitia hic reddenda est