Rough drafts of the Leibniz Scholium in the 2nd Edition of the Principia, and proposed additions to it

When two spherical bodies strike upon one another, not directly as in the foregoing experiment of Pendulums, but in any obliquity: to find the force of their shock & the motion wchwhich they will have afterward, we must first find the ~~plane in they touch one another~~ right line wchwhich passes through their centers at the moment of the shock, & then distinguish the motion of each body into two, the one perpendicular to that ~~plane~~ line the other parallel to it. The ~~parallel~~ perpendicular motions will remain unaltered by the shock, the ~~perpendicular~~ parallel will receive the same alteration ~~by the shock~~ as if the bodies had ~~rem~~ met directly with those motions alone. As for instance

B. Scholium

In literis quæ mihi cum Geometra peritissimo G. G. Leibnitio annis abhinc decem intercedebant, cum significarem me compotem esse methodi determinandi maximas et minimas, ducendi tangentes, quadrandi figuras curvilineas & similia peragendi quæ in terminis surdis æque ac in rationalibus procederet, et methodumque exemplis illustrarem & sed fundamentum ejus literis transpositis hanc sententiam involventalibus [Data æquatione quotcunque fluentes quantitates involvente, fluxiones invenire, et vice versa] Pandercelarem: rescripsit vVir Clarissimus anno proximo, se quoque in ejusmodi methodum incidisse, & methodum suam communicavit a mea vix abludentem præterquam in verborum & notarum formulis. not addito in 2d EditVtriusque fundamentum continetur in hoc Lemmate.

✝ In eadem Epistola subjunxi — — — ~~a consilio detenuerunt~~ simul ederem. Sed antequam Tractatum illum absolvissem, Epistolá ad te missa qua breviter explicui conceptus meos de natura lucis, lites de coloribus on subortæ sunt, quæ me quietis amantem a consilio detenuerunt. Epistola illa data fuit 6 Febr. 16 $\frac{71}{72}$ & in Transactionibus Philosophicis eodem mense impressa. Et in eadem scripsi me initio anni 1666 in Theoriāam novem illam lucis et colorum incidisse. Inveni igitur hanc Theoriam eodem fere tempore cum methodo fluxionum, utramque simul edere constitueram, & ab itriusque editione simul destiti quietis gratia. Tandem vere anno 1704

In Tractatu eadem Epistola 24 Octob. 1676 ad Oldenburgum scripta, dixi me in Tractatu illo fundamentum aliquatenus posuisse solvendi Problemata In my Analysis per series æquat. communicated by Dr B. to Mr C. in Iuly 1669 n. L. inf. I said that my methods of series by series gave the areas of curvilinear figures exactly & generat if it might be done & Mr Collins Mr Collins in his Letter to Mr Strode dated 26 Iuly 1672 afterward the same thing & thence it appears that when I wrote that Analysis I had the Me the Method of fluxions so far at least as it is conteined in the first six Propositions of the Book of Quadratures. In that Analysis Tract I represented time by a line increasing uniformly & a momtmoment of time by a particle of the line generated by in a moment of time & thence called the particle of the a moment of the line, & the particles of all other quantiteies generated in the same moment of time I called the moments of those quantities, & the fact under the re rectangular Ordinate & the Or I & a moment of the Abscissa I called considered as the moment of the Curvilinear area generated described by that Ordinate while it movinesg uniformly upon the Abscissa. And from the fluxion of time came the names of fluents And by con& fluxions. And by considering how to deduce the moments from quantities & quantities from moments I squared Curves & deduced the areas of figures from their areas deduced their Ordinates & their Ordinates from their areas: wchwhich is the same thing with deducing fluxions from fluents & fluents from fluxions. And in the end of the book I applied t deonstrated the first Rule by this calculus. the first of the three Rules set down in the beginning thereof. And by applying this method not only to finite equations but also to converging Series considered as an Equations consisting of an infinite number terms I gave to this Tract the name of Analysis per Ss æquationes numero terminorum infinitas. And in my Letter of 13 Iune 1676 I said with relation to of this Analysis: Ex his videre est quantum fines Analyseos per hujusmodi series infinitas æquationes ampliantur: Quippe quæ earum beneficio, ad omnia pene dixerim problemata (si numeralia Diophanti et similia excipias) sese extendit. [Non tamen omnino universalis evadit nisi per ulteriores quasdam methodos eliciendi series infin] And Mr Leibnitz in his Answer dated 27 Aug. 1676, replied Quod dicere videmini plerasque Difficultates (exceptis Problematibus Diophantæis) ad Series infinitas reduci; id mihi non videtur. Sunt enim multa usque adeo mira et implexa ut neque ab æquationibus pendeant neque ex quadraturis: Qualia sunt (ex multis alijs) Problemata mathodi tangentium inversæ. In the same Letter he placed the perfection of Analysis in another meth not in the Differential Method but in another Method composed of Analytical Tables of Tangents & the Combinatory Art. Nihil est, said he, quod norim . . . . . . cogitationum humarum. This was the top of his skill at that time. Mr Collins in his Letter to Mr Strode dated 26 Iuly 1672 gave this account of the Method. Mense Septembri 1668 — — — — — haud integrum ducit. It appeared therefore to Mr Collins by the testimony of Dr Barrow upon from grounded upon papers which I had communicated to him from time to time that I had the method conteined in the Analysis per Æquationes numero terminorum infinitas that I had this Method some years before the Dr sent that Tract to Mr Collins. And this is sufficient to justify wtwhat I said in the Introduction to the Book of Quadratures. Mr Iames Gregory after a years study found the Method of Series proprio Marte but did not claim it because he knew that he was not the first invenentor Nothing has been said to prove that Mr Leibnitz had the method before he came to London the second time. Then he met with Dr Barrows Lectures, & the Marquess de l'Hospital has said that where the Dr left off Mr Leibnitz proceeded, & that the improvement wchwhich DMr BLeibniz made21 made to the Doctors Methods consisted in shewing how to exclude fractions & surds radicals. But the Marquess did not know that by my Letter of 24 Octob. 1676 & a copy of my Letter of 10 Decem 1672 he had notice of this improvement & that it related to a very general method. He might afterwards find them proprio Marte but by that notice knew that I had them before him. For in his Letter of 21 Iune 1677 wherein he first began to communicate his method he acknowledged that I knew the improvement before him when I wrote my Letter of 24 Octob. 1676. Dr Barrow then read his Lectures about the generation of figures by motion & that might put me upon taking these things into consideration. In the beginning of the year 1666 I found also the Theory of colours. In the year 1671 I was preparing to publish thenis Theory with the methods of Series & fluxions, but for a reason given mentioned in my Letter of 24 Octob. 1676 I desisted till the year 1704 & then printed both together the Tracts of Colours & Quadratures together. I That Tract c was therefore written not only upon a method of finding Series, but upon a general method of Analysitc by applying the method of moments to Equations both finite & infinite & to In the beginning of the year 1665, uppon winter between the years 1664 & 1665 I had the method of TangtsTangents by considering of how to interpose some Series of Dr Wallis. I found the Rule for of Hudden Gregory & Slusius & by considering how Slusius & a method of finding the quantity of crookedness of curve lines att any given point. to And by considering set down in my Letter of 13 Iune 1676 for squar reducing any power or dignity of any Binomium into an approximating series, & in the falling Spring, before the Plague, wchwhich invaded us that summer, forced me from Cambridge, I found how to do the same thing by continual division & extraction of roots, as I mentioned in my Letter of 24 Octob. 1676 & in the following autum when I was absent from the university I found the direct method of fluxions either & soon after I extended the Method to the extraction of affected roots out of equations. In the beginning of the year 1665 I had the method of tangtstangents of GHuddon Gregory & Slusius, & soon after I found about the same time or soon after I found the direct method of first fluxions & I set it down with examples & a demonstration in a paper dated 13 Novem. 1665. In the winter between the years 1664 & 1665 I had a method of Tangets like that of Hudden Gregory & Slusius & a method of finding the crokedness of Curves at any given point, & by considering how to interpole certain series of Dr Wallis I found the Rule set down in my Letter of 13 Iune 1676 for reducing any power or dignity of any Binomium into an approximating Series, & in the following Spring, before the Plague, wchwhich invaded us that summer, forced me from Cambridge, I found how to do the same thing by continual division & extraction of roots, as I mentioned in my Letter of 24 Octob. 1676. And soon after I extended the Method to the extraction of the roots of affected Equations in species. [And in a paper dated 13 Novem 1665 I find the direct Method of first fluxions set down with examples & a Demonstration] And from all this learnt how to deduce the Ordinates or Abscissas of Curvilinear figures from their Areas or Acs given, as well as the Areas & Arcs from the Abscissas & Ordinates. Thus far I proceeded before the plague frorced me from Cambridge. And in a paper dated 13 Nov. 1665 I find the direct Method of first fluxions set down with examples & a Demonstration.

22 A Scholium. In litt Literis Scholium. In literis quæ mihi cum Geometra peritissimo G. G. Leibnitio anno 1676 intercedebant, cum significarem me compotem esse methodi Analyticæ determinandi Maximas & Minimas, ducendi tangentes, quadrandi Figuras curvilineas, conferendi easdem inter se, & similia peragendi, quæ in terminis surdis æque ac in rationalibus procederet, & Tractatus duos de eadem similibus hujusmodi rebus scripsisse alterum anno 1671, ① alterum autē quem Barrovius noster anno 1669 and Collinium misi, cumque literis transpositis hanc sententiam involventibus [Data æquatione quotcunque fluentes quantitates involvente, Fluxiones invenire, et vice versa] fundamentum hujus methodi celarem, specimen vero eusdem in Curvis quadrandis subjungerem & exemplis illustrarem; et cum vir clarissimus sub. finem mensis Octobris in reditu suo e Gallio per Angliam in Germaniam epistolas meas in manu Collinij insuper consuluisset: incidit is in methofinito itinere in metho finito itinere anno proximo anno proximo in metho paulo post in metho finito itinere in methodum similem sub diversis verborum et notarum formulis, Methodi utriusque fundamentum continetur in hoc Lemmate. et mense Iunio anno 1677, sequente specimen ejusdem ad me misit. Methodi utriusque fundamentum continetur in hoc Lemmate. Collinius Epistolam 10 Decem. 1672 datam a me accepisset in quo methodum hanc descripsissemeram et exemplo Tangentium more Slusiano ducendarum illustrssumveram, & hujus Epistolæ exemplar mense Iunio anni 1676 in Galliam ad D. Leibnitium misisset; & Vir Clarissimus sub finem mensis Octobris, in reditu suo & Gallia per Angliam in Germaniam, epistolas meas in manu Collinij insuper consuluisset: incidit is finito itinere tandem in methodum similem sub diversis verborum et notarum formulis, et mense Iunio sequente specimen ejusdem in Tangentibus more Slusiano ducendis ad me misit, & & subjunxit se credere eandemque a methodo mea methodum suam a meam a sua non abludere se credere subjunxit præsertim cum quadraturæ curvarum per utrāamque methodaum faciliores redderentur. Methodi vero utriusque fundamentum continetur in hoc Lemmate. et Tractatum duos de hujusmodi rebus scripsisse alterum quem Barrovius, anno 1669 ad Collinium misit quique jam extat & methodum sea serierum ab momentorum troctat, & alterum anno 1671 in qua hanc methodum fusius exposueram; cumque literis transpositis hanc et Tractatus duas de hujusmodi scripsisse, alterum quem Barrovius anno 16769 ad Collinium misit, & alterum anno 1671 in quo hanc methodum plures fusius exposueratm; cumque literis transpositis fundamentum hujus methodi literis transpositis hanc sententiam involventib [Data — — et vice versa] celarem, specimen vero ejusdem in Curvis quadrandis subjungerem & exemplis illustrarem; et cum Collinius Epistolam 10 Decem 1672 datam a me accepisset, 23 In utique momenta mea quantitatum differentias vocavit & quantitates ipsas summas momentorum, & literis d et s præfixis differentias & summas notavit, & pro fluxionibus nulla habuit symbola: ego vero plenius exponitur in Tractatu Scholium Analysin meam per series et momenta. Barrovius noster anno 1669 ad Collinium misit. & Easdem methodos annis 1665 et 1666 inventas in alio Tractatu plenius explicim anno 1671; et in Epistola 10 Decem 1672 ad Collinium data [methodum meam generalem in hoc Tractatu isto descriptam hocce explicatam verbis generalibus descripsi & exemplo d Tangentium more Slusiano ducendarum illustravi dixique eandem ad quantitates surdas non hærere. Collinius vero exemplar hujus Epistolæ mense Iunio anni 1676 ad D. Leibnitium hinc in Gallia agentem misit.] In literis vero insuper quæ mihi cum D. Leibnitio eodem anno 1676, intercedebant, cum sig Tractatibus prædictis verba facerem & significarem me compotem esse methodi Analyticæ determinandi maximas & minimas, ducendi Tangentes, quadrandi figuras curvilineas, conferendi easdem inter se, et similia peragendi quæ in terminis surdis æque ac in rationalibus procederet, ,[& Tractatum prædictorum etiam miminissem], & literis transpositis hanc sententiam involventibus [Data æquatione quotcunque quantitates involvente, fluxiones invenire, et vice versa] fundamentum hujus methodi celarem, specimen vero ejusdem in curvis curvilineis quadrandis subjungerem et exemplis illustrarem, et Vir clarissimus eadem anno in reditu suo e Gallia per Angliam in Germaniam, sub finem mensis Octobris Epistolas meas in manu Collinij consuluisset consuleret: incidit is non mulo post in methodum similem sub diversis verborum et notarum formulis, et mense Iunio sequente specimen ejusdem in Tangentibus more Slusiano ducendis ad me misit; & subjunxit se credere methodum meam a sua non abludere, præsertim cum quadraturæ curvarum per utramque methodum methodum faciliores redderentur. Methodi vero utriusque fundamentum continetur in hoc Lemmate. methodum momentorum exemplo tangentium more Slusiano ducendarum illustravi dixique eandem dixisse eandem ipsam ad Quæstiones de areis areis tangentibus s curvitatibus, areis, longitudinibus, centris gravitatum Curvarum & Curvilinearum &c se extendere, et esse generalem et ad ad quantitatis surdas non hærere. Et Collinius exemplar — ad Collinium data inter per methodum generalem quam methodo serierum [in Tractatu novissimo] intertexui, intellexi methodum momentorum, in Tractatu eamque verbis generalibus ibi descripsi, et exemplo Tangentium more Slusiano ducendarum illusstravi, dixique eandem ad quantitates surdas non hærere. — ad Collinium data, ha methodum serierum quam in hoc tractatu methodum momentorum [quam in his Tractatibus novissimo intertexi methodo serierūum intertexaum,] vocavi methodum generalem [eamque ad Curvarum eamque vera ad curvarum Tangentes Areas, Curvitates, longitudines, centra gravitis, & similia esse extende stem] & exemplo Tangentium more Slusiano ducendarum illustravi, dixique eandem ad quantitates surdas non hærere. Et Collinius vero exemplar hujus epistolæ mense Iunio anni 1676 ad D. Leibnitium misit tunc in Gallia agentem misit. In literis insuper quæ mihi cum D. Leibnitio anno 1676 intercedebant; Scholium Analyseosin meam Tractatum quendam meum per series et momenta a me scriptam Barrovius noster anno 1669 ad Collinium misit. Eansdem mMethodos ibi expositos an alio Tractatu plenius explicui anno 1671 E nde in Epistola 10 Decem. 1672 ad Collinium data, methodum momentorum vocavi generalem et exemplo tangentium more Slusiano ducendarum illustravi, dixique eandem etiam ad quæstiones de curvitatibus, areis, longitudinibus, centris gravitatum Curvarum & Curvilinearum &c sese extendere et esse generalem & ad quantitates surdas non hærere; et Collinius exemplar hujus epistolæ mense Iunio anni 1676 ad D. Leibnitium tunc in Gallia agentem misit. In literis insuper quæ mihi cum D. Leibnitio anno 1676 intercedebant, cum verba facerem de Tractatibus prædictis, & Significarem me compotem esse methodi Analyticæ determinandi maximas et minimas, ducendi tangentes, quadrandi figuras curvilineas, conferendi easdem inter se, et similia peragendi quæ in terminis surdis æque ac in rationalibus procederet; & literis transpositis hanc sententiam involventibus [Data æquatione quotcunque fluentes quantitates involventes fluentes quantitates fluxiones invenire, et vice versa] fundamentum hujus methodi celarem, specimen vero ejusdem in curvilineis quadrandis subjungerem & exemplis illustrarem; et Vir celeberrimus eodem anno in reditu suo e Gallia per Angliam in Germaniam, sub finem mensis Octobris, epistolas meas in manyu Collinij etiam consuluisseret: incidit is non multo post in methodum similem eandem momentorum sub diversis verborum et notarum formulis, et mense Iunio sequente specimen ejusdem in Tangentibus more Slusiano ducendis ad me misit, & subjunxit se credere methodum meam a sua non abludere, præsertim cum quadraturæ curvarum per utramque methodum faciliores redderentur. Methodi vera uniusque hujus fundamentum continetur in hoc Lemmate, et plen hæc methodus 24 In writing the Book of Principles I made very much use of the following Book of Quadratures & therefore have subjoyned it. This Book is founded upon the method of fluxions, & in the Introduction to it I said that I found the method of fluxions gradually in the years 1665 & 1666, & it may be expected this has been called in question. The accusatio But the accusation is not yet proved as it ought to have been, nor hath any answer been given to the collection of ancient Letters & Papers published by order of the R. Society under the title of Commercim Epistolicum for setting this matter in a fair light. But instead of doing this the accusers have been at work to set up judeges if theie are freeing themselves from calumny Of the contrary, the Accusers have been endeavouring to avoid answering it by telling us that we shall not have the pleasure to see them answer & that to answer it would require a book as big as the Commercium, & by appealing to Iudges of their own setting up, & by proposing Problems to be solved challenging challingchallenging the English Mathematicians to solve their Problems, & by framing many new accusasions for ma to make a squabble & writing Letters to Ladies & other persons unacquainted with these matters & to Ladies for making an interest] However, upon publishing this Book anew, it may be expected that I say some thing in justifationjustification of what I said in the Preface of the former edition. Vpon notice from Mr Iohn Collins in December 1672 that Mr Iames Gregory had impr The Logarithmotechnia of Mr Mercator was published in septemb Autumn 1668 & a few months after Dr Mr Collins Dr Barrow received a copy thereof from Mr Collins & sent back to Mr Collins a small Tract of mine in M.S. entituled Analysis per series numero terminorum infinitas in wchwhich I affe were these words concerning the joint method of series & fluxions: Denique ad Analyticam merito pertinere censeatur cujus beneficio curvarum areæ & longitudines &c (id modo fiat ID MODO FIAT) exacte et Geometrice determinentur: sed ista narrandi non est locus. And by the testimony of Mr Collins Dr Barrow & Mr Collins I had this Method some years before the Logarithmotechnia came abroad: For Mr Collins in a Letter to Mr Strode dated 26 Iuly 1672 & published by order of the R. S. wrote thus Mense Septembri 1668, Mercator Logarithmotechniam edidit suam, quæ specimen hujus methodi (i.e. Serierum) in unica tantum figura, nempe quadraturam Hyperbolæ, continet. Haud multo postquam in publicum prodierat liber, exemplar ejus — Barrovio Cantabrigiam misi, qui quasdam Newtoni chartas — extemplo remisit: e quibus et ALIIS, quæ OLIM ab Auctore cum Barrovio communicata fuerant, patet illam methodum a dicto Newtono aliquot annis antea excogitatam fuisse et modo universali applicatam fuisse: ita ut ejus ope in quavis Figura Curvilinea proposita, quæ una vel pluribus figuris proprietatibus definitur, Quadratura vel Area dictæ Figuræ, accurata si possibile sit ACCVRATA SI POSSIBILE SIT, sin minus infinite vero propinqua, Evolutio vel longitudo lineæ curvæ, centrum gravitatis figuræ, Solida ejus rotatione genita, et eorum superficies,sine ulla radicum extractione, obtineri queant. How this is done I explained in a larger Tract written by me A.C. 1671 as I mentioned in a letter to Mr Oldenburg dated 24 Octob 1676 & p ublished by Dr Wallis. In And in that Letter I set down the foundation of the method in these words Data Æquatione quotcunque fluentes quantitates involvente fluxiones invenire et vice versa, & then added ytthat upon this foundation I had endeavoured to make advance the Theory of Quadratures & obteined some general Theorems for that purpose, & there set down the first of those Theorems & illustrated it with examamplesexamples. This is a Theoreme for squaring Curves whose Ordinates are Binomial & I added that I had other Theorems Rules for Trinomials & some others And figures. And this is the Method for squaring of figures by series wchwhich break of & become finite equations when the Curve admits of an exact & Geometric quadrature. Now to do this requires the knowledge of the method of fluxions so far as it is described in the first five or six Propositions, of the following book following book of Quadratures. And therefore by the testimony of Dr Barrow & Mr Collins, I knew thus much of the method some years before the Logarithmotechnia came abroad & by consequence in the year 1666. And this may suffice to justify what I said in the Preface of this book. Mr Iames Gregory in a Letter to Dr Barrow Mr Collins dated 5 Sept 1670, wrote that by comparing Dr Barrows methods of drawing Tangents with his own he had found a method of drawing Tangents to all Curves without calculation. And upon notice of thereof & that Mr Slusius had such another method wchwhich he intended to communicate to Mr Oldenburg I wrote the to Mr Collins the following Letter dated 10 Decem 1672. Ex animo gaudeo — — — — reducendo eas ad series infinitas. In this Letter you have — — — — in the third volume of his works. For in the For from the Tract wchwhich I wrote in the year 1671, I extracted in the year 1676 the following Book of Quadratures & in my aforesdaforesaid Letter of Octob 24 1676 I set down the first Proposition thereof verbatim, & copied without any alteration the two Tables set down in the Scholium upon the tenth Proposition. And in my Letter to Mr Collins dated 8 Novem. 1676 & published by Mr Iones, I had relation to the tenth Proposition of this Book in saying: Nulla extat Curva — — — — was known to me in the year 1676 when I wrote this Letter. [In a letter dated 12 May 1676 Mr Leibnitz wrote thus to Mr Oldenburg. Cum Georgius Mohr Danus — — — — desiderio meo. Mr Leibnitz therefore knew that Mr L Collins had the Demonstration of thi of this two series that is the & desired a copy of it, that is a copy of the Analysis per series numero terminorum infinitas. For Mr Collins had no other demonstration of those two Series. And thereupon I wrote But Mr Collins & Mr Oldenburg desired that I would send hi chose rather to write to me to send him what he wanted, & thereupon I wrote my to Mr Oldenburg my Letter of 13 Iune 1676 & Mr Oldenburg sent to him 26 Iune 1676 sent to a copy of it to Mr Leibnitz together with c copies a copy of my Letter of 210 abovementioned dated 10 Decem 1672, & copies o extracts of many of Gregories Letters amongst wchwhich were his two Letters abovementioned dated 5 Sept 1670 & another Letter dated 15 Feb. 167

\frac{0}{1}

in wchwhich in wch Mr Gregory was sent to Mr Col there the series

a = t - \frac{t^{3}}{3 r^{2}} + \frac{5^{5}}{5 r^{4}} - \frac{t^{7}}{7 r^{5}} + \frac{t^{9}}{9 r^{7}} - &c

, where r denoting the radius, t the TangtTangent, & a the Arc of a circle. And the same series was sent to Mr Leibnitz b the year before by Mr Oldenburg from Mr Collins in a letter dated 15 Apr. 1675. And Mr Leibnits in answer to all that had been sent him sent back hi a Letter dated 27 Aug. 1676] In the year 1691 this Book in MS was in the hands of Dr Halley & Mr Raphson as the ones has left attested in print & the other still attests. And hte next year Dr Wallis in the second Volume of his works published he printed the first Proposition of theis Book illustrated with examples in finding first & second Differences fluxions And this is at the request of Dr Wallis that I would explain how I found the Theoremes set down in my Letter of 24. Octob. 1676 for squaring Curvilineas wchwhich — — — — — — & is still the shortest the clearest & the best. In the beginning of the year 1695 the March 1695 Dr Wallis upon notice notice from Holland that the the Method of fluxions was celebrated in Holland there by the name of the Differential Method of Fluxions Mr Leibnitz, inserted into the Preface of the first Volume of his works (wchwhich came abroad after the second Volume) the following Preface Paragraph. Quæ in secundo Volumine habentur — — — — ab ipso excogitatam. And soo as soon as the Volume was printed off gave me notice of what he had wrote to me the following Letter. dated Apr. 10th. 1695. 25 And the next year the Editor of the Acta Eruditorum in gaveing an Account of this Volume cited some words out of this Paragraph & Dr Wallis in a Letter dated 1 Decem. 1696 gave notice to Mr Leibnitz of the same Paragraph in the Leters wchwhich ensued between him & Dr W. And yet neither the Editor of the Acta nor Mr Leibnitz denyed what Dr Wallis had affirmed. On the contrary Mr Leibnitz in a letter to me dated the Marquess de L'Hospital in the Preface to his book de Infinite parvis published this year of allowed that the Book of Principles published this year was almost wholy of the Differential calculus, & Mr Leibnitz himself three years before in a letter to me dated

\frac{7}{17}

March 1693 acknowledged the same thing (the Original of wchwhich is in the Archives of the R. S.) acknowledged the same thing in these words Quantum tibi Scientiam rerum Mathematicarum totiusque Naturæ debere arbiter, occasione data etiam publice professus sum professus. Mirifice ampliaveras Geometriam tuis seriebus, sed edito Principiorum opere ostendisti patere Tibi, etiam quæ Analysi receptæ non subsunt. Conatus sum ego quoque Notis commodis adhibitis quæ differentias & summas exhibent, Geometriam illam quam Transcendentem appello Analysi quodammodo subjicere nec res male processit. Here he gives me the Preference. And in the Acta Eruditorum for May 1700, he acknowledged that I had the was the first who had manifested by a specimen made publick that I had the method of maxima & minima in infinitesimals or moments, meaning the specimen in the Scholium upon the 34th Proposition of the Book of Math. Principles. And this method I could not have without the Method of Fluxions. And in general this Book of Principles is the first specimen made publick of applying thise Method of Fluxions & Moments to the difficulter Problems. And Mr Leibnitz in the Acta Eruditorum for 1684 affirmed that such Problems were not to be solved Without the differential method or another like it, meaning the method of fluxions. For when he wrote h to Mr Oldenburg his Letter of 21 Iune 1677 wherein he was first began to make known write of the Differential method, & shewed how it might be deduced it from the method of Tangents of Dr Barrow, he aknowledged that I had such another method. Clarissimi Slusij (saith he) Methodum Tangentium nondum esse absolutam Celeberrimo Newtono assentior. And after he had shewn how to deduce the method of Slusiu Tangents of Slusius from that of Dr Barrow (as Gregory had done before) & how to make this method proceed without taking away surds, he added: Arbitror quæ celare voluit Newtonus de Tangentibus ducendis ab his non abludere. Quod addit, ex hoc edem fundamento quadraturas quoque reddi faciliores, me in sententia hac confirmat. He knew therefore in those days by my Letters that I had a method like the Differential, & by the same Letters he was told that I in the year 1671 I wrote a Tract upon this method & the Method of Fluxions Series together. If it In the second Letmma of the second Book of Principles I wrote a Tract upon demonstrated the Elements of the Method of fluxions synethetically. And because Mr Leibnitz had published those Elements in another form two years before, without acknowledging the correspondence wchwhich had been between us eight years before, I added a Scholium not to give away that Lemma but to put Mr Leibnitz in mind of making a publick acknowledgmtacknowledgment of that correspondence. 26 In composing these Books I was much assisted by the Book of Quadratures. At the request of Dr Wallis I sent to him the first & second Proposition of this Book of Q Aug in the year 1692 Aug. 27, & he printed it in the second Volume of his works before the end of the year & the Book came abroad in Spring 1693. & this was the first time that any rule for finding second third & fourth fluxions was published, & so the Rule is the best. This is a proof that the Book of Quadratures was then in Manuscript. And This Book Dr Halley & Mr Ralpson borrowed it of me saw it in my hands at Cambridge in summer 16961 & carried it with them to London as Mr Ralphson has left attested in print, & Dr Halley a living evidence still attests. And therefore it was in MS in yethe year 1691, & continued in MS fthirteen years at the least before it was published. In my Letter of 24 Octob 1676 I cited many things out of it, Particulary I cited (in an Enigma) the very words of the first Proposition Data æquatione fluentes quotcunque quantitates involventoe invenire fluxiones, [putting them into an Ænigma] I mentioned also the substance of the fift & sixt Propositions & gave a solution of the fift wthwith some examples. These two are the inverse of the third & fourth Propositions & these two are exampels of the second & all of them are deduced from the first & second, & therefore I was in those days no stranger to the first six Propositions of this Book. In the same Letter of 24 Octob. 1676, I set wrote thus. Seriei a D. Leibnitio pro Quadratura Conicarum Sectionum propositæ, affinia sunt Theoremata quædam quæ pro comparatione Conicarum Sectionum Curvarum cum Conicis sectionibus in Catalogum dudum retuli. The series for squaring the Conic sections Mr Leibnitz received had twice from Mr Oldenburg. The Theorems for comparing other Curves with the Conic Sections, I reduced into a Catalogue in the Tract above mentioned wchwhich I wrote in the year 1671, & thenc I copied it into the Book of Quadratures & the Ordinates of the Curves in the more intricate part of the Table I set down in the said Letter in the very same order & in the very same letters & symbols in wchwhich you will now find them in the scholium upon the 10th Proposition of the Book of Quadratures. And therefore that Table was composed before I wrote that Letter; & the 7th 8th 9th 10th Propositions upon wchwhich it depends were then known to me. Between the years 1671 & 1676 I medled not wthwith these studies being tyred with them before: but in the year 1676 I extracted the Book of Quadratures from the Tract wchwhich I wrote in year 1671 & from other older papers. And soon after I had finished it & I wrote to Mr Collins a Letter dated Novem. 8. 1676 in which a part of which was in the following words is here set down. Nulla extat Curva cujus Æquatio ex tribus constat terminis, in qua, licet quantitates incognitæ se mutuo afficiant, vel Indices dignitatum sint surdæ quantitates (v. g.

a x^{λ} + b x^{μ} y^{σ} + c y^{τ} = 0

, x designat basin, y Ordinatam, λ, μ, σ, τ Indices dignitatum ipsius x et y, & a, b, c quantitates cognitas una cum signis suis + vel −) nulla inquam hujusmodi est Curva, de qua, an quadrari possit necne, vel quænam sint figuræ simplicissimæ quibusquam comparari possit, sive sint Conicæ Sectiones sive aliæ magis complicatæ, intra horæ octantem respondere non possim. Deinde methodo directa & brevi, imo methodorum omnium generalium brevissima, eas modo comparari possint, comparo. To do all this is the inverse method of fluxions so far as that method is carried on in the Book of Quadratures. And by the help of this method I composed the Book of Principles, & therefore in this Edition have added the Book of Quadratures to the end of it. [I made use also of the method of maxima & minima in Infinitesimals & by the confession of Mr Leibnitz was the first who shewed by a specimen made publick that I had this method. I made use also of the method by me called the differential Method, & for that reason have annexed theis Method to the Book of Quadratures. In my Letter of Octob. 24 1676 I said that the Tract wchwhich I was writing in the year 1671 I never finished, & that that part of it was wanting in wchwhich I intended to teach the manner of resolving Problems Problems wchwhich cannot be reduced to Quadratures. But what I then intended to write is now gone out of my mind through long disuse of these methods] In the Introduction to the Book of Quadratures wchwhich I wrote I said that I invented the Method of fluxions in the yea gradually in the years 1665 & 1666. & Dr Wallis (who received from Mr Oldenburg copies of my Letters in the year 1676 & corresponded also wthwith Mr Collins in those days & was very inquisitive in things of this nature, wrote affirmed published the same things six years before me nine years before me in the Introduction to the first Volume of his works wchwhich came abroade in the year April 1695. the following Paragraph His words are. Quæ in secundo Volumine [ante biennium edito] habentur, in Præfatione eidem præfixa dicitur. Vbi (inter alia) habetur Newtoni Methodus de Fluxionibus (ut ille loquitur) consimilis naturæ cum Leibnitij (ut hic loquitur) Calculo Differentiali, (quod qui utramque methodum contulerit, satis animadvertat, utut sub loquendi formulis diversis) quam ego descripsi (Algebræ cap. 91 &c præsertim cap. 95) ex binis Newtoni Literis (aut earum alteris) Iunij 13 & Octob. 24 1676 ad Oldenburgum datis, cum Leibnitio tum communicandis (ijsdem fere verbis, saltem leviter mutatis, quæ in illis literis habentur) ubi methodum hanc exponit Leibnitio exponit, tum ante decem annos nedum plures, ab ipso excogitatam. In the year Iuly 1669 Dr Barrow sent to Mr Collins a small Tract written by me under the Title of Analysis per series numero terminorum infinitas. This title implies that the scope of the Book was not only to reduce finite quantites into converging series but also by the help of those series as well as to apply æquations involving such series to resolve as well as by finite æquations to the resolution of Problems by means of the Analysis there proposed And in this Analysis I considered quantities as generated by motion & their parts generated in moments of time I call their moments, & shew how to compute the moments of quantities lines, superficies, & solids & to find as many Curves as I please whose areas wchwhich may be squared wchwhich is (in substance) the first & second Propositions of the Book of Quadratures. And after I had shewed how to find the Ordinate Areas & Lengths of Mechanical lines, I add describe the universality of this method in these words. Nec quicquam hujusmodi scio ad quod hæc methodus, idque varijs modis sese non extendit. Imo tangentes — — — — sed ista narrandi non est locus. These last words relate to the method described in the fift & sixt Book of Propositions of the book of Quadratures. And therefore the pro Methods conteined in the first six Propositions of this book were then known to me. For the fift & sixt Propositions are the inverse of the third & fourth. [The first Rule in the Analysis per Series is this. Figuræ cujusvis curvilineæ Abscissa AB sit x, & Ordinata BD sit y, & sint a, b, c & quantitates datæ, & m, n numbri integri, et si sit

a x^{\frac{m}{n}} = y

, erit

\frac{a x}{m + n} x^{\frac{m + n}{n}} = areæ Curvilineæ ADB

. Et vicissim si Area sit a erit

m + n n x

b x^{\frac{m}{n}}

, oOrdinata erit

\frac{m b}{n} x^{\frac{m - n}{n}}

. which is the first] And] In the beginning of this Tract I call this Method methodūum generalem olim a me quam olim excogitaveram. This Tract is the Compendium methodi serierum wchwhich mentioned in my Letter of Octob. 24. 1719, where I say that it was communicated by Dr Barrow to Mr Collins at that time when Mr Mercators Logarithmotechnia came abroad. And Mr Collins in his Letter to Mr Tho. Strode dated 26 Iuly 1672 has testified that by this & other Papers communicated before to Dr Barrow it appeared that I had invented this method & applied it generally some years before he sent it to Mr Collins. In my Letter of 24 Octob. 1676 at the request of Mr Leibnitz I describe how I fel found out the method of converging series a little before the plague wchwhich raged in London in the year 16 965, & particularly the how I then invented the reduction of any power of any Binomial into such a Series by the Rule set down in teh beginning of my Letter of 13 Iune 1676. Let

{x + o}^{n}

be any line dignity of any binomial

x + o

& by this Rule

{x + o}^{m}

will be equal to

x^{n} + o o x^{n - 1} + \frac{n n - m}{2} o o x^{n - 2} + &c

. And here if x be an increasing quantity of fluents & o its moment, then

n o x^{n - 1}

will be the first moment of the fluent

x^{n}

. And this Theoreme quickly gave me the first Proposition of the Book of Quadratures & the inverse thereof gave me the two first Rules in the Analysis per æquationes numero terminorum infinitas. And this Relation between the Methods of Series & moments made me joyne them together as two branches of a one very general27 general method. And all this I hope may suffice to justify me in saying in the Introduction to the Book of Q. that I found the method of fluxions gradually in the years 1665 & 1666. 28 In writing the Book of Principles I made very much use of the following Book of Quadratures, & therefore have subjoyned it. And because my saying in the Introduction, that I found the Method of fluxions gradually in the years 1665 & 1661666 has been called in question, it will not be amiss to explain that matter justify what I said In December 1672 Upon notice from Mr Collins in Decemb. 1672 that Mr Iames Gregory had improved the method of Tangents of Dr Barrow so as to draw tangents without calculation & that Mr Slusius had such another method wchwhich he intended to communicate to Mr Oldenburg, I wrote the following. Letter to Mr Collins dated 10 Decem 1672. "Ex animo gaudeo D. Barrovij, amici nostri reverendi lectiones Mathematicas exteris adeo placuisse, neque parum me juvat intelligere eos [Slusium & Gregorium] in eandem mecum incidisse ducendi Tangentes Methodum. Qualem eam esse conjiciam, ex hoc exemplo percipies. Pone CB applicatam ad AB in quovis angulo dato, terminari ad quamvis Curvarum AC; et dicatur AB x & BC y, habitudoque inter x et y exprimatur qualibet æquatione, puta

x^{3} - 2 x x y + b x x - b b x + b y y - y^{3} = 0

, qua ipsa determinatur Curva. Regula ducendi Tangentem hæc est: Multiplica æquationis terminos per quamlibet progressionem arithmeticam juxta dimensiones y, puta

\begin{matrix} x^{3} - 2 x x y + b x x - b b x + b y y - y^{3} \\ 010023 \end{matrix}

; ut et juxta dimensiones x, puta

x^{3} - 2 x x y + b x x - b b x + b y y - y^{3} = 0

, qua ipsa determinatur Curva. Regula ducendi Tangentem hæc est: Multiplica æquationis terminos per quamlibet progressionem arithmeticam juxta dimensiones y, puta

\begin{matrix} x^{3} - 2 x x y + b x x - b b x + b y y - y^{3} \\ 322100 \end{matrix}

Prius productum erit Numerator, & posterius Denominator divisum, per x Denominator Fractionis quæ exprimet longitudinem BD æd cujus extremitatem D ducenda est Tangens CD. Est ergo longitudo

BD = \frac{−2 x x y + 2 b y y - 3 y^{3}}{3 x x - 4 x y + 2 b x - b b}

. Hoc est unum particulare vel corollarium potius Methodi generalis quæ extendit se, citra molestum ullum calculum, non ad ducendum Tangentes ad quasvis Curvas, sive Geometricas, sive Mechanicas, vel quomodocunque rectas lineas aliasve Curvas respicientes; verum etiam ad resolvendum alia abstrusiora Problematum genera de Curvitatibus, aAreis, Longitudinibus, centris gravitatis Curvarum &c. Neque (quemadmodum Huddenij methodus de Maximis et Minimis) ad solas restringitur æquationes illas quæ quantitatibus surdis sunt immunes. Hanc methodum intertexui alteri sibi isti, qua Æquationum Exegesin instituo reducendo eas ad Series infinitas. &c. In this Letter you have an example of the Method of Fluxions in drawing of Tangents & a description of the large extent of it in this & other more difficult Problems without stopping at surds; & the last words of the Letter relate to a Tract wchwhich I wrote the year before & in which the method here described as interwoven with another Method wherein æquations are reduced to converging series, both wchwhich methods together constitutde the general Method described in my Letters 14 to M Oldenburg dated 13 Iune & 24 Octob. 1676 & which Letters pub were published by Dr Wallis in the third volume of his works. For in the year 1671 I wrote a Tract concerning the method of converging series & Fluxions joyntly but did not finish it, that part of it being wanting in wchwhich I intended to explain the manner of solving such problems as cannot be reduced to quadratures, as I mentioned in my Letters of 24 Octob. 1676 aforesaid. And from this Tract I extracted in the year 1676 the following Book of Quadratures & therein copied from the former Tract without any alteration all all the two Tables set down in the Scholium upon the tenth Proposition, the one for squaring some Curves & the other for comparing others with the Conick Sections. And In my Letter to Mr Collins dated 8 Novem. 1676 & published by Mr Iones, I had relation to the 9th & 10th Propositions of this Book in saying: Nulla extat Curva cujus Æquatio ex tribus constat terminis, in qua, licet quantitates incognitæ se mutuo afficiant, vel indices dignitatum sint surdæ quantitates (v. g.

a x^{λ} + b x^{μ} y^{σ} + c y^{τ} = 0

, x designat Basin, y Ordinatam & λ, μ, σ, τ indices dignitatum ipsius x & y & a, b, c quantitates cognitas una cum signis suis + vel −) nulla inquam hujusmodi est Curva de qua an quadrari possit, necne, vel quænam sint Figuræ simplicissimæ quibuscum comparari possit, sive sint Conicæ Sectiones sive aliæ magis complicatæ, intra horæ octantem respondere non possim. Deinde methodo directa et brevi, imo methodorum omnium brevissima generalium brevissima [de qua vide Coroll. 2 Prop. 10 Libri sequentis] eas, modo compararei possint, comparo. Affirmatio quidem videri potest temeraria, propterea quod perdifficile sit dictu an Figura quadrari vel cum alia comparari possit, necne; mihi autem manifestum est, ex eo unde deduxi, quanquan id alijs demonstrare:strare in me suscripere nollem. Eodem methodus Æquationes quatuor terminorum aliasque complectitur, haud tamen adeo generaliter. All this relates to the method of squaring figures set down in the Book of Quadratures, & chiefly to the tenth Proposition of that Book: & the methodus directa & brevis compendi f mentioned in this Letter, by wchwhich I compare such trinomial figures as may be compared is that described mentioned in Corol 2 Prop. 10; & this Proposition with its Corollaries is deduced from the 5t 6th 7th 8th & 9th Propositions of this Book & these are deduced from the first 1st 2d 3d & 4th Propositions of the same Book: & therefore the method of Quadratures so far as it is conteined in the ten first Propositions of this Book was known to me in the year 1676 when I wrote this Letter. This Book I made use of in the year 1679 when I found the demonstration of Keplers Proposition that the Planets moved in Ellipses, & again in the year 1684, 1685 & 1686 when I wrote the Book of Mathematical Principles of Philosophy I made much use of it & for that reason I have now subjoyned it to that Book In the year 1691 it was in the hands of Mr Ralpson & Dr Halley as the one of them attested in print before his death & the other still attests. And in the year 1692 at the request of Dr Wallis [I sent to him in a Letter dated the first Proposition of the Book illustrated with examples in finding first & second fluxions & it was printed in the second Volume of his works before the end of the year, & the volume came abroad in Spring following the April 1693. And this was the first time that any Rule was published for finding second third fourth & other fluxions or differences] that I would explain how I found the Theorems set down in my Letter of 24 Octob 1676 for squaring of Curves by series wchwhich break off & become finite equations when the Curves can be squared by finite equations, I sent to him in a Letter dated Aug 27 the fift Proposition of the Book of Quadratures as a more general Theoreme wchwhich comprehended the Theorems set down in Letter of mentioned by him & he at the same time I sent to him also the first Proposition of this book illustrated with examples in sending first & second fluxions. And these two Propositions were printed befor the end of the year in in the second Volume of his works pag. 391, 392, 393 & this which Volume came abroad in August 1693. And this was the first time that any Rule was published for finding second third & fourth & other fluxions or differences & is still the shortest the clearest & the best. But the Book of Quadratures continued in wchwhich I affirmed that I found the Method of fluxions gradually in the years 1665 & 1666. For I thought that I might safely write this because Dr Wallis, in the Preface to the first Volume of his works wchwhich came out after the second & was published in Spring April 1695, had said without ever being contradicted, inserted the following Paragraph. Quæ in secundo Volumine habentur in Præfatione eidem præfixa dicitur. Vbi (inter alia) habetur Newtoni Methodus de Fluxionibus (ut ille loquitur) consimilis naturæ ad cum Leibnitij (ut hic loquitur) cCalculo Differentiali, (quod qui utramque contulerit methodum contulerit, satis animadvertat, utut sub loquendi formulis diversis,) quam ego descripsi (Algebræ cap. 91 &c præsertim cap. 95) ex binis Newtoni literis (aut earum alteris) Iunij 13 & Octob. 24, 1676 ad Oldenburgum datis, cum Leibnitio tum communicandis (ijsdem fere verbis, saltem leviter mutatis, quæ in illis literis habentur;) ubi methodum hanc Leibnitijo exponit tum ante decem annos, nedum plures, ab ipso excogitatam. At the request of Mr Leibnitz I described in my said Letter of 24 Oct 1676, how, before the g plague wchwhich raged in London in the years 1665 & 1666, by interpoling the series of Dr Wallis, I found the method of converging series together with the Rule for resolving the dignities of Binomials into such series. I there I mentioned also that upon the publication of Mercator's Logarithmotechnia Dr Barrow sent abroad to Mr Collins a Compendium of these series. A copy of this Compendium was in the handwriting of Mr Collins was found by Mr Iones in the Archive of Mr Collins in the handwr & published after it had been collated wthwith the original in my custody wchwhich Mr Iones borrowed of me for that purpose. The litle thereof was Analysis per Series numero terminorum infinitas. And in this Tract the method of series is interwoven with that of fluxions. For after I had found the method of Series it quickly led me into the method of fluxions, & their affinity made me write of them both together as composing one general method of Analysis. In this Tract I affirmed that theis method extendesd to all Problemes & that ejus beneficio Curvarūum29 Curvarum areæ et longitudines &c (id modo fiat) exacte et Geometrice determinantur. Which I could not have said without understanding at that time so much of the method as is conteined in the first five or six Propositions of the Book of Quadratures. And by the testimony of Dr Barrow & Mr Collins I understood thus much of the method some years before the Mercators Logarithmotechnia came abroad For Mr Collins in a Letter to Mr Strode dated 26 Iuly 1672 & published by order of the R. Society, wrote thus. Mense Septembri 1668 Mercator Logarithmotechniam edidit suam, quæ specimen hujus methodi (i.e. serierum infinitarum) in unica tantum figura, nempe Quadraturam Hyperbolæ continet Haud multo postquam in publicum prodierat Liber, exemplar ejus Cl. Wallisio Oxonium misi, qui suum de eo judicium in Actis pPhilosophicis statim fecit; aliumque Barrovio Cantabrigiam, qui quasdam Newtoni chartas extemplo remisit: e quibus et ALIIS, quæ OLIM ab Auctore cum Barrovio communicata fuerant, patet illam methodum a dicto Newtona ALIQVOT ANNIS ANTEA excogitatam & modo universali applicatam fuisse: ita ut ejus ope in quavis Figura Curvilinea proposita quæ una vel pluribus proprietatibus definitur Quadratura vel Area dictæ figuræ, ACCVRATA SI POSSIBILE sitSIT, sin minus infinite vero propinqua, Evolutio vel Longitudo Curvæ lineæ curvæ, Centrum gravitatis Figuræ; solida ejus rotatione genita, & eorum superficies; sine ulla radicum extractione, obtineri queant. / And all this may suffice to justify what I said my saying in the Introduction to the Book of Quadratures, that I found this Method gradually in the years 1665 & 1666. However, the Method is capable of improvements, & the improvements are theirs who make them. Now because By the help of this Book of Quadratures & fluxions I found between the end of the years 16798 & 1687 Keplers the demonstration of Keplers Proposition concerning the motion of the Planets in Ellipses, & Method I wrote the Book of Principles & I have therefore have now subjoyned thies book of Quadratures to the end of it. And because the methods of series & fluxions are nearly related to one another & were invented in the same year (the year 1665), & were enjoyned by me in the Tracts wchwhich I wrote above 49 years ago in those days, & jointly ma compose one very general method of Analysis I have here added to the end of this book out of my Letters formerly published, some Propositions for resolvingducing quantities into series, & for resolving Problems by the help of such series quamproxime both methods together. For tho the resolustion of Problems by the into series by continual approximation be not Geometrical, yet it is allowed in Arithmetick, & by consequence also in Algebra; as is manifest by continual approximations in decimal numbers & by perpetual divisions & extractions of square & cube roots & the roots of affected æquations, allowed in Arithmetick & it may sometimes lead to of Problems by finite equations Now since By the help of this Method I wrote the Book of Principles & I have therefore have subjoyned the book of Quadratures to the end of it. And for the same reason I have subjoyned also the Differential Method. And because many Propositions the solution of several Problems require the Quadrature of figures proposed in the Book of Principles to be solved concessis Figurarum Quadraturis I have added to the end of the Book of Quadratures some Propositions taken from my Letters already published for reducing quantities into converging series & thereby squaring the figures. to For Quadratures by such series have the same place in Arithmetick & Algebra with operations in decimal numbers. & the extractions of roots In the Analysis per series abovementioned were several series for instances of squaring the cCircle & Conic Sections & finding the Arc whose sine was given & the sine of any given arc & other Figures & of finding the lengths of Curve lines, & on the contrary of finding the Abscissas & Ordinates of Figures whose Areas or lengths of the Curve lines are given & Mr Collins was very free in communicating to Mathematicians the series there set down. Mr Leibnitz was in London in the begining of yethe year 1673 & conversed wthwith Dr Pell &c above numeral series & carried with him to Paris Mercator's Logarithmotechnia but did not yet understand the higher Geometry. The next year he studied this Geometry & in a two Letters dated 15 Iuly & 26 Octob. he wrote to Mr Oldenburg that he had found the circumference of a circle in rationa a series of rational numbers, & that by the same method, the any arch of a circle whose sine was known might be known exhibited in a like series whose without knowing its proportion to the whole circumference. Which is as much as to say that that he had found a series a series of rational numbers expressing the length of any arch of a circle whose sine was given, wchwhich gave the whole circumference if the its proportion to the Arch was known, or else at least it gave the arch. The next year Mr Oldenburg in a Letter dated 15 Apr. 1675 sent to Mr Leibnitz these two series wchwhich Mr Collins had received from me Posita pro Radio Vnitate, datoque x pro sinu, ad inveniendum z Arcum Series est hæc est:

z = x + \frac{1}{6} x^{3} + \frac{3}{40} x^{5} + \frac{5}{112} x^{7} + \frac{35}{1152} x^{9}

&c et ad invenien si dederis z pro sinu arcu ad inveniendum x sinum series hæc est;

x = z - \frac{1}{6} z^{3} + \frac{1}{120} z^{5} - \frac{1}{5040} z^{7} + \frac{1}{362880} z^{9} - &c

And these two wchwhich Mr Collins had received from Mr Newton Iames Gregory. Pone

formula = r

, Arcum a, Tangentem t; erit

t = a + \frac{a^{3}}{3 r^{2}} + \frac{2 a^{5}}{15 r^{4}} + \frac{17 a^{7}}{315 r^{6}} + \frac{62 a^{9}}{2835 r^{8}} + &c

. Et conversim ex Tangente invenire Arcum ejus

a = t - \frac{t^{3}}{3 r^{2}} + \frac{t^{5}}{5 r^{4}} - \frac{t^{7}}{7 r^{6}} + \frac{t^{9}}{9 r^{8}} - &c

. And Mr Leibnitz acknowledged the receipt of these series by a Letter dated May 20th 1675. Literas tuas multa fruge Algebraica refertas accepi, pro quibus tibi & doctissimo Collinio gratias ago. Cum nunc præter ordinarias Curas Mechanicais imprimis negotijs distrahar, non potui examinare series quas misistis ac cum meis comparare. Vbi fuero perscribam tibi sententiam meam: nam aliquodt jam anni sunt quod inveni meas. The next year Mr Oldenburg Leibnitz wrote the following Letter to Mr Oldenburg dated 12 May 1676. Cum Georgius Mohr Danus in Geometria et Analysi versatissimas, nobis attulerit communicatam sibi a Doctissimo Collinio vestro expressionem relationis inter Arcum et Sinum per infinitas series sequentes: Posito sinu x, arcu z, Radio 1. erit

z = x + \frac{1}{6} x^{3} + \frac{3}{40} x^{5} + \frac{5}{112} x^{7} + \frac{35}{1152} x^{9} + &c

x = z - \frac{1}{6} z^{3} + \frac{1}{120} z^{5} - \frac{1}{5040} z^{7} + \frac{1}{362880} z^{9} - &c

. Hæc inquam Cum nobis attulerit ille, quæ mihi valde ingeniosa videntur, et posterior imprimis series elegantiam singularem habeat, ideo rem gratāam mihi feceris, Vir clarissime, si Demonstrationem transmiseris. Habebis vicissim mea ab his longe diversa circa hanc rem meditata, de quibus jam ante aliquot abhinc annis ad te perscripsisse credo, demonstratione tamen non addita quam nunc polio. Oro ut Cl. Collilnio multam a me Salutem dicas: is facile materiam suppeditabit satisfaciendi desiderio meo. By the dDemonstration of these series Mr Leibnitz meant the Method of finding them by Analysis & since he sent Mr Oldenburg to Mr Collins for the same he meant had heard of the Analysis per series numero terminorum infinitas, & wanted it. But Mr Collins being unwilling to let him have a copy of it he & Mr Oldenburg joyned in desiring that I would send him the Demonstration & thereupon I wrote to Mr Oldenburg my the following letter of dated 13 Iune 1676 to be communicated to Mr Leibnitz. And Mr Leibnitz sent back by way of recompence the series of Mr Gregory above mentioned, vizt

arc = t - \frac{t^{3}}{3} + \frac{5^{5}}{5} - \frac{t^{7}}{7} + \frac{t^{9}}{9} - &c

& five years after published it in the Acta Eruditorum without taking as his own without taking any notice of the correspondence between him & Mr Oldenburgh by wchwhich he had received it from London. And all this may suffice to justify my saying that in the Introduction to the following Book of Quadratures, that I found the method gradually in the years 1665 & 1666 However, the method is capable of improvements & the improvements & theirs who make them. 30 C In the end of the Scholium in Princip. Philos. pag. 227 after the words Vtriusque fundamentum continetur in hoc Lemmate, add Sunto quantitates datæ a, b, c; fluentes x, y, z; fluxiones p, q, r; & momenta op, oq, or: et proponatur æquatio quævis

x^{3} - 2 x x y + b x x - b b x + b y y - y^{3} + c z z = 0

. Et per hoc Lemma, Si sola fluat x, erit fluxio totius

3 x x o p - 4 x o p y + 2 b x o p - b b o p

; si sola fluat y, erit fluxio totius

−2 x x o q + 2 b y o q - 3 y y o q

; si sola fluat z, erit fluxio totius

2 c z o r

; si fluant omnes, erit fluxio totius

3 x x o p - 4 x o p y - 2 b o x p - b b o p - 2 x x o q + 2 b y o q - 3 y y o q + 2 c z o r

. Et quoniam totum semper æquale est nihilo, erit fluxio totius æqualis nihilo. Dividatur totum per momentum o, et prodibit æquatio quæ ex fluentibus dat fluxiones, vizt

3 x x p - 4 x p y + 2 b x p - b b p - 2 x x q + 2 b y q - 3 y y q + 2 c z r = 0

. Exhibet igitur hoc Scholium Lemma solutionem Propositionis præfatæ, Data æquatione fluentes quotcunque quantitates involvente in ti & fluxiones invenire.  Dixi vero in Epistola mea 24 Octob. 1676 ad Oldenburgium missa ut cum Leibnitio communicaretur quo mediante commercium tunc habui cum D. Leibnitio, quod hæc Propositio fundamentum esset methodi d generalis de qua scripseram anno 1671. Et hanc Propositionem esse fundamentum methodi generalis de qua scripseram anno 1671 dixi in Litteris prædictis Et hanc Propositionem esse fundamentum methodi generalis de qua scripseram anno 1671 deixi in Literis præfatis, anno 1671. In Epistola mea 24 Octob 16746 ad Oldenburgium missa & cum D. Leibnitio communicata, posui hanc Epistolam Propositionem ut fundamentum methodi de qua scripseram anno 1671. Et hæc methodus facile colligitur etiam ex Epistola quam ad Collinium 10 Decem 1672 scripsi, et cujus exemplar ad D. Leibnitium missam fuit anno 1676. Sit ACF linea quævis Curva, AB ejus Abscissa, & BE momentum ABbscissæ, et sint BC, EF Ordinatæ duæ ad Curvam in C et F terminatæ, CF momentum Curvæ, & FG momentum Abscissæ existentæ BCGE parallelogrammo. Agatur chorda CF et producatur eadem donec Abscissæ occurrat in D: et similia erunt triangula CGF, DBC. Et ubi momentum CF diminuitur in infinitum, recta CD curvam tanget in C. Hoc omnibus notum est. Iam dicatur AB x, & BC y, et sint earum fluxiones p et q, & habitudo inter x & y exprimatur per qualibet æquatione puta

x^{3} - 2 x x y + b x x - b b x + b y y - y^{3} = 0

. Et per Epistolam illam Regula ducendi Tangentem hæc erit. Multipliæ æquationses terminos per quamlibet progressionem Arithmeticam juxta dimensiones y, puta

\begin{matrix} x^{3} - 2 x x y + 6 x x - b b x + b y y - y^{3} = 0 \\ 010023 \end{matrix}

. ut et juxta dimensiones x, puta

\begin{matrix} x^{3} - 2 x x y + b x x - b b x + b y y - y^{3} \\ 322100 \end{matrix}

. Prius productum erit Numerator & posterius divisum per x denominator fractionis quæ exprimet longitudinem BD ad cujus extremitatem D. ducenda est Tangens. Ducet ur est autem BD ad eandem ptlagam cum Abscissa AB ubi valor ejus affirmativus etst et ad contrariasm partes ubi negativus. Est ergo, per Regulam in hac Epistola positam; Vt summa omnium terminorum multiplicatorum per indices dignitatum x ac divisorum per x ad summam omnium multiplicatorum per indices dignitatum y ac divisorum per y, ita ut est y ad ad — BD, et ita est FG ad — CG seu oq ad — oxp. Ducantur extrema et media in invicem & Æquatio factorum per o dividsa evadet

3 x x p - 4 x p y + 2 b x p - b b p = 2 x x q - 2 b y q + 3 y y q

, seu

3 x x p - 4 x p y + 2 b x p - b b p - 2 x x q + 2 b y q - 3 y y q = 0

. Hæc equatio producetur etiam per Lemma ubi duce tantum sunt æquationes fluentes quantitates. Vbi vero sunt plures, eadem operatio applicari debet ad omnes. Et sic habebitur æquatio fluxiones involvens. Et in hac operatione fundatur methodus generalis, uti dixi in edemeadem Epistola, quæ extendit se citra molestum31 molestum ullum calculum non modo ad ducendum Tangentes ad quasvis Curvas sive Geometricas sive mechanicas vel quomodocunque rectas lineas aliasve Curvas respicientes; verum etiam ad resolvendum alia abstrusiora Problematum genera de Curvitatibus, Areis, Longitudinibus, centris gravitatis Curvarum &c. Neque quemadmodum Huddenij methodus de Maximis et Minimis) ad solas restringitur æquationes illas, quæ sunt quantitatibus surdis sunt immunes. Hanc methodum intertexui alteri isti quæ Æquationum Exegesin instituo, reducendo eas ad series infinitas: sc. in Tractatu illo quem scripsi anno 1671. Et hæc Quæstio est: uVtrum D. Leibnitius sit inventor & methodi, & pro differentijs igitur Leibnitianis Newtonus adhibet semperque [ex quo methodo intellexerit usus est hac methodo] adhibuit fluxiones, quemadmodum Honoratus Fab Fabrius motuum progressus Carallerianæ methodo substituit. Gregorius ad Collinium scripsit 5 Sept. 1670 se ex Barrovij methodis tangentes ducendi dum quibusdam & suis & suis invenisse Methodum generalem & Geometriicam tangentes ducendi cum quibusdā ad omnes Curvas sine calculo. Newtonus ad eundem cCollinium scripsit in hæc 10 Decem 1672 in hæc verba Ex animo gaudeo D. Barrovij nostri reverendi Lectiones Mathematicas exteris adeo placuisse, neque parum me juvat — — — reducendo eas ad quanti series infinitas. Et harum duarum literarum exemplaria ad Leibnitium missa sunt ab Oldenburgo inter excerpta ex Epistolis Gregorij 26 Iunji 1676. Et Leibnitius mense novembri proximo Lectiones Barrovij secum tulit in Germaniam. Sunto jam ipsarum x et y fluxiones pet q & momenta op et oq et in AB producta capiatur

BE = o p

, erigatur ordinata EF parallela BC & occurrens Curvæ in F et compleatur parallelogrammum BCGE et erit

FG = o q

, et ex methodo tangentium Barrovij erit

FG . CG ∷ CB . BD

, seu BD. CB∷CG.FG∷op. oq∷p.q. adeoque BCD

\frac{p}{q}

\frac{−p y}{q} = BD = \frac{−2 x x y + 2 b y y - 3 y^{3}}{3 x x - 4 x y + 2 b x - b b}

. Et facta reductione fit evadit prodit

2 x x q + 2 b y q - 3 y y q -

habetur

3 p x x - 4 p x y + 2 b b x - 2 p b b + 2 x x q - 2 b y q - 3 y y q = 0

. Vnder prodit hujusmodi Regula si detur æquatio fluentes quocumque duas quantitates x et y involvetse, invenientur æquatio fluxiones involvens p et q involvens multiplicando terminos omnesm per æquationes data terminum per indicem dignitatis quantitatis cujusque fluentis quam in volvit et in singulis multiplicationibus mutando dignitatis latus in fluxionem suam. & ponendo Nam factorum omnium aggregatum sub proprijs signis erit æquatio nova. Et si plures sint fluentes, eadem operatio instituenda est in singulis. [Hæc est Propositio prima libri Newtoni de Quadraturis. Hanc Propositionem Newtonus posuit in Epistola 24 Octob 1676.] Et hæc Regula totum illud omne comprehendit quod Leibnitius anno 1677 a de methodo differentiali ad Newtonum remisit rescripsit estque Propositio primæ Libri de quadraturis. In Hæc Regula habetur Algorithmus hujus methodi. Hanc Regulam eamque Wallisius in lucem edidit anno 170 1693 in secundo Volumine operum ejus pag 392. in lucem edidit, anno 1693 a Newtono acceptam a Aug 27 anno 1692 a Newtono acceptam Aug. 27 anno 1692 cum exemplis in fluxionibus primis et secundis inveniendis. Estque Regula antiquissima quæ lucem vidit pro differentialibus differentiandis. Eandemque ut damextum methodi fluxionum Newtonus posuit in Epistola sua 24 Octob 1676 ad Oldenburgum pro Leibnitio datam Eandemque Newtonus eandem demonstravit Synthetice in secundo LemmamateLemmate Libri Principiorum anno 1686 et prop cum Propositionem prius posuisset in Epistola sua ad 24 Octob. 1676, ad Oldenburgum pro Leibnitio missa, et ibi significasset eandem esse fundamentum methodi de qua scripserat tum ante annos quinque id est anno 1671. Demonstratur vero in Lemmate illo in hunc modum.   Sunto quantitates datæ A, B, C a, b, c, fluentes x, y, z fluxiones p, q, r momenta op, oq, or, & proponatur æquatio quævis — — — — — — — exhibet igitur hoc Lemma solutionem Propositionis data æquatione fluentes quotcunque quantitates involvente invenire fluxiones. Et hanc Propositionem esse fundamentum methodi generalis de qua scripserat anno 1671 dixit Newtonus in Epistola literis prædictis. Propositionem quintam — Propositionem quintam libri de Quadraturis, Wallisius edidit anno 1693 in secundo operum suorum Volumine pag. 391. Hac Propositione quadrantur Figuræ curvililne accurate & Geometrice si fieri potest. Partem hujus Propositionis Newtonus posuit in Epistola 24 Octob 1676 ad Oldenburgum scripta. In Analysi sua per series dixit quod Analysen illius beneficio Curvarum areæ & longitudie, &c (id modo fiat) exacte & Geometrice determinetur: ideoque quinta illa Propositio tunc ipsi innotuit. Hanc Analysen Leibnitios videre potuit in secundo suo in Angliam itinere ubi Collinius ips. monstravit plures Newtoni Gregorij & aliorum Literas quæ circa series præcipue versæbantur. Collinius in Epistola sua Collinius in Epistola ad Thomam Strode 26 Iuly 1672 data Scripsit in hæc verba: Mense Septembri 1668 Mercator Logarithmotechniam edidit suam — Haud multo postquam in publicum prodierat liber exemplar ejus Cl. Wallisio Oxonium misi — aliumque Barrovio Cantabrigiam, qui quasdam Newtoni chartas [sc. Analysin per series] extemplo remisit: e quibus et EX ALIIS quæ OLIM q ab auctore communicata fuerant, patet illam Methodum a dicto Newtono aliquot annis antea excogitatam & mod[id est antequam prodiret Logarithmotechnia] illa] excogitatam & modo universali applicatam fuisse: ita ut ejus ope in quavis Faigura Curvilinea proposita quæ una vel pluribus Proprietatibus definitur, Quadratura vel Area: dictæ figuræ ACCVRATA SI POSSIBILE SIT, sin minus infinite vero propinqua &c — obtineri queat: id est accurata si series abrumpitur, sin minus infinite vero propinqua. Testibus igitur Barrovio et Collinio Newtonus methodum habuit aliquot annis antea quam prodiret Logarithmotechnia illa adeoque quadrandi Curvilineas anno 16766 aut antea, methodum habuit per series accurate si series abrumpitur et finita evadit; sin minus infinite vero propinqua. Et haoc fit per Propositionem illam quintam. Hæc autem Propositio pendet a quatuor prioribus: et propterea Method uis serierum et fluxionum quatenus contineantur in Propositionibus quinque primis libri de Quadratura Curvilinearum Newtono innotuere anno 1666 aut antea testibus Barrovio et Collinio; ut et teste etiam Wallisio, qui in Præfatione ad L Volumen primūum operum suorum ubi scripsit quod Newtonus in literis suis 13 Iunij & 24 Octob. 1676 datis methodum hanc [fluxionum vel differentiarlem] Leibnitio exponit tum ante decem annos nedum plures Leib ab ipso excogitatam, i. e. anno 1666 aut antea. plures ab ipsa excogitam; id est anno 1666 aut antea. Hoc Leinitius et Menkenius legerunt & per ea tempora non negarunt, ut ex Actis Eruditorum & Epistolis a Wallisio editis patet. Paulo post D. Nicolas Flio Fatio in Tractatu cui tituli de investigatione solidi minimæ resistentiæ in quod minima fiat resistentia, scripsit se anno 1687 in Calculum differentiali similem incidisse, Newtonum tamen primum ac pluribus annis vetustissimum hujus Calculi inventorem ipsa rerum evidentia se coactum agnoscere, visis scilicet Newtoni manuscriptis codicibus. Et Le pbneitius in Actis Eruditorum respondendo Ad nondum cœpit hoc negare, sed conatus est tantum se defendere quasi anno 1684 cum elementa calculi sui edidit, ne constabat quidem ipsi aliud de Inventis Newtoni in hoc genere quam quod ipse in hoc genere olim significaverat in literis, posse se Tangentes ducere invenire absque non sublatis irrationalibus: quod Hæ quod Hugenius quoque se posse ipsi significavit postea etsi cæterorum istius calculi adhuc expers; & ipsum Fatium agressus est proponendo Problema solvendum. Tandem anno 1703 Wallisius mortuus est et anno prox subinde Leibnitius audacior esse cœpit. Nam cum libri Newtoni de coloribus & de numero cCurvarum secondi generis de quadratura Figurarūum anno proximo proximo prodierent & Newtonus initio anni sequentis s Leibnitijo qui synopses librorum mathematicorum ad Mankenium mittere salebat, Newtonus in Actis Eruditorum pro Ianuario anni 1705 accusatus est quasi Leib accusari cæpit quasi Methodum a Leinitio habuisset stylo Leibnitiano stylo Leibnitij qui synopses librorum mathematicorum ad Menkenium mittere solebat. accusari cæpit quasi Methodum a Leibnitij suffurasset a Leibnitio habuisset dedicisset. Nam Leibnitium enim inventore vocant deinde addant Pro differentijs Ejus Elementa, inquiunt, ab I INVENTORE D. G. GLeibnitio in his Actis sunt tradita, varijsque usus tum ab ipso tum a fr D. D. Fratribus Bernoullijs tum & D. Marchione Hospitalio — sunt ostensi. Pro differentijs igitur Leibnitianis D. Newtonus adhibet semperque [ex quo usus est huc methodo] coluit didisit] adhibuit fluxiones — quemadmodum & Honoratus Fabrius in sua Synopsi Geometrica, motuum progressus Cavallerianæ methodo substituit Quæstio est igitur est utrum Newtonus hanc methodum habuit didicita a Leibnitio. Ad hanc Qu Quæstionem dirimandam editam sunt anno 1712 est ex antiquis monumentis Commercium Epistolicum 1712 & R L & ineunte anno 1615 Recensio ejusdem Libri. anno 1795. Anglice Hæc Recensio edita fuit Anglicæ in Actis Philosophicis R. Societatis, et jam latine versa præponitur Commercio & ad veritatem stabiliendam Commercium citatur Commercio opposuit. Et hæc et his præmissis, et quod pro symbolis

\frac{a a}{64 x}

& oυ, y vel oy quibus Newtonus utitur in Analysi per series Leibnitius utatur symbolis

\int \frac{a a}{64 x}

& d dz, & Analysin illam videre potuit in in secundo suo in Angliam itinere ubi Collinius ipsi monstravit plures Newtoni Gregorij & aliorum Literas quæ circa Series præcipue versabantur: his inquam præmissis, legatur jam Commercium Epistolicum. Scholium In literis quæ mihi cum Geometra peritissimo G. G. Leibnitio s abhinc decem in anno 1676 intercedebant, cum significarem me compotem esse methodi determinandi Maximas et Minimas, ducendi Tangentes, quadrandi Figuras curvilineas, conferendi easdem cum Sectionibus conicis, se mutuo & similia peragendi quæ in terminis surdis æque ac in rationalibus procederet, & Tractatum de eadem anno 1671 scripsisse, ut et Tractatum alium quem Barrovius noster anno 1669 ad Collinium atque literis transpositis hanc sententiam involventibus [Data æquatione quotcunque Fluentes quantitates involvente, Fluxiones invenire, et vice versa] eandem celarem: rescripsit Vir Clarissimus mense Iunio anni proximi 1677 se quoque in ejusmodi methodum incidisse; & methodum suam communicavit a mea vix abludent oem præterquam in verborum & notarum formulis & Idea generationis quantitatum. Mense Iunio anni 1676 Collin Extan Incidi paulætim in hanc methodum, annis 16766 & 166767 Et extant ejusdem specimina quædam hujus methodi in Analysi mea per Series quam Barrovius mense Iunio anni 1669 ad Collinium misit: Et in mense Iunio anni 1676 Collinius exemplar epistolæ meæ ut et in Epistola quam de hac methodo 10 Decem. 1672 ad ipsum datæ Collinium misi, et cujus exemplar Collinius mense Iunio anni 1676, una cum exemplaribus epistolarum plurium Ia. Gregory, epistolarum D. Leibnitium in Gallia tunc agentem misit. Et Mense autem Octobri ejusdem anni ad finem vergente, D. Leibnitius E Gallia in Angliam veniens, ✝ ✝ vidit Epistolam meam Octob. 24 ad Oldenburgum datam ubi iterum admonitus de hac methodo, locutis sum de deque Analysi per series, ad Collinium missa deque Tractatu quem anno 1671 conscripsi, & ac de methodo qua maximarsum et minimarsum, deque quadratura figurarum primam exh et areas curvarum inveniro: Et ubi Theorema etiam posui quadrandi figuras ex methodo hac methodo de inventem et exemplis illustratamvi Quinetiam D. Leibnitius epistolas autographas et Gregorij et meas in manu Collinij de consuluit. Et subinde in Germaniam rediens ut negotijs publicis publicis interesset, incidit in hanc methodum epistolas autographas et Gregorij et meas in manu Collinij consulavist. Et subinde in Germaniam rediens ut negotijs publicis interesset, in Germaniam rediens incidit in hanc methodum anno 1677; et mense Iunio ejusdem anni 1677 specimen ejus ad me misit, ut supra. Ejusmodi Hujus vero methodi fundamentum continetur in hoc Lemmate. et vice versa] eandem Propositionem principalem hujusujus fundamentalem fundamentum hujus methodi hujus celarem], specimen vero s methodi hujus ejusdem subjungerem et exemplis illustravem; res cumque vir clarissimus episto in reditu suo e Gallia per Angliam in Germaniam, epistolas in manu et meas et in manu Collinij insuper etiam consuluisset: incidit is in methodum similem sub diversis verborum et notarum formulis. Methodi utriusque fundamentum continetur in hoc Lemmate. Et cum Collinius noster exemplar Epistolaæm meaæm 10 Decem. 1672 ad ipsum datæ exemplar mense Iunio anni 16 in qua methodum hanc descripsissem & ex exemplar mense Iunio at Leibnitium misisset Et cum Collinius Epistolam 10 Decem 1672 datam a me accepisset in qua methodum hanc descripsissem & exemplo Tangentium ducendarum illustrassem, et hujus Epistola exemplar mense Iunio anni 1676 in Galliam ad D. Leibnitium misisset; & vir clarissimus sub finem mensis Octobris in reditu suo e Gallia per Angliam Et in Germaniam Epistolas meas in manu Collinij insuper consuluisset: incidit is finito itinere, in methodum similem sub diversis verborum et notarum formulis, et mense Iunio sequente specimen ejusdem in Tangentibus ducendis ad me misit, eandemque a methodo mea non abludere subjunxit. Methodi utriusque fundamentum continetur in hoc Lemmate. Pag. 226ad Spatium quod corpus in Medid non resistente e queste cadendo eodem tempore describere potest, ut arearum prædictarum differentia ad

\frac{BD \times V^{2}}{AB}

. In epistola quadam ad D. I. Collinium nostratem 21 10 Decem. 1672 data, cum descripsissem methodum tangentium quam conjicebam suspicabar eandem esse cum methodo Slusij tum nondum p communicata; subjunxi: Hoc est unum particulare, vel corollarium potius Methodi generalis, quæ extendit se, citra molestum ullum calculum, non modo ad ducendum tangentes ad quasvis Curvas sive Geometricas verum etiam ad resolvendum alia abstrusiora problematum genera de curvitatibisus, Areis, longitudinibus, centris gravitatis curvarum &c. Neque (quemadmodum Huddenij methodus de Maximis et Minimis) ad solas restringitur æquationes illas quæ quantitatibus surdis sunt immunes. Hanc methodum intertertexui alteri isti, qua æquationum exegesin instituo, reducendo eas ad series infinitas,. scilicet in tractatum quem tunc ante annum (nempe A.C. 1671) Hactenus Epistola Et Hæc ultima verba spectant ad Tractatum quem anno 1671 id est de his rebus scripseram. Methodi vero hujus generalis fundamentum continetur in Lemmate præcedente. per quod utique resolvitur pars prior cujus problematis, Data æquatione fluentes quotcunque quantitates involvente, invenire fluxiones; & vice versa. Nam Coroll. Quare si longitudo dicatur quæ oritur applicando aream DET ad BD et cujus momentum erat m; lineam BD dicatur M Et si longitudo alia V sumatur in ea ratione ad longit. M quam habet linea DA ad lineam DE [et cujus momentum propterea erit

\frac{DA}{DE} \times m

] Spatium quod corpus — — — illarum differentia ad

\frac{BD \times V^{2}}{AB}

, ideoque ex dato tempore datur. Nam spatium in Medio non resistente est in duplicata ratione temporis sive ut

V^{2}

, et ob datas BD et AB, ut

\frac{BD \times V^{2}}{AB}

. Hæc area æqualis est areæ

\frac{{DA}^{q} \times BD \times M^{2}}{{DE}^{q} \times AB}

et ipsius M momentum erat est m et Augeatur M momento suo m cl et ipsius M momentum est m et propterea hujus areæ momentūum coad etst

\frac{{DA}^{q} \times BD \times {}^{2}M m}{{DE}^{q} \times AB}

. Et hoc momentum est ad momentum differentiæ arearum DET et AbNK vizt ad

\frac{AP \times BD \times m}{AB}

{DA}^{q} \times BD \times M {DE}^{q}

\frac{1}{2} AP \times BD

sive ut

\frac{{DA}^{q}}{{DE}^{q}}

in DET ad DAP adeoque ubi areæ DET et DAP quam minimæ sunt, in ratione æqualitatis. Æqualis est igitur area quam minima

\frac{BD \times V^{2}}{AB}

differetiæ quam minimæ arearum DET & Ab AbNK. Vnde cum spatia in Medio utroque ut

\frac{{DA}^{q} \times BD \times 2 M \times m}{{DE}^{q} \times AB} + \frac{{DA}^{q} \times BD \times m m}{{DE}^{q} \times AB}

\frac{AP \times BD \times m}{AB}

hoc est ut

\frac{{DA}^{q} \times BD \times 2 M + m}{{BE}^{q}}

ad AP, BD. sive ut

\frac{{DA}^{q}}{{DE}^{q}}

{}^{2}{DET} + DTV

2 DAP

; adeqoque ubi areæ DET, DAP quam minimæ sunt, et area DET evanescit, in ratione æqualitatis. [Æqualis est igitur area quam minima] seu

\frac{{DA}^{q}}{{DE}^{q}}

DTV = 2 DAP

U = \frac{2 DET \times DA}{BD \times DE}

\frac{BD, V V}{4 AB} = \frac{DET \times DET \times {DA}^{q} \times BD}{AB \times {DE}^{q} \times {BD}^{q}} = \frac{{DET}^{q} \times {DA}^{q}}{AB, {DE}^{q} \times BD} = \frac{DET \times DET \times {DA}^{q}}{AB \times {DB}^{cub}} = \frac{BD \times V V}{4 AB}

M = \frac{DET}{BD}

\frac{BD \times V V}{4 AB} = \frac{M M \times {DA}^{q}}{AB \times BD}

V = \frac{2 DA \times M}{DE}

\frac{BD, V V}{4 AB} = \frac{4 {DA}^{q} \times M^{q} \times BD}{4 AB, {DE}^{q}} = \frac{{DA}^{q} \times M^{q} \times BD}{AB \times {DE}^{q}}

Momentum

M^{q} = 2 DEV \times DVT + DVT q = \frac{2 DET \times DVT}{D^{q}} + \frac{{DVT}^{q}}{{BD}^{q}} = \frac{2 DET \times DVT}{{BD}^{q}} + \frac{{DVT}^{q}}{{BD}^{q}} = \frac{2 DET \times BD \times m}{{BD}^{q}} + \frac{{BD}^{q} \times m^{q}}{{BD}^{q}}

m = \frac{1}{2} TV

M = \frac{DET}{BD} = \frac{DE \times \frac{1}{2} ET}{DE} = \frac{1}{2} VT

ET = x + o

{ET}^{q} = x x + 2 o x

lin 1. sive huic æqualis

\frac{{DA}^{q} \times BD \times M^{2}}{{DE}^{q} \times AB \times BD}

et ad momentum differentiæ arearum DET et AβNK ut

\frac{{DA}^{q} \times M \times m}{{DE}^{q} \times AB}

AP \times BD \times m AB

, hoc est ut

\frac{{DA}^{q} \times BD \times M}{{DE}^{q}}

\frac{AP \times BD}{2 AB}

, sive ut

\frac{{DA}^{q} \times DET}{{DE}^{q}}

DAP \times {BD}^{q} = BD \times 2 DAP

sive ut

\frac{{DA}^{q}}{{BD}^{q}} DET

ad DAP. P. 252. lin. 1. sive huic æqualis

\frac{{DA}^{q} \times BD \times M^{2}}{{DE}^{q} \times AB}

, ad est ad Momentum differentiæ arearum DET et AβNK, ut

\frac{{DA}^{q} \times BD \times M \times m}{{DE}^{q} \times AB}

AP \times BD \times m AB

, hoc est, ut

\frac{{DA}^{q} \times BD \times M}{{DE}^{q}}

AP \times BD

, sive ut

\frac{{DA}^{q}}{{DE}^{q}}

in DET ad

2 DAP

\frac{{DA}^{q} M^{2}}{AB, BD} \begin{array}{l} = {DA}^{q} \times M^{2} + 2 M o + o o \\ = {BA}^{q} \times M^{2} + M \times m \end{array}

o = \frac{1}{2} VT = \frac{DTV}{2 BD}

\frac{DET}{DB} = M

\frac{DET \times DET}{{DB}^{q}} = MM

. Ejus

M momentum = \frac{DVT}{DB} = m

Corol. Quare si longitudo aliqua V sumatur in ea ratione ad longitudinem M quæ oritur applicando aream DET ad BD quam habet linea DA ad lineam DE: spatium quod corpus ascensu vel descensu tota in Medio resistente describit erit ad spatium quod in Medio non resistente describit eodem tempore describere posset, ut arearum illarum differentia ad

\frac{BD \times V V}{AB}

ideoque ex dato tempore datur. Nam spatium in Medio non resistente est in duplicata ratione temporis sive ut

V V

, et ob datas BD et AB ut

\frac{BD \times V^{q}}{AB}

. Momentum hujus areæ, sive areæ huic æqualis

\frac{{DA}^{q} \times BD \times M^{2}}{{DE}^{q} \times AB}

est ad momentum d 34 Scholium. Analysin per series et momenta a me scriptam, Barrovius noster anno 1669 ad Collinium misit. Methodos ibi expositas in alio Tractatu plenius explicui anno 1671, I sed hos Tractatus in lucem edere distuli quietis gratia et ex hoc Tractatu Tractatum de quadratura Curvilinearum anno 1676 extraxi. deinde Interea Et In Epistola 10 Decem. 1672 ad Collinium data, methodum momentorum Exemplo tangentium more Slusiano ducendarum illustravi, dixique eandem etiam ad quæstiones de curvitatibus & longitutdinibus Curvarum & areis ac centris gravitatis Curvilinearum &c sese extendere, & esse generalem et ad quantitates surdas non hærere, et Collinius exemplar hujus & methodo serierum in scriptis meis intertextam esse: et Collinius exemplar hujus Epistolæ mense Iunio anni 1676 ad D. Leibnitium tunc in Gallia agentem misit. In Literis insuper quæ mihi cum D. Leibnitio anno 1676 intercedebant, cum verba facerem de scriptio Tractatibus prædictis, & significarem me compotem esse methodi. Analyticæ determinandi maximas et minimas, ducendi tangentes, quadrandi figuras curvilineas, conferendi easdem inter se, et similia peragendi quæ in terminis surdis æque ac in rationalibus procederet; et literis transpositis hanc sententiam involventibus [Data æquatione quotcunque fluentes quantitates involvente, fluxiones invenire, et vice versa] fundamentum hujus methodi celelaremcelarem, specimen vero ejusdem in curvilineis per seriem quadrandis subjungerem et exemplis illustrarem; ☉ — et exemplis illustrarem, ☉ sed in lucem vero ere mea me quietis gratia hæc non edere dicerem; et cum Vir clarissimus eodem anno et Vir clarissimus eodem anno in reditu suo e Gallia per Angliam in Germaniam, sub finem mensis Octobris, epistolas meas in manu Collinij consuluisset etiam consuleret: incidit is paulo post non multo post in methodum momentorum sub diversis verborum et notarum formulis, et mense Iunio sequente specimen ejusdem in Tangentibus more Slusiano ducendis idque non obstantibus irrationalibus, ad me misit, et subjunxit se credere methodum suor meam a sua non abludere, præsertim cum quadraturæ perutramque methodum meam faciliores redderentur. Methodi hujus fundamentum continetur in hoc Lemmate, & in Tractatu de Quadratura curvilinearum plenius fusius exponitur. 35 Schol. Per momenta quantitatitatum hic intelligo earu particulas earum momento temporis genitas. Hæ particulæ non sunt minima indivisibiles, sed sunt dibiles in infinitum; non sunt quantitates minimimæ sed sunt indefinite parvæ et vel vedem æqualibus temporis momentis genitæ. Et considerando tempus ut fluxu continuo et æqualili genitum, & alias quantitates ut fluxibus quibuscunque genitas, seu æquabilibusis seu inæquabilibus genitas; expono tempus per quantitatem quancunque uniformiter fluentem, uti per lineam velx lineam v & ejus fluxium vel fluxionem per unitatem, et ejus momentum per literam alias quantitates fluentes (seu indeterminatsos) per alias symbola y, denoto per alia symbola, ut x, y, z: et Schol. Per momenta quantitatum hic intelligo earum incrementa momentanea seu particulas momento tem- to temporis genitas; easque indefinite parvas & in infinitum divisibiles. Et expodesignando fluxionem temporis per unitatem & fluxionis aliarum quantitum ut x y z per x, y, z vel etiam per alia symbola uti p, q, r: denoto harum momenta per momentum temporis per

1 \times o

seu o et fluxiones momenta aliarum quantitatum per po, qo, ro,

\dot{x} o

\dot{y} o

\overset{..}{z} o

&c vel po, qo, ro. Et habita quantitatis fluxione pqr ut

\dot{x}

ut p, q vel r quantitate fluentem, quandoque designo vel quandoque per eadem symbola vel in imponendo lineolam ut in in hunc modum

p'

q'

r'

vel præfigendo aut circumscribendo rectangulum ut in his

▯ p

▯ q

▯ r

& his vel

p

q

. Et hujusmodi symbolis adhibitis Analysin instituo in fluentibus et eorum fluxionibus & momentis adhibendo unicam tantum quantitatem indefinite parvam o. Sed hujusmodi Analysin docere non est hujus loci. Pag. 226. In literis quæ mihi cum Geometra peritissimo G. G. Leibnitio annis abhinc decem Anno 1676 intercedebant, [vizt anno 1676,] cum significarem me compotem esse methodi determinandi Maximas & Minimas, ducendi Tangentes, quadrandi figuras curvilineas conferendi easdem cum Sectionibus Conicis & similia peragendi, quæ in terminis surdis æque ac in rationalibus procederet, & Tractatum de eadem anno 1671 scripsisse; & atque literis transpositis hanc sententiam involventibus [Data æquatione — — — — et vice versa] eandem celarem: rescripsit vir clarissimus mense Iunio anni proximi, se quoque — — — Lemmate quantiatum. [Extant specimina quædam hujus methodi in Analysi mea per Series, a Barovio ad Collinium anno 1669 a Barrovio nostro ad Collinium missita Eandem D. Lei] Incidit is in hanc methodum post reditunt postquam redijt æ Gallia per Angliam in Germaniam i uti negotijs publicas interesset, id est anno 1677. Vtriusque methodi fundamentum continetur in hoc Lemmate.] Incidit autem is in hanc methodum post mensem junium anni 1676 quo Exemplar Epistolæ meæ de hac methodo ad Collinium 10 Decem 1672 datæ acceperat a Collinio acceperat & specimen hujus methodi a Collinio acceperat & postquam e Gallia per Londinum Angliam in Germaniam redijt ut publicis negotijs interesset; id est anno 1677.] Incidit autem is in hanc methodum post Menseum Iuni anni 1676 Collinius exemplar Epistolæ meæ de hac hac methodo 10 Decem 1672 una cum exemplaribus epistolaria pl. G. D. Leibnitium ipsum misit in Gallia agen agentem misit, et postquam e Gallia per Londinu Galliam in Germa & Mense ang Octobri. ejusdem anni D Leibnitius e Gallia m in Angliam venit & veniens Collinium de commercio suo Epistolas Gregorij et meas in hanc Collinij consuluit consuluit, & subinde in Germaniam rediens ut negotijs publicis interessent, incidit in hanc methodum: Ejusque fundamentum continetur in hoc Lemmate. Mense Iunio anni 1676 Collinius exemplar Epistolæ meæ de hac methodo 10 Decem. 1672 ad ipsum datæ, una cum exemplaribus Epistolarum plurium Iacobi Gregorij ad D. Leibnitium in Gallia agentem misit. Mense Octobri ejusdem anni ad finem vergente D. Leibnitius E Gallia in Angliam veniens, epistolas autographas et Gregorij et meas in manu Collinij consuluit. inter alias vidit epistolam meam 13 Iunij 1676 ☉ ☉ Vidit etiam Epistolam meam 129 Iune 16 24 Octob. 1676 ad Oldenburgum datam. Et subinde in Germaniam rediens ut negotijs publicis interesset, & incidit in hanc methodum, et mense Iunio anni & subinde in Germaniam rediens ut negotijs publicis interesset, incidit in hanc methodum: ‡ ‡ Et mense Iunio anni 1677 specimen ejus ad me misit. Ejus vero fundamentum continetur in hoc Lemmate. ique Ejus vero fundamentum continetur in hoc Lemmate. Phænomenon VII. Planetæ circum axes suos uniformi cum motu Iupiter revolvuntur respectu fixarum: Iupiter horis 9. & min primis 56′ Mars horis 24.40′ Venus horis 23, Sol diebus 265

\frac{1}{2}

circiter Luna diebus 27. horis 7. minutis primis 43′; & secundis et Terra horis 23 minutsis primis 56′ Revolutiones Iovis, Martis, Veneris & Solis per revolutionems macularum in eorum corporibus observatæ fuerunt. Maculæ Solis non sunt permanentes. Raro Post mensem unum aut altera evanescere solent, ideoque tempus revolutionis corporis ejus nondum satis accurate definiri potuit. Redeunt ad eundem situm corbq solis in disco Solis Spatio dierum 27

\frac{1}{2}

vel 28 circiter respectu Terræ et inde colligitur quod respectu fixarum redeunt ad eundem situm diebus 25

\frac{1}{2}

ve 26 circiter. Luna eandem faciem in Terram Eadem Lunæ facies Terram semper vertitur respicit; [sed cum motu quodam Libratorio ideoque respectu fixarum Luna revolvitur circa axem suum semel in mense squodie periodico. Eadem vero facies ejus in Terram vertitur semper respicit non accurate sed cum motu quodam libratorio tam in Longitudinem quàm in Latitudinem. Librando in Longitudinem faciem eandem semper vertita in umbilicum superiorem, Orbis sui quamproximendi ex observationibus Hevelianis olim di ex observationibus Hevelianis olim ddice, ideoque revolutio ejus respectu facerum uniformis est quoad motum in longitudinem. Librando in Latitudinem faciem ejus vertit in boream in Latituduine boreali & in asustrum in Latitudine australi ex observationibus Hevelianis etiam di idque in angulo latitudinem æquante (uti ex observationibus Hevelianis etiam didici uti ex Observationibus Hevelianis etiam constat.) [ideoque axis aream quem revolvitur plano Eclipticæ perpendicularis est quamproxime. Hæc omnia, quæ ex observationibus Hevelianis olim d quæ ex observationibus Hevelianis olim dididi, dem a me accepta D. Mercator in Astronomia sua initio anni 1676 edita plenius exposuit a me plenius exposuit, eademque nt Phænomenis sentire D Cassinus agrovit. [Maculæ in corporibus Saturni et Mercurij nondum observatæ sum] Simili motu Satelles extimus Saturnius circa axem suum revolvi videtur, eadem sui facie Saturnum perpetuo respiciens. Nam circa Saturnum revolvendo, quoties ad orbis sui partem orientalem venit accedit ægerrime videtur & plerumque videri cessat: id quod evenire videtur potest per maculas quasdam in ea corporis sui parte quæ Terræ tunc advertitur, ts Simili etiam motu Satelles extimus Ioviales circa axem suum revolvi videtur, propterea quod maculam habeat in p in parte corporis Iovi aversa: maculam habeat, qua tanquam [Maculæ in corporibus Saturni et Mercurij & re nondum observantur] quæ macula tanquam in corpore Iovis apparit quoties cernitur ubi Satelles inter Iovem & oculos nostros transit. Hæc omnia a me anno 1675 accepta D. Mercator in Astronomia sua plurias ex initio anni sequentis edita plenius exposuit — — idque in angulo latidtudinem æquante nisi quatenus hic angulus nunc augetur nunc diminuitur aliquantulum per inclinationem axis Lunæ ad planum Eclipticæ Hæc omnia Librationis rationem a me acceptarem, D. Mercator in Astronomia sua initio anni 1676 edita plenius exposuit. Simili moto Satelles extimus Saturnius circa axem suum revolvi videtur, eadem sui facie — — — In Analysi mea per series, quam Barrovius mense Iunio anni 16569 ad Collinium misit, extant methodi hujus specimina quadam, tum ante annos aliquot inventæ Mense Iunio anni 1676 Collinius 36 Pag. 226. Scholium. Analysin per series et momenta a me scriptam, Barrovius noster anno 1669 ad Collinium misit. Methodos ibi expositos an alio Tractatu plenius explicui anno 1671, ex hoc Tractatu inde Tractatum de Quadratura Curvilinearum anno 1676 extraxi. Interea In Epistola 10 Decem 1672 ad Collinium data, methodum momentorum exemplo tangentium more Slusiano ducendarum illustravi, dixique eandem etiam ad quæstiones de curvitatibus et longitudinibus Curvarum & ac de areis & centris gravitatis Curvilinearum &c sese extendere, & esse generalem, et ad quantitates surdas non hærere, & methodo serierum in scriptis meis intertextam esse: et Collinius exemplar hujus Epistolæ mense Iunio anni 1676 ad D. Leibnitium tunc in Gallia agentem misit. In literis insuper quæ mihi cum D. Leibnitio anno 1676 intercedebant, cum verba facerem de Tractatibus prædictis & significarem me compotem esse methodi Analyticæ determinandi maximas et minimas, ducendi tangentes, quadrandi figuras curvilineas, conferendi easdem inter se, et similia peragendi quæ in terminis surdis æque ac in rationalibus procederet; & literis transpositis hanc sententiam involventibus [Data æquatione quotcunque fluentes quantitates involvente, fluxiones invenire, et vice versa] fundamentum hujus methodi celarem, specimen vero ejusdem in curvilineis per seriem quadrandis subjungerem et exemplis illustrarem, sed in lucem me quietis gratia hæc in lucem non edere dicerem; et cum Vir clarissimus D. Leibnitius eodem anno in reditu suo E Gallia per Angliam in Germaniam, sub finem mensis Octobris, epistolas meas Commertium meum in manu Collinij etiam consuleret: incidit is non multo post in methodum momentorum sub diversis verborum et notarum formulis, et mense Iunio anni sequenteis seq specimen ejusdem in Tangentibus more Slusiano ducendis idque obstantibus irrationaibus ad me misit, et subjunxit methodum suam ad irrationales non hærere & se credere methodum meam a sua non abludere, præsertim cum quadraturæ per methodum meum utramque faciliores redderentur. Methodi hujus fundamentum continetur in hæ Lemmate, et in Tractatu prædictæ de Quadraturis curvilinearum Tractatu fusius exponitur. Fluxiones utique sunt velocitates quibus fluentium momenta generantur, et momenta a D. Leibnitio vocantur differentiæ: et inde Methodus fluentium et Methodus fluxionum & Methodus momentorum & Methodus differentialis dem & eandem methodum significaret Methodus hujus meæ autem fundameturntum continetur in hoc Lemmate, et in prædicto de Quadraturis Tractatum fusius exponitur eandem methodum significare possunt. Methodi hujus fundamentum continetur in Lemmate præcedente, et in prædicto de Quadraturis Tractatu fusius exponitur. Sept. Octob 5. 1722. Apog ☽ in ♎♌ 23

\frac{1}{2}

. ☉ in ♎ 23. ☽ in ♐♑ 16.

\begin{array}{r} 1722 . \\ 574 \end{array}

1656. 3907. 2251. 563.