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 $\boxed{\frac{\mathrm{a}\mathrm{a}}{4\mathrm{x}}}$ 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^rLeibnitz|.| 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 <3r> 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}$\mathrm{d}\frac{1}{2}\times \mathrm{p}\frac{1}{2}$ ${\mathrm{d}}^{\frac{1}{2}}{\mathrm{r}}^{\frac{1}{2}}=\frac{{\mathrm{d}}^{\frac{1}{2}}}{{\mathrm{d}}^{\frac{1}{2}}}.\frac{{\mathrm{p}}^{\frac{1}{2}},\stackrel{.}{\mathrm{x}},{\mathrm{d}}^{\frac{1}{2}}}{2\sqrt{\mathrm{d}\mathrm{x}-\mathrm{x}\mathrm{x}}}$ $\begin{array}{l}\mathrm{3477∟142857}\\ \mathrm{1042∟5}(0299815\\ \underline{\phantom{0}6954285714}\\ \phantom{0}3470714286\\ \phantom{0}\underline{3929428571}\\ \phantom{00}341285715\end{array}$ $\begin{array}{r}341285715\\ \underline{312942857}\\ 28342858\\ \underline{27817143}\\ 525715\\ 347714\end{array}$ $\begin{array}{r}178001\\ \underline{173857}\\ 144\end{array}$ $\begin{array}{r}\mathrm{19∟63}\\ \underline{\phantom{00}126}\\ \phantom{00}\mathrm{7∟56}\end{array}$

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 & $6⁤\frac{1}{2}$ 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

$\begin{array}{rrr}3477143)& \underline{\mathrm{1042∟5}\phantom{00000}}& (0.2998151\\ 0.3010299& 6954286\phantom{00}& \phantom{0.0000000}\\ 0.0752575& 31294287\phantom{0}& \phantom{0.0000000}\\ \mathrm{b\; 1.\; 1∟1892}\phantom{10}\mathrm{a}\phantom{-10\mathrm{b}⁤\frac{1}{2}}& 31294287& \phantom{0.0000000}\\ 0.1892\phantom{00}\mathrm{a}-\mathrm{b}\phantom{10⁤\frac{1}{2}}& & \phantom{0.0000000}\\ 0.0946\phantom{10}5\mathrm{a}-5\mathrm{b}\phantom{1⁤\frac{1}{2}}& 27817144& \phantom{0.0000000}\\ 0.054\phantom{01}6\mathrm{b}-5\mathrm{a}\phantom{1⁤\frac{1}{2}}& 34771& \phantom{0.0000000}\\ 0.162\phantom{0}18\mathrm{b}-15\mathrm{a}\phantom{⁤\frac{1}{2}}& \underline{\phantom{00000}17386}& \phantom{0.0000000}\\ 189\phantom{0}21\mathrm{b}-17⁤\frac{1}{2}\mathrm{a}& 1042499628& \phantom{0.0000000}\\ 18⁤\frac{1}{2}\mathrm{a}-22\mathrm{b}\phantom{021\mathrm{b}-17⁤\frac{1}{2}\mathrm{a}}& \underline{348}& \phantom{0.0000000}\\ 37.44.//5⁤\frac{2}{7}. 6⁤\frac{2}{7}.\phantom{15\mathrm{a}}& 499976& \phantom{0.0000000}\end{array}$ 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 & $6⁤\frac{1}{2}$ 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

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 < insertion from the left margin > 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 < text from f 4v resumes > 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 $\mathrm{p}\mathrm{m}\mathrm{a}+\mathrm{x}-\frac{\mathrm{p}\mathrm{n}}{\mathrm{a}-\mathrm{x}}.\mathrm{y}\colon\colon \stackrel{.}{\mathrm{y}}.\stackrel{.}{\mathrm{x}}\colon\colon \stackrel{.}{\mathrm{y}}.1.\frac{\mathrm{p}\mathrm{m}}{\mathrm{a}+\mathrm{x}}-\frac{\mathrm{p}\mathrm{n}}{\mathrm{a}-\mathrm{x}}=\mathrm{y}\stackrel{.}{\mathrm{y}}$ $\boxed{\frac{\mathrm{p}\mathrm{m}}{\mathrm{a}+\mathrm{x}}}-\boxed{\frac{\mathrm{p}\mathrm{n}}{\mathrm{a}-\mathrm{x}}}=\frac{1}{2}\mathrm{y}\mathrm{y}$.

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.

For S^r Isaac Newton at his house in S^t Martins lan\e/ \street/ nigh Leîsterffields

These

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 <6bis(v)> 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

Aug^t y^e 7^th 1716

Y^r most humble & most obliged Servant Wm Newton

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.

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.

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 proportio <10r> nal. 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 <13v> 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^rIones 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 <16v> 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.