Historia methodi infinitesimalis

It is represented that in the third fourth & fift Centuries

The testimony of the three in heaven (1 Iohn 5. 7) is wanting in all the Greek Manuscripts now extant hitherto examined & in all the old Versions except the vulgar Latin made by Ierome. It was originally wanting in this Version also for it is wanting in most of the oldest MSS of this Version but is got into almost all those which have been made since the twelf Century.

pag 191 lin 19. – quence not before the yeare 1677 or not above a month or two before.

Ib. lin 23. – year 1670, & M^r Ia. Gregory in a letter to M^r Collins dated 5 Sept. 1670 wrote thus. Barrovij Lectiones summa cum voluptate & attentione perlegi; atque omnes qui unquam hisce de rebus scripserunt infinito intervallo superasse comperio. Ex ejusdem [Barrovij] methodis Tangentes ducendi cum quibusdam e proprij collatis inveni Methodum generalem & Geometricam ducendi Tangentes ad omnes Curvas sine calculo. [A copy of this Letter was sent to M^r Tschurnhause in May 1675 & another copy thereof was sent to M^r Leibnitz in Iune 1676 in the collection of M^r Gregorys Letters above mentioned]. M^r Newton Notice of the method & of that of Slusius, being sent to M^r Newton he communicated his method of Tangents in a Letter to M^r Collins dated Decem 10 1672 saying that he took it to be the same with that of Slusius & Gregory, & after he had explained & illustrated it by an example he added: Hoc est unum particulare — — —

Pag 192. lin 7. Copies of M^r Gregories Letter of Sept 5. 1670 & of M^r Newtons of 10 Decem 1672 were sent to M^r Leibnitz by M^r Oldenburg — — — — of Iune 13 1676; & M^r Leibnitz in October following coming to London met there with the Lectures of D^r Barrow & with M^r Newton Letter of 24 Octob 1676, & M^r Newton having in his Letter & in that of 10 Decem. 1672 described that he had a very general Analysis — — — M^r Leibnitz in his returning returning home from London through Holland was meditating upon the improvement of the method of Slusius. For in a Letter to M^r Oldenburg — —

Pag. 193 l. 7. M^r Newton in his Letter dated Octob. 24. 1676.

Pag. 194. l 6. M^r Leibnitz saw this Letter before he left London but not having time to get it copied a copy thereof was sent to him so soon as M^r Oldenburg had notice that he was arrived at Hannover: & M^r Leibnitz soon after viz^t in a Letter dated Iune 21, 1677

Pag. 196 lin. ult. — {ry} of his general method, & M^r Gregory that the method of Barrow might be improved so as to draw Tangents without calculation, that is, so as to give the Rule of Slusius.

Pag 191 lin. 21 & 22. D^rBarrow published his Method of Tangents in the year 1670. M^r Iames Gregory in a Letter to M^r Collins dated 5 Sept 1670 wrote that he had improved the method of Barrow so as to be able to draw Tangents to all Curves without calculation. M^r Collins in autum 1672 gave notice of this method to M^r Newton & that M^r Slusius had offered to <173v> writ to M^r Oldenburg about such another Method. M^r Newton in a letter dated 10 Decem 1672 wrote back to M^r Collins that he took the methods of Gregory & Slusius to be the same with his own & after he had described his own Method & illustrated it with an example he added: Hoc est unum particulare vel Corollarium potius methodi generalis —

pag. 198. lin. 10. Occasio fuit hæc. D. Leibnitius Viennæ agens ad amicum scripsit quod cum probabile esset literas aliquas nondum publicatas inter eas Oldenburgi et Collinij latere, ipse optaret ut hæ sibi a Regia Societate mitterentur. Se enim cum Hanoveram rediret posse Commercium Epistolicum etiam in lucem emittere & velle publicare non minus litteras quæ contra ipsum allegari possent quam illas quæ ipsi faverent. Newtonus autem respondit, se non edidisse Commercium Epistolicum, ne dum literas quas ipse in archivis suis habuit in lucem protulisse ut in Commercio edito pulicarentur, se enim seipsum in propria causa testem facere, nolluisse & D. Leibnitium in propria causa testem admitti non debere. Et subinde in hujus rei testimonium is Epistolas duas protulit, utramque ad seipsum scriptam, unam a D. Wallisio Apr. 10 1695, alteram a D. Leibnitio Martij $\frac{7}{17}$ 1693, quæ postquam examinatæ & pro genuinis agnitæ sunt ab ijs qui utriusque autographa ex scribendi formulis noverant, , lectæ sunt coram Societate Regia et subinde in Societatis Archivis repositæ. Addidit insuper Newtonus quod siquid in Archivis R. Societatis adhuc inveniri posset luce dignum, hoc in Transactionibus Philosophicis imprimi posset, & Et siquid amplius D. Leibnitius haberet quod imprimi vellet, deberent autographa prius examinari. Et examinari quidem possent coram Regia Societate et subinde vel in Phil. Transactionibus vel alias imprimi si Leibnitio placeret Sed nihil missum est ut examinaretur, nihil in archivis R. Societatis nondum editum inventum est quod vel pro Leibnitio vel contra Newtonum faceret.

Pag 207 lin penult. After 393 & 396 add. And in the year 391 D^r Halley & M^r Ralphson borrowed this Book of Quadraturs at Cambridge & carried it with them to London as D^r Halley hath declared before the R. Society & M^r Ralphson has affirmed in his Treatise of              And therefore this Book was in MS in those days & M^r Newton had not then forgotten the method of second fluxions.

Ego vero libentur,, saith he, ingentia Newtoni merita oblatis . . . . . . . mihi mature transmisit. Here M^r Leibnitz allowed that before he published his meth{ors} he understood by M^r Newtons Letter of 1676 that M^r Newton could draw tangents without taking away surds & by his Principia published 1687 that he had gone much further in this method. And M^r Newton had told him in those Letters that his Analysis extended to the quadrature of Curves & to inverse Problemes of tangents & others more difficult: & had demonstrated the elements of his calculus in the Principia, And before he began to lay any claim to the Method M^r Newton had told him that had wrote a tract concerning it five years before the writing of those Letters. And a copy of a Letter dated 10 Decem 1672 concerning the method was sent him by M^r Oldenburg as was said above. Hitherto therefore M^r Leibnitz did not pretend to be the first inventor, but only excused himself for not having made a fuller acknowledgment of what M^r Newton had done. He did not begin to put in a claim of first inventor till after the death of D^r Wallis. the last

The occasion of inserting this intimation D^r Wallis signified also to M^r Leibnitz in a Letter dated Decem 1. 1696 & published in the third Volum of the Doctors Works. Cum Præfationis (præfigendæ) postremum folium erat sub prælo ejusque typos jam posuerant Typothetæ; me monuit amicus quidam (harum rerum grarus) qui peregræ fuerant, tum talem methodum in Belgio prædicari tum illam cum Newtoni methodo fluxionum quasi coincidere. Quod fecit ut (translatis typis jam positis) id monitum in ter seruerim

p. 199 And in a Letter to M^r Leibnitz dated 1 Decem. 1696, he gave this Account of it Cum præfationis – – – – intersererem. And in a Letter to M^r Newton dated Apr 10. 1695, he wrote thus about it. I wish

p. 201 And in his Letter of Aug. 27 1676 he denyed that such problemes what M^r Newton had affirmed concerning the reduction of Problemes to such æquations, & therefore was then an absolute stranger to this method

p. 198. l. 14. And again in his answer to M^r Fatio printed in the Acta Eruditorum of May 1700 pag 203 lin 21 he acknowledged the same thing.

p. 198 l 31 Method, & illustrated it with examples of drawing tangents & squaring curves, that it could not be difficult.

p. 200 l 27 — so early; did not so much as affirm that he had it earlier tho D^r Wallis had affirmed that M^r Newton had it ten years before that time or above; allowed

p. 201. l. 13. [And in making this Defence he pretended that in the year 1684 when he published his method of tangents he knew nothing more of M^r] & published it without knowing what M^r Newton had done before him. And in making this Defense

p. 203 l. 5. – Novices, M^r Leibnitz had prepossessed his country men that he was the first inventor

p. 202 l. 3. fluxions for differences & by consequence taken his method from that of M^r Leibnitz. Hitherto he had upon several occasions acknowledged that M^r Newton had found the Method apart.

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M^r Leibnitz had hitherto acknowledged upon several occasions that M^r Newton had invented the Method apart

p. 202 l. 17 complaining that what M^r Keil had written amounted to a calumny & moved that the R. S. &c    l 21 leave to do so, being now sensible that

allowed that M^r Newton had found the method apart & did not deny – – – then that he also had found the method apart

& when he published it knew little or nothing of what M^r Newton had done. And in making this Defen ce he added —

In calling it injustice to question his candour he

He was bound in candour & justice to make good his accusation against M^r Keill that it might not go for a calumny.

He knew that no man can be a witness in his own cause & yet he calls it injustice to question his candor. He knew that the R. Society could not in justice condemn M^r Keil without some proof against him & yet he insists only upon his own candour. & calls it injustice to expect that he should defend it. He tells the R. Society that his friends know how he came by the method, but he should have told the R. Society how he came by it. He saith that the Editors of the Acta Lipsiensia had given every man his due & in the next words that he & his friends had some times shewn that they were willing to beleive that M^r Newton found the method apart. but without depriving M^r Leibnitz of his right to the invention. He cites the opinion of Hugenius in his behalf tho Hugenius never had an opportunity to examin the matter He saith that

He had seen three of M^r Newtons

M^r Newton had given his opinion in this matter in his Letters of Decem. 10 1672 Iune 13 1676 & Octob 24 1676 copies of all which had been sent to M^r Leibnitz. And therefore M^r Leibnitz gave him seven years time to print his method. & refused to tell how he came by the method

$44⁤\frac{3}{4}.$

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For Sir Isaac Newton

Motum Cometarum Leibnitius non attigit, Planetæ et Cometæ ijsdem legibus revolvuntur apud Newtonum. Vortices alicubi harmonice alibi non harmonice moveri Leibnitius pro lubitu finxit: harmonicæ circulationi vortices satellitum Saturni Iovis ac Terræ plane obstant: Motibus Cometarum Vortices Leibnitianæ non favent. Motum harmonicum Planetarum Leibnitius non probavit sed in Tentaminis Artic 6 assumpsit: Newtonus motum harmonicum corporum omnium in centrum immotum attractorum demonstravit. Motum circulationis et motum paracentricum Leibnitius nunc a diversis causis nunc ab eadem deducit: Newtonus utrumque ab eadem causa semper deducit. Vim centrifugam sinui verso anguli circulationis proportionalem esse Leibnitius in Tentaminis Artic 11 assumpsit contra veritatem; et inde deduxit vim centrifugam mobilis harmonice circulantis esse in ratione radiorum reciproca triplicata (Artic 12) newtonus demonstravit hanc vim mobilis in Ellipsi circa focum harmonice circulantis esse in ratione radiorum reciproca duplicata. Leibnitius Motum paracentricum Planetarum a differentia virium centrifugarum et centripetarum oriri finxit (Artic 15, 21, 25) Newtono differentia illa nulla est. Ex errantibus Articulis 12 et 15 Leibnitius deduxit Artic 19: et Propositiones Mathematicæ sic inveniri non solent.

Dicunt aliqui D. Leibnitium in Tentamine suo Propopsitiones 19 et 20 a Propositionibus falsis (nempe 14, 12 & 15) per calculum deduxisse suum Ac talis calculus ad Propositiones quidem prius inventas aptam potuit non autem invenotrem constituere.

Dicunt aliqui falsas esse Tentaminis Propositiones 11, 13 & 15, & D. Leibnitium ab his per calculum suum deduxisse Propositiones 19 et 20 ejusdem Tentaminis. Talis autem calculus ad Propositiones prius inventas aptari quidem potuit, non autem inventorem constituere.

Gravitas in corpore cadente et tempore $\sqrt{\mathrm{F}}\mathrm{G}$ spatium FG cadendo describente generat velocitatem qua duplum illud spatium eodem tempore describi posset id est velocitatem $\frac{2\mathrm{F}\mathrm{G}}{\sqrt{\mathrm{F}}}$ seu $2\sqrt{\mathrm{F}}\mathrm{G}$; at in corpore &c $\frac{\mathrm{H}\mathrm{n}}{\mathrm{C}\mathrm{N}+\mathrm{F}}-\frac{\mathrm{C}\mathrm{F}}{2\mathrm{F}\mathrm{G}}=\frac{\mathrm{a}\mathrm{e}\mathrm{e}}{2\mathrm{n}\mathrm{e}\mathrm{e}+\mathrm{a}{\mathrm{a}}^3\mathrm{o}}=\frac{\mathrm{a}}{2\mathrm{n}}$. $\frac{\mathrm{F}\mathrm{I}}{\mathrm{C}\mathrm{F}}=\frac{\mathrm{a}}{\mathrm{n}}$ Resistentia ad Gravitatem ut $\frac{\mathrm{a}}{2\mathrm{n}}+\frac{\mathrm{a}}{\mathrm{n}}$ ad 1, seu 3a ad 2n. Densitas Medij ut $\frac{3\mathrm{a}}{2\mathrm{n}}$ in $\frac{\mathrm{F}\mathrm{G}}{\mathrm{C}\mathrm{F}\mathrm{q}}$ id est ut a. Et velocitas ut $\frac{\mathrm{C}\mathrm{F}}{\sqrt{\mathrm{F}\mathrm{G}}}$ id est ut $\sqrt{\mathrm{e}}$.

And by those Letters & Papers it appeared to the Committee that M^r Newton had the Method in or before the year 1669 & it did not appear to them that M^r Leibnitz had it before the year 1677.

M^r Leibnitz began his Second Letter to D^r Sloan with these words. Quæ D. Iohannes Keillius nuper ad te scripsit, candorem meum apertius quam ante oppugnant: quem ut ego hac ætate post tot documenta vitæ, Apologia defendam & cum homine docto, sed novo, & parum perito rerum anteactarum cognitore, nec mandatum habente ab eo cujus interest, tanquam pro Tribunali litigem, nemo prudens æquusque probabit. ⨳ < insertion from f 178v > ⨳ Thus he declined to make good his accusation against M^r Keil as he was bound to do for avoyding the censure of calumny & told the R. Society that they would be unjust if they should question his candor, & refused to contend with any body but M^r Newton or those imployed by him, justifying the Acta Lipsiensia against him in the same letter, & pressing him to declare his mind, that is, to retract what he had published in the Introduction to his Principles & to submit to the judgment of the Editors of those Acta. But those editors have sometimes imployed the pen of M^r Leibnitz himself, & sometimes the pens of other men of less note & by the law of all nations no man can be either Iudge or Witness in his own cause, & the Motto of the R. Society is NVLLIVS IN VERBA. It lies upon M^r Leibnitz as well as upon M^r Keill to prove his assertions. It lies upon him &c < text from f 177r resumes > Thus he declined making good his accusation against M^r Keil as he was bound to do for avoyding the censure of calumny & refused to contend with any body but M^r Newton or those imployed by him pressing M^r Newton in the same Letter to declare his mind & justifying the Acta Lipsiensia against him, &, & tells the R. Society that they are unjust if they question his candor, that is, if they do not allow him to be a witness in his own cause contrary to the la{w o}f all nations. But the Motto of the R. Society is NVLLIVS IN VERBA. It lies upon M^r Leibnitz to prove that he had the Differential Method a multo tempore before the year 1677. It lies upon him to prove that he had the series of Gregory before he received it from M^r Oldenburg A.C. 1675 at which time he did not know it to be his own & even before Gregory sent it to Collins A.C. 1671.  < insertion from f 178v >  He has affirmed (in his Letters of 28 Decem 1675 & 27 Aug. 1676) that this series was communicated by him to his friends at Paris in 1673 & was the series whereof he had sometimes writ to M^r Oldenburg, before December 1675 viz^t in Iuly & October A.C 1674 It lies upon him to prove that the series which he then wrote of to M^r Oldenburg was this series & not the series for finding the Arc by the sine. In his Letter of October 1674 he has affirmed that he had a method of finding the Arc by the sine in a series of rational numbers whether the proportion of the Arc to the whole circumference was known or not, & that the same method gave him a series for the whole circumference & in his Letter of 12 May 1676 he desired M^r Oldenburg to procure from M^r Collins the method of finding this Series: it lies upon him to prove that he had the method in or before the year 1674. At his request M^r Newton in his Letter of 13 {Iu} sent him his method of finding that & such like series direct & inverse & illustrated his method with examples of series & upon the receipt of that Letter M^r Leibnitz pretended that he had found some of those series before it < text from f 177r resumes > It lies upon him to prove that he had a method of finding the series for the Arc by the Sine in the year 1674. It lies upon him to prove that before the receipt of M^r Newtons Letter of 13 Iune 1676 he had some of the series sent him therein with the method of finding them. It lies upon him to prove that he had found out one of M^r Newtons inverse methods of series so early as to have forgot it before the year 1676. He pretends to be the inventor of the method of series by assuming the terms thereof & determining them gradually by the conditions of the Problem: it lies upon him to prove that he had this method before M^r Newton sent it to him in cyphers in his Letter of Octob. 24 1676. A year or two after M^r Newton had published his Principia Philosophiæ M^r Leibnitz published three papers relating to the principal Propositions in that Book pretending that he had found them before the publishing of that book: it lies upon him to prove that he had found them before. He pretended that the Propositions concerning the motion of Bodies in resisting Mediums were for the most part found out by him twelve years before while he was yet at Paris, that is, before he had the Differential method: it lies upon him to prove it. It lies upon him also to prove that by an erroneous Demonstration he could find out the Proposition that a Body revolving in an Ellipsis & with a radius drawn to the lower focus describing equal areas in equal times, is attracted towards that focus by a force which is reciprocally as the square of that radius. ♀ < insertion from f 178v >  For making himself a coinventor of the method of series He has laid claim to it by a Proposition for transmuting any one figure into anothers equal to it. It lies upon him to prove that this method of Series is either general or useful or that any thing can be done by it which cannot be better done without it or that he had it before the year 1675 or that he has done any thing of moment by it which was not done before by other æquations where the indices of dignities were fract or surd: M^r Leibnitz in his reply proposed æquations where the indices of dignities were indeterminate & calling these æquations exponential, & recons then the most perfect of all others. It lies upon him to shew that any use hath been made or can be made of them. < text from f 177r resumes > For making M^r Newtons method of fluxions his own He has represented that M^r Newton at first used the letter o in the vulgar manner for the given increment of x, which destroys the advantages of the differential method, but after the writing of his Principles changed o into $\stackrel{.}{\mathrm{x}}$ substituting $\stackrel{.}{\mathrm{x}}$ for dx: it lies upon him to prove that M^r Newton ever changed o into $\stackrel{.}{\mathrm{x}}$ or used $\stackrel{.}{\mathrm{x}}$ for dx or left off the use of the letter o.

M^r Newton used the letter o in his Analysis & in his Principia Philosophiæ & in his book of Quadratures & still uses it in the very same sense as at first. In his book of Quadratures he used it in conjunction with the symbol $\stackrel{.}{\mathrm{x}}$ & therefore did not use that symbol in ints room. These symbols o & $\stackrel{.}{\mathrm{x}}$ are put for things of a different kind. The one is a moment the other a fluxion or velocity. as has been explained above. When the letter x is put for a quantity which flows uniformly, the symbol $\stackrel{.}{\mathrm{x}}$ is an unit & the letter o a moment & the symbols $\stackrel{.}{\mathrm{x}}\mathrm{o}$ & dx signify the same moment; but $\stackrel{.}{\mathrm{x}}$ without the coefficient o either exprest or understood to make it infinitely little doth not signify a moment. Prickt letters never signify moments, unless when they are multiplied by the moment o either exprest or understood to make them infinitely little, & then the rectangles are put for moments.

^[1]M^r Newton doth not place his method in forms of symbols nor confine himself to any particular sort of symbols for fluents or fluxions. Where he puts the Areas of Curves for fluents he puts the ordinates for fluxions & denotes the fluxions by the symbols of the Ordinates. When he puts lines for fluents, he puts any symbols for the velocities of the points which describe the lines, that is for the first fluxions, & any other symbols for the increases of those velocities, that is, for the second fluxions: instances of which are frequently to be met with in his Principia Philosophiæ: And where he puts the letters x, y, z for fluents, he denotes their fluxions either by other letters as p, q, r or AB, CD, EF, or by the same letters in other forms as X, Y, Z or $\stackrel{.}{\mathrm{x}}$, $\stackrel{.}{\mathrm{y}}$, $\stackrel{.}{\mathrm{z}}$, And this is eviden{t b}y his book of Quadratures where he uses for fluxions prickt letters in the first {Pr}oposition, the Ordintes of Curves in the last Proposition & other symbols in solving several Problems in the Introduction. M^r Leibnitz has no symbols of fluxions in his method, & therefore all M^r Newton's symbols of fluxions are the oldest in the kind. M^r Leibnitz began to use the symbols of moments dx & dy in the year 1677. M^r Newton representes moments by the rectangles under the fluxions & the moment o & has done so ever since the writing of his Analysis, which was at least 45 years ago. M^r Leibnitz has used the symbols $\int \mathrm{x}$, $\int \mathrm{y}$, $\int \mathrm{z}$ for the summs of the ordinates ever since the year 1686: M^r Newton represented the same thing by inscribing the Ordinate in a square or rectangle in his Analysis written 17 years before or above. All M^r Newtons symbols are the oldest in their several kinds by many years.

And whereas M^r Leibnitz has represented that the use of the letter o is vulgar & destroys the advantages of the Differential method: on the contrary the method of fluxions as used by M^r Newton has the advantage of the differential in all respects. It is more elegant because in his calculus there is but one infinitely small quantity signified by the Letter o. It is more natural & gemetrical because founded upon the primæ quantitatum nascentium rationes which have a being in Geometry whilst indivisibles upon which the Differential method is founded have no being either in Geometry or in Nature. There are rationes primæ quantitatum nascentium, but not quantitates primæ nascentes. Nature generates quantities by continual flux or increase, & the ancient Geometers admitted such a generation of areas & solids when they drew one line into another by local motion to generate an area & the area into a line to generate a solid. But the summing up of indivisibles to generate an area or solid was never yet admitted into Geometry. M^r Newtons method is also fo a greater extent being adapted either to the ready finding out of a Proposition or to the demonstrating it: M^r Leibnitz's is only for finding it out. When the work succeeds not in finite equations M^r Newton has recourse to converging series & thereby his method becomes incomparably more universal then that of M^r Leibnitz which is confined to finite equations. And when the law of the fluxions is not known but the fluxions are had only in a few particular cases, M^r Newton finds that law quamproxime by drawing a Curve line through any number of given points, & thence deduces the solution of the Probleme. And to this degree of perfection M^r Newton had brought his method sometime before the year 1676, as appears by his Analysis & his Letters of 10 Decem 1672, 13 Iune 1676 & 24 Octob 1676. And when he wrote his Principia Philosophiæ, he had recourse upon all occasions to this method & most frequently to that part of it which is conteined in his book of Quadratures And because by the law of the Ancients Propositions invented by Analysis were not admitted into Geometry till they were demonstrated synthetically <178r> the Propositions which he found out by his Analysis, he composed that the systeme of the heavens might be founded upon good Geometry. And this makes it now difficult for unskilful men to see the Analysis by which those Propositions were found out. But this Book compounded with the Tentamen de motuum cœlestium causis will prove a lasting {d}emonstration that M^r Newton when he wrote it was master of an Analysis which carried him through many such difficulties as M^r Leibnitz would have stuck at.

Phil. Trans. p. 206.

resist$=\mathrm{A}{\mathrm{B}}^{\mathrm{q}}+\mathrm{C}{\mathrm{B}}^{\mathrm{q}}$. Grav ut AC.

In the 19^th Article of his Tentamen de Motuum cœlestium causis he has pretended to the invention of M^r Newtons Proposition that a body revolving in an Ellipsis & with a radius drawn to the

His differential method of Tangents published at Leip{iu} in the year 1684 is nothing else then D^r Barrows differential method improved so as not to stick at fractions & surds & disguised with new notation & a new name without acknowledging any part of the method to be D^rBarrow's, [It lies upon him in point of justice to tell the word how much of the Differential method is D^r Barrows.] or that the Author had received any light into it from M^r Newtons Letters. It lies upon him to prove that what he then published was intirely his own without being obliged to either of them for any thing

All the pretensions above mentioned tend to diminish the right of invention due to D^r Barrow M^r Greg. & M^r Newton & to let in M^r Leibnitz for a share. M^r Leibnitz has hitherto claimed this share without any proof. His candor cannot make him a witness for himself. By the laws of all nations he must prove his claim or quit it, & this he ought to do without delay in point of candor & justice.

M^r Newton at his request sent him two inverse methods of Series, A.C. 1676 & he understood then with difficulty & then wrote back that he had found one of them so long before as to have forgot it, as he perceived by his old papers: it lies upon him to prove that he had found it & forgot it before he wrote for it, that is before 12 May 1676. He pretends to be the first inventor of that method of series which consists in assuming the terms of a series & determining them &c

In his letter of 27 Aug. 1676 he denyed that M^r Newtons method of series extended to invers Problems of tangents & when M^r Newton replied that it did extend to them he answered that M^r Newton spake of a solution by his method of series whereas he had desired a Geometric solution : this implies that he knew before that it might be solved by the method of series it lies upon him to prove that he knew it.

It seems therefore that as he learnt the differential method by means of M^r Newtons aforesaid three Letters compared with D^r Barrows method of Tangents: so ten years after when the Principia Philosophiæ came abroad he improved his knowledge in these matters by trying to extend this calculus to the principal Propositions in that Book. & by this means composed the said three Tracts. For the Propositions conteined in them (errors & trifles excepted) are M^r Newtons, being published by him in other forms of words before. And yet M^r Leibnitz published them as invented by himself long before. came abroad. For in the end of the first For In the end of the first Tract he represents that he invented them all before M^r Newton's Principia Philosophiæ came abroad & some of them before he left Paris, that is before the end of the year 1676. And the second Tract

Here he allows that the fundamenta Geometrica in quibus maxima consistebat difficultas were now laid by him in what he published in this Tract, but he should have acknowledged that they were more fully laid before by what Newton published in his Principles. Here he allows that he had by this Tract opened vias quasdam novas satis antea impeditas, but he should have acknowledged that M^r Newton had opened them more fully before by his Principles. Here he allows that all that he had now published answered to the calculus summarum & differentiarum, tho it was expressed communibus quoad licuit verbis, in vulgar words without calculations but in candor he should also have acknowledged that what M^r Newton had published in his Principles answered to the calculus fluxionum tho writ communibus quod licuit verbis . It remains therefore in point of candor that he now do justice to M^r Newton by acknowledging that he was the first who laid the fundamenta Geometriæ in quibus maxima consistebat difficultas & opened vias quasdam novas satis antea impeditas, & that the composition of his Principles tho written communibus quoad licuit verbis, answered in all things to the calculus infinitesimalis.

Here he pretends that the fundamenta Geometrica in quibus maxima consistebat difficultas were first laid by him self in this Tract & that he himself had in this other Tract opened vias quasdam novas satis antea impeditas. And yet M^r Newton's Principia Philosophiæ came abroad almost two years before & gave occasion to the writing of this Tract, & conteins all these Principles & all these new ways. And therefore M^r Leibnitz when he published that Tract knew & ought to have acknoledged that M^r Newton was the first who laid the fundamenta Geometrica in quibus maxima consistebat difficultas & opened the vias novas satis antea impeditas. Answer to M^r Fatio he allows if he objects that the Principles were written not Analytically but Communibus quoad licuit verbis, he tells us the same thing of his own Tract & yet affirms that in all things it answers to the new Analysis] He makes no objection against the Book being written not in the way of Analysis but communibus quoad licuit verbis. For his own Tract was written in the same manner & yet in all things answered to the new Analysis. In his Answer to M^r Fatio he allows that No man before M^r Newton specimine publice dato [hanc Methodum] se habere probavit, [proved that he had this method by a specimen thereof made publick. And if he knew that the Principia Philosophiæ were a specimen of this method & that M^r Newton was the first who had published such specimen & thereby proved that had the method, he ought in candor & justice to have acknowledged this when he published those Tracts & ought still to acknowledge it.] For silencing D^r Wallis & M^r Fatio he was forced to allow that M^r Newton was the first who had Invented the the infinitesimal method & the first who had proved by a publick specimen (meaning the Book of Princip) that he had this method: at other times he calls the infinitesimal method his own & at length begins to pretend that M^r Newton had the method from him. What he acknowledged in his Answers to D^r Wallis & M^r Fatio, & in his Letter to M^r Newton dated March $\frac{7}{17}$ 1693 he ought in candor & justice still to acknowledge upon all occasions.

when he wrote those Tracts therefore he was but a learner, & improved his knowledge by M^r Newtons Principles & this he ought in candor to acknowledge.

He pretended indeed to D^r Wallis that he had added some things to the infinitesimal Analysis of M^r Newton, & particularly Differential Æquations: but by their Letters written in the year 1676 its abundantly manifest that M^r Leibnitz did not then know how to reduce Problems to differential Equations & M^r Newton did then know how to do it, tho he did not use the differential characters which M^r Leibnits invented afterwards.

M^r Newton in his Letter of Octob 24 1676, wrote – – – – wrote his said Letter.

D^r Wallis interpoled the indices of Dignities 0, 1, 2, 3, 4, 5 &c by this series $0.\frac{1}{2}.1.1⁤\frac{1}{2}.2.2⁤\frac{1}{2}.3.3⁤\frac{1}{2}\mathrm{\&c}$. And M^r Newton introduced into his Analytical computations the fract, surd, indefinite & negative indices of Dignities; & in – – – – – its usefulness to the world

It lies upon M^r Leibnitz also to make a publick acknowlegment of his Receipt of M^r Oldenburghs letter of Apr. 15. 1675 – – – – – – series as his own

It lies upon him also to make – – – – Method of fluxions.

It lies upon him also to tell the World – – – – from M^r Collins

It lies upon him also to tell the world what was the Method – – – from M^r newton.

And whereas M^r Newton sent him at his own request – – – – – – – Letter of 10 Decem. 1672 concerning it.

The Editors of the Acta Eruditorum for Iune 1696 – – – – – beg his pardon.

When M^r Leibnitz first published – – – – of candor & justice.

It lies upon him also to satisfy the world why in his Answers to D^r Wallis & M^r Fatio – – – Guilt of that crime

It has been said that the Royal Society – – – – by Persons of Note. And in the mean time – – – – – expulsion to defame them.

The Philosophy of M^r Newton – – – – part of Europe against him.

It is true – – – – rail without arguing.

21 M^r Leibnitz in his Letter to D^r Sloan dated 29 Decem 1711 returned this answer to D^r Keill. Quæ D. Ioannes Keillius nuper ad te scripsit, candorem meum apertius quam ante oppugnant: quem ut Ego hac ætate, post tot documenta vitæ, Apologia defendam, & cum homine docto, sed novo, et parum perito rerum ante octarum cognitore, nec mandatum habente ab eo cujus interest, tanquam pro Tribunali litigem, nemo prudens æquusque probabit. &c. Thus M^r Leibnitz endeavoured to strip M^r Newton of his friends, & engage him in person & then to set his own friends upon him, as appeared by the consequence. D^r Wallis & M^r Fatio had writen in behalf of M^r Newton, but they did it voluntarily without being desired to write & M^r Leibnits could not fall out with him for that D^r Keill did the like by two Letters & in such a manner as made M^r Leibnitz cry out both times: But the Doctor wanted authority from M^r Newton, & therefore M^r Leibnits would not engage with him, but challenged M^r Newton himself, & when the Commercium Epistolicum came abroad withdrew, pretending that he had not seen that Book, & employed (or pretended to employ) a very great & candid Mathematician to examin it , & another nameless Here to assault M^r Newton with hard words. And by these Arts & the tot documenta vitæ above described you may judge whether any prudent & just man will place him above all humane judicature & allow him to be both witness & judge in his owne cause contrary to the prudence justice & laws of all nations, or not rather expect that he should defend his candor.

Pag. 208. lin. 8. after the words [will vanish] add this sentence. But the like Objection made against M^r Leibnitz by D^r Keil, namely that before the publishing of the Principia Philosophiæ M^r Leibnitz was unskilled in the method of second & third differences, & has not yet sufficiently learnt it wants an Answer By the Letters of M^r Newton & the method of Tangents of D^r Barrow he fell into the Calculus differentialis, & by the Principia Philosophiæ received further light into it, but is not yet a Master therein.

In the Commercium – – – – without the assistance of M^r Newton. 3 2 By his errors in this Proposition D^r Keil has shewed that when he wrote thise three Tracts he did not well understand the ways of working in second Differences. And this is further manifest by his 10^th 11^th & 12^th Propositions of this third Tract. For these he lays down as the foundation of his infinitesimal Analysis about centrifugal forces, & proposes the first of them with relation to the center of curvity but uses it in the two next with relation to the center of circulation. And by confounding these two centers with one another in the fundamental Propositions of this calculus, it is manifest that he was hitherto unacquainted with the manner of arguing about centrifugal & centripetal forces by the infinitesimal method, & was now attempting to learn it. by trying to adapt a calculus to M^r Newtons Propositions And this thing is further confirmed by the sixt Article of the second of these his three Tracts. For that Article is an erroneous Proposition & the error arises from his not knowing how to argue well about second & third differences. <180r> & further described in the second Lemma of the second book of these Principles & in the Scholium upon that Lemma, & illustrated with an example in demonstrating the 14^th Proposition of that book & with another example in solving the 31^th Probleme of the first Book & with another in the Scholium upon the 103^d Proposition of the same Book. And since he has not yet acknowledged these things but on the contrary has of late began to deny them: it lyes upon him in point of candor & justice to make an express & full acknowledgment thereof.

4

15 The Editors of the Acta Eruditorum, in Iune 1696 in giving account of the two first Volumes of the mathematical works of D^r Wallis, wrote thus in the style of M^r Leibnitz. Cæterum ipse Newtonus non minus candore quam præclaris in rem mathematicam meritis insignis, publice et privatim agnovit, Leibnitium tum cum (interveniente celeberrimo Viro Henrico Oldenburgio Bremensi, Societatis Regiæ Anglicanæ tunc Secretario) inter ipsos (ejusdem jam tum Societatis Socios) Commercium intercederet, id est jam fere ante annos viginti et amplius Calculum suum differentialem, seriesque infinitas, et pro ijs quoque Methodos generales habuisse; quod Wallisius, in Præfatione operum, factæ inter eos communicationis mentionem faciens, præterijt, quoniam de eo fortasse non satis ipsi constabat. Cæterum Differentiarum consideratio Leibnitiana cujus mentionem facit Wallisius (ne quis scilicet, ut ipse ait, causaretur de Calculo Differentiali nihil ab ipso dictum fuisse) meditationes aperuit, quæ aliunde non æque nascebantur. &c. ‡ < insertion from the bottom of the page > ‡ M^r Leibnitz had therefore seen the Preface of D^r Wallis where M^r Newton is said to have explained to M^r Leibnitz) in the year 1676 the method of fluxions found by him ten years before or above & in answer thereunto, pretended only that M^r Newton had acknowledged & D^r Wallis should have acknowledget had he known it that he (M^r Leibnitz) had the differential method in the year 1676 or before: whereas M^r Newton acknowledged nothing more then what he still acknowledges, viz^t that M^r Leibnitz had it when he wrote his Letter dated 21 Iune 1677. And this confirms me in the a suspicion that M^r Leibnitz in that Letter used the words above mentioned Et jam a multo tempore &c, with a designe to make the Germans beleive that he had found the differential method long before the writing of that Letter, & by consequence before the writing of the Letters which passed between him & M^r Newton in the year 1676, & that he has been ever since carrying on a designe to make this be beleived And yet its very certain by what we shewed above that he himself knew that he did not invent it before his return from France through England & Holland into Germany in the end of the year 1676. In his Answer to M^r Fatio printed in Acta Eruditorum anni 1700 pag 203 he writes: Ipse [Newtonus] scit unus omnium optime, satisque indicavit publice Cum sua Mathematica Naturæ Principia publicaret Anno 1687, nova quædam inventa Geometrica quæ ipsi communia mecum fuere, NEVTRVM LVCI AB ALTERO ACCEPTÆ, sed meditationibus quemque suis debere, et a me jam decennio ante [i.e. anno 1677] exposita fuisse. And yet M^r Newton in his Letter of 24 Octob. 1676 did represent that the ground of his method was suff < insertion from f 179v > iciently obvious by what he had written, & < text from the bottom of the page resumes > never did allow that M^r Leibnitz found out that Method without receiving light into it from the three Letters above mentioned sent to him by M^r Oldenburg, in the years 1676 & 1677. M^r Newton indeed out of an aversness from disputes forbore to complain & M^r Leibnitz pretended not only that he found the method apart without receiving any light into it from M^r Newton & even before he received any of M^r Newton's Letters; but also that M^r Newton himself has allowed all this. And for abusing M^r Newtons candor in this manner, he ought in justice to beg his pardon. < text from f 180r resumes > ...

$\begin{array}{c}\phantom{-}4.10\phantom{\mathrm{.}}0\\ -\phantom{4.}11.3.\\ +\phantom{4.}1.1⁤\frac{1}{2}\end{array}$ and the result, not related in any clear visual way to the sum: $3.19.10⁤\frac{1}{2}$

Philos. Tra{nsacti} p. 210 after lin. 20 add

9 p. 215 And whereas M^r Newton sent him at his own request a method of Regression, which upon the first reading he did not known to be his own, nor understood it; but so soon as he understood it, he claimed it as his own by pretending that he had found it long before & had forgot it, as he perceived by his old papers: it lies upon him in point of candor & justice, either to prove that he was the first inventor of this method, or to renounce his claim to it for preventing future disputes

10 p. 215 M^r Leibnitz in his Letter to M^r Oldenburg dated 3 Feb. 167$\frac{2}{3}$, claimed a right to a certain property of a Series of numbers natural triangular pryamidal, triangulo-triangular &c: & to make it his own represented that he wondred that M^r Paschal in his book entituled Triangulum Arithmeticum should omit it. That book was published in the year 1665 & conteins this property of the series & M^r Leibnitz has not yet done him the justice to renounce his claim to this property & acknowledge M^r Paschal the first inventor.

11 p     He is also to renounce all right to the Differential Method of Mouton as second Inventor. For second Inventors have no right. The sole right is in the first Inventor untill another finds out the same thing apart. In which case to take away the right of the first Inventor & divide it between him & that other would be an Act of injustice.

12 p. 215 In his Letter to D^r Sloan dated 29 Decemb. 1711, he has told us that his friends know how he came by the Differential Method. It lies upon him in point of candor openly & plainly & without further hesitation to satisfy the world how he came by it.

13 p. 216 In the same Letter he has told us that he had this method above nine years before he published it, & it follows from thence that he had it in the year 1675 or before. And yet its certain that he had it not when he wrote his Letter to M^r Oldenburgh dated 27 Aug. 1676 wherein he affirmed that Problems of the Inverse Method of Tangents & many others could not be reduced to infinite series nor to Equations or Quadratures. It lies upon him therefore in point of candor to tell us what he means by pretending to have found the Method before he had found it.

14 p. 216We have shewed that M^r Leibnitz in the end of the year 1676 in returning home from France through England & Holland was meditating how to improve the Method of Slusius for Tangents & extend it to all sorts of problems, & for this end proposed the making of a general Table of Tangents, & therefore had not yet found out the true improvement: but about half a year after, when he was newly fallen upon the true improvement, wrote back; Clarissimi Slusij methodum tangentium nondum esse absolutam Celeberrimo Newtono assentior. Et jam a multo tempore rem tangentium generalius tractavi, scilicet per differentias Ordinatarum. Which is as much as to say, that he had this improvement long before those days. It lies upon him in point of candor to make us understand that he pretended to this antiquity of his invention with some other designe then to rival & supplant M^r Newton & to make us beleive that he had the Differential Method before M^r Newton explained it to him by his <181v> Letters of 13 Iune & 24 Octob 1676, & before M^r Oldenburgh sent him a copy of M^r Newton's Letter of 10 Decem. 1672 concerning it.

16. p. 219 220 When M^r Leibnitz first published his Differential method he ought in candor to have acknowledged what he knew of M^r Newtons method for doing the same things. [All that he then explained of his own method, was how to draw Tangents & determin maxima & minima without taking away fractions or surds. He certainly knew that M^r Newton's method would do all this & therefore ought in candor to have acknowledged it. After he had thus far explained his own method, he added, that what he had laid down were the Principles of a much sublimer Geometry reaching to the most difficult & valuable Problems which were scarce to be resolved without the differential calculus AVT SIMILI or another like it. What he meant by the words AVT SIMILI was impossible for the Germans to understand without an interpreter. He ought to have done M^r Newton justice in plain intelligible language & told his Readers that the Method which he there published extended to such difficult Problemes as were not to be resolved without his Calculus Differentialis or another calculus invented by M^r Newton some notice of which he had received by his correspondence with M^r Oldenburg & which by reason of performances he took to be like his own & which was invented some years before the year 1676. But on the contrary in his Answer to M^r Fatio published in the Acta Eruditorum for May 1700, he denyes all this. His words are: Certe cum elementa Calculi mea edidi anno 1684, ne constabat quidem mihi aliud de inventis ejus [sc. Newtoni] in hoc genere quam quod ipse olim significaverat in literis, posse se tangentes invenire non sublatis irrationalibus, quod Hugenius quoque se posse mihi significavit postea, etsi cæterorum ejus calculi adhuc expers. Sed majora multo consecutum Newtonum, viso demum libro Principiorum ejus, satis intellexi. And he is since gone back even from what he acknowledged here, & tells us now that the book of Principles is written in the manner of the Ancients, & hath nothing of the new Analysis in it nor makes it appear that M^r Newton understood these methods when he wrote it. And yet that Book is full of such Problemes as M^r Leibnitz himself, when he first published his method, called difficillima et pulcherrima etiam mistæ Matheseos Problemata quæ sine Calculo Differentiali AVT SIMILI non temere quisquam pari facilitate tractabit. And M^r Leibnitz himself in his Letter of 21 Iune 1677 after he had explained his method & shewed how it readily gave the method of Tangents of Slusius & proceeded without taking away surds, declared himself of opinion that since M^r Netons method did the same things it was of the same kind; especially since both the methods also facilitated Quadratures. Arbitror, saith he, quæ celare voluit Newtonus de Tangentibus ducendis ab his non abludere. Quod addit, ex hoc eodem fundamento Quadraturas quoque reddi faciliores, me in sententia hac confirmat, nimirum semper figuræ illæ sunt quadrabiles quæ sunt ad Æquationem differentialem. And M^r Newton had told him further in his three Letters above mentioned that his Analysis (which proceeded in equations both finite & infinite) extended to problems about the lengths & curvatures of curves & to inverse Problemes of Tangents & to almost all sorts of Problemes & represented it so general that M^r Leibnitz himself (in his letter of Iune 21, 1676) exprest his disbeleif of it. It lies upon him therefore in candor & justice to acknowledge this & to give an account why he was silent about all this when he first published <182r> the Differential method. For it was not enough to mention a methodus SIMILIS without saying whose it was & of what extent & antiquity (according to the notices he had received from England,) & acknowledging that his own method was not so ancient. Nothing elss than this could fully deserve the name of Candor & Iustice.

Notes & emendations upon the Account of the Commercium Epistolicum published in the Transactions for Ian. & Feb. 171$\frac{4}{5}$.

Pag. 175. lin.       Add. The Logarithmotechnia came out in September 1662 & therefore by the testimony of D^r Barrow the Analysis per series was invented & generally applied before Septem. 1666. The Exercitationes Geometricæ came out towards the end of the yeare 1668 & D^r Barrow sent the said Compendium to M^r Collins in the Iuly following, as appears &c.

Pag. 176. lin 20 After the word conferatur add this Paragraph.

In the first of these two citations the words Quadratura vel Area dictæ figuræ, accurata si possibile sit, sin minus infiniè vero propinqua relate to the words of the Analysis: cujus [Analyseos] beneficio Curvarum areæ & longitudines &c (id modo fiat) exacte et Geometrice determinantur. Sed ista narrandi non est locus. How this {is} done is explained in the first six Propositions of the Book of Quadratures. And without the method there explained it is not to be done. And therefore by the testimonies of D^r Barrow & M^r Collins Newton some years before Iuly 1669 (that is two years before the Logarithmotechnia came abroad, or in the year 1666, as was exprest above) had improved this method of Analysis so far at least as it is described in the first or six Propositions of the said book of Quadratures.

Pag. 194. lin. 5. Add this Paragraph.

This was the furst Rule made publick finding second third & fourth differences & is the best.

Rixarum scopus fuit ut

Ad Lectorem

The ancients had their method of Analysis but admitted nothing into Geometry before it was demonstrated by Composition. The moderns are intent only upon Analysis, & admit analytical inventions into Geometry before they are demonstrated by Composition. Synthetical Demonstrations are easier to be read render Propositions more certain, & convey them better to posterity. For the symbols used in Analysis are apt to be changed from time to time. M^r Newton for these reasons & that the Propositions in his Principia Philosophiæ might be fit to be received into Geometry, after he had invented them by Analysis, demonstrated them by composition. But the Analysis is so conspicuous through the composition that the Marquess de l'Hospital said that this Book was almost wholy of the Infinitesimal calculus, & M^r Leibnitz in a Letter to M^r Newton dated 7 Mart. 1693 wrote thus of it. Mirifice ampliaveras Geometriam tuis seriebus, sed edito Principiorum opere ostendisti patere tibi etiam quæ Analysi receptæ non subsunt. Conatus sum Ego quoque notis commodis adhibitis quæ differentias & summas exhibent, Geometriam illam quam transcendentem appello, Analysi quodam modo subjicere, nec res male successit.

The Elements of this method are conteined in the first Proposition of the Book of Quadratures & also in the second Lemma of the second book of Principles, with the Scholium thereupon. The Proposition is in the Scholium & the solution therof in the Lemma. The same Elements are also conteined in M^r Newton's Letter to M^r Collins dated 10 Decem 1672: a copy of which was sent to M^r Leibnitz by M^r Oldenburg among the extracts of Gregorius Letters 26 Iune 1676. For there M^r Newton described a method of Tangents which he conjectured to be the same – – – so is the Ordinate $\mathrm{y}=\mathrm{B}\mathrm{C}$ to the subtangent $-\mathrm{B}\mathrm{D}$, & so is the increase or fluxion of the Ordinate $\stackrel{.}{\mathrm{y}}$ to the fluxion of the Abscissa $-\stackrel{.}{\mathrm{x}}$. And by multiplying the extreames & means you have the Equation $3\mathrm{x}\stackrel{.}{\mathrm{x}}-4\mathrm{x}\mathrm{y}\stackrel{.}{\mathrm{x}}+2\mathrm{b}\mathrm{x}\stackrel{.}{\mathrm{x}}-\mathrm{b}\mathrm{b}\stackrel{.}{\mathrm{x}}=\mathrm{+2}\mathrm{x}\mathrm{x}\stackrel{.}{\mathrm{y}}-2\mathrm{b}\mathrm{y}\stackrel{.}{\mathrm{y}}++3\mathrm{y}\mathrm{y}\stackrel{.}{\mathrm{y}}$, or $3\mathrm{x}\mathrm{x}\stackrel{.}{\mathrm{x}}-4\mathrm{x}\mathrm{y}\stackrel{.}{\mathrm{x}}+2\mathrm{b}\mathrm{x}\stackrel{.}{\mathrm{x}}-\mathrm{b}\mathrm{b}\stackrel{.}{\mathrm{x}}-2\mathrm{x}\mathrm{x}\stackrel{.}{\mathrm{y}}+2\mathrm{b}\mathrm{y}\stackrel{.}{\mathrm{y}}-3\mathrm{y}\mathrm{y}\stackrel{.}{\mathrm{y}}=0$. And this equation is the same with that which is produced by the first Proposition of the Book of Quadratures when there are but two unknown Quantities proposed. If there be more then two the same operation applied to them all gives the solution of that Proposition.

In M^r Newtons Letters of 13 Iune & 24 October 1676 there was a further description of this method which gave occasion to D^r Wallis in the Preface to the two first volumes of his works, to write Quam [Methodum] ego descripsi (Algebræ cap. 91 &c præsertim cap 95) ex binis Newtoni Literis (aut earum alteris) Iunij 13 &c Octob. 24 1676 ad Oldenburgium datis, cum Leibnitio tum communicandis (ijsdem fere verbis, saltem leviter mutatis quæ in illis literis habentur;) ubi methodum hanc Leibnitio exponit, tum ante decem annos nedum plures ab ipso excogitatam.

By M^r Newtons Letters M^r Leibnits was put upon thinking how to improve the Method of Tangents of Slusius & in his way from England into Germany wrote to M^r Oldenburg 18 Novem 1676 st. v. that it might be done by a Table of Tangents, And that he had discoursed with Hudden at Amsterdam & that the method of Tangents of Slusius was known to him long ago, & was more ample then that which Slusius had published. And we understand since that Hudens method did not stop at surds.

was seen by M^r Leibnitz in London & a copy thereof came to his hands at Hanover in the beginning of Spring following. Pag 194. lin. 6.                So in his Analysis he uses the symbol $\boxed{\frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}}$ in the same sense in which M^r Leibnitz uses the symbols $\int \frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}$. pag. 25. lin 16.

M^r Newtons Rule for finding second third & fourth Differences published by D^r Wallis in the second Volume of his works which came abroad in Spring 1693, was the first Rule made public for doing this. And it doth not appear that M^r Leibnitz had any Rule for second & third differences before. p. 208. lin. 8.

Philosoph. Transact. for Ian & Feb. 171$\frac{4}{5}$

Pag 174. lin. 21 for ten write fifteen.

P. 176. lin 21 for being write from a copy

Ib. l. 24. After of M^r Newton add: The impression was finished in December 1710.

P. 184. lin 3. dele and M^r Collins

Pag 191 lin 22 After year 1670 add From this Method & his own M^r Gregory Deduced a method of Tangents without computation, & notified it to M^r Collins in a Letter dated 5 Novem 1670.

Ib. between lin 23 & 24 insert, conjecturing that it was the same with that of Gregory & Slusius.

Pag 194 lin 6 for [came to the hands of M^r Leibnitz in the end of winter or] write was seen by M^r Leibnitz in London in November, & a copy thereof came to his hands at Hanover in the beginning of Spring following.

Pag 195 lin 23. after 1696 add none of them being printed before the year 1699.

Pag 198 lin penult.

Pag 199 lin 14 after [he gave the] add following, & put an asterisk after the word Letter.

Pag. 201 lin 6 for his write the

Ib. lin 28 after he added insert concerning a branch of this method by which M^r Newton A.C. 1686 found the solidum minimæ resistentiæ: Quam

Pag. 203 lin. 4 after Acta Leipsica insert had not in the least detracted from anybody but

Ib. lin. 9 for Novice write new man

Pag. 205 lin 16. after Rectangle add For there he uses the symbol $\boxed{\frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}}$ in the same sense in which M^r Leibnitz uses the symbol $\int \frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}$.

Pag. 207 lin 33. after & 396, add. This was the first Rule made publick for finding second third & fourth differences & is the best.

Pag. 209 lin. 21. after compared insert with Gregories Letter of 5 Novem. 1670 &

Pag. 212 lin 199. for many years after write in the Acta Eruditorum for Aug. 1693.

Ib. lin 21. after thereof add, & calling it Methodus universalissima pro seriebus.

Pag. 224. lin 34. after Arguments for insert the being of

Of the Notes.

Annotationes quæ in Epistolas scriptæ sunt nullius sunt authoritatis nisi quam ab ipsis epistolis derivant. De his facta est quærela quasi malignæ essent sed nondum ostensum est quod aliquid contra D. Leibnitz in ipsis sit quod non sit in Epistolis.

Commercium hocce Epistolicum lucem vidit sub finem anni 1712, et subinde D. Leibnitius, cum eidem respondere non posset, prætendit per biennium se librum non vidisse, sed ad judicium primarij Mathematici provocasse, cum ipse per occupationes diversas rem tunc discutere non satis posset. Et sententia hujus Mathematici 7 Iunij 1713 datam in charta scurrili die 29 Iulij data descripsit, et per Europam sparg{a} curavit, sine nomine vel Mathematici vel Impressoris vel urbis in qua impressa fuit. Et anno proximo per Epistola m ad D. Chamberlayne 25 Aug. 1714 datam, postulavit ut Societas Regia Epistolas nondum editas ad ipsum mitterent. Nam, cum Hanoveram inquit rediero possum etiam in lucem emittere Commercium aliud Epistolicum quod Historiæ Litterariæ inservire possit. Et litteras quæ contra me allegari possunt non minus publici juris faciam quam quæ pro me faciunt. Hæc Leibnitius. Sed ficta erat accusatio cum totum inter ipsum et oldenburgum Commercium continua serie impressum fuit, ut jam dictum est. præter Epistolas duas quæ non extant. Sub finem anni proximi cum D. Leibnitius probare vellet quod Concessus a R. Societate constitutus omnia omissent quæ contra Newtonum facerent, scripsit is in Epistola sua prima ad Abbatem de Comitibus, quod in secundo ejus in Angliam itinere, Collinius ostendit ipsi – – – ad ipsum Leibnitium 8 Decem 1674.

In eadem ad Abbatem de Comitibus Epistola, scripsit D. Leibnitius quod $\stackrel{..}{\mathrm{y}}$ qui contra ipsum scripsissent (id est Concessus a Societate Regia constitutus) candorem ejus aggressi essent per interpretationes duras & male fundatas, et quod hi illi voluptatem non habebunt videndi responsa ejus ad parvas rationes eorum qui ipsum tam male tractassent. Si hoc ita esset; ostendere debuisset in casibus aliquot quod eorum interpretationes essent duræ et male fundatæ. Notæ in Epistolas nullius sunt authoritatis nisi quam ab ipsis Epistolis derivant, et facile confutantur si male fundatæ sunt. Sed neque Epistolis responsum fuit neque Notis in easdam: ideoque utroque jam denuo impressæ sunt ut quibus responderi non potest.

Newtonus in Epistola sua ad Abbatem de Comitibus 26 Feb 171$\frac{5}{6}$ data, D. Leibnitij Literis ita respondebat. D. Leibnitius hactenus respondere recusavit, bene intelligens impossibile esse factis respondere. Silentium suum hac in re excusat allegando impræsentia quod librum nondum vidisset, et quod otium illi non esset ad examinandum, sed quod orasset Mathematicum celebrem ut hoc negotium in se susciperet &c Et D. Leibnitius in proxima sua ad Abbatem de Comitibus Epistola 9 Apr. 1716 data, pergebat se excusare ne responderet, dicendo: Vt operi contra me edito perfecte respondeam, opus erit alio opere non minore quam hoc est. opus erit percurrere corpus magnum minutorum ante annos 30 vel 40 præteritorum, quorum perparvum reminiscor, opus erit examinare veteres epistolas quarum plurimæ sunt perditæ, præterquam quod maxima ex parte non conservavi minuta mearum, & reliquæ sepultæ sunt in maximo chartarum acervo quem non possum sine tempore et patientia sigillatim examinare. Sed otium minime mihi suppetit alijs negotijs alterius prorsus generis occupato. Hactenus Leibnitius. Attamen post mortem ejus, quæ accidit proximo mense Septembri, scriptum est in ejus Elogio quod Commercio Epistolico Anglorum aliud quoddam suum, idemque amplius opponere decreverat.

Mathematicus Iohannem Bernoullium tanquam a se diversum citaverat. Set post annos duos cum semisse Leibnitius citationem omisit & Bernoullium esse Mathematicum illum prætendere cœpit.

Sed accusationem Leibnitius probare non potuit ut jam vidimus, Et Totum inter ipsum et Oldenburgum Commercium continua serie jam ante impressum fuit præter Epistolas duas quæ non extant.

Elementa methodi fluxionum & momentorum in prima Propositione libri Anno 1691 Halleus & Ralphsonus Librum MS Newtoni de Quadraturis figurarum manibus suis inverunt ut Regula inveniendi fluxiones primas secundas tertias cæterasque in infinitum in Propositione prima Libri de Quadraturis habetur. Hanc Propositionem exemplis in fluxionibus primis et secundis illustratam Wallisius in secundo operum Volumine in lucem edidit ineunte anno 1693 et hæc fuit Regulæ omnium prima pro fluxionibus secundis tertijs alijsque in infinitum inveniendisquæ lucem vidit. Propositionem quintam libri de quadraturis edidit etiam Wallisius in eodem Volumine pag.      Et hæc Propositio continet methodum quadrandi Curvilineas ubi fieri potest. Hanc methodum anno 1669 Newtono innotuisse patet ex Ejus Analysi per Series quam Barrovius eo anno ad Collinium misit, in qua utique dicitur quod hujus Analyseos beneficio Curvarum areæ et longitudines &c (id modo fiat) exacte et geometrie determinantur. Sed et aliquot annis antequam Mercatoris Logarithmotechnia prodiret hæc methodus Newtono innotuit Barrovio et Collinio testibus. Sic enim Collinius ad D. Thomam Strode 26 Iulij 1672 scripsit. Mense septembri 1668, Mercator Logarithmotechniam edidit suam, quæ specimen – – – obtineri queant. Quinta autem Propositio Libri de Quadraturis pendet a quatuor prioribus. Et propterea methodus fluxionum quatenus continetur in Propositionibus quinque primis libri de quadraturis Newtono innotuit anno 1666 aut antea. Hoc Wallisius in Præfatione ad operum suorum volumina duo prima asseruit nemine per ea tempora contradicente. Newtonus in Introductione ad librum de Quadraturis nihil amplius asseruit. Et tamen inde nata est hæc controversia.

In Analysi per series Newtonus utitur symbol $\boxed{\frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}}$ in eodem sensu quo Leibnitius symbolo $\int \frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}$, et symbolis o, ov, oy in eodem quo Leibnitius symbolis dx dy dz.

In methodum inversam fluxionum ingressum tantum dedit &

Aliud quoddam Commercium Epistolicum non habuit. Inventum novum in Analysi ad rem nil spectat.

Mathematicus autem Bernoullium citabat tanquam a se diversum, Leibnitius vero sub finem anni 1715 citationem delevit et Mathematicum esse Bernoullium ipsum scripsit; eum tamen Bernoullius judex constitui non posset nisi ipse juri omni in methodum infinitesimalem prius renunciasset. Huic judice opponendus est Wallisius vir antiquus qui litras Newtoni anno 1676 ab Oldenburgo acceperat & rem intellexerat ab initio & in Præfatione ad operum suorum volumina duo prima anno 1695 judicium pro Newtono tulit Dominis Leibnitius & Menkenio per ea tempora non mussitantibus.

Porro D. Leibnitius, anno 1714 per Epistolam ad D. Chamberlayn 25 Aug. datam postulavit ut Societas Regia – – – quæ non extant. Et ne respondeat me aggreditur insuper proponendo disputationes novas Philosophicas & Problemata solvenda quæ duo ad rem nil spectant.

Scribendis literis quam plurimis, disputationibus philosophicis instituendis, & Problematibus tractandis vacabat, Commercio Epistolico confutando non vacabat.

— Dixit utique Newtonum 1676 Methodum infinitessimalem Leibnitio per Literas exposuisse tum ante decem annos nedum plures ab ipso excogitatam. Sed et in secundo ejus Volumine quod lucem vidit anno 1693, Propositionem primam et quintam libri de Quadraturis in lucem adidit, Vnde manifestum est quod Liber ille tunc in MS extabat. Nam et Halleius et Ralphsonus nostrates anno 1691 librum illum in MS manibus suis triverunt ut alter eorum testatum reliquit, alter adhuc testatur.

In Propositione illa prima habetur Regula inveniendi fluxiones primas secundas tertias aliasque omnes in infinitum. Et hæc fuit Regula omnium prima quæ lucem vidit pro fluxionibus secundis tertijs et superioribus inveniendis estque omnium optima.

In Propositione quinta habetur methodus illa cujus beneficio Curvarum areæ (id modo fiat) exacte et Geometrice determinentur, uti Newtonus in Analysi sua per series anno 1669 a Barrovio ad Collinium missa locutus est Hæc Propositio pendet a quautor prioribus: ideoque methodus fluxionum quatenus in Propositionibus quinque primis Libri de Quadraturis continetur anno 1669 Newtono innotuit: imo et annis aliquot ante quam Mercatori Logarithmotechnia prodiret [mense septembris anni 1668] testibus Barrovio Et Collinio in Epistola Collinij ad Thomam Strode inferius impressa, id est anno 1666 aut antea, ut Wallisius affirmavit.

Dixit Iudex ille literas punctis notatas lucem non vidisse antequam tertium Wallisij Volumen prodiret quod fuit anno 1699. Lucem autem viderunt in secunde ejus Volumine quod prodijt anno 1693. In hoc secundo Volumine pag. 391 392, 393 habetur Propositio prima Libri Newtoni de Quadraturis , constructa et exemplis in fluxionibus primis & secundis illustrata. Et hoc fuit Regula omnium prima in lucem edita pro fluxionibus & momentis secundis tertijs quartis cæterisque in infinitum inveniendis, estque omnium optima. In eodem secundo Volumine edita pag. 391 etiam fuit Propositio quinta libri de Quadraturis. Et In hac Propositione habetur methodus illa cujus beneficio

– pro dz Newtonus utitur symbolo ov & pro $\int \frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}$ symbolo $\boxed{\frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}}$. In Introductione ad

After 1693 write. Et anno 1669 Newtonus in ejus Analysi per series usus est symbolis ov & $\boxed{\frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}}$ in eodem sensu quo Leibnitius postea usus est symbolis dz & $\int \frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}$.

neque unquam comparavit; sed priusquam Epistolam 12 Maij 1676 datam ad Oldenburgium scripsit oblitus est se series illas accepisse

— Leibnitius vero sub finem anni 1715, in versione Gallica in Hollandia edita citationem illam (nescio qua fide) omisit et Mathematicum esse Bernoullium ipsum scripsit, & problemata Bernoullij Analystis Anglis solvenda proposuit, & chartam illam volantem denuo dispersit, et ad sententiam Bernoullij appellando, amicos suos rerum . . . . conatus est

Post tres annos Marchio Hospitalius Regulam similem edidit

Eodem spectat quod Newtonus eandem Regulam demonstraverit synthetice in Lemmate secundo — id est anno 1671. Sunto – – – fluxiones invenire.

puta ${\mathrm{x}}^3-4\mathrm{x}\mathrm{y}\mathrm{z}+\mathrm{a}\mathrm{y}\mathrm{y}-\mathrm{z}$

Et ejusdem Propositionis Solutionem Newtonus demonstravit synthetice in Lem 2 Lib 2 Principiorum anno 1686. Sunto quantitates datæ a, b, c, fluentes x, y, z, fluxiones p, q, r, & momenta op, oq, or. et proponatur æquatio quævis fluentes involvens, puta ${\mathrm{x}}^4-\mathrm{a}\mathrm{x}\mathrm{y}\mathrm{y}+\mathrm{b}{\mathrm{y}}^3-{\mathrm{z}}^4+\mathrm{b}\mathrm{b}\mathrm{c}\mathrm{c}=0$. Et per Lemma prædictum si sola fluat x, momentum totius erit $4{\mathrm{x}}^3\mathrm{o}\mathrm{p}-\mathrm{a}\mathrm{y}\mathrm{y}\mathrm{o}\mathrm{p}$: si sola fluat y, momentum totius erit $\mathrm{-2}\mathrm{a}\mathrm{x}\mathrm{y}\mathrm{o}\mathrm{q}+3\mathrm{b}\mathrm{y}\mathrm{y}\mathrm{o}\mathrm{q}$. si sola fluat z, momentum totius erit $\mathrm{-4}{\mathrm{z}}^3\mathrm{o}\mathrm{r}$: si fluant omnes, momentum totius erit $4{\mathrm{x}}^3\mathrm{o}\mathrm{p}-\mathrm{a}\mathrm{y}\mathrm{y}\mathrm{o}\mathrm{p}-2\mathrm{a}\mathrm{x}\mathrm{y}\mathrm{o}\mathrm{q}+3\mathrm{b}\mathrm{b}\mathrm{y}\mathrm{y}\mathrm{o}\mathrm{q}-4{\mathrm{z}}^3\mathrm{o}\mathrm{r}$. Et quoniam totum semper est æquale nihilo, momentum totius erit æquale nihilo. Hæc est æquatio invelvens fluentium momenta. Si Eadem dividatur per o , habebitur æquatio involvens fluxiones. Sic Propositio, Data æquatione fluentes quotcunque quantitates involvente invenire fluxiones Solvitur per hoc Lemma, et in hujus solutione fundatur Methodus {Flu}x uti di{a}t{illeg} est in Scholio quod eidem Prop. subjungitur. Eadem Propositio extat in Epistola Newtoni ad Oldenburgum 24 Octob. 1676, et ibi dicitur esse fundamentum methodi generalis de qua Newtonus Tractatum scripserat tum ante annos quinque, id est anno 1671. Hujus autem solutio exhibet Algorithmum seu calculum Arithmeticum Methodi ejusdem, ideoque Algorithmus ille Newtono innotuit anno 1671.

Anglice edita in Actis Regiæ Soc. A.D. 1715,          et A.D. 1712 publicatas est et ex Anglico in Latinum versa.

Introductio ad Recensionem Libri de Commercio Epistolico Collinij et aliorum.

Historia Methodi quam Newtonus Methodum fluxionum Leibnitius methodum differentialem vocavit ex Commercio Epistolico Collinij et aliorum jussu Regiæ Societatis collecto anno 1712 & Recensione Commercij in Actis Philosophicis ejusdem Societatis anno 1715, (anno et aliquot mensibus ante obitum Leibnitij) edita

Historia Methodis Analyseos quam Newtonus Methodum fluxionum, Leibnitius methodum differentialem vocavit, in Commercio Epistolico Collinij et aliorum & Recensione Commercij contenta; quorum prius ex antiquis Literis jussu Regiæ Societatis fuit collectum & editum anno 1712, altera in Actis Philosophis ejusdem Societatis anno 1715 (anno et aliquot mensibus ante obitum Leibnitij) lucem vidit.

Historia Methodi Analyseos
Quam Newtonus Methodum Fluxionum & Momentorum Leibnitius Methodum differentialem vocavit,
in Commercio Epistolico Collinij et aliorum & Recensione Commercij
contenta:
Quorum prius ex antiquis Literis jussu Societatis Regiæ
collectum fuit et editum anno 1712
Altera in Actis Philosophicis ejusdem Societatis anno 1715
(anno et aliquot mensibus ante obitum Leibnitij)
lucem vidit.

Historia Methodi Analyseos
per fluxiones et momenta
in Commercio Epistolico Collinij et alio
Methodus Differentialis vocatæ

Historia Methodi Analyseos
per Fluxiones et Momenta Augmenta, auctus Differentias
ex scriptis antiquis
eruta.

Historia Methodi Differentialis
ex antiquis Litteris
in Commercio Epistolico Collinij et aliorum jussu R. Societ < insertion from f 188r > as tempore postquam alijs fuisset familiaris. < text from f 187v resumes >
editis anno 1712
et est Recensione Commercij anno 1715 (anno et aliquot mensibus ante
obitum Leibnitij) in Actis Philosophicis ejusdem societatis

Historia Methodi Analyseos
per fluxiones & momenta a D. Newtono inventæ,
a D. Leibnitio Differentialis nominatæ
ex literis antiquis deducta.

Historia methodi fluxionum
in Commercio Epistolico
et Recensione Commercij
tradita:
contenta:
Quorum prius jussu Regiæ Societatis
ex antiquis Litteris collectum fuit
Anno 1712.
Altera in Actis Philosophicis ejusdem Societatis
Anno 1715
(anno et aliquot mensibus ante obitum Leibnitij)
lucem vidit.

Historia methodi Fluxionem | Differentialis
In Commercio Epistolico et Recensione Commercij
contenta:
Quorum prius jussu R. Societatis editum fuit
Anno 1712
Altera in Actis Philosophicis ejusdem Societatis
Anno 1715
(Anno et aliquot mensibus ante obitum Leibnitij)
lucem vidit.

Epistolæ Leibnitij ad Oldenburgium datæ sunt 3 Feb. 20 Feb. 30 Mar. 26 Apr. 24 Maij, & 8 Iunij 1673; 15 Iulij & 26 Octob. 1674; 30 Mar. 20 Maij, 12 Iul. & 28 Decem 1675; 12 Maij, 27 Aug & 18 Novem 1676; 21 Iun. & 12 Iulij 1677 & harum omnium Autographa adhuc asservantur si duas tantum excipias 27 Aug. & 18 Novem 1676 scriptas et in tertio operum Wallisij volumine impressas ubi etiam eæ 15 Iul. & 26 Octob. 1674; 12 Iul. & 28 Decem 1675 & 21 Iun. & 12 Iul. 1677 leguntur. Harum etiam septendecim Epistolarum Apographa (si tertiam et ultimas quinque excipias) extant in libris antiquis Epistolicis Regiæ Societatis Numero 6, pag 35, 34, * 101, 115, 137 & N^o 7, pag 93, 110, 213, 235, 149, 189. Et

Sententiam futilem voce quia Iudex methodos collocas in characteristica; & fictam quia finxit literas punctis notatas primum publicatas fuisse in tertio volumine Operum Wallisij Et Newtonum pro secundis differentijs Regulam falsam dedisse & Newtono rectam methodum differentiandi differentialia non innotuisse longo, cum tamen literæ punctis notatæ lucem vide in secundo Volumine pag     fuissit familiaris, & Regula Newtoni pro secundis differentijs in eadem pagina ejusdem Volumine impressa fuit et est verissima.

To the Right Honourable the Lords Commissioners of his Majesties Treasury

May it please your Lordships

It being three years since the Pix was tryed & There having been coyned since the last tryal above seventeen hundred thousand pounds I humbly pray that a triall of the Pix of these moneys may be appointed this summer.

Et res longe antea a Wallisio judicata sit | fuit Leibnitio & Menkenio tum non mussitantibus.

Quæstio tota ad Epistolas antiquas referri debet.

Vt Commercium Epistolicum postpone

Eodem denique spectat 1674 scripsit ad Oldenburgum se seriem numerorum rationalium invenisse quorum summa exacte æquaretur circulo et eadem methodo valorem Arcus cujuslibet cujus sinus datur exhiberi posset licet ratio ejus ad integram circumferentiam non innotescaret : sed anno 1676 postulavit ab Oldenburgo ut is Demonstrationem ej{ic}{illeg}{a}m, id est Methodum inveniendi impetraret a Collinio et ad se mitteret. & non amplius seriem illam sibi arrogavit. Quinetiam Leibnitius seriem pro Arcu inveniendo ex Tangente data quam bis ab Oldenburgio acceperat & anno 1682 edidit in Actis eruditorum ut suam, tandem vero a Gregorio inventam fuisse agnoscere coactus est. Nam in Epistola sua ad Cometissam de Kilmansegg hac de re verba facians: Inventum est, inquit, postea, quod Gregorius quidem eandem etiam seriem invenerat; sed hoc didici tarde.

Eodem denique spectat quod seriem pro arcu inveniendo ex tangente data, Leibnitius anno 1682 ut suam edidit, non obstante quod eandem ab Oldenburgo bis acceperat. Et quod seriem pro arcu inveniendo ex sinu dato, se invenisse in Literis ad Oldenburgium 26 Octob. 1674 datis affirmabat, licet methodum inveniendi tunc non haberet sed Literis 12 Maij 1676 ad Oldenburgium datis eandem eni{illeg}e rogabat.

① Historia Methodi Fluxionum et Methodi Differentialis ex Epistolis antiquis Præfato^[2]

Concessus Arbitrorum a Regia Societate constitutus Commercij subsequentis Epistolici exemplaria tantum pauca anno 1712 imprimi curavit, et ad Mathematicos mitti qui soli de his rebus judicare possent. Cum vero D. Leibnitius huic Libro minime responderet Sed Quæstiones Metaphysicas aliasque ad hanc rem nihil spectantes & sine fine tractandas proponeret, et ejus amici adhuc rixentur: visum est hunc Librum una cum ejus Recensione quæ in Transactionibus Philosophicis initio anni 1715 impressa fuit, in lucem iterum mittere, ut Historia vera ex antiquis monumentis deducta ad posteros absque rixis perveniat, & sic finis imponatur huic controversiæ. Nam res non digna est de qua ulterius dis{p}utetur.

Epistolæ D. Leibnitij ad Oldenburgium scriptæ quæ hic imprimuntur,^[3] datæ sunt 3 Feb. 20 Feb. 30 Mar. 26 Apr. 24 Maij, & 8 Iunij 1673; 15 Iulij & 26 Octob. 1674; 30 Mar. 20 Maij, 12 Iulij, 28 Decem. 1675; 12 Maij, 27 Aug & 18 Novem 1676; 21 Iunij & 12 Iulij 1677. Et hæ omnes Epistolæ, si tertiam & ultimas quinque excipias, descriptæ extant in libris antiquis epistolicis Regiæ Societatis N^o 6, pag. 35, 34, *, 101, 115, 137, & N^o 7, pag. 93, 110, 213, 235, 149, 189. Et omnium etiam autographa asservantur, si duas tantum excipias 27 Aug. & 18 Novem. 1676 scriptas & in tertio operum Wallisij Volumine impressas. Epistolæ prædi ct æ 15 Iulij et 26 Octob. 1674, 12 Iulij et 28 Decem 1675, & 21 Iunij & 12 Iulij 1677, in tertio etiam operum ejus volumine impressæ sunt. Et hæ omnes Leibnitij epistolæ, una cum Epistolis mutuis Oldenburgij ad Leibnitium quarum Exemplaria adhuc asservantur, perpetuum constituunt inter eos per epistolas commercium a die 3 Feb 1673 ad usque mortem Oldenburgij, si modo epistolæ duæ excipiantur, quarum altera Leibnitius postulavit Excerpta ex epistolis Gregorij ad se mitti, in altera Oldenburgius excerpta illa misit Epistolæ Leibnitij versabantur circa Numeros ad usque 8 Iunij 1673. dein Leibnitio Geometriam addiscente, commercium aliquamdiu intermissum est. 1 & 15 Iulij 1674 renovatum est a Leibnitio sic scribente; Diu est quod nullas a me habuisit literas. Commercium igitur quod Leibnitius cum Oldenburgo Collinio et Newtono habuit; hic integrum imprimitur, præter dictas duas epistolas quæ interciderunt. Nam Collinius & Newtonus nullum cum Leibnitio commercium præterquam per <190r> Oldenburgium habuerunt. De fide epistolarum impressarum minime dubitatur apud Anglos.

Vbi primum Commercium epistolicum lucem vidit, D. Leibnitius Viennæ agens, ne libro responderet, causabatur per biennium se librum non vidisse,^[4] sed ad judicium primarij Mathematici et a partium studio alieni provocasse, cum ipse per occupationes diversas rem tunc discutere non satis posset. Et sententiam hujus Mathematici 7 Iunij 1713 datam, in alia Schedula contumelijs referta die 29 Iulij data describi, & utramque per Europam spargi curavit sine nomine vel Mathematici, vel Impressoris, vel Vrbis in qua impressa fuit; adjuvante, ni fallor, Mankenio Mathematicus in hoc scripto suo edito Bernoullium citavit tanquam a se diversum: Leibnitius vero sub finem anni 1715 citationem illam (nescio qua fide) delevit, et Mathematicum esse Bernoullium ipsum scripsit, & problemata Bernoullij Analystis Anglis solvenda proposuit, & chartam illam volantem denuo dispersit, et Gallice in Hollandia imprimi curavit & auctoritate Bernoullij amicos suos rerum mathematicarum inscios in Newtonum per literas impellere conatus est: cum tamen Bernoullius judex consistu jure nullo posset, nisi ipse juri omni in methodum infinitesimalem prius renunciasset.

In charta illa volante 29 Iulij 1713 data,^[5] Leibnitius Epistolam 15 Apr. 1675 scriptam (qua Oldenburgius series aliquot ad Leibnitium misit, et inter alias seriem Gregorij quam Leibnitius postea ut suam edidit,) suspectam reddere conatus est; Tale quiddam, inquiens, Gregorium habuisse ipsi Angli et Scoti, Wallisius, Hookius, Newtonus & junior Gregorius ultra triginta sex annos ignoraverunt, & Leibnitij esse inventum crediderunt. Verum hæc Epistola in Libro epistolico Regiæ Societatis asservata, ut et Epistola autographa Leibnitij se series missas accepisse agnoscentis, cum ijsdem Epistolis in Commercio editis, coram Comite de Kilmansegg, Abbate de Comitibus, Ministris aliquot publicis exterorum Principum, & alijs exteris non paucis, Anno 1715 in domo Regiæ Societatis collatæ sunt, & Impressionis fides probata: Sed et Leibnitius ipse anno proximo in Epistola sua ad Comitissam de Kilmansegg 18 Apr. data,^[6] idem agnovit, dum narrat, ut cum ipse de serie quam pro circulo invenerat, ad Oldenburgium scriberet (viz^t per Epistolas 15 Iulij & 26 Octob. 1674;) Oldenburgius responderit (viz^t 8 Decem 1674)^[7] Newtonum quendam Cantabrigiensem jam ante similia dedisse non solum pro circulo sed etiam pro omni figurarum aliarum genere et ipsi miserit serierum ✝ ^[8] specimina. His verbis Leibnitius agnoscit se Epistolam Oldenburgij 15 Apr. 1675 accepisse. Nam specimina illa erant in hac Epistola. Et in eadem erat series Gregorij, ut in Commercio Epistolico videre licet.^[9] Sed pergit Leibnitius. Hoc non obstante, ait, series mea satis laudata fuit per Newtonum ipsum. Postea inventum est Gregorium quendam eandem etiam seriem invenisse: sed hoc didici tarde. Hæc Leibnitius. Literas utique multa fruge Algebraica refertas acceperat, sed tunc præter ordinarias curas Mechanicis imprimis negotijs distractus non potuit examinare series quas Oldenburgius miserat, <191r> ac cum suis comparare (ut ipse tunc rescripsit) neque unquam comparavit. Et Newtonus, Wallisius, & junior Gregorius, hanc seriem a Gregorio seniore ad Collinium & ab Oldenburgio ad Leibnitium fuisse missam,^[10] per ea tempora ignorarunt.

Contra fidem Epistolarum in Commercio editarum scripsit insuper Leibnitius, Arbitrorum Consessum a R. Societate constitutum omnia edidisse quæ contra ipsum facerent, omnia omisisse quæ contra Newtonum:^[11] et præterea per Epistolam 25 Aug. 1714 ad D. Chamberlain datam,^[12] postulavit ut Societas Regia Epistolas nondum editas ad ipsum mitterent. Nam cum Hannoveram, inquit, rediero, possum etiam in lucem mittere Commercium aliud Epistolicum, quod Historiæ Literariæ inservire possit; et Literas quæ contra me allegari possunt, non minus publici juris faciam, quam quæ pro me faciunt. Hæc Leibnitius. Sed omnes inter ipsum et Oldenburgium Epistolæ, quatenus ad hanc rem spectant, continua serie jam ante in Commercio edito impressæ sunt, præter duas quæ non extant (uti jam dictum est), et nullius est momenti videntur.

Attamen ut accusationem suam confirmaret, scripsit Leibnitius sub finem anni 1715, in Epistola sua prima ad Abbatem de Comitibus^[13] per Galliam missa quod in secundo ejus in Angliam itinere Collinius ostenderit ipsi partem Commercij sui, in qua Newtonus agnoscebat ignorantiam suam in pluribus, dicebatque (inter alia) quod nihil invenisset circa dimensiones Curvilinearum quæ celebrantur præter dimensionem Cissoidis; sed Consessus hoc totum suppressit. Et newtonus in Epistola sua ad dictum Abbatem 26 Feb. 171$\frac{5}{6}$, respondit hoc non fuisse omissum sed extare in epistola sua ad Oldenburgium 24 Octob 1676 missa,^[14] et impressum fuisse in Commercio Epistolico pag. 74. lin. 10, 11. Et subinde Leibnitius in Epistola sua proxima ad Abbatem de Comitibus Apr. 9, 1716^[15] agnovit se errasse; sed, inquit, exemplum dabo aliud. Newtonus in una epistolarum ejus ad Collinium, agnovit se non posse invenire magnitudinem sectionum secundarum (vel segmentorum secundorum) sphæroidum & corporum similium, sed Consessus hunc locum vel hanc Epistolam in Commercio Epistolico minime edidit. Newtonus autem in Observationibus suis quas in hanc Leibnitij epistolam scripsit, respondit;^[16] Si Concessus hoc omisisset, recte omnino omissum fuisse; cum hujusmodi cavillationes ad Quæstionem de qua agitur nil spectent; Consessum hoc minime omisisse: Collinius in Epistola ad D. Gregorium 24 Decem 1670, et in altera ad D. Bertet 21 Feb. 1671^[17] (utrisque impressis in Commercio p. 24, 26) scripsit quod methodus Newtoni se extenderet ad secunda solidorum segmenta quæ per rotationem generantur. Et Oldenburgius idem, scripsit ad Leibnitium ipsum 8 Decem. 1674, ut videre est in Commercio, pag. 39.

Cæterum Leibnitius in prima sua ad Abbatem de Comitibus epistola^[18] scripsit, eos qui contra ipsum scripsissent (id est Con <192r> sessum a Regia Societate constitutum) candorem ejus aggressos esse per interpretationes duras & male fundatas; et voluptatem non habituros esse videndi Responsa ejus ad pusillas rationes eorum qui ijs tam male utuntur. Interpretationes illæ nullius quidem sunt authoritatis, nisi quam ab Epistolis derivant: at male fundatas esse Leibnitius nunquam ostendit.

Subinde vero Newtonus in prima sua ad Abbatem Epistola, 26 Feb. 171$\frac{5}{6}$,^[19] ita rescripsit. D. Leibnitius hactenus respondere recusavit, bene intelligens impossibile esse res factas refutare silentium suum hac in re excusat; prætexens se librum nondum vidisse, & otium illi non esse ad examinandum, sed se orasse Mathematicum celebrem ut hoc negotium in se susciperet. — Vtitur & novo prætextu ne respondeat, dicendo quod Angli qui Commercium ediderunt voluptatem non habebunt videndi responsum ejus ad pusillas eorum rationes; et proponendo disputationes novas philosophicas ineundas, & Problemata solvenda: quæ duo ad rem nil spectant.^[20]

D. Leibnitius autem in proxima sua ad Abbatem Epistola 9 Apr. 1716 data,^[21] & per Galliam in Angliam missa, pergebat se excusare ne respondeat, dicendo. Vt operi contra me edito sigillatim respondeam, opus erit alio opere non minore quam hoc est; percurrendum erit corpus magnum minutorum ante annos 30 vel 40 præteritorum, quorum perparvum reminiscor; examinand{æ} erunt veteres epistolæ quarum plurimæ sunt perditæ, præterquam quod maxima ex parte non conservari Minuta mearum, & reliquæ sepultæ sunt in maximo chartarum acervo quem non possum sine tempore et patientia discutere. Sed otium minime mihi suppetit, alijs negotijs alterius prorsus generis occupato. Hæc Leibnitius.

Attamen post ejus mortem (qua contigit proximo mense Septembri,) in Elogio ejus quod in Actis Eruditorum pro mense Iulio anni 1717 impressum fuit, amici ejus scripserunt eum Commercio Epistolico Anglorum aliud quoddam suum, idemque amplius, opponere decrevisse; et paucis ante obitum diebus Cl. Wolfio significasse se Anglos famam ipsius lacessentes reipsa refutaturum: quamprimum enim a laboribus historicis vacaturus sit, daturum se aliquid in Analysi prorsus inexpectatum, et cum inventis quæ hactenus in publicum prostant, sive Newtoni sive aliorum nihil quicquam affine habens. Hæc illi. Verum ex jam dictis patet eum aliud nullum cum Oldenburgio Commercium Epistolicum habuisse. Et inventum novum his nihil affine habens, ad rem nihil spectat. Missis, egrorum somnijs, Quæstio tota ad Epistolas antiquas referri debet. Et hæc Quæstio est <193r> secundum compilatores Actorum Leipsiensium) Vtrum Leibnitius sit inventor methodi de qua disputatur, & "pro Differentijs igitur Leibnitianis Newtonus adhibet semperque [ex quo usus est hac methodo] adhibuit fluxiones, quemadmodum Honoratus Fabrius motuum progressus Cavallerianæ methodo substituit." Quæritur, non quis methodum totam invenit (nam tota nondum inventa est:) sed quis methodum invenit quatenus in scriptis a Newtono editis habetur.

Ad hanc Quæstionem spectat quod D. Wallisius Professor Oxoniensis celeberrimus, Propositionem primam libri de Quadraturis, exemplis inveniendi fluxiones primas et secundas illustratam ediderit anno 1693 in volumina secundo operum suorum pag. 392. [Et hæc fuit Regula omnium prima quæ lucem vidit pro fluxionibus secundis tertijs, quartis, cæterisque in infinitum inveniendis. Estque Regula verissima & optima.] Eandem Newtonus demonstravit syntheticè in Lemmate secundo Libri secundi Principiorum anno 1686: cum Propositionem sine demonstratione prius posuisset in Epistola ad Oldenburgium 24 Octob. 1676 scripta, & ibi significasset eandem esse fundamentum methodi generalis de qua scripserat tum ante annos quinque, id est anno 1671. ☉ < insertion from f 188v > ☉ Sunto quantitates datæ a,b,c; fluentes x, y, z; fluxiones p, q, r; & momenta op, oq, or: et proponatur æquatio quævis fluentes involvens, puta ${\mathrm{x}}^3-2\mathrm{x}\mathrm{x}\mathrm{y}+\mathrm{b}\mathrm{x}\mathrm{x}-\mathrm{b}\mathrm{b}\mathrm{x}+\mathrm{b}\mathrm{y}\mathrm{y}-{\mathrm{y}}^3+\mathrm{c}\mathrm{z}\mathrm{z}=0$. Et per hoc Lemma prædictum, si sola fluat x, erit momentum totius $\mathrm{-3}\mathrm{x}\mathrm{x}\mathrm{o}\mathrm{p}-4\mathrm{x}\mathrm{o}\mathrm{p}\mathrm{y}+2\mathrm{b}\mathrm{x}\mathrm{o}\mathrm{p}-\mathrm{b}\mathrm{b}\mathrm{o}\mathrm{p}$; si sola fluat y erit momentum totius $\mathrm{-2}\mathrm{x}\mathrm{x}\mathrm{o}\mathrm{q}+2\mathrm{b}\mathrm{y}\mathrm{o}\mathrm{q}-3\mathrm{y}\mathrm{y}\mathrm{o}\mathrm{q}$; si sola fluat z, erit momentum totius $\mathrm{+2}\mathrm{c}\mathrm{z}\mathrm{o}\mathrm{r}$; si fluant omnes, erit momentum totius $3\mathrm{x}\mathrm{x}\mathrm{o}\mathrm{p}-4\mathrm{x}\mathrm{o}\mathrm{p}\mathrm{y}+2\mathrm{b}\mathrm{x}\mathrm{o}\mathrm{p}-\mathrm{b}\mathrm{b}\mathrm{o}\mathrm{p}-2\mathrm{x}\mathrm{x}\mathrm{o}\mathrm{q}+2\mathrm{b}\mathrm{y}\mathrm{o}\mathrm{q}-3\mathrm{y}\mathrm{y}\mathrm{o}\mathrm{q}+2\mathrm{c}\mathrm{z}\mathrm{o}\mathrm{r}$. Et quoniam totum semper est æquale nihilo, erit momentum totius æqualis >nihilo. Dividatur momentum totius per momentum o et prodibit æquatio quæ ex fluentibus dat fluxiones, viz^t $3\mathrm{x}\mathrm{x}\mathrm{p}-4\mathrm{x}\mathrm{p}\mathrm{y}+2\mathrm{b}\mathrm{x}\mathrm{p}-\mathrm{b}\mathrm{b}\mathrm{p}-2\mathrm{x}\mathrm{x}\mathrm{q}+2\mathrm{b}\mathrm{y}\mathrm{q}-3\mathrm{y}\mathrm{y}\mathrm{q}+2\mathrm{c}\mathrm{z}\mathrm{r}=0$. Exhibet igitur hoc Lemma solutionem Propositionis hujus: Data æquatione fluentes quotcunque quantitates involvente fluxiones invenire. Per hujus Propositiones solutionem inveniuntur omnes fluxiones (primæ sundæ tertiæ &c) utin Propositione prima Libri de quadraturis ostenditur, et in eadem solutione habetur Algorithmus seu calculus Arithmeticus methodi fluxionum.

Eodem spectat quod Propositio secunda libri de Quadraturis extet soluta in Analysi per series quam Barrovius anno 1699 ad Collinium misit pag. 19. Nam hæc Propositio pendet a Propositione prima ejusdem libri, ideoque Propositiones duæ primæ hujus Libri de Quadraturis Newtono innotuere anno 1669. Propositio autem tertia et quarta sunt exempla tantum Propositionis secundæ ut ibi dicitur et propterea methodus fluxionum quatenus in Propositionibus quatuor primis libri de Quadraturis habetur, Newtono innotuit anno 1669.

< text from f 193r resumes > In hujus Propositionis solutione habetur Algorithmus methodi differentialis.

Eodem spectat quod Propositionem quintam libri de Quadraturis Wallisius ediderit anno 1693 in secundo operum suorum Volumine pag 391. Et Newtonus eandem invenerit per Methodum fluxionum, uti patet per ejus Epistolam Octob 24, 1676 ad Oldenburgum missam. Hac Propositione quadrantur Figuræ accurate et Geometrice si fieri potest. Et hoc artificium Newtono innotuit anno 1669, uti affirmatur in Analysi per series quam Barrovius eo anno ad Collinium misit, imo et annis aliquot antequam Mercatoris Logarithmotechnia prodijt, testibus Barrovio et Collinio in Epistola Collinij ad D. Strode. Atqui Propositio illa quinta pendet a Propositionibus quatuor prioribus. Ideoque methodus fluxionum, quatenus continetur in Propositionibus quinque primis Libri de Quadraturis, Newtono innotuit annis aliquot antequam prodiret Mercatoris Logarithmotechnia, id est, anno 1666 aut antea. Id quod testatus est etiam Wallisius in Præfatione ad Operum suorum Volumen primum.

Eodem spectat quod in Libro de Analysi per Series, Fluxiones, ac Differentias, anno 1711 a Ionesio edito,^[22] pag. 38 extet Fragmentum Epistolæ Newtoni ad Collinium Nov. 8, 1676 datæ, his verbis. Nulla extat Curva cujus Æquatio ex tribus constat terminis, in qua, licet quantitates incognitæ se mutuo afficiant & indices dignitatum sint surdæ quantitates (v.g. $\mathrm{a}{\mathrm{x}}^{\mathrm{\lambda }}+\mathrm{b}{\mathrm{x}}^{\mathrm{\mu }}{\mathrm{y}}^{\mathrm{\sigma }}+\mathrm{c}{\mathrm{y}}^{\mathrm{\tau }}=0$ ubi x designat basin; y ordinatam; λ, μ, σ, τ indices dignitatum ipsarum x et y; & a, b, c quantitates cognitas una cum signis suis + & −) nulla, inquam, hujusmodi est Curva, de qua, an <194r> 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 generalium brevissima, eas comparare queo. Quinetiam si duæ quævis figuræ per hujusmodi æquationes expressæ proponentur, per eandem Regulam eas, modo comparari possint, comparo. — Eadem methodus æquationes quatuor terminorum aliasque complectitur, haud tamen adeo generaliter. Hactenus Newtonus. Hæc autem fiunt per Propositionem decimam Libri de Quadraturis, & absque Methodo Fluxionum fieri non possunt: indicant vero methodum quadrandi curvilineas in Libro de Quadraturis expositam, & methodum fluxionum in qua methodus altera fundatur, eousque promotas fuisse ante 8 Novem. 1676.

Eodem spectat etiam quod in Epistola Newtoni ad Oldenburgium 24 Octob. 1676^[23] descriptæ habentur Ordinatæ Curvilinearum, quarum collationes cum Conicis sectionibus Newtonus in Catalogum tunc olim retulerat, id est anno 1671 aut antea. Nam anno 1676 Newtonus annos quinque ab hac methodo promovenda abstinuerat, ut ipse ibidem refert. Earundem Curvilinearum et eodem ordine & modo descriptarum Collationes cum Conicis Sectionibus, ponuntur in Tabula posteriore duarum quæ in Scholio ad Propositionem decimam libri de Quadraturis habentur: ideoque Tabula illa composita fuit, et methodus quadrandi Curvilineas eousque producta, annis minimum quinque ante annum 1676. Id quod absque methodo Fluxionum fieri non potuit.  < insertion from f 188v >  Propositio decima pendat a Propositionibus quinta, sexta septima, octava, & nona Libri de Quadraturis, ideoque Propositiones primæ decem hujus libri Newtono innotuere anno 1676 < text from f 194r resumes >

Eodem denique spectat quod Newtonus in Epistola sua prædicta ad Oldenburgium 24 Octob. 1676 data, ubi Problematum genera quædam nominasset quæ per methodum suam solverentur et methodum Tangentium Slusij inde fluere idque absque æquationum Reductione dixisset; subjungit: Fundamentum harum operationum, satis OBVIVM quidem, quoniam jam non possum explicationem ejus prosequi, sic potius celavi. 6accdæ &c. Celavit igitur ut obvium, ne subriperetur. Quàm vero fuit obvium et quàm facile subripi potuit, sic patebit. Gregorius scripsit ad Collinium 5 Sept. 1670^[24] se ex Barrovij Methodis Tangentes ducendi invenisse methodum generalem et Geometricam ducendi Tangentes ad omnes Curvas sine calculo. Slusius se ejusmodi methodum Tangentium habere mense Octobri 1672 scripsit ad Oldenburgum. Et Newtonus 10 Decem. 1672 scripsit ad Collinium in hæc verba.^[25] Ex animo gaudeo D. Barrovij nostri Reverendi Lectiones Mathematicas exteris adeo placuisse, neque parum me juvat intelligere eos [Slusium et Gregorium] in eandem mecum incidisse ducendi Tangentes methodum, &c. Et subinde Newtonus in eadem Epistola methodum suam ducendi Tangentes descripsit, & addidit hanc methodum esse partem vel corollarium potius methodi suæ generalis solvendi abstrusiora Problemata, et non hærere ad quantitates surdas. Epistolas totas Gregorij et <195r> Newtoni habes infra in Commercio Epistolico, et earum Exemplaria Oldenburgius 26 Iunij 1676 misit ad Leibnitium^[26] inter Excerpta ex Gregorij Epistolis; & Leibnitius incidit in Prælectiones Barrovij in Anglia mense Octobri anni 1676, ut ipse asserit in Epistola sua ad Abbatem de Comitibus 9 Apr. 1716.

Sunto jam, ut in Epistola Newtoni, quantitates datæ a, b, c, Abscissa $\mathrm{AB}=\mathrm{x}$, Ordinata $\mathrm{BC}=\mathrm{y}$, et linea curva AC; & proponatur æquatio quævis quantitates illas duas fluentes x et y involvens, puta ${\mathrm{x}}^3-2\mathrm{x}\mathrm{x}\mathrm{y}+\mathrm{b}\mathrm{x}\mathrm{x}-\mathrm{b}\mathrm{b}\mathrm{y}+\mathrm{b}\mathrm{y}\mathrm{y}-{\mathrm{y}}^3=0$, ut in eadem Epistola; et ducenda sit recta CD quæ Curvam tangat in C., & Abscissam utrinque productam secat in D. Multiplicetur omnis æquationis terminus per indicem dignitatis x; & productum divisum per x (videlicet $3{\mathrm{x}}^2-4\mathrm{x}\mathrm{y}+2\mathrm{b}\mathrm{x}-\mathrm{b}\mathrm{b}$) vocetur R. Multiplicetur omnis æquationis terminus per indicem dignitatis y; et productum, divisum per y (videlicet $\mathrm{-2}\mathrm{x}\mathrm{x}+2\mathrm{b}\mathrm{y}-3\mathrm{y}\mathrm{y}$) vocetur S. Et per Regulam in Epistola illa Newtoni traditam, erit subtangens $\mathrm{BD}=\frac{\mathrm{S}\mathrm{y}}{\mathrm{R}}$, vel potius $=-\frac{\mathrm{S}\mathrm{y}}{\mathrm{R}}$, propterea quod AB et BD ducantur ad partes contrarias. Et hæc est Regula ducendi Tangentes quam Newtonus in Epistola illa posuit, ut partem aliquam vel specimen vel Corollarium Methodi suæ generalis. Methodus vero tota ex hac ejus parte, et Propositio generalis ex hoc ejus Corollario sic deducitur.

Agatur, secundum methodum Tangentium Barrovij & Gregorij, Ordinata nova EF, priori BC proxima, & compleatur parallelogrammum BCGE, et pro differentijs vel momentis BE et GF, scribantur p et q: et erit FG ad CG ut est CB ad BD, id est, q ad p ut y ad BD, seu $\frac{\mathrm{p}\mathrm{y}}{\mathrm{q}}=\mathrm{BD}=\frac{\mathrm{-S}\mathrm{y}}{\mathrm{R}}$, et facta reductione, $\mathrm{R}\mathrm{p}+\mathrm{S}\mathrm{q}=0$. Hæc æquatio, ubi duæ tantum sunt fluentes, involvit earum differentias. Et ubi plures sunt fluentes, operatio similis ad omnes applicata, dabit æquationem involventem omnium differentias. Et Theorema hocce, quod sic ex Newtoni Epistola consequitur, illud omne comprehendit quod Leibnitius ad Newtonum Anno 1677 rescripsit, ut et illud omne quod in Actis Eruditorum Anno 1684 in lucem edidit. Nam solutionem comprehendit Propositionis primæ libri de Quadraturis.

+☉ < insertion from f 188v >

+☉ Eodem denique spectat quod Leibnitius per Literas 12 Maij 1676 petijt ut Oldenburgius demonstrationem serierum duarum Newtoni, id est methodum easdem inveniendi in Analysi per series descriptam, postularet a Collinio & ad se mitteret: et quod <188r> sub finem mensis Octobris ejusdem anni Leibnitius viderit in manibus Collinij Epistolam Newtoni ad Oldenburgium 24 Octobris ejusdem datam, ubi Newtonus Analysin illam sic describit: [Eo ipso tempore — illustraveram diversis seriebus:] Collinius utique eodem tempore ostendit Leibnitio Epistolas plures Newtoni Gregorij et aliorum quæ præcipue de seriebus scriptæ erant [Recueil. p. 5, 33, 56.] & nondum probatum fuit quod Leibnitius eo tempore non viderit Analysin illam Newtoni per series

[Et his præmissis legatur jam Recensio Commercij Epistolici, & consulatur Commercium ipsum sicubi de factis dubitatur.] Et non scripserit dx et dy pro o ov, oy, et $\int \frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}$ pro $\boxed{\frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}}$.

Et his præmissis legatur jam Recensio — —

ubi Newtonus exponit fluentes per areas et utitur Symbolis l, v, y &c pro fluxionibus earum & symbolis o, ov, oy pro mom

vbi Newtonus exponit fluentes per areas & utitur symbolis o, ov, oy, $\boxed{\frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}}$ eodem sensu quo Leibnitius postea usus est symbolis dz, dy, dx, dv, $\int \frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}$.] vbi symbola o, ov, oy, $\boxed{\frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}}$ idem significant cum symbolis dz dy, dx, dv, $\int \frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}$ quæ Leibnitius postea adhibuit.

et seriem quam anno 1682 pro sua edidit Gregorium quendam invenisse tarde didicit. < text from f 195r resumes > Et his præmissis legatur jam Recensio Commercij Epistolici, & consulatur Commercium ipsum sic ubi de factis dubitatur

Historia Methodi Fluxionum et Methodi Differentialis ex Epistolis antiquis erictæ. Præfatio.^[27]

Consessus Arbitrorum a Regia Societate constitutus Commercij subsequentis Epistolici exemplaria tantum pauca Anno 1712 imprimi curavit, et ad Mathematicos mitti qui soli de his rebus judicare possent. Cum verò D. Leibnitius huic Libro minime responderet, sed Quæstiones Metaphysicas aliasque ad hanc rem nihil spectantes, & sine fine tractandas proponeret, et editores Actorum Lipsiensium & eorum amici adhuc rixentur: visum est hunc Librum una cum ejus Recensione quæ in Transactionibus Philosophicis ac Diario Literario Anno 1715 (anno et septem vel octo mensibus ante obitum D. Leibnitij) impressa fuit, in lucem iterum mittere, ut Historia vera ex antiquis monumentis deducta ad posteros absque rixis ac disputationibus ad rem nil spectantibus perveniat, et sic finis imponatur huic controversiæ. Nam depulso plagij crimine res non digna est de qua ulterius disputetur.

Epistolæ D. Leibnitij ad Oldenburgum scriptæ quæ hic imprimuntur, datæ sunt 3 Feb. 20 Feb. 30 Mar. 26 Apr. 24 Maij, & 8 Iunij 1673; 15 Iulij & 26 Octob. 1674; 30 Mar. 20 Maij, 12 Iulij & 28 Decem. 1675; 12 Maij, 27 Aug. & 18 Novem 1676; 21 Iunij & 12 Iulij 1677. Et harum omnium Apographa (si tertiam et ultimas quinque excipias) extant in libris antiquis epistolicis Regiæ Societatis N^o. 6. pag. 35, 34, *, 101, 115, 137; & N^o 7, pag. 93, 110, 213, 235, 149, 189. Et omnium etiam Autographa asservantur, si duas tantum excipias 27 Aug. & 18 Novem. 1676 scriptas & in tertio operum Wallisij volumine impressas: ubi etiam hæ 15 Iulij & 26 Octob. 1674, 12 Iulij & 28 Decem. 1675 & 21 Iunij & 12 Iulij 1677 leguntur. Et hæ omnes Leibnitij Epistolæ una cum Epistolis mutuis Oldenburgij ad Leibnitium quarum exemplaria adhuc asservantur, perpetuum constituunt inter eos per Epistolas commercium a die 3 Feb. 1673 ad usque mortem Oldenburgij, si modo epistolæ duæ excipiantur, in quarum altera Leibnitius postulat Excerpta ex epistolis Gregorij ad se mitti, in altera Oldenburgius excerpta illa misit. Epistolæ Leibnitij versabantur circa Numeros ad usque 8 Iunij 1673: dein Leibnitio Geometriam addiscente, Commercium aliquamdiu intermissum est, et 15 Iulij 1674 renovatum est a Leibnitio sic scribente: Diu est quod nullas a me habuisit literas. Et ab hoc tempore, Commercium quod Leibnitius cum Oldenburgio, Collinio, et Newtono habuit, circa series et altiorem Geometriam versabatur & hic integrum imprimitur, præter dictas duas epistolas quæ interciderunt. Nam Collinius et Newtonus <200r> nullum cum Leibnitio commercium præterquam per Oldenburgium habuerunt. De fide Epistolarum impressarum minime dubitatur, certe non apud Anglos.

Vbi primum Commercium epistolicum lucem vidit, D. Leibnitius Viennæ agens, ne libro responderet, causabatur per biennium se librum non vidisse, sed ad judicium primarij Mathematici et a partium studio alieni provocasse, cum ipse per occupationes diversas rem tunc non satis discutere posset. Et sententiam propria nomine hujus Mathematici 7 Iunij 1713 datam in schedula contumelijs referta, die 29 Iulij data describi, & utramque per Europam spargi curavit, sine nomine vel Mathematici, vel Impressoris, vel Vrbis in qua impressa fuit; adjuvante, ni fallor, Menkenio. Mathematicus in hoc scripto latine edito Bernoullium citavit tanquam a se diversum: Leibnitius vero sub finem anni 1715, in versione Gallica in Hollandia edita, citationem illam (nescio qua fide) omisit & Mathematicum esse Bernoullium ipsum scripsit, et literis ad Abbatem de Comitibus aliosque scriptis Newtonum totis viribus aggressus est, & problemata Bernoullij Analystis Anglis solvenda proposuit, & chartam illam volantem denuo dispersit, et ad sententiam Bernoullij appellans amicos suos rerum mathematicarum inscios in Newtonum per literas impellere conatus est: cum tamen Bernoullius judex consistui jure nullo posset, nisi ipse jure omni in methodum infinitesimalem prius renunciasset; et de hac re longe antea a Wallisius contrarium judicam talis ut Leibitio et Menkenio tum non mussitantibus.

In charta illa volante 29 Iulij 1713 data, Leibnitius Epistolam 15 Apr. 1675 scriptam (qua Oldenburgius series aliquot ad Leibnitium miserat, et inter alias seriem Gregorij quam Leibnitius postea ut suam edidit,) suspectam reddere conatus est; Tale quiddam, inquiens, Gregorium habuisse ipsi Angli et Scoti, Wallisius, Hookius, Newtonus & junior Gregorius ultra triginta sex annos ignoraverunt & Leibnitij esse inventum crediderunt. Verum hæc Epistola in Libro Epistolico Regiæ Societatis asservata, ut et hæc Epistola autographa Leibnitij se series missas accepisse agnoscentis, cum ijsdem Epistolis in Commercio editis, coram Comite de Kilmansegg, Abbate de Comitibus, Ministris aliquot publicis exterorum Principum, & alijs exteris non paucis, Anno 1715 in domo Regiæ Societatis collatæ sunt, & impressionis fides probata. Sed et Leibnitius ipse anno proximo in Epistola sua ad Comitissam de Kilmansegg 18 Apr. data idem agnovit dum narrat, ut cum ipse de serie quam pro circulo invenerat, ad Oldenburgium scriberet (viz^t per Epistolas 15 Iulij & 26 Octob. 1674;) Oldenburgius responderit (viz^t 8 Decem 1674) Newtonum quendam Cantabrigiensem jam ante similia dedisse non solum pro circulo sed etiam pro omni figurarum aliarum genere, et ipsi miserit serierum ✝^[28]specimina. His verbis Leibnitius agnoscit se epistolam Oldenburgij 15 Apr. 1675 accepisse. Nam specimina illa erant in hac epistola. Et in eadem erat series Gregorij, ut in Commercio Epistolico videre licet.^[29] Sed pergit Leibnitius. Hoc non obstante, ait, series mea satis laudata fuit per Newtonum ipsum. Postea inventum est Gregorium quendam eandem etiam seriem invenisse: sed hoc didici tarde. Hæc Leibnitius. Literas utique multa fruge Algebraica refertas acceperat, sed tunc præter <201r> ordinarias curas Mechanicis imprimis negotijs distractus non potuit examinare series quas Oldenburgius miserat,^[30] ac cum suis comparare, (ut ipse tunc rescripsit,) neque unquam comparavit; sed priusquam epistolam 12 Maij 1676 datam^[31] ad Oldenburgium scripsit, oblitus est se series anno superiore missas accepsse, et seriem quam anno 1682 pro sua edidit Gregorium quendam invenisse tarde didicit. Newtonus autem et Wallisius et junior Gregorius, hanc seriem a Gregorio seniore ad Collinium & ab Oldenburgio ad Leibnitium fuisse missam, per ea tempora multo magis ignorarunt. Leibnitius itaque Epistolam Oldenburgij seriebus refertam accepit, sed series, si fas est, credere nunquam contulit cum suis.

Contra fidem Epistolarum in Commercio editarum scripsit insuper Leibnitius, Arbitrorum Consessum a R. Societate constitutum omnia edidisse quæ contra ipsum facerent, omnia omisisse quæ contra Newtonum. Et præterea per Epistolam 25 Aug. 1714 ad D. Chamberlain datam, postulavit ille ut Societas Regia Epistolas nondum editas ad ipsum mitterent. Nam cum Hanoveram, inquit, rediero, possum etiam in lucem mittere Commercium aliud Epistolicum, quod Historiæ Literariæ inservire possit; et literas quæ contra me alligari possunt, non minus publici juris faciam quam quæ pro me faciunt. Hæc Leibnitius. Sed omnes inter ipsum et Oldenburgium Epistolæ, quatenus ad hanc rem spectant, continua serie jam ante in Commercio edito impressæ sunt, præter duas quæ non extant (uti jam dictum est), et nullius esse momenti videntur.

Attamen ut accusationem suam confirmaret, scripsit Leibnitius sub finem anni 1715, in epistola sua prima ad Abbatem de Comitibus per Galliam missa, quod in secundo ejus in Angliam itinere Collinius ostenderit ipsi partem Commercij sui, in qua Newtonus agnoscebat ignorantiam suam in pluribus, dicebatque (inter alia) quod nihil invenisset circa dimensiones Curvilinearum quæ celebrantur, præter dimensionem Cissoidis; sed Consessus hoc totum suppressit. Et Newtonus in Epistola sua ad dictum Abbatem 26 Feb. 171$\frac{5}{6}$, respondit, hoc non fuisse omissum sed extare in epistola sua ad Oldenburgium 24 Oct. 1676 missa, & impressum fuisse in Commercio Epistolico pag. 74. lin. 10, 11. Et subinde Leibnitius in Epistola sua proxima ad Abbatem de Comitibus Apr. 9, 1716 agnovit se errasse: sed, inquit exemplum dabo aliud. Newtonus in una Epistolarum ejus ad Collinium, agnovit se non posse invenire magnitudinem sectionum secundarum (vel segmentorum secundorum) sphæroidum et corporum similium, sed Consessus hunc locum vel hanc Epistolam in Commercio Epistolico minime edidit. Newtonus autem in Observationibus quas in hanc Leibnitij epistolam scripsit, respondit: Si Consessus hoc omisisset, recte omnino omissum fuisse, cum hujusmodi cavillationes ad Quæstionem de qua agitur nil spectent; sed Consessum hoc minime omisisse. Collinius in Epistola ad D. Gregorium 24 Decem. 1670, et in altera ad D. Bertet 21 Feb. 1671 (utrisque impressis in Commercio p. 24, 26) scripsit quod methodus Newtoni se extenderet ad secunda solidorum segmenta quæ per rotationem generantur. Et Oldenburgius idem scripsit ad Leibnitium ipsum 8 Decem. 1674, ut videre est in Commercio, pag. 39. Leibnitius igitur accusationem finxit.

Cæterum Leibnitius in prima sua ad Abbatem de Comitibus epistola scripsit, eos qui contra ipsum scripsissent (id est Consessum <202r> a Regia Societate constitutum) candorem ejus aggressos esse per interpretationes duras & male fundatas; et voluptatem non habituros esse videndi Responsa ejus ad pusillas rationes eorum qui ijs tam male utuntur. Interpretationes >illæ nullius quidem sunt authoritatis, nisi quam ab Epistolis derivant: at male fundatas esse Leibnitius nunquam ostendit.

Subinde vero Newtonus in prima sua ad Abbatem Epistola, 26 Feb. 171$\frac{5}{6}$, ita rescripsit. D. Leibnitius hactenus respondere recusavit, bene intelligens impossibile esse res factas refutare Silentium suum hac in re excusat, prætexens se librum nondum vidisse & otium illi non esse ad examinandum, sed se orasse Mathematicum celebrem ut hoc negotium in se susciperet. — Vtitur et novo prætextu ne respondeat, dicens quod Angli qui Commercium ediderunt voluptatem non habebunt videndi responsum ejus ad pusillas eorum rationes; et proponens disputationes novas philosophicas ineundas & Problemata solvenda: quæ duo ad rem nil spectant.

D. Leibnitius autem in proxima sua ad Abbatem Epistola 26 Feb. 171$\frac{5}{6}$, ita rescripsit; D. Leibnitius hactenus respondere recusavit, bene intelligens impossibile esse res factas refutare

Duplicate of part of p 4

a Regia Societate constitutum) candorem ejus aggressos esse per interpretationes duras et male fundatas; et voluptatem non habituros esse videndi Responsa ejus ad pusillas rationes eorum qui ijs tam male utuntur. Interpretationes illæ nullius quidem sunt auctoritatis nisi quam ab Epistolis derivant; at male fundatas esse Leibnitius nunquam ostendit.

Subinde vero Newtonus in prima sua ad Abbatem Epistola 26 Feb. 171$\frac{5}{6}$, ita rescripsit. D. Leibnitius hactenus respondere recusavit, bene intelligens impossibile esse res factas refutare Silentium suum hac in re excusat, prætexens se librum nondum vidisse, & otium illi non esse ad examinandum, sed se orasse Mathematicum celebrem ut hoc negotium in se susciperet. — Vtitur et novo prætextu ne respondeat; dicens quod Angli qui Commercium ediderunt voluptatem non habebunt videndi responsum ejus ad pusillas eorum rationes; et proponens disputationes novas philosophicas ineundas, & Problemata solvenda: quæ duo ad rem nil spectant.

D. Leibnitius autem in proxima sua ad Abbatem Epistola 9 Apr. 1716 data, & per Galliam in Angliam missa pergebat se excusare ne respondeat. Vt operi, inquit, contra me edito sigillatim respondeam, opus erit alio opere non minore quam hoc est; percurrendum erit corpus magnum minutorum ante annos 30 vel 40 præteritorum, quorum perparvum reminiscor; examinandæ erunt veteres Epistolæ, quarum plurimæ sunt perditæ, præterquam quod maxima ex parte non conservavi minuta mearum, et reliquæ sepultæ sunt in maximo chartarum acervo quem non possum sine tempore et patientia discutere. Sed otium minime mihi suppetit, alijs negotijs alterius prorsus generis occupato. Hæc Leibnitius.

Attamen post ejus mortem (quæ contigit proximo mense Septembri,) in Elogio ejus quod in Actis Eruditorum pro mense Iulio anni 1717 impressum fuit, amici ejus scripserunt eum Commercio Epistolico Anglorum aliud quoddam suum, idemque amplius opponere decrevisse; et paucis ante obitum diebus Cl. Wolfio significasse se Anglos famam ipsius lacessentes reipsa refutaturum: quamprimum enim a laboribus historicis vacaturus sit, daturum se aliquid in Analysi prorsus inexpectatum, et cum inventis quæ hactenus in publicum prostant, sive Newtoni sive aliorum nihil quicquam affine habens. Hæc illi. Verum ex jam dictis patet illum aliud nullum cum Oldenburgio Commercium Epistolicum habuisse. Et inventum novum his nihil affine habens, ad rem nihil spectat. Missis ægrorum somnijs, Commercium Epistolicum quod Leibnitius, et ejus amici eludere absque responso oblivioni dare conati sunt, in lucem revocare debet, et Quæstio tota ad Epistolas antiquas referri. Et hæc Quæstio est (secundum Compilatores Actorum Leipsiensium,) Vtrum Leibnitius sit inventor methodi de qua disputatur, & pro Differentijs igitur Leibnitianis Newtonus adhibet semperque [ex quo usus est hac methodo] adhibuit Fluxiones, quemadmodum Honoratus Fabrius motuum progressus Cavallerianæ methodo substituit. Quæritur, non quis methodum totam invenit (nam tota nondum <205r> inventa est:) sed quis methodum invenit quatenus in scriptis a Newtono editis habetur.

^✝ < insertion from f 204v > ^✝ Ad hanc quæstionem spectat quod Leibnitius differentias & methodum differentialem vocat quas Newtonus momenta & methodum momentorum, et quod methodus momentorum & methodus fluxionum una et eadem sit methodus. Momenta sunt partes quas Leibnitius differentias vocat, fluxiones sunt velocitates quibus partes generantur. In methodo Leibnitij considerantur partes, in ea Newtoni considerantur etiam velocitates. Newtoni methodus est amplior & Leibnitij methodum complectitur, nisi forte Leibnitius methodo inversæ aliquid addiderit. < text from f 205r resumes > Eodem spectat quod D. Wallisius Professor Oxoniensis celeberrimus, Propositionem primam libri de Quadraturis, exemplis inveniendi fluxiones primas et secundas illustratam ediderit Anno 1693 in volumine secundo operum suorum, pag. 392. Et ejusdem Propositionis solutionem Newtonus demonstravit synthetice in Lem. 2 Lib. 2 Principiorum anno 1686. Sunto quantitates datæ a, b, c, fluentes x, y, z, fluxiones p, q, r, & momenta op, oq, or. Et proponatur æquatio quævis fluentes involvens, puta ${\mathrm{x}}^4-\mathrm{a}\mathrm{x}\mathrm{y}\mathrm{y}+\mathrm{b}{\mathrm{y}}^3-{\mathrm{z}}^4+\mathrm{b}\mathrm{b}\mathrm{c}\mathrm{c}=0$. Et per Lemma prædictum, si sola fluat x, momentum totius erit $4{\mathrm{x}}^3\mathrm{o}\mathrm{p}-\mathrm{a}\mathrm{y}\mathrm{y}\mathrm{o}\mathrm{p}$; si sola fluat y, momentum totius erit $-2\mathrm{a}\mathrm{x}\mathrm{y}\mathrm{o}\mathrm{q}+3\mathrm{b}\mathrm{y}\mathrm{y}\mathrm{o}\mathrm{q}$; si sola fluat z, momentum totius erit $-4{\mathrm{z}}^3\mathrm{o}\mathrm{r}$; si fluant omnes momentum totius erit $4{\mathrm{x}}^3\mathrm{o}\mathrm{p}-\mathrm{a}\mathrm{y}\mathrm{y}\mathrm{o}\mathrm{p}-2\mathrm{a}\mathrm{x}\mathrm{y}\mathrm{o}\mathrm{q}+3\mathrm{b}\mathrm{y}\mathrm{y}\mathrm{o}\mathrm{q}-4{\mathrm{z}}^3\mathrm{o}\mathrm{r}$. Et quoniam totum semper est æquale nihilo, momentum totius erit æquale nihilo. Hæc est æquatio involvens fluentium momenta. Si eadem dividatur per o, habebitur æquatio involvens fluxiones. Per hoc Lemma igitur solvitur Propositio: Data æquatione fluentes quotcunque quantitates involvente fluxiones invenire: & in hujus solutione fundatur methodus fluxionum uti dictum est in Scholio quod eidem Propositioni subjungitur. Eadem Propositio extat in Epistola Newtoni ad Oldenburgium 24 Octob. 1676,^[32] et ibi dicitur esse fundamentum methodi generalis de qua Newtonus Tractatum scripserat tum ante annos quinque, id est, anno 1671. Hujus autem solutio exhibet Algorithmum seu calculum Arithmeticum Methodi ejusdem; ideoque Algorithmus ille Newtono innotuit anno 1671.

Eodem spectat quod Propositio secunda Libri de Quadraturis extet soluta in Analysi per series quam Barrovius anno 1699 ad Collinium misit,^[33] pag 19 ubi docetur Inventio Curvarum quæ quadrari possunt. Nam hæc Propositio pendet a Propositione prima libri ejusdem; ideoque Propositiones duæ primæ Libri de Quadraturis Newtono innotuere anno 1669. Propositio autem tertia et quarta sunt exempla tantum Propositionis secundæ, ut ibi dicitur. Et propterea Methodus fluxionum, quatenus in Propositionibus quatuor primis Libri de Quadraturis habetur, Newtono innotuit Anno 1669.

Eodem spectat quod Propositionem quintam Libri de Quadraturis Wallisius ediderit anno 1693 in secundo operum suorum volumine pag 391. Hac Propositione quadrantur Figuræ accurate et Geometrice si fieri potest. Et hoc artificium Newtono innotuit anno 1676, uti patet per Epistolam ejus 24 Octob. ejusdem anni,^[34] ad Oldenburgium missam: Vt et anno 1669 uti affirmatur in Analysi per series quam Barrovius eo anno ad Collinium misit:^[35] Imo et annis aliquot antequam Mercatoris Logarithmotechnia prodijt, testibus Barrovio et Collinio in Epistola Collinij^[36] <206r> ad D. Strode. Atqui Propositio illa quinta pendet a Propositionibus quatuor prioribus. Ideoque methodus fluxionum quatenus continetur in Propositionibus quinque primis Libri de Quadraturis Newtono innotuit annis aliquot antequam prodiret Mercatoris Logarithmotechnia, id est, anno 1666 aut antea testibus Barrovio et Collinio. Id quod testatus est etiam Wallisius in Præfatione ad Operum suorum Volumen primum.

Ad eandem Quæstionem spectat quod in Libro de Analysi per Series Fluxiones ac Differentias, anno 1711 a Ionesio edito pag. 38, extet Fragmentum Epistolæ Newtoni ad Collinium Nov. 8, 1676 datæ, his verbis. Nulla extat Curva cujus Æquatio ex tribus constat terminis, in qua, licet quantitates incognitæ se mutuo afficiant, & indices dignitatum sint surdæ quantitates (v.g. $\mathrm{a}{\mathrm{x}}^{\mathrm{\lambda }}+\mathrm{b}{\mathrm{x}}^{\mathrm{\mu }}{\mathrm{y}}^{\mathrm{\sigma }}+\mathrm{c}{\mathrm{y}}^{\mathrm{\tau }}=0$: ubi x designat basin; y ordinatam; λ, μ, σ, τ indices dignitatum ipsarum x et y; & a, b, c quantitates cognitas una cum signis suis + et −) 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 generalium brevissima, eas comparare queo. Quinetiam si duæ quævis figuræ per hujusmodi æquationes expressæ proponantur, per eandem Regulam eas, modo comparari possint, comparo. — Eadem methodus æquationes quatuor terminorum aliasque complectitur, haud tamen adeo generaliter. Hactenus Newtonus. Hæc autem absque methodo fluxionum fieri non possunt: indicant vero Methodum quadrandi Curvilineas in Libro de Quadraturis expositam, et methodum fluxionum in qua methodus altera fundatur, eousque promotas fuisse ante 8 Novem. 1676.

Eodem spectat etiam quod in Epistola Newtoni ad Oldenburgium 24 Octob. 1676 data,^[37] descriptæ habentur Ordinatæ Curvilinearum, quarum collationes cum Conicis sectionibus Newtonus in Catalogum tunc olim retulerat, id est, anno 1671 aut antea. Nam anno 1676^[38] Newtonus annos quinque ab hac methodo promovenda abstinuerat, ut ipse ibidem refert. Earundem Curvilinearum et eodem ordine & modo ijsdemque literis descriptarum Collationes cum Conicis Sectionibus ponuntur in Tabula posteriore duarum quæ in Scholio ad Propositionem decimam Libri de Quadraturis habentur: ideoque Tabula illa composita fuit, et methodus quadrandi Curvilineas eousque producta annis minimum quinque ante annum 1676. Id quod absque methodo Fluxionum fieri non potuit. Iam vero Propositio decima Libri de Quadraturis pendet a Propositionibus novem primis ejusdem Libri: ideoque Propositiones decem primæ hujus Libri Newtono innotuere anno 1676, vel potius anno 1671.

Ad eandem Quæstionem spectat quod Newtonus in Epistola sua prædicta ad Oldenburgium 24 Octob. 1676 data,^[39] ubi Problematum genera quædam nominasset quæ per Methodum suam solveren <207r> tur, et methodum Tangentium Slusij inde fluere, idque absque æquationum Reductione, dixisset; subjungit: Fundamentum harum operationum satis OBVIVM quidem, quoniam jam non possum explicationem ejus prosequi, sic potius celavi. 6accdæ &c. Celavit igitur ut obvium, ne subriperetur. Quam vero fuit obvium et quam facile subripi potuit, sic patebit. Gregorius scripsit ad Collinium 5 Sept. 1670,^[40] se ex Barrovij Methodis Tangentes ducendi invenisse methodum generalem et Geometricam ducendi: Tangentes ad omnes curvas sine calculo. Slusius se ejusmodi methodum Tangentium habere mense Octobri 1672 scripsit ad Oldenburgum Et Newtonus 10 Decem 1672 scripsit ad Collinium in hæc verba.^[41] Ex animo gaudeo D. Barrovij nostri Reverendi Lectiones Mathematicas exteris adeo placuisse, neque parum me juvat intelligere eos [Slusium et Gregorium] in Eandem mecum incidisse ducendi Tangentes methodum &c. Et subinde Newtonus in eadem Epistola methodum suam ducendi Tangentes descripsit, & addidit, hanc methodum esse partem vel Corollarium potius methodi suæ generalis solvendi abstrusiora Problemata, et non hærere ad quantitates surdas. Epistolas totas Gregorij et Newtoni habes infra in Commercio Epistolico, et earum apographa Oldenburgius 26 Iunij 1676 misit ad Leibnitium^[42] inter Excerpta ex Gregorij Epistolis, et Leibnitius incidit in Prælectiones Barrovij in Anglia mense Octobri anni 1676, ut ipse scripsit in Epistola ad Abbatem de Comitibus 9 Apr. 1716.

Sunto jam ut in Epistola Newtoni quantitates datæ a, b, c, Abscissa $\mathrm{AB}=\mathrm{x}$, Ordinata $\mathrm{BC}=\mathrm{y}$, et Linea Curva ACF; & proponatur Æquatio quævis quantitates illas duas fluentes x et y involvens, puta ${\mathrm{x}}^3-2\mathrm{x}\mathrm{x}\mathrm{y}+\mathrm{b}\mathrm{x}\mathrm{x}-\mathrm{b}\mathrm{b}\mathrm{y}+\mathrm{b}\mathrm{y}\mathrm{y}-{\mathrm{y}}^3=0$, ut in eadem Epistola; et ducenda sit recta CD quæ Curvam tangat in C, & Abscissam utrinque productam secet in D. Multiplicetur omnis æquationis terminus per indicem dignitatis x, et productum divisum per x (videlicet $3\mathrm{x}\mathrm{x}-4\mathrm{x}\mathrm{y}+2\mathrm{b}\mathrm{x}-\mathrm{b}\mathrm{b}$) vocetur R. Multiplicetur omnis æquationis terminus per indicem dignitatis y, et productum divisum per y (videlicet $\mathrm{-2}\mathrm{x}\mathrm{x}+2\mathrm{b}\mathrm{y}-3\mathrm{y}\mathrm{y}$) vocetur S. Et per Regulam in Epistola illa Newtoni traditam, erit subtangens $\mathrm{BD}=\frac{\mathrm{S}\mathrm{y}}{\mathrm{R}}$, vel potius $=-\frac{\mathrm{S}\mathrm{y}}{\mathrm{R}}$ propterea quod AB et BD ducantur ad partes contrarias. Et hæc est Regula ducendi Tangentes quam Newtonus in Epistola illa posuit, ut partem aliquam vel specimen vel Corollarium Methodi suæ generalis. Methodus vero tota ex hac ejus parte, et Propositio generalis ex hoc ejus Corollario sic deducitur.

Agatur secundum methodum Tangentium Barrovij & Gregorij, Ordinata nova EF, priori BC proxima, et compleatur parallelogrammum BCGE, et pro differentijs vel momentis BE et GF, scribantur p et q: et erit FG ad GC ut est CB ad BD, id est, q ad p ut y ad BD, seu $\frac{\mathrm{p}\mathrm{y}}{\mathrm{q}}=\mathrm{BD}=-\frac{\mathrm{S}\mathrm{y}}{\mathrm{R}}$, et facta reductione, <208r> $\mathrm{R}\mathrm{p}+\mathrm{S}\mathrm{q}=0$. Hæc æquatio, ubi duæ tantum sunt fluentes, involvit earum Differentias. Et ubi plures sunt fluentes, operatio similis ad omnes applicata dabit æquationem involventem omnium Differentias. Et Theorema hocce, quod sic ex Newtoni Epistola facillime colligitur, illud omne comprehendit quod Leibnitius ad Newtonum Anno 1677 rescripsit, ut et illud omne quod in Actis Eruditorum Anno 1684 in lucem edidit. Nam solutionem comprehendit Propositionis primæ libri de Quadraturis.

Eodem denique spectat quod Leibnitius per Literas 12 Maij 1676 datas,^[43] peteret ab Oldenburgio ut is demonstrationem serierum duarum Newtoni, id est, methodum easdem inveniendi in Analysi per series descriptam, postularet a Collinio, et ad se mitteret: et quod sub finem mensis Octobris ejusdem anni Leibnitius videret in manibus Collinij Epistolas plures Newtoni Gregorij et aliorum quæ præcipue de seriebus scriptæ erant et inter alias Epistolam Newtoni ad Oldenburgium 24 Octobris ejusdem datam,^[44] ubi Newtonus Analysin illam per series Compendium methodi serierum vocat. Et nondum probatum fuit quod Leibnitius eo tempore non viderit hoc Compendium vel hanc Analysin per series; ubi symbola o, ov, oy, $\boxed{\frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}}$ idem significant cum symbolis dz, dy, dx, dv, $\int \frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}$ et similibus quæ D. Leibnitius postea adhibuit.

Et his præmissis legatur jam Recensio Commercij Epistolici, et consulatur Commercium ipsum sicubi de factis dubitatur,

Præfatio.^[45]

Consessus Arbitrorum a Regia Societate constitutus Commercij subsequentis Epistolici exemplaria tantum pauca Anno 1712 imprimi curavit, et ad Mathematicos mitti qui soli de his rebus judicare possent. Cum vero D. Leibnitius huic Libro minime responderet, sed Quæstionem de primo Inventore desereret & ad Quæstiones Metaphysicas aliasque ad hanc rem nihil spectantes & sine fine tractandas, id est ad rixas confugeret , et ejus amici quidam adhuc rixentur: visum est hunc Librum una cum ejus Recensione quæ in Transactionibus Philosophicis ac Diario Literario Anno 1715 (anno et septem vel octo mensibus ante obitum D. Leibnitij) impressa fuit, in lucem iterum mittere, ut Historia vera ex antiquis monumentis deducta ad posteros absque rixis perveniat, et sic finis imponatur huic controversiæ. Nam depulso plagij crimine res non digna est de qua ulterius disputetur.

[Epistolas ad Oldenburgium Leibnitius scripsit 3 Feb. 20 Feb. 30 Mar. 26 Apr. 24 Maij & 8 Iun. 1673; 15 Iul. & 26 Octob. 1674; 30 Mar. 20 Maij, 12 Iul. & 28 Decem. 1675; 12 Maij, 27 Aug. & 18 Novem 1676; 21 Iun. & 12 Iul. 1677. Et harum omnium Autographa adhuc asservantur si duas tantum excipias 27 Aug & 18 Novem. 1676 datas et in tertio Operum Wallisij Volumine impressas: ubi etiam eæ 15 Iul. & 26 Octob. 1674; 12 Iul. & 28 Decem. 1675; 21 Iun. & 12 Iu. 1677 leguntur. Harum etiam septendecim Epistolarum Apographa (si tertiam et ultimas quinque excipias) extant in libris antiquis Epistolicis Regiæ Societatis Num^o. 6. pag. 35, 34, *, 101, 115, 137, & N^o 7, pag. 93, 110, 213, 235, 149, 189. Et hæ omnes Leibnitij Epistolæ una cum Epistolis mutuis Oldenburgij perpetuum inter eos constituunt per Epistolas commercium a die 3 Feb. 1673 ad usque mortem Oldenburgij, præter Epistolam qua Leibnitius postulabat Excerpta ex epistolis Gregorij ad se mitti & epistolam qua Oldenburgius excerpta illa misit. Epistolæ Leibnitij versabantur circa numeros ad usque 8 Iunij 1673: dein Leibnitio Geometriam addiscente, Commercium aliquamdiu intermissum est, et 15 Iulij 1674 renovatum est a Leibnitio sic scribente: Diu est quod nullas a me habuisit literas. Et ab hoc tempore Commercium quod Leibnitius cum Oldenburgio Collinio et Newtono habuit, circa series et altiorem Geometriam versabatur, et hic integrum (quoad hanc disputationem) imprimitur, præter dictas duas epistolas quæ interciderunt. Nam Collinius & Newtonus nullum cum Leibnitio Commercium habuerunt præterquam per Oldenburgium. De fide Epistolarum minime dubitatur, certe non apud Anglos.

Vbi primum Commercium epistolicum lucem vidit, D. Leibnitius <210r> Viennæ agens, ne libro responderet, causabatur per biennium se librum non vidisse, sed ad judicium primarij Mathematici et a partium studio alieni provocasse, cum ipse per occupationes diversas rem tunc non satis discutere posset. Et Iudicium mirabile nomine hujus Mathematici 7 Iunij 1713 datum in schedula contumelijs referta die 29 Iulij data describi, & utrumque per Europam spargi curavit, sine nomine vel Mathematici, vel Impressoris, vel Vrbis in qua impressa fuit; adjuvante ni fallor Menkenio. Author schedulæ utitur voce illaudabili quæ Leibnitio fere propria fuit, et narrat quæ inter Hugenium & Leibnitium Parisijs ante annos 37 vel 38 privatim gesta fuerant; et quæ de serie Gregorij habet, de Commercio Epistolico desumpta sunt. Annon Leibnitius hæc scripserit, & Commercium viderat? Sententiam in schedula descriptam voco, mirabilem. Sententiam voco mirabilem quia Iudex methodum collocat in characteristica, in methodo synthetica desiderat symbola analytica, contra Newtonum disputat ex usu literæ o, literas punctis notatas lucem primo vidisse ait in tertio volumine operum Wallisij id est anno 1699, et Newtonum Regulam falsam dedisse pro differentijs secundis, et rectam methodum differentiandi differentialia non cognovisse nisi longo tempore postquam alijs fuisset familiaris: cum tamen characteristica mutari possit non mutata methodo, in methodo synthetica nulla sit occasio utendi symbolis analyticis, Newtonus adhuc utatur litera o eodem sensu quo prius, & literæ punctis superimpositis notatæ methodusque Newtoni differentiandi differentialia lucem viderint in secundo operum Wallisij volumine, pag. 392, id est anno 1693; annis tribus antequam Marchio Hospitalius quæ a Bernollio didicerat in lucem edidit. & methodus illa verissima sit, et optima.

Mathematicus ille Iudex in scripto prædicto latine edito Bernoullium citabat tanquam a se diversum: Leibnitius vero sub finem anni 1715, in ejusdem versione Gallica, citationem illam (nescio qua fide) delevit et Mathematicum esse Bernoullium ipsum scripsit, et literis ad Abbatem de Comitibus datis Problemata Bernoullij Analystis Anglis solvenda proposuit, & chartam illam volantem denuo dispersit, et ad sententiam Bernoullij appellans, amicos suos rerum mathematicarum inscios in Newtonum totis viribus per literas impellere conatus est: cum tamen Bernoullius judex consistui jure nullo posset nisi ipse jure omni in methodum differentialem prius renunciasset; et Wallisius longe antea de hac re judicium contrarium tulisset, Leibitio et Menkenio per ea tempora non mussitantibus.

In charta illa volante Leibnitius Epistolam 15 Apr. 1675 scriptam (qua Oldenburgius series aliquot ad Leibnitium miserat, et inter alias seriem Gregorij quam Leibnitius postea ut suam edidit) suspectam reddere conatus est; Tale quiddam, inquiens, Gregorium habuisse ipsi Angli et Scoti, Wallisius, Hookius, et Newtonus & junior Gregorius ultra triginta sex annos ignoraverunt et Leibnitij esse inventum crediderunt. Verum hæc Epistola in Libro Epistolico Regiæ Societatis asservata, ut et Epistola autographa Leibnitij se series missas accepisse agnoscentis, cum ijsdem epistolis in Commercio editis, coram Comite de Kilmansegg, Abbate de Comitibus, Ministris aliquot publicis exterorum Principum, & alijs exteris non paucis, Anno 1715, in domo Regiæ Societatis collatæ sunt, et impressionis fides probata. Sed et Leibnitius ipse anno <211r> proximo in Epistola sua ad Comitissam de Kilmanseg 18 Apr. data, idem agnovit dum narrat, ut cum ipse de serie quam pro circulo invenerat, ad Oldenburgium scriberet (viz^t per epistolas 15 Iulij & 26 Octob. 1674;) Oldenburgius responderit (8 Decem. 1674) Newtonum quendam Cantabrigiensem jam antea similia dedisse, non solum pro circulo sed etiam pro omni figurarum aliarum genere, et ipsi miserit serierum ✝^[46]specimina. His verbis Leibnitius agnoscit se epistolam Oldenburgij 15 Apr. 1675 datam accepisse. Nam specimina illa erant in hac epistola. Et in eadem erat series Gregorij, ut in Commercio epistolico videre licet. Sed pergit Leibnitius: Hoc non obstante, ait, series mea satis laudata fuit per Newtonum ipsum. Postea, inventum est Gregorium quendam eandem etiam seriem invenisse: sed hoc didici tarde. Hæc Leibnitius. Literas utique multa fruge Algebraica refertas acceperat, sed tunc præter ordinarias curas Mechanicis imprimis negotijs distractus non potuit examinare series quas Oldenburgius miserat ac cum suis comparare (ut ipse tunc rescripsit) neque unquam comparavit, sed priusquam Epistolam 12 Maij 1676 datam ad Oldenburgium scripsit, oblitus est se series anno superiore missas accepsse, et seriem quam anno 1682 pro sua edidit, Gregorium quendam invenisse didicit tarde. Newtonus autem et Wallisius et junior Gregorius hanc seriem a Gregorio seniore ad Collinium et ab Oldenburgio ad Leibnitium fuisse missam, per ea tempora multo magis ignorarunt. Leibnitius itaque epistolam Oldenburgij seriebus refertam accepit, sed series illas, si fas est credere, nunquam contulit cum suis.

Contra fidem epistolarum in Commercio editarum scripsit insuper Leibnitius, Arbitrorum Consessum a Regia Societate constitutum omnia edidisse quæ contra ipsum facerent, omnia omisisse quæ contra Newtonum. Et præterea per epistolam 25 Aug. 1714 ad D. Chamberlain datam postulavit ille ut Societas Regia epistolas nondum editas ad ipsum mitterent. Nam cum Hanoveram, inquit, rediero, possum etiam in lucem mittere Commercium aliud Epistolicum, quod historiæ Literariæ inservire possit; et literas quæ contra me allegari possunt non minus publici juris faciam quàm quæ pro me faciunt. Hæc Leibnitius. Sed omnes inter ipsum et Oldenburgium epistolæ, quatenus ad hanc rem spectant continua serie jam antea in Commercio edito impressæ sunt præter duas quæ non extant (uti jam dictum est), et nullius esse momenti videntur.

Attamen ut accusationem suam confirmaret, scripsit Leibnitius sub finem anni 1715, in Epistola sua prima ad Abbatem de Comitibus per Galliam missa, quod in secundo ejus in Angliam itinere Collinius ostenderit ipsi partem Commercij sui in qua Newtonus agnoscebat ignorantiam suam in pluribus, dicebatque (inter alia) quod nihil invenisset circa dimensiones Curvilinearum <212r> quæ celebrantur, præter dimensionem Cissoidis; sed Consessus hoc totum suppressit. Et Newtonus in epistola sua ad dictum Abbatem 26 Feb. 171$\frac{5}{6}$, respondit, hoc non fuisse omissum sed extare in epistola sua ad Oldenburgium 24 Octob. missa, & impressum fuisse in Commercio Epistolico pag. 74. lin. 10, 11. Et subinde Leibnitius in Epistola sua proxima ad Abbatem de Comitibus Apr. 9. 1716 agnovit se errasse. Sed, inquit, exemplum dabo aliud. Newtonus in una Epistolarum ejus ad Collinium agnovit se non posse invenire magnitudinem sectionum secundarum (vel segmentorum secundorum) sphæroidum & corporum similium sed Consessus hunc locum vel hanc Epistolam minime edidit. Newtonus autem in Observationibus quas in hanc Leibnitij Epistolam scripsit, respondit: Si Consessus hoc omisisset, recte omnino omissum fuisse, cum hujusmodi cavillationes ad Quæstionem de qua agitur nil spectent; sed Consessum hoc minime omisisse. Collinius in epistola ad D. Gregorium 24 Decem. 1670, et in altera ad D. Bertet 21 Feb. 1671 (utrisque impressis in Commercio p. 24, 26) scripsit quod methodus Newtoni se extenderet ad secunda solidorum segmenta quæ per rotationem generantur. Et Oldenburgius idem scripsit ad Leibnitium ipsum 8 Decem. 1674, ut videre est in Commercio, pag. 19. Leibnitius igitur accusationem finxit.

Cæterum Leibnitius in prima sua ad Abbatem de Comitibus epistola dixit, eos qui contra ipsum scripsissent (id est Consessum a Regia Societate constitutum) candorem ejus aggressos esse per interpretationes duras et male fundatas, & voluptatem non habituros esse videndi Responsa ejus ad pusillas rationes eorum qui ijs tam male utuntur. Interpretationes illæ nullius quidem sunt autoritatis nisi quam ab Epistolis derivant; at male fundatas esse Leibnitius nunquam ostendit.

Subinde vero Newtonus in prima sua ad Abbatem Epistola 26 Feb. 171$\frac{5}{6}$ ita rescripsit. D. Leibnitius hactenus respondere recusavit, bene intelligens impossibile esse res factas refutare Silentium suum hac in re excusat, prætexens se librum nondum vidisse, & otium illi non esse ad examinandum, sed se orasse Mathematicum celebrem ut hoc negotium in se susciperet. — Vtitur et novo prætextu ne respondeat, dicens quod Angli voluptatem non habebunt videndi responsum ejus ad pusillas eorum rationes; & proponens disputationes novas philosophicas ineundas, & Problemata solvenda: quæ duo ad rem nil spectant.

D. Leibnitius autem in proxima sua ad Abbatem Epistola 9 Apr. 1716 data, et per Galliam in Angliam missa, pergebat se excusare ne respondeat. Vt operi, inquit, contra me edito sigillatim respondeam, opus erit alio opere non minore quam hoc est; percurrendum erit corpus magnum minutorum ante annos 30 vel 40 præteritorum quorum perparvum reminiscor; examinandæ erunt veteres epistolæ quarum plurimæ sunt perditæ, præterquam quod maxima ex parte non conservavi minuta mearum, et reliquæ sepultæ sunt in maximo chartarum acervo quem non possum sine tempore et patientia discutere. Sed otium minime mihi suppetit, alijs negotijs alterius prorsus generis occupato. Hæc Leibnitius.

Attamen post ejus mortem (quæ contigit proximo mense Septembri,) in Elogio ejus quod in Actis Eruditorum pro mense Iulio anni 1717 impressum fuit, amici ejus scripserunt eum Commercio epistolico Anglorum aliud quoddam suum idemque amplius opponere decrevisse; et paucis ante obitum diebus Cl. Wolfio significasse se Anglos famam ipsius lacessentes reipsa rfutaturum: quamprimum enim a laboribus historicis vacaturus sit, daturum se aliquid in Analysi prorsus inexpectatum, et cum inventis quæ hactenus in publicum prostant, sive Newtoni sive aliorum nihil quicquam affine habens. Hæc illi. Verum ex jam dictis patet illum aliud nullum cum Oldenburgio Commercium epistolicum habuisse. Et inventum novum his nihil affine habens, ad rem nihil spectat. Missis ægrorum somnijs Quæstio tota ad Epistolas antiquas referri debet. Et hæc Quæstio est (secundum Compilatores Actorum Leipsiensium) Vtrum Leibnitius sit inventor methodi de qua disputatur, et pro Differentijs igitur Leibnitianis Newtonus adhibet semperque [ex quo usus est hac methodo] adhibuit Fluxiones, quemadmodum Honoratus Fabrius motuum progressus Cavallerianæ methodo substituit. Quæritur, non quis methodum totam invenit (nam tota nondum inventa est) sed quis Methodum invenit quatenus in scriptis a Newtono editis habetur. Quæstionem aliæ omnes dimittendæ sunt & hæc sola discutienda.

Ad hanc Quæstionem spectat quod Leibnitius differentias & methodum differentialem vocat quas Newtonus momenta & methodum momentorum, et quod methodus momentorum et methodus fluxionum una et eadem sit methodus. Momenta sunt partes quas Leibnitius differentias vocat, fluxiones sunt velocitates quibus partes generantur. In methodo Leibnitij considerantur partes, in ea Newtoni considerantur etiam velocitates. Newtoni methodus est amplior & Leibnitij methodum complectitur. Sed Newtonus methodum inversam reliquit imperfectam, et quæritur quida alij addiderint.

Eodem spectat quod D. Wallisius Propositionem primam Libri de Quadraturis, exemplis inveniendi fluxiones primas et secundas illustratam edidit anno 1693 in Volumine secundo operum suorum, pag. 392, ut supradictum est. Et ejusdem Propositionis solutionem Newtonus demonstravit synthetice in Lem. 2 Lib. 2 Principiorum anno 1686. Sunto quantitates datæ a, b, c, fluentes x, y, z, fluxiones p, q, r, & momenta op, oq, or. Et proponatur æquatio quævis fluentes involvens, puta ${\mathrm{x}}^4-\mathrm{a}\mathrm{x}\mathrm{y}\mathrm{y}+\mathrm{b}{\mathrm{y}}^3-{\mathrm{z}}^4+\mathrm{b}\mathrm{b}\mathrm{c}\mathrm{c}=0$. Et per Lemma prædictum, si sola fluat x, momentum totius erit $4{\mathrm{x}}^3\mathrm{o}\mathrm{p}-\mathrm{a}\mathrm{y}\mathrm{y}\mathrm{o}\mathrm{p}$; si sola fluat y, momentum totius erit $-2\mathrm{a}\mathrm{x}\mathrm{y}\mathrm{o}\mathrm{q}+3\mathrm{b}\mathrm{y}\mathrm{y}\mathrm{o}\mathrm{q}$; si sola fluat z, momentum totius erit $-4{\mathrm{z}}^3\mathrm{o}\mathrm{r}$; si fluant omnes, momentum totius erit $4{\mathrm{x}}^3\mathrm{o}\mathrm{p}-\mathrm{a}\mathrm{y}\mathrm{y}\mathrm{o}\mathrm{p}-2\mathrm{a}\mathrm{x}\mathrm{y}\mathrm{o}\mathrm{q}+3\mathrm{b}\mathrm{y}\mathrm{y}\mathrm{o}\mathrm{q}-4{\mathrm{z}}^3\mathrm{o}\mathrm{r}$. Et quoniam totum semper est æquale nihilo, momentum totius erit æquale nihilo. Hæc est æquatio involvens fluentium momenta. Si eadem dividatur per o, habebitur æquatio involvens fluxiones. Per hoc Lemma igitur solvitur Propositio: Data æquatione fluentes quotcunque quantitates involvente, fluxiones invenire. Et in hujus Solutione fundatur methodus fluxionum uti dictum est in Scholio quod eidem Pro <214r> positioni subjungitur. Eadem Propositio extat in Epistola Newtoni ad Oldenburgium 24 Octob. 1676, et ibi dicitur esse fundamentum methodi generalis de qua Newtonus Tractatum scripserat tum ante annos quinque, id est, anno 1671. Hujus autem Solutio exhibet Algorithmum seu calculum Arithmeticum Methodi ejusdem, ideoque Algorithmus ille Newtono innotuit anno 1671.

Eodem spectat quod Propositio secunda Libri de Quadraturis extet soluta in Analysi per series quam Barrovius anno 1699 ad Collinium misit, pag. 19, ubi docetur Inventio Curvarum quæ quadrari possunt. Nam hæc Propositio secunda pendet a Propositione prima Libri ejusdem; ideoque Propositiones duæ primæ Libri de Quadraturis Newtono innotuere anno 1669. Propositio autem tertia et quarta sunt exempla tantum Propositionis secundæ, ut ibi dicitur. Et propterea Methodus fluxionum quatenus in Propositionibus quatuor primis Libri de Quadraturis habetur, Newtono innotuit Anno 1669.

Eodem spectat quod Propositionem quintam Libri de Quadraturis Wallisius edidit anno 1693 in secundo operum suorum volumine pag. 391. Hac Propositione quadrantur figuræ accuratè et Geometricè si fieri potest. Et hoc artificium Newtono innotuit anno 1676, uti patet per Epistolam ejus 24 Octob. ejusdem anni ad Oldenburgium missam pag       Vt et anno 1669 uti affirmatur in Analysi per series quam Barrovius eo anno ad Collinium misit, pag     Imò et annis aliquot antequam Mercatoris Logarithmotechnia prodijt, testibus Barrovio & Collinio in Epistola Collinij ad D. Strode p 119. Atqui Propositio illa quinta pendet a Propositionibus quatuor prioribus. Ideoque methodus fluxionum quatenus continetur in Propositionibus quinque primis Libri de Quadraturis Newtono innotuit annis aliquot antequam prodiret Mercatoris Logarithmotechnia, id est anno 1666 aut antea, testibus Barrovio et Collinio. Id quod testatus est etiam Wallisius in Præfatione ad Operum suorum Volumen primum.

Ad eandem Quæstionem spectat quod in Libro de Analysi per Series Fluxiones ac Differentias, anno 1711 a Ionesio edito, pag. 38, extet Fragmentum epistolæ Newtoni ad Collinium Nov. 8 datæ, his verbis: Nulla extat Curva cujus Æquatio ex tribus constat terminis, in qua, licet quantitates incognitæ se mutuo afficiant, & indices dignitatum sint surdæ quantitates (v.g. $\mathrm{a}{\mathrm{x}}^{\mathrm{\lambda }}+\mathrm{b}{\mathrm{x}}^{\mathrm{\mu }}{\mathrm{y}}^{\mathrm{\sigma }}+\mathrm{c}{\mathrm{y}}^{\mathrm{\tau }}=0$: ubi x designat basin; y ordinatam; λ, μ, σ, τ indices dignitatum ipsarum x & y; & a, b, c quantitates cognitas una cum signis suis + & −) 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 generalium brevissima, eas comparare queo. Quinetiam si duæ quævis figuræ per hujusmodi æquationes expressæ proponantur, per eandem Regulam eas modo comparari possint, comparo. — Eadem methodus æquationes quatuor terminorum aliasque complectitur, haud tamen adeo generaliter. Hactenus Newtonus. Hæc autem absque methodo fluxionum fieri non possunt; indicant vero Methodum quadrandi Curvilineas in Libro de Quadraturis expositam, et methodum fluxionum in qua methodus altera fundatur, eousque promotas fuisse ante 8 Novem. 1676.

Eodem spectat etiam quod in Epistola Newtoni ad Oldenbur <215r> gium 24 Octob. 1676 data, descriptæ habentur Ordinatæ Curvilinearum, quarum collationes cum Conicis sectionibus Newtonus in Catalogum tunc olim retulerat, id est anno 1671 aut antea. Nam anno 1676 Newtonus annos quinque ab hac methodo promovenda abstinuerat, ut ipse ibidem refert. Earundem Curvilinearum et eodem ordine et modo ijsdemque literis descriptarum collationes cum Conicis sectionibus ponuntur in Tabula posteriore duarum quæ in Scholio ad Propositionem decimam Libri de Quadraturis habentur, ideoque Tabula illa composita fuit et methodus quadrandi Curvilineas eousque producta annis minimum quinque ante annum 1676. Id quod absque methodo Fluxionum fieri non potuit. Iam vero Propositio decima Libri de Quadraturis pendet a Propositionibus novem primis ejusdem Libri: ideoque Propositiones decem primæ hujus Libri Newtono innotuere anno 1676, vel potius anno 1671.

Ad eandem Quæstionem spectat quod Newtonus in Epistola sua prædicta ad Oldenburgium 24 Octob. 1676 data, ubi Problematum genera quædam nominasset quæ per Methodum suam solverentur, et methodum Tangentium Slusij inde fluere, idque absque æquationum Reductione dixisset; subjungit: Fundamentum harum operationum satis OBVIVM quidem, quoniam jam non possum explicationem ejus prosequi, sic potius celavi. 6accdæ &c. Celavit igitur ut obvium, ne subriperetur. Quam vero fuit obvium et quam facile subripi potuit, sic patebit.

Gregorius scripsit ad Collinium 5 Sept. 1670 se ex Barrovij Methodis Tangentes ducendi invenisse methodum generalem et Geometricam ducendi: Tangentes ad omnes Curvas sine Calculo. Slusius se ejusmodi methodum Tangentium habere, scripsit ad Oldenburgium mense Octobri vel Novembri 1672. Et Newtonus 10 Decem 1672 scripsit ad Collinium in hæc verba. Ex animo gaudeo D. Barrovij amici nostri Reverendi Lectiones Mathematicas exteris adeo placuisse, neque parum me juvat intelligere eos [Slusium et Gregorium] in eandem mecum incidisse ducendi Tangentes methodum &c. Et subinde Newtonus in eadem Epistola methodum suam ducendi Tangentes descripsit, et addidit hanc methodum esse partem vel Corollarium potius methodi suæ generalis solvendi abstrusiora Problemata, et non hærere ad quantitates surdas. Epistolas totas Gregorij et Newtoni habes infra in Commercio Epistolico, et earum Apographa Oldenburgius 26 Iunij 1676 misit ad Leibnitium inter excerpta ex Gregorij Epistolis, et Leibnitius incidit in Prælectiones Barrovij in Anglia mense Octobri anni 1676, ut ipse scripsit in Epistola ad Abbatem de Comitibus 9 Apr. 1716. Sunto jam, ut in epistola Newtoni quantitates datæ a, b, c, Abscissa $\mathrm{AB}=\mathrm{x}$, Ordinata $\mathrm{BC}=\mathrm{y}$, & Linea curva ACF; et proponatur Æquatio quævis quantitates illas duas fluentes x et y involvens, puta ${\mathrm{x}}^3-2\mathrm{x}\mathrm{x}\mathrm{y}+\mathrm{b}\mathrm{x}\mathrm{x}-\mathrm{b}\mathrm{b}\mathrm{y}+\mathrm{b}\mathrm{y}\mathrm{y}-{\mathrm{y}}^3=0$, ut in eadem Epistola: et ducenda sit recta CD quæ Curvam tangat in C et Abscissam utrinque productam secet in D. Multiplicetur omnis Æquationis terminus per indicem dignitatis x et productum divisum per x (videlicet $3\mathrm{x}\mathrm{x}-4\mathrm{x}\mathrm{y}+2\mathrm{b}\mathrm{x}-\mathrm{b}\mathrm{b}$) <216r> vocetur R. Multiplicetur omnis æquationis terminus per indicem dignitatis y, et productum divisum per y (videlicet $\mathrm{-2}\mathrm{x}\mathrm{x}+2\mathrm{b}\mathrm{y}-3\mathrm{y}\mathrm{y}$) vocetur S. Et per Regulam in Epistola illa Newtoni traditam, erit subtangens $\mathrm{BD}=\frac{\mathrm{S}\mathrm{y}}{\mathrm{R}}$, vel potius $=-\frac{\mathrm{S}\mathrm{y}}{\mathrm{R}}$ propterea quod AB et BD ducantur ad partes contrarias. Et hæc est Regula ducendi Tangentes quam Newtonus in Epistola illa posuit ut partem aliquam vel specimen vel Corollarium Methodi suæ generalis. Methodus vero tota ex hac ejus parte, et Propositio generalis ex hoc ejus Corollario sic deducitur.

Agatur secundum methodum Tangentium Barrovij & Gregorij, Ordinata nova EF, priori BC proxima, et compleatur parallelogrammum BCGE, et pro differentijs vel momentis BE et GF, scribantur p et q: et erit FG ad GC ut est CB ad BD, id est, q ad p ut y ad BD, seu $\frac{\mathrm{p}\mathrm{y}}{\mathrm{q}}=\mathrm{BD}=-\frac{\mathrm{S}\mathrm{y}}{\mathrm{R}}$, et facta reductione, $\mathrm{R}\mathrm{p}+\mathrm{S}\mathrm{q}=0$. Hæc Æquatio ubi duæ tantum sunt fluentes involvit earum differentias. Et ubi plures sunt fluentes, operatio similis ad omnes applicata dabit æquationem involventem omnium Differentias. Et Theorema hocce quod sic ex Newtoni Epistola facillime colligitur, illud omne comprehendit quod Leibnitius ad Newtonum Anno 1677 rescripsit, ut et illud omne quod in Actis Eruditorum Anno 1684 in lucem edidit. Nam solutionem comprehendit Propositionis primæ Libri de Quadraturis.

Eodem denique spectat quod Leibnitius per Literas 12 Maij 1676 datas, peteret ab Oldenburgio ut is demonstrationem serierum duarum Newtoni, id est, methodum easdem inveniendi in Analysi per series descriptam, postularet a Collinio et ad se mitteret; et quod sub finem mensis Octobris ejusdem anni Leibnitius videret in manibus Collinij epistolas plures Newtoni Gregorij et aliorum quæ præcipue de seriebus scriptæ erant, et inter alias Epistolam Newtoni ad Oldenburgium 24 Octobris ejusdem datam, ubi Newtonus Analysin illam per series Compendium methodi serierum vocat. Et nondum probatum fuit quod Leibnitius eo tempore non viderit hoc Compendium vel hanc Analysin per series; ubi Methodus fluxionum describitur, & symbola o, ov, oy, $\boxed{\frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}}$, idem significant cum symbolis dz, dy, dx, $\int \frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}$, & similibus, quæ D. Leibnitius postea adhibuit.^[47]

Appendix to 2^d Edit. of Com. Epist

Vbi Commercium Epistolicum prodijt; cum Leibnitius Viennæ agens, per occupationes diversas, per occupationes diversas (ut ipse scripsit) rem tunc discutere non satis posset, ad judicium primarij Mathematici, et harum rerum peritissimi, & a partium studio alieni recurrendum sibi putavit. Is vero ita pronunciavit literis 7 Iunij 1713 datis:

Videtur ^aN . . . . us occasionem nactus, serierum opus multum promovisse per Extractiones Radicum quas primus in usum adhibuit, et quidem in ijs excolendis ut verisimile est, ab initio omnes suuam studium posuit, nec credo tunc temporis vel somniavit adhuc de ^bcalculo suo fluxionum & fluentium ad instar ^bAlgorithmi vel Regularum Arithmeticarum aut Algebraicarum. Ejusque meæ conjecturæ [primum] validissimum indicium est quod de literis x vel y punctis, uno, duobus tribus &c punctis superpositis, quas pro $d\mathrm{x}$, $dd\mathrm{x}$, $d^3\mathrm{x}$; $d\mathrm{y}$, $dd\mathrm{y}$, &c nunc adhibet; ^cin omnibus istis Epistolis [Commercij Collinsiani, unde argumenta ducere volunt] nec colam nec vestigium invenias. Imo ne quidem in Principijs Naturæ Mathematicis N . . . . i ubi calculo suo fluxionum utandi tam ^dfrequentem habuisset occasionem, ejus vel verbulo fit mentio, aut notam hujusmodi unicam cernere licet, sed omnia fere per lineas figurarum sine certa Analysi ibi peraguntur more non ipsi tantum, sed et Hugenio, imò jam antea [in nonnullis] dudum Torricellio Robervallio, Cavallerio, alijs, usitato. Prima vice hæ literæ punctatæ comparuerunt in ^etertio Volumine operum Wallisij, multis annis postquam Calculus differentialis jam ubique locorum invaluisset. Alterum indicium quo conjicer licet Calculum fluxionum non fuisse natum ante Calculum differentialem, hoc est, differentiandi differentialia, N . . . . us nondum cognitam habuerit, quod patet ex ipsis Principijs Phil. Math. ^fubi non tantum incrementum constans ipsius x quod nunc notaret per x punctatum uno puncto, designat per o [more vulgari qui calculi differentialis commoda destruit,] sed etiam Regulam circa gradus ulteriores ^ffalsam dedit [quemadmodum ab eminente quodam Mathematico dudum notatum est.] . . . . . . . . . . . . Saltem apparet N . . . . o ^h rectam methodum differentiandi differentialia non innotuisse longo tempore postquam alijs fuisset familiaris. &c. Hæc ille.

Annotationes in præcedentem sententiam.

a Hæc Epistola, cum celetur et Iudicis et Iudicat nomen, Libellum defamatorium magis sapit, quam probi Iudicis sententiam.

b Algorithmus habetur in Vol. 2 Operum Wallisij, pag. 392, ut <222r> et in Lem. 2 Lib. 2 Princip. et in Epist ad Collins 10 Decem. 1672 & antea in Analysi per series pag. 19, ubi Ordinatim Applicata Curvæ ({ccu} momentum areæ fluentis) ex æquatione Abscissam et Aream involvente deducitur.

c In Analysi per series Newtonus utitur symbolis ov, oy, $\boxed{\frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}}$. In Lem 2. Lib. 2 Princip. utitur symbolis A, B, C; a, b, c. In Introductione ad librum de Quadraturis methodum docet et exempli illustrat absque symbolis. In ipso libro puncta superponit literis.

d Nullam habuit occasionem. Propositiones invenit quidem per Analysin sed inventas demonstravit & edidit synthetice more veterum qui nihil in Geometriam prius admittebant quam demonstratum esset synthetice.

e Volumen tertium prodijt anno 1699, literæ punctatæ comparuerunt in secundo Volumine anno 1693, annis duobus antequam Wallisius audivit calculum differentialem apud exteros celebrari, ideoque antequam calculus differentialis invaluit.

f Eodem sensu Newtonus utitur litera o in Introductione ad Librum de Quadraturis ubi methodum fluxionum expresse docet.

g In Principijs Phil. Math. Newtonus nullam dedit Regulam differentiandi fluentes præterquam in Lem. 2 Lib. 2. Et Regula illa verissima est, et ad gradus ulteriores facile applicatur.

h Recta methodus differentiandi differentialia habetur in Propositione prima Libri de Quadraturis. Hanc Propositionem exemplis in differentijs primis et secundis illustratam Wallisius anno 1693 in secundo operum suorum Volumine pag. 392 edidit. Post tres annos Marchio Hospitalius Regulam similem edidit, et tum demum Methodus differentiandi differentialia cœpit alijs esse familiaris.

Commercium epistolicum prodijt, Prodijt hic Liber sub finem Anni 1712 Ex tempore D. Leibnitius Viennæ agens prætendebat se per occupationes diversas rem tunc discutere non satis posse, sed Mathematicum primarium rerum orasse ut is librum examinaret; & Sententiam vero hujus Mathematici 7 Iun 1713 datam chartæ volanti inclusit & perorbem dispersit; & nunquam postea adduci potuit ut huic libro ulterius responderet. [ Hanc sententiam igitur subjungimus ut ut Responsionem quam D. Leibnitius finalem esse voluit] sententia Iudicis quam Leibnitius finalem esse voluit, hæc erat.

c In Scholio ad Prop. XXXIV Lib. II Princip. probavit absque symbolis aut calculo se partem nobilem methodi inversæ tunc habuisse, fatente Leibnitio. Vide Acta Erudit. pro      1700 pag.

{i} Algorithmus habetur in solutione Prop. 1 Libri de Quadraturis, quam Wallisius edidit anno 1693 in Vol. 2 Operum suorum pag. 392, Newtonus Demonstravit anno 1686 in Lem 2 lib 2 Princip. & adumbravit in Epist. ad Collins 10 Decem 1672, & fundamentum consistuit hujus methodi in libro quem scripsit anno 1671, & exposuit anno 1669 in Analysi sua per series pag 19 ubi Ordinatim Applicata Curvæ (ceu momentum areæ fluentis) ex æquatione Abscissam et Aream involvente deducitur.

c In Analysi per series Newtonus utitur symbolis ov, oy, $\frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}$ eodem sensu quo Leibnitius symbolis dz, dx, $\int \frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}$. In Lem. 2. lib. 2 Princip. utitur symbolis A, B, C; a, b, c. In Introductione – – – superponit literis. In scholio ad Prop. XXXIV Lib. II Princip. per ea quæ ibi dixit de solido minimæ resistentiæ absque calculo, probavit, fatente D. Leibnitio, quod habuit partem quandam nobilem methodi differentialis.

Præfatio.

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, & inde tractatum subsequentem de Quadratura curvilinearum anno 1676 extraxi. In epistola 10 Decem 1672 ad Collinium data methodum momentorum exemplo tangentium more Slusiano ducendarum illustravi, dixique eandem etiam ad abstrusiora problematum genera de Curvitatibus & longitudinibus Curvarum, deque areis & centris gravitatis &c sese extendere, et esse generalem et ad quantitates surdas non hærere, & methodo serierum in scriptis meis intertextam esse: et Collinius exemplar hujus epistolæ mense Iuniori anni 1676 ad D. Leibnitium tunc in Gallia agentem misit. Tandem in componendo libro præcedente de Philosophiæ naturalis principijs mathematicis, tractatu de quadratura curvilinearum plurimum usus sum, et eundem propterea subjungere visum est. ut et Tractatum de methodo differentiali cujus specimen in Lemmate V Lib. III. Princip habetur.

Epistola 10 Decem 1672 ad Collinium data

Tractatus de quadratura Curvlinearum.

Methodus differentialis

Præfatio.

Analysin per series & momenta a me scriptam Barrovius noster anno 1669 ad Collinium misit. Methodos ibi expositas in alio tractatu plenius explicui anno 1671. In Epistola 10 Decem 1672 ad Collinium data methodum momentorum exemplo tangentium more Slusiano ducendarum illustravi, dixique eandem etiam ad abstrusiora Problematum genera de curvitatibus & longitudinibus Curvarum deque areis & centris gravitatis & sese extendere et esse generalem & ad quantitates surdas non hærere: et Collinius exemplar hujus Epistolæ ad D. Leibnitium tunc in Gallia agentem misit idque mense Iunio anni 1676. Et hoc anno sub autumno ex tractatu prædicto quem scripseram anno 1671, tractatum de quadratura curvilinearum extraxi, eodemque plurimum usus sum in componendo libro præcedente de Philosophiæ naturalis principijs mathematicis. Et in lib III de Principijs Philosophiæ Lem. V, specimen dedi methodi cujusdem differentialis. Quapropter tractatum prædictum de quadratura figurarum subjunge visum est, ut et tractatum de methodo illa differentiali.

Corrigenda in Philosop. Transact. pro Ian. et Feb. 171$\frac{4}{5}$.

Pag. 174. lin 21. for ten write fifteen

Pag 176 lin 21. for being write from a copy

Pag 176 lin 24 after of M^r Newton add. The impression was finished in Decem 1710.

Pag 184 lin 3 blot out & M^r Collins.

Pag 191 lin 22 after year 1670 add. From this method & his own M^r Gregory deduced a method of Tangents without computation, & notified it to M^r Collins in a Letter dated 5 Novem. 1670.

Ib. between lin. 23 & 24 insert, conjecturing that it was the same with that of Gregory & Slusius.

Pag. 192

Pag. 194 lin. 6 for [came to the hands of M^r Leibnitz in the end of Winter or] write was seen by M^r Leibnitz in London in November, & a copy thereof came to his hands at Hanover in the beginning of Spring following

Pag 195 lin 23 after 1696 add none of them being printed before the year 1699

Pag 198 lin. penult. for at that time write four years after

Pag 199 lin. 14 after he gave the add following, & put an asterisk after the word letter.

Pag 201 lin. 6 for his write the.

Ib. lin. 28 after he added, insert concerning a branch of this method by which M^r Newton A.C. 1686 found the solidum minimæ resistentiæ

Pag 203 lin 4 after Acta Leipsica insert had not in the least detracted from any body but

Pag 203 lin 9 for novice write new man

Pag 205 lin 16 after rectangle add for there he uses the symbol $\frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}$ in the same sense in which M^r Leibnitz uses the symbol $\int \frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}$

Pag. 207 lin 33 after et 396 add. This was the first Rule made publick for finding second third & fourth differences &c, & is the best:

Pag. 209 lin 21 after compared insert & with Gregories Letter of 5 Novem. 1670 &c

Pag. 212 lin 19 for many years after write in the Acta Eruditorum for Aug. 1693.

Ib. lin. 21 after thereof add & calling it Methodus universalissima pro seriebus.

Pag. 224. lin. 34. after Arguments for in sert the being of –

Pag. 206 lin 32 after Book add published by D^r Wallis in the year 1693 (Vol. 2 pag 392)

Historia Methodi infinitesimalis ex Epistolis antiquis eruta.

Consessus Arbitrorum a Regia Societate constitutus Commercij subsequentis Epistolici exemplaria tantum pauca anno 1712 imprimi curarunt, et ad Mathematicos mitti qui soli de his rebus judicare possent. Cum vero D. Leibnitius huic Libro minime responderet, sed Quæstiones Metaphysicas aliasque ad hanc rem nihil spectantes confugeret, et ejus amici adhuc rixentur: visum est hunc librum una cum ejus Recensione quæ in Transactionibus Philosophicis ante annos impressam fuit, in lucem iterum emittere, ut historia vera ex antiquis monumentis deducta sepositis rixis ad posteros perveniat.

Epistolæ D. Leibnitij ad Oldenburgium scriptæ quæ hic imprimuntur, datæ sunt 3 Feb. 20 Feb. 30 Mar. 26 Apr. 24 Maij & 8 Iunij 1673; 15 Iulij & 26 Octob. 1674; 30 Mar. 20 Maij, 12 Iulij & 28 Decem. 1675; 12 Maij, 27 Aug. & 18 Novem. 1676; 21 Iunij, & 12 Iulij 1677. Et hæ omnes Epistolæ, si tertium et ultimas quinque excipias, descriptæ extant in Libris antiquis epistolicis Regiæ Societatis Numero 6, pag. 35, 34, *, 101, 115, 137; & N^o 7, pag. 93, 110, 213, 235, 149, 189. Et omnium etiam autographa etiam asservantur si duas tantum excipias 27 Aug et 18 Novem 1676 datas & in tertio operum Wallisij volumine impressas. Epistolæ 15 Iulij & 26 Octob. 1674 12 Iulij et 28 Decem. 1675 & 21 Iunij & 12 Iulij 1677 in tertio etiam Operum volumine impressæ sunt. Et hæ omnes Leibnitij Epistolæ una cum Epistolis mutuis Oldenburgij ad Leibnitium, quarum Exemplaria adhuc asservantur, perpetuum constituunt inter eos per Epistolas commercium a die 3 Feb. 1673 ad usque mortem Oldenburgij, si modo epistolæ duæ excipiantur quarum alterâ postulavit Leibnitius Excerpta Epistolarum Gregorij ad se mitti, alteram Oldenburgius cum Excerptis misit. Epistolæ Leibnitij versabantur circa numeros ad usque 8 Iunij 1673. Dein Leibnitio Geometriam altiorem addiscente, Commercium aliquamdiu quievit, et 15 Iulij 1674 renovatum est a Leibnitio sic scribente. Diu est quod nullas a me habuisti Literas. Commercium igitur quod Leibnitius cum Oldenburgio, Collinio et Newtono habuit hic integrum imprimitur præter dictas duas Epistolas. Nam Collinius & Newtonus nullum cum Leibnitio commercium præterquam per Oldenburgium habuerunt. De fide epistolarum impressarum minime dubitatur apud Anglos. Contra fidem earum scripsit Leibnitius quod Arbitrorum Consessus a R. Societate constitutus omnia edidissent quæ contra ipsum facerent, omnia omisissent quæ contra Newtonum. ♂ < insertion from f 224v > ♂ Et per epistolam 25 Aug. 1714 ad D. Chamberlayn datam postulavit ut Societas Regia epistolas nondum editas ad ipsum mitterent. Nam cum Hanoveram, qinuit, rediero, possum etiam in lucem immittere Commercium aliud Epistolicum quod Historiæ Litterariæ inservire possit; & Litteras quæ contra me allegari possunt, non minus publici juris faciam quam quæ pro me faciunt. Hæc Leibnitius. Sed omnes inter ipsum et Oldenburgium Epistolæ, quatenus ad hanc rem spectant continua serie jam ante impressæ fuerunt præter duas quæ non extant et nullius esse momenti videntur.

Attamen ut accusationem suam probaret, scripsit Leibnitius sub finem anni 1715 in Epistola sua prima ad < text from f 224r resumes > sub finem anni 1715 in Epistola sua prima ad Abbatem de Comitibus, quod in secundo ejus in Angliam itinere, Collinius ostendit ipsi partem Commercij sui in qua Newtonus agnoscebat ignorantiam suam in pluribus, dicebatque (inter alia) quod nihil invenisset circa dimensiones Curvilinearum quæ celebrantur præter dimensionem Cissoidis; sed Consessus hoc totum suppressit. Et Newtonus in Epistola sua ad Abbatem 26 Feb. 171$\frac{5}{6}$ respondit hoc non fuisse omissum, sed extare in epistola sua ad Oldenburgium 24 Octob. 1676 missa, & impressum fuisse in Commercio Epistolico pag. 74. lin. 10, 11. Et subinde Leibnitius in Epistola sua ad Abbatem de Comitibus Apr. 9, 1676 agnovit se deceptum fuisse; sed, inquit, exemplum dabo aliud. Newtonus in una Epistolarum ejus ad Collinium agnovit se non posse invenire magnitudinem Sectionum secundarum (vel segmentorum secundorum) sphæroidum & corporum similium; sed Consessus hunc locum vel hanc Epistolam in Comercio epistolico minime edidit. Newtonus autem in Observationibus quas in hanc Leibnitij Epistolam <225r> scripsit, respondit quod si Consessus hoc omisisset, recte omnino fecisset, cum hujusmodi cavillationes ad Quæstionem de qua agitur nil spectent; sed quod Consessum hoc omisit. Collinius in Epistola ad D. Gregorium 24 Decem 1670, & in altera ad D. Bertet 21 Feb. 1671 (utrisque impressis in Commercio pag. 24, 26,) scripsit quod Methodus Newtoni se extenderet ad secunda solidorum segmenta quæ per rotationem generantur. Et Oldenburgius idem scripsit ad ipsum Leibnitium 8 Decem. 1674, ut videre est in Commercio, pag. 39.

Vbi primum Commercium Epistolicum lucem vidit, D. Leibnitius ne libro responderet, prætendit per biennium se librum non vidisse sed ad judicium primarij Mathematici & a partium studio alieni provocasse cum ipse per occupationes diversas rem tunc discutere non satis posset. Et sententiam hujus Mathematici 7 Iunij 1713 datam, in alia charta maxime scurrili die 29 Iulij data descripsit, & utramque per Europam spargi curavit sine nomine vel Mathematici vel Impressoris vel Vrbis in qua impressa fuit. Mathematicus in scripto illo edito Bernoullium ipsum scripsit, cum tamen Bernoullius judex æquus constitui non posset nisi ipse juri omni in methodum infinitesimalem prius renunciasset.

Cæterum Leibnitius in prima sua ad Abbatem de Comitibus Epistola scripsit quod ij qui contra ipsum scripsissent, (id est, Consessus a Regia Societate constitutus) candorem ejus aggressi essent per interpretationes duras et male fundatas; et quod illi voluptatem non habebunt videndi Responsa ejus ad pusillas rationes eorum qui ijs tam male utuntur. Interpretationes illæ nullius quidem sunt authoritatis nisi quam ab Epistolis derivant: at male fundatas esse D. Leibnitius nunquam ostendit.

Subinde Newtonus in prima sua ad Abbatem Epistola, 26 Feb. 171$\frac{5}{6}$, his ita respondebat. D. Leibnitius hactenus respondere recusavit, bene intelligens impossibile esse facta confutare. Silentium suum hac in re excusat allegando impræsentia quod librum non vidisse, & quod otium illi non esset ad examinandum, sed quod orasset Mathematicum celebrem ut hoc negotium in se susciperet — Vtitur et novo prætextu ne respondeat dicendo quod Angli, qui Commercium ediderunt, voluptatem non habebunt videndi responsum ejus ad pusillas eorum rationes, et proponendo disputationes novas philosophicas ineundas et Problemata solvenda; quæ duo ad rem nil spectant.

D. Leibnitius autem in proxima sua ad Abbatem Epistola 9 Apr. 1716 data, pergebat se excusare ne responderet, dicendo: Vt operi <226r> contra me edito sigillatim respondeam, opus erit alio opere non minore quam hoc est; percurrendum erit corpus magnum minutorum ante annos 30 vel 40 præterritorum quorum perparvum reminiscor; examinandæ erunt veteres Epistolæ quarum plurimæ sunt perditæ, præterquam quod maxima ex parte non conservari minuta mearum, et reliquæ sepultæ sunt in maximo chartarum acervo quem non possum sine tempore et patientia discutere. Sed otium minime mihi suppetit, alijs negotijs alterius prorsus generis occupato. Hactenus Leibnitius.

Attamen post ejus mortem (quæ contigit mense proximo Septembri) in Elogio ejus quod in Actis Eruditorum pro mense Iulio anni 1717 impressum est, amici ejus scripserunt quod Commercio Epistolico Anglorum aliud quoddam suum idemque amplius opponere decreverat; et quod paucis ante obitum diebus Cl. Wolfio significavit se Anglos famam ipsius lacessentes reipsa refutaturum; quamprimum enim a laboribus historicis vacaturus sit, daturum se aliquid in Analysi prorsus inexpectatum, et cum inventis quæ hactenus in publicum prostant, sive Newtoni sive aliorum, nihil quicquam affine habens. Hæc illi. Verum ex jam dictis patet quod is aliud quoddam cum Oldenburgio Commercium Epistolicum non habuit. Et inventum novum his nihil affine habens ad rem nihil spectat. Missis somnijs, Quæstio tota ad Epistolas antiquas referri debet.

Ad hanc Quæstio

Historia Methodi Infinitesimalis ex Epistolis antiquis erata Præfatio.

Consessus Arbitrorum a Regia Societate constitutus Commercij subsequentis Epistolici exemplaria tantum pauca anno 1712 imprimi curarunt et ad Mathematicos mitti qui soli de his rebus judicare possent. Cum vero D. Leibnitius & ejus amici huic Libro minime responderent, sed ad rixas Metaphysicas aliasque confugerent quæ ad hanc rem nihil spectant, atque adhuc (D. Leibnitio licet mortuo) rixentur: visum est missis rixis hunc librum una cum ejus Compendio quod in Transactionibus Philosophicis pro mense Ianuario anni 171$\frac{4}{5}$ impressam est & cui etiam nullum responsum adhuc datum est, in lucem iterum emittere, ut historia vera ad posteros perveniat.

Epistolæ D. Leibnitij ad D. Oldenburgium scripta quæ hic imprimuntur, datæ sunt 3 Feb. 20 Feb. 30 Mar. 26 Apr. 24 Maij, & 8 Iunij, 1673; 15 Iulij, & 26 Octob. 1674; 30 Mar. 20 Maij, 12 Iulij & 28 Decem 1675; 12 Maij, 27 Aug. & 18 Novem. 1676; 21 Iun ij, & 12 Iulij 1677. Et hæ omnes Epistolæ, si tertium & ultimas quinque excipias descriptæ extant in Libris Epistolicis Regiæ Societatis N^o 6, pag. 35, 34, *, 101, 115, 137; & N^o 7 pag. 93, 110 213, 235, 149, 189. Et omnium etiam autographa adhuc asservantur, Si duas tantum excipias 27 Aug & $\frac{18}{28}$ Novem 1676 datas & in tertio operum Wallisij volumine impressas. Epistolæ 15 Iulij & 26 Octob. 1674, 12 Iulij & 28 Decem 1675, & 21 Iunij & 12 Iulij 1677 in tertio etiam Operum Wallisij Volumine impressæ sunt. Et hæ omnes Leibnitij Epistolæ una cum Epistolis mutuis Oldenburgij ad Leibnitium quarum exemplaria adhuc asservantur, perpetuum constituunt inter eos per Epistolas commercium a die 3 Feb 1673 ad usque mortem Oldenburgij, si modo Epistolæ duæ excipiantur quarum alterâ Leibnitius postulavit Excerpta Epistolarum Gregorij ad se mitti, alterâm Oldenburgius cum Excerptis illis misit. Epistolæ Leibnitij versabantur circa numeros adusque 8 Iunij 1673. Deinde Leibnitio Geometriam altiorem addiscente, commercium per literas aliquandiu quievit, Et 15 Iulij 1714 renovatam est a Leibnitio sic incipiente: Diu est quod nullas a me habuisti literas. Commercium igitur quod Leibnitius cum Oldenburgio, Collinio et Newtono habuit hic integrum imprimitur præter dictas duas Epistolas. Nam Collinius & Newtonus nullum cum Leibnitio commercium præterquam per Oldenburgium habuerunt. De fide Epistolarum impressarum minime disputatur apud Anglos.

Contra fidem earum scripsit Leibnitius quod Arbitrorum Consessus a R. Societate constitutus omnia edidissent quæ contra Leibnitium facerent, omnia omisissent quæ contra Newtonum. Et ut hoc probaret scripsit sub finem anni 1715, in Epistola sua prima ad Abbatem de Comitibus, quod in secundo ejus in Angliam itinere, Collinius ostendit ipsi partem Commercij sui in qua Newtonus agnoscebat ignorantiam suam in pluribus, dicebatque (inter alia) quod nihil invenisset circa dimensiones Curvilinearum quæ celebrantur præter dimensionem Cissoidis; Sed Consessus hoc totum suppressit. Et Newtonus in Epistola sua ad Abbatem 26 Feb. 171$\frac{5}{6}$ respondit hoc non omissum fuisse sed extare in Epistola sua ad Oldenburgium 24 Octob. 1676 missa & impressum fuisse in Commercio Epistolico pag. 74. lin. 10, 11. Et subinde D. Leibnitius in Epistola sua ad Abbatem de Comitibus Apr. 9, 1676 agnovit se deceptum fuisse; sed, inquit, exemplum dabo aliud. Newtonus in una Epistolarum ejus ad Collinium agnovit se non posse invenire magnitudinem Sectionum secundarum (vel segmentorum secundorum) sphæroidarum et corporum similium; sed Consessus hunc locum vel hanc Epistolam in Commercio Epistolico minime edidi. Newtonus autem in observationibus suis quas in hanc Leibnitij Epistolam fecit, respondit quod si Consessus hoc omisisset, recte omnino fecisset, cum hujusmodi cavillationes ad questionem de qua agitur, nil spectent: Sed quod Consessus hoc non omisit. Collinius enim in Epistola ad Ia. Gregorium 24 Decem. 1670, & in altera ad D. Bertet 21 Feb. 1671 (utrisque impressis in Commercio pag 24, 26,) scripsit quod Methodus Newtoni se extenderet ad secunda solidorum segmenta quæ per rotationem generantur. Et Oldenburgius idem scripsit ad ipsum Leibnitium 8 Decem. 1674. Vide Commerc. p. 39

Vbi primum Commercium Epistolicum lucem vidit, D. Leibnitius ne libro responderet prætendit per biennium se librum non vidisse, sed ad judicium primarij Mathematici & a partium studio alieni provocasse cum ipse per occupationes diversas rem tunc discutere non satis posset. Et sententiam hujus Mathematici 7 Iunij 1713 datam in alia charta maxime scurrili die 29 Iulij data descripsit et per Europam spargi curavit sine nomine vel Mathematici vel Impressoris vel Vrbis in qua impressa fuit. ♀ < insertion from f 227v > ♀ Mathematicus in scripto suo edito Bernoullium citaverat tanquam a se diversum. Leibnitius vero sub finem anni 1715 citationem delevit et Mathematicum esse Bernoullium ipsum scripsit cum tamen Bernoullius judex æquus constitui non posset nisi ipse juri omni in methodum infinitesimalem prius renunciasset. Huic Iudici contrarius et Wallisius Oxoniensis, vir antiquus qui Literas Newtoni anno 1676 ab Oldenburgo acceperat et rem intellexerat ab initio, & is anno 1695 in Præfatione ad operum suorum Volumina duo prima sententiam pro Newtono tulit, Dominis Leibnitio & Menkenio per ea tempora non mussitantibus. Dixit utique

Cæterum D. Leibnitius ut testimonia antiqua eludaret, anno 1714 per Epistolam ad D. Chamberlain 25 Aug. datam postulavit ut Societas Regia < text from f 228r resumes > Et anno proximo, per Epistolam ad D. Chamberlain 25 Aug 1714 datam, postulavit ut Societas Regia Epistolas nondum editas ad ipsum mitterent. Nam cum Hannoveram, inquit, rediero, possum etiam in lucem emittere Commercium aliud Epistolicum quod Historiæ Litterariæ inservire possit; et Literas quæ contra me allegari possunt non minus publici juris faciam quam quæ pro me faciunt. Hæc Leibnitius. Sed accusationem probare non potuit ut jam vidimus; Et omnes inter ipsum & Oldenburgium Epistolæ quatenus ad hanc rem spectant continua seriæ jam ante impressæ fuerunt præter duas quæ non extant, & quantum sentio nullius sunt momenti.

Cæterum In prima sua ad Abbatem de Comitibus Epistola dixit D. Leibnitius quod ij qui contra ipsum scripsissent (id est Concessus a Societate Regia constitutus) candorem ejus aggressi essent per interpretationes duras et male fundatas; et quod illi voluptatem non habebunt videndi responsa ejus ad pusillas rationes eorum qui ipsum tam male tractassêt. . ✝ < insertion from f 227v > ✝ Interpretationes illæ nullius quidem sunt authoritatis nisi quam ab Epistolis derivant; at male fundatas esse D. Leibnitius nunquam ostendit. < text from f 228r resumes > Interpretationes illæ nullius quidem sunt authoritatis nisi quam ab ipsis Epistolis derivant: at male fundatas esse D. Leibnitius nunquam ostendit.

Subinde In priima sua ad Abbatem Epistola, 26 Feb. 171$\frac{5}{6}$ data, Newtonus his ita respondebat. D. Leibnitius hactenus respondere recusavit, bene intelligens impossibile esse facta confutare. Silentium suum hac in re excusat allegando impræsentia quod librum nondum vidisset, et quod otium illi non esset ad examinandum, sed quod orasset mathematicum celebrem ut hoc negotium in se susciperet &c. — Vtitur < insertion from f 227v > Vtitur et novo novo prætextu ne respondeat, dicendo quod Angli, qui Commercium ediderunt voluptatem non habebunt videndi responsum ejus ad pusillas eorum rationes, et proponendo disputationes novas Philosophicas ineundas et Problemata solvenda; quæ duo ad rem nil spectant. < text from f 228r resumes >

D. Leibnitius autem in proxima sua ad Abbatem Epistola 9 Apr. 1716 data, pergebat se excusare ne responderet, dicendo: Vt operi contra me edito perfecte respondeam, opus erit alio opere non minore quam hoc est; percurrendum erit corpus magnum minutorum ante annos 30 vel 40 præterritorum quorum perparvum reminiscor; examinandæ erunt veteres Epistolæ quarum plurimæ sunt perditæ, præterquam quod maxima ex parte non conservari minuta mearum, et reliquæ sepultæ sunt in maximo chartarum acervo quem non possum sine tempore et patientia discutere . Sed otium minime mihi suppetit alijs negotijs alterius prorsus generis occupato. Hactenus Leibnitius.

Attamen post ejus mortem (quæ contigit mense proximo Septembri) in Elogio ejus quod in Actis Eruditorum pro mense Iulio anni 1717 impressum est, amici ejus scripserunt quod Commercio Epistolico Anglorum aliud quoddam suum idemque amplius opponere decreverat; et quod paucis ante obitum diebus Cl. Wolfio significavit se Anglos famam ipsius lacessentes reipsa refutaturum; quamprimum enim a laboribus historicis vacaturus sit, daturum se aliquid in Analysi prorsus in expectatum, et cum inventis quæ hactenus in publicum prostant, sive Newtoni sive aliorum, nil quicquam affine habens. Verum ex jam dictis patet quod is Aliud quoddam cum Oldenburgio Commercium Epistolicum habere non potuit. Et non habuit. Et Inventum novum his nihil affine habens, ad rem nihil spectat. Missis somnijs disputatio tota ad Epistolas antiquas referri debet.

Quo tempore Commercium epistolicum prodijt, D. Leibnitius Viennæ agebat et pretendebat se per occupationes diversas rem tunc discutere non satis posse , sed Mathematicum primarium & harum rerum peritissimum & a partium studio alienum orasse ut is librum examinaret, et cum literis 7 Iunij 1713 datis ita pronunciasse.

a. Hæc Epistola – – – – sententiam. Sed cum Leibnitius nullum aliud huic Libro responsum unquam dedit, visum est

Et per epistolam 25 Aug. 1714 ad D. Chamberlain missam postulavit ut Societas Regia Epistolas — nullius sunt momenti.

Attamen ut accusationem suam probaret scripsit Leibnitius sub finem anni 1715 in epistola sua prima ad Abbatem de Commitibus — — quamprimum is audivit methodum infinitesimalem apud exteros Leibnitij nomine celebrari, in Præfatione

1693. Hæc autem symbola ad methodum non sunt necessaria. omnium optima. In eadem Propositione habetur Algorithmus methodi, ut et in Lemmate secundo libri Principiorum. Sit æquatio $\mathrm{a}\mathrm{x}\mathrm{x}-\mathrm{x}\mathrm{x}\mathrm{y}+{\mathrm{y}}^3-\mathrm{b}\mathrm{z}\mathrm{z}=0$. Fluat sola x et per Lemma illud fluxio totius erit $2\mathrm{a}\mathrm{x}\stackrel{.}{\mathrm{x}}-2\mathrm{x}\stackrel{.}{\mathrm{x}}\mathrm{y}$ Fluat sola y et fluxio totius erit $-\mathrm{x}\mathrm{x}\stackrel{.}{\mathrm{y}}+3\mathrm{y}\mathrm{y}\stackrel{.}{\mathrm{y}}$ Fluat sola z et fluxio totius erit $2\mathrm{b}\mathrm{z}\stackrel{.}{\mathrm{z}}$. Fluant omnes & cum summo fluentium $\mathrm{a}\mathrm{x}\mathrm{x}-\mathrm{x}\mathrm{x}\mathrm{y}+{\mathrm{y}}^3$ $-\mathrm{b}\mathrm{z}\mathrm{z}$ semper sit $=0$, summa fluxionum $2\mathrm{a}\mathrm{x}\stackrel{.}{\mathrm{x}}-2\mathrm{x}\stackrel{.}{\mathrm{x}}\mathrm{y}-\mathrm{x}\mathrm{x}\mathrm{y}+3\mathrm{y}\mathrm{y}\stackrel{.}{\mathrm{y}}+2\mathrm{b}\mathrm{z}\stackrel{.}{\mathrm{z}}$ erit $=0$. Sed et idem Algorithmus ex Epistola Newtoni 10 Decem 1672 ad Collinium data facillime colligitur. In eodem secundo volumine pag. 391, edita – – – Wallisius affirmavit.

Sunto quantitates datæ a, b, c; fluentes x, y, z; fluxiones p, q, r; & momenta op, oq, or: & proponatur æquatio quævis ${\mathrm{x}}^3-2\mathrm{x}\mathrm{x}\mathrm{y}+\mathrm{b}\mathrm{x}\mathrm{x}-\mathrm{b}\mathrm{b}\mathrm{x}+\mathrm{b}\mathrm{y}\mathrm{y}-{\mathrm{y}}^3+\mathrm{c}\mathrm{z}\mathrm{z}=0$. Et per hoc Lemma, si sola fluat x erit fluxio totius $3\mathrm{x}\mathrm{x}\mathrm{o}\mathrm{p}-4\mathrm{x}\mathrm{o}\mathrm{p}\mathrm{y}+2\mathrm{b}\mathrm{x}\mathrm{o}\mathrm{p}-\mathrm{b}\mathrm{b}\mathrm{o}\mathrm{p}$ : si sola fluat y erit fluxio totius $\mathrm{-2}\mathrm{x}\mathrm{x}\mathrm{o}\mathrm{q}+2\mathrm{b}\mathrm{y}\mathrm{o}\mathrm{q}-3\mathrm{y}\mathrm{y}\mathrm{o}\mathrm{q}$, si sola fluat z erit fluxio totius $2\mathrm{c}\mathrm{z}\mathrm{o}\mathrm{r}$ si fluant omnes erit fluxio totius $3\mathrm{x}\mathrm{x}\mathrm{o}\mathrm{p}-4\mathrm{x}\mathrm{o}\mathrm{q}\mathrm{y}+2\mathrm{b}\mathrm{x}\mathrm{o}\mathrm{q}-\mathrm{b}\mathrm{b}\mathrm{o}\mathrm{q}-2\mathrm{x}\mathrm{x}\mathrm{o}\mathrm{q}+2\mathrm{b}\mathrm{y}\mathrm{o}\mathrm{q}-3\mathrm{y}\mathrm{y}\mathrm{o}\mathrm{q}+2\mathrm{c}\mathrm{z}\mathrm{o}\mathrm{r}$. 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 viz $3\mathrm{x}\mathrm{x}\mathrm{p}-4\mathrm{x}\mathrm{p}\mathrm{y}+2\mathrm{b}\mathrm{x}\mathrm{p}-\mathrm{b}\mathrm{b}\mathrm{p}-2\mathrm{x}\mathrm{x}\mathrm{q}+2\mathrm{b}\mathrm{y}\mathrm{q}-3\mathrm{y}\mathrm{y}\mathrm{q}+2\mathrm{c}\mathrm{z}\mathrm{r}=0$. Continent igitur hoc Scholium solutionem Propositionis prædictæ Data æquatione fluentes quotcunque quantitates involvente invenire fluxiones.

Hanc Propositionem posui in Epistola mea 24 Octob 1676 ad Oldenburgum data ut fundamentum methodi de qua scripseram anno 1671. Eademque methodus facile colligitur ex epistola quam ad Collinium 10 Decem 1672 scripsi & cujus exemplar ad D. Leibnitium missa fuit anno 1676. Sit ACc Linea quævis curva, AB ejus Abscissa & Bb momentum Abscissæ, Et sint BC, bc ordinatæ duæ, Cc momentum Curvæ & ce momentum Abscissæ existente BCeb parallelogrammo. Et Cc producatur donec Abscissæ occurrat in D et erunt triangula Cce, DCB similia. Et ubi momentum Cc in infinitum diminuitur recta CeD Curvam tanget in C. Hoc omnibus notum est. Iam dicatur AB x et BC y et sint earum fluxiones p & q, & habitudo inter x et y exprimatur qualibet æquatione puta ${\mathrm{x}}^3-{}^2\mathrm{x}\mathrm{x}\mathrm{x}\mathrm{y}+\mathrm{b}\mathrm{x}\mathrm{x}-\mathrm{b}\mathrm{b}\mathrm{x}+\mathrm{b}\mathrm{y}\mathrm{y}-{\mathrm{y}}^3$. Et per Epistolam illam Regula ducendi Tangentem hæc est Multiplica æquationis terminos per quamlibet progressionem Arithmeticam juxta dimensiones y, puta $\begin{array}{c}{\mathrm{x}}^3-2\mathrm{x}\mathrm{x}\mathrm{y}+\mathrm{b}\mathrm{x}\mathrm{x}-\mathrm{b}\mathrm{b}\mathrm{x}+\mathrm{b}\mathrm{y}\mathrm{y}-{\mathrm{y}}^3\\ 010023\end{array}$ ut et juxta dimensiones x, puta $\begin{array}{c}{\mathrm{x}}^3-2\mathrm{x}\mathrm{x}\mathrm{y}+\mathrm{b}\mathrm{x}\mathrm{x}-\mathrm{b}\mathrm{b}\mathrm{x}+\mathrm{b}\mathrm{y}\mathrm{y}-{\mathrm{y}}^3\\ 322100\end{array}$. Prius productum erit Numerator, & posterius divisum p ex x denominator fractionis quæ exprimet logitudinem BD, ad cujus extremitatem D, ducenda est Tangens CD Est perg 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 y ad −BD et ita est ce ad−eC seu oq ad−op. Ducantur extrema et media in seinvicem et æquatio per o divisa evadet $3\mathrm{x}\mathrm{x}\mathrm{p}-4\mathrm{x}\mathrm{p}\mathrm{y}+2\mathrm{b}\mathrm{x}\mathrm{p}-\mathrm{b}\mathrm{b}\mathrm{p}=2\mathrm{x}\mathrm{x}\mathrm{q}-2\mathrm{b}\mathrm{y}\mathrm{q}+3\mathrm{y}\mathrm{y}\mathrm{q}$, seu $3\mathrm{x}\mathrm{x}\mathrm{p}-4\mathrm{x}\mathrm{p}\mathrm{y}-2\mathrm{b}\mathrm{x}\mathrm{p}-\mathrm{b}\mathrm{b}\mathrm{p}-2\mathrm{x}\mathrm{x}\mathrm{q}+2\mathrm{b}\mathrm{y}\mathrm{q}-3\mathrm{y}\mathrm{y}\mathrm{q}=0$. This is the same equation with that produced by the Lemma when there are but two fluent quantities. And when there are more then two, the same operation applied to them all will give the Equation desired involving the fluxions. Et in hac operatione fundatur methodus generalis, uti dixi in Epistola illa, quæ extendit se, citra molestum ullum calculum, non modo ad ducendum tangentes — eas ad series infinitas, &c. id est in Tractatu illo quem scripsi anno 1671, uti etiam dixi in Epistola mea prædicta 24 Octob. ad Oldenburgum data.