Extract from Bernoulli's letter of 7 June 1713, with Newton's Observations on it

Remarques sur la dispute entre Monsieur de Leibnitz et Monsieur Newton, touchant l'invention de l'Arithmetique differentiale.

Ce que bon trouve sur ce sujet dans le Iournal literaire de la Hage, et dans nos actes allemans p: 587 ne s'accorde pas en ses circonstances, et celuy qui l'a conou n'a pas êté bien instruit.

Monsieur de Liebnitz, et Monsieur Newton jamais n'ont eu des disputes entre eux sur aucune matiere; Car Monsieur Newton n'a jamais fait connoitre vouloir s attribuer à luy, et disputer à Monsieur de Leibnitz l'invention de l'Arithmetique differentiale.

Monsieur de Leibnitz n'a su, que par le rapport de quelques-uns, qui ont lu le Commercium Epistolicum imprimé à Londres il y a quelque tems, que Monsieur Newton eut part à une telle injustice, que Monsieur de Leibnitz êtant encore à Viénne, et qui n'a pas-encoure vu ce traité attribue plûtôt à la mauvaise volunté de les envieux ou des gens in|g|norans. Aussy n'en a t-il jamais porté ses plaintes à la Societè Royale en Angleterre, comme superflu dans une cause si juste: seulement a-t-il pris occasion d'écrire à Monsieur le Secretaire de la Societé, qu'il ne doutoit point, que la Societé A Monsieur Newton même desaprouveroit entieérement un tel procedé. En sorte que la Societé n'a pas pu examiner les argumens des deux parties, et prononcer un arrêt definitif.

Voicy la verité du fait: Il y a environ quarante ans, plus ou moins, que Monsieur de Leibnitz, Oldenbourg, Newton, Collin's et autres ont eu un Commerce des lettres entre eux, dont il se trouve quelque chose d'imprimé <487v> dans la troìsiéme partie des Oeuvres Mathematiques de Wallisius. Il paroit par les lettres publieês par Wallisius que Monsieur Newton faìsoìt mystere d'une invention laquelle il a ensuite debité être l'Arithmetique differentiale, au lieu que Monsieur de Leibnitz luy communiquoit de bonne foi le fonds de cette Arithmétique, quoy qu'il ait paru du depuis, que Monsieur Newton ne l'avoit pas bien compris, sur tout ce qui regarde differentias differentiarum; Aprés cela on a trouve encore d'autres lettres ecrites entre Collins et Ses Amis, que l'on a fait imprimer à Londres avec des notes, par lesquelles on pretend prouver par des conjectures mal fondèes, et par des suppositions faus, Ses, que Monsieur Newton a inventé l'Arithmetique differentiale, et \que/ Monsieur de Leibnitz l'a aprise de luy, non obstant que le contraire paroit clairement par les lettres de ces deux Messeurs publiées par Monsieur Wallisius.

L'auteur de ces remarques a jugé trop temerairement de choses qui me luy ont pas êté assés connues; et il a mal reussi à deviner, {illeg}|p|ar quelle voye Monsieur de Leibnitz est parvenu à cette invention. Il s'est trouve de plus que Monsieur Newton n'a pas bién compris la vraye Arithmetique differentiale, lors qu'en 1687 il mit au jour Ses Principia Philosophiæ naturalis Mathematica, vu qu'il ne s'en est point servi, lorsqu'il en avoit la meilleux occasion du monde; mais a même commis des fautes capitales, tout-à-fait contraires aux principes de l'Arithmetique differentiàle, comme un Mathematicién entierement impartial a remarqué le premier. Aussy Monsieur Newton, aprés en avoir été a verti, a corrigé l|c|es fautes, en faisant changer quelques feilles dans la nouvelle impression, qui à paru l'année passée. Monsieur de Leibnitz avoit déja auparant publié son Arithmetique en l'annee 1685|4|: mais Monsieur Newton n'a rien donné sur cette matiere, jusqu'à ce qu'en 1693 le 2^me Tome des Oeuvres Mathematiques de Wallisius vit le <488r> jour, lorsque le systeme de Monsieur de Leibnitz êtoit déja en reputation par tout, et que partientierement les deux fréres Messieurs Iaques et Iean Bernoulli en eussent fait publiquement l'essais avec beaucoup d'aplau{illeg}|d|issement; ce qui rendoit sans doute Monsieur Newton /quoyqu'un peu trop tard:/ Si temeraire pour vouloir y avoir part.

L'en voyoit d'abord ches Wallisius l'invention de Monsieur de Leibnitz se presenter, mais en d'autres figures et termes plus impropres. Cependant Monsieur Newton n'any alors, ny longtems aprés, eu l'audace, de troubler Monsieur de Leibnitz dans la possession de son invention Et tant que Hugenius et Wallisius, qui El{illeg}|êtoient| que|des| Iuges impartiaux, vivoient, lesquels avount des connoissances â fond de cette affaire, il a bién jugé qu'il ne reussiroìt pas; C'est pourquoy il a attendu jusqu'à ce qu'il n'y eut plus personne de ceux, qui puissent être témoins du progrés de cette science, et qui mêmes y avoient beaucoup de part, Mais maintenant il a recours à des novices, qui ignorent ce qui s'est passe autrefois, et qui jugent selon leurs preoccupations et leurs sottes passions Un certain Novice s'est voulu mettre en reputation, en attaquant Monsieur de Leibnitz et luy a{illeg}|d|sersant une espece de defi; Mais Monsieur de Leibnitz voyant, que cet homme n'êtoit pas d'humour à se laisser detromper, il n'a pas voulu s'engager avec luy. Et il a fort bièn fait, car sans cela il auroit eu un pretexte |p|de|ou||r| dire, que l'on avoit les argumens de deux parties, sur lesquels on pouvoit decider; au lieu que les Iuges pretendus n'ont que les argumens d'un coté seul.

Dans cette vuë l'on a fait imprimer le Commercium Epistolicum, dans lequel on croit avoir trouvé sur quoy se fonder, bién qu'il ne s'y trouve rién du tout <488v> qui puisse decider la dispute d{illeg}|u| veritable Inventeur de l'Arithmetique differentiale. Et Monsieur Newton a commis la foiblesse de se laisser entrai{t} sur ces fausses apparences. S'il avoit gardé le silence, il seroit demeuré participant de l'invention attendu que Monsieur de Leibnitz avoit cru sur sa parole, qu'il avoit trouvé quelque chose, qui aprochat en quelque façon de l'Arithmetique differentiale Mais presentement se trouve le contraire.

Des Gens entendus et impartiaux ont ris d'une pretension si tardive et si mal fondée; et l'on a déja fait imprimer l'opinion d'un celebre Mathematicièn fondée {h}on seulement sur le long silence, mais /: ce qui plus est./ sur les fautes de Monsieur Newton, lesquelles prouvent evidemment, qu'il n'a pas seulement compris ce qu'il pretend avoir inventé avant Monsieur de Leibnitz il y a 40 ans.

Remarques upon the dispute between M^rons^r Leibnitz & Mons^r Newton touching the invention of the differential characters Arithmeti

That w^ch is found upon this subject in the Iournal literaire a|o|f the Hague & in our German Acts p. 587 do not agree in their circumstances, & he that conceived it was not well instructed.

Mons^r Leibnitz & Mons^r Newton never had any disputes about any between them about any matter. For Mons^r Newton never made it known that he \would/ attributed to himself & dispute with M^r Leibnitz the invention of the differential Arithmeti.

M^r Leibnitz knew not but by the report of some who had read the Commercium Epistolicum printed at London some time ago that M^r Newton had a share in such a piece of injustice, seing that M^r Leinitz was still at Vienna & had not seen the book treatise attributed for the most part to the ill will of thi|e| enemies \envious/ & or of ignorant people. So he has not yet sent his complaint to y^e Society Royal Society in England as being superfluous in a cause so just. He only took occasion to write to the Secretary of the Society that he doubted not at all but the Society & M^r Newton himself would entirely disapprove such a proceeding. So that the Society has not been able t{illeg} at all to examine the arguments on both sides & to pronounce a definitive sentence

See the truth of the fact. It is about 40 years more or less that M^rs Leibnitz Oldenburg Newton Collins & others had a commerce of Letters with one another^[1]

Commercium Epistolicum Collinij et aliorum prodijt anno ineunte 1713. Et post mensis sex vel septem prodijs Resposum {sic} subsequens in Germania sine nomine vel autoris vel Typographi vel Vrbis in qua impressa fuit.

29 Iulij 1713.

L . . . . us nunc Viennæ Austriæ agens ob distantiam locorum nondum vidit libellum in Anglia nuper editum &c — nec vitium paucorum genti imputari debet.

In scripto hocce diffamatorio ma{illeg}lico me|a|ledico, fin dicitur Leibnitiū nondum vidisse Commercium Epistolicum sed a Mathematico pr quodam primario postulasse ut is re examinata judicium suum proferret, & Mathematicum literis 7 Iunij datis resp \D./ Leibnitio respondisse. Vnde patet Leibnitium ipsum curasse ut hæc ederentur. Dicitur etiam in hoc scripto quod Modum quo L . . . . us invenit seriem Gregorio ascriptam ipse statim Hugenio B. Lutetiæ agenti communicavit qui et per Epistolam laudavit. Et quid L . . . . . us & Hugenius ante annos \feret/ quadraginta Lutetiæ æ{illeg}|eg|issent, solo Leibnitio innotescere potuit. Ideo L . . . . us scriptum hocce diffamatorium composuit. Nam et \phrasis/ [illaudabili laudis amore] Leibnit stylum Leibnitianum sapit. Vtrum vero Mathematicus ille primarius sit L . . . . us ipse {illeg}lius quisquam nondum constat.

In Diario Literario \Iohnsoni/ pro mensibus Novembri ac Decembri anni 1713 impressa|u|m fuit hoc scriptum Gallice versum cum Præfatio Epistola sequente vel a L . . . . o vel ab aliqua præfixa

Pag. 2. l. 21. N . . . . us n|N|on differentias sed Fluxiones quæ differentiæ non sunt N . . . . us quando literis punt|c|tatis quando alijs notis pro lubitu designat. L . . . . us pro differe fluxionibus nullas habet notas. N . . |. . us| pro differentijs ponit rectangula sub fluxionibus & momento temporis. Hoc fecit in Analysi anno 1669 ad Collinium missæ. Hoc fecit in Ad p 2 l. 16 Iudex noster Tractatur de Qua Analysi ad Collini Analysi {illeg}|a|nno 1669 ad Collinium missæ. Hoc facit us hodie

Ad p. 2 l. 16. Quamvis Newtonus \anno 1669/ specimen dedit hujus Analy methodi in Anno anno 1669 in Analysi prædicta;

Ad p 2. l. 16 Quamvis Newtonus in Epistola sua ad Oldenburgum anno \24 Octob./ 1676 data Methodum fluxionum hac sententia complexus sit: [Data æquatione fluentes quotcun quantitates involvente fluxiones invenire et vice versa,] at iterum hacce, t{illeg} [{Ea} [fluentem ex æquatione fluxione m|s| involvente extrahe{illeg}/re\] & librum de hac methodo et methodo serierum cujus anno 1671 \se/ scripsisse testatus sit & in Epistola 10 Decem 1672 ad Collinium data se universalitatem hujus methodi descripsit & in Analysi \anno 166{9}/ ad Collinium missæ specimen dedit hujus calculi, tamen Iudex a L . . . . o constitutus credit Newtonum per ea tempora de Calculo fluxionum ne quidem somniasse

Ad p. 2. l. 20 Eodem argumento probare potuit {illeg}|N|ewtonum non habuisse methodum fluxionum ubi scripsit Introductionem ad Quadraturam Curvarum quia in {illeg} Introductione illa non utitur literis punctatis.

Ad l. 2. Annon vidit?

Ad l. 7. Pri{illeg}|m|am a Barrovio

Ad l. 8, 9. \Cum amicis ex{c}o{illeg} c{illeg}ta est anno 1690/ Newtonus edidit Anno 1687 in Schol. ad Lem II Lib. II Princip. I

Ad l. 13, 14. Communicavit Collini{illeg}|o| Anno 1669 & Leibniti\. . . ./ o anno 1676.

Ad pag. 2. l. 6, 7. Calculum fluxionum ad imitationem calculi differentialis fo{illeg}|rm|atum fuisse finxit mu fingere cœpit in Actis Eruditorum pro mense Ianuario anni 16 1705 & inde nata est hæc controversia Figmentum vero confirmavit in Epistola sua ad D. Sloane 29 Decem 1711 data: {illeg} in Et inde nata est hæc controversia.

Ad l. 4 Vox [illaudabilis] {illeg}|a| L . . . . o solo usurpari solet.

Ad l. 9 Iudicem anonymus|m| L . . . . us constituit, id est, vel seipsum vel amicum cui maxime potuit confidere.

Ad l. 16, 19. Mathematicus tantum c{illeg}|o|njectatur, non probat.

Ad l. 21. N . . . . us literas punctatas non adhibet \non/ pro diffentijs {sic} \vel momentis/ sed \adhibet/ pro fluxionibus quæ differentiæ non sunt. Et hujusmodi pro fluxionib|es|us his quando literas punctatas quando alia|j|s nota|i|s adhibet \designat/ L . . . . us pro fluxionibus nullas habet Notas.

Ad l. 22. In Analysi ad Collinsium anno 1669 \missæ/ fluxiones \arearum/ designantur per literas ordinatas curvarum & hæ ordinatæ areæ et ordinatæ et {areæ} designtur {sic} per literas z et v, o] quantitates et earum fluxiones designatur per literas, momenta vero per rectangula sum|b| fluxionibus et litera o qua momentum temporis designatur. Et hujusmodi rectangulis pro momentis Newtonus usque hodie usurpat. In libro Scholio ad Lib. II Lem II Lib. II Princip. \anno 1686 conscripto/ demonstrantur elementa methodi fluxionum \synthetica/ & pro fluentibus et fluxionibus ponuntur literæ majusculæ et minusculæ, & pro momentis eædem literæ minusculæ subintellecto factore o. Mathematicus vero fradulentur fingiat methodum fluxionum abs literis punctatis invenire non potuisse. Porro in Libro de Quadratura Curvarum \ante annum 1676 composito/ pro fluxionibus quando adh litteræ punctatæ quando aliæ notæ adhibentur.

Ad l. \24/ 26, 27. Inventæ sunt Propositiones in libro illo per med|t|hodum fluxionū {P}{illeg} {illeg} Sed demonstratæ \sunt/ synthetice ut in Geometriam admitterentur. Nam Propositiones (ob certitudinem Geometriæ) non prius in Geometriam admittendæ sunt quam \{illeg}/ demonstrantur synthetice. Proinde nulla erat occasio \in hoc Libro/ utendi calculo fluxionum.

Ad l. 29. Prima vice liter{illeg} \Imo/ In secundo Volumine pag anno 1693 impresso pag 392 393. Newtonus uti \anno 1692/ postulante Wallisio anno 1692 Extractionem fluentis ex æquat expicationem {sic} epistolæ suæ \24 Octob/ ann{e}|{o}| 1676 scriptæ, et extractionem fluentis ex æquatione fluxionem {illeg} involventæ cum Wallisio communicavit. Et calculus differentialis non|d|inval{illeg}dum invalu{it}\erat/ ubi

Ad p. 3. |l. 2.| Vbi n Falso dicitur quod Newtonus N . . . . us incrementum constans ipsius x nunc notat per x punctatum uno pun\c/to. {illeg} Ponitur x punctatum non pro incremento sed pro fuxione {sic} ipsius x. Et incrementum \constans ipsius x/ us hodie designatur per $1\times$o seu o (id est per $\stackrel{.}{\mathrm{x}}\mathrm{o}$ seu $1\times \mathrm{o}$, subintellecto factore $\stackrel{.}{\mathrm{x}}$ seu 1) Et hæc Notatio magis commoda est et quam ea calculi differentialis

Ad l. 3, 4 Falso etiam dicitur quod N . . . . us Regulam circa gradus ulteriores falsam dedit: Regulam dedit verissimam in Prop. I. Libri de Quadratura curvarum. |Et| Hanc Regulam Wallisius publicavit anno 1693 in secundo Volumine Operum suorum pag. 392. Erravit etiam ergo eminens

Ad l. 4 Erravit ergo eminens ille Mathematicus \nemp {sic} Ba|e|rnoullius/ ut etiam ab alio eminente mathematico \nuper/ demonstratum est. Cæterum hic observandum est \venit/ quod author hujus Epistolæ qui Bernoullium hic citat hæc Epistola nuper Gallice edita fuit sub tanquam a Bernoullio scripta & ut Bernoulliu|o|m {illeg}|sc|ripta videretur, om{j}|i|ssa sunt verba \ad Bernoullium spectantia/ [quemadmodum ab eminente quodam me|a|thematico dudum notatum est.]

Ad l. 11 Vult Author noster L . . . . um Calculum differentialem in numeris primum invenisse a Moutono prius inventam; & {illeg} (excogitata Analysi infinitesimalium) ad Geometriam transtulisse. Cet|r|te Leibni\ . . . ./ us anno 1677 ubi primum incidit in methodum infinitesimalem, methodum differentialem Barrovij <491r> mutatis symbolis misisse ad N|O|ldenburgum ut suam.

L. 16|5|, 16, 17 Rixatur L . . . . us cum N . . . . o

L 20. Disputat L . . . . us contra Epistolam propria manu scriptam.

L. 31. Hanc mehtodum generalem Newton Ne . . . . us invenit

P. 4. l. 5. Inventioni Newtonus ausam dedit. Vide ejus Epistolas Newtoni 24 Oct. 1676 et Iun L . . . . i 21 Iunij 1677.

P. 2. l. 22.

Observations upon the \a paper entituled/ Remarks concerning the difference between M^r Leibnitz & M^r Newton & upon the translation of a Latine piece where this judgment is found.

Obs 1. We The Latine piece was printed in Germany in the form of a defamatory Libel in August last w^thout any the name of any author, the Letters L—s bein & N—s being put for Leibnitius & Newtonus. But|And| being void of argument & full of reflexions without any proof was judged too mean a piece to deserve any answer. But being sent prin sent to y^e Hague to be reprinted in yo^r Iournal, I send you the following Observations upon it. Th

|1| Obs. 1. The author of y^e Latin \piece/ tells you us that M^r Leibnitz {illeg}a|be|ing at Vienna had not yet seen the Commercium Epistolicum. And this he could not know without keeping a correspondence w^th M^r Leibnitz. He tells us further that \M^r Leibnitz/ not being at leisure to examin this affair himself he had referred it to the judgment of a primary {illeg}|M|athematician \of the first rank/ & very skilful in these thin{illeg}|g|s, & very free from partiality. So then M^r Leibnitz himself was the first mover this paper was writ by the correspondents of M^r Leibnitz & M^r Leibnitz himself was the first mover. And therefore it must be looked upon as the ablest defence that he & his correspondents were able to make. We are indeed told that he had not yet seen the Commercium. But a Copy of it was sent to him by the Resident of y^e Elector of Hannover & another was sent to Lipsic for him & an answer came back \thence/ to the {illeg} Secretary of y^e R. S. that it was sent to him. And those th{illeg} He knew how to write to a great Mathematician who had the book And those in England who have seen the Latin Paper generally conclude from the style & humour of it that it was writ by M^r Leibnitz himself. Now this author tells us that this|e| great Mathematician having dicussed all things gave judgment in his Letters of 7 Iune 1713 as follows

Obs. 2. The whole Latin paper is full of assertions & reflexions without any proof as if M^r whereas M^r whereas M^r Leibnitz by the laws of all nations M^r Leibnitz cannot be a witness for him self. And as He & his friends ought to prove what they assert. For there cannot be a greater argument of a bad cause then to affirm & reflect w^thout being able to prove any thing.

Obs. 3. The author of the Latin paper saith that M^r Hook complained of M^r Newton about the Hypothesis of the Planets & M^r Flamsteed about the use of his Observations. And indeed M^r Hook claimed the invention of y^e        \Demonstration of the/ Proposition of the 1^st book of the Principia but could never produce his Demonstration & M^r Leibnitz has produced an erroneous demonstration of the same Proposition to make it his own. Whether M^r Flamsteed treated M^r Newton w^th candour is better known here than in Germany & whether M^r Leibnitz or M^r Tschurnhause were in the right when they fell out about an invention is not material to us in England.

Obs. 4. The great Mathematician observes that M^r Newton gave a false rule about the higher degrees of moments or differences But it seems this great Artist had not skill enough \in these matters/ to mend a press fault. For in the Scholium at the end of the book of Quadratures had he where M^r Newton saith Hæ fluxiones sunt ut termini serierum <492v> infinitarum convergentium, had he but repeated the word ut in the next sentence as the sense requires, the Objection would have been at an end.

|2| Obs. 5. But the great Mathematician objects represents \He conjectures/ that M^r Newton spent his first years in cultivating the method of {i}{illeg} series without thinking of the calculus of fluxions, or reducing it to general rules. That is, he will not allow plain matter of fact. In the Analysis w^ch D^r Barrow w{illeg} in the year 1669 communicated to M^r Bar Collins the method of fluxions is described with examples of the calculus. M^r Leibnitz knew nothing of the method before the year 1677 & & there is no proof that M^r Leibnitz knew any thing of the method before the year 1677.

|3| But the great Mathematician brings two arguments for his conjecture 1 first saith \he/ in all the Letters published in the Commercium Epistolicum there is nothing to be \& in all his Principia Principia Philosophiæ/ the letters with pricks are not to be met with w^ch M^r Newton now uses are not to be met with. M^r Newton {illeg} indeed \{illeg}/ uses those letters in his book of Q for fluents w^th pricks for fluxions & \And by the same argumenthe may conclude that the Ancients had no Analysis because they wrote by composition./ M^r Leibnitz indeed confines his method to the symbols dx & dy so that if you take away his symbols you take away his method, M^r Newton doth not so. And whether he uses Letters with pricks or other symbols his method is still the same In his Letter of 24 Octob 1676 he represents that he had a method of extracting fluents out of equations involving their fluxions. Will M^r Leibnitz say that he had no such method because unless he \then/ used letters with pricks. If so his letters with pricks are ol \are/ as old \at least/ as the year 1676 & by consequence older then the differential notes of M^r Leibnitz.

|4| But its to be observed that fluxions & \moments or/ differences are \not/ quantities of the same kind. Fluxions are finite quanties {sic} velocities, moments \differences/ are infinitely small parts of things generated by fluxion in moments of time. Fluxions are finite quantities differences are infinitely little. The M^r Newton frequently \sometimes/ uses prickt letters \sometimes other marks/ for fluxions, M^r M^r Leibnitz uses no symbols for fluxions The symbols of M^r to this day. The symbols of fluxions therefore used by M^r Newton are the oldest in the kind. These he multiplies by the letter o to make them infinitely little moments or differences.

|5| But He|i|s second reason is that M^r Newton understood not the the differences of differences or second differences till after the writing of his Principia. For there, saith he, the constant increase of the letter x he represents not by a prickt letter as at present but by the letter o after the vulgar manner w^ch destroys the advantages of the differential calculus. But that M^r Newton understood it some yea many years before is manifest by his Letter dated 10 Decem 1672 wherein he represents that his method extended to the determining {illeg} \Questions about/ the Curvature of Curves. a {P} And as for the use of the letter o, M^r Newton used it in his Analysis communicated to M^r Collins in the year 1669, \&/ in his book of Quadratures where he uses let represents fluxions by prickt letters & still uses it as the best way of notation. And I recommend it to be still used in honour to the memory of M^r Fermat who made the first step towards this way of \sort of/ calculus. Know therefore that M^r Newton usuall puts a known quantity & most commonly an unit for the fluxion of time & considered as the exponent of time. [And for the moments of this fluents quantities w^ch M^r Leibnitz called their differences he puts their fluxions multiplied by the moment o, that is $1\times \mathrm{o}$ or o for the moment of the exponent] the letter o or $1\times \mathrm{o}$ he puts for its moment, & for y^e moments of other flowing quantities he puts others the rectangle|s| under their fluxions & the moment o.|,| And borrowing the names of fluxions & moments from the fluxion & moments of time. |And| This sort of Notation M^r Newton used in|w|hen he wrote his Analysis above mentioned.|,| This s{illeg} & {sic} when he wrote his Treatise of Quadratures & uses it to this day w^ch was but D^r Barrow in his method of Tangents \published A C. 1670/ put the letters a & e for the differences of the Abscissas & Ordinates & m^r Leibnitz seven years after changed the letters a & followed D^r B{illeg}|a|rrows meth \so make this method his own in/ changed the letters a & e into y^e symbols dy & dx, And this way of Notation is not capable of the advantages of M^r Newtons way & gave the name of the differential method to D^r Barrow's method of Tangents \& called it the differential method/ beginning where D^r Barrow left off as that candid Gentleman the Marquess de L'{illeg}Hospital has <493r> observed long since in this|e| p|P|reface to his book Analysis. And whereas our Geometer represents that M^r Newton uses the letter o more vulgari in a vulgar manner w^ch destroys the advantages of the differential calculus the method of M^r Newton has all the advantages of M^r Leibnitz & is more universal & more Geometrical.

The grand Geometer adds that M^r Newton has given a false rule about the higher degrees of Differences. But it seems this great Artist had not skill enough to in these matters to mend a press-fault. For in the Scholium at y^e end of the book of Quadratures where {illeg}|M|^r Newton saith: Hæ fluxiones sunt ut termini serierum infinitarum convergentium, had he but repeated the word ut in the next sentence \as the sence requires/ the Objection would have been at an end. And thus much in answer to the judgment of the great Mathematician |[All w^ch amounts to| But this skilful, & \this/ impartial Mathematician could not see a word of the method of fluxions in the Analysis & other nothing more then this that M^r Leibnitz appeals from th{illeg} to himself from the records published by order of the R. S. In his Letter to {illeg} their Secretary dated 29 Decem 16111711 he 1711 he wrote that M what M^r Keil wrote opposed his candor w^ch that he should defend \at that age/ after so many documents of his life no prudent or just man would approve of: w^ch is as much as to say that the Royal Society were unjust unless they would admitt him to be a witness in his own cause, contrary to the laws of all nations. The By]

In The Author of the paper pretends that M^r Leibnitz never communicated his reasons to y^e Society. And so the Society did not examin the reason on both sides for giving judgment. Whereas M^r Leibnitz refused to give calling it unjust to expect that one of his age & credit should defend his can his reasons calling it unjust to expect that one of his age & reputation should defend his candour. That is, he insisted upon it to be \being/ a witness for himself, contrary to y^e laws of all nations. And besides And when he would give no reasons, the Society appointed a Committee to search records & make a report the Abo of the matter how they found it & mo thereof. And now to talk of reasons against Record matter of fact, & to do it w^thout producing any \those/ reasons is very trifling. The Report of the Committe must stand till M^r Leibnitz produces his reasons \against it/ if he has any.

But The author of the paper in the next place pretends to give his a true report of what passed. And But begins his report w^th what passed in the years before|.| & told how M^r Newton had the method of fluxions in year 1669 {illeg} & given account of the method of fluxions set down in the Analysis w^ch D^r Ba

And now we have brought down to our own times the history of the Greek Empire with the e|E|mpiress of the Saracens & Turks w{illeg}|hi|ch invaded {itag} the {illeg} by the Little hor king w^ch doth according to his will

D^r Barrow printed his method of Tangents in the year 1670 &

That candid Gentleman the Marquess de l'Hospital, in the Preface to his Analysis represents that \D^r Barrow stopt at fractions & surds &/ where D^r Barrow left off M^r Leibnitz began. His method of Tangents is the same w^th D^r Barrows except that he has changed the letters a & e used by D^r Barrow into y^e symbols dy & dx. Let the advertism^ts w^ch M^r Leibnitz received in M^r Newtons Letters of 10 Decem 1672, 13 Iune 176 {sic} & 24 Octob 1676 be added to D^r Barrows method of Tangents & you have the Differential method.

And thus much in answer to the great Mathematician.

M^r Leibnits in his Letter of      Iune 1677

The Elements of his method of fluxions he described in his second

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$\begin{array}{c}6999\phantom{.6}\\ \phantom{6}233.6\\ \overline{7232.6}\end{array}$

I have seen in yo^r journal the translation of a Latin piece dated 29 Iuly 1713 & published in Germany, & the Remarks upon it. These pieces are full of assertions without proof & w^thout the name of the author & so are of no authority.

The author of the Latine piece represents that M^r Leibnitz had not then seen the Commercium Epistolicum, & this he could not know without keeping a correspondence with M^r Leibnitz. But a copy of this book was sent to M^r Leibnitz by the Resident of the Elector of Hannover above a year ago & several other copies were \then/ sent to Lipic one of w^ch was for him.

That Author tells us further that M^r Leibnitz not being at leasure to examin this affair himself had referred it to the judgment of a Mathematician of the first ra{k}|n|k very skilfull in these things & very free from partiality. So then this paper was writ by the correspondents of M^r Leibnitz & for these reasons M^r Leibnitz himself \desiring the judgm^t of the great Mathematician & sending it to his correspondent to be published/ was the first mover & |y^t| the credit of the Mathematician for candor & ability depends upon the credit of M^r Leibnitz. And for these reasons this paper must be looked upon as the best defence that he & his correspondents were able to make: especially if this paper be writ in the stile of M^r Leibnitz himself as some think. By his Letters against M^r Keill it appears that he is too much concerned to neglect this matter, & his appealing from a numerous Committee of the Royall Society to a nameless Mathematician of his own chusing is no better then appealing to himself. For he has wrote to the Society that it would be injustice to question his candour, that is, to deny him to be |both| witness \& Iudge/ in his own cause.

Now this great Mathematician conjectures that M^r Newton spent his first years in cultivating the method of series without thinking of the method \{ca}lculus/ of fluxions or reducing it to general rules. That is he will not allow the Analysis communicated by D^r Barrow to M^r Collins in the year 1669 to be a genuine piece. And he brings too arguments for his conjecture.

First, saith he, in all the Letters published in the Commercium Epistolicū & in all the Principia Philosophiæ, the letters with pricks w^ch M^r Newton now uses are not to be met with. But in all those Letters (except the Analysis) & in all the Principia there was no occasion to make use of that Analysis the fluxional calculus. And D^r Keill hath given a further Answer to this argument long since in his Letter dated 24 May 1711. Observo ipsum Newtonum, saith he, sæpius mutassæ nomen et notationem calculi. In tractatu de Analysi Æquationum per series infinitas, incrementum abscissæ per literam o designat, et in principijs Philosophiæ, Fluentem quantitatem Genitam vocat ejus incrementum Momentum appellat: illam literis majoribus A vel B, hoc minusculis a et b designat. M^r Leibnitz confines his Method to the symbols dx & dy, so that if you take away his symbols you take away \{a} {sic}/ his method. M^r Newton doth not confine his method in such a manner. As he uses \a propriety of/ any symbols for fluents so he uses any others for fluxions, & whether he uses letters w^th pricks or other symbols for fluxions his method is still the same. In his Letter of 24 Octob. 1676 he represents that he had a \pro/ method of extracting Fluents out of equations involving their fluxions. Will M^r Leibnitz say that he had no such method unless he then used letters with pricks? If so, his letters with pricks must be allowed as old at least as the year 1676, & by consequence older then the differential Notes of M^r Leibnitz.

But its \further/ to be observed that Fluxions & Differences are not quantities of the same kind. Fuxions {sic} are velocities, & Differences are not quantities small parts of things generated by fluxion in moments of time: fluxions are always finite quantities & differences are infinitely little. M^r Newton uses sometimes prickt letters sometimes other symbols for fluxions, M^r Leibnitz <494v> uses no symbols for fluxions to this day. The symbols of fluxions used by M^r Newton whether w^th pricks or without, are the oldest therefore the oldest in the kind. These he multiplies by the moment o to make them ^[2]infinitely little & puts the rectangles for moments or differences \of y^e Moments/, & without the moment o either exprest or understood they never signify moments or differences, {illeg}|b|ut are always finite quantities & signify velocities. The fluxion of time or of any exponent of time he usually represents by an unit, & the moment thereof by the letter o.

The second reason of the great Mathematian {sic} for his conjecture is that M^r Newton understood not the differences of differences till after the writing of his Principia. For there, saith he, the constant increase of the letter x he represents not by a prickt letter as at present but by the letter o after the vulgar manner, w^ch destroys the advantages of the differential method. Here o^r great Mathematician commits two mistakes; one by supposing that M^r Newton represents differences by prickt letters, another by supposing that the method used in the tenth Proposition of the second Book of the Principia is M^r Newtons method of fluxions. Tis only a branch of his method of converging series. In his Letter dated 10^th Decemb. 1672, where he speaks of a method whereof the method of Tangents there described is a branch or Corollary, he represents that this method (w^ch is the method of fluxions) extended to Questions about the curvature of curves; & thence it is manifest that he then understood the second fluxions or differences of differenc{illeg}|e|s.

The letter o was used by M^r Newton in the manner above ^[3]mentioned in his Analysis communicated by D^r Barrow to M^r Collins in the year 1669, & in his Book of Quadratures & is still used by him in the very same manner. And as it is the oldest notation for moments or differences so it is the best, the method being thereby more convenient \{commodi} {sic}/ more elegant & more suitable to Geometry then \by/ the differential notation & as universal, & does justice to the memory of M^r F{e}rmat who first \a {inmadnit}/ brought in the use of this letter o.

To signify \{din}{illeg}/ the summ of the ordinates or area of a Curve M^r Leibnit{z} prefixes \propo{s}{illeg}/ the letter s to y^e Ordinate & M^r Newton in his Analysis communicated to M^r Collins in the year 1669, inclosed the Ordinate in a square. M^r Newtons notation of this kind is also much the oldest.

D^r Barrow published his method of Tangents in the year 1670, & ^[4]that very candid Gentleman the Marquess de l'Hospital, in the Preface to his Analysis, represents that D^r Barrow stopt at fractions & surds, & where D^r Barrow left off M^r Leibnitz began. His method of Tangents is the same with D^r Barrows except that he has changed the|i||s| letters a & e into y^e symbols dx & dy, & taught (being admonished by M^r Newtons Letters of 10 Decem. 1672, 13 Iune 1676 & 24 Octob. 1676) ^[5]taught how to avoyd fractions & surds.

As to what the Author of the Latin paper saith of M^r Hook & M^r Flamsteed: M^r Hook indeed claimed one of M^r Newton's Propositions but could never produce a Demonstration thereof, M^r Leibnitz claimed it also but the Demonstration by w^ch he claimed it is erroneous. M^r Leibnitz claimed also an Invention from M^r Tschurnhause \& who is in y^e right may be a question/, but M^r Newton always acknowledged the use of M^r Ha{illeg}ks /Flamsteads\ Observations.

This Author in the next place complains of the Committee of the R. Society for representing that M^r Leibnitz had from M^r Iames Gregory the series for finding the Arc of a circle by the Tangent given, that is, he represents that the Letter of M^r Gregory, M^r Collins, M^r Oldenburg & M^r Leibnits examined & approved by a numerous Committee <495r> of the Royal Society were fourged. The Letter of M^r Gregory dated 15 Feb. 161|7|$\frac{0}{1}$ is still extant \su{b}{illeg}/ in his own hand-writing & conteins the|i||s| series with several others then sent to M^r Collins. That of M^r Oldenburg dated 15 Apr. 1675 is extant in the Letterbook of the Royal Society left by M^r Oldenburg in their Archives & conteins this series with several others then sent from M^r Collins by M^r Oldenburg to M^r Leibnitz at Paris. The answer of M^r Leibnitz dated |f|at|rom| Paris May 20^th 1675 was found in the same Letter-book; & the original Letter in the hand-writing of M^r Leibnitz was also found in the Archives of the R. Society & conteins his acknowledgment of the Receipt of M^r Oldenburghs Letter above mentioned. It begins thus. Literas tuas multa fruge Algebraica refertas accepi pro quibus tibi et doctissimo Collinio gratias ago. Cum nunc præter ordinarias curas Mechanicis imprimis negotijs distrahar, non potui examinare series quas misistis, ac cum meis comparare. By these words its plain that M^r Leibnitz had not yet {illeg}|a|t this time knew none of the series then sent him to be his own, tho before the end of the end of the year he communicated to his friends at Paris one of those series then sent him as his own, viz^t that of Gregory then dead, & by vertue of that communication has ever si{illeg}|nce| claimed it as his own. The collection of the papers of M^r Gregory made by M^r Collins after the death of that Gentleman, is still extant in the hand-{w}{illeg}twriting of M^r Collins, & at the request of M^r Leibnitz was sent to Paris in Iune 1676, & conteins \{illeg}/ a copy of the aforesaid Letter of M^r Gregory. But upon the death of that Gentleman M^r Leibnitz pretended in his Letter dated 28 Decem. 1675 that he had communicated it at Paris above two years before & that it was the series whereof he had wrote before to M^r Oldenburg, that is, in his Letter of 15 Iuly & 24 Octob. 1674. An{illeg}|d| under this pretence he sent it back to M^r Oldenburg as his own in his Letter dated 27 Aug. 1676. \{illeg}/ And yet the Series w^ch he wrote of in his said two Letters dated 15 Iuly & 24|7| Aug 1676 24 Octob 1676|4| was not this Series for finding the arc by the tangent but a Theoreme or Method for finding the Arc by the sine. This Theoreme or Series M^r Collins had received from M^r Newton in Iuly 1669 & communicated it soon after to his friends very freely. M^r Leibnitz was in London in the years 1671, 1672 & the beginning of 1673 & having met with this series either in London or soon after in France pretended in his said Letters of 15 Iuly & 24 Octob 1674 to have found it himself, & yet in his Letter dated 12 May 1676 desired M^r Oldenburg to procure from M^r Collins the Demonstration thereof, that is, the method of finding it. And when he had received the method with some of M^r Newtons series, he pretended to have found three of those series before, tho he did not yet understand the method of finding them. For in his Letter of 27 Aug. 1676, he wrote back for a further explication of the method. M^r Newton therefore in his Letter of ^[6] 24 Aug. 1676 explained it further & added another method of the same kind, & M^r Leibnitz in his Letter dated 21 Iune 1677 still desired a further explication|,| of but so soon as he understood it, he wrote in his Letter dated {illeg} 24 Iuly 1677 12 Iuly 1677 that he found by his old papers, that he had used one of those methods before. And by the same power \{illeg} faculté/ of invention \d'{illeg}/, when he had newly found the Differential method (w^ch he might do by the help of Gregories & Barrows methods of Tangents & Newtons Letters) he 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. And yet its very certain that he had but newly found it. For in his Letter dated 27 Aug. 1676, he wrote: Quod dicere videmini pleras difficultates (exceptis Problematibus Diophantæis) ad series infinitas reduci, id mihi non videtur. Sunt enim multa us adeo mira et implexa ut ne ab æquationibus pendeant ne ex quadraturis. Qualia sunt ex multis alijs Problemata methodi tangentium inversæ. Quæ etiam Cartesius in potestate non esse fassus est These words are a Demonstration that he did not then understand the Differential method. He was then composing & polishing his Quadrature of the Circle vulgari more, & left of that way of writing as soon as he found the Differential method. I pass by his claiming the Inventions of Mouton & Paschal & a considerable part of M^r Newton's Principia Philosophiæ.

Our Author tells us further that M^r Leibnitz published in the Acta Eruditorum a general method for finding the Ordinates of transcendent Curves not by <495v> extraction of roots but deduced from a profounder foundation of the differential calculus, by w^ch the business of Series was brought to a greater degree of perfection. But M^r Newton many years before (viz^t in his Letter to M^r Oldenburg dated 24 Octob. 1676, communicated the very same method in this sentence. Altera [methodus consistit] tantum in assumptione seriei pro quantitate qualibet incognita ex qua cætera commode derivari possint, & in collatione terminorum homologorum æquationis resultantis ad eruendos terminos assumptæ seriei. M^r Leibnitz therefore has no title to any part of the method of converging series.

Our Author tells us further that M^r Leibnitz was the first who used the exponential calculus while M^r Newton knew nothing thereof. Certainly M^r Newton was \one of/ the first who introduced into Analysis fractions radicals & negative quantities for the indices \or exponents/ of dignities & thereby very much enlarged Analysis & laid the \i{illeg}/ foundation of making it universal. In his Letter dated 24 Octob. 1676 he mentioned such Exponents of Dignities \& thereupon M^r Leibnitz in his Answer dated 21 Iune 1677, proposed indeterminate exponents of Dignities./. {sic} And this seems to have been the Original of the Exponential Calculus. But such a calculus has hitherto been of no use. |a {illeg}la{r}t s'a{n} trouvé n'a{illeg}in ana{mar}age|

Our Author tells us also that the English & Scotch, Wallis, Hook, Newton & Gregory junior, acknowledged 36 years ago the series for finding the arc of a circle by the Tangent to be the invention of M^r Leibnitz That is, he complains of M^r Oldenburg for not letting the English & Scotch know that he had sent this series with several others to M^r Leibnitz in April 1675. Tis sufficient that the Letters between M^r Oldenburg & M^r Leibnitz were left by M^r Oldenburg in the Letters-book of the R. Society & that the Original Letters of M^r Leibnitz is still extant in his own hand-writing.

And thus much concerning the printed paper.

In the Remarks its represented that M^r Leibnitz never communicated his reasons to y^e Royal Society of England & so the So{ciet}y hath not examined the reasons on both sides for giving judgment. But the truth is, M^r Leibnitz refused to give any reasons at all, representing it injustice that he to expect that he should defend his candor, detracting from the candor of M^r Keil & pressing the R. Society to give judgment \prononcer/ without hearing reasons: & the Committee of the R. Society grounded their Report not upon plausible reason & slippery reasons but upon the matter of fact conteined in the Letters & Papers found in the Adversaria of the R. Society & in those of M^r Collins & in the Acta Eruditorum; & published those Letters & Papers in the Commercium Epistolicum ,|t|hat all the world might see the grownd \& justice/ of their Report. And those Records are sufficiently plain to any man that considers them impartially.

probabilite{illeg} conj{ectures}

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I have seen in your Iournal Literaire the translation of a Latin piece dated 29 Iuly 1713 & published in Germany & the Remarks upon it These pieces are full of assertions without any proof & without the name of the Author & so are of no credit or authority.

The Author of the Latine piece represents that M^r Leibnitz had not then seen the Commercium Epistolicum: & this he could not know without keeping a correspondence with M^r Leibnitz. Certainly a Copy of this book was sent to him by the Resident of the Elector of Hannover above a year ago, & several other Copies were sent to Leipsic one of w^ch |was| for him, & he knew how to write to a friend who had a copy

For this Author tells us further that M^r Leibnitz not being at at {sic} leasure to examin this affair himself had referred it to the judgment of a Mathematician of the first rank very skilfull in these things & very free from partiality. So then this paper was writ by the Correspondents of M^r Leibnitz, & M^r Leibnitz himself was the first mover, & the credit of the Mathematician for candor & ability depends upon the credit of M^r Leibnitz. And for these reasons this paper must be looked upon as the best defence that he & his correspondents were able to make, especiall {sic} if the Latin paper be writ in the style of M^r Leibnitz himself as some think. By his Letters against M^r Keil it appears that he is too much concerned to neglect this matter, & his appealing from the Report of a numerous Committee of y^e Royal Society to a nameless Mathematician of his own chusing is no better then appealing to himself He has told the R. Society that it would be injustice to question his candor that is, to deny him to be a witness in his own cause, & now his {sic} for makeing himself also the Iudge.

Now the great Mathematician conjectures that M^r Newton spent his first years in cultivating the method of series without thinking of the calculus of fluxions or reducing it to general Rules. That is, he will not allow the Analysis communicated by D^r Barrow to D|M|^r Collins in the year 1669 to be a genuine piece. And he brings too Arguments for his conjecture

First, saith he, in all the Letters published in the Commercium Epistolicum, & in all the Principia Philosophiæ, the Letters w^th pricks w^ch M^r Newton now uses are not to be met with. But in all those Letters except the Analysis, & in all the Principia there was no occasion to make use of the fluxional calculus. And D^r Keill hath given a further Answer to this argument long since in his Letter dated 24 May 16 1711. Observo ipsum Newtonum, saith he, sæpius mutasse nomen et notationem calculi. In Tractatu de Analysi {illeg}|Æq|uationum per series infinitas, incrementum Abscissæ per literam o designat, et in Principijs Philosophiæ, Fluentem quantitatem Genitam vocat, ejus incrementum Momentum appellat: illam literis majoribus A vel B, hoc minusculis a et b designat. M^r Leib I may add that in one & the same book, the Book of Quadratures he sometimes uses prickt letters sometimes not. For in the Introduction to that book where he describes his Method of fluxions & illustrates it will|t|h examples he makes no use of prickt letters. M^r Leibnitz confines his method to the symbols dx & dy, so that if you take away his symbols you take away the characteristick of his method. M^r Newton doth not so. Whether you he uses Letters with pricks or other symbols for fluxions, his method is still the same. In his Letter of 24 Octob. 1676, he represents that he had a method of extracting fluents out of Equations involving their fluxions. Will M^r Leibnitz say that he had no such method unless he then <496v> used letters with pricks? If so, his letters with pricks are as old at least as the year 1676, & by consequence older then the differential notes of M^r Leibnitz.

But its to be observed that Fluxions & Differences are not quantities of the same kind. Fluxions are velocities: Differences are small parts of things generated by fluxion in moments of time. Fluxions are always finite quantities: Differences are infinitely little. M^r Newton uses sometimes prickt letters, sometimes other symbols for fluxions: M^r Leibnitz uses no symbols for fluxions to this day. The symbols of fluxions used by M^r Newton whether with pricks or without are therefor{e} the oldest in the kind. These he multiplies by the moment o to make them infinitely little, & puts the rectangles for moments of fluent quantities: & without the moment o either exprest or understood the prickt letters never signify moments or differences but are always finite quantities & signify velocities. The fluxion of time or of any exponent of time he usually represents by an unit & the moment thereof by the Letter o w^ch is equipllollent to y^e rectangle $1\times \mathrm{o}$. In his Analysis above mentioned he represents fluents f by the areas of curves, fluxions by their ordinates & moments for by the rectangles under the ordinates & the moment of the common Abscissa. And these rectangles he uses instead of the Indivisibles of Cavallerius, & thereby makes his method Geometrical. For in Geometry there are no Indivisibles. When he is demonstrating any Proposition he always writes down the moment o & takes it in the sense of y^e vulgar for an indefinitely small moment part of time & performs the whole calculation in finite figures by the Geometry of the Ancients without any approximation, & so soon as the calculation is ended & the Equation reduced he supposes the moment o to decrease in infinitum & vanish. Examples of this you have in the end of his Analysis & in the first Proposition of his Book of Quadratures. But when he is not demonstrating but only investigating a Proposition, he supposes the moment o to be infinitely small & usually for making dispatch neglects to write it down & proceeds in the calculation by any approximations w^ch he thinks will create no error in the conclusion. But this last way (to w^ch the Differential method is equipollent) is not Geometrical. For Geometry admits not of approximations nor of \lines & figures/ quantities \& quantities/ infinitely little.

The second reason of the great Mathematician for his conjecture is that M^r Newton understood not the differences of differences till after the writing of his Principia. For there saith he the constant increase of the letter x he represents not by a prickt letter as at present but by the letter o after the vulgar manner w^ch destroys the advantages of the differential method. But here o^r great Mathematician commits two mistakes: one in supposing that M^r Newton represents Differences by prickt letters, the other by supposing that the method used in the 10^th Scholium the method used in the 10^th Proposition of the second Book of y^e Principia, is M^r Newtons method of Fluxions. Tis only a branch of his method of converging series. The elements of his method of fluxions he described in the second Lemma of the second book of his Principles & subjoyned this Scholium. In literis quæ mihi cum Geometra peritissimo G. G. Leibnitio annis abhinc decem intercedebant, cum significarem me compotem esse methodi determinandi maximas et minimas, ducendi Tangentes et similia peragendi quæ in terminis surdis æque ac in rationalibus procederet, & literis transpositis hanc sententiam involventalibus [Data æquatione quotcun Fluentes quantitates involvente, Fluxiones invenire & vice versa] eandem celarem; rescripsit Vir Clarissimus se quo in ejusmodi methodum incidisse & methodum suam communicavit a mea vix abludentem præterquā in verborum et notarum formulis. Vtrius fundamentum continetur in hoc Lemmate. The Letter here referred unto is that of 24 Octob. 1676 printed by D^r Wallis. In this Letter M^r Newton distinguishes between the Method of infinite series {illeg}|&| that of fluxions & represents that he had writ a treatise of both these methods five years <497r> before & that the method of fluxions readily gives the method of Tangents of Slusius & sticks not at equations affected with radicals|.| & that he writ had writ {illeg} t And in a Letter to M^r Collins dated 10 Decem 1672 M^r Newton writing of the method whereof the Method of tangents of Slusius is a branch or Corollary & which sticks not at Curves surds, & w^ch by consequence was the Method of fluxions, represented it a very general method reaching to y^e abstruser sorts of Problemes & among others to the determining of the Curvature of Curves, a Probleme w^ch requires the consideration of the second fluxions. And therefore he had then extended the method to the second fluxions or fluxions of fluxions. And it is further observable that M^r Newton in the 2^d Book of his Principles makes frequent mention of the increase of the velocities wherewith lines are described. The lines are the fluents, the velocities their fluxions & the increase of the velocities their fluxions of their fluxions or second fluxions. And particularly in demonstrating the 14^th Proposition of the second Book of his Principles, he has these words. Est igitur differentia momentorum, id est momentum differentiæ arearum &c Where differentia arearum is the first difference & momentum differentiæ is the second difference of the areas. So then M^r Newton, when he wrote his Principles of Philosophy & a great many years before, had extended his method to the consideration of the second fluxions of quanties {sic}. And indeed, to say that he then understood not second fluxions is all one as to say that he understood not how to consider motion as a quantity increasing & decreasing.

And whereas the great Mathematician represents that M^r Newton uses the Letter o in the vulgar manner w^ch destroys the advantages of the vulgar \Differential/ method: he uses it, & has used it ever since the writing of his Analysis, in such a manner as makes his method more beautiful more universal Geometrical & more advantageous then the Differential, & (by joyning the methods of series & fluxions together) much more universal.

The Differential method is nothing else then the method of Tangents published by D|M|^r Gregory in y^e year 1668 & by D^r Barrow in the year 1670, disguised by changing D^r Barrows symbols a & e into dy & dx, improved by the instructions w^ch M^r Leibnitz received by the Letters of M^r Newton, & taken from them all by pretending that M^r Leibnitz found them it long before he did. For in his Letter dated 21 Iuly 1677 he received it pretended to have found it jam a multo tempore, & yet he had not found it the year before. For in his Letter dated 27 Au{illeg}|g|. 1676 he wrote that there were many Problems w^ch could not be reduced to Equations or Quadratures such as were those of the inverse method of Tangents & many others. This method without the use of the Letter o is not demonstrative, without the method of series is not universal, non {sic} has any advantages w^ch are not to be found in the method of Fluxions, nor has M^r Leibnitz added any thing to it of his own besides a new name & a new notation. And thus much in answer to the great Mathematician.

As to what the Author of the Latine paper said|t|h of M^r Flamsteed & M^r Hook: M^r Newton always acknowledged the use of M^r Flamsteeds Observations; M^r Hook could never produce a Demonstration of the Proposition claimed by him tho often asked to produce one; M^r Leibnitz pretended to y^e same Proposition by an erroneous Demonstration; & whether he or M^r Tschurnhause were in the right about an erroneous Proposition claimed by them both, I leave to be examined.

This Author in the next place complains of the Committee of the R. Society for representing that M^r Leibnitz had from M^r Iames Gregory the series for finding the Arc of a circle by the Tangent, that is, he confesses that he has no better way other way of defending M^r Leibnitz then by laying aside the records relating to this matter examined & approved by a numerous Com̄ittee of the |R.| Society & still ready to be produced

The same Author in the next place ascribes a \certain/ general method of series to M^r Leibnitz, tho this method was found many years before by M^r Newton as <497v> appears by his Letter of 24 Octob. 1676. And in the next place he magnifies an Invention called the Exponential Calculus, w^thout considering that M^r Leibnitz had the hint from M^r Newton & that this Calculus hath hitherto been of no use.

In the last place o|O|ur Author tells us \also/ that the English & Scotch, Wallis, Hook, Newton, Gregory junior, acknowledged 36 years ago that the series for finde|i|ing the Arc by the Tangent of a circle by the Tangent was the invention of M^r Leibnitz \& that M^r Hugens commended it/. But he should have complained of M^r Oldenburg \& M^r Leibnitz/ for not letting the English & Scotch \& M^r Hugens/ know that this series with many others was sent by him \M^r Oldenburg/ to M^r Leibnitz in April 1675; And that a Collection of Gregories Papers with this series in it, was also sent by him to M^r Leibnitz the next year.

In the Remarks it is represented that M^r Leibnitz never communicated his reasons to the R. Society of England, & so the Society has not examined the reasons on both sides for giving judgment. And upon this pretence, the Author of the Remarks gives a judgment contrary to that of the Committee of the R. Society. But the truth is, M^r Leibnitz absolutely refused to give any reasons, calling it injustice to expect that he should defend his candor. And the Committee of the R. Society grownded their Report upon ancient & unquestionable Records, & published the Records, that the justice of their Report might be appear to the world. But the Author of the Remarks has laid aside the Records of the first seven years w^ch make for M^r Newton & begins his Report w^th the years 1676 & 1677, & thereby confesses that he has no way to defend M^r Leibnitz but by laying aside the Records w^ch make against him.

In the Remarks its said further that M^r Newton did not speak of this matter till after the death of M^r Hugens & D^r Wallis who were well informed & able to judge thereof: Which is as much as to say \implyes/ that M^r Newton began this dispute. Whereas M^r Leibnitz began it \nine years ago/ by giving an abusive \reflecting/ account of his \M^r Newton/ book De Quadratura Figurarum, & M|D|^r Keil retorted the charge upon M^r Leibnitz before M^r Newton knew what M^r Leibnitz had done. As for M^r Hugens he never was well informed about this matter nor doth it appear that he gave any judgment about it. And as for D^r Wallis, he gave his judgment 19 ye against M^r Leibnitz 19 years ago in his Preface to the first Volume of his Mathematical works published A.C. 1695. For there he saith that M^r L Newton in his Letters of Iune 13 & Octob. 24 1676 methodum hanc [de Fluxionibus] Leibnitio exponit, tum ante decem annos nedum plures ab ipso excogitatam, explained to M^r Leibnits the method of Fluxions invented by him ten years before or above, that is, in the year 1666 or before \1665/. And in a Letter dated from Oxford Apr. 20 1695 & extant in the Archives of the R. Society D^r Wallis represented that he had intimation from Holland that M^r Newton's Papers relating to the Method of Fluxions should be printed because his notions of fluxions passed there w^th great applause under the name of the Differential method,|.| &|A|nd tho M^r Newton has \in this matter/ neglected his reputation abroad, yet in the second book of his Principles written 28 years ago, he mentioned the method of Fluxions as known to him in the year 1676, & M^r Leibnitz has hitherto allowed it without being able \going about/ to make it appear that the Differential method was known to him before the year 1677.

But before M^r Leibnitz & his correspondents or some of them have composed & published the in Germany the Latin Paper without a name whereby they defame M^r Newton accuse the R committee of y^e R. Society of partiality, affirm & deny things without proof, & endeavour to set aside Records & bring things to a wrangle: I intend to give you hereafter a fuller account of these matters out of the Records themselves.

I have seen in your journal the piece sent you out of Germany conteining remaks {sic} upon the difference between M^r Leibnitz & M^r Newton \{m}ade/ with a translation of a Latin treatise piece dated 29 Iuly 1713 & published in Germany \with |& the| Remarks thereupon it./. {sic} These pieces are full of assertions \& reflections/ without any proof & without the name of the author & so are of no authority.

The author of the Latine piece represents that M^r Leibnitz had not then seen the Commercium Epistolicum: w^ch \& this/ he could not know without keeping a correspondence with M^r|him| Leibnitz \M^r Leibnits./ But{illeg} a copy of this book was sent to him \M^r Leibnitz |him|/ by the Resident of the Elector of Hannover above a year ago & another several other copies being \were/ sent to Lipsic, notis|c|e was sent back that one of them was sent to him \M^r Leibnitz/ from there. \one of w^ch was for him./ He \The Author/ tells us further that M^r Leibnitz not being at Leasure to examin this affair himself had refe{illeg}|r|red it to the judgment of a Mathematician of the first rank very skillfull in these things & very free from partiality. So then this paper was writ by the correspondents of M^r Leibnitz, & M^r Leibnitz himself was the first moover & credit of the Mathematician depen for candor & ability depends upon the credit of M^r Leibnitz {illeg}: & for these reasons this paper must be looked upon as the ablest \best/ defence that he & his correspondents were able to make, especially if this paper be writ in the style of M^r Leibnitz himself as some think. By his Letters against M^r Keil it appears that he is very \too/ much concer\n/ed about \to neglect/ this matter: & therefore his referring it to another person as if he himself was not at leasure is but a pretence |& his appealing from the Report of a numerous Committe of the R. Society to a {illeg} nameless Mathematician of his own chusing is no better then appealing to his|m|sf. For he has told the Society that it would be injustice to question his candor, that is, to deny him to be a witness in his own cause.|

Now thi|e|s great Mathematician conjectures that M^r Newton spent his first years in cultivating the method of series without thinking of the calculus of fluxions or reducing it to general rules. That is, he will not allow the Analysis communicated by D^r Barrow to be genuine M^r Collins in the year 1669 to be a genuine piece. And he brings too arguments for his conjecture.

First, saith he, in all the Letters published in the Commercium Epistolicum & in all this|e| Principia Philosophiæ, the L \l/etters with pricks w^ch M^r Newton now uses are not to be met with. But in all those Letters (except the An{illeg}|a|lysis) & in all the Principia, there was no occasion to make use of the fluxional calculus. And D^r Keil hath given a further answer to this argument long since in his Letter dated 24 May 16 1711. Observo ipsum Newtonum, saith he, sæpius mutasse nomen & notationem calculi. In tractatu de Analysi Æquationum per series infinitas, incrementum Abscissæ per litteram o designat, et in Principijs Philosophiæ, Fluentem quantitatem Genitam vocat ejus incrementum Momentum appellat: illam literis majoribus A vel B, hoc minusculis a et b designat. M^r L{eibnitz} {con}f{ines} {his} method to the symbols {dx and} dy, so that if you take away his symbols you take away his {method.} M^r Newton doth not so. And whether he uses Letters with pricks or other symbols for fluxions his method is still the same. In his Letter of 24 Octob. 1676 he represents that he had a method of extracting fluents out of Equations involving their fluxions. Will M^r Leibnitz say that he had no such method unless then he then used letters with pricks? If so, his Letters w^th pricks are as old at least as the year 1676, & by consequence older then the differential Notes of M^r Leibnitz.

But its to be observed that fluxions & Differences are not quantities of the same kind. Fluxions are velocities & Differences are small parts of things generated by fluxion in moments of time: fluxions are always finite quantities & differences are infinitely little. M^r Newton uses sometimes prickt letters sometimes other symbols for fluxions: M^r Leibnitz uses no symbols for fluxions to this day. The symbols of fluxions \used by M^r Newton,/ whether with pricks or without, <498v> therefore the oldest in the kind. These he multiplies by the letter \moment/ o to make them infinitely little & puts the infinitely little rectangles for moments & without the letter \mo{illeg}moment/ o either exprest or understood they \prickt letters/ never signify moments or differences, but are always finite quantities & signify velocities. The fluxion of time or of any exponent of time he usually represents by an unit & the moment thereof by the letter o.

The second reason of the great Mathematician for his conjecture is that M^r Newton understood not the differences of differences till after the writing of his Principia. For there, saith he, the constand|t| increase of the letter x he represents not by a prickt letter as at present but by the letter o after the vulgar manner w^ch destroys the advantages of the differential method. Here o^r great Mathematician commits two mistakes: one by supposing that M^r Newton represents differences by a prickt line prickt letters, another by supposing that the method used in the 10^th Proposition of the second book of the Principia is M^r Newtons method of fluxions. Tis only a branch of his method of converging series. In his Letter dated 10 Decem 1672 where he speaks of a method whereof the method of Tan{illeg}|g|ents there described is a branch or Corollary, he represents that this method (w^ch is the method of fluxions) extended to Questions about the curvature of curves; & thence it is manifest that he then understood the second fluxions or differences of Differences.

The Letter o was used by \by M^r Newton/ in the manner above described in the|his| Analysis communicated by M^r Collins Barrow to M^r Collins in the year 1669 & in his book of Quadratures, & is still used by him in the \very/ same manner. And \as/ it is the oldest notation \for moments or differences/ so it is the best, the method thereby being more convenient, more elegant, \&/ more suitable to Geometry then the differential \notation/, & as universal, & does justice to the memory of M^r Fermat who first brought in the use of this letter o. And thus much in answer to the great Mathematician.

As to what the Author of the Latin pa{illeg}|p|er saith of M^r Hook & M^r Flamsteed. M^r Hook indeed claimed one of M^r Newtons Propositions but could never produce a demonstration: M^r Leibnitz claimed it also but the Demonstration by w^ch he claimed it wa is erroneous. M^r \newton |Newton always| acknowledge{s}|d|/ Flamsteed refused \acknowledged/ the use of his \M^r Flamsteeds/ observations, & M^r Leibnitz \also/ claimed an invention from M^r Tschurnhause \But M^r Newton always acknowledged y^e use of M^r Flamsteeds observations./. {sic}

Our Author in the next place complains of the Committee of the R. Society for representing that M^r Leibnitz had from M^r Iames Gregory the series for finding the Arc of a circle by the tangent; that is, he \confesses that he has no other way of defending M^r Leibnitz then by/ represents|i|/ng\ that the Letters of M^rs Gregory Collins Oldenburg & Leibnitz examined & approved by a Committee numerous Committee of the R. Society were fourged. The Letter of M^r Gregory dated 15 Feb. 167$\frac{0}{1}$ is still extant in his own hand-writing & conteins this series with several others then sent to M^r Collins. That of M^r Oldenburg dated 15 Apr. 1675 is extant in the Letter-book left by of the R. Society left by M^r Oldenburge in their Archives & conteins this series w^th several others then sent by M^r Oldenburgh from M^r Collins to M^r Leibnitz then \{illeg}/ at Paris. The answer of M^r Leibnitz dated from Paris May 20^th 1675 was found a{illeg} in the same letter-book; & the original letter in the hand-writing of M^r Leibnitz was also found in the Archives of the R. Society & conteins his acknowledgment of the receipt of M^r Oldenburgs Letter above mentioned with the series conteined therein. It begins thus. Litteras tuas multa fruge Algebraica refertas accepi pro quibus tibi et doctissimo Collinio gratias \ago/. Cum nunc præter ordinarias curas Mechanicis imprimis negotijs distrahar, non potui examinare Series quas misistis ac cum meis compar{illeg}|a|re. By w^ch \these/ words its manifest that he did not not at that time \in his answer/ claim any of \its plain \that/ M^r Leibnitz at this time knew/ \distinguished all none of/ the series then sent him \to be from his own, tho before/, {sic} then he communicated the same ye{illeg}|a|r \the end of the year he communicated/ to his friends at Paris one of them as his own, viz^t that of Gregory \then dead/, & by vertue of that communication has ever since claimed it as his own. The collection of the papers of M^r Gregory made by M^r Collins after the death of that Gentleman, is still extant in the hand-writing of {illeg} M^r Collins & at the request of M^r Leibnits was sent to Paris in Iune 1676, & conteins <499r> a copy of the aforesaid Letter of M^r Gregory. But upon {the d}eath of that Gentleman M^r Leibnitz to make this series his {own} {p}retended in his Letter dated 27 Aug. 1676 28 Decem 1675 that he h{ad co}mmunicated it at Paris above two years before, & that it was the ser{ies} w^ch he had wrote of before to M^r Oldenburg, that is in his Letters {of} 15 Iuly & 24 Octob. 1674. And under this pretence he sent it back to M^r Oldenburg as his own in his Letter dated 27 Aug. 1676. And yet the series w^ch he wrote of in his said two Letters dated 15 Iuly & 24 Octob 1676|4| {illeg}|C|alling \it/ a Theoreme or method for finding the arc of a circle or any part thereof. was not this series but another for finding the arc by the tangent, but another \Theorem or Method/ for finding the arc by the sine. This \Theorem or/ series M^r Collins had received from M^r Newton in Iuly 1669, {illeg}|&| communicated it soon after to his friends very freely. M^r Leibnitz was in London in the years 1671, 1672 & the beginning of 1673, & having met with this series \either in London or soon after in France/ pretended in his said Letters of 15 Iuly & 24 Octob. 1674 to have found it himself, & yet in his Letter dated 12 May 1676 desired M^r Oldenburge to procure from M^r Collins the Demonstration thereof, that is, the method of finding it. And when he had received the method w^th some of M^r Newtons series he pretended to have found some \three/ of those series before, tho he did not yet understand the method of finding them. For in his Letter of 27 Aug. 1676 he wrote back for a further explication of the method. M^r Newton therefore in his Letter of 24 Octob. 1676 explained it further & added another method of the same kind, & M^r Leibnitz in his Letter dated 12 Iuly \21 Iune/ 1677 still desired a further explication, but so soon as he understood it, he wrote in his Letter dated 12 Iuly 1676|7| that he found by his old papers that he had used one of those methods before. And by the same power of invention when \he had newly found the differential method, (w^ch he might do/ by the help of Newtons Gregories & Barrows methods of Tangents & Newtons Letters \of 10 Dec. 1672, 13 Iune 1676 & 24 Octob 1676) to {se}/ he had newly found the differential method, to make that method his own he 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. And yet its very certain that he had but newly found it. For in his Letter dated 27 Aug 1676 \he wrote/: Quod dicere videmini pleras diffi{illeg}|c|ultates ad (exceptis Problematibus Diophantæis) ad Series Infinitas reduci; id mihi non videtur. Sunt enim multa us adeo mira et implexa ut ne ab æquationibus pendeant ne ex Quadraturis. Qualia sunt ex multis alijs Problemata methodi tangentium inversæ; quæ etiam Cartesius in potestate non esse fassus est. These words are a demonstration that he did not then understand the differential method. |He was then composing & polishing the Quadrature of the circle vulgari more & left off that way of writing as soon as he found the differential method.| I pass by his claiming the inventions of Mouton & Paschal, & a considerable part of M^r Newtons Principia Philosophiæ.

Our Author tells us further that M^r Leibnitz published in the Acta Eruditorum a general method for finding the Ordinates of transcendent Curves not by extraction of roots but deduced from a profounder foundation of the differential calculus, by w^ch the business of series was brought to a greater degree of perfection. But M^r Newton many years before (viz^t in his Letter to M^r Oldenburge dated 24 Octob. 1676) communicated the \very/ same method in this sentence. Altera [methodus consistit] in assumptione seriei pro quantitate qualibet incognita ex qua cætera commode derivari possint & in collatione terminorum homologorum æquatione|i|s resultantis ad eruendos terminos assumptæ seriei. M^r Leibnitz therefore has no title to any part of the method of converging series.

Our Author tells us further that M^r Leibnitz \{illeg}/ was the first who used the exponential calculus \M^r Newton knowing nothing thereof/. Certainly M^r Newton was the first who |introd|us|c|ed \into Analysis/ fractions radicals & negative quantities for the indices \or exponents/ of d|D|ignities & thereby very much enlarged Analysis & laid the foundation of making <499v> it {unive}rsal. In his Letter dated 24 Octob. 1676 he mentioned such exponents {of D}ignities, & thereupon M^r Leibnitz in his Answer dated 21 Iune 1677 propose{d in}determinate Exponents of Dignities. |And| This was \might be seems to have been/ the Original of such the exp{on}ential calculus: but such a calculus has hitherto been of no use.

B{ut} o|O|^r author tells us that y^e English & Scotch, Wallis, Hook, Newton & Gregory junior, acknowledged 36 years ago the series for finding the arc of a circle by the Tangent to be the invention of Leibnitz. That is, he complains of M^r Oldenburg for not letting the English & Scotch know that he had communicated this series to with several others to M^r Leibnitz in April 1675. Tis sufficient that the Letters between M^r Oldenburg & M^r Leibnitz were left by M^r Oldenburg in the Letter-book of the R. Society & that the Original Letter of M^r Leibnitz is still extant in his own handwriting. And thus much concerning the printed |Tis as good an argument for M^r Newton \I may add that/ that M^r Leibnitz in his Letter dated 21 Iune 1677 acknowledged that M^r Newton had a method like the differential. And thus much concerning y^e printed| paper.

In the Remarks it's represented that M^r Leibnitz never communicated his Reasons to y^e Royal Society of England & so the Society has not examined the reasons on both sides for giving judgment. Whereas \But the truth is/ M^r Leibnits refused to give any reasons at all, calling it injustice to expect that he should defend his candour \detracting from y^e candor of \M^r Keil/ & pressing the R Society to give judgment without hearing his reasons/; & the Committee of the R. Society grounded their Report \not upon plausible reasons but/ upon the matter of fact conteined in the Letters & Papers found in their {illeg} Adversaria of the R. Society & \in/ those of M^r Collins \with C/, & published those Letters & Papers in the Commercium Epistolicum that all the world might see the grownd of their Report. When M^r Leibnitz produces his reasons against matter of fact it will be time enough to consider them |And those records are sufficiently s|p|lain to any man that {illeg} \consider/ them impartially. authentic them {illeg} impartially considers them impartially.| |records| to prove that he had the Differential method before the year 1677 or that Newton had not the method of fluxions before that year \or that he invented the series of Gregory before he received it from Oldenburg/ h{illeg} may deserve to be heard \he may be heard/. All other pretenses are trifling. And when he produces reasons to prove that he invented the series of Gregory before he received it from Oldenburg, he may be heard also upon that head. And the like for \all/ the rest of his pretended inventions above mentioned. In short he |has| the character of being too loquacious & noisy to be a good inventor & too vainglorious to forbear assuming In the mean time it is to be observed y^t the authors of this|e| paper is too much Latin paper & the remarks \upon it/ have not been able to defend M^r Leibnitz without d laying aside the Analysis & several of the \ancient/ Letters examined & approved by a numerous Committee of the R. Society. And that their Ar

M^r Leibnitz in y^e Acta Eruditorum \of Febr./ 1705 {illeg} accused M^r Newton of {illeg} deriving his Method of flxuions from the differentiall {illeg} method & in the differen his Letters against M^r Keil {illeg} under the colour of appealing has pressed M^r Newton to give his judgment in \& still persist in the accusation in affirming it,/ & therefore ought in justice to \prove his accusation &/ defend his own candor \& prove his accusation/. For no man is a witnesse in his own cause, & To accuse without To accuse without proof is calumny.

And they have this further weight that M^r Leibnitz is known to be of a temper too loquacious, boystero & vainglorious to be admitted a witness in his own cause.

S^r

I have seen in your journal the piece sent you from Germany conteining remarks upon the difference between M^r Leibnitz & M^r Newton with a translation of a piece a {illeg} transla a Latine piece dated 29 Iuly 1713 & published in Germany All th These pieces are full of assertions without proof & w^thout the name of the author & so are of no authority. {illeg}

The author of y^e Remarks pretends that M^r Leibnitz has not yet seen the Commercium Epistolicum, & therefore is one of this|e| \he keeps a/ correspondents|ce| of He pretends also that N whereas \with M^r Leibnitz/ \But/ a copy thereof was sent to him by the Resi last \M^r Leibnitz/ above a year ago by the Resident of the Elector of Hanover & several other copies being sent him by to Leipsick, notice was sent back that one of them was sent \to him/ from thence.

This \nameless/ Author pretends that M^r Leibnits never communicated his reasons to M^r Leibnitz the R. Society of England & so the Society has not examined the reasons on both {illeg} sides for giving judgment. {illeg} {H} When D^r Keil sent le to M^r When the secretary of the R. Society sent to M^r Leibnitz a Letter of D^r Keil a Letter of conteining such reasons against him as were unanswerable, M^r Leibnitz refused to answer {illeg}|t|hem pro answer them or produce any reasons for himself & {illeg} & cried out that M^r Keil impugned his candour, w^ch that he should d at such an age & \after/ so many documents of his life he should defend, would be injustice to expect That is, he told the Society that they were would be unjust unless they admitted him to be a witness for himself, contrary to the laws of all nations. Whereupon the Society whose motto is Nullius in verba appointed a Committee to examin Records & the Report of the Committee being grounded upon Records is s|v|alid, & {illeg} ought \upon questionable/ Records is as valid as the Records themselves, For the Records are plain & evident, & being published may be {we}{illeg} as a{n}{illeg} A{illeg} & the \pretended/ reasons of M^r Leibnitz are not to be regarded till he produces them.

The said Author proceeds to make a contrary Report \contrary to that of y^e R. Society/, as if the Report of a nameless person could be of any validity. He begins his report of the w^th the Letters between M^rs Leibnitz Oldenburgh & Newton & dr & {illeg} the & drops the Analysis communicated by D^r Barrow to M^r Collins in 1669 \some years before/ in w^ch M^r Newtons method of fluxions is plainly descri{illeg}|b|ed & therefore his report is partial. Let the Records themselves published in the Commercium Epistolicum be consulted.

The Author of the Latine piece \dated 29 Iuly 1713/ represents also that he M^r Leibnitz had not yet seen the Commercium Epistolicum & therefore he {illeg} also keeps of M^r Leibnitz \a/ correspondents|ce| w^th M^r Leibnitz.

The author complains that M^r of M^r Newton's long silence but before he complained of this he ought to have shewed that {illeg} M^r Newton brake silence now \began this dispute/ It's almost 40 years that since he left of wr corresponding in Mathematicks & almost twenty since he left of these studies. M^r Leibnitz by what he printed against M^r Newton in the Acta Eruditorum \in Ianuary/ Anno 1707, Th begun this dispute, M^r Keil answered M^r Leibnitz & M^r Leibnitz replied upon \wrote his first Letter against/ M^r Keil before M^r Newton knew what was printed against him as is here well known in the Acta Eruditorum as is here well known. And M^r Newton must be allowed to give his friends leave to repell injuries.

The Author of the Latine piece dated 29 Iuly 1713 represents that M^r Leibnitz had not then seen the Commercium Epistolicum & therefore he also keeps a correspondence with M^r Leibnitz. He tells us further that M^r Leibnitz not being at leasure to ex{illeg}|a|min this affair himself, he had referred it to the judgment of a Mathematician of the first rank very skilful in these things & very free from partiality. So then this paper was writ by the correspondents of M^r Leibnits, & M^r Leibnitz \himself/ was the first mover: & therefore it must be looked upon as the ablest defence that he & his correspondents were able to \could/ make, especially if this paper be writ in the style of M^r Leibnitz \himself/ as some think. By his Letters ag^t M^r Keill it appears that he is much

Now this great Mathematician tells us conjectures that M^r Newton <500v> spent his first years in cultivating the method of series without thinking of the calculus of fluxions or reducing it to general Rules. That is, he will not allow the Analysis communicated by D^r Barrow to M^r Collins in the year 1669 to be a genuine piece.

But|And| he brings two arguments for his conjecture. First saith he in all the Letters published in the Commercium Epistolicum & in all his Principia Philosophiæ the letters with pricks w^ch M^r Newton now uses are not to be met with. But D^r Keill hath answered this argument long since in his Letter dated 24 May 16 1711. Illud Observo, saith he, Observo ipsum Newtonum saith he sæpius mutasse nomen & notationem calculi. In tractatu de Analysi Æquationum per series infinitas, incrementum Abscissæ per literam o designat. Et in Principijs Philosophiæ, Fluentem, quantitatem Genitam vocat, ejus incrementum momentum appellat: illam literis majoribus A vel B, hoc minusculis a et b designat. M^r Leibnitz confines his Method to the symbols dx & dy, so that if you take away his symbols you take away his method. M^r Leibnit M^r Newton doth not so. And whether he uses Letters with pricks or other symbols for fluxion his method is still the same. In his Letter of 24 Octob. 1676 he represents that he had a method of fluxions extracting fluents out of Equations involving their fluxions. Will M^r Leibnitz say that he had no such method unless he then used letters with pricks? If so, his letters with pricks are as old at least as the year 1676 & by consequence older then the differential Notes of M^r Leibnitz.

But its to be observed that fluxions & differences are not quantities of the same kind. Fluxions are velocities & differences are small parts of things generated by fluxion in moments of time. Fluxions are fi always finite quantities differences are infinitely little. M^r Newton sometimes uses prickt letters sometimes other letters or symbols for fluxions: M^r Leibnitz uses no symbols for fluxions to this day. The symbols of fluxions therefore used by M^r Newton are the oldest in the kind. These he multiplies by the Letter o to make them infinitely little moments or differences, & without the letter o either exprest or understood they never signify differences but are always finite quantities & signify veloci{illeg}ties.

The second reason of our Mathematician \for his conjecture/ is that M^r Newton understood not the differences of differences till after the writing of his Principia. For there, saith he, the constant increase of the letter x he represents not by a prickt letter but by as at present but by the letter o after the vulgar manner w^ch destroys the advantages of the differential method. Here o^r great Mathematician commits two mistakes: one by supposing that M^r Newton represents differences by a prickt letter|s|; another by supposing that the method used in the Principia is M^r Newtons method of fluxions. Tis only \a branch of his method of converging series,/ a method of resolving quantities into converging series & applying those series to the solution of Problemes \without considering fluxions/. In his Letter dated 10 Decem 1676|2| where he is speaking of fluxions \a method/ whereof the method of Tangents there spoken of \described/ is a branch or corollary, he represents that this method \(w^ch is the method of fluxions)/ extends|e||d| to Questions about the Quadrature of Curvature of curves, & thence its manifest that he then understood the \second fluxions or/ differences of differences. |In the end of his book of Quadratures|

And as for the use of the letter o in the method of fluxions M^r Newton used it in his Analysis communicated to M^r Collins by D^r Barrow in the year 1669, & in his book of Quadratures where he represents fluxions by by {sic} prickt letters, & it is \he/ still uses|d| \it/ in the same sence. And as it is the oldest notation so it is the best, the method thereby being more convenient more \elegant & more/ Geometrical & more universal \then the Differential & as universal./ D^r Barrow in his method of Tangents published A.C. 1670, put the letters a & e for the differences of the Abscissas & Ordinates, & M^r Leibnits seven years after changed these letters a & e into the symbols dx & dy & gave the method a new name; beginning where <501r> M^r Leibnitz D^r Barrow left off as that candid Gentleman the Marquess de L'Hospital in the Intr observed long since in the Preface to his book Analysis. And this was the original of the differential method, M^r Leibnitz learning by the Letters of M^r Newton how to improve the differential method of D^r Barrow. M^r {O} Leibnitz in his Letter to the M^r When M^r Leibnitz wrote \in/ his Letter to M^r Oldenburgh dated 27 Aug. 1676 he affirmed he affirmed \wrote/: Quod dicere videmini pleras difficultates ad (exceptis Problematibus Diophantæis ad series infinitas reduci id mihi non videtur. Sunt enim multa us adeo mira et implexa ut ne ab æquationibus pendeant ne ex Quadraturis. Qualia sunt (ex multis alijs) problemata methodi tangentium inversæ. And these words make it demonstratively certain that M^r Leibnitz did not then understand the differential method. And yet in his Letter dated 21 Iune 1677 he wrote Clarissimi|e| Slusij Newtono assentior methodum tangentium nondum esse absolutam Celeberrimo Newtono assentior. Et jam a multo tempore rem tangentium generalius tractavi, scilicet per differentias Ordinatarum. He had newly {illeg} found the differential method: & to make it his own pretends|e||d| that he had found it jam a multo tempore But if he would have us beleive that he found it before the year 1677 lies it lies upon him to prove it. For no man is a witness its against the law of all nations to allow any many to be a witness in his own cause. |And thus much in Answer to the great Mathematician|

As to what o^r |y^e| Author \of the Latin paper/ saith of M^r Hook & M^r Flamste He \indeed/ claimed one of M^r Newtons Propositions but could never produce a demonstration. M^r Leibnitz has claimed it also but {p}{illeg} thi|e|s Demonstration \by w^ch he endeavoured to make it his own is erroneous/ is erroneous {sic}. And for as for \if/ M^r Flamsteed how that matter stands is better known in London then in Germany. \denyed M^r Newton the use of his his {sic} Observations, M^r Tschurnhause denyed M^r Leibnitz an invention/ And whether M^r Leibnitz or M^r Tschurnhause were in the right when they fell out about an invention is not material to us in England.

Our Author in the next place that complains of the Committee of the Royal Society for attributing to representing that M^r Leibnitz had from M^r Iames Gregory the Series for finding the arc of a Circle by the tangent & of M^r Newton for approving this accusation, that is he & of M^r Newton for approving this accusation, that is, he represents that y^e letters of Gregory Collins & Oldenburg & Leibnitz were put \examined &/ approved by the Committee & published in |y^e| Commercium were spurious fourged. The L Wherea The Letter of M^r Gregory dated 15 Feb 167$\frac{0}{1}$ is still extant in his own hand writing & conteins this Series w^th several others then sent to M^r Collins. That of M^r Oldenburg dated 15 Aprril 1675 is extant in the Letterbook left by M^r Oldenburg in the Archives of the R. Society & conteins this series w^th several others then sent from M^r Collins to|by| M^r Oldenburg to M^r Leibnitz The Letter \Answer/ of M^r Leibnitz dated 20 May 1675 from Paris 167 20 May 1675 {illeg} was found in the Archives of the R. S. & is still kept by their Secretaries & is still extant in the very \very/ hand \writing/ of M^r Leibnitz & conteins his {illeg} acknowlegment of the {illeg} receipt of M^r Oldenburgs letter above mentioned w^th the series therein conteined therein. The Collection of the papers of M^r Gregory wer by M^r Collins after the death of that Gentleman {illeg} is still extant in the handwriting of M^r Collins & {illeg} was sent to at y^e request of M^r Leibnitz was sent to Paris hi{illeg} \Paris/ in Iune 1676 & conteins the aforesaid Let a copy of the aforesaid Letter of M^r Gregory. The But \upon the death of M^r Gregory/ M^r Leibnitz to make this series his own pretended in his Letter dated 28 Decem 1675 that he had found it \communicated it at Paris/ above two years before & that it was the series which he had wrote of to M^r Oldenburg before, that is in his Letters of 15 Iuly & 24 Octob 1674 & under this pretence \he/ sent it back to M^r Oldenburg as his own in his Letter dated 27 Aug. 1676. Whereas the series which he wrote of to {illeg} in his said <501v> two Letters dated 15 Iuly & 24 Octob 1674 was not this series but another for finding the Arc of a circle by the sine. This series M^r Collins had received from M^r Newton in 16 in Iuly 16{illeg}|6|9 & communicated it \soon after/ to some of his friends. M^r Collins Leibnitz was in London 16 in the years 1670, 1671 1672 & 1673, & having met with this series pretended in his said Letters of 15 Iuly & 24 Octob 1674 to have found it himself & yet in his Letter dated 16 12 May 1676 desired M^r Oldenburg to procure him \from M^r Collins/ the Demonstration thereof, that is, the method of finding it. And when he had received the method w^th some examples thereof \of M^r Newtons series/, he endeavoured pretended to have found those some of them \M^r Newtons |those|/ series in \all/ those \in those/ series before, though he did not yet understand the method of finding them. For in his Letter of 27 Aug 1676 he wrote back for a further explication of the method. And when M^r Newton in his Letter of 24 Octob. 1676 had explained a double method he \& added another method of regression M^r Leibnits/ wrote back \in his Letter dated 12 Iuly 1677 for a further explication. {or} so soon as he understood it he wrote back/ that he had found by his old papers that |he| had found \used/ one of those methods before. And by the same spirit, of {illeg} \of invention/ when he had found by the help of M^r Newtons Letters & D^r Barrows \Gregories & Barrows/ methods of Tangents he had newly found the differential method, to make that method his own, he wrote back {illeg} Et jam a multo tempore rem tangentium generalius tractavi scilicet per differentias Ordinatarum. |And yet its very certain that he had but newly found it. For in his Letter dated|

But our Author tells us that the English & Scotch |(|\viz^t/ Wallis, Hook, Newton & Gregory junior,|)| &|k|new not acknowledged for above 36 years together the the {sic} Series for finding the arc of a circle by the Tangent to be M^r the invention of Leibnits. As for Hoos is dead long since & medled not w^th these matters. Wallis \& Gregory/ lived {illeg}|a|t Oxford & Newton at Cambridge \and first at Edinburgh & then at Oxford/ & knew not \had not opportunity of knowing/ what passed between Oldenburg & Leibnitz. Gregory the younger knew compla in a treatise published 1684, complained that nothing of his Vnkles adversaria relating to this method of series came to his hands except a few examples destitute of y^e method of finding them. Oldenburg died soon after he had sent Gregories series to Leibnitz & when Leibnitz sent him word that he had found this series a{b}o{illeg} \some years/ before, there was no reason fa|o|r Oldenburg to fall out w^th him|s| \friend & countriman/ about it. It was no bodies interest to trouble th but Gregories to trouble themselves about it, & the matter might have lain some years {illeg}|s|till in the dark had not M^r Leibnitz himself put the R. Society to \upon/ appointing a Committe to se{illeg}|a|rch their Archives & those of M^r Collins {illeg} about what related to {illeg}thi|e|s correspondence w^th M^r Oldenburg & M^r Collins between Oldenburg Collins & Leibnitz. And since M^r Leibnitz & his correspondents question the credit of the Letters found in the in \upon this search/ the R. Society have ordered affadavits to be taken thereof before a publick Notary & are ready to shew \them Originals & affadavits/ to any publick ministers of the Emperor King of Prussia or Elector of Hannover \Empire France or Holland/ at the request of M^r Leibnitz or his friends.

Our Author tells us that M^r Leibnitz shewed the series for the arc by the Tangent to M^r Hugens|i||us| who applauded the same & that M^r Newton also acknowledged the method by w^ch he found it to be a new one but he does not tell us that {illeg} {illeg} M^r Leibnitz let either M^r Hygens or M^r Newton know that he had received this series from M^r Oldenburg. |And his method is only a Proposition for transmuting of Figures & deserves not to be called a method of series.|

Our Author adds that M^r Leibnitz published in the Acta Leipsica a new method general method for finding the Ordinates of transcendent curves not by extraction of roots but from deduced from a profounder foundation of the differential Caculus {sic}, by w^ch the business of series was brought to a greater degree of perfection. But M^r Newton many years before, (viz in his Letter dated 24 Octob. 1676) proposed \communicated/ the same method in this sentence. Altera [methodus consistit] in assumptione seriei pro quantitate qualibet incognita ex qua cætera commode derivari possint, & in collatione terminorum homologorum æquationis resultantis ad eruendos terminos {illeg}|a|ssumptæ seriei. M^r Leibnitz therefore has no title to any part of the method of converging series.

Our author tells us further that M^r Leibnitz was the first who used the exponential calculus. Certainly M^r Newton was the first who {p} used fractions radicals & negative quantities for the indices of Dignities & by this thereby Analysis has been \very/ much enlarged & improved \Analysis & laid the foundation of making it universal./ In his Letter dated 24 Octob. 1676 he mentioned such indices of Dignities & thereupon M^r Leibnitz in his Letter \Answer/ dated 21 Iune 1677 proposed inde{t} æquations w^th indeterminate dignities, But the use of such æquations is not yet discovered. \such æquations have hitherto/ been of no use.

The Author of the Remarks makes a Report in opposition to \the Report of/ y^e Committe of y^e R. Society, but begins his Report with what passed in the year 1676 whereas he should have begunn it w^th what passed seven years before. \What he has omitted we may take occasion to supply hereafter./ He saith further that M^r Newton did not speak of this matter till after the death of M^r Huygens & D^r Wallis who were \well/ informed & able to judg thereof: whereas neither of them had seen the Analysis & Letters published since their death, & M^r Newton claimed it {sic} \the method/ in his Principia Philosophiæ, \as for/ & D^r Wallis in the Præface to the first Volume of his Mathematical works published A.C. 1695 saith that M^r Newton in his Letters of Iune 13 & Octob 24 1676, methodum hanc exponit [de Fluxionibus] Leibnitio exponit, tum ante decem annos, nedum plures, ab ipso excogitatam.

But since M^r Leibnitz began these disputes, & detracts from the candor of those who oppose him & in opposition to them represents it unjust to question his candor, making himself a witness in his own cause contrary to the laws of all nations, & appeals from the Report of a large Committee of y^e R. Society, to the judgment of a nameless Mathematician chosen by himself, w^ch is {all}|th|e same thing as to make himself a Iudge as well as a witness in his own cause; & since his correspondents endeavour to set aside the consideration of the ancient Letters & Papers & bring matters to a wrangle: I desire you to print (in French) the Letter of M^r Iames Gregory dated 15 Febr. 1671 to the words secundum vulgaris Algebræ præcepta, a copy of w^ch Letter was sent to Paris in Iune 1676 to be communicated to M^r Leibnitz. I desire you to print also the two Letters of M^r Leibnitz dated 15 Iuly & 26 Octob 1674 concerning a Theoreme or method ofor finding the Sector or Arc of a Circle whose sine is given; & M^r Oldenburgs Letter of 15 Apr. 1675 wherein he sent several series to M^r Leibnitz amongst w^ch was the series of Gregory; & the Answer of M^r Leibnitz dated 20 May 1675 wherein he acknowledged the receipt of those series; & the latter part of his Letter dated 28 Decem. 1675 beginning with the words, Habebis & a me instrumentum &c. All w^ch five Letters were left entered in the Letter books of y^e R. Society by M^r Oldenburg. Then print the letter of M^r Leibnitz dated 12 May 1676 w^ch is still extant in his own hand writing, & that part of his Letter of 27 August 1676 w^ch begins with these words, Sit QAIF Sector duabus rectis &c, & ends with these maxime afficiens mentem. And then leave it to y^e Reader to make his judgment upon those Letters concerning the pretence of M^r Leibnitz to the series of M^r Newton for finding the Arc by the sine & to that of M^r Gregory for finding the Arc by the Tangent & to some other series sent to him by M^r Newton. After w^ch the Reader will be better able to make a judgment of his pretence to the original invention of the method of moments &|o|r differences.

To the R

The author of y^e Remarks makes a Report in opposition to y^e Committee of y^e R. S. but begins his Report with what passed in the year 1676 whereas he should have begunn it with what passed seven years before. He saith further that M^r M^r {sic} Newton {illeg}|d|id not speak of this {in}|m|atter till after the death of M^r Huygens & D^r Wallis who were well informed & able to judge impartially of this matter. When as neither of them had seen the \Analysis &/ Letters published since their death \thereof. Whereas neither of them \M^r Hugens/ was not (were fully informed/, & D^r Wallis in the Preface to y^e first Volume of his mathematical works published A.C. 1695 |saith| that M^r Newton in his Letters of Iune 13 & Octob 24 17|6|76, methodum hanc [de Fluxionibus] Leibnitio exponit, tum ante decem annos nedum plures, ab ipso excogitatam. And M^r Newton in his Principia Philosophiæ \written 28 years ago, spake of this matter/ represented|in||g| that he had the method of

But since M^r Leibnitz in y^e Act has t{ax}ed t has began these disputes & & dectracts from the candor of every body \those/ who opposes him & in opposition to them represents it unjust to question his candor & making himself both witness \making himself a witness in his own/ \cause contrary to the laws of all nations/ & appeales from the Report of a large Committee of y^e R. Society to a {sic} the judgment of a nameless Mathematician of his own chusing, w^ch is the same thing as to make himself both {illeg}|W|itness & |a| Iudge \as well as a Witness/ in his own cause, {illeg} & we \since/ his correspondents endeavour to set aside the consideration of the original \ancient/ Letters & Papers & brig bring the matters to a wrangle, I desire you to print the Letter of M^r Iames Gregory dated 15 Febr. 1671 to y^e words secundum vulgaris Algebræ præcepta, a copy of w^ch Letter was sent to M^r sent to Paris in Iune 1676 to be communicated to M^r Leibnitz. I desire you to print also the two Letters of M^r Leibnitz dated 15 Iuly & 26 Octob. 1674 concerning a serie Theoreme or Method of finding the sector or ac|r|c whose sine is given. And M^r Oldenburgs Letter of 15 Apr. 1675 \wherein the \several/ series of M^r Gregory {illeg} sen |wherin several series were sent| to M^r L:/ & the Answer of M^r Leibnitz dated 20 May 1675 \wherein he acknowledged the receipt of those series/ & the latter part of his the Letter of M^r Lei his Letter dated 28 Decemb. 1675 beginning with the words Habebis & a me Instrumentum &c. All w^ch five Letters are {illeg} were entre left entered in the Letter books of y^e R. Society by M^r Oldenburg. Then print the letter of M^r Leibnitz dated 12 May 1676 w^ch is still extant in his own had|n|d writing & that part of his Letter of 27 Aug. 1676 w^ch beginns w^th these words Sit QAIF Sector, duabus rectis &c & ends with these maxime afficiens mentem. A{l}|n|d \then/ leave it to y^e Reader to make his judgment upon them|ose| Letters concerning this|e| pretence of M^r Leibnitz to the series of M^r Gregory Newton for finding the Arc by the sine & to that of M^r Gregory for finding the Arc by the tangent, & to \some other series sent to him by M^r Newton/ After w^ch the Reader will be better able to make a judgment of his pretence to y^e original invention of the method of moments & different|c|es.

\written to M^r Leibnits concerning the method of fluxions/ fluxions in y^e ye at least ten years before that time but did not acknowledged that M^r Leibnitz had it b above nine years before the differential method so long ago & M^r before the next year] Where{as} b|B|ut M^r Leibnitz did not pretend to the differential method till the \year/ after the receipt of M^r Newtons Letters. And this is \was/ affirmed by M^r Newton himself partly in his P Princip the second book of his Principles. And M^r Newton also in the second Book of his Principles Commentio written 28 years ago claimed the method of fluxions as known to him in the year 1676 & his claim M^r Leibnitz has allowed his hitherto allowed this|a|t claim without being able to prove \make it apper/ that the Differential method was known to him before the year 1677. But because And \D^r Wallis also/ in a letter dated from Oxford Apr. 20 1695 & still extant in the Archives of the R. Society in the hand writing of the author, complained that M^r Newtons notions of Fluxions \represented that/ he had intimation from Holland that M^r Newtons notions of Fluxions friends in Holland {illeg} Letters {illeg} or papers relating {illeg} to y^e Method of Fluxions were \should be/ printed because his notions of Fluxions passed there with great applause under the name of the d|D|ifferential method, & thereupon he complained that M^r Newton in neglecting this matter was not so kind to him|s|self \reputation/ as he ought to be.

But because the Author of the Paper published in Germany & of y^e Remarks {sic} upon it \& \But because/ M^r Leibnitz & his correspondents have published in Germany a Paper whereby they endeavour/ endeavours to set aside the {C} ancient Records & run the Dispute Le into a squabble \defame M^r Newton & the Committee of the R. S. & misrepresent the whole affair/. I intend to take an occasion of g{illeg}|i|ve|i|ng you an Account of these matters out of Originals \records/ themselves. But because M^r Leibnitz by his correspondents have published \somewhere/ in Germany a Paper \without a name/ whereby tha|e|y endeavour to defame M^r Newton & the Committee of the R. Society, & M^r Newton, \&/ to set aside Re{illeg} to make it a dispute between England & Germany, & \&/ \& to/ set aside Records & bring y^e matter to a squabble & make it a dispute between England & Germany (all w^ch are {illeg}wicked evil /very dishonest\ practises) I intend to give you hereafter a fuller account of these matters out of the Records themselves.

M^r Newton in a Letter written to M^r Collins 10 Decemb. 1672, that is, some weeks before Slusius sent his Method of Tangents into England, described the same method of tangents as a Corollary or branch of his general method – – – – extended it to the consideration of the second fluxions.

And in his Letter to M^r Oldenburg dated 24 Octob. 1676 he represented that five years before, viz A.C. 1671, he wrote a treatise of the method of infinite series & of another method w^ch readily gave the method of Tangents of Slusius & stuck not at surds & w^ch was founded in this sentence. Data æquatione quotcun fluentes quantitates involvente, invenire fluxiones & vice versa. Which sentence relating to y^e 2^d 3^d & following fluxions as well as to the first: it must be allowed that in the year 1671 he had extended his method to all the|o||se| fluxions, especially since the method is one & the same in them all. The method which being applied to y^e first equation gives a new equation involving the first fluxions, if applied to this new \least new/ equation will give a \another/ new one involving the second fluxions, & so on perpetually.

The sentence Data æquatione \quotcun/ fluentes quantitates involvente invenire fluxiones & vice versa, being the foundation of the method upon w^ch he wrote in the year 1671 & extending \relating/ to y^e 2^d 3^d & following fluxions as well as to y^e first, sufficiently shews that in & being one & the same method in them all, it must be allowed that \his method/ in y^e year 1671 he had extended his method to all the fluxions. For after the \very/ same manner that this method being applied to any æquations gives th{illeg} a new æquation involving the first fluxions of the fluents, if it be applied to this new one it will gives another new one involving their second fluxions & so on perpetually

I had almost forgot to observe that the author of the Remarks complains of the Royal Society for giving judgment without hearing both sides, & thereupon represents their sentence voyd. M^r Leibnitz complained to them against M^r Keil /me\, proposing that they should make him \me/ publickly retract what he I had written. And would they make justly make him retract without acknowledging \{illeg} examining/ the matter? Have they not as much authority over M^r Leibnitz as over M^r Leibnitz as over M^r Keill? O And could they with any colour of justice make M^r Keill condemn M^r Keil \me/ without examining into the matter? But M^r L But M^r Leibnitz thought the matter so clea{illeg}|r| as to need no examination & did therefore demanded justice against M^r Keil \me/ without submitting the matter to the{ir} examination of the R. Society. And are So then he So then M^r Leibnitz For he told them {illeg} in his last letter letter that M^r Keil attackt his candor w^ch that he sho So there|n| And was it either just or decent for M^r Leibnitz to make himself Iudge in his own cause & d{illeg} order the R. Society to put his sentence in execution. And But M^r Leibnitz wrote to the Society that M^r Keil I impugned his candor w^ch that he at such so great an age should defend & after so many documents of his life should defend, w^th an Apology no prudent or just man would & contend \as it were before a tribunal/ with a novice & one unaccquainted with the matter \things done formerly/ no man prudent or just would approve of. And have not I as much a right to complain of M^r Leibnitz as M^r Leibnitz has to cop|m|plain before the R. Society as M^r Leibnitz has to complain of me? Or have they not an equal authority over us both? Or if they could not condemn M^r Leibnitz without a hearing, {illeg} If they could it be just in M^r Leibnitz to condemn me w^th write to the|m| Society to condemn me without a hearing? Certainly M^r Leibnitz in complaining to the R. Society against me gave them authority to examin the matter between us, And if But the R. Society have given judgment without hearing both parties & therefore their sentence is voyd. No sure And Certainly M^r Leibnitz upon complaining \to them/ against M^r Keil was bound to give his reasons \might justly look {as his imply}/. And the R. Society upon his giving us refusing to give his reasons against M^r Keill \me/, and (as he has did in his last letter) & still can had authority to censure him as guilty of calumny \& obliged him\self/ to give his reasons again against me for justifying his accusation {illeg} [least he should be condemned of calumny And upon the writing of his last letter {illeg} dated 29 Decem 16|7|11 wherein he declined giving {illeg} his reasons against me had sufficient ground to {c}en censure him for calumny.] & upon declining to {illeg} justify his accusation made himself liable to be condemned of calumny./ However the R. Society have only appointed a Committee to seach {sic} out old {illeg} records & \to/ given their opinion upon to the Society upon them & con ordered the papers \Records/ & Report \upon them/ to be pl published.|,| {illeg} & they that compare the Report with the Records will find them agree. But M^r Leibnitz after all this \still/ declines to enter into the merits of the cause, {illeg} \&/ pretends that he being has not yet seen the Commercium Epistolicum & that he {illeg} \being/ not at leasure has writ to another to examin it, & a great Mathematician to examin it \that book/ & {the} Mathematici \give his judgment upon the matter/ & sent the answer \judgment/ of the great Mathematician to his correspondent in Germany to be published, & {illeg} \dated 29 Iuly 16 1713/ dated 7 Iune 1713 to his correspendent {sic} in Germany \a nameless friend \correspondent/ in Germany/ to be \there/ published|.| in Germany. And his correspondent has published it withou {illeg} inserted it into \published it in Germany together with/ a railing scurrilous letter dated 29 Iuly 1713, & published it \in Germany/ without setting either his own \without setting/ any man's name to it|.| [& \the same or/ another nameless person has added Remarks upon it in the same stile.] And while M^r Leibnitz \instead of {illeg}|a| sober answer defence has/ has set on foot the wri set on foot the writing of this \scurrilous/ paper has has & either writ it himself & sent to {illeg} it to his correspondents to be published or knows the names of them who writ it, he has made himself answerable for the whole untill he shall discover the names of his Accomplices.

I had a{ml} almost forgotten to observe that the the Author of the Remarks complains of the Committee of the R. S. for giving judgm^t without hearing both parties. But|And| \have/ I have {sic} \not/ as much \more as much/ reason to complain that he wrote to the{illeg} Secretary of \M^r Leibnitz desired/ the R. Society to condemn me without a hearing. If If & refused to give any reasons against me If he thought matters t|s|o{o} plaine against me as to need no examination, I think think them plainer against him. {illeg}|I|f By comcomplaining {sic} to the R. S. against me he gav & pressing his complaint by a second letter he gave them authority to appoint a Committee to examin the matter \between us/ & obliged himself to produce his justify his accusation produce his reas justify \produce his reasons & for justifying/ his accusation before the Society, {illeg} by least it should go for a calumny.