The occasion of publishing thise Let this Collection of Letters will be understood by the Letters of Mr Leibnits & Mr Keil published in the end thereof. Mr Leibnitz taking offence at a passage published by Mr Keil in a discours of Mr Keil published in the transactions about years ago, wrote a Letter to the Secretary of the R. Society complaining thereof as a calumny, & the Society would desiring a remedy therefof from the R. Society, namely that they would do justice in this matter, which he beleived & suggesting that he beleived they would judge it equal that he Mr Keil should make a publick acknowledgmtacknowledgment of his fault. Mr Keil chose chose rather to returnedreturn an answer in writing, wherein he explained his meaning in those words that passage & justified that meaning. Mr Leibnitz not meeting wthwith that satisfaction he desired wrote a second Letter to the R. Soci Secretary of the R. S. wherein he complainged still of Mr Keill, representing him a young man not acquainted wthwith what was done before his time nor authorized by the person concerned; And appealinedg to the equity of the Society to checque the unjust clamours of Mr Keil this person. And And uVpon this double repeated appeale M the R. Society used the appointed a Committee to examin the ancient Letters concer & w & Records relating to this matter, & upon the Report of the Committee ordered them to be published with the Report of the Committee. ordered the Report of the Committee to be published wthwith so much of the Letters & papers as related to this matter.

The question is about the invention of the method called by Mr Leibnitz the Differential method by SrSir Isaac Mr Newton yethe method of fluxions. Mr Leibnitz is contends that he found it by himself wthwithout the sassistance of any other person, & in his last letter allows the like to SrSir Isaac Newton, but yet claims the title of Inventor & justifies a Paper printed in the Acta Leipsica whereinby he makes himself the first inventor. Whether the se The first inventor is the inventor & whether the second Inventor foutnd it by himself or not is a question of no consequ moment. He blames Mr Keill for medling with this matter without authority from SrSir I. Newton [& desires that SrSir Isaac would give his opinion — whereas every d man hath a right to repell injuries from his neighbour &] but the R. Society have not found fault wthwith him on that account, it being every mans right to repell injuries from his neighbour, & a right very necessary to be preserved among learned men least this dilemma be put upon Inventors, that they must either lose their inventions to pretenders to noisy pretenders or writing books against noisy pretenders against them. to noisy sticking pretenders or spend their time in controversies & run the hazzard of being censured for valuing themselves upon things of that nature.

The method of fluxions being described in I in a letter of LMr Newton to Mr Collins A.C. 1672 concerning the method of Tangents ascribed to M Slusius, it was thought fit to print that letter in the following papers that Letter wthwith two or three others relating to it. And it may not be amiss to observe that what was said in those letters concerning that method of tangents as known to Mr Slusius before he printed his Mesolabium, was grownded upon a a mistake of Mr Oldenburg, for rectifying of wchwhich it may not be amiss to reprint here a Letter of Mr Collins to Mr Newton dated 18th Iune 1673 & printed by Dr Wallis in the third volume of his works. The Letter runs thus.

Quod ad Slusij methodum . . . . . . . . . . ab ipso propediem expectare debeant. And this is that method of Thangents wchwhich Slusius when he published his Mesolabium relinquished to forbore to write of least he should prevent his friend Riccius & wchwhich Riccius afterwards declining mathematical studies desired Slusius to publish, & Slusius thereupon promised to send to Mr Oldenberg to be published in the Transactions. & M Vpon this notice Mr Oldenburgh & Mr Collins contin became of opinion that the Method of Tangents wchwhich by the leave of M Sl Riccius they expected from Slusius was known to Slusius when he published his Mesolabium But Slusius sent them another method & they not very different from that wchwhich Riccius & Mr O & not derivable from his principles ButAnd Mr Oldenburgh & Mr Collins not being aware of understanding the difference applied to this new method what Slusius had written to Mr Oldenburgh concerning the method wchwhich Riccius had given him leave to publish & accordingly wrote to Mr Newton that it was known to Slusius before he published his Mesolabium. Now hHow Slusius came to send Mr Oldenberg another Method of tangents then that wchwhich Riccius gave him leave to publish, doth not appeare: but its observable that [Mathematicians have not yet been able to derive theis new method of Tangents from the three Lemmas upon wchwhich Slusius pretended to found it, founded it, saying that it flowed from those Lemmas very easily. Let Mathematicians consider whether it can be derived from those Le three Lemmas, or from what other principles Slusius might derive it.] Slusius founded it upon] when Slusius was to demonstrate this new method, he proposed three Lemmas saying representing that it flowed very easily from them without further explication, but Mathematicians have not yet been able to tells us how it flows from them. Let Mathematicians therefore consider whether this method of Tangensts can be derived from demonstrated by the said three Lemmas from wchwhich Slusius pretended to have derived it or whether Slusius had a better demonstration.

In Mr Newtons Letter of A 24 Octob 1676 he uses letters set out of order to conceale his method of fluxions. Galileo Hugenius & Hook upon some occasions used the same method, but Mr Newton seems to have used it at that time only to avoid explaining a method wchwhich he had touched upon & wchwhich was besides his buisineto explo But Neither doth it appearse that Slusius could not demonstrate the method wchwhich he sent. For in the end of his letter in wchwhich he described it, he said Addo tantum me Regulæ meæ Demonstrationem facilem habere facilem et quæ solis constaet Lemmatibus; quod mirum Tibi forte videbitur. And a little after he sent the following three Lemmas

Idem fecerat Slusium ip

Tractabāant etiam Slusius Barrovius, Gregorius & Fermatius rem tangentiusm per differentias Ordinatarum Et idem fecit Slusius ut ex tribus ejus Lemmatibus, in quibus hanc methodum suam fundavit, manifestum est. Nam per ejus Lemmata duo prima, difentiadifferentia dignitatum homologarum applicata ad differentiam laterum producit dignitate infinite parvam producit factum ex dignitate et indice suo applicatum ad latus. Per differentias laterum et dignitatum Slusius hic frgit exponit differentias abscissarum et ordinatarum. Vtrum Leibnitius

The Question is about a method of Analysis wchwhich Mr Leibnitz claims & Mr Keil attributes to SrSir Isaac Newton. Mr Leibnitz was in England in the beginning of the year 1673 & again in the latter end of the year 1676 & in all the intervall of time kept a correspondence wthwith Mr Iohn Collins by means of Mr Oldenburg but & some years after published in the Acta Leipsica some

1. Mr Leibnitz was in England in the beginning of yethe year 1673 & in the end of yethe year 1676 & in the intervall in france & all that time kept a correspondence wthwith Mr Col Iohn Collins by means of Mr Oldenberg & what he learnt of from the English in all that time is the question by that correspondence is the main Question, & this Ques Mr Oldenburg Mr Collins being long since dead for deciding it the R. Society found it necessary to search out & publish what could be met with of that correspondence relating to these about in writing relating to the things in Question.

2 By the Analysis published in the beginning the reader will find that Mr Newton was acquainted with the method of infinite series Anno 1669 & by the help of the method of fluent quantities & their incrementa momentanea or moments had had then applied that method very generally to the solution of Th Problemes before the communication of that meth Analysis of Dr Barrow to Mr Collins wchwhich was in the year 1669 And it appears not by any that Mr Leibnits was aquainted theat method of moments or differences as he calls ymthem before the year 1677. 2B And by the Letters next following ymthem Analysis That Mr Collins Gregory in ymthem end of the year 1670 fell into ymthem same method but did not claim inventoris jura, because he had notice from Mr Coll received one of Mr Newtons series from Mr Collings & with notice that Mr Collin Newton had a general method for finding such series at pleasure And That Mr Collins was thenceforward very free in communicating the series wchwhich he had received from Mr Newton & Mr Collings, & that Mr Leibnitz soon after his being in London began to talk of his having two such series both of them found by one & the same method, that both these series could not be found by the Method of transmutations, & that Mr Leibnits in those days pretended to no other method, & therefore had only the series wthwithout a method of finding them.

3 You will find also that in April or May 1675 Mr Leibnitz received from Mr Collins by the letters of Mr O. eight or nine series & knew none of them to be his own: that before ymthem end of the year Mr Gregory died & Mr Leibnitz had communicated the last of those serieds as his own to his friends at Paris without letting them know that he had received it from Mr Collins 4 That two others of those series being given sent by Mr Collins to Paris by one Mr Mohr a Dane, Mr Leibnitz in May 1676 desired of Mr O. to procure him the demonstration of those two series therefore & had not yet the method of finding them, [& that one of those two series was for finding the arch by the sine, & either (as was one of the two which he wrote of soon after his going th from London to Paris) & the other for finding the sine by the arch]. ☉ ☉ That he then represented his own meditations or inventions (of wchwhich had writ to Mr O. some years before) very different from these two d series, but never produced them: that he then admired these two series & especially the second for its elegance, &but knew not how to derive it from the first, tho For he wrote the next year to Mr Oldenburg that he had found in his old papers a method of Mr Newtons for such purposes, but but had neglected ifit for some time for want of an elegant instance of its use. It this The two series When he had series wchwhich would have given him elegant examples of this method he had forgot the method & when he had the method he wanted series to give him an elegant example of its use it.] The His own series wchwhich he had in yethe years 1674; 17675 & 1676 afforded him no elegant examples & therefore were different from a those communicated to him by Mr Oldenburgh & Mr Mohr. And these gave him no elegant examples tho they were all of them elegant examples of such a method & therefore he had forgot the Method before he received Mr Oldenburghs Letter of April 1675. An&d therefore the method is old enough to make Mr L. the first inventor. And hence it follows also that the series wchwhich he had before that time are not yet published: for those hitherto published by Mr Leibnitz give elegant examples. 6 That when Mr Newton at yethe request of Mr O & Mr C sent his method of fi series to be communicated t by Mr O. to Mr Leibnitz he sent back h the last of the above mentioned series as his own wchwhich he received of Mr had received of Mr O. & [communicated at Paris as his own] & endeavoured claimed also some other seiries as his own wchwhich by some sl small alterations he had derived from the series sent him by Mr Newton, & wchwhich he was not able to find out before he received the Mr Newtons method for wchwhich he had written which were of the same kind wthwith theose of wchwhich he wanted the Demonstration till he recived Mr & wchwhich therefore he could not be found then out by him before he received Mr Newtons Letter.

5 That in his answer to Mr Newtons Letter he sent his method of transmutations to Mr Newton as a general method of Series , wchwhich was not a general method before Mr Newton made it so by the methods wchwhich he sent him communicating his method of extracting roots, & that this method of series is tedious & useless, it being easier to find series by Mr Newtons methods alone.

That whe in his last letter to Mr Oldenburge dated 12 Iuly 1677 he represents that in his old papers he used a Mr Newtons method of transforming one series into another like that of Mr Newton, & yet but not meeting wthwith an elegant instance of its use rejected it.

$\frac{1}{\mathrm{x}+\mathrm{c}\mathrm{x}\mathrm{x}}=\mathrm{y}$. $\frac{1}{\mathrm{x}+\mathrm{c}\mathrm{x}\mathrm{x}+\mathrm{o}+2\mathrm{c}\mathrm{o}\mathrm{x}+\mathrm{c}\mathrm{o}\mathrm{o}}=\frac{1}{\mathrm{x}+\mathrm{c}\mathrm{x}\mathrm{x}}-\frac{\mathrm{o}+2\mathrm{c}\mathrm{o}\mathrm{x}+\mathrm{c}\mathrm{o}\mathrm{o}}{\mathrm{x}+\mathrm{c}\mathrm{x}\mathrm{x}\times \mathrm{x}+\mathrm{c}\mathrm{x}\mathrm{x}}+\frac{\mathrm{c}\mathrm{o}+4\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{x}+2\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{x}\mathrm{x}+2\mathrm{c}{\mathrm{o}}^3+4{\mathrm{c}}^2{\mathrm{o}}^3\mathrm{x}}{{\overline{)\mathrm{x}+\mathrm{c}\mathrm{x}\mathrm{x}}}^3}-\frac{{\mathrm{o}}^3+{}^6\mathrm{c}{\mathrm{o}}^3\mathrm{x}+12\mathrm{c}\mathrm{c}{\mathrm{o}}^3\mathrm{x}\mathrm{x}+8{\mathrm{c}}^3{\mathrm{o}}^3{\mathrm{x}}^3}{{\overline{)\mathrm{x}+\mathrm{c}\mathrm{x}\mathrm{x}}}^4}+\mathrm{\&c}$.