Mr Leibnitz by his Letter of 29 Decem 1711 justified the Passage in the Acta Eruditorum for Ianuary 1705 pag. 34 & 35, & thereby made it his own, & now endeavours in vain to excuse it, pretending that the words adhibet semperque adhibuit are malitiously interpreted by the Word substituit. But in the interpretation which he would put upon the place he omitts the words igitur & quemadmodum, the first of which makes the words semperque adhibuit a consequence of what went before & the latter makes them equipollent to substituit neither of which can be true in the sense which Mr Leibnitz endeavours now to put upon the words.

In his Letter of 4 March st. n. 1711 he pressed the R. Society to condemn Dr Keill without hearing both parties, & when the DrDoctor put in his Answer, Mr Leibnitz refused to give his reasons against the DrDoctor & called it injustice to expect it from him, & thereby put the R. Society upon a necessity yet persisted in pressing them against him & thereby put them upon a necessity of appointing a Committee to search out old papers & give their opinion upon them. If they did it without him it was his own fault: he was for over-ruling them, and called it injustice to expect that he should defend his candor & plead before them. If they gave him no opportunity to except against any of the Committee it was because he refused to be heard & they had sufficient authority to appoint a Committee without him, & he had no right to except against what they did for their own satisfaction. If they have not yet given judgment against him; it is because the Committee did not act as a Iury, nor the R. Society as a formal Court of justice. The Committe examined old Letters & Papers & gave their opinion upon them alone, & left room for Mr Leibnitz to produce further evidence for himself. And it is sufficient that the Society ordered their Report with the Papers upon which it was grownded to be published, & that Mr Leibnitz in all the three years & four months wchwhich are since elapsed has not been able to produce any further proof against Dr Keill then what was then before them.

Mr Leibnitz saith that the Letter wchwhich I call defamatory being no sharper than that wchwhich has been published against him I have no reason to complain. But the sharpness of the Letter lies in accusations & reflexions without any proof wchwhich way of writing is unlawful & infamous: the sharpnes of the Commercium lies in facts wchwhich are lawfull to be produced & fit to be produced. The Letter was published in a clandestine back-biting manner (as defamatory papers use to be) without the name of the author or Mathematician or Printer or City where it was printed, & was dispersed above two years before we were told that the Mathematician was Iohn Bernoulli, the Commercium was printed openly at London by order of the R. Society.

The Mathematician to whom Mr Leibnitz appealed from the R. Society, I called a Mathematician or pretended Mathematician, not to disparage the skill of Mr Bernoulli, but because the Mathematician in his Letter of yethe 7th of Iune 1673 cited Mr Bernoulli as a person distinct from himself, & Mr Leibnitz lately caused that Letter to be reprinted without the citation & tells us that the Mathematician was Mr Bernoulli himself, & whether the Mathematician was or Mr Leibnitz is to be believed I do not know.

He complains that the Committee have gone out of the way in falling upon the method of Series: but he should consider that both methods are but two branches of one general method. I joyned them together in my Analysis. I interwove them in the Tract which I wrote in the year 1671 as I said in my Letters of the 10 Decem 1672 & 24 Octob 1676. In my Letter of 13 Iune 1676, I said that my method of series extended to almost all Problems but became not general without some other other methods, meaning (as I said in my next Letter) the method of fluxions, and the method of arbitrary series, & now to take those other methods from me is to restrain and stint the method of series & make it cease to be general. In my Letter of 24 Octob. 1676 I called all these methods together my general method. See the Commercium Epistolicum pag. 86. lin. 16. And if Mr Leibnitz has been tearing this general method in pieces & taking from me first one part & then another part whereby the rest is maimed, he has given a just occasion to the Committee to consider the whole. It is also to be observed that Mr Lei he is perpetually giving testimony for himself, & its allowed in all Courts of justice to speak to the credit of the witness.

He represents that the Committee of the R. Society have omitted things wchwhich made against me & printed every thing wchwhich could be turned against him by strained glosses, & to make this appear he produced in his last Letter but one an instance of somy ignorance omitted by them but confesses now that he was mistaken in saying that it was omitted & saith that he will cite another instance. In one of my Letters to Mr Collins (he means that of 1672 not yet printed) he saith that I owned that I could not find the second segments of Sphæroids & that the Committee have omitted this. If they had omitted such a passage I think they would have done right it being nothing to the purpose. But on the contrary in that Letter I gave the dimensions of the second segments of the Spheroid in a series & only said that the series was not fit for the gauging of vessels. And such a series I set down also in my Letter of 13 Iune 1676 wchwhich was published in the Commercium Epist. p. 55 And Mr Collins in a Letter to Mr Iames Gregory 24 Decem. 1670 & in another to Mr Bertet 21 Feb 1671 both printed in the Commercium Epistolicum p. 24, 26 wrote that my method extended to second segments of round solids. And Mr Oldenburg wrote the same thing to Mr Leibnitz himself 8 Decem. 1674. See the Commer. Epist. p. 39. So you see that Mr Leibnitz hath accused the Committee of the R. Society without knowing the truth of his accusation & therefore is guilty of calumny clamour a misdemeanour. The Committee were so far from acting corruptly against Mr Leibnitz that omitted his ignorance of Geometry in those days & several other things wchwhich made strongly against him, such as were the two Letters in my custody, & the Paragraph in the Preface to the two first Volumes of Dr Wallis's works relating to this matter & that a copy of Gregories Letter of 5 Septem. 1670 was sent to Mr Leibnitz in Iune 1676 amongst the extracts of Gregories Letters.

Mr Leibnitz acknowledges that when he was in London the second time he saw some of my Letters in the hands of Mr Collins, & he has named two of those wchwhich he then saw, vizt thosate dated 20 Aug. 1672 & 24 Octob 1676, & that in which he pretends that I confessed my ignorance of second segments & no doubt he would principally desire to see the Letter which conteined the chief of my series & particularly those two for finding the Arc by the signe & the sine by the Arc with the Demonstration thereof wchwhich a few months before he had desired Mr Oldenburgh to procure from Mr Collins, that is, the Analysis per Æquationes numero terminorum infinitas. But yet he tells us that he never saw where I explained my method of fluxions & that he finds nothing of it in the Commercium Epistolicum where that Analysis & my Letters of 10 Decem. 1672, 13 Iune 1676 & 24 Octob. 1676 are published.

He saith also that he never saw where I explain the method claimed by me in which he assumes an arbitrary series: If he pleases to look into the Commercium Epistolicum pag. 56 & 86 he will there see that I had that Method in the year 1676 & five years before. Mr Leibnitz might find it himself but not so early, & second Inventors have no right.

He pretends that in my book of Principles pag. 253, 254, I allowed him the invention of the calculus differentialis independently of my of my own, & that to attribute this invention to my self is contrary to my knowledge there avowed. But in the Paragraph there referred unto, I do not find one word to this purpose. On the contrary I there represent that I sent notice of my method to Mr Leibnitz before he sent notice448 notice of his method to me, & left it him to make it appear that he had found his method before the date of my Letter, that is, eight months at the least before the date of his own. And by referring to the Letters which passed between Mr Leibnitz & me ten years before, I left the Reader to consult those Letters & interpret the Paragraph thereby. For by those Letters he would see that I wrote a Tract of that method & the method of Series together five years before the writing of those Letters or in the year 167 that oris in the year 1671. And these hints were as much as was proper in that short Paragraph, it being besides the designe of that Book to enter into disputes about these matters.

He saith that when he was in London the first time wchwhich was in Ian & Feb 1673 he knew nothing of the higher Geomet infinite series nor of the avanced Geometry nor was then acquainted with the higher Geometry Mr Collins as some have maliciously affirmed feigned. But who hath feigned this or what need there was of feigning it I do not know. At that time Dr Pell gave him notice of Mercators series for the Hyperbola & he carried Mercators Book with him to Paris tho he did not yet understand the higher Geometry. And any of those to whom Mr Collins had communicated mine & Gregories Series might give him notice of them without his being acquainted with Mr Collins.

He saith that after his coming from London to Paris his first Letters were of other matters then Geometrical till Mr Hugygens had instructed him in these things & that he found the Arithmetical Quadrature of the Circle towards the end of the year 1673 & began to write of it to Mr Oldenburg the next year, & found the general method by arbitrary series a little after & the differential Calculus in the year 1676 deducing it from the series of numbers, & that in his Letter of 27 Aug. 1676 by the words certa Analysi he meant the differential Analysis. And am not I as good a Witness that I found the methods of fluxions in the year 1665 & improved it in the year 1666 & that before the end of the year 1666 I wrote a small Tract on this subject which was the grownd of that larger Tract wchwhich I wrote in the year 1671 (both which are still in my custody,) & that in this smaller Tract tho I generally put letters for fluxions as Dr Barrow in his Method of Tangents put Letters for differences, yet in giving a general Rule for finding the Curvature of Curves I put the letter x with one prick for first fluxions drawn into their fluents & with two pricks for second fluxions drawn into the square of their fluents, & that when I wrote the larger of those two Tracts I had made my Analysis composed of those two methods so universal as to reach to almost all sorts of Problemes as I mentioned in my Letter of 13 Iune 1676.

In the year 1684 Mr Leibnitz published only the Elements of the Calculus differentialis & applied them to questions about tangents Tangents & Maxima & minima, but proceeded as Fermat Gregory & Barrow had done before, & shewed how to proceed in these Questions without taking away surds, but proceeded not to the higher Problemes. The Principia mathematica gave the first instances made publick of applying this Calculus to the higher Problemes & I understood Mr Leibnitz in this sense in what I said concerning the Acta Eruditorum for May 1700 pag. 206. But Mr Leibnitz observes that what was there said by him relates only to a particular artifice de maximis et minimis with which he there allowed that I was acquainted when I gave the figure of my vessel in my Principles. But this Artifice depending npon the differential method as an improvement thereof, & being the artifice by which they solved the Problemes which they value themselves most upon (those of the linea celerrimi descensus & the linea Catenaria & Velaria) & which Mr Leibnitz there calls a method of the highest moment & greatest extent, I content my self with his acknowledgement that I was the first that proved by a specimen made publick, that I had this artifice.

In the year 1689 Mr Leibnitz published the principal Propositions of this Book as his own in three papers called Epistola de Lineis Opticis, Schediasma de de resistentia Medij & motu projectilium gravium in Medio resistente, et Tentamentamen de motuum cœlestium causis, pretending that he had found them all before that book came abroad. And to make the principal Proposition his own adapted to it an erroneusous Demonstration. And this was the second specimen made publick of applying the method to the higher Problemes. Hitherto this method made no noise, but within a year or two it began to be celebrated.

Dr Barrow printed his differential method of Tangents in the year 1670. Mr Gregory from this method compared with his own deduced a general method of Tangents without calculation & by his Letter of 5 Sept. 1670 gave notice thereof to Mr Collins. Slusius in November 1672 gave notice of the like method to Mr Oldenburgh. In my Letter of 10 Decem. 1672 I sent the like method to Mr Oldenburgh Collins & added that I found mentioned it to Dr Barrow when he was printing his Lectures, & that I took the method of Gregory & Slusius to be the same with mine, & that it was but a branch or Corollary of a general method wchwhich without any troublesome calculation extended not only to tangents but also to other abstruser sorts of Problemes concerning the crookedness, areas, lengths, centers of gravity of Curves &c & did all this even without freeing equations from surds; & I added that I had interwoven this method with that of infinite series, meaning in the Tract wchwhich I wrote in the year 1671. Copies of these two Letters were sent to Mr Leibnitz by Mr Oldenburge in the Extracts of Gregories Letters in Iune 1676, & Mr Leibnitz in his Letter 21 Iune 1677 sent nothing more back then what he had notice of by these two Letters, namely, Dr Barrows Differential Method of Tangents disguised by a new notation, & extended to the method of Tangents of Gregory & Slusius, & to Equations involving surds, & to Quadratures. But this is not the case between me and Dr Barrow. He saw my Tract of Analysis in the year 1699 & was pleased with it. And before his Lectures came abroad I had deduced the method of Tangents of Gregory & Slusius from my general method. But Mr Leibnitz in those days knew nothing of the higher Geometry nor was yet acquainted with the higher the vulgar Algebra.

In his Letter of 27 Aug. 1676, he wrote thus. Quod dicere videmini plerasque difficultates (exceptis problematibus Diophantæis) ad series infinitas reduci, id mihi non videtur. Sunt enim multa usque adeo mira et implexa ut neque ab æquationibus pendeant neque ex Quadraturis. Qualia sunt ex multis alijs Problemata methodi tangentium inversæ. And when I answered that such Problems were in my power he replied (in his Letter of 21 Iune 1677) that he conceived that I meant by infinite series but he meant by vulgar equations. See the Answer to this in the Commercium Epistolicum p. 92.

He saith that one may judge that when he wrote his Letter of 27 Aug. 1676, he had some entrance into the differential Calculus because be he said there that he had solved the Probleme of Beaune certa Analysi by a certain Analysis. But what if that Probleme may be solved certa Analysi without the Differential method? For no further Analysis is requisite then this; That the Ordinate of the Curve desired increases or decreases in Geometrical Progression when the Abscissa increases in Arithmetical, & therefore the Abscissa & Ordinate have the same relation to one another as the Logarithm & its Number. And to infer from this that Mr Leibnitz had entrance into the differential method is as if one should say that Archimedes had entrance into it, because he drew tangents to the Spiral, Squared the Parabola & found the proportion between the sphere and the cylynder, or that Cavallerius Fermat & Wallis had entrance into it because they did many more things of this kind.

Mr Leibnitz by telling his own story in his Letter of 9 April 1716 put me upon doing the like in my my Remarks upon it. And since in his Letters to the Comtesse of Kilmansegger & Baron the Comte de Bothmaer he has told it at large, I will tell the rest of my story & & leave you to beleive what you please.

Mr Leibnitz was in London in the beginning of the year 1673, & going from thence to Paris in February, corresponded with Mr Oldenburg by Letters about Arithmetical matters till Iune following, being not yet acquainted with the higher Geometry. In April 1673 the Horologium oscillatorium of Mr Huygens came abroad, & this was the first book wchwhich he studied in learning the higher Geometry, Mr Huygens introducing him. In the year 1674 Mr Leibnitz after a years intermission renewed his correspondence with Mr Oldenburg & began to write to him about series for finding the Area or Circumference of a circles or any Arc whose sine was given. And in the year 1675 Mr Oldenburg sent from Mr Collins to Mr Leibnitz several of mine & Gregories series for the same purpose; & Gregory dying in the end of the year, Mr Collins at the request of Mr Leibnitz collected the Letters wchwhich he had receivinged from Gregory, & Mr Oldenburg in Iune 1676 sent the Collection to Paris to be perused & returned, & it is now in the Archives of the R. Society In this Collection wasere a copy of Mr Gregories Letter of 5 Sept. 1670 & & a Copy of my Letter of 10 Decem 1672 to Mr Collins: and by these Letters Mr Leibnitz had notice that Mr Barrow's method of Tangents was capable of improvement so as to give tmy general Analysis mentioned in my said Letter & that this Analysis proceeded without sticking at surds, & that I had written interwoven it with the Method of Series, vizt in my Analysis per series abovementioned & in another Tract wchwhich I wrote upon itthem in the year 1671. Mr Leibnitz wrote also to Mr Oldenburg for the demonstration of some of my series, that is for the method of finding them, & promised him a Reward, & told him that Mr Collins could help him to it; & therefore he had heard that Mr Collins had my method of series, that is, my Analysis per series above mentioned. For I had sent my method of series to Mr Collins in no other Paper then that. But Mr Collins instead of sending a copy of that Tract joyned with Mr Oldenburg in solliciting me to write an Answer to Mr Leibnitz's Letter. And thereupon I wrote my Letter of 13 Iune 1676, wchwhich was sent to Mr Leibnitz at the same time with the aforesaid Collection. And Mr Leibnitz in his Answer dated 27 Aug. 1676, replied that he did not beleive that my Analysis was so general as I represented, there being many Problemes, & particulary the inverse Problemes of Tangents, not reducible to equations or quadratures. And in the same Letter he placed the perfection of Analysis not in the differential calculus as he did after he found it, but in another method founded on Analytical Tables of Tangents & the Combinatory Art. Nihil est, saith he, quod norim in tota Analysi momenti majoris. And a little after: Ea verò non differt ab Analysi illa suprema SVPREMA ad cujus intima Cartesius non pervenit. Est enim ad eam constituendam opus Alphabeto cogitationum humanarum. Mr Leibnitz never pretended to have found the differential calculus Analysis before this end of this year, & these circumstances satisfy me that he(2458 he did not find it till after the writing of this Letter.

In October following he came to London & there met wthwith Dr Barrows Lectures & saw Mry Newton's Letter of 24 Octob. 1676 & therein had fresh notice of the said method & of Compendium of series sent by Dr Barrow to Mr Collins in the year 1669 under the title of Analysis per series &c & consulting Mr Collins saw in his hadnds several of mine & Gregories Letters, especially those relating to series; & in his way home from London was meditating how to improve the method of Tangents of Slusius as appears by his Letter to Mr Oldenburgh dated from Amsterdam $\frac{18}{28}$ Novem 1676. And the next year upon his arrival at Hannover he a Copy of my Letter of 24 Octob. 1676 was sent after him, & in a letter to Mr Oldenburg dated 21 Iune 1677 he sent us his new Method with this Introduction. Clarissimi Slusij methodum tangentium nondum esse absolutam celeberrimo Newtono assentior. And in describing this method he abbreviated Dr Barrow's method of Tangents, & shewed how it might be improved so as to give the Method of Slusius & to proceed in æquations involving surds, & then subjoyned: Arbitror quæ celare voluit Newtonus de tangentibus ducendis ab his non abludere: Quod addit, ex eodem fundamento quadraturas reddi faciliores me in sententia hac confirmat. And after seven years vizt in October 1684 he published the elements of this method of as his own without mentioning the correspondence which he had formerly had with the English about these matters. He mentioned indeed a Methodus similis, but whose that method was & what he knew of it, he did not say, as he should have done. And this [his silence put me upon a necessity of writing the Scholium upon the second Lemma of the second Book of Principles least it should be thought that I borrowed that Lemma from Mr Leibnitz. In my Letter of 24 Octob. 1676 when I had been speaking of the method of Fluxions I added: Fundamentum harum operationum, satis obvium quidem, quoniam non possum explicationem ejus prosequi sic potius celavi 6accdæ13eff7i3l9n404qrr4s9t12vx. And in the said Scholium I opened this ænigma, saying that it conteined the sentence Data æquatione quotcunque fluentes quantitates involvente, fluxiones invenire, et vice versa; & was written written in the year 1676. For I looked upon this as a sufficient security without entring into a wrangle: but Mr Leibnitz was of another opinion.]