Several Drafts of an Intended Preface to the Commercium Epistolicum

① Ad lectorem of {Com Epist}

The occasion of publishing thise Let this Collection of Letters will be understood by the Letters of M^r Leibnits & M^r Keil published in the end thereof. M^r Leibnitz taking offence at a passage published by M^r Keil {illeg} in a discours of M^r 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 theref|o|f 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 \M^r Keil/ should make a publick acknowledgm^t of his fault. M^r Keil chose \chose rather to/ returned {sic} an answer in writing, wherein he explained his meaning in those words \that passage/ & justified that meaning. M^r Leibnitz not meeting w^th that satisfaction he desired wrote a second Letter to the R. Soci Secretary of the R. S. \wherein he/ complaing|ed| still of M^r Keill, {illeg} representing \him/ a young man not acquainted w^th what was done before his time nor authorized by the person concerned; And appealin|ed|g \to/ the equity of the Society to chec the unjust clamours of M^r Keil \this person./ And \And/ u|V|pon this double \repeated/ app{illeg}|e|ale M the R. Society {illeg}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 w^th so much of the Letters & papers as related to this matter./

The question is about the {illeg} invention of the method called by M^r Leibnitz the Differential method by S^r Isaac M^r Newton y^e method of fluxions. M^r Leibnitz {illeg}{is} contends that he found it \by himself/ w^thout the s{illeg}|a|ssistance of any other person, {illeg} & in his last letter allows the like to S^r Isaac Newton, but \yet claims the title of Inventor &/ justifies a Paper printed in the Acta Leipsica wherein|by| he makes himself the first inventor. Whether the se The first inventor is the inventor & whether the second Inventor fout|n|d it by himself or not is a question of no consequ moment. He blames M^r Keill for medling with this matter without authority from S^r I. Newton [& desires that S^r Isaac would give his opinion — whereas every {illeg}d man hath a right to repell injuries from his neighbour &] but the R. Society have not found fault w^th him on that account, it being every mans right to repell injuries from his neighb{illeg}|o|ur, & 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 \{stick}ing/ 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 {L}|M|^r Newton to M^r 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 w^th 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 {illeg} as known to M^r Slusius before he printed his Mesolabium, was grownded upon <539v> a mistake of M^r Oldenburg, for rectifying of w^ch it may not be amiss to reprint here a Letter of M^r Collins to M^r Newton dated 18^th Iune 1673 & printed by D^r Wallis in the third volume of his works. The Letter runs thus.

Quod ad Slusij methodum . . . . . . . . . . ab ipso propediem expectar{e} debeant. And this is that method of Th|a|ngents w^ch Slusius when he published his Mesolabium relinquished to \forbore to write of least he should prevent/ his friend Riccius & w^ch Riccius afterwards declining mathematical studies desired Slusius to publish, & Slusius thereupon promised to send to M^r Oldenberg to be published in the Transactions. & {illeg} M Vpon this notice M^r Oldenburgh & M^r Collins contin{illeg} became of opinion that the Method of Tangents w^ch 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 w^ch Riccius & M^r O & not derivable from his principles But|And| M^r Oldenburgh & M^r Collins not being aware of \understanding/ the difference applied to this new method what Slusius had written to M^r Oldenburgh concerning the method w^ch Riccius had given him leave to publish & accordingly{illeg} wrote to M^r Newton that it was known to Slusius before he published his Mesolabium. Now h|H|ow Slusius came to send M^r Oldenberg another Method of tangents then that w^ch Riccius gave him leave to publish, doth not appeare: but its observable that [Mathematicians {illeg} have not yet been able to derive the|i|s new method of Tangents from the three Lemmas upon w^ch 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 Tangens|t|s can be derived from \demonstrated by/ the said three Lemmas from w^ch Slusius pretended to have derived it or whether Slusius had a better demonstration.

In M^r 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 M^r Newton seems to have used it at that time only to avoid explaining a method w^ch he had touched upon & w^ch was besides his buisineto explo But \Neither doth/ it appears|e| that Slusius could not demonstrate the method w^ch he sent. For in the end of his letter in w^ch he described it, he said Addo tantum me Regulæ meæ Demonstrationem facilem habere facilem et quæ solis consta|e|t Lemmatibus; quod mirum Tibi forte videbitur. And a little after he sent the following three Lemmas

Idem fecerat Slusium ip

Tractabāt etiam \{illeg}/ Slusius \Barrovius, {illeg}|G|regorius & Fermatius/ {illeg}|re|m tangentius|m| 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, \difentia {sic}/ 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 {illeg}{f}{illeg}rgit \exponit/ differentias {illeg} abscissarum et ordinatarum. Vtrum Leibnitius

The Question is about a method of Analysis w^ch M^r Leibnitz claims & M^r Keil attributes to S^r Isaac Newton. M^r 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 w^th M^r Iohn Collins by means of M^r Oldenburg {illeg} but & some years after published in the Acta Leipsica some

1. M^r Leibnitz was in England in the beginning of y^e year 1673 & in the end of y^e year 1676 & in the intervall \in france & all that time/ {illeg} kept a correspondence w^th M^r Col Iohn Collins by means of M^r Oldenberg & what he learnt of \from/ the English in all that time is the question by that correspondence is the \main/ Question, & this Ques |M^r Oldenburg M^r 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 {illeg} about \in writing relating to/ the things in Question.

2 By the Analysis published in the beginning the reader will find that M^r 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 D^r Barrow to M^r Collins w^ch was in the year 1669 {illeg} \And it appears not \by any/ that M^r Leibnits was aquainted the|a|t method of moments \or differences as he calls y^m/ before the year 1677. 2B And by the Letters next following y^m Analysis/ That M^r Collins \Gregory/ \in y^m end of the year 1670/ fell into y^m same method but did not claim inventoris jura, because he had notice from M^r Coll received one of M^r Newtons series from M^r Colling|s| & with notice that M^r Collin Newton had a general method for finding such series at pleasure And That M^r Collins was thenceforward very free in communicating the series w^ch he had received from M^r Newton & M^r Colling|s|, & that M^r Leibnitz \soon after his being in London/ began to ta{illeg}|lk| 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 M^r Leibnits in those days pretended to no other method, & therefore had only the series w^thout a method of finding them.

3 You will find also that in April or \May/ 1675 M^r Leibnitz received from M^r Collins \by the {illeg}|letters| of M^r O./ eight or nine series & knew none of them to be his own: that before y^m end of the year M^r Gregory died & M^r Leibnitz had communicated the last of those seried|s| as his own to his friends at Paris without letting them know that he had received it from M^r Collins \4/ That two others of those series being given \sent/ by M^r Collins to \Paris by/ one M^r Mohr a Dane, {illeg}|M|^r Leibnitz in May {illeg} \1676/ desired of M^r 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/]. \☉/ < insertion from the bottom of the page > ☉ That he then represented his own meditations or inventions (of w^ch had writ \to M^r 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, &|b|ut knew not how to derive <540v> it from the first, tho \For/ he wrote the next year to M^r Oldenburg that he had \found/ \in his old papers/ a method of M^r Newtons for such purposes, but but had neglected if {sic} for some time for want of an elegant instance of its use. It th{is} The two series {illeg} When he had series w^ch would have{illeg} 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 w^ch he had in y^e years 1674; 17|6|75 & 167{illeg}|6| afforded him no elegant examples & therefore were different from a{illeg} those communicated to him by M^r Oldenburgh & M^r 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 M^r Oldenburghs Letter of {illeg} April 1675. |An|&|d| \therefore the method is old enough to make M^r L. the first inventor. |And| hence it follows also that/ the series w^ch he had before that time are not yet published: for those hitherto published by M^r Leibnitz give elegant examples. < text from f 540r resumes > 6 That when M^r Newton at y^e request of M^r O & M^r C sent his method of fi series to be communicated t by M^r O. to M^r Leibnitz {illeg}|h|e sent back h the last of the \above mentioned/ series \{illeg}|as| his own/ w^ch he received of M^r \had received of M^r O. &/ [communicated at Paris as his own] & endeavoured claimed also some other seiries as his own w^ch by some {s}l small alterations he had derived from the series sent him by M^r Newton, & \w^ch he/ was not able to find out before he received the M^r Newtons method for w^ch he had written which were of the same kind w^th the|o|se of w^ch he wanted the Demonstration till he rec{iv}ed M^r & \w^ch/ therefore \he/ could not \be/ fo\u/nd \then/ out \by him/ before he received M^r Newtons Letter.

5 That in his answer to M^r Newtons Letter he sent his method of transmutations to M^r Newton as a general method of Series {illeg}, w^ch was not a general method before M^r Newton made it so by the methods w^ch he sent him \communicating/ his method of extracting roots, & that this method \of series/ is tedious & useless, it being easier to find series by M^r Newtons methods alone.

That whe in his last letter to M^r Oldenburge dated 12 Iuly 1677 he represents that in his old papers he used a \M^r Newtons/ method of transforming one series into another like that of M^r Newton, & yet {illeg} \but not/ meeting w^th an elegant instance of its use rejected it.

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

$\frac{1}{\mathrm{a}+\mathrm{b}}=\frac{1}{\mathrm{a}}-\frac{\mathrm{b}}{\mathrm{a}\mathrm{a}}+\frac{\mathrm{b}\mathrm{b}}{{\mathrm{a}}^3}-\frac{{\mathrm{b}}^3}{{\mathrm{a}}^4}$

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

Are $\mathrm{Area}=\frac{1}{\mathrm{x}+\mathrm{c}\mathrm{x}\mathrm{x}}\mathrm{o}-\frac{1+2\mathrm{c}\mathrm{x}}{2\times {\overline{)\mathrm{x}+\mathrm{c}\mathrm{x}\mathrm{x}}}^2}{\mathrm{o}}^2+\frac{\mathrm{c}+3\mathrm{c}\mathrm{x}+3\mathrm{c}\mathrm{c}\mathrm{x}\mathrm{x}}{3\times {\overline{)\mathrm{x}+\mathrm{c}\mathrm{x}\mathrm{x}}}^3}{\mathrm{o}}^3-\mathrm{\&c}$

$\mathrm{e}\mathrm{S}=\mathrm{R}\sqrt{1+\mathrm{Q}\mathrm{Q}}$

$\frac{\sqrt{1+\mathrm{x}\mathrm{x}+2\mathrm{c}{\mathrm{x}}^3+\mathrm{c}\mathrm{c}{\mathrm{x}}^4}}{{\overline{)\mathrm{x}+\mathrm{c}\mathrm{x}\mathrm{x}}}^2}$ in $\frac{1+2\mathrm{c}\mathrm{x}}{2\times {\overline{)\mathrm{x}+\mathrm{c}\mathrm{x}\mathrm{x}}}^2}=\frac{\mathrm{e}\mathrm{c}+3\mathrm{e}\mathrm{c}\mathrm{x}+3\mathrm{e}\mathrm{c}\mathrm{c}\mathrm{x}\mathrm{x}}{3\times {\overline{)\mathrm{x}+\mathrm{c}\mathrm{x}\mathrm{x}}}^3}$

$\mathrm{m}\mathrm{n}+3\times 2{\mathrm{m}}^2{\mathrm{n}}^2+6\mathrm{m}\mathrm{n}+42{\mathrm{m}}^3{\mathrm{n}}^3+\underset{6}{6}\mathrm{m}\mathrm{m}\mathrm{n}\mathrm{n}+\underset{18}{4}\mathrm{m}\mathrm{n}\underset{+12}{\phantom{000}}$

$\frac{{\mathrm{x}}^{\mathrm{m}}}{{\overline{)\mathrm{x}1+\mathrm{x}}}^{\mathrm{n}}}=$

${\overline{){\mathrm{x}}^{\mathrm{m}}+\mathrm{c}{\mathrm{x}}^{\mathrm{m}+1}}}^{\mathrm{n}}={\mathrm{x}}^{\mathrm{m}\mathrm{n}}+\mathrm{n}\mathrm{c}{\mathrm{x}}^{\mathrm{m}\mathrm{n}+1}+\frac{\mathrm{m}\mathrm{n}-\mathrm{n}}{2}\times {\mathrm{c}}^2{\mathrm{x}}^{\mathrm{m}\mathrm{n}+2}+\frac{{\mathrm{n}}^3-3\mathrm{n}\mathrm{n}+2\mathrm{n}}{6}{\mathrm{c}}^3{\mathrm{x}}^{\mathrm{m}\mathrm{n}+3}$.

$\mathrm{Ord}=\frac{1}{\mathrm{m}\mathrm{n}+1}{\mathrm{x}}^{\mathrm{m}\mathrm{n}+1}+\frac{\mathrm{n}\mathrm{c}}{\mathrm{m}\mathrm{m}\mathrm{n}\mathrm{n}+3\mathrm{m}\mathrm{n}+2}{\mathrm{x}}^{\mathrm{m}\mathrm{n}+2}+\frac{\mathrm{m}\mathrm{n}\mathrm{n}{\mathrm{c}}^3-\mathrm{n}\mathrm{n}{\mathrm{c}}^3}{2{\mathrm{m}}^3{\mathrm{n}}^3+12\mathrm{m}\mathrm{m}\mathrm{n}\mathrm{n}+22\mathrm{m}\mathrm{n}+12}{\mathrm{x}}^{\mathrm{m}\mathrm{n}+3}$

$\sqrt{1}+\frac{1}{\mathrm{m}\mathrm{m}\mathrm{n}\mathrm{n}+2\mathrm{m}\mathrm{n}+1}=\frac{\sqrt{\mathrm{m}\mathrm{m}\mathrm{n}\mathrm{n}+2\mathrm{m}\mathrm{n}+1}\text{ in }{\mathrm{x}}^{2\mathrm{m}\mathrm{n}+2}+\mathrm{e}\mathrm{e}}{\mathrm{m}\mathrm{n}+1}\times 2\frac{\mathrm{m}\mathrm{n}\mathrm{n}{\mathrm{c}}^3-\mathrm{n}\mathrm{n}{\mathrm{c}}^3}{\mathrm{m}\mathrm{n}+2\times \mathrm{m}\mathrm{n}+2\times \mathrm{m}\mathrm{n}+3}=\frac{\frac{4}{3}\mathrm{n}\mathrm{n}\mathrm{c}\mathrm{c}}{\mathrm{m}\mathrm{n}+1\times \mathrm{n}\mathrm{n}+1}$

Cum Resistentia sit ad gravitatem ut $3\mathrm{S}\sqrt{1+\mathrm{Q}\mathrm{Q}}$ ad 4RR: Sit hac resistentia ut Medij densitas et Velocitatis V potestas ${\mathrm{V}}^{\mathrm{n}}$: & Medij densitas erit ut resistentia directe & velocitatis potestas ${\mathrm{V}}^{\mathrm{n}}$ inverse, id est ut $\frac{3\mathrm{S}\sqrt{1+\mathrm{Q}\mathrm{Q}}}{4\mathrm{R}\mathrm{R}{\mathrm{V}}^{\mathrm{n}}}$ pro V scribatur $\sqrt{\frac{1+\mathrm{Q}\mathrm{Q}}{\mathrm{R}}}$, et Densitas erit ut $\frac{3\mathrm{S}\sqrt{1+\mathrm{Q}\mathrm{Q}}}{4\mathrm{R}\mathrm{R}\times {\frac{1+\mathrm{Q}\mathrm{Q}}{\mathrm{R}}}^{\frac{\mathrm{n}}{2}}}=\frac{\mathrm{S}{\mathrm{R}}^{\frac{\mathrm{n}-4}{2}}}{{1+\mathrm{Q}\mathrm{Q}}^{\frac{\mathrm{n}-1}{2}}}=\frac{\mathrm{S}{\mathrm{R}}^{\frac{\mathrm{n}-4}{2}}}{{1+\mathrm{Q}\mathrm{Q}}^{\frac{\mathrm{n}-1}{2}}}$. Sit $\mathrm{n}=1$, et Densitas erit ut $\frac{\mathrm{S}}{{\mathrm{R}}^{\frac{3}{2}}}$. Sit Densitas ut $\mathrm{n}=3$ et Densitas erit ut $\frac{\mathrm{S}}{\overline{1+\mathrm{Q}\mathrm{Q}}\text{ in }{\mathrm{R}}^{\frac{1}{2}}}$. Sit Resiste $\mathrm{n}=5$ et Densitas erit ut $\frac{\mathrm{S}{\mathrm{R}}^{\frac{1}{2}}}{{\overline{)1+\mathrm{Q}\mathrm{Q}}}^2}$ Sit $\mathrm{n}=4$ et densitas Medij erit ut $\frac{\mathrm{S}}{{\overline{)1+\mathrm{Q}\mathrm{Q}}}^{\frac{3}{2}}}$. Sit $\mathrm{n}=2$ et Dens. erit ut $\frac{\mathrm{S}}{\mathrm{R}\sqrt{1+\mathrm{Q}\mathrm{Q}}}$. Sit $\mathrm{n}=0,1,2,3,4,5,6$ \7/ et Densitas Medij erit ut $\frac{\mathrm{S}\sqrt{1+\mathrm{Q}\mathrm{Q}}}{\mathrm{R}\mathrm{R}},\frac{\mathrm{S}}{{\mathrm{R}}^{\frac{3}{2}}},\frac{\mathrm{S}}{\mathrm{R}\sqrt{1+\mathrm{Q}\mathrm{Q}}},\frac{\mathrm{S}}{{\mathrm{R}}^{\frac{1}{2}}\times \overline{1+\mathrm{Q}\mathrm{Q}}}$ $\frac{\mathrm{S}}{{\mathrm{R}}^{\frac{1}{2}},{\overline{)1+\mathrm{Q}\mathrm{Q}}}^1},\frac{\mathrm{S}}{{\mathrm{R}}^0\times {\overline{)1+\mathrm{Q}\mathrm{Q}}}^{\frac{3}{2}}},\frac{\mathrm{S}}{{\mathrm{R}}^{-\frac{1}{2}}\times {\overline{)1+\mathrm{Q}\mathrm{Q}}}^2},\frac{\mathrm{S}}{{\mathrm{S}}^{-1}\times {\overline{)1+\mathrm{Q}\mathrm{Q}}}^{\frac{5}{2}}},\frac{\mathrm{S}}{{\mathrm{R}}^{\frac{\mathrm{-3}}{2}}\times {\overline{)1+\mathrm{Q}\mathrm{Q}}}^3},\mathrm{\&c}$. Et universaliter, Velocitas ut ${\overline{)1+\mathrm{Q}\mathrm{Q}}}^{\frac{1}{2}}\times {\mathrm{R}}^{-\frac{1}{2}}$, & Densitas ut $\mathrm{S}\times${illeg} Resistentia ut ${\overline{)1+\mathrm{Q}\mathrm{Q}}}^{\frac{1}{2}}\times \mathrm{S}\times {\mathrm{R}}^{-2}$, Densitas ut $\mathrm{S}\times {\mathrm{R}}^{\frac{\mathrm{n}-4}{2}}\times {\overline{)1+\mathrm{Q}\mathrm{Q}}}^{\frac{1-\mathrm{n}}{2}}$.

intellexit demonstrationem Slusij ad|{n}| animum intendit ad methodos differentiales Fermatij Gregorij & Slusij Barrovij] Sed |et| Barrovius Gregorius & {illeg} Fermat{illeg}|i|us {illeg} {illeg} rem tangentium per differentias Ordinatarum tracta{illeg}runt \& Leibnits|i|us eorum vestigijs insistendo/: Leibnitius autem a Newtono adm Newtonus autem anno 1672 scr admonit methodum Slusij esse corollarium tantum methodi suæ generalis \suam/ tangentium communicand{illeg} s{illeg} communicando \describend {sic}/ quam Slusius etiam mox communicavit, scripsit hanc esse unum particulare vel corollarium tantum methodi suæ generalis. Et Literæ Newtoni cum Leibnitio communicatæ sunt sunt {sic} ut supra.

Ad Lectore{m}. {I. Can. Epis}

1 The occasion of publishing this Collection of Letters will be understood by the Letters of M^r Leibnitz & M^r Keil in the end thereof. M^r Leibnitz taking offence at a passage in a discourse of M^r Keil published in the Transactions about A C. 16 1708, wrote a Letter to the Secretary of the R. Society complaining thereof as a calumny, desiring a remedy from y^e Society & suggesting that he beleived they would judg it equal that M^r Keil should make a publick recantation acknowledgm^t of his fault. M^r Keil chose rather to return an answer in writing wherein he explained his meaning in that passage & defended it. M^r Leibnitz not meeting with that satisfaction he desired wrote a second Letter to the Secretary of the R. Society, wherein he still complained of M^r Keil, representing him a young man not acquainted w^th what was done before his time nor authorized by the person concerned, & appealed to the equity of the Society to che the his unjust clamours. Vpon this repeated Appeal the R. Society appointed a Committee to examin the ancient Letters & Records relating to this matter |& report their opinion|, & ordered the Report of the Committee with so much of the Letters & papers as related to this matter to be published.

2 M^r Ol|Le|ibnitz was in England in the beginning of the year 1673 & again in y^e end of the year 1676 & in France all the time between |&| kept a correspondence all that time w^th M^r Oldenburgh & by his means with M^r Iohn Collins persons long since dead, & what he learnt from the English by means of that correspondence is the main Question. M^r Leibnitz appeals from young men to those \them |old{illeg} ones| who/ know what passed in those days, {illeg} refuses to let any man \man/ meddle w^thout authority from S^r Isaac Newton & desires that S^r Isaac would give his judment. S^r Isaac lived then at Cambridge & knew not \very little of/ what passed between M^r Ol M^r Olden. in the said correspondence, between M^r Oldenberg & M^r L|C|ollins on the one hand & M^r Leibnits on the other \& knew only his own correspondence printed by D^r Wallis/ |&| M^r Oldenbergh & M^r Collins are \long since/ dead & so are D^r Barrow M^r Gregory & D^r Wallis who corresponded w^th D M^r Collins. The |R.| Society therefore being \twice/ pressed by M^r Leibnitz & having no other means of enquiring into this matter, g{illeg} a ordered the antient Letters \Letter books/ & papers left by M^r Oldenburg in the hands of the R. S. & those found amongst the papers of M^r Collins to be search {illeg}|r|elating to the matters in dispute between M^r Keil Leibnitz & M^r Keil to be searched out \&/ {illeg}ied examined & published. And the substance of the Letters & papers is as follows \by {illeg} written & {illeg} & appointed a Committe to do it & ordered the Report of the Committe to be published together with the extracts of the Letters & Papers presented \to/ them by the Committee. And the substance of the Letters & Papers is as follows./

M^r Leibnitz was in England in the beginning of the year 1673 & again towards the end of the year 1676 & {illeg} in the middle time in France,

3 By the Analysis published in the beginning \of this Collection/ the Reader will find that M^r Newton was acquainted in the year 1669 with the method of infinite series & by the help of \the method of/ fluent quantities & their incrementa momentanea or moments had then applied it very generally to the solution of problemes, |& by the method of series \had/ demonstrated the method of fluents, in w^ch demonstration he considers the first term of the series as the fluent, the rest as the augmentum of the fluent & the second alone as the augmentum momentaneum or moment \& the moment as the exponent of the velocity of increase or fluxion/. Whence it is that in this & a following tract he joyned these two methods together as two parts of one general method for solving of Problems: w^ch method proceed by resolving finite æquations into series when it is necessary & applying both finite & infinite equations to the solution of Problemes by the method of fluents.|

M^r Newton in his letter of Octob 1676 sent to M^r Leibnitz by M^r Oldenburgh, being desired by M^r Leibnitz to g relate the Original <541v> \M^r Newton being desired by M^r Leibnits to relate the original of his Theorem/ of his Theoreme for turning bino the dignities of binomials into infinite series, related in his letter of Octob 24 1676 that he found it \by t|i|nterpoling a series of D^r Wallis/ a little before the plague w^ch raged in England in the years 1665 & 167|6|6 & in the year 1671 at the request of some friends wrote it a Treatise of the methods of infinite series together & fluent quantities together. But This being \being was was/ written to M^r Leibnitz \long/ before any \the present/ disputes arose & was never yet questioned by M^r Leibnitz nor excepted against no dispute has hitherto risen about it. As M^r Newton conjoyned the two methods in his Analysis, \as one general method/ so he conjoyned them again two years after \more at large/ \afterwards/ in this Tract \written more at large upon the same subject./ For they are really but two branchis of one & the same universal method. For in the end of his Analysis he con < insertion from f 542r > sidered the first term < text from f 541v resumes >

4: M The Analysis being sent to M^r Collins in the year 1669, he sent one of the series to M^r I. Gregory with an acc^t notice that M^r Newton had a general method of finding such solving probles|m|es by such series, & M^r Gregory after much search found out the method in the end of the year 1670 & in the next year sent several series to M^r Collins (one of w^ch was that claimed afterwards by M^r Leibnitz for y^e circle sector of the circle & Hyperbola), & gave leave to M^r Collins to communicate his series freely but left it to S^r I N M^r Newton to publish the Method as y^e first inventor.

5 In December 1672 M^r Newton at the request of M^r Collins sent him his method of Tangents & represented that it was but one particular or rather a Corollary of general & easy method of solving difficulter Problems without sticking at radical quantities |meaning the method of fluent quantities & their moments|. And about a month after M^r|on| Slusius sent his method \of Tangents/ to M^r Oldenburgh w^ch proved to be the same w^th M^r Newtons but not so perfect For it stuck at radicals \& was not applicable to mechanical curves/. M^r Oldenburgh represented \to M^r Newton Collins & both to M^r/ < insertion from f 542r > Newton < text from f 541v resumes > that it was found by Slusius some years before he printed his Mesolabium \found it first/ & so it goes under the name of Slusius's method.

6 Hitherto nothing was met w^th about by w^ch it appeared that M^r Leibnitz knew any thing of the {illeg} of these methods of Series {illeg}|&| moments. In the beginning of the next year he | M^r Leibnits was at London & pretended to Moutons differential method & in such a manner as makes it evident that he knew then of no other differential method. The next year \A.C. 166|7|4/ being at Paris he began to write to M^r Oldenburgh of two series found by one & the same method \But it doth not appear that he had the method. For/ one of the series was for finding the arc \arc/ by the sine, & the method a little \sometime/ after he wrote to M^r Oldenburgh for the method of finding two series, one \of w^ch was/ for the arc by the sine the other for the sine by the arc, & his method of transmutations (the only method w^ch he then pretended to be his own) doth not extend to the invention of either of these series without the help of M^r Newtons method of extracting roots at that time not known to him M^r Leibnitz.

7 In May 1675 M^r Oldenburgh sent him \from M^r Collins/ four of M^r Newtons series & four or five of M^r Collins's \Gregories/, & he owned the receipt of them & kno|e|w none of them to be his own. For he promised to compare them with his own|.| as soo But M^r Gregory died before the end of the year & M^r Leibnitz forgot that he had received these series from M^r Oldenburgh & communicated the last of them in writing to his friends at Paris as his own & the next \series & the next year/ sent it back to M^r Oldenburgh & M^r Newton as his own|.| series. <542r> year sent it back to M^r Oldenburgh as his own & six years after published it in the Acta Lipsica as his own

9 In the winter between 166|7|5 & 166|7|6 M he received two series from one M^r Mohr w^ch M^r Mohr had received from M^r Collins & having \as if he had/ forgotten that he had received them before with \several others/ from M^r Oldenburg \& taken time to consider them/ he wrote to M^r Oldenburgh to procure \from M^r Collins/ the Demonstration of those two series \shewed him by M^r Mohr, that is the method/ from M^r Collins. Whereupon \of finding them. And thereupon/ M^r Oldenbe|u|rgh & M^r Collins \least he should \also/ forget the receipt of the Demonstration/ wrote pressingly to M^r Newton to writ to describe his own Method himself. & M^r Leibnits receiving M^r Newtons method w^th some examples of series, {illeg}|e|ndeavoured by small alterations to lay claim to some of s|t|hose series tho he had no method for finding them before he received M^r Newtons Letter. And at the same time he sent \M^r Newton/ a method of his own for finding series pretending that \by transmutation of figures/ representing it a general method tho it was not \far from being/ general \& deserved not the name of a method/ till M^r Newton made it s{illeg}|o| by communit|c|ating his method of extracting roots, & then it was of no use till

10 M^r Gregory had but one series sent him w^th notice that it was y^e result of a general method & therby within the space of a year found out the method.|,| This method being altioris indaginis M^r Leibnits could M^r Leibnits \but left it to M^r Newton to publish the Method as the first/ inventor. M^r Leibnitz had eight series sent him w^th the same notice, but this method being altioris indaginis he could not find it out but after a years consideration was forced to desire \it of/ M^r Oldenburg to send it to him {th} it him \to procure him the method/, & yet he continues to this day to number himself among the inventors of this|e| methods of infinite series \of infinite series/

11 M^r Newton in his first Letter had said that {illeg}|the| Analysis \of the Moderns/, by the help of those series \or infinite æqu. it self/ extended to almost all Problemes, (except numeral ones like those of Diophants|u|s) meaning \here the by these æquations/ the method of series \together/ w^th the consequence \thereof/ the method of fluents & moments. For the series afte without considering the second terms as moments of the first terms are but of not of so large extent \in the solving of Problems/. |in his next Letter he explained himself in this manner by extending this method to inverse problems of tangents & to the resolution of Equations involving fluxions.| M^r Leibnitz in his answer had disputed this assertion & said that \there were/ many Problemes, & amongst others the Problemes of the inverse method of Tangents, w^ch depended not on the inverse \neither on/ Equations nor on Quadratures. By w^ch answer its {sic} manifest \concluded appears/ that he knew nothing yet of the method of fluents & their moments w^ch he has since called the differential method. And the same thing is manifest \concluded/ also by \from/ an Opusculum w^ch M^r Leip|b|nits composed composed in a vulgar method & put about befo concerning the series of M^r Gregory \of M^r Gregory/ above mentioned & put communicated to his friends at Paris before the end of y^e year 1675 & continued to \adorn &/ polish in the year 1676 in order to send it to M^r Oldenburg in recompence for the Demonstration of the two series of M^r Mohr. but left of \ceased left of/ to polish after he fell into other business & after the invention of his new Analysis or Differential method did not think worth publishing.

8 After the death of M^r |I.| Gregory was known to the Mathematicians at Paris, M^r Leibnitz & some others there desired that his letters & papers might be <542v> collected into a body & preserved. The collection was made by M^r Collins at the pressing desire of M^r Oldenburgh & sent by him to Paris to be communicated to M^r Ol Leibnitz & returned .|b||ack| And to {illeg} London. And in this collection there is an Epistle of M^r Gregory dated 15 Feb. 1671 conteining several series one of w^ch is that above mentioned {c}laimed by M^r L w^ch M^r Leibnitz p{illeg} commu received from M^r O. & published as communicated at Paris as his own. [But M^r Leibnitz soon forgot that he had seen \either M^r Oldenburghs Letter \or// this collection & \having/ sent the series back to M^r Oldenburgh \& M^rNewton/ in his Lett answer to S^r Isaac Newtons letter as his own series & published it at \published/ six years after published it as his own in the fi{illeg}t Acta Lipsica, & continues to this day to claim it as to {illeg} claim it for his own. In the same Collection of Letters a Copy of M^r Newtons |was]| \And another is M^r Newtons/ Letter to M^rCollins dated 15 Dec. 1672 concerning his method of Tangents \which extended to Curves whether Geometrical to or Mechanical Curves & others \or howsoever related to right lines// & \concerning/ his general method of w^ch \fluents & moments whereof/ /{of} solving Problemes whereof\ the method of Tangents was but a particular or Corollary.

12. But after M^r Leibnits had seen the said Collection & was there informed that the Method of Tangents the described in M^r Newton's aforesaid Letter was but a \one/ particular or rather a Corollary of a general \& easy/ method of solving Problems w^ch stuck not at surd quantities & knew that this method was the same w^th that of M^r Slusius but more general because it stuck not at Tangen surds, [he considered the method of th Tangents Slusius & how to improve it, & make it as appears by his Letter of 18 Novem 1676 & the beginning of his Letter of 21 Iune 1677. A [Slusius had set down three Lemmas as the foundation of his Method, & the two first of them gave the ratio of the difference of powers or Ordinates to the difference of the Latera or Abscissas. If M^r Leibnits understood these Lem the meaning of these Lemmas he was thereby put upon considering rem tangentium per differentias Ordinatarum & might take the name of Differential method from thence But if he understood not Slusius, he might understand Archimedes who began the solution of Problems by the proportion \{illeg}|&| summs of/ their small parts, he might understand the application of {illeg}|t|his|e| notions by Cavallerius & Fermat \the moderns/ to equations.] And being further told in M^r Newtons Letter of Iune 1676 that the method of Series was was so general as to extend to the solution of almomst all Problemes exep numeral ones like those of Diophantus,] And after he had also received M^r Newtons \afor/ Letter of I 13 Iune 1676 wherein he was told that the method of series extended to the solution of almost all Problemes except th numeral ones like those of Diophantus: He considered the method of Slusius & how to improve it as appears by his Letters of 18 Novem 1676 & the beginning of his letter of his Letter of 21 Iune 1677, & {illeg} tried to improve it by considering the differences of the Ordinates, that is by considering the second |termes| of the series \representing/ when the Ordinates are turned into series. For this consideration would presently lead into a more general method of Tangents extending even to met|c|hanical curves & not sticking at surd quantities. And if he had not such a method before the receipt of M^r Newtons second Letter {illeg} there was so much said \M^r Newton said enough/ in that Letter of M^r Newtons \his/ general methos|d|s|:| {illeg} & their use in \altho to fo{illeg} avoiding enlarging upon it/ he concealed the fundamental Propositions by ænigmas as Galilæo & Hugens in other cases had done before. And \now/ M^r Leibnitz in his answer described this|e| de|i|fferential method & its use in drawing of tangents without as in the method of Slusius but without sticking at surds, & in squaring of Curves, & declared \allowed/ that in these things it resembled M^r Newtons \the/ method w^ch M^r Newton endeavoured to conceal, & that it extended also to the solution of inverted Problems of th|a|ngents by Equations & Quadratures. But he forgot to acknowledge that <543r> he ha{illeg}|d| invented the|i|s method since his Letter of 27 Aug. 1676 wherein he reprimanded M^r Newton for making his method too general & objected that may Problemes were so intricate & particularly the inverse probleme of Tangents as not to depend upon æquations & quadratures. He was convinced now that M^r Newtons methods were very general & that understood that M^r Newton had writ a treatise of these methods above five year {sic} before & it becomes not men of candor & modesty to interrupt one anothers proceedings & snatch away one anothers inventions.

M^r Leibnitz \in his Letter of 27 Aug 1676 in his Letter of 21 Iune 1677/ having desired M^r Newtons \to explain his/ method of {illeg}|d|eriving reciprocal series from one another {illeg}|s|ent a second Letter wrote again to M^r Oldenburgh Iul 12 Iuly 1677 that upon reading {illeg}|M|^r Newton's Letter over agin he saw how it was done not only by extracting of roots by his other method described in y^e end of his Letter, qua me \quo/, saith he, aliquando usum est|s|e in veteribus meis schedis reperio: sed cum in exemplo quod forte in manus meas sumpseram, nihil prodisset elegans, solita impatientia eam porro adhibere neglexisse.

M^r Leibnitz in his Letter of 27 Aug 1676 having desired M^r Newtons method of deriving reciprocal series from one another, & received M it, wrote again in his Letter, of 21 Iune 1677 {fo} to know how M^r Newton derived a certain series from its reciprocal. And I 12 Iuly 1677 {illeg}|w|rote again that upon he saw how it was to be done by the ex both by the extraction of roots & by the method in thi|e|{s} end of M^r Newtons Letters w^ch method saith he I me quo aliquando usum in veteribus meis schedis reperio; sed cum in exemplo quod forte in manus meas sumpseram, nihil prodijsset elegans solita impatientia eam porro adhibere neglexisse. He had therefore forgot this series before the writing of date of his Letter of 27 Aug 1676, & before that of his Letter of 12 May 1676 in w^ch he desired the demonstrion of two reciprocal series, & before that of M^r Oldenburgh Letter of May 15 May 1675 in w^ch he received several good instances of his method if he had then remembred it, & before he had the series of M^r Gregory w^ch he published as his own, whether he first received that series from M^r Oldenburgh or had it before. For that series would have helped him to a good example of his method. if

After this, M^r Leibnitz forgot that M^r Newton's method {illeg} was genera had any method extended further had {illeg} method was general untill or that had a general method, or any other method then that of M^r Slusius for drawing Tangents a little improved so as not to stick at radicals: |& so he published his differential method in the Acta Leipsica without mentioning that M^r Newton had \found/ a method of the same kind before the year 1671.| {illeg} But so soon as he understood by M^r Newtons Principia, that his method was much more general \then that of Slusius for Tangents/ & by D^r Wallis's works that it had great affinity w^th the method differential method he acknowledged the same in the Acta Lipsica, but continued still of opinion that the differential method was the originall,|.| & M^r Keil represented |t|him|at|self of another opinion & the Report of the Committee & is on his side. For the truth of this narrative & of Report the Letters & ex extracts of Letters & Papers in this Collection are to be consulted. \{t}he method of fluents was y^e Orig. & M^r Leibnitz complained of him to the R. S./ This is the state of the case collected out of the ensuing papers. And the Report of the Committe upon them in|s| in favour of M^r Keil.

When M^r Leibnitz could not procure M^r Newton's \the/ Method of series from M^r O. & M^r C. without the knowledge of M^r Newton he desired M^r Newton to communicate his method of deducing reciprocal series from one another & M^r Newton sent it. M^r Leibnits understood it with difficulty but so soon as he understood it he wrote back to M^r O. that he had found it \long/ before as he perceived by his old papers but not meeting w^th a good example of it {sic} use had neglected it. It seems he had found it & forgot it again before he had the series of M^r Gregory w^ch would have helped him to a very good example.

And|t| length M^r Oldenburgh \Leibnitz/ published the differential method in the Acta Leipsica A.C. 1684, but made no mention of M^r Newtons having found the like method \long before/. For he had then forgot that M^r Newtons had a general method of that {g} kind for of that kind, & wrote afterwards that \in the year A.C. 1684/ when he published this|e| elements of his calculus he knew nothing more of M^r Newtons inventions of this kind then what he had signified in his Letters namely that he could draw Tangents w^thout taking away irrational quantities. But when he saw M^r Newtons Principles he perceived this method was of much larger extent & when he D^r Wallis's works came abroad he but perceived \understood but knew/ not that it was so like the differential method till before D^r Wallis's works came abroad, & still contends that y^e Differential method was the original, representing soemtimes that y^e method of fluents was came in the room of the method differential method sometimes that it w \M^r/ M^r Newton substituted fluxions for differences soemtimes allowing that he found out the method of fluents by himself. M^r Keil on the contrary represents that the method of fluents was the original|.| & M^r This \is/ the state of the m{illeg}|a|tter is drawn up from y^e following papers, as you will find by reading them.

12 And after M^r Leibnitz had seen the said Collection & M^r Newtons Letter of 13^th Iune 1676 wherein he was told that the method of Series extended to y^e solution of almost \all/ Problemes except nueral {sic} ones like those of Diophantus: his thoughts were upon improving the methods of Tangents as appears by his Letters of 18 Nov. 1676 & the beginning of his Letter of 21 Iune 1677. And the drawing of Tangents to Mechanical Curves & reducing the Ordinates of any other Curves into series would naturally lead him to consider the second terms of the series as differences of the Ordinates For he tells us thh|a|t he fell into the differential method by considering the differences of the Ordinates in drawing of Tangents

12 After M^r Leibnitz had seen the aforesaid Collection of M^r |was told by M^r Newton in his Letter of 13 Iune 1676 that his method of applying series to the solution of Problems was very general, & he had also seen the aforesaid Collection of M^r| Gregories Letters, w^th M^r Newtons {illeg} & therein fr amongst w^ch was M^r Newtons Letter of to M^r Collins dated 15 Decemb. 1672, & therby understood that M^r Newton had a general method of solving Problems whereof his method of Tangents was but a particular or Corollary & & {sic} that it exted|n|ded to Mechanical Curves & by consequence proceeded upon the augmenta momentanea of the right lines by w^ch Mechanical Curves are defined & determined, & (for there is no \other/ way yet known of drawing tangents to mechanical line Curves;) & after he understood also that th M^r Slusiuss {illeg} Method of Tangents so far as it extended was the same with M^r Newtons but was capa & by consequence was capable of being improved into \a/ general method: his mind ran upon improving the method of Tangents by the incrementa momentanea of lines, & particularly the method of Slusius by the incrementa momentanea of the Ordinates w^ch he called their differences. For as is manifest by his Letters of 18 Nov 1676 & the beginning of his Letter of 21 Iune 1677. {illeg} These incrementa momentaneas \of quantities/ Slusius in the first of {the th{illeg}|e|} three Lemmas \on which he founded his method/ called differences & M^r Leibnitz retained the name{illeg}.{]} And while he had these things under consideration he received further light into the method by M^r Newtos Letter of 24 Octob. 1676. And falling into M^r Newton's method described it in his answer, Dated 21 Iune 1677 gav & \with Slusius/ gave the name of Differences to the quantities w^ch Newton {illeg} incrementa momentanea \of quantities by New/ For Slusius called them Differences in the first of the three Lemmas upon w^ch he founded his method of Tangents. And now M^r Leibnitz acknowledged the extent of M^r Newtons general method for drawing Tangents without Sticking at surds, For squaring of fig{ures} <544r> & solving of inverse Problemes of Tangents.

The Question is Who was the first author of the method called by M^r Leibnits the differential method. M^r Leibnitz claims Inventoris jura, M^r Keil denies y^t same he was y^e first Inventor represents S^r Isaac Newton y^e first inventor & allows that S^r Isaac Newton might also find it apart. M^r Keill asserts S^r Isaac to be the first Inventor & the Report of the Committe is on his side.

The papers & Letters till y^e year 1677 are to shew that S^r Isaac had a general method of solving Problemes by resolving finite equations into infinite ones when it was is \was/ necessary, & \deducing fluents & their moments from one ano/th\{er}/ by the help of Equations {illeg} finite or infinite. {illeg} The Analysis first printed \in the beginning of this Collection/ shews that he had such a method in y^e year 1669, \& that it was very general/ \And/ some letters shew that \in y^e year 1671/ he composed a larger treatise of this method: [others that at the request of M^r Leibnitz M^r Newton sent him a large & distinct description of y^t part of his method w^ch concerned series w^th] A A Letter of his writ in the year 1672 was copied & sent to M^r Leibnitz in y^e year 1676 in w^ch he described this method to be general] And by the Letters w^ch follow it appears that {illeg} in y^e year 1671 he composed a \larger/ treatise of this metho{t}|d| |&| that it was very general & easy without sticking at surds & extended to mechanical curves |& extended to problems of tangents diret {sic} & inverse & to finding y^e areas lengths, centres of gravity & curvature of curves & oto {sic} other more difficult problems & y^t in mechanical curves as well as others|, & by consequence was founded upon the consideration of the indefinitely small particles of quantities called indivisibles by Cavallerius, Augmenta momentanea & moments by M^r Newton & differences by Slusius & Leibnitz. that it was so general as to extend to the solution of almost all Problemes except the numeral ones of Diophantus, & reched to & particularly to those quadratures of curves centers of tangents direct & inverse, the quadratures of curves, {illeg} \& to the finding the areas/ the curvity \segments lengths/ centers of gravity & curvatures & others more segments \of curves/ \&/ the first & second segments of solids & \determining/ other problems more difficult \For there are no other ways of drawing tangents to these \mechanical/ curves or of squaring any curves then by considering the particles of Quantities. That this method was so general as to extend to the solution of almost all Problemes except the numeral ones of Diophantus/ & M^r Newton giv|a|ve examples of this \his/ method in drawing tangents squaring of Curves & solving inverse Problemes \of Tangents, all w^ch was communicated to M^r Leibnits/ & at y^e request of M^r Leibnitz \M^r N./ communicated \to him also/ that part of the method w^ch consisted in the reduction of all \finite/ equations to series infinite series, & thereby enabled him to find the Ordinates of Mechanical curves w^th their Differences which are the second terms of the series expressing the Ordinate. M^r |I.| Gregory by {are} having one of M^r Newtons series with notice that it was the result of a general \method found out the method/ in a|t|he space of a year. M^r Leibnitz had eight{illeg} series sent him by M^r Oldenburg besides two other w^ch he pretended to have a year or two before: but this method being altioris indaginis M^r Leibnits could not find it out but after M^r Leibnits had forgot the receipt of y^e right series th at length wrote to M^r Oldenburg to procure him the Method from M^r Colling. But o{illeg} M^r Leibnitz having forgot the receipt of the eight series they forbore to send him the method without M^r Newton's knowledge, {illeg}|&| desired M^r Newton to describe his own method himself, w^ch he did \at their importun{ing}/ But And M^r Leibnits not yet \fully/ understanding the extent {illeg} methods sent him desired M^r Newton to explain how he derived reciprocal it further & how he derived reciprocal series from one another.

M^r Collins in y^e year 1670 commun

Hitherto{illeg} M^r Leibnitz continued to compose things in the vulgar way of writing, w^ch after he fell into the Differential method he did not think worth publishing. Hitherto he continued of opinion that M^r Newton had made too large a description of the extent of his method & that many problems & particularly the inverse problems of Tangents depended not on {equ}ations nor on quadratures: but at le after these descriptions & examples of M^r Newtons general method had been sent to him & one half of the method at {illeg} {his} request had been \{illeg}/ described to him {illeg} he fell into the rest of the method & began to describe it in his letter of 21 {Iun}{e} 16{77} <544v> & saw now that it extended to the drawing of Tangents w^thout sticking at radica{ls} & to quadratures of {illeg} & inverse Problemes of Tangents, & that the solution of y^e example of the {sic} inverse problems of Tangents flowed f \in the Example/ w^ch M^r Newton ha{d} sent him to convice {sic} him that these Probll|e|mes were to be solved by æquations & Quadratures, was practicable & flowed from his new Arts as well as from Newtons. But he forgot to acknowledge that his Arts were but just found out. For without acknowledgi{illeg}|n|g this, he ught not to have intermedled w^th M^r Newton method. Fo \Candid/ Men are not to interrupt one anothers in their proceedings, not t & snatch away one anothers inventions. M^r D^r Pell reprehended him for pretending to Moutons Method. M^r Collins reprehended him for intermedling with what Gregory & Tschurnhause were about as appears by a letter not yet published. He deserves to be reprehended for pretending \should not have pretended/ to two series in y^e year 168|7|4, when he wanted the method of finding them, for pretending to the invention of the series w{illeg} he {res} forgetting \He should not have have {sic} forgot/ the receipt of the eight series w^ch {illeg} Oldenburgh sent \him/ & publishing \nor have published/ one of them as his own after he knew | w^thout mentioning that he had received it from Oldenburg & knew that Gregory had sent it to Collins in y^e beginning of the year 1671, for endeavouring \He should not have endeavoured/ to get M^r Newtons method of series from M^r Oldenburgh & M^r Collins w^thout M^r Newtons knowledge when he will not allow M^r Keil to asser{t} M^r Newtons right w^thout M^r Newtons authority. He deserves to be reprehended for making \should not/ recconing himself among the inventors of infinite series & of the methods of finding them when he has not produced one series of note invented by himself nor has any general method of finding them besides what he received from M^r Newton. H.

of the Series as the th a fluent quantity & the second with all those that follow as its incrementum, & by this|e| incrementu second term infinitely small so that all the following th|e|rms may vanish {illeg}|he| as being infinitely smaller he considers the second term alone as the incrementum momentaneum of the first. And by this means he there derives & demonstrates this|e| method of Series fluents from the method of series.

And \that is, the method of finding them &|A|nd thereupon/ M^r Oldenburg & M^r Collins seing that he wanted the demonstration of \method of finding w^ch he had boasted of & the two series w^ch he had boasted of & the/ the eight series w^ch they had sent him & |he| pretended to have forgot receipt of them \{illeg}te as if he had forgot them/, forbore to send him the method w^thout M^r Newtons knowledg{e} & wrote \very/ pressingly to M^r Newton to describe his own Method himself|.| least he should lo And M^r Leibnitz.

– are not of so large extent. Without the method of fluents they do not amount to a general method for solving of Problems. By the method of series finite equations are to be resolved into infinite ones when there is occasion & both sorts of equations are to be applied to y^e solution of Problems by the method of fluents to make the general method here spoken of.

Ad {sic} Newtonus scripserat

|1| Scripserat Newtonus se methodum generalem habere solvendi problema, & hanc |2| methodum in re tangentium ad curvas mechanicas extendi (p. 30, 47,    ) quod perinde |3| est ac si dixisset methodum suam \generalem Augmentis momentaneis/ in momentis quantitatum fundari. Tangentes |4| enim ad Curvas mechanicas abs considerationem momentorum duci non possunt. [Resp{illeg} Leibnitius \[Cum Slusio]/ nomen differentiarum imponit augmentis illis & responde se jam a multo tempore rem tangentium longi|e| generalius [quam Slusius] tractasse scilicet per differentias] Ordinatarum]|]| significaverat etiam methodum suam generaliorem esse quam ea Slusij Newtonus methodum suam Ordinatæ in Curvis mechanicis sunt series infinitæ & ostenderat Newtonus inventionem {tali} & serierum |5| differentiæ sunt eo momentane \momenta/ sunt earum termini secundi. Newtonus autem |6| inventionem talium serierum, Scripserat etiam Newtonus methodum suam |7| ad omnia pene problemata præter numeralia Diophantæis\i/ \&/ similia se |8| extendere (p. 55) etiam ad inversa tangentium problemata & his difficiliora (p. 85, |9| 86) & Leibnitius hæc intellexerat de method serib|e|bus et augmentis momentaneis |$\frac{1}{2}$| \momentis/ {illeg}|in| unam methodum generalem conjunctis \conspirantibus./ (p 92, 93.) [Problemata ad æquationes reducuntur seu finitas seu infinitas reducuntur & \per has/ æquatines per augmenta momentanea \solvuntur/ solutiones exhibentes & contra] Hæc momenta differetias vocat Prop|b|lemata \{illeg} scriberat/ ad æquationes seu finitas seu infinitas reducuntur & \per has/ æquationes & momenta \fluentium/ solvuntur. {illeg} \Tractantur uti Problemata per {illeg}|æ|quationes seu finitas seu infinitas & fluentium momenta conjunctim./ Leibnitius igitur de tali methodo admonitus \& nomen differen{illeg}tiarum imponendo momentis/ rescribit se quo rem tangentiū {illeg} ampliasse tractando per generali{ui}s tractasss \ampliasse tractando scilicet generalius tractass/ scilicet per differentias Ordinatarum. Sic enim vocat Or Differentias enim vocat quæ Newtonus|u|s momenta {illeg} \Sed/ et methodum mox extendit a q|Q|uadraturas |$15⁤\frac{1}{2}$| \Curvarum/ & inversa tangentium Problemata, Newtoni vestigijs insistendo

Ad Lectorem of Commercium 2|3|

The occasion of publishing this Collection of Letters & Papers will be understood by the Letters of M^r Leibnitz & M^r Keil in the end thereof. M^r Leibnitz taking offence at a passage in a discourse of M^r Keil published in the Transactions A.C. 1708, wrote a Letter to the Secretary of the R. Society complaining thereof as a {illeg}t{illeg}tary calumny, desirin|ed|g a remedy from the Society, & suggestin|ed|g I that he beleived they would judge it equal that he should make a publick acknowledgment of his fault. M^r Keil chose rather to return an answer in writing, wherein he explained his meaning in that passage & defended it. M^r Leibnitz not meeting with that satisfaction he desired wrote a second Letter to the Secretary of the R. Society, wherein he still complained of M^r Keil, representing him a young man & not authorised acquainted w^th what was done before his time, nor authorized by the person concerned, & appealed to the equity of the Society to che his unjust clamours.

M^r Leibnitz was in England in the beginning of the year 1673 & again in the year October 1676, & during the intervall in France & all that time kept a correspondence with M^r Oldenburg, & by his means w^th M^r Iohn Collins & sometimes w^th M^r Is. Newton, & what he might learn from the English \either in London or by that correspondence {illeg}/ is the main Question. M^r Oldenburg & M^r Collins are long since dead & M^r Newton lived then at Cambridge & knew little more then his own correspondence since published by D^r Wallis. M^r Newton can be no witness for M^r Keill nor M^r Leibnitz for himself, & there appears no other living evidence. The R. Society therefore being twice pressed by M^r Leibnitz against M^r Keill, appointed a Committee to search out & examin the Letters Letter-bo\o/ks & papers left by M^r Oldenburg in the hands of the Society & those found among the papers of M^r Ioⁿ. Collins relating to the matters in dispute, & to report their opinion thereupon & ordered the Report of the Committee w^th the extracts of the Letters & Papers to be printed. published.

When M^r Newton wrote the Analysis printed in the beginning of this collection, he had a method of resolving finite equations into infinite ones & of applying both finite & infinite equations to the solution of Problemes by meanes of the \f{illeg}/ proportions of the incrementa momentanea of growing or increasing quantities. These incrementa M^r Newton calls particles & moments & M^r Leibnitz infinitesimals indivisibles & differences. The increasing quantities M^r Newton calls fluents & M^r Leibnitz summs, & the velocities of increase M^r Newton calls fluxions & exposes these <546r> fluxions by the moments of the flowing quantities. That part of the method w^ch consists in resolving finite equations into infinite ones, was at the request of M^r Leibnitz communicated to him by M^r Newton in his Letters of Iune 13^th & Octob 24^th 1676. And M^r Newton having so far touched upon the other part as to reckon that it was become sufficiently obvious (p. 72 lin. 1,) to secure it from being taken from him before he should have occasion to explain it, he expressed it in cyphre after the manner used by Galilæo & Hugens|i|us in like cases upon other like occasions. And M^r Leibnitz the next year in his Letter of Iune 21 communicated his \now claimed the/ invention of \this/ other part & now claims \the first invention/ of & M^r Keil asserts it to M^r Newton & is favoured in his opinion by the Report of the Committee. But there is nothing in these papers w^ch can affect other persons abroad who have received the method from M^r Leibnitz. They were strangers to the correspondence between M^r Leibnitz & M^r Oldenburg. They found the method useful & are much to be commended for the use & improvements that they have ma{illeg}|d|e of it.

Some Notes are added to the Letters to enable such Readers as want leasure, to compare them with more ease & see the sense of them at one reading.

Ad Lectorem of Comm Epis 3 4

The occasion of publishing this Collection of Letters \& Papers/ will be understood by the Letters of M^r Leibnitz & M^r Keill in the end thereof. M^r Leibnitz taking offence at a passage in a discourse of M^r Keil published in the Transactions A.C. 1708, wrote a Letter to the Secretary of the R. Society complaining thereof as a calumny, desiring a remedy from the Society & suggesting that he beleived they would judge it equal they|at| \he/ should make a publick acknowledgment of his fault. M^r Keil chose rather to return an answer in writing, wherein he explained his meaning in that passage & defended it. M^r Leibnitz not meeting with that satisfaction he desired wrote a second Letter to the Secretary of the R. Society, wherein he still complained of M^r Keil, reprentingsenting him a young man not acquainted w^th what was done before his time nor authorized by the person concerned \nor provoked/ & appealed to the equity of the Society to che his unjust clamours.

M^r Leibnitz was in England in the beginning of the year 1673 & again in the year October 1676 & during the interval in France, & all \that/ time kept a correspondence with M^r Oldenburg, & by his means w^th M^r I. Collins, & sometimes w^th M^r Newton, & what he \might/ learnt|e| from the English by that correspondence either in London or by that correspondence is the main Quæstion. M^r Oldenburg & M^r Collins are long since dead, M & of what passed between them & M^r Leibnitz there appear{illeg} no other & M^r Newton lived then at Cambridge & knew little more then \{or les} little more then/ his own correspondence since published by D^r Wallis. {illeg} He {&} \M^r Newton can be no witness for M^r Keill, nor/ M^r Leibnitz can be no witnesses for themselves \for himself/ & {illeg} there appear no other living evidence. The R. Society therefore being twice pressed by M^r Leibnitz against M^r Keil, appointed a Committee to search out & examin the Letters Letters-books & papers left by M^r Oldenburgh in the hands of the Society & those found among the papers of M^r Iohn Collins relating to the matters in dispute & \to/ shew them to such as knew the hands & report their opinion thereupon & ordered the Report of the Committee to be published w^th the extracts of the Letters & Papers to be published

✝ The Question is about the first invention of the infinitesimal method, & the opinion of M^r Keil \against M^r Leibnitz/ is favoured by the Committee {[}But there is nothing in these papers that can affect other persons abroad who have used \received/ the method \from M^r Leibnitz/; as the late Marquess de L'Hospital the Mons^r Varignon, \&/ the two brothers Iohn & Iames & Iohn Bernoulli. They were strangers to the correspondence between M^r Leibnitz & M^r Oldenburg, \They found the method usefull/ & are much to be commended for the use \& improvements/ that they have made of this infinitesimal Method |it|

‡ The Analysis she printed in the first place shews that M^r Newton had the method in 1669; the first instance of M^r Leibnitz his knowing any thing of the method is in his Letter dated 21 Iune 1677. In the times between there are Letters w^ch shew that M^r Newton had the method & that M^r Newton \Leibnitz/ had it not & received light into it from M^r Newton. M^r Newton considered the method of series & the infinitesimal method as two methods called by him the method of fluents fluxions & moments as two {l}{illeg} methods neatly related to one another & conspiring into one general method for the solut resolution of almost all sorts of Problemes. At the request of M^r Leibnitz he communicated \to him/ one half of this general method by his Letter of 13 Iune 1676, & & {sic} it doth not appear that M^r Leibnitz found out the other half till some time after this communication.

✝ The Question is about the first invention of a|t|he infinitsimal method called by M^r Newton the method of fluents, fluxions & moments the exponents of {illeg} & by M^r Leibnitz the method of differences & indivisibles The Analysis printed in the first place shews that M^r Newton had the method in 1669: the first instance of M^r Leibnitz's knowing the method is in his Letter dated 21 Iune 1677. In the times between there are Letters w^ch shew that M^r Newton had the method & M^r Leibnitz had it not, & that M^r Leibnitz received light into it from M^r Newton.

The infinitesimal method & the method of series were in those days considered by M^r Newton as one general two methods nearly related to one another \& subservient to one another & {illeg}/ & conspiring into one general method for the solution of almost all sorts of problems B as appears by the Letters. At the request of M^r Leibnitz he communicated to M^r Newton communicated to him one half of this general method by his Letter of 13 Iune 1676, & & {sic} it doth not appear that M^r Leibnitz found out the other half till some time after this communication. When he received the first half of the method he put in for coinventor. When he had light into the second half he put in for coinventor & when M^r Newtons Principia philosophiæ came abroad he put in for coinventor. But the Gentlemen of the Committee \upon examining the Letters & Papers think/ think that M^r Keill has done him no wrong in representing M^r Newton the first inventor in the second case as well as in the first & third.

As for other Gentlemen who have used the differential method (as the Marquss {sic} de L'Hospital, the brother Iames & Iohn Bernoulli & Mons^r Varignon \M^r C{illeg}|r|aig &c/) there is nothing in these papers that can affect them. They were strangers to the correspondence between M^r Oldenburg & M^r Leibnitz, they found the method usefull & \they/ are much to be commended for the use & improvements they have made of it.

$\begin{array}{ccc}\frac{1}{1-\mathrm{x}}=1+\mathrm{x}+\mathrm{x}\mathrm{x}+{\mathrm{x}}^3& \mathrm{x}+\frac{1}{2}{\mathrm{x}}^2+\frac{1}{3}{\mathrm{x}}^3+\frac{1}{4}{\mathrm{x}}^4& \frac{1}{36}{\mathrm{x}}^6\\ \phantom{\frac{1}{1-\mathrm{x}}}\mathrm{x}+\frac{{\mathrm{x}}^2}{2}+\frac{{\mathrm{x}}^3}{3}+\frac{{\mathrm{x}}^4}{4}& \phantom{\mathrm{x}}-{\mathrm{x}}^2-\frac{1}{2}{\mathrm{x}}^3-\frac{1}{3}{\mathrm{x}}^4& \phantom{\frac{1}{36}{\mathrm{x}}^6}\\ \phantom{\frac{1}{1-\mathrm{x}}=}\phantom{1+\mathrm{x}+\mathrm{x}\mathrm{x}+{\mathrm{x}}^3}& \mathrm{x}-\frac{1}{2}{\mathrm{x}}^2-\frac{1}{6}{\mathrm{x}}^3-\frac{1}{12}{\mathrm{x}}^4& \phantom{\frac{1}{36}{\mathrm{x}}^6}\end{array}$

$\frac{3\mathrm{b}\mathrm{b}}{{\mathrm{a}}^4}\sqrt{1+\mathrm{Q}\mathrm{Q}}.\frac{4\mathrm{b}4}{\mathrm{a}6}.\colon\colon 3\sqrt{\phantom{0}} . \frac{4\mathrm{b}\mathrm{b}}{\mathrm{a}\mathrm{a}} \colon\colon 3\mathrm{a}\mathrm{a}\sqrt{\phantom{0}}.4\mathrm{b}\mathrm{b}\colon\colon \frac{\mathrm{G}\mathrm{T}}{\mathrm{D}\mathrm{N}}.\frac{4\mathrm{b}\mathrm{b}}{3\mathrm{a}\mathrm{a}}$

Schol. Fingere lice\re/t quod corpora in projectil{illeg} \projectile/ perge{illeg}t\eret/ in arcuum GH, HI, IK chordis, et in solis punctis G, H, I, K, per vim gravitatis & vim resistentiæ agitaretur, perinde ut in Propositione prim{illeg}|a| libri primi corpus per vim centripetam {illeg} intermittentem agitabatur; Et Solutio Problematis D|d|einde chordas in infinitum diminui ut vires redderentur continuæ. Et solutio Problematis hac ratione facillima redderetur.

p 268. lin. 10, lege [gravitatem ut 3XY ad 2YG

p 269. lin. 8. lege ut ${}^3\mathrm{S}$ in $\frac{\mathrm{X}\mathrm{X}}{\mathrm{A}}$ ad 4RR id est ut XY ad $\frac{2\mathrm{n}\mathrm{n}+2\mathrm{n}}{\mathrm{n}+2}\mathrm{V}\mathrm{G}$

p 270. lin 9, 14. $\frac{2\mathrm{n}\mathrm{n}+2\mathrm{n}}{\mathrm{n}+2}$.

injuriam Newtono illatam repellendo

Ad Lectorem of Comm Episto 2|3| 4 5

The occasion of publishing this Collection of Letters & Papers will be understood by the Letters of M^r Leibnitz & M^r Keill in the end thereof. M^r Leibnitz taking offence at a passage in a discourse of M^r Keill published in the Transactions A.C. 1708, wrote a Letter to y^e Secretary of the R. Society complaining thereof as a calumny, desirin|ed|g a remedy from the Society, & suggested that he beleived they would judge it equal that he should make a publick acknowledgment of his fault. M^r Keil chose rather to return an answer in writing, wherein he explained his meaning in that passage & defended it. M^r Leibnitz not meeting with that satisfaction he desired wrote a second Letter to the Society, wherein he still complained of M^r Keill, representing him a young man & not acquainted with things done before his time & nor authorized by the person concerned, & appealed to the equity of the Society to che his unjust clamours.

M^r Leibnitz was in England in the beginning of the year 1673 & again in October 1676, & during y^e intervall in France, & all that time kept a correspondence w^th M^r Oldenburg, & by his means w^th M^r Iohn Collins & sometimes with M^r Is. Newton, & what he might learn from the English either in London or by that correspondence is the main question. M^r Oldenburg & M^r Collins are long since dead, & M^r Newton lived then at Cambridge & knew little more then his own correspondence since published by D^r Wallis. M^r Newton can be no witness for M^r Keill nor M^r Leibnitz for himself, & there appears no other living evidence. The R. Society therefore being twice pressed by M^r Leibnitz against M^r Keill, appointed a Committee to search out & examin the Letters Letter-books & papers left by M^r Oldenburg in the hands of the Society & those found among the papers of M^r Iohn Collins relating to the matters in dispute & report their opinion thereupon & ordered the Report of the Committee with the extracts of the Letters & Papers to be published.

When M^r Newton wrote the Analysis printed in the beginning of this Collection, he had a method of resolving finite equations into infinite ones, & of applying both finite & infinite equations to the solution of Problemes by meanes of <549v> the proportions of the augmenta momentanea of growing or increasing quantities. These {illeg}|aug|menta M^r Newton calls particles & moments, & M^r Leibnitz infinitesimals indivisibles & differences. The increasing quantities M^r Newton calls fluents & M^r Leibnitz summs, & the velocities of increase M^r Newton calls fluxions & exposes these fluxions by the moments of the flowing quantities. That part of the method w^ch consists in resolving finite equations into infinite ones, was at the request of M^r Leibnitz communicated to him by M^r Newton in his Letters of Iune 13^th & Octob^r 24^th, 1676. And M^r Newton having so far touched upon the other part as to reckon that it was become^a[1] sufficiently obvious, to secure it from being taken from him before he should have occasion to explain it, he expressed it in cyphre after the manner used by Galilæo and Hugenius upon other like occasions. M^r Leibnitz claims the invention of this other part, & M^r Keill asserts it to M^r Newton & is favoured in his opinion by the Report of the Committee. But there is nothing in these Papers w^ch can affect other persons abroad who have received the method from M^r Leibnitz. They were strangers to the correspondence between M^r Leibnitz & M^r Oldenburg. They found the method usefull & are much to be commended for the use & improvements that they have made of it.

Some Notes are added to y^e Letters to enable such Readers as want leasure, to compare them with more ease & see the sense of them at one reading.

6 Ad Lectorem of the Commercium

The occasion of publishing this Collection of Letters will be understood by the Letters of M^r Leibnitz & M^r Keil in the end thereof. M^r Leibnitz taking offence at a passage in a discourse of M^r Keil published in the Transactions A.C. 1708, wrote a Letter to the Secretary of the R. Society complaining thereof as a calumny, desiring a remedy from the Society & suggesting that he beleived they would judge it equal that M^r Keil should make a publick acknowledgment of his fault. M^r Keil chose rather to return an answer in writing, wherein he explained his meaning in that passage & defended it. M^r Leibnitz not meeting with that satisfaction he desired wrote a second Letter to the Secretary of the R. Society, wherein he still complained of M^r Keil, representing him a young man not acquainted with what was done before his time nor authorized by the person concerned & appealed to the equity of the Society to che his unjust clamours.

M^r Leibnitz was in England in the beginning of the year 1673 & again in October 1676 & during the interval in France, & all that time kept a correspondence with M^r Oldenburgh & by his means with M^r Iohn Collins, & what he learnt from the English by that correspondence is the main Question. M^r Leibnitz appeals from young men to old ones who knew what passed in those days, refuses to let any man be heard against him w^thout authority from S^r Isaac Newton & desires that S^r Isaac himself would give judgment. S^r Isaac lived then at Cambridge, & knew only his own correspondence printed by D^r Wallis. M^r Oldenburgh & M^r Collins are long since dead & so are D^r Barrow M^r Gregory & D^r Wallis who corresponded with M^r Collins, & D^r Wallis gave judgment against M^r Leibnitz in a Letter dated Apr 10^th 1695 & not yet printed. The R. Society therefore being twice pressed by M^r Leibnitz & having no other means of enquiring into this matter by living evidence, appointed a Committee to search out & examin the pap Letters Letter-books & papers left by M^r Oldenburgh in the hands of the {illeg} Society & those found among the papers of M^r Collins relating to the matters in dispute, & ordered the Report of the Committee with the extracts of the Letters & Papers to be published.

The Question is, Who was the first author of the method called by M^r Leibnitz the infinitesimal method {illeg}|t|he Analysis of infinitesima indivisibles & infinites, & the Differential & summatory method & by M^r Newton the Method of fluents fluxions & moments. M^r Leibnitz claims inventoris jura, & allo sometimes allows that M^r Newton might also find it apart. M^r Keil asserts M^r Newton to be the first inventor & the Report of the Committe is o{illeg}|n| his|the| side same opinion.

The Letters & Papers till y^e year 1676 inclusively shew that M^r Newton had a general method of solving Problemes by {illeg}|r|educing them to equations finite or infinite \whether those equations/ includi|e|ng moments (the exponents of fluxions) or \do/ not includi|e|ng the|m,| {illeg}ts, & by deducing moments fluents & their moments from one another by means of those equations.

The Analysis printed in the beginning of this Collection shews that he had such a \general/ method in y^e year 1669. And \by/ the Letters & Papers w^ch follow, it it appears that in the year 168|7|1, at the desire of his friends he composed a larger Treatise upon this method (p. 27. l. 10, 27 & p. 71. l. 4, 26) that it was very general & easy without sticking at surds \or mechanical curves/ (p. 37) & extended to Problemes of Tangents direct & inverse (p 34, {illeg}) & to the finding the \tangents/ areas lengths centers of gravity & curvatures of Curves {illeg} (p. 27 30, 85) & solving other more difficult Problems \&c/ (p. 27, 30, 85) that in Problems reducible to Quadratures it proceeded by the Propositions since printed in the book of Quadratures \w^ch Propositions are there founded upon the method of fluents/ (p. 72, 74, 76) that it extended to the extracting of fluents out of æquations involving their fluxions (p. 8 & proceeded in difficulter cases by assuming the terms of a series & determining them by the conditions of the Probleme (p. 86) that it \determined the curve by the length thereof p 24 &/ extended to inverse Problems of tangents & others more difficult & was so general as to reach almost all Problemes except the nume <551v> ral ones like those of Diophantus (p. 55, 85, 86) And all this was found out by \represented known to/ M^r Newton before M^r Leibnitz knew \understood/ any thing of the method as appears by the dates of their Letters.

For \in the year 1673 he was upon another differential method p. 32./ in May 1676 he desired M^r Oldenburgh to procure him the method of infinite series (p. 45) & in his Letter of 27 Aug 1676 he wrote that he did not beleive M^r Newton's method to be so general as M^r Newton had described it. For, said he, there are many Problemes & particularly the inverse Problems of Tangents w^ch cannot be reduced to æquations {illeg}|o|r Quadratures (p. 65) w^ch words make it evident that M^r Leibnitz had not yet the Differential method of differential equations. And in the year 1675 he wrote \communi/ a piece in a vulgar he {sic} communicated to his friends at Paris \a tract/ written in a vulgar manner about a series w^ch he received from M^r Oldenburgh, & continued to polish in the year 1676 w^th intention to print it; but it swelling in bulk he left off polishing it after other business came upon him, & afterwards finding the Differential Analysis he did not think it worth publishing because written in a vulgar manner. p. 42, 45. In all these Letters & Papers there appears nothing of his knowing the Differential method before y^e year 1677: It is first mentioned by him in his Letter of 21 Iune 1677, & there he began the description of it w^th these words Hinc nominando IN POSTERVM dy differentiam duarum proximarum y &c. p. 88.

M^r Newton was therefore the first inventor, & whether M^r Leibnitz invented it afterwards proprio Marte afterwards or not, is a question of no consequence. The first inventor is the inventor, & inventoris jura are due to him alone. He has the sole right till another finds it out, & then to take his right from him without his consent & share it with another would be an Act of injustice, & an endless encouragement to pretenders. But however, there are great reasons to beleive that M^r Leibnitz did not invent it proprio marte, but received some light from M^r Newton.

For it is to be observed that wherever M^r Newton in his Letters spake of his general method, he understands his method of series & fluents taken together as two parts of one general method. In his Analysis the series are applied to the solution of problemes by the method of fluents & thereby give new series, & the method of fluents is demonstrated by the method of series (p. 14, 15, 18, 19) & in the year 1671 he wrote of both together, p. 71. The series in his book of Quadratures are derived from the method of fluents & were derived from it before the year 1676, p. 72. The method of extracting fluents out of equations involving their fluxions comprehends both together, & the method of assuming the terms of a series & determining them by the conditions of the probleme proceeds by means of the {illeg} method of fluents, p 86. When M^r Newton represented that his method of series extended to the solution of almost all Problemes except numeral ones like those of Diophantus, he included inverse problemes of Tangents (p. 55, 56, 85) & those problems are not tractable w^thout the method of fluents. And sometimes series are considered as fluents & their second terms as moments. And M^r Newton sometimes derives his method of fluents from the series into w^ch the power of a binomium is resolved, p 19. lin. 19, 20.

In the next place it must be observed that M^r Newton at the request of M^r Newton Leibnitz communicated to him freely & plainly the method of series w^ch was one half of this general method, namely the method of series p. 45, 49. M^r Gregory by the help of but one series, with notice that it was the result of a general method, found out the method within the space of a year, p. 22, 23, 24. M^r Leibnitz pretended to have two series in the year 1674, & had eight others sent him by M^r Oldenburg in April 1675 & took as the result of a general method {illeg}|&| took a years time to consider them p. 38, 40, 41, 42: but \this/ method being altioris indaginis he could not find it out but at length requested M^r Oldenburg to procure it from M^r Collins (p. 45) & at the request of M^r Oldenburg & M^r Collins, M^r Newton sent it to him. And when he had it he understood it with difficulty & desired M^r Newton to explain thing some things further p. 49, 63.

And \as/ for the other half of the method, M^r Leibnitz had a general description of it in M^r Newton's Letters of 10 Decemb. 1672, 13 Iune 1676 & 24 Octob 1676, with examples in drawing of Tangents (p. 30)|,| 47) squaring of curves p 42 & solving of inverse problemes of Tangents p. 86. & understanding by the same Letters that the method of tangents printed by Slusius was a branch & Corollary of M^r Newtons general method (p. 30) he set his mind upon improving his method of Tangents so as to bring it to a general method of solving problemes. For in his journey home from Paris by London & Amsterdam, he was upon a project of extending it to the solution of all sorts of problemes by calculating a T certain Table of Tangents as the most easy & useful th{illeg}|in|g he could then think of, \{illeg} spake to M^r Collins to help him to a calculator |for a calculator w^ch he wanted|/ & wrote |also| of this designe to M^r Oldenburg in a Letter dated at Amsterdam 28 Novemb. 1676 (pag. 87.) & therefore he had not then invented the differential method but was \only/ endeavouring to {sic} find out such a general method as M^r Newton had described \improved the method of Slusius into a general method of solving all sorts of problems but was only endeavouring to do it./ Now M^r Newton had told him that his method extended to Tangents of mechanical Curves & to Quadratures centers of gravity & Curvatures of Curves in general & to inverse problemes of Tangents, & he was thereby sufficiently informed that this method was founded upon the consideration of the small particles of quantity called augmenta momentanea & moments by M^r Newton, & infinitesimals indivisibles & differences by M^r Leibnitz. For there is no other way of resolving any of those sorts of Problemes, then by considering these particles of quantity. This consideration therefore might make him \lay aside his project of a Table of Tangents &/ begin to think upon the methods of Fermat Gregory Barrow & Slusius, who drew tangents by the proportion of the particles of lines. For he tells us that he found out the Differential method by considering how to draw Tangents by the differences of the Ordinates & \how/ thereby to render the method of Slusius more general (p. 88) & considered that as the summs of the Ordinates gave the area so their differences gave the tangents, & thence received the first light into the differential method (p. 104) And with Slusius he gave the name of differences to the moments of dignities. ({illeg}|T|ransact. Philosoph Num. 95.) And when he had found the Method, he saw that it answered <552v> to the description w^ch M^r Newton had given of his method in drawing of Tangents, in rendring Problems of Quadratures more general easy, & in bringing inverse Problems of Tangents to Equations & Quadratures, w^ch p. 88, 89, 90, 91, 93 in his letter of 27^th August 1676 he had represented impossible. p. 65, 88, 89, 90, 91, 93.

But when he sent his method to M^r Newton he forgot to acknowledge that he had but newly found it, & that the wand|t| of it had made him of opinion the year before that inverse problemes of Tangents & such like could not be reduced to Equations & quadratures. He forgot to acknowledge that by means of this invention he now perceived that M^r Newtons method w^ch extended to such Problems was much more general then he had could beleive the year before. He forgot to acknowledge that in the collection of Gregories Letters & Papers w^ch at his own request were sent to him at Paris by M^r Oldenburg & M^r Collins, he found the copy of M^r Newtons Letter of Decem. 10^th 16{illeg}|7|2, conteining his method of Tangents & representing it a branch or corollary of a general method of solving all sorts of problems & that the agreement of this method of Tangents w^th that of Slusius, put him upon considering how to enlarge the method of Slusius, {illeg}|b|y the Differences of the Ordinates. He forgot to acknowledge that M^r Newtons Letters of 13 Iune & 24 Octob. 1676, gave him any light into the method. And those things are now so far out of his memory, that he has told the world that when he published the elements of his differential method he knew nothing more of M^r Newtons inventions of this sort then what M^r Newton had formerly signified in his Letters, namely that he could draw tangents without taking away irrationals: w^ch Hugenius had signified that he could also do before he understood the infinitesimal method. p. 104, 107.

When M^r Newton was desired by M^r Leibnitz to tell him the original of his Theoreme {illeg}|f|or reducing binomials into series, he gave him an historical account of the invention but tooke care in the same Letter that the his narrative should not prejudice M^r {illeg} Mercator. [M^r Leibnitz on the contrary makes himself a w insists upon his own testimony {illeg} against M^r Keil & calls it imprudence & injustice to expect that a man of his age & credit & candour should defend himself against M^r Keil (p    ) & without producing any other testimony for himself then that his own would have the Society condemn him] M^r Leibnitz is in like manner to forbeare expecting that his own testimony for himself will be taken in evidence against M^r Keil. to the prejudice of M^r Keil. If he would have it beleived that he found the differentia differential method before the winter between the years 166|7|6 & 1677 he must bring better evidence then there is in these Letters & papers to the contrary, & forbeare to call those men imprudent & unjust that will not take his bare word for it.

But M^r Leibnitz denyes now that he had all this light from M^r Newton, or in the time of this correspondence learnt any thing more that M^r Newton had a general method or any further method of this kind then to draw tangents without sticking at surds (p. 104, 107) And in like manner after he had conversed w^th the Mathematicians at London, he wrote from Paris as if he had never heard of M^r Newton's method of series, & pretended to be the first inventor of the method of series two series for the circle (p. 38) & the next year when he received eight series for the circle from M^r Oldenburg \& M^r Collins/ & knew none of them to be his own, he forgot the receipt of them before the end of the year & communicated \to his friends at Paris, an opusculum upon/ one of them as his own \series/ to his (p. 42) & the spring following \he/ endeavoured to get the method from M^r Oldenburg & M^r Collins without the knowledge of M^r Newton (p. 45) tho by his own rule, if he should afterwards have forgot the receipt of the method & taken it for his own, M^r Oldenburg & M^r Collins were not to contradict him without authority from M^r Newton (p. 118.) And in recompence for M^r Newtons method, he promised \to send them/ the said opusculum, having forgotten that it was written upon one of the eight series w^ch he received from them the year before (p. 42, 45, 61) And when at his own request he received from M^r Newton the method of deriving reciprocal series from one another, tho he understood it with difficulty, yet he wrote back that he had found it before as he perceived by his old papers, but not meeting with with an elegant example of its use, had neglected it, p. 63, 96. And when he published the above mentioned series in the Acta Lipsiensia as his own (p. 97) he had not only forgot that he had received that series from M^r Oldenburg & M^r Collins but also that the collection of Gregories Letters had been sent him at his own request, by one of w^ch dated 15 Feb. 1671 Gregory had |the {sic}| sent that series to Collins, p. 25, 47.

As for those Gentlemen who have used the differential method & particularly the Marquess de l'Hospital, Mons^r Varignon, & fratres \the brothers/ Iacobus & Ioannes Bernoullius, there is nothing in these Letters & Papers which can affect them. They were strangers to the correspondence between M^r Leibnitz & M^r Oldenburg, they found \it was before their time, M^r Leibnitz handed/ the infinitesimal method \to them, they found it/ very usefull & they are much to be commended for the use & improvements that they have made of it.

that he & his friends allow that found the method above nine years before that is this is in y^e year 1674 before October 1675, & that he & his friends allowed M^r Newton to have invented the like principles by himself meaning perhap{s} before the date o writing of his Letter of Nov Octob 24. 1676. And but before M^r Leibnitz. And here his memory seems to \have/ fail|d| him again. For in the years 95 & 96 he was polishing his opusculum & when he wrote his letter of Aug 27 1696 (w^ch was but a month or two \six weeks/ before he left Paris) he was of opinion that inverse problems of tangents were not reducible to æquations or quadratures. {illeg} {illeg}t When M^r Newton was desired by M^r Newton Leibnitz to expla tell him the original of his binomial Theoreme for reducing Binomials into series, he repres gave him an historical acc^t of of {sic} the invention but tooke car at the same time \letter/ that Mercator who had printed bef{ore} him should not be prejudiced by the narration: M^r Leibnitz is in like manner to

When M^r Newton wrote the Analysis printed in the beginning of these papers \this Collection/, he had a method of resolving \finite/ equations into infinite series \ones/ & \of/ applying w|t|hose series \both finite & infinite æquations/ to the solution of Problemes by another method w^ch he calle|s|d the inf \means of/ the proportions of the incrementa momentanea or moments of \s growing or/ increasing quantities. These moment incrementa {illeg} M^r Newtons {sic} calls \particles &/ moments, & |M^r| Leibnitz Differences infinitesimals & indivisibles & differences. The increasing quantities M^r Newtons calls fluents & M^r Leibnitz summs, & the velocities of increase M^r Newton calls fluxions & exposes these fluxions by the moments of the flowing quantities. And whe in his That part of the method w^ch consists in resolving finite æquations into infinites ones & a{illeg} & trans M^r Newton at the request of M^r Leibnitz communicated \plainly/ to him in his Letters of 13 Iune & 24 Octob. 1676. ✝ < insertion from the bottom of the page > ✝ And having also \so far/ touched upon the second \other/ part so as to reckon \that/ the invention the invention thereof to be thereby made \was rendred \become/ sufficiently/ obvious, {illeg} \(pag 72 lin. 1)/ to secure it from being taken from him \ |till there| till|should| \{be}/ another /a\ occasion of explaining it should offer it self,/, {sic} he expressed it in cyphres ænigmati in cyphres after the manner used by Galilæus & h|H|ugenius upon other \like/ occasions. M^r Leibnitz contends for the first invention of this other part; M^r Keill that M^r Newtō M^r Leibnitz the next year put in for the invention of the|is| other part & now claims it, & M^rKeil asserts it to M^r Newton & {illeg} is favoured in his opinion by M^r the Committe. But there is nothing &c And having so far touched upon the other part as to reccon that {illeg} as to reccon that {sic} it was become sufficiently obvious to be found (p. 72 lin 1;) to secure it from being taken from him before there occasion should be offered of explaining it, he expressed it in cyphre after the manner used by Galilæus & Huygenius < text from f 555r resumes > M^r Keil represents M^r Leibnitz contends for y^e first invention of y^e other part of the method w^ch he calls the differential method: M^r Keil that M^r Newton was the first inventor thereof, & the opinion of th{illeg} C M^r Keil is favoured by the Committee. But there is nothing in these Papers w^ch can affect other persons abroad, who have received the method from M^r Leibnitz. They were strangers to the correspondence between M^r Leibnitz & M^r Oldenburg. They found the method usefull, & are much to be commended for y^e use & improvements that they have made of it.

Some notes are added \to the Letters/ to enable such Readers as want leasure, to compare the|m| Letters with more ease & see the sense of them at one reading.

p. 119 l 11. ad verbam [\non/ properavi] In epistola Aug. 27. 1676 \{illeg}/ properavit assere se coinventorem methodi serierum asserere. In epistola 21 Iunij 1677 \{illeg}/ properavit |methodū| infinitesimal{illeg} Newto|no| er{illeg}pere \{ca}pere de qua Newtonus tractatum ante annos quin scripserat./ In schedis tribus anno 1689 impressis properavit Principia Philosophiæ deflorare.

ib ad verba [plusquam nonum] Probandum est.

Pag. 119 lin 11. Ad verba [non properavi] notetur. In Epistola Aug. 27 1676 properavit se coinventorem methodi serierum proponere. In Epistola Iunij 21 1677 properavit methodum ut suam describere de qua Newtonus tractatum ante annos quin scripserat. In schedis tribus anno 1689 impressis properavit Propositiones principales Principiorum Philosophiæ ad calculum differentialem revocatas in lucem edere ut in Inventoris jura veniret.

Ib. Ad verba [plusquam nonum] notetur. Probandum est.