Copy of letter from Newton to Henry Oldenburg, dated 13 June 1676

|Fragmentum|

|Misi apographū {Suj} ad Dn. Leibnitium per Samuelem Regiū Vratislaviensem julij 26. 1676.|

Dignissime D^ne

Quanquam D^{illeg}. Leibnitij modestia, in excerptis quæ ex epistola ejus ad me nuper misisti, \nostratibus/ multùm tribuat Mathematicis nostræ gentis circa speculationem quandam infinitarum serierum \de qua jam cœpit esse rumor/: nullus dubito tamen quin ille, non tantùm (quod asserit) methodum reducendi quantitates quascunqꝫ in ejusmodi series, sed et varia compendia, fortè nostris similia, si non et meliora, adinvenerit. Quoniam tamen ea scire pervelit quæ ab Anglis hac in re inventa sunt, et ipse ante annos aliquot in hanc speculationem inciderim: ut votis ejus aliqua saltern ex parte satisfac{illeg}erem, nonnulla eorum quæ mihi occurrerunt, ad te transmisi.

Fractiones in infinitas {illeg}|s|eries reducuntur per divisionem et quantitates radicales per extractionem radicum, perinde instituendo operationes istas in speciebus ac institui solent in decimalibus numeris. Hæc sunt fundamenta harum reductionum; sed extractiones radicum,{illeg} et divisiones, multum abbreviantur per hoc Theorema.

${\left.\overline{\mathrm{P}+\mathrm{P}\mathrm{Q}}\right\}}^{\frac{\mathrm{m}}{\mathrm{n}}}={\mathrm{P}}^{\frac{\mathrm{m}}{\mathrm{n}}}+\frac{\mathrm{m}}{\mathrm{n}}\mathrm{A}\mathrm{Q}+\frac{\mathrm{m}-\mathrm{n}}{2\mathrm{n}}\mathrm{B}\mathrm{Q}+\frac{\mathrm{m}-2\mathrm{n}}{3\mathrm{n}}\mathrm{C}\mathrm{Q}+\frac{\mathrm{m}-3\mathrm{n}}{4\mathrm{n}}\mathrm{D}\mathrm{Q}+\text{\&c}$ Ubi $\mathrm{P}+\mathrm{P}\mathrm{Q}$ significat quantitatem cujus radix vel \etiam/ dimensio quævis vel radix dimensionis investiganda est, P primum terminum quantitatis ejus, Q reliquos terminos divisos per primum, & $\frac{\mathrm{m}}{\mathrm{n}}$ numeralem indicem dimensionis ipsius $\mathrm{P}+\mathrm{P}\mathrm{Q}$ sive dimensio illa integra sit, sive (ut ita loquar) fracta, sive affirmativa, sive negativa. Nam sicut Analystæ pro $\mathrm{a}\mathrm{a}$, $\mathrm{a}\mathrm{a}\mathrm{a}$, &c s{illeg}|c|ribere \solent/ ${\mathrm{a}}^2$, ${\mathrm{a}}^3$, sic ego pro $\sqrt{}\mathrm{a}$, $\sqrt{}{\mathrm{a}}^3$, $\sqrt{}c.{\mathrm{a}}^5$ &c {illeg} scribo ${\mathrm{a}}^{\frac{1}{2}}$, ${\mathrm{a}}^{\frac{3}{2}}$, ${\mathrm{a}}^{\frac{5}{3}}$, & pro $\frac{1}{\mathrm{a}}$, $\frac{1}{\mathrm{a}\mathrm{a}}$, $\frac{1}{{\mathrm{a}}^3}$ scribo ${\mathrm{a}}^{-1}$, ${\mathrm{a}}^{-2}$, ${\mathrm{a}}^{-3}$. Et sic pro $\frac{\mathrm{a}\mathrm{a}}{\sqrt{}\mathrm{c:}\overline{{\mathrm{a}}^3+\mathrm{b}\mathrm{b}\mathrm{x}}}$ scribo $\mathrm{a}\mathrm{a}\times {\left.\overline{{\mathrm{a}}^3+\mathrm{b}\mathrm{b}\mathrm{x}}\right\}}^{-\frac{1}{3}}$, & pro $\frac{\mathrm{a}\mathrm{a}\mathrm{b}}{\genfrac{}{}{0ex}{}{\phantom{.}}{\sqrt{}\mathrm{c:}\stackrel{‾}{\overline{{\mathrm{a}}^3+\mathrm{b}\mathrm{b}\mathrm{x}}\times \overline{{\mathrm{a}}^3+\mathrm{b}\mathrm{b}\mathrm{x}}}}}$ scribo $\mathrm{a}\mathrm{a}\mathrm{b}\times {\left.\overline{{\mathrm{a}}^3+\mathrm{b}\mathrm{b}\mathrm{x}}\right\}}^{-\frac{2}{3}}$: in quo ultimo casu si ${\left.\overline{{\mathrm{a}}^3+\mathrm{b}\mathrm{b}\mathrm{x}}\right\}}^{-\frac{2}{3}}$ conciapiatur es{illeg}|s|e ${\left.\overline{\mathrm{P}+\mathrm{P}\mathrm{Q}}\right\}}^{\frac{\mathrm{m}}{\mathrm{n}}}$ in Regula; erit $\mathrm{P}={\mathrm{a}}^3$, $\mathrm{Q}=\frac{\mathrm{b}\mathrm{b}\mathrm{x}}{{\mathrm{a}}^3}$, $\mathrm{m}=-2$ & $\mathrm{n}=3$. Deniqꝫ pro terminis inter operandum inventis in Quoto, usurpo A, B, C, D &c nempe A pro primo termino ${\mathrm{P}}^{\frac{\mathrm{m}}{\mathrm{n}}}$, B pro secundo $\frac{\mathrm{m}}{\mathrm{n}}\mathrm{A}\mathrm{Q}$, & sic deinceps. Cæterùm usus Regulæ patebit exemplis.

Exempl: 1. Est $\sqrt{}\overline{\phantom{|}\mathrm{c}\mathrm{c}+\mathrm{x}\mathrm{x}\phantom{|}}\left(\text{seu}{\left.\overline{\mathrm{c}\mathrm{c}+\mathrm{x}\mathrm{x}}\right\}}^{\frac{1}{2}}\right)=\mathrm{c}+\frac{\mathrm{x}\mathrm{x}}{2\mathrm{c}}-\frac{{\mathrm{x}}^4}{8{\mathrm{c}}^3}+\frac{{\mathrm{x}}^6}{16{\mathrm{c}}^5}-\frac{5{\mathrm{x}}^8}{128{\mathrm{c}}^7}$$+\frac{7{\mathrm{x}}^{10}}{256{\mathrm{a}}^9}+\text{\&c.}$ |c|. Nam in hoc casu est $\mathrm{P}=\mathrm{c}\mathrm{c}$, $\mathrm{Q}=\frac{\mathrm{x}\mathrm{x}}{\mathrm{c}\mathrm{c}}$, $\mathrm{m}=1$, $\mathrm{n}=2$, $\mathrm{A}\left(={\mathrm{P}}^{\frac{\mathrm{m}}{\mathrm{n}}}={\left.\overline{\mathrm{c}\mathrm{c}}\right\}}^{\frac{1}{2}}\right)=\mathrm{c}$. $\mathrm{B}\left(=\frac{\mathrm{m}}{\mathrm{n}}\mathrm{A}\mathrm{Q}\right)=\frac{\mathrm{x}\mathrm{x}}{2\mathrm{c}}$. $\mathrm{C}\left(=\frac{\mathrm{m}-\mathrm{n}}{2\mathrm{n}}\mathrm{B}\mathrm{Q}\right)=\frac{-{\mathrm{x}}^4}{8{\mathrm{c}}^3}$, & sic deinceps.

Exempl: 2. Est {illeg} $\sqrt{}⑤\overline{{\mathrm{c}}^5+{\mathrm{c}}^4\mathrm{x}-{\mathrm{x}}^5}\left(\text{i.e.}{\left.\overline{{\mathrm{c}}^5+{\mathrm{c}}^4\mathrm{x}-{\mathrm{x}}^5}\right\}}^{\frac{1}{5}}\right)=\mathrm{c}+\frac{{\mathrm{c}}^4\mathrm{x}-{\mathrm{x}}^5}{5{\mathrm{c}}^4}$ $\frac{-2{\mathrm{c}}^8\mathrm{x}\mathrm{x}+4{\mathrm{c}}^4{\mathrm{x}}^6-2{\mathrm{x}}^{10}}{25{\mathrm{c}}^9}+\text{\&c}$, ut patebit substituendo in allatam Regulam 1 pro m, 5 pro n, ${\mathrm{c}}^5$ pro P, & $\frac{{\mathrm{c}}^4\mathrm{x}-{\mathrm{x}}^5}{{\mathrm{c}}^5}$ pro Q. Potest etiam $-{\mathrm{x}}^5$ substitui pro P, & $\frac{{\mathrm{c}}^4\mathrm{x}+{\mathrm{c}}^5}{-{\mathrm{x}}^5}$ pro Q, et tunc evadet $\sqrt{}⑤\overline{{\mathrm{c}}^5+{\mathrm{c}}^4\mathrm{x}-{\mathrm{x}}^5}=-\mathrm{x}+\frac{{\mathrm{c}}^4\mathrm{x}+{\mathrm{c}}^5}{5{\mathrm{x}}^4}+\frac{2{\mathrm{c}}^8\mathrm{x}\mathrm{x}+4{\mathrm{c}}^9\mathrm{x}+{\mathrm{c}}^{10}}{25{\mathrm{x}}^9}+\text{\&c}$. Prior modus eligendus est si {illeg} x valde parvum sit, posterior si valde magnum.

<1v>

Exempl: 3. Est $\frac{\mathrm{N}}{\sqrt{}③\overline{{\mathrm{y}}^3-\mathrm{a}\mathrm{a}\mathrm{y}}}\left(\text{hoc est}{\mathrm{N}\times \left.\overline{{\mathrm{y}}^3-\mathrm{a}\mathrm{a}\mathrm{y}}\right\}}^{-\frac{1}{3}}\right)=\mathrm{N}\times \stackrel{‾}{\frac{1}{\mathrm{y}}+\frac{\mathrm{a}\mathrm{a}}{3{\mathrm{y}}^3}+\frac{2{\mathrm{a}}^4}{9{\mathrm{y}}^5}+\frac{14{\mathrm{a}}^6}{81{\mathrm{y}}^7}}+$ Nam $\mathrm{P}={\mathrm{y}}^3$. $\mathrm{Q}=\frac{-\mathrm{a}\mathrm{a}}{\mathrm{y}\mathrm{y}}$. $\mathrm{m}=-1$. $\mathrm{n}=3$. $\mathrm{A}\left(={\mathrm{P}}^{\frac{\mathrm{m}}{\mathrm{n}}}={\mathrm{y}}^{3\times \frac{-1}{3}}\right)={\mathrm{y}}^{-1}$. hoc est $\frac{1}{\mathrm{y}}$. $\mathrm{B}\left(=\frac{\mathrm{m}}{\mathrm{n}}\mathrm{A}\mathrm{Q}=\frac{-1}{3}\times \frac{1}{\mathrm{y}}\times \frac{-\mathrm{a}\mathrm{a}}{\mathrm{y}\mathrm{y}}\right)=\frac{\mathrm{a}\mathrm{a}}{3{\mathrm{y}}^3}\text{. \&c}$

Exempl: 4. Radix cubica ex quadrato-quadrato ipsius $\mathrm{d}+\mathrm{e}$ est (hoc est $\left.{\left.\overline{\mathrm{d}+\mathrm{e}}\right\}}^{\frac{4}{3}}\right)$ est ${\mathrm{d}}^{\frac{4}{3}}+\frac{4\mathrm{e}{\mathrm{d}}^{\frac{1}{3}}}{3}+\frac{2\mathrm{e}\mathrm{e}}{9{\mathrm{d}}^{\frac{2}{3}}}-\frac{4{\mathrm{e}}^3}{81{\mathrm{d}}^{\frac{5}{3}}}+\text{\&c}$. Nam $\mathrm{P}=\mathrm{d}$. $\mathrm{Q}=\frac{\mathrm{e}}{\mathrm{d}}$. $\mathrm{m}=4$. $\mathrm{n}=3$. $\mathrm{A}\left(={\mathrm{P}}^{\frac{\mathrm{m}}{\mathrm{n}}}\right)={\mathrm{d}}^{\frac{4}{3}}$ &c.

Eodem modo simplices etiam potestates eliciuntur. Ut si quadrato-cubus ipsius $\mathrm{d}+\mathrm{e}\left(\text{hoc est}{\left.\overline{\mathrm{d}+\mathrm{e}}\right\}}^5\text{, seu}{\left.\overline{\mathrm{d}+\mathrm{e}}\right\}}^{\frac{5}{1}}\right)$ desideretur: erit juxta Regulam $\mathrm{P}=\mathrm{d}$. $\mathrm{Q}=\frac{\mathrm{e}}{\mathrm{d}}$. $\mathrm{m}=5$ & $\mathrm{n}=1$; adeoqꝫ $\mathrm{A}\left(={\mathrm{P}}^{\frac{\mathrm{m}}{\mathrm{n}}}\right)={\mathrm{d}}^5$, $\mathrm{B}\left(=\frac{\mathrm{m}}{\mathrm{n}}\mathrm{A}\mathrm{Q}\right)=5{\mathrm{d}}^4\mathrm{e}$, & sic $\mathrm{C}=10{\mathrm{d}}^3\mathrm{e}\mathrm{e}$, $\mathrm{D}=10\mathrm{d}\mathrm{d}{\mathrm{e}}^3$, $\mathrm{E}=5\mathrm{d}{\mathrm{e}}^4$, $\mathrm{F}={\mathrm{e}}^5$, & $\mathrm{G}\left(=\frac{\mathrm{m}-5\mathrm{n}}{6\mathrm{n}}\mathrm{F}\mathrm{Q}\right)=0$. Hoc est ${\left.\overline{\mathrm{d}+\mathrm{e}}\right\}}^5={\mathrm{d}}^5+5{\mathrm{d}}^4\mathrm{e}+10{\mathrm{d}}^3\mathrm{e}\mathrm{e}+10\mathrm{d}\mathrm{d}{\mathrm{e}}^3+5\mathrm{d}{\mathrm{e}}^4+{\mathrm{e}}^5$.

Quinetiam Divisio, sive simplex sit, sive repetita, per eandem Regulam perficitur. Ut si $\frac{1}{\mathrm{d}+\mathrm{e}}$, $\left(\text{hoc est}{\left.\overline{\mathrm{d}+\mathrm{e}}\right\}}^{-1}\text{sive}{\left.\overline{\mathrm{d}+\mathrm{e}}\right\}}^{\frac{-1}{1}}\right)$ in seriem simplicium terminorum resolvendum sit: erit juxta Regulam $\mathrm{P}=\mathrm{d}$. $\mathrm{Q}=\frac{\mathrm{e}}{\mathrm{d}}$. $\mathrm{m}=-1$. $\mathrm{n}=1$. & $\mathrm{A}\left(={\mathrm{P}}^{\frac{\mathrm{m}}{\mathrm{n}}}={\mathrm{D}}^{\frac{-1}{1}}\right)={\mathrm{d}}^{-1}$ seu $\frac{1}{\mathrm{d}}$. $\mathrm{B}\left(=\frac{\mathrm{m}}{\mathrm{n}}\mathrm{A}\mathrm{Q}=-1\times \frac{1}{\mathrm{d}}\times \frac{\mathrm{e}}{\mathrm{d}}\right.$$=-\frac{\mathrm{e}}{\mathrm{d}\mathrm{d}}$, & sic $\mathrm{C}=\frac{\mathrm{e}\mathrm{e}}{{\mathrm{d}}^3}$, $\mathrm{D}=\frac{-{\mathrm{e}}^3}{{\mathrm{d}}^4}$ &c Hoc est $\frac{1}{\mathrm{d}+\mathrm{e}}=\frac{1}{\mathrm{d}}-\frac{\mathrm{e}}{\mathrm{d}\mathrm{d}}+\frac{\mathrm{e}\mathrm{e}}{{\mathrm{d}}^3}-\frac{{\mathrm{e}}^3}{{\mathrm{d}}^4}+\text{\&c}$

Sic et {sic} ${\left.\overline{\mathrm{d}+\mathrm{e}}\right\}}^{-3}$ (hoc est unitas \ter/ divisa per cubum ipsius $\mathrm{d}+\mathrm{e}$ vel semel per cubum ejus,) evadit $\frac{1}{{\mathrm{d}}^3}-\frac{3\mathrm{e}}{{\mathrm{d}}^4}+\frac{6\mathrm{e}\mathrm{e}}{{\mathrm{d}}^5}-\frac{10{\mathrm{e}}^3}{{\mathrm{d}}^6}+\text{\&c}$

Et $\mathrm{N}\times {\left.\overline{\mathrm{d}+\mathrm{e}}\right\}}^{-\frac{1}{3}}$ hoc est N divisum per radicem cubicam ipsius $\mathrm{d}+\mathrm{e}$ evadit $\mathrm{N}\times \stackrel{‾}{\frac{1}{{\mathrm{d}}^{\frac{1}{3}}}-\frac{\mathrm{e}}{3{\mathrm{d}}^{\frac{4}{3}}}+\frac{2\mathrm{e}\mathrm{e}}{9{\mathrm{d}}^{\frac{7}{3}}}-\frac{14{\mathrm{e}}^3}{81{\mathrm{d}}^{\frac{10}{8}}}+\text{\&c}}$

Et $\mathrm{N}\times {\left.\overline{\mathrm{d}+\mathrm{e}}\right\}}^{-\frac{3}{5}}$ (hoc est N divisum per radicem quadrato-cubicam ex cubo ipsius $\mathrm{d}+\mathrm{e}$, sive $\left.\frac{\mathrm{N}}{\sqrt{}⑤\overline{{\mathrm{d}}^3+3\mathrm{d}\mathrm{d}\mathrm{e}+3\mathrm{d}\mathrm{e}\mathrm{e}+{\mathrm{e}}^3}}\right)$ evadit $\mathrm{N}\times \frac{1}{{\mathrm{d}}^{\frac{3}{5}}}-\frac{3\mathrm{e}}{5{\mathrm{d}}^{\frac{9}{5}}}+\frac{12\mathrm{e}\mathrm{e}}{25{\mathrm{d}}^{\frac{13}{5}}}$$-\frac{52{\mathrm{e}}^3}{125{\mathrm{d}}^{\frac{18}{5}}}+\text{\&c.}$

Per eandem Regulam etiam Geneses Potestatum, Divisiones \per Potestates aut \per/ quantitates radicales/, et extractiones radicum {illeg} \altiorum/ in numeris etiam commodè instituuntur.

Extractiones Radicum affectarum in speciebus imitantur earum extractiones in numeris, sed Methodus Vietæ et Oughtredi nostri huic negotio minùs idonea est, Quapropter aliam excogitare adactus sum cujus specimen exhibent sequentia Diagrammata ubi dextra columna prodit substituendo in media columnâ valores ipsorum y, p, q, r {illeg} &c in sinistra columnâ expressos. Prius Diagramma exhibet resolutionem hujus literalis \numeralis/ æquationis ${\mathrm{y}}^3-2\mathrm{y}-5=0$; et {illeg} \hic/ in supremis numeris pars negativa Radicis subducta de parte affirmativa relinquit absolutam Radicem $2\underset{‾}{|09455148}$: et posterius Diagramma exhibet resolutionem hujus litera{illeg}|r|iæ æquationis ${\mathrm{y}}^3+\mathrm{a}\mathrm{x}\mathrm{y}+\mathrm{a}\mathrm{a}\mathrm{y}-{\mathrm{x}}^3-2{\mathrm{a}}^3=0$.

< insertion from f 2v >

$\begin{array}{l}\begin{array}{r}\phantom{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}\\ & \phantom{.\_\_\_\_\_\_\_\_\_\_\_\_\_.}& \begin{array}{r}\left(\begin{array}{cc}+& \mathrm{2,10000000}\\ -& \mathrm{0,00544852}\end{array}\right.\\ \begin{array}{cc}+& \mathrm{2,09455148}\end{array}\end{array}\hfill \end{array}\\ \begin{array}{rr}\phantom{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}& \phantom{.\_\_\_\_\_\_\_\_\_\_\_\_\_.}\\ \begin{array}{c}2+\mathrm{p}=\mathrm{y}\end{array}& \begin{array}{r}{\mathrm{y}}^3\\ -2\mathrm{y}\\ -5\end{array}& \begin{array}{llllllll}+& 8& +& 12\mathrm{p}& +& 6\mathrm{p}\mathrm{p}& +& {\mathrm{p}}^3\\ -& 4& -& \phantom{0}2\mathrm{p}\\ -& 5\end{array}\hfill \\ \begin{array}{c}\end{array}& \begin{array}{c}\\ \phantom{{\mathrm{p}}^3}\text{summa}\phantom{{\mathrm{p}}^3}\end{array}& \begin{array}{l}\\ -& 1& +& 10\mathrm{p}& +& 6\mathrm{p}\mathrm{p}& +& {\mathrm{p}}^3\end{array}\hfill \end{array}\\ \begin{array}{rr}\phantom{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}& \phantom{.\_\_\_\_\_\_\_\_\_\_\_\_\_.}\\ \begin{array}{r}+\mathrm{0,1}+\mathrm{q}=\mathrm{p}\end{array}& \begin{array}{r}+{\mathrm{p}}^3\\ +6\mathrm{p}\mathrm{p}\\ +10\mathrm{p}\\ -1\end{array}& \begin{array}{llllllll}+& \mathrm{0,001}& +& \phantom{0}\mathrm{0,03}\mathrm{q}& +& \mathrm{0,3}\mathrm{q}\mathrm{q}& +& {\mathrm{q}}^3\\ +& \mathrm{0,06}\phantom{1}& +& \phantom{0}\mathrm{1,2}\phantom{0\mathrm{q}}& +& 6\phantom{\mathrm{,3}\mathrm{q}\mathrm{q}}\\ +& 1\phantom{\mathrm{,001}}& +& \mathrm{10,}\phantom{0\mathrm{q}}\\ -& 1\end{array}\hfill \\ \begin{array}{c}\end{array}& \begin{array}{c}\\ \phantom{{\mathrm{q}}^3}\text{summa}\phantom{{\mathrm{q}}^3}\end{array}& \begin{array}{r}\\ \phantom{+}& \mathrm{0,061}& +& \mathrm{11,23}\mathrm{q}& +& \mathrm{6,3}\mathrm{q}\mathrm{q}& +& {\mathrm{q}}^3\end{array}\hfill \end{array}\\ \begin{array}{rr}\phantom{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}& \phantom{.\_\_\_\_\_\_\_\_\_\_\_\_\_.}\\ \begin{array}{c}-\mathrm{0,0054}+\mathrm{r}=\mathrm{q}\end{array}& \begin{array}{r}+{\mathrm{q}}^3\\ +\mathrm{6,3}\mathrm{q}\mathrm{q}\\ +\mathrm{11,23}\mathrm{q}\\ +\mathrm{0,061}\end{array}& \begin{array}{lllllll}-& \mathrm{0,0000001}& +& \phantom{0}\mathrm{0,000}\mathrm{r}& \phantom{+}& \text{\&c}& \phantom{{\mathrm{q}}^3}\\ +& \mathrm{0,0001837}& -& \phantom{0}\mathrm{0,068}& \phantom{\mathrm{q}}\\ -& \mathrm{0,060642}\phantom{0}& +& \mathrm{11,23}& \phantom{\mathrm{q}}\\ +& \mathrm{0,061}\end{array}\hfill \\ \begin{array}{c}\end{array}& \begin{array}{c}\\ \phantom{{\mathrm{q}}^3}\text{summa}\phantom{{\mathrm{q}}^3}\end{array}& \begin{array}{r}\\ -& \mathrm{0,0005416}& +& \mathrm{11,162}\mathrm{r}\end{array}\hfill \end{array}\\ \begin{array}{rr}\phantom{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}& \phantom{.\_\_\_\_\_\_\_\_\_\_\_\_\_.}\\ \begin{array}{c}-\mathrm{0,00004852}+\mathrm{s}=\mathrm{r}\end{array}& \phantom{.\_\_\_\_\_\_\_\_\_\_\_\_\_.}& \end{array}\\ \end{array}$

$\begin{array}{c}\begin{array}{l}\\ \begin{array}{r}& \phantom{.}\\ \phantom{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}& \phantom{.\_\_\_\_\_\_\_\_\_\_\_\_\_.}& \left(\mathrm{a}-\frac{\mathrm{x}}{4}+\frac{\mathrm{x}\mathrm{x}}{64\mathrm{a}}+\frac{131{\mathrm{x}}^3}{512\mathrm{a}\mathrm{a}}+\frac{509{\mathrm{x}}^4}{16384{\mathrm{a}}^3}\text{\&c}\right.\hfill \\ \end{array}\\ \begin{array}{rr}\phantom{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}& \phantom{.\_\_\_\_\_\_\_\_\_\_\_\_\_.}\\ \begin{array}{c}\mathrm{a}+\mathrm{p}=\mathrm{y}\end{array}& \begin{array}{c}{\mathrm{y}}^3\end{array}& \begin{array}{llllllll}\phantom{+}& {\mathrm{a}}^3& +& 3\mathrm{a}\mathrm{a}\mathrm{p}& +& 3\mathrm{a}\mathrm{p}\mathrm{p}& +& {\mathrm{p}}^3\end{array}\hfill \\ \begin{array}{c}\end{array}& \begin{array}{c}+\mathrm{a}\mathrm{x}\mathrm{y}\end{array}& \begin{array}{llll}+& \mathrm{a}\mathrm{a}\mathrm{x}& +& \mathrm{a}\mathrm{x}\mathrm{p}\end{array}\hfill \\ \begin{array}{c}\end{array}& \begin{array}{c}+\mathrm{a}\mathrm{a}\mathrm{y}\end{array}& \begin{array}{llll}+& {\mathrm{a}}^3& +& \mathrm{a}\mathrm{a}\mathrm{p}\end{array}\hfill \\ \begin{array}{c}\end{array}& \begin{array}{c}-{\mathrm{x}}^3\end{array}& \begin{array}{l}-{\mathrm{x}}^3\end{array}\hfill \\ \begin{array}{c}\end{array}& \begin{array}{c}-2{\mathrm{a}}^3\end{array}& \begin{array}{l}-2{\mathrm{a}}^3\end{array}\hfill \\ \phantom{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}& \phantom{.\_\_\_\_\_\_\_\_\_\_\_\_\_.}\end{array}\\ \begin{array}{rr}\phantom{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}& \phantom{\_\_\_\_\_\_\_\_\_\_\_\_}\\ \begin{array}{c}-\frac{1}{4}\mathrm{x}+\mathrm{q}=\mathrm{p}\end{array}& \begin{array}{c}{\mathrm{p}}^3\end{array}& \begin{array}{llllll}-& \frac{1}{64}{\mathrm{x}}^3& +& \frac{3}{16}\mathrm{x}\mathrm{x}\mathrm{q}& & \text{\&c}\end{array}\hfill \\ \begin{array}{c}\end{array}& \begin{array}{c}+3\mathrm{a}\mathrm{p}\mathrm{p}\end{array}& \begin{array}{llllll}+& \frac{3}{16}\mathrm{a}\mathrm{x}\mathrm{x}& -& \frac{3}{2}\mathrm{a}\mathrm{x}\mathrm{q}& +& 3\mathrm{a}\mathrm{q}\mathrm{q}\end{array}\hfill \\ \begin{array}{c}\end{array}& \begin{array}{c}+\mathrm{a}\mathrm{x}\mathrm{p}\end{array}& \begin{array}{llll}-& \frac{1}{4}\mathrm{a}\mathrm{x}\mathrm{x}& +& \mathrm{a}\mathrm{x}\mathrm{q}\end{array}\hfill \\ \begin{array}{c}\end{array}& \begin{array}{c}+4\mathrm{a}\mathrm{a}\mathrm{p}\end{array}& \begin{array}{l}-\mathrm{a}\mathrm{x}\mathrm{x}+4\mathrm{a}\mathrm{a}\mathrm{q}\end{array}\hfill \\ \begin{array}{c}\end{array}& \begin{array}{c}+\mathrm{a}\mathrm{a}\mathrm{x}\end{array}& \begin{array}{l}+\mathrm{a}\mathrm{a}\mathrm{x}\end{array}\hfill \\ \begin{array}{c}\end{array}& \begin{array}{c}-{\mathrm{x}}^3\end{array}& \begin{array}{l}-{\mathrm{x}}^3\end{array}\hfill \\ \phantom{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}& \phantom{.\_\_\_\_\_\_\_\_\_\_\_\_\_.}\end{array}\\ \begin{array}{rr}\phantom{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}& \phantom{.\_\_\_\_\_\_\_\_\_\_\_\_\_.}\\ \begin{array}{c}+\frac{\mathrm{x}\mathrm{x}}{64\mathrm{a}}+\mathrm{r}=\mathrm{q}\end{array}& \begin{array}{c}3\mathrm{a}\mathrm{q}\mathrm{q}\end{array}& \begin{array}{llll}+& \frac{3{\mathrm{x}}^4}{4096\mathrm{a}}& & \text{\&c}\end{array}\hfill \\ \begin{array}{c}\end{array}& \begin{array}{c}+\frac{3}{16}\mathrm{x}\mathrm{x}\mathrm{q}\end{array}& \begin{array}{llll}+& \frac{3{\mathrm{x}}^4}{1024\mathrm{a}}& & \text{\&c}\end{array}\hfill \\ \begin{array}{c}\end{array}& \begin{array}{c}-\frac{1}{2}\mathrm{a}\mathrm{x}\mathrm{q}\end{array}& \begin{array}{llll}-& \frac{1}{128}{\mathrm{x}}^3& -& \frac{1}{2}\mathrm{a}\mathrm{x}\mathrm{r}\end{array}\hfill \\ \begin{array}{c}\end{array}& \begin{array}{c}+4\mathrm{a}\mathrm{a}\mathrm{q}\end{array}& \begin{array}{llll}+& \frac{1}{16}\mathrm{a}\mathrm{x}\mathrm{x}& +& 4\mathrm{a}\mathrm{a}\mathrm{r}\end{array}\hfill \\ \begin{array}{c}\end{array}& \begin{array}{c}-{\mathrm{x}}^3\end{array}& \begin{array}{l}-{\mathrm{x}}^3\end{array}\hfill \\ \begin{array}{c}\end{array}& \begin{array}{c}-\frac{65}{64}{\mathrm{a}}^3\end{array}& \begin{array}{l}-\frac{65}{64}{\mathrm{a}}^3\end{array}\hfill \\ \begin{array}{c}\end{array}& \begin{array}{c}-\frac{1}{16}\mathrm{a}\mathrm{a}\mathrm{x}\end{array}& \begin{array}{l}-\frac{1}{16}\mathrm{a}\mathrm{a}\mathrm{x}\end{array}\hfill \\ \phantom{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}& \phantom{.\_\_\_\_\_\_\_\_\_\_\_\_\_.}\end{array}\\ \end{array}\\ \left.\phantom{\frac{0}{0^0}}+4\mathrm{a}\mathrm{a}-\frac{1}{2}\mathrm{a}\mathrm{x}\right)+\frac{131}{128}{\mathrm{x}}^3-\frac{15{\mathrm{x}}^4}{4096\mathrm{a}}\left(+\frac{131{\mathrm{x}}^3}{512\mathrm{a}\mathrm{a}}+\frac{509{\mathrm{x}}^4}{16384{\mathrm{a}}^3}\right.\text{.}\phantom{\frac{0}{0^0}}\end{array}$

In priori Diagrammate primus terminus valoris ipsorum p, q, r in prima columna invenitur dividendo primum terminum Summæ proximè superioris per coefficientem secundi termini ejusdem Summæ: Et idem terminus eodem ferè modo invenitur in secundo Diagrammate. Sed \hic/ præcipua difficultas est in inventione primi termini radicis: id quod methodo generali perficitur, sed hoc brevitatis gratia jam prætereo, ut et alia quædam quæ ad concinnandam operationem spectant. Neqꝫ hic compendia tradere vacat, sed dicam tantum in genere, quod radix cujusvis æquationis semel extracta pro regula resolvendi consimiles æquationes asservari possit; {illeg}|&| quod ex pluribus ejusmodi regulis, regulam generaliorem plerumqꝫ efformare liceat; quodqꝫ radices omnes, sive simplices sint sive affectæ, modis infinitis extrahi possint, de quorum simplicioribus itaqꝫ semper consulendum est.

< text from f 1v resumes > <2r>

Quomodo ex æquationibus, sic ad infinitas series reductis, areæ & longitudines curvarum, contenta et superficies solidorum, vel quorumlibet segmentorum figurarum quarumvis eorumqꝫ centra gravitatis determinantur, et quomodo etiam Curvæ omnes {illeg}|M|echanicæ ad ejusmodi æquationes infinitarum serierum reduci possint, indeqꝫ Problemata circa illas resolvi perinde ac si geo{m}etricæ essent, nimis longum foret describere. Sufficiat specimina quædam talium Problematum recensuisse: inqꝫ ijs brevitatis gratia literas A, B, C, D &c pro terminis seriei, sicut sub initio, nonnunquam usurpabo.

1 Si ex dato sinu recto vel sinu verso arcus desideretur: sit radius r et sinus rectus x eritqꝫ arcus $=\mathrm{x}+\frac{{\mathrm{x}}^3}{6\mathrm{r}\mathrm{r}}+\frac{3{\mathrm{x}}^5}{40{\mathrm{r}}^4}+\frac{5{\mathrm{x}}^7}{112{\mathrm{r}}^6}+\text{\&c}$: hoc est $=\mathrm{x}+\frac{1\times 1\times \mathrm{x}\mathrm{x}}{2\times 3\times \mathrm{r}\mathrm{r}}\mathrm{A}+\frac{3\times 3\mathrm{x}\mathrm{x}}{4\times 5\mathrm{r}\mathrm{r}}\mathrm{B}+\frac{5\times 5\mathrm{x}\mathrm{x}}{6\times 7\mathrm{r}\mathrm{r}}\mathrm{C}+\frac{7\times 7\mathrm{x}\mathrm{x}}{8\times 9\mathrm{r}\mathrm{r}}\mathrm{D}+\text{\&c}$. Vel sit d diameter {illeg} ac x sinus versus, et erit arcus $={\mathrm{d}}^{\frac{1}{2}}{\mathrm{x}}^{\frac{1}{2}}+\frac{{\mathrm{x}}^{\frac{3}{2}}}{6{\mathrm{d}}^{\frac{1}{2}}}+\frac{3{\mathrm{x}}^{\frac{5}{2}}}{40{\mathrm{d}}^{\frac{3}{2}}}+\frac{5{\mathrm{x}}^{\frac{7}{2}}}{112{\mathrm{d}}^{\frac{5}{2}}}+\text{\&c}$ hoc est $=\sqrt{}\mathrm{d}\mathrm{x}in1+\frac{\mathrm{x}}{6\mathrm{d}}+\frac{3\mathrm{x}\mathrm{x}}{40\mathrm{d}\mathrm{d}}+\frac{5{\mathrm{x}}^3}{112\mathrm{d}\mathrm{d}\mathrm{d}}+\text{\&c}$.

2 Si vicissim ex dato arcu desidere\n/tur sinus: {illeg} sit radius r et arcus z, eritqꝫ sinus rectus $=\mathrm{z}-\frac{{\mathrm{z}}^3}{6\mathrm{r}\mathrm{r}}+\frac{{\mathrm{z}}^5}{120{\mathrm{r}}^4}-\frac{{\mathrm{z}}^7}{5040{\mathrm{r}}^6}+\frac{{\mathrm{z}}^9}{36288{\mathrm{r}}^8}-\text{\&c}$, hoc est $=\mathrm{z}-\frac{\mathrm{z}\mathrm{z}}{2\times 3\mathrm{r}\mathrm{r}}\mathrm{A}-\frac{\mathrm{z}\mathrm{z}}{4\times 5\mathrm{r}\mathrm{r}}\mathrm{B}-\frac{\mathrm{z}\mathrm{z}}{6\times 7\mathrm{r}\mathrm{r}}\mathrm{C}-\text{\&c}$; Vel sit Et sinus versus $=\frac{\mathrm{z}\mathrm{z}}{2\mathrm{r}}-\frac{{\mathrm{z}}^4}{24{\mathrm{r}}^3}+\frac{{\mathrm{z}}^6}{720{\mathrm{r}}^5}-\frac{{\mathrm{z}}^8}{4032{\mathrm{r}}^7}+\text{\&c}$, hoc est $\frac{\mathrm{z}\mathrm{z}}{1\times 2\mathrm{r}}-\frac{\mathrm{z}\mathrm{z}}{3\times 4\mathrm{r}\mathrm{r}}\mathrm{A}-\frac{\mathrm{z}\mathrm{z}}{5\times 6\mathrm{r}\mathrm{r}}\mathrm{B}-\frac{\mathrm{z}\mathrm{z}}{7\times 8}\mathrm{C}$.^q[1]

3 Si arcus capiendus sit in ratione datâ ad \alium/ arcum: {illeg}|es|to radius \diameter/ $=\mathrm{d}$, {illeg} \chorda/ arcus dad|t|i $=\mathrm{x}$, & arcus quæsitus ad arcum illum datum ut n ad 1; eritqꝫ sinus arcûs quæsiti {illeg} \chorda/ $=\mathrm{n}\mathrm{x}+\frac{1-\mathrm{n}\mathrm{n}}{2\times 3\mathrm{d}\mathrm{d}}\mathrm{x}\mathrm{x}\mathrm{A}$$+\frac{9-\mathrm{n}\mathrm{n}}{4\times 5\mathrm{d}\mathrm{d}}\mathrm{x}\mathrm{x}\mathrm{B}+\frac{25-\mathrm{n}\mathrm{n}}{6\times 7\mathrm{d}\mathrm{d}}\mathrm{x}\mathrm{x}\mathrm{C}+\frac{36-\mathrm{n}\mathrm{n}}{8\times 9\mathrm{d}\mathrm{d}}\mathrm{x}\mathrm{x}\mathrm{D}+\frac{49-\mathrm{n}\mathrm{n}}{10\times 11\mathrm{d}\mathrm{d}}\mathrm{x}\mathrm{x}\mathrm{E}+\text{\&c}$. Ubi nota quod cùm n est numerus impar, series desinet esse infinita, et evadet eadem quæ prodit per vulgarem Algebram ad multiplicandum datum angulum per istum numerum n.

4 Si in axe {illeg} \alterutro/ AB {illeg} Ellipseos ADB (cujus centrum C & se{illeg} {illeg} \alter/ {illeg} E axis alter DH) detur punctum aliquod E circa quod recta EG occurrens Ellipsi in G {illeg} motu angulari feratur, et ex data area sectoris Ellipticæ BEG quæratur recta GF quæ a puncto G ad axem AB normaliter demittitur: esto $\mathrm{BC}=\mathrm{q}$, $\mathrm{DC}=\mathrm{r}$, $\mathrm{EB}=\mathrm{t}$, ac duplum areæ $\mathrm{BEG}=\mathrm{z}$; et erit $\mathrm{GF}=\frac{\mathrm{z}}{\mathrm{t}}-\frac{\mathrm{q}{\mathrm{z}}^3}{6\mathrm{r}\mathrm{r}{\mathrm{t}}^4}+\frac{10\mathrm{q}\mathrm{q}-\mathrm{q}\mathrm{q}\mathrm{t}}{120{\mathrm{r}}^4{\mathrm{t}}^7}{\mathrm{z}}^5\frac{-280{\mathrm{q}}^3+504\mathrm{q}\mathrm{q}\mathrm{t}-225\mathrm{q}\mathrm{t}\mathrm{t}}{5040{\mathrm{r}}^6{\mathrm{t}}^{10}}{\mathrm{z}}^7+\text{\&c}$. Sic itaqꝫ Astronomicum illud Kepleri Problema resolvi potest.

5. In eâdem Ellipsi si statuatur $\mathrm{CD}=\mathrm{r}$, {illeg} $\frac{{\mathrm{CB}}^q}{\mathrm{CD}}=\mathrm{c}$, et $\mathrm{CF}=\mathrm{x}$, \erit/ arcus Ellipticus $\begin{array}{lllllllllllllll}\mathrm{DG}& =& \mathrm{x}& +& \frac{1}{6\mathrm{c}\mathrm{c}}{\mathrm{x}}^3& +& \frac{1}{10\mathrm{r}{\mathrm{c}}^3}{\mathrm{x}}^5& +& \frac{1}{14\mathrm{r}\mathrm{r}{\mathrm{c}}^4}{\mathrm{x}}^7& +& \frac{1}{18{\mathrm{r}}^3{\mathrm{c}}^5}{\mathrm{x}}^9& +& \frac{1}{22{\mathrm{r}}^4{\mathrm{c}}^6}{\mathrm{x}}^{11}& +& \text{\&c}\\ & & & & & -& \frac{1}{40{\mathrm{c}}^4}& -& \frac{1}{28\mathrm{r}{\mathrm{c}}^5}& -& \frac{1}{24\mathrm{r}\mathrm{r}{\mathrm{c}}^6}& -& \frac{1}{22{\mathrm{r}}^3{\mathrm{c}}^7}\\ & & & & & & & +& \frac{1}{112{\mathrm{c}}^6}& +& \frac{1}{48\mathrm{r}{\mathrm{c}}^7}& +& \frac{3}{88\mathrm{r}\mathrm{r}{\mathrm{c}}^8}\\ & & & & & & & & & -& \frac{5}{1152{\mathrm{c}}^8}& -& \frac{5}{352\mathrm{r}{\mathrm{c}}^9}\\ & & & & & & & & & & & +& \frac{7}{2816{\mathrm{c}}^{10}}\end{array}$
|6|
Hic numerales coefficientes {illeg}|s|upremorum terminorum $\left(\frac{1}{6}.\frac{1}{10}.\frac{1}{14}\text{\&c}\right)$ sunt in musica progressione, & numerales coefficientes omnium inferiorum in unaquaqꝫ columna prodeunt multiplicando continuò numeralem coefficientem <3r> supremi termini per terminos hujus progressionis $\frac{\frac{1}{2}\mathrm{n}-1}{2}.\frac{\frac{3}{3}\mathrm{n}-3}{4}.$$\frac{\frac{5}{4}\mathrm{n}-5}{6}.\frac{\frac{7}{5}\mathrm{n}-7}{8}.\frac{\frac{9}{6}\mathrm{n}-9}{10}.\text{\&c}$: ubi n significat numerum dimensionum ipsius {illeg} c in denominatore \istius/ supremi termini. E.g. ut terminorum infra $\frac{1}{22{\mathrm{r}}^4{\mathrm{c}}^6}$, numerales coefficientes inveniantur, pono $\mathrm{n}=6$, ducoqꝫ $\frac{1}{22}$ (numeralem coefficientem ipsius $\frac{1}{22{\mathrm{r}}^4{\mathrm{c}}^6}$) in $\frac{\frac{1}{2}\mathrm{n}-1}{2}$ hoc est $in1$; et prodit $\frac{1}{22}$ numeralis coefficiens termini proximè inferioris; dein duco hunc $\frac{1}{22}in\frac{\frac{3}{3}\mathrm{n}-3}{4}$ sive $in\frac{\mathrm{n}-3}{4}$ hoc est $in\frac{3}{4}$ & prodit $\frac{3}{88}$ numeralis coefficiens tertij termini in ista columna. Atqꝫ ita $\frac{3}{88}\times \frac{\frac{5}{4}\mathrm{n}-5}{6}$ facit $\frac{5}{352}$ num. coeff. q^rti termini |6.| & $\frac{5}{352}\times \frac{\frac{7}{5}\mathrm{n}-7}{8}$ facit $\frac{7}{2816}$ numeralem coefficientem infimi termini Idem in alijs ad infinitum {illeg} usqꝫ columnis præstari potest, adeoqꝫ valor ipsius {illeg} \DG/ per hanc Regulam pro lubitu produci.

Adhæc si BF dicatur x, sitqꝫ r latus rectum Ellipseos & $\mathrm{e}=\frac{\mathrm{r}}{\mathrm{AB}}$; erit arcus Ellipticus {illeg}
$\begin{array}{ccc}\begin{array}{c}\\ \begin{array}{c}\mathrm{BG}=\sqrt{}\mathrm{r}\mathrm{x}in\end{array}\end{array}& \begin{array}{c}\\ \begin{array}{ccccccccccc}\begin{array}{c}\begin{array}{lll}1& +& 2& \phantom{\sqrt{}}\\ & -& \frac{3}{2}\mathrm{e}\end{array}\}\\ 3\mathrm{r}\end{array}& \mathrm{x}\phantom{\sqrt{}}& & \begin{array}{c}\begin{array}{ll}-& 2\phantom{\sqrt{}}\\ +& 3\mathrm{e}\\ -& \frac{5}{8}\mathrm{e}\mathrm{e}\end{array}\}\\ 5\mathrm{r}\mathrm{r}\end{array}& \mathrm{x}\mathrm{x}\phantom{\sqrt{}}& & \begin{array}{c}\begin{array}{ll}+& 4\phantom{\sqrt{}}\\ -& 9\mathrm{e}\\ +& \frac{23}{4}\mathrm{e}\mathrm{e}\\ -& \frac{7}{16}{\mathrm{e}}^3\end{array}\}\\ 7{\mathrm{r}}^3\end{array}& {\mathrm{x}}^3\phantom{\sqrt{}}& & \begin{array}{c}\begin{array}{ll}-& 10\phantom{\sqrt{}}\\ +& 30\mathrm{e}\\ -& \frac{123}{4}\mathrm{e}\mathrm{e}\\ +& \frac{91}{8}{\mathrm{e}}^3\\ -& \frac{45}{128}{\mathrm{e}}^4\end{array}\}\\ 9{\mathrm{r}}^4\end{array}& {\mathrm{x}}^4\phantom{\sqrt{}}\end{array}\end{array}& \begin{array}{c}\\ \begin{array}{c}+\text{\&c.}\phantom{\sqrt{}}\end{array}\end{array}\end{array}$
Quare si ambitus totius Ellipseos desideretur: b{illeg}|is|eca CB in F, & quære arcum DG per prius Theorema et arcum BG per posterius.

6. Si vice versa ex dato arcu Elliptico DG quæratur sinus ejus CF, tum dicto $\mathrm{CD}=\mathrm{r}$, $\frac{{\mathrm{CB}}^q}{\mathrm{CD}}=\mathrm{c}$, & arcu illo $\mathrm{DG}=\mathrm{z}$ erit
$\begin{array}{lllllllllll}\mathrm{CF}& =& \mathrm{z}& -& \frac{1}{6\mathrm{c}\mathrm{c}}{\mathrm{z}}^3& -& \frac{1}{10\mathrm{r}{\mathrm{c}}^3}{\mathrm{z}}^5& -& \frac{1}{14\mathrm{r}\mathrm{r}{\mathrm{c}}^4}{\mathrm{z}}^7& -& \text{\&c.}\\ & & & & & +& \frac{13}{120{\mathrm{c}}^4}& +& \frac{71}{420\mathrm{r}{\mathrm{c}}^5}\\ & & & & & & & -& \frac{493}{5040{\mathrm{c}}^6}\end{array}$

Quæ autem de Ellipsi d{illeg}|i|cta sunt, omnia facilè accommodantur ad Hyperbolam: mutat{illeg}|i|s tantum signis ipsorum c et e ubi sunt imparium dimensionum.

7. Præterea si sit CE Hyperbola cujus Asymptoti AD, AF rectum angulum \FAD/ constituant et ad AD erigantur utcunqꝫ perpendicula BC DE occurrentia Hyperbolæ in C et E, & AB dicatur a, BC b, & area BCED z, erit $\mathrm{BD}=\frac{\mathrm{z}}{\mathrm{b}}+\frac{\mathrm{z}\mathrm{z}}{2\mathrm{a}\mathrm{b}\mathrm{b}}+\frac{{\mathrm{z}}^3}{6\mathrm{a}\mathrm{a}{\mathrm{b}}^3}+\frac{{\mathrm{z}}^4}{24{\mathrm{a}}^3{\mathrm{b}}^4}+\frac{{\mathrm{z}}^5}{120{\mathrm{a}}^4{\mathrm{b}}^5}\text{\&c}$: Ubi coefficientes denominatorum prodeunt multiplicando terminos hujus arithmeticæ progressionis, $1,2,3,4,5,\text{\&c}$ in se continuò. Et hinc ex Logarithmo dato potest numerus ei competens inveniri.

8 Esto VDE Quadratrix cujus vertex V, existente A centro et AE^[2] \semi-/diametro circuli ad q{illeg}|ue|m aptatur, et angulo VAE recto. Demissoqꝫ ad AE perpendiculo quovis DB et acta \Quadratricis/ tangente DT occurrente axi ejus AV in T: dic $\mathrm{AV}=\mathrm{a}$, & $\mathrm{AB}=\mathrm{x}$, eritqꝫ $\mathrm{BD}=\mathrm{a}-\frac{\mathrm{x}\mathrm{x}}{3\mathrm{a}}-\frac{{\mathrm{x}}^4}{45{\mathrm{a}}^3}-\frac{2{\mathrm{x}}^6}{945{\mathrm{a}}^5}-\text{\&c.}$ Et {illeg} $\mathrm{VT}=\frac{\mathrm{x}\mathrm{x}}{3\mathrm{a}}+\frac{{\mathrm{x}}^4}{15{\mathrm{a}}^3}+\frac{2{\mathrm{x}}^6}{189{\mathrm{a}}^5}+\text{\&c.}$ Et area $\mathrm{AVDB}=\mathrm{a}\mathrm{x}-\frac{{\mathrm{x}}^3}{9\mathrm{a}}-\frac{{\mathrm{x}}^5}{225{\mathrm{a}}^3}-\frac{2{\mathrm{x}}^7}{6615{\mathrm{a}}^5}-\text{\&c}$ <3v> $\mathrm{BD}=\mathrm{a}-\frac{\mathrm{x}\mathrm{x}}{3\mathrm{a}}-\frac{{\mathrm{x}}^4}{45{\mathrm{a}}^3}-\frac{2{\mathrm{x}}^6}{945{\mathrm{a}}^5}-\text{\&c.}$ Et $\mathrm{VT}=\frac{\mathrm{x}\mathrm{x}}{3\mathrm{a}}+\frac{{\mathrm{x}}^4}{15{\mathrm{a}}^3}+\frac{2{\mathrm{x}}^6}{189{\mathrm{a}}^5}+\text{\&c}$ area $\mathrm{AVDB}=\mathrm{a}\mathrm{x}-\frac{{\mathrm{x}}^3}{9\mathrm{a}}-\frac{{\mathrm{x}}^5}{225{\mathrm{a}}^3}-\frac{2{\mathrm{x}}^7}{6615{\mathrm{a}}^5}-\text{\&c.}$ Et arcus $\mathrm{VD}=\mathrm{x}+\frac{2{\mathrm{x}}^3}{27\mathrm{a}\mathrm{a}}$$+\frac{14{\mathrm{x}}^5}{2025{\mathrm{a}}^4}+\frac{604{\mathrm{x}}^7}{893025{\mathrm{a}}^6}+\text{\&c}$. Unde vicissim ex dato BD, vel VT, aut areâ AVDB arcuve VD, per resolutionem affectarum æquationum erui potest x seu AB.

9 Esto deniqꝫ AEB Sphæroides, revolutione Ellipseos AEB circa axem AB genita, et secta planis quatuor, AB per axem transeunte, DC parallelo AB, CDE perpendiculariter bisecante axem, et FC parallel{illeg}|o| CE: sitqꝫ recta $\mathrm{CB}=\mathrm{a}$. $\mathrm{CE}=\mathrm{c}$. $\mathrm{CF}=\mathrm{x}$. & $\mathrm{FG}=\mathrm{y}$; et Sphæroideos segmentum CDFG dictis quatuor planis comprehensum erit
$\begin{array}{rrrrrrrrrrrrrrrr}+2\mathrm{c}\mathrm{x}& \mathrm{y}& & -\frac{\mathrm{x}}{3\mathrm{c}}& {\mathrm{y}}^3& & -\frac{\mathrm{x}}{20{\mathrm{c}}^3}& {\mathrm{y}}^5& & -\frac{\mathrm{x}}{56{\mathrm{c}}^5}& {\mathrm{y}}^7& & -\frac{5\mathrm{x}}{576{\mathrm{c}}^7}& {\mathrm{y}}^9& & \hfill -\text{\&c.}\hfill \\ -\frac{\mathrm{c}{\mathrm{x}}^3}{3\mathrm{a}\mathrm{a}}& & & -\frac{{\mathrm{x}}^3}{18\mathrm{c}\mathrm{a}\mathrm{a}}& & & -\frac{{\mathrm{x}}^3}{40{\mathrm{c}}^3\mathrm{a}\mathrm{a}}& & & -\frac{5{\mathrm{x}}^3}{336{\mathrm{c}}^5\mathrm{a}\mathrm{a}}& & & \hfill -\text{\&c}\hfill \\ -\frac{\mathrm{c}{\mathrm{x}}^5}{20{\mathrm{a}}^4}& & & -\frac{{\mathrm{x}}^5}{40\mathrm{c}{\mathrm{a}}^4}& & & -\frac{3{\mathrm{x}}^5}{160{\mathrm{c}}^3{\mathrm{a}}^4}& & & \hfill -\text{\&c}\hfill \\ -\frac{\mathrm{c}{\mathrm{x}}^7}{56{\mathrm{a}}^6}& & & -\frac{5{\mathrm{x}}^7}{336\mathrm{c}{\mathrm{a}}^6}& & & \hfill -\text{\&c}\hfill \\ -\frac{5\mathrm{c}{\mathrm{x}}^9}{576{\mathrm{a}}^7}& & & \hfill -\text{\&c}\hfill \\ \hfill -\text{\&c}\hfill \end{array}$
|56| Ubi \/ numerales coefficientes supremorum terminorum
$\left[2,\frac{-1}{\phantom{-}3},\frac{-1}{20},\frac{-1}{56},\frac{-5}{576}\text{\&c}\right]$ \in/ infinitum producuntur multiplicando numerum binarium \primum coefficientem 2/ continuò per terminos hujus progressionis $\frac{-1\times 1}{\phantom{-}2\times 3}.\frac{1\times 3}{4\times 5}.\frac{3\times 5}{6\times 7}.\frac{5\times 7}{8\times 9}.\frac{7\times 9}{10\times 11}.\text{\&c}$. Et \numerales/ coefficientes terminorum in unaquaqꝫ coluna descendentium in infinitum producuntur multiplicando continuo coefficientem supremi termini in prima colu{illeg}|m|na per eandem progressionem in secunda autem per terminos hujus $\frac{1\times 1}{2\times 3}.\frac{3\times 3}{4\times 5}.\frac{5\times 5}{6\times 7}.\frac{7\times 7}{8\times 9}.\text{\&c}$, in tertia per terminos hujus $\frac{3\times 1}{2\times 3}.\frac{5\times 3}{4\times 5}$$\frac{7\times 5}{6\times 7}.\frac{9\times 7}{8\times 9}.\text{\&c}$, in quarta per terminos huius $\frac{5\times 1}{2\times 3}.\frac{7\times 3}{4\times 5}.\frac{9\times 5}{6\times 7}.\text{\&c}$, in quinta per terminos huius $\frac{7\times 1}{2\times 3}.\frac{9\times 3}{4\times 5}.\frac{11\times 5}{6\times 7}.\text{\&c}$ Et sic in infinitum. |[| Et eodem modo segmenta aliorum solidorum designari, et valores eorum \aliquando commodè/ per series quasdem numerales in infinitum produci possunt.

Ex his videre est quantum fines Analyseos per hujusmodi infinitas æquationes ampliantur: qu{illeg}|ippe| quæ earum beneficio, ad omnia, pene dixerim, problemata (si numeralia Diophanti et similia excipias) sese extendit |[| Non tamen omninò universalis evadit, nisi per {illeg} ulteriores quasdem methodos eliciendi series infinitas. Sunt enim quædam Problemata in quibus non liceat ad series infinitas per divisionem vel extractionem radicum simplicium affectarumve pervenire: sed quomodo in istis casibus procedendum sit jam non {illeg}|v|acat dicere, ut neqꝫ alia quædam tradere quæ circa reductionem infinitarum serierum in finitas, ubi rei natur{a} tulerit, excogitavi. Nam {hisce} quanquam paucis scribendis fatigor, utpote \parciùs scribo, quòd {horum} {illeg}/ {illeg} hæ speculationes diu \mihi/ fastidio e{illeg}|s|se cœperunt, adeò ut ab ijsdem jam per quinqꝫ ferè annos abstinuerim. |[| Unum tamen addam: quòd postquam Problema aliquod ad infinitam æquationem deducitur, possint inde variæ approximationes in usum Mechanicæ nullo ferè negotio formari, quæ per alias methodos quæsitæ, multo labore temporisqꝫ dispendio constare solent. |[| Cujus rei exemplo esse possunt Tractatus Hugenij aliorumqꝫ de Quadrat{ura} circuli. Nam ut ex data arcûs chorda A, & dimidij arcûs chorda {B} arcum {illeg} illum proxime assequa{illeg}|ris|, finge arcum illum esse z, et circul{i} radium r; juxtaqꝫ superiora erit A (nempe duplum sinus dimidij z) <4r> $=\mathrm{z}-\frac{{\mathrm{z}}^3}{4\times 6\mathrm{r}\mathrm{r}}+\frac{{\mathrm{z}}^5}{4\times 4\times 120{\mathrm{r}}^4}-\text{\&c}$. Et $\mathrm{B}=\frac{1}{2}\mathrm{z}-\frac{{\mathrm{z}}^3}{2\times 16\times 6\mathrm{r}\mathrm{r}}+\frac{{\mathrm{z}}^5}{2\times 16\times 16\times 120{\mathrm{r}}^4}-\text{\&c}$. Duc jam B in numerum fictitium n et a producto aufer A, et residui secundum terminum $\left(\text{nempe}-\frac{\mathrm{n}{\mathrm{z}}^3}{2\times 16\times 6\mathrm{r}\mathrm{r}}+\frac{{\mathrm{z}}^3}{4\times 6\mathrm{r}\mathrm{r}}\text{,}\right)$ eo ut evanescat, pone $=0$, indeqꝫ emerget $\mathrm{n}=8$, et erit $8\mathrm{B}-\mathrm{A}=3\mathrm{z}\ast -\frac{3{\mathrm{z}}^5}{64\times 120{\mathrm{r}}^4}+\text{\&c}$: hoc est $\frac{8\mathrm{B}-\mathrm{A}}{3}=\mathrm{z}$ errore tantum existente $\frac{{\mathrm{z}}^5}{7680{\mathrm{r}}^4}-\text{\&c}$ in excessu. Quod est {illeg} Theorema Hugenianum.

Insuper si in arcûs Bb sagittâ AD \indefinitè/ productâ quæratur punctum G a quo actæ rectæ GB, Gb abscindant tangentem Ee quam proximè æqualem arcui ist{illeg}|i|: esto circuli \centrum C,/ diameter $\mathrm{AK}=\mathrm{d}$, et erit DB sagitta $\mathrm{AD}=\mathrm{x}$ et erit $\mathrm{DB}\left(=\sqrt{}\overline{\mathrm{d}\mathrm{x}-\mathrm{x}\mathrm{x}}\right)$ $={\mathrm{d}}^{\frac{1}{2}}{\mathrm{x}}^{\frac{1}{2}}-\frac{{\mathrm{x}}^{\frac{3}{2}}}{2{\mathrm{d}}^{\frac{1}{2}}}-\frac{{\mathrm{x}}^{\frac{5}{2}}}{8{\mathrm{d}}^{\frac{3}{2}}}-\frac{{\mathrm{x}}^{\frac{7}{2}}}{16{\mathrm{d}}^{\frac{5}{2}}}-\text{\&c}$ Et AE $\left(=\mathrm{AB}\right)={\mathrm{d}}^{\frac{1}{2}}{\mathrm{x}}^{\frac{1}{2}}+\frac{{\mathrm{x}}^{\frac{3}{2}}}{6{\mathrm{d}}^{\frac{1}{2}}}+\frac{3{\mathrm{x}}^{\frac{5}{2}}}{40{\mathrm{d}}^{\frac{3}{2}}}+\frac{5{\mathrm{x}}^{\frac{7}{2}}}{112{\mathrm{d}}^{\frac{5}{2}}}+\text{\&c}$. Et $\mathrm{AE}-\mathrm{DB}.\mathrm{AD}\colon\colon \mathrm{AE}.\mathrm{AG}$. \Quare AG/ $=\frac{3}{2}\mathrm{d}-\frac{1}{5}\mathrm{x}-\frac{12\mathrm{x}\mathrm{x}}{175\mathrm{d}}$\$-\text{vel}+\text{\&c}$/. Finge ergo $\mathrm{AG}=\frac{3}{2}\mathrm{d}-\frac{1}{5}\mathrm{x}$, et vicissim erit $\mathrm{DG}\left(\frac{3}{2}\mathrm{d}-\frac{6}{5}\mathrm{x}\right).\mathrm{DB}\colon\colon \mathrm{DA}.\mathrm{AE}-\mathrm{DB}$. Quare $\mathrm{AE}-\mathrm{DB}=\frac{2{\mathrm{x}}^{\frac{3}{2}}}{3{\mathrm{d}}^{\frac{1}{2}}}+\frac{{\mathrm{x}}^{\frac{5}{2}}}{5{\mathrm{d}}^{\frac{3}{2}}}+\frac{23{\mathrm{x}}^{\frac{7}{2}}}{300{\mathrm{d}}^{\frac{5}{2}}}+\text{\&c}$. Adde AB et prodit $\mathrm{AE}={\mathrm{d}}^{\frac{1}{2}}{\mathrm{x}}^{\frac{1}{2}}+\frac{{\mathrm{x}}^{\frac{3}{2}}}{6{\mathrm{d}}^{\frac{1}{2}}}+\frac{3{\mathrm{x}}^{\frac{5}{2}}}{40{\mathrm{d}}^{\frac{3}{2}}}+\frac{17{\mathrm{x}}^{\frac{7}{2}}}{1200{\mathrm{d}}^{\frac{5}{2}}}+\text{\&c}$. Hoc aufer de valore ipsius AE supra habito et restabit error $\frac{16{\mathrm{x}}^{\frac{7}{2}}}{525{\mathrm{d}}^{\frac{5}{2}}}+\mathrm{vel}-\text{\&c}$. Quare in AG cape AH quintam partem DH, et $\mathrm{KG}=\mathrm{HC}$; et actæ GBE, Gbe abscindent tangentem Ee quam proximè æqualem arcui Bab errore tantum existente $\frac{16{\mathrm{x}}^3}{525{\mathrm{d}}^3}\sqrt{}\mathrm{d}\mathrm{x}+\mathrm{vel}-\text{\&c}$; multò minore scilicet quam in Theoremate Hugenij. Quod si fiat $7\mathrm{AK}.3\mathrm{AH}\colon\colon \mathrm{DH}.\mathrm{n}$, et capiatur $\mathrm{KG}=\mathrm{CH}-\mathrm{n}$ erit error adhuc multò minor.

Atqꝫ ita si circuli segmentum aliquod BAb per Mechanicam designandum esset: primò reducerem aream istam in infinitam seriem; puta hanc $\mathrm{BbA}=\frac{4}{3}{\mathrm{d}}^{\frac{1}{2}}{\mathrm{x}}^{\frac{3}{2}}-\frac{2{\mathrm{x}}^{\frac{5}{2}}}{5{\mathrm{d}}^{\frac{1}{2}}}-\frac{{\mathrm{x}}^{\frac{7}{2}}}{14{\mathrm{d}}^{\frac{3}{2}}}-\frac{{\mathrm{x}}^{\frac{9}{2}}}{36{\mathrm{d}}^{\frac{5}{2}}}-\text{\&c}$; dein quærerem constructiones me{illeg}|c|hanicas quibus hanc serie{illeg}|m| proximè assequere{illeg}|t|ur; cujusmodi sunt hæc. Age rectam AB, et erit segm: $\mathrm{BbA}=\overline{\frac{2}{3}\mathrm{AB}+\mathrm{BD}}\times \frac{4}{5}\mathrm{AD}$ proximè, existente scilicet errore tantum $\frac{{\mathrm{x}}^3}{70\mathrm{d}\mathrm{d}}\sqrt{}\mathrm{d}\mathrm{x}+\text{\&c}$, in defectu: vel proximiùs erit segmentum illud, (bisecto AD in F et acta recta BF,) $=\frac{4\mathrm{BF}+\mathrm{AB}}{15}\times 4\mathrm{AD}$, existente errore solummodo $\frac{{\mathrm{x}}^3}{560\mathrm{d}\mathrm{d}}\sqrt{}\mathrm{d}\mathrm{x}+\text{\&c}$: qui semper minor est quàm $\frac{1}{1500}$ totius segmenti, {illeg} \dummodo/ segmentum \illud non/ ponatur semicirculus sit major semicircula etiamsi segmentum illud ad usqꝫ semicirculum augeatur.

Sic et in Ellipsi BAb cujus vertex A, axis alteruter AK, et latus rectum AP, cape $\mathrm{PG}=\frac{1}{2}\mathrm{AP}+\frac{19\mathrm{AK}-21\mathrm{AP}}{10\mathrm{AK}}\times \mathrm{AP}$; in Hyperbola verò cape {illeg} $\mathrm{PG}=\frac{1}{2}\mathrm{AP}+\frac{19\mathrm{AK}+21\mathrm{AP}}{10\mathrm{AK}}\times \mathrm{AP}$: et acta recta GBE abscindet tangentem AE quamproximè æqualem arcui Elliptico vel Hyperbolico AB, dummodo ar{cus} ille non sit nimis magnus. Et pro area segmenti Hyperbolici BbA, cape $\mathrm{PT}=\mathrm{Lat:\; recto}$, $\mathrm{PS}=\mathrm{Lat:\; transverso}$, $\mathrm{PQ}=\mathrm{AD}$, et \in DP cape {illeg}/ $\mathrm{DM}=\frac{3{\mathrm{AD}}^q}{4\mathrm{AK}}$ et ad {illeg} \D/ et {illeg} \M/ erige perpendicula \Dβ, MN/ occurrentia semicirculo {illeg} super diametro AP {illeg} descripto, eritqꝫ $\frac{4\mathrm{AN}+\mathrm{A\beta }}{15}\times 4\mathrm{AD}=\mathrm{BbA}$ proximè vel proximius erit $\frac{21\mathrm{AN}+4\mathrm{A\beta }}{75}\times 4\mathrm{AD}=\mathrm{BbA}$, si modoò capiatur $\mathrm{DM}=\frac{5{\mathrm{AD}}^q}{7\mathrm{AK}}$.