Letter from Isaac Newton to Henry Oldenburg, dated 24 October 1676

<1r>

vir Dignissime

Quanta cum voluptate legi Epistolas Clarissimorū virorū D. Leibnitij & D. Tschurnhausij vix dixerim. Perelegans sane est Leibnitij methodus perveniendi ad series convergentes, & satis ostendisset \{sola satis ostendit}/ ingenium authoris |[|etsi nihil aliud scripsisset |]| sed \{et}/quæ alibi ꝑ Epistolam sparguntur suo nomine dignissima, efficiunt etiam ut ab eo speremus maxima. Diversitas modorū quibꝫ e{illeg}|od|em tenditur, eò magis placuit quòd mihi \cum/ tres methodi ꝑveniendi ad ejusmodi series \{nobis}/ innotuerant, |[| adeo ut novam nobis communicandam vix expectarem |]| |{illeg} illa Leibnitiano illa ab {illeg} {isti{illeg}} omnibus {plane} diversa est| unam e me{illeg}|i|s prius descripsi jam addo aliam, illam scilicet quâ primū incidi in has series: nam incidi in eas antequam scirem divisiones et extractiones radicū quibꝫ jam utor et hujus explicatone {sic} p{illeg}|e|ndendum est fundamentum Theorematis sub initio Epistolæ prioris positi quod D. Leibnitius a me desiderat. |[| sub initio studiorum meorum Mathematicorum ubi incideram in opera Celeberrimi Wallisij nostri, considerando series quarum intercalatione ipse exhibet aream Circuli et Hyperbolæ, utpote quod in serie curvarū quarū basis sive axis comunis sit x, et ordinatim applicatæ, ${\overline{1-\mathrm{x}\mathrm{x}|}}^{\frac{0}{2}}.{\overline{1-\mathrm{x}\mathrm{x}|}}^{\frac{1}{2}}.{\overline{1-\mathrm{x}\mathrm{x}|}}^{\frac{2}{2}}.$ ${\overline{1-\mathrm{x}\mathrm{x}|}}^{\frac{3}{2}}.{\overline{1-\mathrm{x}\mathrm{x}|}}^{\frac{4}{2}}$ ${\overline{1-\mathrm{x}\mathrm{x}|}}^{\frac{5}{2}}\text{\&c}$ . Si areæ alternarū quæ sunt
$\mathrm{x}.\mathrm{x}-\frac{1}{3}{\mathrm{x}}^3.\mathrm{x}-\frac{2}{3}{\mathrm{x}}^3+$ $\frac{1}{5}{\mathrm{x}}^5.\mathrm{x}-\frac{3}{3}{\mathrm{x}}^3+\frac{3}{5}{\mathrm{x}}^5-\frac{1}{7}{\mathrm{x}}^7.\text{\&c}$ int{illeg}|e|rpolari possent, haberemus areas intermed{illeg}|i|arū quarū prima ${\overline{1-\mathrm{x}\mathrm{x}|}}^{\frac{1}{2}}$ est circulus: ad has interpolandas notabam, quod in omnibꝫ primus terminus esset x, quodqꝫ secundi termini $\frac{0}{3}{\mathrm{x}}^3.\frac{1}{3}{\mathrm{x}}^3.\frac{2}{3}{\mathrm{x}}^3.\frac{3}{3}{\mathrm{x}}^3\text{\&c}$ essent in Arithmeticâ progressione, et proinde quod duo primi termini serierū intercalandarū deberent esse $\mathrm{x}-\frac{\frac{1}{2}{\mathrm{x}}^3}{3}.\mathrm{x}\frac{\frac{1}{2}{\mathrm{x}}^3}{3}.$ |$\mathrm{x}-\frac{\frac{1}{2}{\mathrm{x}}^3}{3}$| |$\mathrm{x}-\frac{\frac{3}{2}{\mathrm{x}}^3}{3}$| $\mathrm{x}-\frac{\frac{3}{2}{\mathrm{x}}^3}{3}$ $\mathrm{x}-\frac{\frac{5}{2}{\mathrm{x}}^3}{3}\text{\&c}$ ad reliquas intercalandas considerabam quod denominatores $1.3.5.7\text{\&c}$ erant in Arithmeticâ progressione, adeoqꝫ solæ numeratorū coefficientes numerales restabant investigandæ hæ autem in alterius \alternis/ datis areis erant figuræ potestatum numeri 11 ^H[1]; nempe harum ${\overline{11|}}^0.{\overline{11|}}^1.{\overline{11|}}^2.{\overline{11|}}^3.{\overline{11|}}^4.$ hoc est primo \1/, deinde $1,1$. {illeg} terti{illeg}|o| $1.2.1$. quarto $1.3.3.1$. quinto $1.4.6.4.1$. quærebam itaqꝫ quomodo in his seriebꝫ, ex datis duabꝫ primis figuris reliquæ derivari possent, et inveni quod posita secunda figura, m, reliquæ producerentur per continuam multiplicationem terminorum hujus seriei $\frac{\mathrm{m}-0}{1}\times \frac{\mathrm{m}-1}{2}\times \frac{\mathrm{m}-2}{3}$ $\times \frac{\mathrm{m}-3}{4}\text{\&c}$. Exempli gratiâ sit $\mathrm{m}=4$ et erit $4\times \frac{\mathrm{m}-1}{2}$ <1v> hoc est 6 tertius terminus, et $6\times \frac{\mathrm{m}-2}{3}$ hoc est 4, quartus, et $4\times \frac{\mathrm{m}-3}{4}$, hoc est, 1, quintus {illeg} et, $1\times \frac{\mathrm{m}-4}{5}$, hoc est, 0, sextus, quo series in hoc casu terminatur. hanc regulam itaqꝫ applicui ad series interserendas et cùm pro circulo secundus terminus esset $\frac{\frac{1}{2}{\mathrm{x}}^3}{3}$, posui $\mathrm{m}=\frac{1}{2}$, et prodierunt termini $\frac{1}{2}\times \frac{\frac{1}{2}-1}{2}$ sive \id est/ $-\frac{1}{8}$, $-\frac{1}{8}\times \frac{\frac{1}{2}-2}{3}$ sive \id est/ $+\frac{1}{16}$, $+\frac{1}{16}\times \frac{\frac{1}{2}-3}{4}$ sive \id est/ $-\frac{5}{128}$ & sic in infinitum. unde cognovi desideratam aream segmenti circularis {illeg} esse $\mathrm{x}-\frac{\frac{1}{2}{\mathrm{x}}^3}{3}-\frac{\frac{1}{8}{\mathrm{x}}^5}{5}-\frac{\frac{1}{16}{\mathrm{x}}^7}{7}-\frac{\frac{5}{128}{\mathrm{x}}^9}{9}\text{\&c}$. et eadem ratione prodierunt etiam interserendæ areæ reliquarū curvarū, ut et area Hyperbolæ et cæterarū alternarū in hac serie ${\overline{1+\mathrm{x}\mathrm{x}|}}^{\frac{0}{2}},{\overline{1+\mathrm{x}\mathrm{x}|}}^{\frac{1}{2}}$ ${\overline{1+\mathrm{x}\mathrm{x}|}}^{\frac{2}{2}}.{\overline{1+\mathrm{x}\mathrm{x}|}}^{\frac{3}{2}}$ \&c/ et eadem est ratio intercalandi alias series idqꝫ ꝑ intervalla duorum pluriumve terminorū simul deficientiū. hic fuit primus meus ingressus in has meditationes: qui memoria sanè exciderat nisi oculos in adversaria quædam ante paucas septimanas retulissem. |[| {illeg}|U|bi vero hæc didiceram mox considerabam terminos ${\overline{1-\mathrm{x}\mathrm{x}|}}^{\frac{0}{2}}.{\overline{1-\mathrm{x}\mathrm{x}|}}^{\frac{2}{2}}.{\overline{1-\mathrm{x}\mathrm{x}|}}^{\frac{4}{2}}.{\overline{1-\mathrm{x}\mathrm{x}|}}^{\frac{6}{2}}\text{\&c}$. hoc est, $1.1-\mathrm{x}\mathrm{x}.1-2\mathrm{x}\mathrm{x}+{\mathrm{x}}^4.1-3\mathrm{x}\mathrm{x}$ $+3{\mathrm{x}}^4-{\mathrm{x}}^6\text{\&c}$: \hoc est {$1,1-\mathrm{x}\mathrm{x},1-2\mathrm{x}\mathrm{x}+{\mathrm{x}}^4,1-3\mathrm{x}\mathrm{x}+3{\mathrm{x}}^4-{\mathrm{x}}^6\text{\&c.}$}/ eodem modo interpolari posse ac areas ab ipsis generatas: et ad hoc nihil aliud requiri quam omissionem denominatorū $1,3,5,7\text{\&c.}$ in terminis exprimentibꝫ areas; hoc est coefficientes terminorum quantitatis intercalandæ ${\overline{1-\mathrm{x}\mathrm{x}|}}^{\frac{1}{2}}$, vel ${\overline{1-\mathrm{x}\mathrm{x}|}}^{\frac{3}{2}}$ vel generaliter ${\overline{1-\mathrm{x}\mathrm{x}|}}^{\mathrm{m}}$, prodire per continuam multiplicaconem terminorū hujus seriei $\mathrm{m}\times \frac{\mathrm{m}-1}{2}\times \frac{\mathrm{m}-2}{3}$ $\times \frac{\mathrm{m}-3}{4}\text{\&c}$, Adeoqꝫ e.g. ${\overline{1-\mathrm{x}\mathrm{x}|}}^{\frac{1}{2}}$ valeret $1-\frac{1}{2}{\mathrm{x}}^2-\frac{1}{8}{\mathrm{x}}^4-\frac{1}{16}{\mathrm{x}}^6\text{. \&c}$ Et ${\overline{1-\mathrm{x}\mathrm{x}|}}^{\frac{3}{2}}$ valeret $1-\frac{3}{2}\mathrm{x}\mathrm{x}+\frac{3}{8}{\mathrm{x}}^4+\frac{1}{16}{\mathrm{x}}^6\text{\&c}$ Et ${\overline{1-\mathrm{x}\mathrm{x}|}}^{\frac{1}{3}}$ valeret $1-\frac{1}{3}\mathrm{x}\mathrm{x}$ $-\frac{1}{9}{\mathrm{x}}^4-\frac{5}{81}{\mathrm{x}}^6\text{\&c}$, Sic itaqꝫ innotuit mihi generalis reductio radicalium in infinitas series per regulam illam quam posui initio epistolæ prioris, antequam scirem extractionem radicū. sed hâc cognitâ non potuit altera me, dia latere nam ut probarem has operationes multiplicavi $1-\frac{1}{2}{\mathrm{x}}^2-\frac{1}{8}{\mathrm{x}}^4-\frac{1}{16}{\mathrm{x}}^6\text{\&c}$ in se, & factū est $1-\mathrm{x}\mathrm{x}$ terminis reliquis in infinitum evanescentibꝫ per continuationem seriei. Atqꝫ ita $1-\frac{1}{3}\mathrm{x}\mathrm{x}-\frac{1}{9}{\mathrm{x}}^4-\frac{5}{81}{\mathrm{x}}^6\text{\&c}$ bis in se ductum produxit etiam $1-\mathrm{x}\mathrm{x}$. Quod ut certa fuit harum conclusionum demonstratio sic me manu duxit ad tentandum è converso, nam \{utnam}/ hæ series quas sic <2r> constitit esse radices quantitatis $1-\mathrm{x}\mathrm{x}$ non possent inde extrahi more Arithmetico, et res benè successit. Operationis {for}ma in quadraticis radicibus hæc erat his perspectis neglexi penitus $\begin{array}{ll}1-\mathrm{x}\mathrm{x}(\phantom{\frac{1}{1}}& 1-\frac{1}{2}\mathrm{x}\mathrm{x}-\frac{1}{8}{\mathrm{x}}^4-\frac{1}{16}{\mathrm{x}}^6\text{\&c}\\ & \begin{array}{ll}\underset{‾}{1\phantom{-\mathrm{x}\mathrm{x}}}& \\ 0-\mathrm{x}\mathrm{x}& \\ \underset{‾}{\phantom{0}-\mathrm{x}\mathrm{x}+\frac{1}{4}{\mathrm{x}}^4}& \\ \phantom{0-\mathrm{x}\mathrm{x}}-\frac{1}{4}{\mathrm{x}}^4& \\ \phantom{0-\mathrm{x}\mathrm{x}}\underset{‾}{-\frac{1}{4}{\mathrm{x}}^4+\frac{1}{8}{\mathrm{x}}^6+\frac{1}{64}{\mathrm{x}}^8}\text{.}& \\ \phantom{0-\mathrm{x}\mathrm{x}}\phantom{-}\phantom{\frac{1}{4}}{\phantom{0}}^{\phantom{4}}-\frac{1}{8}{\mathrm{x}}^6-\frac{1}{64}{\mathrm{x}}^8\text{,}& \end{array}\end{array}$

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Unde et simul patefactus est ad resolutionem affectarum æquationum. Nam {faceo} Divisiones quarum \utiqꝫ/ specimen {præclarum} N. Mercator sub idem tempus excogitavit & mox \cum laude omnium/ ædidit.

< text from f 2r resumes >

\His perspectis neglexi penitus/ interpolationem serierū et has operationes tanqua fundamenta magis genuina solummodo adhibui nec latuit reductio per divisionem res utiqꝫ facilior Sed et resolutionem affectarum ǽquationū mox aggressus sum eamque obtinui. Unde \qua simul/ simul ordinatim applicatæ \vicissi{illeg}|m|/ segmenta axium aliæqꝫ quælibet rectæ ex areis curvarum vel arcubꝫ datis innotuere. nam regressio ad hæc nihil indigeat præter resoluconem æquationū quibꝫ areæ vel arcus ex datis rectis da{illeg}|b|antur: eo tempore pestis ingruens coegit me h{illeg} \hinc/ fugere et alia cogitare, addidi tamen subinde condituram quandam Logarithmorum ex areâ hyperbolæ, quam hic subjungo Sit dFD Hyperbola cujus centrum C, vertex F, & quadratū interjectum $\mathrm{CAFE}=1$ cape AB, Ab hinc inde $=\frac{1}{10}$ sive 0,1, et erectis perpendiculis BD, bd ad Hyperbolam terminatis, erit semisuma spatiorum AD et $\mathrm{Ad}=\mathrm{0,1}+\frac{0.001}{3}+\frac{0.00001}{5}+\frac{0.0000001}{7}\text{\&c}$ et semidifferentia $=\frac{\mathrm{0,01}}{2}+\frac{0.0001}{4}+\frac{0.000001}{6}+\frac{0.00000001}{8}\text{\&c}$ quaæ reductæ {illeg}||ic se habent
$\begin{array}{ccc}\begin{array}{r}\begin{array}{r}0.100000000000\\ \phantom{0.100}333333333\\ \phantom{0.10000}2000000\\ \phantom{0.100000}142857\\ \phantom{0.10000000}1111\\ \phantom{0.10000000000}9\end{array}\\ \begin{array}{r}0.100335477310\end{array}\end{array}& \phantom{00000}& \begin{array}{r}\begin{array}{r}0.0050000000000\\ \phantom{0.1000}250000000\\ \phantom{0.100000}1666666\\ \phantom{0.10000000}12500\\ \phantom{0.1000000000}100\\ \phantom{0.100000000000}1\end{array}\\ \begin{array}{r}0.0050251679267\end{array}\end{array}\end{array}$
horū suma 0.1053605156577 est Ad — et differentia 0.0953101798043 est AD. et eadem ratione positis AB, Ab hinc inde $=\mathrm{0,2}$, obtinebitur $\mathrm{Ad}=0.2231435513142$, et $\mathrm{AD}=$ $0.1823215567939$ habitis sic Logarithmis Hyperbolicis numerorū quatuor decimaliū 0.8, 0,9, 1,1, & 1,2. cum sit $\frac{1.2}{0.8}\times \frac{\mathrm{1,2}}{\mathrm{0,9}}=2$, et 0,8 et 0.9 sint minores unitate, adde Logarithmos illorum ad duplum Logarithmi <2v> $\mathrm{1,}\underset{‾}{|2}$, et habebis 0,6931471805597 Logarithmū Hyperbolicū numeri 2. cujus triplo adde log {illeg} $\mathrm{0,}\underset{‾}{|8}$ (siquidem sit $\frac{2\times 2\times 2}{\mathrm{0,8}}=10$) et habebis 2,3025850929933 logarithmū numeri 10. indeqꝫ per additionem simul prodeunt Logarithmi numerorum 9 et 11; adeoqꝫ omniū primorū horum $3,5,5,11$ Logarithmi in promptu sunt. Insuper ex solâ depressione numerorū superioris computi ꝑ loca decimalia et additione obtinentur Logarithmi decimalium 0.98, 0 99, 1.01, 1.02 ut et horū 0.998, 0.999, 1.001 1.002, et inde ꝑ additionem et substractionem prodeunt Logarithmi primorum $7.13.17.37.$ {xc {sic}} qui una cū superioribꝫ per Log. numeri 10 divisi evadunt veri Logarithmi in Tabulam inserendi sed hoc postea propiùs obtinui, pudet dicere ad quot figurarū loca hæ computationes otiosus eo tempore perduxi, nam tunc sanè nimis delectabar inventis hisce, sed ubi prodiit ingeniosa illa Nicolai Mercatoris Logarithmoteh{illeg}cnia {sic} (quem suppono sua primum invenisse) cœpi ea minus curare suspicatus, vel eum nosse extractionem radicum æquam ac divisione fractionū, vel alios saltem divisione patefacta inventuros reliqua, priusquam ego ætatis essem maturæ ad scribendum, eo ipso tamen tempore quo liber iste prodiit comunicatum est per amicum (D. Barrow tunc matheseos Professore) ad D. Collinsio compendiū quoddam methodi harum serierum in quo significaveram areas et Longitudines curvarum omnium et solidorum superficies et contenta, ex datis rectis vice versâ ex his datis rectas determinari posse et methodum ibi judicatâ \indicatâ/ illustraveram diversis seriebus |[| Suborta deinde inter nos epistolari consuetudine D. Collinsius vir in rem mathematicam promovendam natus non destitit suggerere ut hæc publici juris facerem & ante annos quinqꝫ cum suadentibꝫ amicis, consilium ceperam edendi Tractatū de refractione Lucis et coloribus quem tunc in promptu habed|b|am; cœpi de his seriebꝫ iterū cogitare & tractatū de iis etiam conscripsi ut utrumqꝫ simul ederem, sed ex occasione Telescoptii {sic} catadioptrici epistolâ ad te missâ quâ breviter explicui conceptus meos de naturâ lucis inopinatū <3r> quiddam effecit ut mei interesse sentirem ad te festinanter scribere de impressione istius Epistolæ et subortæ statim ꝑ diversor{illeg}|um| {illeg}|ep|istolas objectionibus aliisqꝫ refertas, crebræ interpellationes me prorsus a consilio deterruerunt et effecerunt ut me arguerem imprudentiæ quod umbram captando eatenus ꝑdideram quietem meam, rem prorsus substantialem. |[| Sub id tempus Gregorius ex unicâ tantâ serie quadam è meis quam D. Collinsius ad eū transmiserat, post multam consideraconem (ut ad Collinsium rescripsit) ꝑvenit ad eandem methodum, & tractatū de eâ reliquit quem speramus ab amicis ejus editū iri. Siquidū {sic} pro ingenio quo pollebat non potuit non adjiciere adjicere de suo multa nova quæ rei mathematicæ interest ut non pereant. ipse autem tractatum meum non penitus absolveram ubi destiti à proposito, n{illeg}|e|qꝫ in hunc usqꝫ diem mens rediit ad reliqua adjicienda Deerat quippe pars ea qua decreveram explicare modum solvendi Problemata quæ ad Quadraturas reduci nequeunt licet aliquid de fundamento ejus posuissem.

Cæterum in tractatu isto series infinitæ non magnam partem obtinebant alia haud pauca congressi \congessi/ intꝰ quæ erat methodus ducendi tangentes quam Solertissimus Slusius ante annos duos tresve tibi communicavit de quâ tu (suggerente Collinsio) rescripsisti eandem mihi etiam innotuisse diversâ ratione in eā incidimus. nam res non eget demonstratione prout ego operor. habito meo fundamento nemo potuit tangentes aliter ducere nisi volens de recta viâ deviaret. Quinetiam non hic hæretur ad æquationes radicalibꝫ unam vel utramqꝫ indefinitam quantitatem involventibꝫ utcunque affectas, sed absqꝫ aliquâ taliū æquationū reductione (quæ opus plerumqꝫ redderet immensū) tangens confestim ducitur: et eodem modo se res habet in quæstionibꝫ de maximis et minimis alijsqꝫ quibusdam de quibꝫ jam non loquor fundamentum harū oꝑationum satis obvium quidam: quoniam jam non optimum possum {illeg}|ex|plicationem ejus prosequi sic potius celavi 6accd et 13eff.713l9n4o4orr4s8t1{illeg}12vx hoc fundamento conatus Sum etiam reddere speculationes de Quadraturâ curvarū simpliciores ꝑerveniqꝫ ad Theoremata quædam generalia. et ut candide agam ecce primū Theorema. |[| Ad curvā aliquam sit $\mathrm{d}{\mathrm{z}}^{\mathrm{\theta }}\times {\overline{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{\eta }}|}}^{\mathrm{\lambda }}$ ordinatim applicata termino diametri <3v> Seu basis z normaliter insistens ubi litteræ d, e, f denotant quaslibet quantitatꝰ datas, et θ, η, λ indices potestatum sive dignitatū quantitatū Quibꝫ affixæ sunt. fac $\frac{\mathrm{\theta }+1}{\mathrm{\eta }}=\mathrm{r}$, $\mathrm{\lambda }+\mathrm{r}=\mathrm{s}$, $\frac{\mathrm{d}}{\mathrm{\eta }\mathrm{f}}\times {\overline{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{\eta }}|}}^{\mathrm{\lambda }+1}=\mathrm{Q}$, et $\mathrm{r}\mathrm{\eta }-\mathrm{\eta }=\mathrm{\pi }$, et area curvæ erit $\mathrm{Q}in\frac{{\mathrm{z}}^{\mathrm{\pi }}}{\mathrm{s}}-\frac{\mathrm{r}-1}{\mathrm{s}-1}\times \frac{\mathrm{e}\mathrm{A}}{\mathrm{f}{\mathrm{z}}^{\mathrm{\eta }}}+\frac{\mathrm{r}-2}{\mathrm{s}-2}\times \frac{\mathrm{e}\mathrm{B}}{\mathrm{f}{\mathrm{z}}^{\mathrm{\eta }}}-\frac{\mathrm{r}-3}{\mathrm{s}-3}\times \frac{\mathrm{e}\mathrm{C}}{\mathrm{f}{\mathrm{z}}^{\mathrm{\eta }}}+\frac{\mathrm{r}-4}{\mathrm{s}-4}\times \frac{\mathrm{e}\mathrm{D}}{\mathrm{f}{\mathrm{z}}^{\mathrm{\eta }}}\text{\&c}$. literis $\mathrm{A},\mathrm{B},\mathrm{C},\mathrm{d},\text{\&c}$ denotantibꝫ terminos proxime antecedentes nempe A terminū $\frac{{\mathrm{z}}^{\mathrm{\pi }}}{\mathrm{s}}$, B terminū $-\frac{\mathrm{r}-1}{\mathrm{s}-1}\times \frac{\mathrm{e}\mathrm{A}}{\mathrm{f}{\mathrm{z}}^{\mathrm{\eta }}}$ &c hæc series ubi r fractio est vel numerus negativus continuatur in infinitum ubi vero r integer est et affirmativus continuatur ad tot terminos tantꝰ quot sunt unitates in eodem r et sic exhibet Geometricam Quadraturam curvæ rem exemplis illustro.

Exemp. 1. proponatur Parabola cujus ordinatim applicata sit $\sqrt{}\mathrm{a}\mathrm{z}$. hæc in formam regulæ reducta fit ${\mathrm{z}}^0\times {\overline{0+\mathrm{a}{\mathrm{z}}^1|}}^{\frac{1}{2}}$. Quare est d $=1$. {illeg}$\mathrm{\theta }=0$. $\mathrm{e}=0$. $\mathrm{f}=\mathrm{a}$. $\mathrm{\eta }=1$ \$\mathrm{\lambda }=\frac{1}{2}$/ adeoqꝫ $\mathrm{r}=1$. $\mathrm{s}=1\frac{1}{2}$. $\mathrm{Q}=\frac{1}{\mathrm{a}}\times {\overline{\mathrm{a}\mathrm{z}|}}^{\frac{3}{2}}$. $\mathrm{\pi }=0$ et area quæsita $\frac{1}{\mathrm{a}}\times {\overline{\mathrm{a}\mathrm{z}|}}^{\frac{3}{2}}in\frac{1}{1\frac{1}{2}}$, hoc est, $\frac{2}{3}\mathrm{z}${illeg}\$\sqrt{}\mathrm{a}\mathrm{z}$/. et sic in genere si $\mathrm{c}{\mathrm{z}}^{\mathrm{\eta }}$ ponatur ordinatim applicata, prodibit area $\frac{\mathrm{c}}{\mathrm{\eta }+1}{\mathrm{z}}^{\mathrm{\eta }+1}$.

Exemplum Secundum Sit ordinatim applicata $\frac{{\mathrm{a}}^4\mathrm{z}}{{\mathrm{c}}^4-2\mathrm{c}\mathrm{c}\mathrm{z}\mathrm{z}+{\mathrm{z}}^4}$ hæc ꝑ reductionem fit ${\mathrm{a}}^4\mathrm{z}\times {\overline{\mathrm{c}\mathrm{c}-\mathrm{z}\mathrm{z}|}}^{-2}$, vel etiam ${\mathrm{a}}^4{\mathrm{z}}^{-3}\times {\overline{-1+\mathrm{c}\mathrm{c}{\mathrm{z}}^{-2}|}}^{-2}$ in priori casu est $\mathrm{d}={\mathrm{a}}^4$. $\mathrm{\theta }=1$. $\mathrm{e}=\mathrm{c}\mathrm{c}$. $\mathrm{f}=-1$. $\mathrm{\eta }=2$. $\mathrm{\lambda }=-2$. Adeoqꝫ $\mathrm{r}=1$ |{potius $2.\pm 0.2$} Erratum {propter simplum} $\mathrm{r}=1$ cum sit $\mathrm{r}=2$| $\mathrm{s}=-1$ . $\mathrm{Q}=\frac{{\mathrm{a}}^4}{-2}\times {\overline{\mathrm{c}\mathrm{c}-\mathrm{z}\mathrm{z}|}}^{-1}$ hoc est $=\frac{-{\mathrm{a}}^4}{2\mathrm{c}\mathrm{c}-2\mathrm{z}\mathrm{z}}$. $\mathrm{\pi }=0$. et area curvæ = $\mathrm{Q}in\frac{{\mathrm{z}}^0}{-1}$ id est $=\frac{-{\mathrm{a}}^4}{2\mathrm{c}\mathrm{c}-2\mathrm{z}\mathrm{z}}$. I\n/ secundo autem casu est $\mathrm{d}={\mathrm{a}}^4$. $\mathrm{\theta }=-3$ $\mathrm{e}=-1$. $\mathrm{f}=\mathrm{c}\mathrm{c}$. $\mathrm{\eta }=-2$. $\mathrm{\lambda }=-2$. $\mathrm{r}=1$. $\mathrm{s}=-1$ id est $\mathrm{Q}=\frac{{\mathrm{a}}^4}{-2\mathrm{a}}\times {\overline{-1+\mathrm{c}\mathrm{c}{\mathrm{z}}^{-2}|}}^{-1}$ |$-2\mathrm{c}\mathrm{c}$| $=\frac{-{\mathrm{a}}^4\mathrm{z}\mathrm{z}}{2{\mathrm{c}}^4-2\mathrm{c}\mathrm{c}\mathrm{z}\mathrm{z}}$. $\mathrm{\pi }=0$. et Area $=\mathrm{Q}in\frac{{\mathrm{z}}^0}{1}$ hoc est $=\frac{{\mathrm{a}}^4\mathrm{z}\mathrm{z}}{2{\mathrm{c}}^4-2\mathrm{c}\mathrm{c}\mathrm{z}\mathrm{z}}$ area his casibꝫ diversimodè exhibetur Quatenus computatur a diversis finibꝫ, Quorum assignatio ꝑ hos inventos valores arearum facilis est.

Exemp: 3. Sit ordinatim applicata $\frac{{\mathrm{a}}^5}{{\mathrm{z}}^5}\sqrt{\mathrm{b}\mathrm{z}}+\mathrm{z}\mathrm{z}$ hoc est ꝑ reductionem ad debitam formam vel ${\mathrm{a}}^5{\mathrm{z}}^{-\frac{9}{2}}\times {\overline{\mathrm{b}+\mathrm{z}|}}^{\frac{1}{2}}$ |vel| |${\mathrm{a}}^5{\mathrm{z}}^{-4}\times {\overline{1+\mathrm{b}{\mathrm{z}}^{-1}|}}^{\frac{1}{2}}$| et erit in priori casu $\mathrm{d}={\mathrm{a}}^5$. $\mathrm{\theta }=-\frac{9}{2}$. $\mathrm{e}=\mathrm{b}$. $\mathrm{f}=1$. $\mathrm{\eta }=1$. $\mathrm{\lambda }=\frac{1}{2}$. Adeoqꝫ $\mathrm{r}=-\frac{7}{2}$ quare cum r non sit numerus affirmativus procedo ad alterū casū hic est $\mathrm{d}={\mathrm{a}}^5$. $\mathrm{\theta }=-4$. $\mathrm{e}=1$. $\mathrm{f}=\mathrm{b}$. $\mathrm{\eta }=-1$. $\mathrm{\lambda }=\frac{1}{2}$. adeoqꝫ $\mathrm{r}=3$^...[2]. s s$=3\frac{1}{2}$. $\mathrm{Q}=\frac{{\mathrm{a}}^5}{-\mathrm{b}}\times {\overline{1+\mathrm{b}{\mathrm{z}}^{-1}|}}^{\frac{3}{2}}$, seu $=-\frac{{\mathrm{a}}^5\mathrm{z}+{\mathrm{a}}^5\mathrm{b}}{\mathrm{b}\mathrm{z}\mathrm{z}}\sqrt{\mathrm{z}\mathrm{z}+\mathrm{b}\mathrm{z}}$, $\mathrm{\pi }=-2$. Et area $\mathrm{Q}in\frac{{\mathrm{z}}^{-2}}{3\frac{1}{2}}-\frac{2}{2\frac{1}{2}}\times \frac{{\mathrm{z}}^{-1}}{3\frac{1}{2}\mathrm{b}}+\frac{1}{1\frac{1}{2}}\times \frac{\mathrm{z}}{2\frac{1}{2}}\times \frac{{\mathrm{z}}^0}{3\frac{1}{2}\mathrm{b}\mathrm{b}}$ hoc est $=\frac{-30\mathrm{b}\mathrm{b}+24\mathrm{b}\mathrm{z}-16\mathrm{z}\mathrm{z}}{105\mathrm{b}\mathrm{b}\mathrm{z}\mathrm{z}}\times \frac{{\mathrm{a}}^5\mathrm{z}+{\mathrm{a}}^5\mathrm{b}}{\mathrm{b}\mathrm{z}\mathrm{z}}\sqrt{\mathrm{z}\mathrm{z}+\mathrm{b}\mathrm{z}}$.

Exemp: 4. Sit deniqꝫ Ordinatim applicata $\frac{\mathrm{b}{\mathrm{z}}^{\frac{1}{3}}}{\sqrt{⑤{\mathrm{c}}^3-3\mathrm{a}\mathrm{c}\mathrm{c}{\mathrm{z}}^{\frac{2}{3}}}}$ $+3\mathrm{a}\mathrm{a}\mathrm{c}{\mathrm{z}}^{\frac{4}{3}}-{\mathrm{a}}^3\mathrm{z}\mathrm{z}$ hæc ad formam regulæ reducta fit $\mathrm{b}{\mathrm{z}}^{\frac{1}{3}}\times {\overline{\mathrm{c}-\mathrm{a}{\mathrm{z}}^{\frac{2}{3}}|}}^{-\frac{3}{5}}$ Indeqꝫ est $\mathrm{d}=\mathrm{b}$. $\mathrm{\theta }=\frac{1}{3}$. $\mathrm{e}=\mathrm{c}$. $\mathrm{f}=-\mathrm{a}$. $\mathrm{\eta }=\frac{2}{3}$. $\mathrm{\lambda }=-\frac{3}{5}$. $\mathrm{r}=2$^...[3]. $\mathrm{s}=\frac{7}{5}$ $\mathrm{Q}=\frac{3\mathrm{b}}{-2\mathrm{a}}\times {\overline{\mathrm{c}-\mathrm{a}{\mathrm{z}}^{\frac{2}{3}}|}}^{\frac{2}{5}}$. $\mathrm{\pi }=\frac{2}{3}$ et Area $=\mathrm{Q}\times \stackrel{‾}{\frac{5{\mathrm{z}}^{\frac{2}{3}}}{7}-\frac{5}{2}\times \frac{5\mathrm{c}}{-7\mathrm{a}}}$, id est $-\frac{30\mathrm{a}\mathrm{b}{\mathrm{z}}^{\frac{2}{3}}+75\mathrm{b}\mathrm{c}}{28\mathrm{a}\mathrm{a}}\times {\overline{\mathrm{c}-\mathrm{a}{\mathrm{z}}^{\frac{2}{3}}|}}^{\frac{2}{5}}$.

<4r>

$=-\frac{{\mathrm{a}}^5\mathrm{z}+{\mathrm{a}}^5\mathrm{b}}{\mathrm{b}\mathrm{z}\mathrm{z}}\sqrt{\mathrm{z}\mathrm{z}+\mathrm{b}\mathrm{z}}$, $\mathrm{\pi }=-2$. Et area $\mathrm{Q}in\frac{{\mathrm{z}}^{-2}}{3\frac{1}{2}}-\frac{2}{2\frac{1}{2}}\times \frac{{\mathrm{z}}^{-1}}{3\frac{1}{2}\mathrm{b}}+\frac{1}{1\frac{1}{2}}\times \frac{\mathrm{z}}{2\frac{1}{2}}\times \frac{{\mathrm{z}}^0}{3\frac{1}{2}\mathrm{b}\mathrm{b}}$, hoc est $=\frac{-30\mathrm{b}\mathrm{b}+24\mathrm{b}\mathrm{z}-16\mathrm{z}\mathrm{z}}{105\mathrm{b}\mathrm{b}\mathrm{z}\mathrm{z}}\times \frac{{\mathrm{a}}^5\mathrm{z}+{\mathrm{a}}^5\mathrm{b}}{\mathrm{b}\mathrm{z}\mathrm{z}}\sqrt{\mathrm{z}\mathrm{z}+\mathrm{b}\mathrm{z}}$ . Exemp: 4. Sit deniqꝫ Ordinatim applicata $\sqrt{⑤{\mathrm{c}}^3-\mathrm{a}\mathrm{c}\mathrm{c}{\mathrm{z}}^{\frac{2}{3}}+3\mathrm{a}\mathrm{c}{\mathrm{z}}^{\frac{4}{3}}-{\mathrm{a}}^3}$ $\mathrm{z}\mathrm{z}$ hæc ad formam regulæ reducta sit $\mathrm{b}{\mathrm{z}}^{\frac{1}{3}}\times {\overline{\mathrm{c}-\mathrm{a}{\mathrm{z}}^{\frac{2}{3}}|}}^{-\frac{3}{5}}$ indeqꝫ est $\mathrm{d}=\mathrm{b}$. $\mathrm{\theta }=\frac{1}{3}$. $\mathrm{e}=\mathrm{c}$. $\mathrm{f}=-\mathrm{a}$. $\mathrm{\eta }=\frac{2}{3}$ $\mathrm{\lambda }=-\frac{3}{5}$. $\mathrm{r}=2$^...[4]. $\mathrm{s}=\frac{7}{5}$ $\mathrm{Q}=\frac{3\mathrm{b}}{-2\mathrm{a}}\times {\overline{\mathrm{c}-\mathrm{a}{\mathrm{z}}^{\frac{2}{3}}|}}^{\frac{2}{5}}$. $\mathrm{\pi }=\frac{2}{3}$ et area $\mathrm{Q}\times \stackrel{‾}{\frac{5{\mathrm{z}}^{\frac{2}{3}}}{7}-\frac{5}{2}\times \frac{5\mathrm{c}}{-7\mathrm{a}}}$, id est $-\frac{30\mathrm{a}\mathrm{b}{\mathrm{z}}^{\frac{2}{3}}+75\mathrm{b}\mathrm{c}}{28\mathrm{a}\mathrm{a}}$ $\frac{-2\mathrm{a}}{\times {\overline{\mathrm{c}-\mathrm{a}{\mathrm{z}}^{\frac{2}{3}}|}}^{\frac{2}{5}}}$

Quod si res non successisset in hoc casu existente r vel fractione vel numero negativo, tunc tentassem alterum casum purgando terminum $-\mathrm{a}{\mathrm{z}}^{\frac{2}{3}}$ in ordinatim applicatâ a coefficiente ${\mathrm{z}}^{\frac{2}{3}}$ hoc est reducendo ordinatim applicatam ad hanc formam $\mathrm{b}{\mathrm{z}}^{-\frac{1}{15}}\times {\overline{-\mathrm{a}+\mathrm{c}{\mathrm{z}}^{-\frac{2}{3}}|}}^{-\frac{3}{5}}$ et si {illeg} in neutro casu fuisset numerus integer et affirmativus conclusiss{illeg}|em| curvam ex earum numero esse quæ non possunt Geometricè quadrari nam quantum animadverto, hæc regula {illeg}|e|xhibet in infinitis |finitis| æquationibus areas omniū Geometrica quadratura admittentium Curvarum quarum ordinatim applicatæ constant ex potestatibꝫ radicibus vel quibuslibet quanti|digni|tatibus dignitatibus binomij cujuscunqꝫ licèt non directe ubi index dignitatis est numerus integer: |[| At quando hujusmodi curva aliqua non potest Geometricè quadrari sunt ad manus alia Theoremata pro comparatione ejus cum conicis sectionibus vel saltem cum aliis figuris simplicissimis quibuscū potest comparari ad quo{illeg}|d| sufficit {illeg} \etiam/ hoc ipsū {illeg}|u|nicū jam descriptum Theorema si debite continuatur. pro Trjnomijs etiam et alijs quibusdam, regulas \{nonullas}/ quasdem continuari sed in simplicioribꝫ vulgoqꝫ celebratis figuris vix aliquid relatu dignū reperi quod evasit aliorum conatus nisi forte longitudo Cissoidis ejusmodi censeatur. Ea|^a| sic construitur \determina|o|tur/. Sit {illeg} VD, Cissois, AV, diameter circuli ad quem aptatur V, vertex, AF Asymptoton ejus, ac DB perpendiculare quodvis ad AV demissum cum semiaxe $\mathrm{AF}=\mathrm{AV}$ et semiparametro $\mathrm{AG}=\frac{1}{3}\mathrm{AV}$ describatur Hyꝑbola FkK, et inter AB et AV sumpta AC: media proportionali erigantur ad C et V ꝑpendicula Ck et VK <4v> hyperbolæ occurrentia in k & K et agantur rectæ KT et kt tangentes hyperbolam in eisdem K & k et occurrentes AV, in T, ac, t, et ad AV constituatur rectangulum AVnm æquale spatio TK,kt et cissoidis \VD/ longitudo erit sextupla altitudinis Vn demonstratio perbrevis est |[| sed ad infinitas series reddeo redeo.

Quamvis multa restent investiganda circa modos approximandi, & diversa serierū genera quæ possunt ad id conducere, tamen vix cum D. Tschurnhausio speraverim dari posse aut simpliciora aut magis generalia fundamenta reducendi quantitates ad hoc genus serierū de quo agimus quam sunt divisiones et extractiones radicū quibus Leibnitius et ego utimur saltem ut non generaliora quia pro Quadraturâ et ενθυνεσ \ὲνθυνσ/ curvarum ac similibꝫ nullæ possunt dari series ex hisce simplicibus terminis Algebraicis (unicam tantꝰ indefinitam quantitatꝰ involventibꝫ) constantes quas non licet hac methodo colligere nam non possunt esse plures hujusmodi convergentes series ad idem determinandꝰ quam sunt indefinitæ quantitates ex quarum potestatibus series conflentur et ego quidem ex adhibitâ quacunqꝫ indefinitâ quantiitate {sic} seriem novi colligere et idem credo Leibnitio in potestate esse, nam quamvis meâ methodo liberum sit eligere pro conflandâ serie quantitatꝰ quandam |quamlibet| indefinitam a quâ quæsitū dependeat, et methodum |us| quam ipse nobis communicavit determinata videatur ad electionem taliū indefinitarū quantitatū quibꝫ opus comodè deduci potest ad fractiones quæ ꝑ solam divisionem evadant series infinitæ, tamen aliæ quæcunqꝫ indefinitæ quantitates pro seriebus conflandis adhiberi possunt per methodum istam quâ affectæ æquationes resolvuntur in proprijs terminis, hoc est conficiendo seriem ex solis terminis quæ æquatio involvit, præterea non video cur dicitur his divisionibꝫ et extractionibus problemata resolvi per accidens siquidem hæ operationes eodem modo se habeant ad hoc genus Algebræ, ac vulgares operationes Arithmeticæ ad Algebra{illeg}|m| vulgo notam |[| quod autem ad simplicitatem methodi attinet nolim fractiones <5r> et radicales absqꝫ præviâ reductione semper resolvi in series infinitas. Sed ubi perplexæ quantitates occurrunt tentandæ sunt omnimodæ reductiones, sive id fiat, augendo, minuendo, multiplicando, vel dividendo quantitates indefinitas sive per methodum transmutatoriam Leibnitii aut alio quocunqꝫ modo qui occurrat, et tunc resolutio in series ꝑ divisionem et extractionem opportune adhibebitur \hic/ hic autem præcipuè intendum est, ut denominatores fractionū & quantitates in vinculo radicū reducantur ad quam paucissimas & minime compositas, et ad tales etiam quæ in serie abeant citissimè convergentem, et si radices neqꝫ convertantur in fraction{illeg}es neqꝫ deprimantur nam ꝑ regulam initio alterius epist{illeg}|o|læ, extractio altissimarū radicū æque simplex et facilis est ac extractio radicis Quadraticæ vel divisio, et series quæ ꝑ divisionem eliciuntur solent minimè omniū convergere, |[| hactenus de seriebꝫ unicam indefinitam quantitatem involventibꝫ locutus sum sed possunt etiam perspecta methodo series ex duabus vel pluribus assignatis indefinitis quantitatibus pro arbitrio confici |[| Quinetiam beneficio ejusmodi methodi possunt series ad omnes figuras efformari Gregorianis ad circulū et hyperbolam editis affines hoc est quarū ultimo|u|s termino|u|s exhibebit quæsitam aream |[| sed calcul{illeg}|ū| hic onerosiorem nolim lubens subire. |[| possunt deniqꝫ series ex terminis compositis eadem methodo constitui Quemadmodum si sit $\sqrt{\mathrm{a}\mathrm{a}-\mathrm{a}\mathrm{x}+\frac{{\mathrm{x}}^3}{\mathrm{a}}}$ ordinatim applicata curvæ alicujus pone $\mathrm{a}\mathrm{a}-\mathrm{a}\mathrm{x}=\mathrm{z}\mathrm{z}$ et ex binomio $\mathrm{z}\mathrm{z}-\frac{{\mathrm{x}}^3}{\mathrm{a}}$ extractâ radice prodibit $\mathrm{z}+\frac{{\mathrm{x}}^3}{2\mathrm{a}\mathrm{x}}-\frac{{\mathrm{x}}^6}{8\mathrm{a}\mathrm{a}{\mathrm{z}}^3}$ &c. cujus seriei omnes termini quadrari possunt per Theorema jam ante descriptū, sed hæc minoris facio quod ubi series simplices non sunt satis tractabiles aliam nondum comunicatam methodum habeo qua pro lubitu acceditur ad quæsitū. {illeg}|E| jus fundamentum est comoda expedita et general{illeg}|i|s solutio hujus problematis c|C|urvam Geometrica describere quæ per data quotcunqꝫ puncta transibit. docuit Euclides descriptionem Circuli per tria data puncta potest etiam conn\i/ca {sic} Sectio describi per quinque data puncta. et curva triū dimensionū per octo \septem/ data puncta hæc statim{illeg} Geometricè fiunt nullo calculo intꝰposito, sed superius problema est alterius generis, et quamvis primâ fronte intractabile videatur tamen res aliter se habet est enim ferè ex pulcherrimis quæ solvere desiderem |[| Seriei à D. Leibnitio pro quadraturâ Curvarū Sectionū proposita affinia sunt Theoremata quædam quæ pro comparatione curvarū cum Conicis sectionibꝫ in catalogum dudum retuli possū utiqꝫ cum conicis sectionibꝫ Geometricè comparare curvas omnes quarū ordinatim <5v> applicatatæ {sic} sunt $\frac{\mathrm{d}{\mathrm{z}}^{\mathrm{\eta }-1}}{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{\eta }}+\mathrm{g}{\mathrm{z}}^{2\mathrm{\eta }}}$ vel $\frac{\mathrm{d}{\mathrm{z}}^{2\mathrm{\eta }-1}}{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{\eta }}+\mathrm{g}{\mathrm{z}}^{2\mathrm{\eta }}}$ &c aut $\frac{\mathrm{d}{\mathrm{z}}^{\frac{1}{2}\mathrm{\eta }-1}}{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{\eta }}+\mathrm{g}{\mathrm{z}}^{2\mathrm{\eta }}}$ $\frac{\mathrm{d}{\mathrm{z}}^{\frac{3}{2}\mathrm{\eta }-1}}{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{\eta }}+\mathrm{g}{\mathrm{z}}^{2\mathrm{\eta }}}$ vel $\frac{\mathrm{d}}{\mathrm{z}}\sqrt{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{\eta }}+\mathrm{g}{\mathrm{z}}^{2\mathrm{\eta }}}$, vel, $\mathrm{d}{\mathrm{z}}^{\mathrm{\eta }-1}\times \sqrt{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{\eta }}+\mathrm{g}{\mathrm{z}}^{2\mathrm{\eta }}}$ aut $\frac{\mathrm{d}{\mathrm{z}}^{\mathrm{\eta }-1}}{\sqrt{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{\eta }}+\mathrm{g}{\mathrm{z}}^{2\mathrm{\eta }}}}$ vel $\frac{\mathrm{d}{\mathrm{z}}^{2\mathrm{\eta }-1}}{\sqrt{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{\eta }}+\mathrm{g}{\mathrm{z}}^{2\mathrm{\eta }}}}$ &c aut $\frac{\mathrm{d}{\mathrm{z}}^{\mathrm{\eta }-1}\times \sqrt{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{\eta }}}}{\mathrm{g}+\mathrm{h}{\mathrm{z}}^{\mathrm{\eta }}}$ aut $\frac{\mathrm{d}{\mathrm{z}}^{2\mathrm{\eta }-1}\times \sqrt{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{\eta }}}}{\mathrm{g}+\mathrm{h}{\mathrm{z}}^{\mathrm{\eta }}}$ &c vel $\frac{\mathrm{d}{\mathrm{z}}^{\mathrm{\eta }-1}}{\overline{\mathrm{g}+\mathrm{h}{\mathrm{z}}^{\mathrm{\eta }}}\times \sqrt{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{\eta }}}}$ vel $\frac{\mathrm{d}{\mathrm{z}}^{2\mathrm{\eta }-1}}{\overline{\mathrm{g}+\mathrm{h}{\mathrm{z}}^{\mathrm{\eta }}}\times \sqrt{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{\eta }}}}$ |aut $\frac{\mathrm{d}}{\mathrm{z}}\genfrac{}{}{0ex}{}{\sqrt{\phantom{|}}}{\phantom{|}}\frac{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{\eta }}}{\mathrm{g}+\mathrm{h}{\mathrm{z}}^{\mathrm{\eta }}}$ vel $\mathrm{d}{\mathrm{z}}^{\mathrm{\eta }-1}\times \genfrac{}{}{0ex}{}{\sqrt{\phantom{|}}}{\phantom{|}}\frac{\mathrm{e}+\mathrm{f}{\mathrm{z}}^{\mathrm{\eta }}}{\mathrm{g}+\mathrm{h}{\mathrm{z}}^{\mathrm{\eta }}}$ &c.|
|Hic d, e, f, g| significant quasvis datas quantitates cum suis signis + & \−/et affectas, z, axem vel basem curvæ, et η, $2\mathrm{\eta }$, $\frac{1}{2}\mathrm{\eta }-1$, $\frac{3}{2}\mathrm{\eta }-1$, $\mathrm{\eta }-1$, $2\mathrm{\eta }-1$ indices potestatum vel dignitatum z, sive sint affirmativi vel negativi sive integri vel fracti, et singula bina Theoremata sunt duo primi termini seriei in infinitū progredientis in tertio et quarto \$4.\mathrm{e}.\mathrm{g}$/ exem gratia debet esse non majus quam \$\mathrm{f}\mathrm{f}$/ nisi \e/ et \g/ sint contrarii signi in cæteris nulla est limitatio. (harū aliqua nempe secundum, tertium, quartum, quintum & decimum tertium) ex areis duarum conicarū Sectionum co{illeg}|n|junctis constant; aliæ quædam (ut nonum decimum et duodecimum) sunt aliter satis compositæ, et omnia quidem in continuatione progressionum cito evadunt compositissima; adeo ut vix ꝑ transmutationes figurarū quibus Gregorius et alii usi sunt absqꝫ ulteriori fundamento inveniri posse putem. ego quidem haud quicquam generale in his obtinere potui antequam abstraherem a contemplatione figurarum, et rem totam ad simplicem consideraconem solarum ordinatim applicatarum reducerem; sed cum hæc et his generaliora sint in potestate, non dubitabitur credo de binomialibus longè facilioribus quæ in his continentur, et prodeunt ponendo tantum literam aliquam e, vel f vel $\mathrm{g}=0$, et $\mathrm{\eta }=1$ vel 2 et si serie{illeg}|s| in quas ista resolvuntur non posuerim in Epistolâ priori, intentus non in omnia particularia enumeranda, sed in illustranda methodum per unam et alteram in singulis rerum generibus instantiam, quæ ad ostendendam ejus generalitatem sufficere videbatur |[| cæterū hæc Theoremata dant series plusquam uno modo nam primum si ponatur $\mathrm{f}=0$, et $\mathrm{\eta }=1$ evadit $\frac{\mathrm{d}}{\mathrm{e}+\mathrm{g}\mathrm{z}\mathrm{z}}$: unde prodit series nobis communicata, sed si ponatur $2\mathrm{e}\mathrm{g}=\mathrm{f}\mathrm{f}$ et $\mathrm{\eta }=1$ inde tandem obtinemus hanc seriem $1+\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\frac{1}{9}+\frac{1}{11}-\frac{1}{13}-\frac{1}{15}$ pro longitudine quadrantalis arcus cujus chorda est unitas vel quod perinde est hanc $\frac{1}{2}+\frac{1}{15}-\frac{1}{63}+\frac{1}{143}\text{\&c}$ pro longitudine dimidii ejus, et has forte quia æque simplices sunt ac alteræ, et magis convergunt <6r> non repudiabitis. sed ego rem aliter æstimo illud enim melius quod utilius est, et problema minori labore solvit, sic quamvis hæc æquatio ${\mathrm{x}}^3-\mathrm{x}=1$ appareat simplicior hacce $\mathrm{y}\mathrm{y}-2\mathrm{y}\sqrt{\frac{81}{25}-\sqrt{}20}=\sqrt{}20${illeg} tamen in confesso est posteriorem revera simpliciorem esse, propterea quod radicem ejus Geometra facilius eruit; et ob hanc rationem series pro obtinendis arcubus circuli, vel (quod eodem recidit) pro obtinendis sectoribꝫ conicarū Sectionū, pro optimis habeo quæ componuntur ex potestatibꝫ sinuum. nam siquis vellet ꝑ simplex computū hujus seriei $1+\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\frac{1}{9}+\text{\&c}$ colligere longitudinem Quadrantis ad viginti figurarum loca decimalia, opus esset 5000000000 terminis seriei circiter, ad quorum calculū milleni anni requirerentur et res tardius obtineretur ꝑ tangentem {illeg}|45| gradiū. sed adhibito sinu recto $45$ grad. quinquaginta quinque vel sexaginta termini hujus seriei $\sqrt{}\frac{1}{2}\ast \overline{1+\frac{1}{12}+\frac{3}{160}+\frac{5}{896}}\text{\&c}$, sufficerent, quorum computatio tribus ut opinor vel quatuor diebus absolvi posset; et tamen hic non est optimus modus computandi totam peripheriam; nam series ex sinu recto triginta grad. vel ex sinu verso $60$ graduum conflata multo citius dabit arcum suum, cujus sextuplum vel duodecuplum est tota peripheria: neque majori labore eruitur area totius circuli ex segmento cujus sagitta est quadrans diametri ejus computi{illeg} specimen, siquidem ad manus est, visū fuit apponere, et una adjungere aream hyperbolæ quæ eodem calculo prodit. posito axe transverso $=1$, & sinu verso seu segmenti sagitta $=\mathrm{x}$ erit semisegmentum
$\begin{array}{l}\text{Hyperbolæ}\\ \text{Circuli}\end{array}\}={\mathrm{x}}^{\frac{1}{2}}in\frac{2}{3}\mathrm{x}\pm \frac{\mathrm{x}\mathrm{x}}{5}-\frac{{\mathrm{x}}^3}{28}\pm \frac{{\mathrm{x}}^4}{72}$ \&c/ hæc autem series sic in infinitum producitur sit $2{\mathrm{x}}^{\frac{3}{2}}=\mathrm{a}$, $\frac{\mathrm{a}\mathrm{x}}{2}=\mathrm{b}$, \$\frac{\mathrm{b}\mathrm{x}}{4}$/$=\mathrm{c}$, $\frac{3\mathrm{c}\mathrm{x}}{6}=\mathrm{d}$. $\frac{5\mathrm{d}\mathrm{x}}{8}$$=\mathrm{e}$, $\frac{7\mathrm{e}\mathrm{x}}{10}=\mathrm{f}\text{\&c.}$ et erit semisegmentum $\begin{array}{l}\text{hyperbolæ}\\ \text{circuli}\end{array}\}=\frac{\mathrm{a}}{3}\pm \frac{\mathrm{b}}{5}-\frac{\mathrm{c}}{7}\pm \frac{\mathrm{d}}{9}$$-\frac{\mathrm{e}}{11}\pm \frac{\mathrm{f}}{13}\text{\&c}$ eorumqꝫ semisu^ma $\frac{\mathrm{a}}{3}-\frac{\mathrm{c}}{7}-\frac{\mathrm{e}}{11}-\text{\&c.}$ et semidifferentia $\frac{\mathrm{b}}{5}+\frac{\mathrm{d}}{9}+\frac{\mathrm{f}}{13}+\text{\&c.}$ his ita præparatis suppono \assumo/ $\mathrm{x}=\frac{1}{4}$ $\mathrm{x}\mathrm{x}$ quadrantem nempe axis, et prodit $\mathrm{a}\left(\frac{1}{4}\right)=0.25$. $\mathrm{b}\left(=\frac{\mathrm{a}\mathrm{x}}{2}=\frac{0.25}{1\times 8}\right)$$=0.03125$. $\mathrm{c}\left(=\frac{\mathrm{b}\mathrm{x}}{4}=\frac{0.03125}{2\times 8}\right)=0.001953125$. $\mathrm{d}\left(=\frac{3\mathrm{c}\mathrm{x}}{6}=\right.$$\frac{0.001953125}{8}=0.000244140625$ et sic procedo usqꝫ dum venero ad tꝰminū depressissimum qui potest ingredi opus, deinde hos terminos per $3,5,79,$$11,$ respectivè divisos, dispono in duas tabulas, ambiguos cum primo in unam et neg{illeg}|a|tivos in aliam, et addo ut hic vides <6v> $\begin{array}{ccc}\begin{array}{r}\begin{array}{r}\mathrm{0,0833333333333333}\\ \phantom{\mathrm{0,00}}62500000000000\\ \phantom{\mathrm{0,0000}}271267361111\\ \phantom{\mathrm{0,000000}}5135169396\\ \phantom{\mathrm{0,0000000}}144628917\\ \phantom{\mathrm{0,000000000}}4954581\\ \phantom{\mathrm{0,0000000000}}190948\\ \phantom{\mathrm{0,000000000000}}7963\\ \phantom{\mathrm{0,0000000000000}}352\\ \phantom{\mathrm{0,00000000000000}}16\\ \phantom{\mathrm{0,000000000000000}}1\end{array}\\ \begin{array}{r}\mathrm{0,0896109885646618}\end{array}\end{array}& \phantom{00000000}& \begin{array}{r}\begin{array}{r}\mathrm{0,0002790178571429}\\ \phantom{\mathrm{0,00000}}34679066051\\ \phantom{\mathrm{0,0000000}}834465027\\ \phantom{\mathrm{0,00000000}}26285354\\ \phantom{\mathrm{0,0000000000}}961296\\ \phantom{\mathrm{0,00000000000}}38676\\ \phantom{\mathrm{0,000000000000}}1663\\ \phantom{\mathrm{0,00000000000000}}75\\ \phantom{\mathrm{0,000000000000000}}4\end{array}\\ \begin{array}{r}\mathrm{0,0002825719389575}\end{array}\end{array}\end{array}$
tunc a priori suma aufero posteriorem et restat 0,0893284166257043 area semisegmenti hyperbolici. Addo etiam ejusdem sumas & aggregatū aufero à primo termino duplicato 0,1666666666666666, et restat 0.0767731061630478 |3| {&} area semisegmenti circularis huic addo triangulū istud quo completur in sectorem hoc est $\frac{1}{32}\sqrt{}3$, seu 0,0541265877365274 et habeo sectorem sexaginta graduum 0,1308996938995747 cujus sextuplum 0,7853981633974482 est area totius circuli, quæ divisa ꝑ $\frac{1}{4}$ quadrantem diametri dat totam peripheriam 3,1415926535897928 |[| Si artes alias adhibuissem potui per eundem numerum terminorum seriei pervenisse ad multa plura loca figurarū puta viginti quinqꝫ ut amplius sed animus fuit hic ostendere quid per simplex seriei computū præstari posset Quod sane haud difficile est cum in omni opere multiplicatores ac divisores magnâ ex parte non majores quam {illeg}|11| & nunquam majores quam 41 adhibere opus sit. per seriem Leibnitii etiam si ultimo loco dimidiū tꝰmini adjiciatur et alia quædam similia artificia adhibeantur potest computum produci ad multas figuras; ut et ponendo su^mam terminorū $1-\frac{1}{7}+\frac{1}{9}-\frac{1}{15}+\frac{1}{17}-\frac{1}{23}+\frac{1}{25}-\frac{1}{31}+\frac{1}{33}\text{\&c}$ esse a{illeg}|d| totam seriem $1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}\text{\&c}$, ut $1+\sqrt{}2$ ad 2, Sed optimus ejus usus videtur esse quando vel conjungitur cū duabꝫ alijs persimilibus et citissime convergentibus seriebus, vel sola adhibetur ad computandum arcum 30 grad. posita tangente $\sqrt{}\frac{1}{3}$ tunc enim series illa evadit $1-\frac{1}{3\times 3}+\frac{1}{5\times 9}-\frac{1}{7\times 27}+\frac{1}{9\times 81}\text{\&c}$ quæ cito convergit vel si conjunges cum alijs seriebꝫ, pone circuli diametrum $=1$, et $\mathrm{a}=\frac{1}{2}$, et area totius circuli erit $\frac{\mathrm{a}}{1}-\frac{{\mathrm{a}}^3}{3}+\frac{{\mathrm{a}}^5}{5}-\frac{{\mathrm{a}}^7}{7}+\text{\&c}+\frac{\mathrm{a}\mathrm{a}}{1}+\frac{{\mathrm{a}}^5}{3}-$$\frac{{\mathrm{a}}^8}{5}-\frac{{\mathrm{a}}^{11}}{7}+\frac{{\mathrm{a}}^{14}}{9}+\frac{{\mathrm{a}}^{17}}{11}-\text{\&c}+\frac{{\mathrm{a}}^4}{1}-\frac{{\mathrm{a}}^{10}}{3}+\frac{{\mathrm{a}}^{16}}{5}-\frac{{\mathrm{a}}^{22}}{7}+\frac{{\mathrm{a}}^{28}}{9}+\text{\&c}$ hic consideravimus series quatenus adhibentur ad computandum totum circulum, sed quando computandæ sunt partes ejus tunc quælibet series <7r> habet proprium usum et in suo genere optima est si datur tangens satis parva, vel satis magna, non recurrendum erit ad sinum aliquem ut inde computetur arcus, neqꝫ vice versa series dato congruens est æquatio pro solvendo proprio problemate credo Cl. Leibnitium dum posuit seriem pro determinacone {illeg}|co|sinus \{complementi}/ ex arcu dato, vix animo \ū/ advertisse \ad/ seriem meam pro determinacone sinus versi ex eodem arcu siquidem hæc idem sunt, \{hæ eædem} sunt./ neque observasse videtur morem meum generaliter usurpandi literas pro quantitatibus cum signis suis + et − affectis dum dividit hanc seriem $\frac{\mathrm{z}}{\mathrm{b}}+\frac{\mathrm{z}\mathrm{z}}{2\mathrm{a}\mathrm{b}\mathrm{b}}+$$\frac{{\mathrm{z}}^3}{6\mathrm{a}\mathrm{a}{\mathrm{b}}^3}+\frac{{\mathrm{z}}^4}{24{\mathrm{a}}^3{\mathrm{b}}^4}+\text{\&c}$ nam cum area hyperbolicâ BE hic significata per z sit affirmativa vel negativa prout jaceat ex unâ vel alterâ parte ordinatim applicatæ BC
Si area illa in numeris data sit l, & l substituatur in serie pro z orietur vel $\frac{\mathrm{l}}{\mathrm{b}}+\frac{\mathrm{l}\mathrm{l}}{2\mathrm{a}\mathrm{b}\mathrm{b}}+\frac{{\mathrm{l}}^3}{6\mathrm{a}\mathrm{a}{\mathrm{b}}^3}+$$\frac{{\mathrm{l}}^4}{24{\mathrm{a}}^3{\mathrm{b}}^4}\text{\&c}$, vel $-\frac{\mathrm{l}}{\mathrm{b}}+\frac{\mathrm{l}\mathrm{l}}{2\mathrm{a}\mathrm{b}\mathrm{b}}-\frac{{\mathrm{l}}^3}{6\mathrm{a}\mathrm{a}{\mathrm{b}}^3}+\frac{{\mathrm{l}}^4}{24{\mathrm{a}}^3{\mathrm{b}}^4}\text{\&c}$. prout sit affirmativa vel negativa hoc est posito $\mathrm{a}=1=\mathrm{b}$. Et l Logarithmo hyperbolico numerus ei correspondens erit $1+\frac{\mathrm{l}}{1}+\frac{\mathrm{l}\mathrm{l}}{2}+\frac{{\mathrm{l}}^3}{3}+\frac{{\mathrm{l}}^4}{4}\text{\&c}$, si l sit affirmativus et $1-\frac{\mathrm{l}}{1}+\frac{\mathrm{l}\mathrm{l}}{2}-\frac{{\mathrm{l}}^3}{6}+\frac{{\mathrm{l}}^4}{24}\text{\&c}$ si l sit negativus hoc modo fugio multiplicationem Theorematꝰ quæ alias in nimiam molem crescerent, nam v.g. illud unicū Theorema quod supra posui pro quadratura curvarū resolvendꝰ esset in 32 Theoremata si pro signorum varietate multiplicaretur. pr{illeg}|æ|terea quæ habentur de inventione numeri unitate majoris per datum Logarithmū hyperbolicū ope seriei $\frac{\mathrm{l}}{1}-\frac{\mathrm{l}\mathrm{l}}{1\times 2}+\frac{{\mathrm{l}}^3}{1\times 2\times 3}-\frac{{\mathrm{l}}^4}{1\times 2\times 3\times 4}+\text{\&c}$ potius quam ope seriei $\frac{\mathrm{l}}{1}+\frac{\mathrm{l}\mathrm{l}}{1\times 2}+\frac{{\mathrm{l}}^3}{1\times 2\times 3}+\frac{{\mathrm{l}}^4}{1\times 2\times 3\times 4}+\text{\&c}$. nondum \satis/ ꝑcipio. nam si unus terminus adjiciatur amplius ad seriem posteriorem quam ad priorem posterior magis appropinquabit et minor est labor computare unam vel duas primas figuras adjecti hj|u|jus termini, quam dividere unitatem per prodeuntem Logarithmum hyperbolicum ad multa figurarū loca extensū ut inde habeatur Logarithm{illeg}us hyperbolicus Quæsitus, utraqꝫ igitur series (si duas di{illeg}cere fas est \sit/) officio suo fungatur, potest tamen $\frac{\mathrm{l}}{1}+\frac{{\mathrm{l}}^3}{1\times 2\times 3}+$$\frac{{\mathrm{l}}^5}{1\times 2\times 3\times 4\times 5}+\text{\&c}$ series ex dimidiá parte tꝰminorū constans optimè adhibetur\ri/ siquidem hæc dabit semidifferentiam duorum numerorum, ex quâ et rectangulo dato, uterqꝫ datur, sic et ex serie $\frac{\mathrm{l}}{1}+\frac{\mathrm{l}\mathrm{l}}{1\times 2}+\frac{{\mathrm{l}}^4}{1\times 2\times 3\times 4}$ &c datur semisuma numerorū, indeqꝫ etiam numeri. unde prodit relatio serierū inter se, quâ ex unâ datâ dabitur altera Theorema de <7v> inventione arcus ex dato cosin{illeg}o ponendo radiū 1, cosinū c, et arcum $\sqrt{6-\sqrt{24\mathrm{c}+12}}$ minus appropinquat quam primâ fronte videtur posito quidem sinu verso v, error erit $\frac{{\mathrm{v}}^3}{90}+\frac{{\mathrm{v}}^4}{194}+\text{\&c}$ potest fieri ut $120-27\mathrm{v}$ ad $120-17\mathrm{v}$. Ita chorda $\left(\sqrt{}2\mathrm{v}\right)$ ad arcum, et error erit tantum $\frac{61{\mathrm{v}}^3\sqrt{}20}{44800}$ circiter qui semper minor est, quam $5\frac{1}{4}$ minuta secunda, dum arcus non sit major quam $45\mathrm{gr}$, et singulis etiam bisectionibꝫ diminuitur 128 vicibus

Series $\frac{{\mathrm{a}}^3}{1\times 2\times 3}-\frac{{\mathrm{a}}^5}{1\times 2\times 3\times 4\times 5}+\frac{{\mathrm{a}}^7}{1\times 2\times 3\times 4\times 5\times 6\times 7}+\text{\&c}$ applicari posset ad computationem tabulæ segmentorū ut observat vir clarissimus; sed res optimè absolvitur ꝑ canonem sinuū utpote cognitâ quadrantis areâ ꝑ continuas additiones nonæ partis ejus, habebis sectores ad singulos decem gradus in semicirculo, deinde ꝑ continuam additionem decimæ partis hujus habebis sectores ad gradus, et sic ad decimas partes graduum et ultra procedi potest, tunc radio existente 1 ab uno quoqꝫ sectore et ejus complemento ad 180 grad. aufer dimidium communis hujus|sinûs| recti & relinquentur segmenta in tabulam referenda cæterū quamvis series hic non prosint in aljis tamen locum obtinent, et quoniam hoc ad earum usū spectat, non gravabor in aliquibꝫ attingere: |[| constructionem Logarithmorū non aliunde peti debere credetis forte, ex hoc simplici processu qui ab istis pendet, ꝑ methodum supra traditam quærantur Logarithmi hyperbolici numerorum 10, 0.98, 0.99, 1.01, 1.02. id quod fit spatio unius et alterius horæ, deinde divisis Logarithmis quatuor posteriorum ꝑ Logarithmū numeri 10, et addito indice 2 prodibunt veri Logarithmi numerorū 98, 99, 100, 101, 102 in tabulam inserendi, hi ꝑ dena intꝰvalla interpolandi sunt, et exibunt Logarithmi omnium numerorū intꝰ 980 et 1020 et omnibus intꝰ 980 et 1000 iterū ꝑ dena intꝰvalla interpolatis {illeg}|h|abebitur tabula eatenus constructa, tunc ex his colligendi erunt Logarithmi omniū primorū numerorū, et eorum multipliciū minorū quam 100 ad quod nihil requiritur præter additionem et substractionem siquidem sit.
$\sqrt{}⑩\frac{9984\times 1020}{9945}=2$. $\sqrt{}④\frac{8\times 9963}{984}=3$. $\frac{10}{2}=5$. $\frac{\sqrt{}98}{2}=7$. $\frac{99}{9}$ $=11$. $\frac{1001}{7\times 11}=13$. $\frac{102}{6}=17$. $\frac{988}{4\times 13}=19$. $\frac{9936}{16\times 27}=23$. $\frac{986}{2\times 17}=29$. $\frac{992}{32}=31$. \$\frac{999}{27}=37$/. $\frac{984}{24}=41$. $\frac{989}{27}=47$. |$\frac{987}{21}=47$|. $\frac{9911}{11\times 17}=53$. $\frac{9971}{13\times 13}=$ 59. $\frac{9882}{2\times 81}=61$. $\frac{9849}{3\times 49}=67$. $\frac{994}{14}=71$. $\frac{9928}{8\times 17}=73$. $\frac{9954}{7\times 18}$ $=79$. $\frac{996}{12}=83$. $\frac{9968}{7\times 16}=89$. $\frac{9894}{6\times 17}=97$. <8r> et habitis sic Logarithmis omniū numerorū minorū quam 100 restat tantū hos etiam semel atqꝫ iterum ꝑ dena intervalla interpolare |[| constructionis tabulæ sinuū à quâ pendet tota tres trigonometrica fundamentū optimū est continua additio dati anguli ad seipsum vel ad alium datum utpote in angulo addendo bAE inscribantur HI, IK, KL, LM, MN, NO, OP, &c.

æquales radio AB: et ad opposita latera demittantur ꝑpendicula BE, HQ, IR, KS, LT, MV, NX, OY, &c. et angulorū HIQ, IKH, KLI, LMK &c. differentiæ erunt angulus A, sinus HQ, IR, KS &c. et cosinus IQ, KR, LS, &c. detur jam aliquis eorum LMK, et cæteri sic eruentur, Ad SV et MV: demitte perpendicula Ta et Kb, et (propter similia triangula ABE, TLa, KMb, ALT, AMV &c) erit $\mathrm{AB}.$$\mathrm{BE}\colon\colon \mathrm{TL}.\mathrm{La}\left(=\frac{\mathrm{SL}-\mathrm{LV}}{2}\right).\mathrm{KT}\left(\frac{1}{2}\mathrm{KM}\right).\frac{1}{2}\mathrm{Mb}\left(=\frac{\mathrm{MV}-\mathrm{KS}}{2}\right)$. Et $\mathrm{AB}.\mathrm{AE}\colon\colon \mathrm{KT}.\mathrm{Sa}\left(=\frac{\mathrm{SL}+\mathrm{LV}}{2}\right)\colon\colon \mathrm{TL}.\mathrm{Ta}\left(=\frac{\mathrm{KS}+\mathrm{MV}}{2}\right)$ unde dantur sinus et cosinus KS, MV, SL, LV, et simul patet ratio continuandi progressiones, nempè $\mathrm{AB}.2\mathrm{AE}\colon\colon \mathrm{LV}.\mathrm{TM}+\mathrm{MX}\colon\colon \mathrm{MX}.\mathrm{VN}+\mathrm{NY}$ {illeg} &c $\colon\colon \mathrm{MV}.\mathrm{TL}+\mathrm{XN}\colon\colon \mathrm{XN}.\mathrm{VN}-\mathrm{NY}$ &c. Et retrò $\mathrm{AB}.2\mathrm{AE}\colon\colon \mathrm{LS}.$$\mathrm{KT}+\mathrm{RK}$ &c. ponè ergo $\mathrm{AB}=1$, et fac $\mathrm{BE}\times \mathrm{TL}=\mathrm{La}$, $\mathrm{AE}\times \mathrm{KT}=\mathrm{Sa}$. $\mathrm{Sa}-\mathrm{La}=\mathrm{LV}$. $2\mathrm{AE}+\mathrm{LV}-\mathrm{TM}=\mathrm{MX}$, &c. Sed nodus est inventio sinus et cosinus anguli A, et hic subveniunt series nostræ, utpote cognito \ex/ superioribꝫ Quadrantalis arcus longitudine 1.57079, &c et simul quadrato ejus 2.4694 &c divide quadratum hoc per quadratum numeri exprimentis rationem 90 grad. ad angulum A et quoto dicto z, tres vel quatuor primi termini huius seriei $1-\frac{\mathrm{z}}{2}+\frac{\mathrm{z}\mathrm{z}}{24}-\frac{{\mathrm{z}}^3}{720}+\frac{{\mathrm{z}}^4}{40320}$ &c dabunt cosinum istius anguli A, sic primo quæri potest angulus 5 grad et inde tabulam computari ad quinos gradus \ac deinde interpolari ad gradus/ vel dimidios gradus ꝑ eandem methodum, nam non convenit progredi ꝑ nimios saltus, duæ tertiæ partes tabulæ sic computatæ dant reliquam tꝰtiam partem ꝑ additionem vel substractionē more noto, siquidem posito KT cosinu 60 grad. sit $\mathrm{AE}=\mathrm{SV}$, et $\mathrm{BE}=\mathrm{Mb}$, tunc ad decimas, et centesimas partes graduū pergendū est per aliam methodum substitutis tamen prius Logarithmis sinū inventorū, si ejus generis tabula desideretur ad computum tabularū Astronomicarū repleri {sic} pos{illeg}|u|i fundamentum quoddam in alterâ Epistolâ ejus seriei tres primi termini <8v> et aliquando duo sufficiunt, sed ad diversas partes Ellipseos diversæ ejusm
odi series aptari debent, vel potius tales series computandæ sunt quæ ex datâ areâ Sectoris Elliptici BGE, immediatè exhibeant aream sectoris circuli cujus angulus est BEG, radius CB, et habitis hisce, computū earum ad duos tres vel fortè quatuor terminos beneficio Logarithmorū haud gravius erit quam solita resolutio tot triag|n|gulorum in aliis hypothesibus imo forte minus grave si series prius debitæ concinn{illeg}|e|ntur siquidem unus Logarithmus è tabulâ petitus determinet omnes istos terminos, addendo ipsum et ejus multiplices ad Logarithmos datorū coefficientiū in promptu habitos, quæ de hoc genere tabularū dicuntur ad alias transferri possunt ubi la ratiocinia Geometri{illeg}|c|a locum non obtinet sufficit autem per has series computare triginta vel viginti aut fortè pauciores tꝰminos tabulæ in debitis distantijs, tꝰmini intermedii facilè interseruntur ꝑ methodum quandam quam in usum calculatorū ferè hic descripsissem, sed pergo ad alia |[| quæ Cl. Leibnitius à me desiderat explicanda ex parte supra descripsi Quod vero attinet ad invenconem terminorū p, q, r, in extractione radicis affectæ primum p sic eruo, descripto \angul {sic}/ recto BAC, latera ejus BA, AC. divido in partes æquales \{quo{illeg}}/ et inde normales erigo distribuentes angulare spatiū in æqu\a/lia parallelogram̄a vel quadrata quæ concipio denominata esse à dimensionibus duarum indefinitarum specierum puta x et y, regulariter ascendentiū a tꝰmino A prout vides in fig. 1 inscriptas \eas in fig. 1/ ubi \ubi/ y denotat radicem extrahendam et \x/ alterum indefinitam quantitatem ex cujus potestatibꝫ series constituenda est. Deinde cum æquatio aliqua proponitur, parallelogram̄a singulis ejus terminis {illeg}respondentia insigni{illeg}|o| notâ aliquâ, et regulâ ad duo vel forte plura ex insignitis parallelogram̄is applicata, quorū unū sit humillimū \infimum/ in columnâ sinistra \{seu}/ juxta \{latū}/ AB, et aliud ad regulam dextrorsū sitū \{alterū sit infimum} in columna quavis {alia} dextrorsum reperiatur,/ cæteraqꝫ omnia non contingentia regulam \ad quaæ regula non applicatur/ supra eam jaceant seligo tꝰminos <9r> æquationis ꝑ parallelogram̄a contingentia regulam designatos et inde \ex {his tanquam} nihilo æqualibus/ quæro quantitatem quotienti addendam |Quotienti.|

Sic ad extrahendam radicem y ex ${\mathrm{y}}^6-5\mathrm{x}{\mathrm{y}}^5+\frac{{\mathrm{x}}^3}{\mathrm{a}}{\mathrm{y}}^4-7{\mathrm{a}}^2{\mathrm{x}}^2{\mathrm{y}}^2+6{\mathrm{a}}^3{\mathrm{x}}^3$ $+\mathrm{b}\mathrm{b}{\mathrm{x}}^4=0$ parallelogram̄a hujus tꝰminis respondentia signo notâ aliquâ ∗ ut vides in fig 2. Dein applico regulam DE ad inferiorem è locis signatis in sinistrâ columnâ, eamqꝫ ab inferioribꝫ ad superiora dextrorsū gyrare facio donec alium similiter vel fortè plura è reliquis signatis locis coeperit attingere videoqꝫ loca sic attacta esse ${\mathrm{x}}^3$, $\mathrm{x}\mathrm{x}\mathrm{y}\mathrm{y}$ et ${\mathrm{y}}^6$ è terminis itaqꝫ ${\mathrm{y}}^6-7\mathrm{a}\mathrm{a}\mathrm{x}\mathrm{x}\mathrm{y}\mathrm{y}+6{\mathrm{a}}^3{\mathrm{x}}^3$ tanquam nihilo æqualibꝫ (et insuper si placet reductis ad ${\mathrm{v}}^6-7\mathrm{v}\mathrm{v}+6=0$: ponendo $\mathrm{y}=\mathrm{v}\sqrt{}\mathrm{a}\mathrm{x}$), quæro valorem y et invenio quadruplicem $+\sqrt{}\mathrm{a}\mathrm{x}$, − $\sqrt{}\mathrm{a}\mathrm{x}$, $+\sqrt{}2\mathrm{a}\mathrm{x}$: {illeg} \et $-\sqrt{}2\mathrm{a}\mathrm{x}$. Horum valorem/ {quorum} quemlibet pro primo termino Quotientis accipere licet prout è radicibꝫ quampiam extrahere decretū est. sic æquatio ${\mathrm{y}}^3+\mathrm{a}\mathrm{x}\mathrm{y}+\mathrm{a}\mathrm{a}\mathrm{y}-{\mathrm{x}}^3-2{\mathrm{a}}^3=0$ quam resolvebam in priori epistolâ dat $-2{\mathrm{a}}^3+\mathrm{a}\mathrm{a}\mathrm{y}+{\mathrm{y}}^3=0$. et inde $\mathrm{y}=\mathrm{a}$: proximè cum itaqꝫ a sit primus tꝰminus valoris y pono p pro cæteris omnibꝫ in infinitū et substituo $\mathrm{a}+\mathrm{p}=\mathrm{y}$ obvenient hic aliquando dificultates {sic} nonnullæ, sed ex ijs credo D. Leibnitius se proprio marte extricabit subsequentes vero termini \p/ $\mathrm{q}.\mathrm{r}.\mathrm{s}$, &c. eodem modo ex æquationibus secundis tertiis cæterisqꝫ eruuntur quo primus p \y/ $.\mathrm{s}.$ prima, sed cura leviori quia cæteri tꝰmini valoris y solent prodire dividendo tꝰminū involventem infimam potestatem indefinitæ quantitatis \x/ ꝑ coefficientem radicis \lateris/ $\mathrm{p}.\mathrm{q}.\mathrm{r}$ aut s.

Intellex{illeg}ti credo ex superioribꝫ regressionem ab areis curvarū ad lineas rectas fieri ꝑ hanc extractionem radicis affectæ sed duo alii sunt modi quibꝫ idem ꝑficio, eorū unus affinis est computationibꝫ quibus colligebam approximationes sub finem alterius Epistolæ et intelligi potest ꝑ hoc exemplum, proponatur æquatio ad aream hyperbolæ $\mathrm{z}=\mathrm{x}$$+\frac{1}{2}\mathrm{x}\mathrm{x}+\frac{1}{3}{\mathrm{x}}^3+\frac{1}{4}{\mathrm{x}}^4+\frac{1}{5}{\mathrm{x}}^5\text{\&c}$, et partibꝫ ejus multiplicatis in se emerget $\mathrm{z}\mathrm{z}={\mathrm{x}}^2+{\mathrm{x}}^3+\frac{11}{12}{\mathrm{x}}^4+\frac{5}{6}{\mathrm{x}}^5\text{\&c}$. ${\mathrm{z}}^3={\mathrm{x}}^3+\frac{3}{2}{\mathrm{x}}^4+\frac{7}{4}{\mathrm{x}}^5\text{\&c}$. ${\mathrm{z}}^4={\mathrm{x}}^4$$+2{\mathrm{x}}^5\text{\&c}$. ${\mathrm{z}}^5={\mathrm{x}}^5\text{\&c}$. Iam de z aufero $\frac{1}{2}\mathrm{z}\mathrm{z}$ et restat $\mathrm{z}-\frac{1}{2}\mathrm{z}\mathrm{z}=\mathrm{x}-$$\frac{1}{6}{\mathrm{x}}^3-\frac{5}{24}{\mathrm{x}}^4-\frac{16}{60}{\mathrm{x}}^5\text{\&c}$ huic addo $\frac{1}{6}{\mathrm{z}}^3$ et fit $\mathrm{z}-\frac{1}{2}\mathrm{z}\mathrm{z}+\frac{1}{6}{\mathrm{z}}^3=\mathrm{x}+\frac{1}{24}$${\mathrm{x}}^4+\frac{3}{40}{\mathrm{x}}^5\text{\&c}$. \#/^[5] et fit $\mathrm{z}-\frac{1}{2}\mathrm{z}\mathrm{z}+\frac{1}{6}{\mathrm{z}}^3-\frac{1}{24}{\mathrm{z}}^4+\frac{1}{120}{\mathrm{z}}^5=\mathrm{x}$ quamproxime. |$\frac{13}{60}{\mathrm{x}}^4$| Sive $\mathrm{x}=\mathrm{z}-\frac{1}{2}\mathrm{z}\mathrm{z}+\frac{1}{6}{\mathrm{z}}^3-\frac{1}{24}{\mathrm{z}}^4+\frac{1}{120}{\mathrm{z}}^5\text{\&c}$.

|[| eodem modo series de una indefinita quantitate in aliam transferri <9v> possunt quemadmodum si posito r radio circuli, x sinu recto arcus z, et $\mathrm{x}+\frac{{\mathrm{x}}^3}{6\mathrm{r}\mathrm{r}}+\frac{3{\mathrm{x}}^5}{40{\mathrm{r}}^4}+\text{\&c}$ longitudine arcus istius, atqꝫ hanc seriem è sinu recto ad tangentem vellem transferre, quæro longitudinē tangentis $\frac{\mathrm{r}\mathrm{x}}{\sqrt{\mathrm{r}\mathrm{r}-\mathrm{x}\mathrm{x}}}$ et reduco in infinitam seriem $\mathrm{x}+\frac{{\mathrm{x}}^3}{2\mathrm{r}\mathrm{r}}+$$\frac{3{\mathrm{x}}^5}{8{\mathrm{r}}^4}$ quâ dictâ t, colligo potestati|e|s ejus ${\mathrm{t}}^3={\mathrm{x}}^3+\frac{3{\mathrm{x}}^5}{2\mathrm{r}\mathrm{r}}+\text{\&c}$. ${\mathrm{t}}^5={\mathrm{x}}^5$$+\text{\&c}$. aufero autem t de z, et restat $\mathrm{z}-\mathrm{t}=-\frac{{\mathrm{x}}^3}{3}-\frac{3{\mathrm{x}}^5}{10}-\text{\&c}$ Addo $\frac{1}{3}{\mathrm{t}}^3$, et fit $\mathrm{z}-\mathrm{t}+\frac{1}{3}{\mathrm{t}}^3=\frac{1}{5}{\mathrm{x}}^5+\text{\&c}$. aufero $\frac{1}{5}{\mathrm{t}}^5$, et restat $\mathrm{z}-\mathrm{t}+\frac{1}{3}{\mathrm{t}}^3-\frac{1}{5}{\mathrm{t}}^5=0$ quamproximè quare est z$==$ {sic}$\mathrm{t}-\frac{1}{3}{\mathrm{t}}^3+\frac{1}{5}{\mathrm{t}}^5-\text{\&c}$. Sed siquis in usus trigonometricos me jussisset exhibere expressionem arcus ꝑ tangentem, eam non hoc circuitu sed directâ methodo quæsivissem per hoc genus computi colliguntur etiam series ex duabꝫ vel pluribus indefinitis quantitatibꝫ constantes, et radices affectarū æquationū magnâ ex parte extrahuntur, sed ad hunc posteriorem usum adhibeo potius methodum in alterâ Epistolâ descriptam tanquam generalior{illeg}|e|m, et (regulis pro Elisione superfluorū terminorū habitis) paulo magis expeditam, pro regressione vero ab areis ad lineas rectas & similibꝫ, possunt hujusmodi Theoremata adhiberi.

Theorema 1 sit $\mathrm{z}=\mathrm{a}\mathrm{y}+\mathrm{b}\mathrm{y}\mathrm{y}+\mathrm{c}{\mathrm{y}}^3+\mathrm{d}{\mathrm{y}}^4+\mathrm{e}{\mathrm{y}}^5\text{\&c}$ et vicissim erit $\mathrm{y}=\frac{\mathrm{z}}{\mathrm{a}}-\frac{\mathrm{b}\mathrm{z}\mathrm{z}}{{\mathrm{a}}^3}+\frac{2\mathrm{b}\mathrm{b}-\mathrm{a}\mathrm{c}}{{\mathrm{a}}^5}{\mathrm{z}}^3+\frac{5\mathrm{a}\mathrm{b}\mathrm{c}-5{\mathrm{b}}^3-\mathrm{a}\mathrm{a}\mathrm{d}}{{\mathrm{a}}^7}{\mathrm{z}}^4+\frac{3{\mathrm{a}}^2{\mathrm{c}}^2-21\mathrm{a}\mathrm{b}\mathrm{b}\mathrm{c}+6\mathrm{a}\mathrm{a}\mathrm{b}\mathrm{d}+14{\mathrm{b}}^4-{\mathrm{a}}^3\mathrm{e}}{{\mathrm{a}}^9}{\mathrm{z}}^5+\text{\&c}$. ex gratia proponatur æquatio ad aream hyperbolæ $\mathrm{z}=\mathrm{y}-\frac{\mathrm{y}\mathrm{y}}{2}+\frac{{\mathrm{y}}^3}{3}\frac{{\mathrm{y}}^4}{4}+\frac{{\mathrm{y}}^5}{5}\text{etc}$ et substitutis in regulâ 1 pro a, $-\frac{1}{2}$ pro b, $\frac{1}{3}$ pro c, $-\frac{1}{4}$ pro d, et $\frac{1}{5}$ pro e, vicissim exurget $\mathrm{y}=\mathrm{z}+\frac{1}{2}\mathrm{z}\mathrm{z}+\frac{1}{6}{\mathrm{z}}^3+\frac{1}{24}{\mathrm{z}}^4+$\&c/ Theorema 2 sit $\mathrm{z}=\mathrm{a}\mathrm{y}+\mathrm{b}{\mathrm{y}}^3+\mathrm{c}{\mathrm{y}}^5+\mathrm{d}{\mathrm{y}}^7$$+\mathrm{e}{\mathrm{y}}^9\text{\&c}$. et vicissim erit $\mathrm{y}=\frac{\mathrm{z}}{\mathrm{a}}-\frac{\mathrm{b}{\mathrm{z}}^3}{{\mathrm{a}}^4}+\frac{3\mathrm{b}\mathrm{b}-\mathrm{a}\mathrm{c}}{{\mathrm{a}}^7}{\mathrm{z}}^5+\frac{8\mathrm{a}\mathrm{b}\mathrm{c}-\mathrm{a}\mathrm{a}\mathrm{d}-12{\mathrm{b}}^3}{{\mathrm{a}}^{10}}$${\mathrm{z}}^7+\frac{55{\mathrm{b}}^4-55\mathrm{a}\mathrm{b}\mathrm{b}\mathrm{c}+10\mathrm{a}\mathrm{a}\mathrm{b}\mathrm{d}+5\mathrm{a}\mathrm{a}\mathrm{c}\mathrm{c}-{\mathrm{a}}^3\mathrm{e}}{{\mathrm{a}}^{13}}{\mathrm{z}}^9+\text{\&c}$. ex gra /proponatur æquatio\ ad arcum circuli $\mathrm{z}=\mathrm{y}-\frac{{\mathrm{y}}^3}{6\mathrm{r}\mathrm{r}}+\frac{3{\mathrm{y}}^5}{40{\mathrm{r}}^4}+\frac{5{\mathrm{y}}^7}{112{\mathrm{r}}^6}+\text{\&c}$ et substitutis in regulâ 1 pro a, $\frac{1}{6\mathrm{r}\mathrm{r}}$ pro b, $\frac{3}{40{\mathrm{r}}^4}$ pro c, $\frac{5}{112{\mathrm{r}}^6}$ pro d &c orietur $\mathrm{y}=\mathrm{z}-\frac{{\mathrm{z}}^3}{6\mathrm{r}\mathrm{r}}+\frac{{\mathrm{z}}^5}{120{\mathrm{r}}^4}-\frac{{\mathrm{z}}^7}{5040{\mathrm{r}}^6}+\text{\&c}$ |[| alterum modum regrediendi ab areis ad lineas rectas celare statui |[| ubi dixi omnia pene Problemata solubilia existere, volui de ijs præsertim intelligi circa quæ mathematici se hactenus occuparunt vel saltem in quibꝫ <10r> ratiocinia mathematica locum aliquem obtinere possunt nam alia sane adeo perplexis co{illeg}|n|ditionibꝫ implicata excogitare liceat ut non satis comprehendere valeamus et multo minus tantarum computationū onus sustinere quod ista requirerent attamen ne nimiū dixisse videor inversa de tangentibꝫ problemata sunt in potestate aliaqꝫ illis difficiliora ad quæ solvenda usus sum duplici methodo una concinniori altera generaliori utramqꝫ visum est impræsentia literis transpositis consignare ne{illeg} propter alios idem obtinentes institutum in aliquibꝫ mutare cogerer 5accd et 10effh1114t3m9x6oqqr8snt{illeg}|9|v3x: 11ab3cdd10e et g10ill4m7n6o3p\3q/6rsSnt8vx.3ac et 4egh5i4tmsn8oq4r386t4vaadd et eeeeeiiimmnnoopr\rrsssssttuu/ Inversum hoc problema de tangentibꝫ quando tangens inter punctum contactus et axem figuræ est datæ longitudinis, non indiget his methodis, est tamen curva illa mechanica cujus determinatio pendet ab ar{illeg}|e|a \{et pꝰiferia}/ hyperbolæ. ejusdem generis, est etiam problema quando pars axis inter tangentem et ordinatim appli
catam datur longitudine, sed hos casus vix numeraverim inter ludos natur{illeg}|æ| nam quando \Q{illeg} et nam si/ in triangulo rectangulo quod ab illa axis parte, et \ac/ tangente ac|&| ordinatim applicata constituitur, relatio duorū quorūlibet laterum per æquationem quamlibet defini\a/tur, problema solvi potest\rit/ absqꝫ meâ methodo generali sed ubi pars axis ad punctū aliquod positione datum terminata ingreditur vinculū tunc res aliqua aliter se habere solet.

Communicatio Resolutionis affectarum æquationum ꝑ methodū Leibnitii pergrata erit juxta et explicatio quomodo \hæc/ se gerat ubi indices potestatis sunt fractiones ut in hac æquatione $20+{\mathrm{x}}^{\frac{3}{7}}-{\mathrm{x}}^{\frac{6}{5}}{\mathrm{y}}^{\frac{2}{3}}$$-{\mathrm{y}}^{\frac{7}{11}}=0$, aut surdæ quantitates ut in hac ${\overline{{\mathrm{x}}^{\sqrt{}2}+{\mathrm{x}}^{\sqrt{}7}|}}^{\sqrt{}③\frac{2}{3}}=\mathrm{y}$, ubi $\sqrt{}2$ & $\sqrt{}7$ non designant coefficientes ipsius x, sed indices potestatis seu dignitatis ejus; et $\sqrt{}③\frac{2}{3}$ indicem dignitatis binomii ${\mathrm{x}}^{\sqrt{}2}+{\mathrm{x}}^{\sqrt{}7}$. res credo mea{illeg} methodo patet aliter descripsissem sed meta tandem prolixæ huic epistolæ ponenda est, literæ sane excellentissimi Leibnitii valdè dignæ erant quibꝫ fusius hoc responsū darem et volui hac vice copiosior esse, quia credidi amœniora tua negotia severiori hocce scribendi genere non debere a me cred|b|\r/o interpellari.

Tui Studiosissimus
Is. Newton