[illeg] When I said in my Letter of 13 Iune 1676 that Analysis by the help of
infinite æquations extended to almost all sorts of Problems \except perhaps some numeral ones like those of Diophantus/ but \yet/ became not uni-
versal without some further methods of reducing Problems to infinite series then
by division & extraction of roots, & M^r Leibnitz in his answer replied Quod
dicere videmini pleras difficultates (exceptis Problematibus Diophantæis) ad
series infinitas reduci; id mihi non videtur. Sunt enim mult [illeg] |a| us adeo
mira et implexa ut ne ab æquationibus pendeant ne [illeg] |e|x quadraturis; qualia
sunt (ex multis alijs) Problemata methodi tangentium inversæ. Its certain
that we both spake of [illeg] [d] resolving Problems by reducing them to infinite
series. And yet ye would now persuade us that he spake of reducing
them to vulgar equations, & did not intend to deny that they useth
any thing more then that they could be resolved by reducing them to
vulgar equations or quadratures other equations then the vulgar|.| , [but
knew himself how to reduce them to differential equations. [illeg] And that
a few lines after where he told us that he had solved a Problem by certa
Analysis he meant this method.] He I spake of reducing almost all problems
to converging series by methods w^ch I had not yet described, he denied the possi
bility of this & now would make us beleive that he did not deny it.

M^r Leibnitz in his Letter of 27 Aug. 1676 wrote \thus/: Quod dicere
videmini pleras difficultates (exceptis Problematibus Diophantæis) ad
Series infinitas reduci; id mihi non videtur. Sunt enim multa us adeo
mira et implexa ut ne ab æquationibus pendeant ne ex quadraturis
Qualia sunt ex multis alijs Problemata methodi tangentium inversæ. [sq] [illeg]
And when I answered that the such Problemes were in my power he
replied \(in his Letter of 21 Iune 1677)/ that he [s] |c|o [c] |n|ceived that I meant by infinite series but he meant
Geometrically. And now p |h|e persists in the same reply saying that he me [illeg] |a|nt
|by| vulgar equations. See [illeg] I See the Answer to this in the Commercium
Epistolicum pag 92.

He saith that one may judge that when he wrote the his Letter
of Aug 27. 1676 he had some entrance into the differential calculus
because he said there that he \had/ solved the Probleme of Beaune certa Analy
si a certain Analysis. But what if that Probleme may be solved certa
Analysi without the differential method. For no further analysis is requisite
then this, That th as the Numbers [illeg] are in Geometrical progression when
their Logarithms are in Arithmetical progression, so the Ordinate of the Curve
desired increases or decreases in Geometrical progression when the Abscissa
increases in Arithmetical, & therefore the Abscissa & Ordinate have the
\same/ relati ō to one another as the Logarithm & its number. And to infer fro this
that M^r Leibnitz had the entrance into y^e metho differentias |l| method is as
if one should \say/ that Archimedes had entrance into it when \because/ he \he drew tang^ts to y^e spiral/ squared the
Prabola [sic], & \found/ the proportion between the sphere & C y |i|lyn̄der, or that D^r Walli Caval
lerius \Fermat/ & Wallis had entrance into it because they did many more things of this
kind.

Let x be the Abscissa, y the Ordinat, [q] |p| the sub perpendicular cut off by the
Abscissa, s the subperpendicula \r the radius of curvature/ & $\stackrel{.}{\mathrm{x}}\mathrm{o}$ the moment of the Axis & $\frac{\mathrm{s}\mathrm{o}}{\mathrm{y}}\stackrel{.}{\mathrm{x}}$ will be the
mom first moment of the ordinate \ $\stackrel{.}{\mathrm{y}}\mathrm{o}$ / & p [y] [illeg] $\frac{\mathrm{r}\mathrm{y}}{\mathrm{p}}$ . & $\mathrm{o}\mathrm{o}+\frac{\mathrm{s}\mathrm{s}\mathrm{o}\mathrm{o}}{\mathrm{y}\mathrm{y}}=\frac{2\mathrm{r}\mathrm{y}}{\mathrm{p}}\times 2\mathrm{y}\stackrel{..}{\mathrm{y}}\mathrm{o}\mathrm{o}=\frac{\mathrm{p}\mathrm{p}\mathrm{o}\mathrm{o}}{\mathrm{y}\mathrm{y}}$.
& $\frac{{\mathrm{p}}^3\mathrm{o}\mathrm{o}}{4\mathrm{r}{\mathrm{y}}^3}$ the second moment y $\stackrel{..}{\mathrm{y}}\mathrm{o}\mathrm{o}$. $\frac{{\mathrm{p}}^3\mathrm{o}\mathrm{o}}{2\mathrm{r}{\mathrm{y}}^3}=2\stackrel{..}{\mathrm{y}}\mathrm{o}\mathrm{o}$. $\stackrel{.}{\mathrm{x}}\mathrm{s}=\mathrm{y}\stackrel{..}{\mathrm{y}}$. ${\mathrm{p}}^3\stackrel{.}{\mathrm{x}}\stackrel{.}{\mathrm{x}}=4\mathrm{r}{\mathrm{y}}^3\stackrel{..}{\mathrm{y}}$

In my Letters of 13 Iune & 24 Octob. 1676 I I spake of my methods
of Series & Fluxions interwoven with [illeg] one another, by the name of my
general method & said that it extended to almost all sorts of Problems
& particularly to inverse Problemes of Tangents, & to the squaring of
Curves \by series/ w^ch breake [illeg] |o|f & become finite t |w|hen the Curve can be squared
by a finite equation & I set down such a series & illustrated it with
examples. I gave instansces also of the inv the inverse method of tangents saying
that when the relation of any two sides of the right angled triangle were
conteined was de conteined under the tangent subtangent & ordinate were [illeg] |w|as