References to the original letters contained, or intended to be contained, in the Commercium Epistolicum

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Epistola D. Leibnitij ad D. Newtonum 17 Mar. 1693 st. n. data cujus Autographum in Archivis Societatis Regiæ asservatur

Ex Præfatione D. Wallisij in Operum suorum Volumina duo prima Oxonj|i|j ineunte anno 1695 scripta.

Ex Epistola D. Wally|i|sij ad D. Newtonum, Oxonij, Apr. 10, 1695, script{a} et in Archivis Regiæ Societatis asservata.

Ex Actis Eruditorum pro mense Iunio Anni 1696 pag. 257 Calculo quo differentiali Leibnitij affinem esse methodum fluxionum Newtoni - - - - - - - - - - non æque noscebantur

Ex Epistola D. Wallisij ad D. Leibnitium Oxonij Decem. 1. 1696 scripta et in tertio Volumine o|O|perum Wallisij edita. p. 654.

Ne Calculi Differentialis – – – interseruerim.

Qa

NB \Wallisius/ Epistolas Newtoni \olim/ acceperat \{et}/ ab Oldenburgo, eas Leibnitij accepit tandem e scrinijs Collinij. In Actis eruditorum verba Wallisij [nequis causetur de Calculo Differentiali nihil a me dictum fuisse] recitantur ide\o/ Editores Wallisium legerant; Newtonum tamen methodum suam anno 1676 methodum suam Leibnitio exposuisse, {illeg} tum ante annos decem nedum plures ab ipso excogitatam, non negant Dicunt tantum Leibnitium ubi rescripsit [id est anno 1677 mense Iunio] methodum habuisse, & Newtonium hoc in Principijs suis methodum {illeg} Ne\w/tonum in Scholio prædicto agnov Citant tantum Scholium Newtoni Contendunt tantum quod Newtonus publice et privatim agnovit Leibnitium tum cum inter ipsos, intercedente Oldenburgo, commercium intercederet, id est anno 1677, method \mense Iunio/ ubi Leibnitius rescripsit, methodum suam communicavit rescripsit, hanc methodum suam habuisse. Et quæ Leibnitius in \Newtonus {illeg} quidem/ Leibnitium quidem quæ in Epistola sua {illeg}|21| Iunij 1677 data descripsit \posuit/ tunc habuisse, nunquam negavit; quid amplius habu per ea tempora habuerit, no minime novit. Scholium vero ad Lemma secundum libri secundi Principiorum posuit, ut Lemma illud in quo methodus fluxionum fundatur, sibi {ass} anno 167 ante annum 1676 sibi cognitum fuisse assereret

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Schediasma {illeg}

Apographum Schediasmatis

Ex Tractatu parvo sub finem anni 1666 conscripto.

Ex Analysi per æquationes \numero terminorum infinitas/ anno 1669 a Barrovi ad Collinium missa.

Ex Epistola Newtoni 10 Decem 1672 ad Collinium missa

Ex Tractatu{illeg} de methodo|i|s serierum et fluxionum anno 1671 composito. /Fluentes designabo finalibus literis v, x, y, {illeg}|z| et celeritates quibus singulæ a motu generante fluunt et augentur designabo literis l, m, n, r respective.\

Prop|b|. 1. Relatione quantitatum fluentium inte {sic} se data, fluxionum relationem determinare.

Solutio. Equationem qua data {illeg} relatio exprimitur dispona secundum dimensionis alicuius fluentis quantitatis puta x, ac terminos ejus multiplica per quamlibet Arithmeticam Progressionem ac deinde per $\frac{\mathrm{m}}{\mathrm{x}}$. Et hoc opus in qualibet quantitate fluente quantitate seorsim institue. Dein omnium factorum summam pone nihilo æqualem et habebis æquationem desideratam.

Prob. 2. Exposita æquatione fluxiones quantitatum involvente invenire qu relationem quantitatum inter se.

Ex Epistola Newtoni 10 Decem. 1672 ad Collinium missa

Ex Epistola Newtoni 13 Iune {sic} 1676 ad Oldenburgum missa

Ex Epistola Leibnitij 27 Aug 1676 ad Oldenburgum missa

Ex Epistola Newtoni 24 Octob 1676 ad Oldenburgum missa

Ex Epistola Leibnitij 21 Iunij 1677 ad Oldenburgum missa

Ex Schediasmate Leibnitij in Actis Eruditorum pro mense Octobri 1684 edita Et hæc quidem initia sunt – – – pari facilitate tractabit.

Ex Lem. II Lib. II Principiorum Newtoni|.| et ejus Scholio.

Epistola D. Leibnitij ad D. Newtonum

|NB.| Cum D Leibnitius in Actis Eruditorum, edendo \ubi edidit/ Elementa methodi differentialis, Commercium illud silentio præteriret quod \quod/ annis 1676 & 1677 cum Newtono mediante Oldenburgo habuit\erat/, \silentio præteriret, Newtonus/ ne Lemma hocce non suum esse videretur, Newtonus \huic Lemmate/ Scholium sequens subjunxit.

Scholium

Epistola D. Leibnitij ad D. Newtonum 17 Mar 1693 st. n data, cujus Autographum in Archivis Societatis Regiæ asservatur.

Ex Præfatione Wallisij in operum suorum Volumina duo prima

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Extracts of Letters found in the custody of the R. Society & among the papers of M^r Iohn Collins, relating to the Collection of Mathematical Letters published by D^r Wallis in the third volume of his works.

|– 1| Out of a Letter of M^r Isaac Barrow to M^r Collins dated th at Cambridge y^e 20^th of Iuly 1669, & extant in the original hand-writing of the author.

A friend . . . . . . . by the next.

|– 2| Out of a Letter of M^r Barrow to M^r Collins dated y^e 31^th of Iuly 1669 & extant in the hand-writing of y^e author.

I send . . . . . . . . : your desire:

|– 3| Out of a Letter of M^r Barrow to M^r Collins dated y^e 20^th of August 1669 & extant in the hand-writing of the Author.

I am glad . . . . . . . Lord Brounker.

|– 4| Out of a Copy written in the hand of M^r Collins, found amongst his papers & collated w^th a MS borrowed of S^r Is. Newton supposed to be the original, the Analysis per æquationes infinitas hath been printed.

|– 5| Out of a Paper written in the hand of M^r Collins & directed to M^r Oldenburgh to be communicated to M^r Collins Slusius in y^e year 1669.

D^r Barrow hath communicated an universal Analytical method imparted to him by M^r Isaac Newton his Collegiate for the mensuration of the Areas of all such Curves & their Perimeters . . . . . . . . . . . . . . . . non opus est quicquam de ijs adjungere.

|– 6| Out of a Letter of M^r Collins to M^r Iames Gregory dated y^e 25^th of November 1669, a coyp of w^ch is extant in the hand writing of M^r Collins.

M^r Barrow hath . . . . . . . . . . . . . in the Hyperbola.

|– 7| Out of a Letter of M^r Iames Gregory to M^r Collins dated at S^t Andrews the 20^th of April 1670, & extant in the original hand of the author.

I cannot understand the series ye sent me of the circle, viz^t $2\mathrm{R}\mathrm{B}-\frac{{\mathrm{B}}^3}{3\mathrm{R}}-\frac{{\mathrm{B}}^5}{20{\mathrm{R}}^3}-\frac{{\mathrm{B}}^7}{56{\mathrm{R}}^3}-\frac{5{\mathrm{B}}^9}{576{\mathrm{R}}^7}-\mathrm{\&c}$. If this be the original, I take it to be no series.

|– 8| Out of an original Letter of M^r Iames Gregory to M^r Collins dated at S^t Andrews the 5^t of September 1670

I have read over M^r Barrow's . . . . . . . . . . . . not above twelve Propositions.

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|9 –| Out of an original Letter of M^r Iames Gregory to M^r Collins dated at S^t Andrews y^e 19^th of December 1670.

In my last to you I had not taken notice . . . . . . . . . . . . but perchance ye know more of them then my self.

|10 –| Out of a Letter of M^r Collins to M^r Iames Gregory dated the 24^th of December 1670, a copy of w^ch is extant in the hand writing of M^r Collins

When M^r Davy . . . . . . . . . . . as also the finding of the Area.

|11 –| Out of an original Letter of M^r Iames Gregory to M^r Collins dated at S^t Andrews the 15^th of February 1671.

As for M^r Newtons universal method, I imagin I have some knowledge of it both as to geometrick & mechanick curves. However I thak|n|k you for the series ye sent me & send you these following in requital. Sit radius r arcus a, tangens t, secans s erit $\mathrm{a}=\mathrm{t}-\frac{{\mathrm{t}}^3}{3{\mathrm{t}}^2}+\frac{{\mathrm{t}}^5}{5{\mathrm{r}}^4}-\frac{{\mathrm{t}}^7}{7{\mathrm{r}}^6}+\frac{{\mathrm{t}}^9}{9{\mathrm{r}}^8}$ erit $\mathrm{t}=\mathrm{a}+\frac{{\mathrm{a}}^3}{3{\mathrm{r}}^2}+\frac{2{\mathrm{a}}^5}{15{\mathrm{r}}^4}+\frac{17{\mathrm{r}}^7}{315{\mathrm{r}}^6}+\frac{3233{\mathrm{a}}^9}{181440{\mathrm{r}}^8}$, et $\mathrm{s}=\mathrm{r}+\frac{{\mathrm{a}}^2}{2\mathrm{r}}+\frac{5{\mathrm{a}}^4}{24{\mathrm{r}}^3}+\frac{61{\mathrm{a}}^6}{720{\mathrm{r}}^5}+\frac{277{\mathrm{r}}^8}{8064{\mathrm{r}}^7}$ . . . . . . . . . . . . . . . . . for I am little concerned if they be published |u|in|nd||er| any others name or not.

|12 –| Out of a Letter of M^r Collins to M^r Bertet at Paris, dated the 21^th of February 1670, a copy of the Letter being extant in the hand writing of M^r Collins.

To compose . . . . . . . . . . faln into the same method

|– 13| – Out of a Letter of M^r Collins sent into Italy to Borellius (the author of the Book de motionibus naturalibus a gravitate pendentibus) dated in December 1671, a copy of w^ch is extant in the hand writing of M^r Collins.

Kin{k}|c|khuysen's Introduction . . . . . . . . inserting the series.

|– 14| Out of a Letter of M^r Collins to M^r Vernon at Paris dated the 26^th of December 1671, a copy of w^ch is extant in the hand writing of M^r Collins.

D^r Barrow tells me . . . . . . . deductions can be the same.

|– 15| Out of a Letter of M^r Collins to M^r Strode dated y^e 26^th day of Iuly 1672 a copy of w^ch is extant in the handwriting of M^r Collins

As to curvilinear . . . . . . . . the first inventor thereof.

|– 16| Out of a Letter of M^r Collins to M^r Newton dated y^e 30^th of Iuly 1672 a copy of w^ch is extant in the hand \writing/ of M^r Collins

In preparing . . . . . . . . . . . . . he may be intent.

|– 17| Out of a Letter of M^r Newton to M^r Collins dated y^e 10^th of December 1672 found amongst the papers of M^r Collins in the original handwriting of the author, with a copy in the hand-writing of M^r Collins.

I am heartily glad . . . . . . . . . . . from describing it to him

I suppose Slusius his method \of Tangents/ will shortly appear abroad. When it comes over I'l{l} beg of you the trouble of transmitting a copy to me

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|– 18| A Letter of M^r Slusius to M^r Oldenburgh dated at Leige y^e 17^th of Ianuary 1673, & entred in the Books of the R. Society Num. 6 pag 11 was printed in the Transactions Num 90, & conteins his method of drawing Tangents.

|– 19| Out of a Letter of M^r Oldenburgh to M^r Slusius dated the 29^th of Ianuary 1673 in answer to the preceding Letter of Ian 17, & copied in the books of the R. Society Num 6 pag. 27

Statui Deo dante . . . . . . . . . . ut cum novissimis tuis comparare possis.

|– 20| A Letter of M^r Slusius to M^r Oldenburgh dated at Leige y^e 3^d of May 1673 & entred in the books of y^e R. Society Num. 6, pag. 111 was printed in the Transactions Num 95 & conteins the foundation of his method.

|– 21| A Letter of M^r Collins to M^r Newton dated 18^th Iune 1673 was printed by D^r Wallis.

|– 22| Out of a Letter of M^r Oldenburgh to M^r Slusius dated the 10^th of Iuly 1673 & entred in the books of the R. Society. Num. 6. p. 196.

Ex tibi, Vir illustrissime, impressum modum tuum demonstrandi Methodum tuam ducendi Tangentes . . . . . . . . a te acceptum iri confida

|– 23| Out of a Letter \of M^r/ Leibnitz to M^r Oldenburgh dated at London y^e 3^d of February 167$\frac{2}{3}$, extant in the hand-writing of the Author & entred in the Letter-books of y^e R. Society N. 6, pag 35|4|

Cum heri apud illustrissimum Boylium incidissem in clarissimum Pellium . . . . . . . . . . . . . . aut Triangulo-triangulares &c.

There are five other Letters of M^r Leibnitz to M^r Oldenburgh dated in the year 1673, the first of them at London y^e $\frac{10}{20}$ \of/ Febr. the rest at Paris \the/ 30 \of/ March |y^e| $\frac{16}{26}$ \of/ April, y^e $\frac{14}{24}$ \of/ May, & the 8^th of Iune, all of them except the last in the hand writing of the Author, & the first & three last entred in the books of the R. Society Num. 6, pag. 34, 101,        & 137.

Two other Letters of M^r Leibnitz to M^r Collins Oldenburgh dated at Paris y^e 15^th of Iuly 1674, & y^e 26 \16/ of October 1674, the first entred in the books of the R. Society, h{illeg} Num 6|7| pag 93 & 110, have been printed by D^r Wallis.

|24 –| Out of a Letter of M^r Leibnit Oldenburgh to M^r Leibnitz dated y^e 8^th of December 1674, extant in the hand of the author, & entred in the books of the R. Society Num. 7 p. 119, & written in answer to M^r Leibnitz Letter of Octob 16 1674

Quod de profectu in curvilinearum dimensione memoras . . . . . . . . . . . . . de eo quidem tibi gratulor.

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|25 –| Out of a Letter of M^r Leibnitz to M^r Oldenburgh dated at Paris y^e 30^th of March 1675, extant in the hand of the author & entred in the Books of the |R.| Society Num 7 p 213, & written in answer to M^r Oldenburghs Letter of Decemb. 8, 1674.

Scribis Cl. Newtonum vestrum habere methodum exhibendi quadraturas omnes . . . . . . . . . . . dignam. Addis tale quid Gregorio innotuisse.

|26 –| Out of a Letter of M^r Leibnitz Oldenburgh to M^r Leibnitz dated the 15^th of April 1675, & entred in the books of y^e R. Society N. 7, p. 216, & writ in answer to M^r Leibnitz's Letter of y^e 30^th of March. It was first writ in English by M^r Collins & is extant in his hand writing, beind {sic} dated Apr. 10 & sent to M^r Oldenburgh to be translated into Latin & sent to M^r Leibnitz

Dⁿ. Collinus, præmissa salute, quæ sequuntur remittit. Rimò Cl. Gregorium in postrema sua ad Illustrem Hugenium responsione seriem suppeditasse ad semicircumferentiam circuli inveniendam, quæ talis. Pone radium . . . . . . . . . . . . . . . . inventionem orbi mathematico patefaciat.

|27 –| Out of a Letter of M^r Leibnitz to M^r Oldenburgh dated at Paris y^e 20^th of May 1675 extant in the handwriting of the author entred in the books of the R. Society Num 7 p 235 & written in answer to that of M^r Oldenburgh of y^e 15^th of April 1675.

Out of a Letter of M^r Oldenburgh to M^r Leibnitz dated y^e 24^th of Iune 1675 & entred in the Books of the R. Society N 7 p. 243 being in answer to the former.

|28 –| Out of a Paper written in the hand of M^r Collins & found amongst his papers & corrected in some places in the hand of M^r Oldenburgh in Order to be copied \translated into Latin/ & sent to M^r Tschurnhause, whose answer was received by M^r Oldenburgh the 8^th of Iune 1675 & entituled Responsum ad scriptum D. Collinij de Cartesij inventis.

Des Cartes was accused of arrogance, as asserting that of all ways . . . . . . . . . of y^e 10^th of December 1672.

[Read here the Letter above]

|29 –| Out of a Letter of M^r Oldenburgh to M^r Leibnitz dated the 24^th of Iune 1675 & entred in the Books of the R. Society N 7 p 243 being in answer to y^e forme Letter of M^r Leibnitz dated y^e 20^th of May preceding.

Dominus Newtonus beneficio Logarithmorum . . . . . . . . potestas pura radicis quæsitæ

|30 –| Out of a|A| Letter of M^r Leibnitz to M^r Oldenburgh dated at Paris y^e 12^th of Iuly 1675, extant in the hand writing of the author, entred in the books of the R. Society N. 7 p 149 & written in answer to the M^r Oldenburghs Letter of the 24^th of Iune, hath been printed by D^r Wallis. In this Letter Parius is corruptly written for Darius.

|31 –| Out of a Letter of M^r Oldenburgh to M^r Leibnitz dated y^e 30^th of September 1675, extant in the hand of M^r Oldenburgh, & entred in the Books of the R. Society N. 7, p. 1{illeg} 159 & writ in answer to the former.

Tschurnhausius nuper Parisios hinc profectus est a te sine dubio salutavit. Scire cupis an dare nostratas geometrice possint dimensionem \Scriptum quoddam Belgicum Belga quidam Georgius Moor vocatus — \. . . . . . . . .// . . . . . . . . . . . . . . Collinius ipsi communicavit

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|32|1| —| Out of a Letter of M^r Oldenburgh to M^r Leibnitz dated y^e 30^th of September 1675, a copy of w^ch is extant in the hand of M^r Oldenburgh, \& w^ch was/ entred in the books of the R. Society N. 7 p 159 & was written in answer to the former.

Scriptum quoddam Belga quidam Georgius Mohr vocatus . . . . . . . . . . . . . . . . . . . . . Collinius ipsi communicavit.

|32 –| A Letter of M^r Leibnits to M^r Oldenburgh dated at Paris y^e 28^th of December 1675 extant in the handwriting of the author & entred in the books of the R. Society Num. 7 p. 189, hath been printed by D^r Wallis.

|33 –| Out of a Letter of M^r Leibnitz to M^r Oldenburgh dated at Paris y^e 12^th of May 1676 & found amongst the papers of the R. Society in the original hand of the author, with Notes on the backside in the hand of M^r Oldenburgh.

Cum Georgius Mohr Danus . . . . . . . . . . . . . . . . . satisfaciendi desiderio meo.

|34 –| Out of a Letter of M^r Collins to M^r Oldenburgh to be sent to M^r Leibnitz at Paris; a Copy of which dated the 14^th day of Iune 1676 was found amongst the papers of M^r Collins & in his handwriting.

In answer to M^r Leibnitz Letter of the 12^th of May . . . . . . . . . . . . . . . . . was found amongst but as dawning to noon day.

|35 –| Out of a Letter of M^r Collins to M^r David Gregory the brother of M^r Iames Gregory newly deceased, dated the 11^th of August 1676, a copy of w^ch is extant in the handwriting of M^r Collins.

I have drawn up an account of the Letter commerce . . . . . . . . . . . . . . . as himself acknowledgeth in his Letter of the 19^th of December 1670.

|–| Vpon the receipt of the Letter of M^r Leibnitz dated y^e 12^th of May, M^r Oldenburgh & M^r Collins sollicited M^r Newton for an account of his method of infinite series: which occasioned his two Letters dated 13 Iune & 24 October 1676, with M^r Leibnitz answers the first at Paris y^e $\frac{17}{27}$^th of August 1676, the second at Hanover the 21^th of Iune 1677, & a supplement to the second answer dated also at Hanover y^e 12^th of Iuly 1677 & a Letter of M^r Collins to M^r Newton dated the 5^t of March 167$\frac{6}{7}$, all printed by D^r Wallis.

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36. – Out of a Letter of M^r Tschurnhause dated at Paris 1 Sept. 1676, a copy of which is extant in the hand-writing of M^r Collins.

Expectabam cum desiderio Responsum . . . . . . . . . . . . operam navabunt.

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|1| Out of a Letter of M^r Leibnitz to M^r Oldenburgh dated at London 3^d Feb 167$\frac{2}{3}$ & extant in y^e hand of y^e author & \entred/ in y^e Letter books of y^e |R.| Society. N 6 pag 3. It concerens the differential method ascribed by D^r Pell to Mouton.

|2| Out of a Letter of M^r Leibnitz to M^r Oldenburgh dated at London $\frac{10}{20}$ Feb 1673 & extant in y^e hand of the author & \entred/ in the Letter book of y^e |R.| Society Num. 6. pag 34. |Desires to be a Fellow of y^e R. Society.|

|3| Out of a Letter of M^r Leibnitz to M^r Oldenburgh dated at Paris March 30^th 1675 & extant in y^e hand of y^e author, & copied in the books of the Society N 6

|4| Out of a Letter of M^r Leibnitz to M^r Oldenburgh dated at Paris $\frac{16}{26}$ Apr. 1673 & extant in y^e hand of y^e author & copied in the books of y^e R. Society Num 6. p. 101. Subtilissimo Collinio tam præclara communicanti obligatum me profiteor

|5| Out of a Letter of M^r Leibnitz to M^r Oldenburgh dated at Paris $\frac{14}{24}$ May 1673 & extant in y^e hand of the author & entred in the books of y^e R Society May {illeg} N 6

|6| Out of a Letter of M^r Leibnitz to M^r Oldenburgh dated at Paris 1 Iune 167$\frac{2}{3}$ returning thanks for his being elected fellow of y^e |R| Society. It is entred in y^e books of y^e Society N 6 p 137.

NB. D|M|^r Leibnitz was proposed to y^e R Society            & elected

|7| Out of a|A| Letter of M^r Leibnits to M^r Oldenburgh dated at Paris & extant 15^th 1674, extant in the hand of the Author & entred in y^e books of y^e R Society N 7 p 93. & \was/ printed by D^r Wallis in the 3^d Volume of his works

|8| Out of a \And another/ Letter of M^r Leibnits to M^r Oldenburgh dated at Paris $\frac{16}{26}$ Octob 1674, \&/ entred in y^e books of y^e R. Society N 7 p 110, & \was|er|e/ printed by D^r Wallis in y^e 3^d volume of his works.

|9| Out of a Letter of M^r Leibnitz to M^r Oldenburgh to M^r Leibnitz dated at 20^th of May \8^th of December 1674 extant in the hand of the author/ 1675|4| {sic} & entred in y^e books of y^e |R.| Society Num 7 p 235 p. 119. It conteins an answer to M^r Leibnitz Letter of Octob 16 1674.

|10| Out of a Letter of M^r Leibnitz to M^r Oldenburg dated at Paris 30^th March 1675, extant in the hand of the author {illeg}|&| entred in the books of y^e |R| Society Num 7 p. 213.|,| It conteins \& written/ a|i|n answer to M^r Oldenburghs Letter of 8^th Decemb. 1674.

|11| Out of M^r Oldenburghs \a/ Letter of M^r Oldenburgh to M^r Leibnits dated 15^th {illeg}|April| 1675 {illeg} {p} entred \extant in the hand of M^r Collins in English & entred in L./ /Latin\ in the books of y^e |R| Society N. 7, p. 216. It was writ in answer to M^r Leibnitz's Letter of 30 March {illeg}|167|5|, the English copy being dated 10^th April & the Latin 15^th Apr.|

|12| Out of a Letter of M^r Leibnitz to M^r Oldenburgh dated at Paris 20^th May 1675, extant in y^e hand of y^e author entred in the books of y^e |R.| Society Num 7 p. 27|3|5, & written in answer to y^e of M^r Oldenburgh of 15 Apr. 1675.

|13| Out of at {sic} Letter of M^r Leibnitz \Oldenburg/ to M^r Oldenburgh \Leibnitz/ dated [I think at Paris] {124} Iuly 1675 24 Iune 1675, & entred in the books of the |R.| Society N. 7 p. 243. being I think in answer to the former.

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|14| Out of a Letter of M^r Leibnitz to M^r Oldenburgh dated [at Paris] 12 Iuly 1675, extant in the original hand, entred in y^e Books of y^e R Society N 7 p 149 printed by D^r Wallis & written in answer to the M^r Leibnitz \Oldenbergs/ Letter of 24 Iune. In this Letter Parius is corruptly written for Darius.

|15| Out of a Letter of M^r Oldenburgh to M^r Leibnits 30 Sept 1675 \extant in the hand of M^r Oldenburgh &/ entred in the books of y^e R Society Num 7 p 159. It was writ in answer to y^e former. M^r Gregory was then Tschurnhause was then newly gone from London to Paris having received at Lo of M^r Collins at London one of Gregories series for the circle.

|16| Out of a Paper written in the hand of M^r Collins & found among\st/ his papers & corrected in some places in the hand of M^r Oldenburgh in Order to be copied & sent to M^r Tschurnhaus.

NB The answer \of M^r Tschurnhau/ to this paper was received by M^r Oldenburgh y^e 8^th of Iune 1675 & entituled Responsum ad scriptum Dⁿⁱ Collinij de Cartesij inventis.

|17| Out of a paper Letter tof M^r Leibnitz to M^r Oldenbergh dated at Paris 28 Decemb 1675, extant in the hand of the author, entred in the books of y^e Society Num 7 p 189 & printed by D^r Wallis.

|18| Out of a Letter of M^r Leibnitz to M^r Oldenberg dated at Paris 12 May 1676 extant in the found amongst the papers of the R. Society in the original hand of M^r Leibnitz with notes on the back side in the hand of M^r Oldenburgh, for returning an answer. This Letter occasioned the Letters of M^r Newton.

|19| Out of a Letter of M^r Collins to M^r Oldenberg to be sent to M^r Leibnitz in answer to his Letter of the 12^th of May. This A copy of this was found amongst the papers of M^r Collins & in his hand, & dated 14^th Iune 1676

|20| Out of M^r Newtons Letter of Iune {illeg}|1|3 1676.

|21| Out of the Answer of M^r Leibnitz dated 27 A at Paris $\frac{17}{27}$ Aug. 1676

|22| Out of a Letter of M^r Tschurnhause dated at Paris 1 Sept 1676 N.S

Out of M^r Newtons second Letter

|23| Out of {a} Letter of M^r Collins to M^r Newton dated 31 Aug. 1676 O.S.

|24| Out of M^r Collins Letter to M^r Newton dated 9 Sept 1676.

|25| Out of M^r Newtons Answer to M^r Leibnitz dated Octob 24. 1676.

|26| Out of M^r Leibnitz Letter to M^r Oldenberg dated at Amsterdam $\frac{18}{28}$ Novemb 1676. Printed by D^r Wallis in a Letter of M^r Collins dated 5 March 167$\frac{6}{7}$.

|27| Out of a Letter of M^r Leibnitz to M^r Oldenberg dated from Hannover 28 Ian 1677. Extant in y^e Original

|29| Out of a|n| \Original/ Letter of M^r Leibnitz to M^r Oldenb. dated from Hannover $\frac{20}{30}$ March 1677. Ou

30 Out of an original Letter of M^r Leibnitz to M^r Oldenburgh dated from Hannover 30 May 1677.

|28| Out of a Letter of M^r Collins to D^r Wallis dated y^e 14^th of February 167$\frac{6}{7}$ a copy of w^ch is extant in the hand writing of M^r Collins.

Out of a Letter of M^r Collins {illeg} to M^r Leibnits d Newton dated 5 March 167$\frac{6}{7}$ & published by D^r Wallis. Aderat hic . . . . . . prope diem missurum. R P.S. Exemplar Epistol{illeg}|{a}| . . . . . . laturus est.

<272r>

Out of an ancient paper \written/ in y^e hand of S^r I. N. dated written in y^e hand of M^r Newton & dated May the 16^th 1666.

|1| Out of a Letter of D^r Barrow to M^r Collins dated y^e 20^th of Iuly 1669 & extant in tha|e| \original/ hand of y^e author.

|2| Out of a Letter of D^r Barrow to M^r Collins dated y^e 31^th of Iuly 1669 & extant in y^e original hand.

|3| Out of a Letter of D^r Barrow to M^r Collins dated y^e 20^th of August & extant in y^e original hand.

|{illeg} |7|| Out of a|n| \original/ Letter of M^r Iames Gregory \to M^r Collins/ dated at S^t Andrews y^e 20^th of April 1670 & extant in the original hand of M^r Gregory.

|9 |8|| Out of a \original/ Letter of M^r Iames Gregory to M^r Collins dated at S^t Andrews y^e 15|9|^th of December & extant in y^e original hand

|6 9 |10|| Out of a|n| \original/ Letter of M^r Iames Gregory to M^r Collins dated at S^t Andrews y^e 15^th of February 1671

|4 |5|| Out of a|n| \original/ Letter of M^r Collins to M^r Barrow Iames Gregory dated y^e 25^t of November 1669 & \a copy of w^ch is/ extant in y^e hand \writing/ of M^r Collins.

|5|6|| Out of a|{n}| \original/ Letter of M^r Iames Gregory to M^r Collins dated y^e 5^t of September 1670

|9| Out of a|n| \original/ Letter of M^r Collins to M^r Iames Gregory dated y^e 24^th of December 1670, a copy of w^ch is extant in y^e hand \writing/ of M^r Collins.

|{illeg} |11|| Out of a Letter of M^r Collins to M^r Bertet dated y^e 21^th of Feb. 1670 M^r Bertet being then at Paris & a copy of y^e Letter being extant in the hand \writing/ of M^r Collins.

|{illeg} 12| Out of a Letter of M^r Collins \sent into Italy/ to Borellius (the author of the book de motionibus naturalibus a gravitate pendentibus) dated in December 1671, & extant in a copy of w^ch is extant in the hand \writing/ of M^r Collins.

|13| Out of a Letter of M^r Collins to M^r Vernon dated y^e 26^th of December 1671 M^r Vernon being then at Paris. A copy of y^e Letter is extant in y^e hand \writing/ of M^r Collins.

|15|4|| Out of a Letter of M^r Collins to M^r Strode dated y^e 26^th of Iuly 1672, a copy of w^ch is extant in the hand \writing/ of M^r Collins.

|14|5|| Out of a Letter of M^r Collins to M^r Newton dated y^e 30^th of Iuly 1672 & \a copy of w^ch is/ extant in the hand of M^r Collins

|1{illeg}|6|| Out of a Letter of M^r Newton to M^r Collins dated y^e 10^th of December 1672 found amongst y^e papers of M^r Collins in y^e original hand \writing/ of the Author {illeg} with a copy in the hand \writing/ of M^r Collins.

|16 17|8|| Out of a Letter of M^r Oldenburgh to M|on|^sr |Fr.| Slusius dated \at Leige/ y^e 29^th of Ianuary 1673 in answer to the Letter dated y^e 17^th of Ian 167$\frac{2}{3}$. This Letter is copied in y^e Letterbook of the R. Society N. 6. p. 27.

|1{illeg}|20|| Out of a Letter of {illeg} M^r Oldenburgh to M^r Slusius dated y^e 10^th of Iuly 1673 extant in y^e books of y^e R. Society. N 6 p 196.

|17| Out of a|A| Letter of M^r Slusius to M^r Old. dated at Leige 17^th Ian 1673 extant in y^e books of the Society N 6. p. 11 {illeg} |& printed in y^e Transactions, \Num 90/ conteins his method of drawing Tangents.|

|19| Out of a Letter of M^r Slusius to M^r Oldenb. dated at Leige 3^d May 1673 extant in the books of y^e Society. {illeg}|N|um 111 6. pag 111, & printed in y^e <272v> Transaction Num 95 p 6059, nisi quod Newtoni nomen subticetur

Out of a Letter of \M^r Collins to/ M^r David Gregory the brother of M^r Iames Gregory newly deceased dated 11^th August 1676, a copy of w^ch is extant in the hand of M^r Collins.

|4.| The treatise \A Copy of the Analysis per æquationes infinitas \book// mentioned in these Letters being \taken copied in the hand of M^r Collins/ found \in the hand of M^r Co/ < insertion from f 271r > llins < text from f 272v resumes > among|st| thes the |his| {illeg}|p|apers of M^r Collins & published by M^r Iones has been \of M^r Collins/ & in his hand writing & published by M^r Iones &|a|fter he had collated it with another MS borrowed of M^r Newton: by M has been \is/ to be examined, & compared w^th the print.

|4| Out of an original Copy written in the hand of M^r Collins & found amongst his papers & collated w^th \the original/ MS borrowed of M^r Newton by M^r Iones the Analysis per æquationes infinitas has been published by y^e said M^r Iones.

And the general method {illeg} {illeg} \Tract here mentioned conteining the general method/ for finding such series, is that w^ch M^r Newton {mentione}d \spake of/ in his Letter of Octob 24 1675 as written five years before

There are \five/ other Letters of M^r Old Leibnitz to M^r Oldenburgh \in the year 1673/ dated \the first/ at London $\frac{10}{20}$ Feb. 1673 & afterwards \the rest/ at Paris \30/ March 30, 16{illeg} Apr $\frac{16}{26}$, May $\frac{14.}{24.}$ & Iune 1^st 1673 all of them \(except the last)/ in the hand writing of the author & the first & {illeg}|the| last entred in the Letter books of the Society Num 6 pag 34, 101,       137

<273r>

|1.| Out of a Letter of D^r Is. \M^r Isaac/ Barrow to M^r Collins dated \at Cambridge/ y^e 20^th of Iuly 1669 & extant in the original hand of the Author.

|2.| Out of a Letter of D|M|^r Barrow to M^r Collins dated y^e 31^th of Iuly 1669 & extant in the hand-writing of the Author.

|3.| Out of a Letter of D|M|^r Barrow to M^r Collins dated the 20^th of August 1669 & extant in the hand-writing of the Author.

|4.| Out of a Copy written in the hand of M^r Collins & found amongst his papers & collated w^th a MS borrowed of S^r Is. Newton by M^r Iones & supposed to be the original, the Analysis per æquationes infinitas hath been printed.

|6| Out of a Letter of M^r Collins to M^r Iames Gregory dated y^e 25^th of November 1669, a copy of w^ch is extant in the hand writing of M^r Collins.

|6 |7|8||| Out of an original Letter of M^r Iames Gregory to M^r Collins dated at S^t Andrews y^e 5^th of September 1670.

|6 |7|| Out of an original Letter of M^r Iames Gregory to M^r Collins dated at S^t Andrews y^e 20^th of April 1670

|9| Out of an original Letter of M^r Iames Gregory to M^r Collins dated at S^t Andrews y^e 19^th of December 1670

|10| Out of a Letter of M^r Collins to M^r Iames Gregory dated y^e 24^th of December 1670 a copy of w^ch is extant in the hand writing of M^r Collins

|10|1|| Out of an original Letter of M^r Iames Gregory to M^r Collins dated at S^t Andrews y^e 15^th of February 1671.

|5| Out of a Paper written in the hand of M^r Collins & directed to M^r Oldenburgh to be communicated to M^r Slusius in y^e year 1669 before

|12| Out of a Letter of M^r Collins to M^r Bertet \at Paris/ dated y^e 21^th of Feb 1670, a copy of the Letter being extant in the hand writing of of {sic} M^r Collins.

|13| Out of a Letter of M^r Collins sent into Italy to Borellius (the author of the Book de motionibus naturalibus a gravitate pendentibus) dated in December 1671, a copy of w^ch is extant in the hand writing of M^r Collins.

|14| Out of a Letter of M^r Vernon Collins to M^r Vernon at Paris dated y^e 26^th of December 1671, M^r Vernon being then at Paris a copy of w^ch is extant in the hand writing of M^r Collins.

|15| Out of a Letter of M^r Collins to M^r Strode dated y^e 26^th day of Iuly 1672, a copy of w^ch is extant in the hand writing of M^r Collins.

|16| Out of a Letter of M^r Collins to M^r Newton dated the 30^th of Iuly 167{illeg}|2|, a copy of w^ch is extant in the hand \writing/ of M^r Collins

|17| Out of a|n| \{ori}g/ Letter of M^r Newton to M^r Collins dated the 10^th of December 1672 found amongst y^e papers of M^r Collins in the original hand writing of the author, {illeg} w^th a copy in the hand \writing/ of M^r Collins.

|18| A Letter of M^r Slusius to M^r Oldenburgh dated at Leige 17^th Ian 1673, ex|n|tant|red| in the books of y^e R. Society Num. 6 pag. 11, & \conteini|s|ng his method of drawing Tangents & was/ printed in the Transactions Num. 90. conteins his method of drawing Tangents \It conteins his method of drawing Tangents/

|19| Out of a Letter of M^r Oldenburgh to Mons^r Slusius dated y^e 29^th of Ianuary 1673 in answer to y^e \preceding/ Letter of Ian 17 & copied in the Books of the R. Society Num. 6, pag. 27.

<274r>

|20| Out of a|A| Letter of M^r Slusius to M^r Oldenberg dated at Leige y^e 3^d May 1673, entred in the books of y^e R. Society. Num 6 pag 111 & printed in the Transactions Num 95 p 6059 {sic}, conteins|i||ng| the Demonstration of his method was printed in the Transactions Num 95 p 6059.

|21| Out of a|A| Letter of M^r Collins to M^r Newton dated 18 Iune 1673 {illeg}|w|as printed by D^r Wallis.

|22| Out of a Letter of M^r Oldenburgh to M^r Slusius dated y^e 10^th of Iuly 1673, & exta entred in the books of y^e R. Society. Num 6. p 196.

|23|4|| Out of a Letter of M^r Collins to M^r David Gregory the brother of M^r Iames Gregory newly deceased, dated 11^th August 1676, a copy of w^ch is extant in the hand of M^r Collins.

|23| Out of a paper written in the hand of M^r Collins & corrected in some places in the hand of M^r Oldenburgh in order to be copied & sent to M^r Tschunhaus

<275r>

A Letter of M^r Leibnitz to M^r Oldenburgh dated at Paris the 28^th of December 1675 extant in the handwriting of the author & entred in the books of the R. Society Num 7 p 189, hath been printed by D^r Wallis.

Out {illeg} \of a/ Letter of M^r Leibnitz to M^r Oldenbergh dated at Paris the 12^th of May 1676 & found amongst the papers of the R. Society in the original hand of M^r Leibnitz with notes on the back side in the hand of M^r Oldenburgh

Cum Georgius Mohr Danus . . . . . . . . . . satisfaciendi desiderio meo.

Vpon the receipt of this Letter M^r Oldenburgh & M^r Collins sollicited M^r Newton . . . . . . . . . . . . . all printed by D^r Wallis.

Out of a Letter of M^r Collins to M^r Oldenberg to be sent to M^r Leibnitz in answer to his Letter of y^e 12^th of May. A copy of this Letter was found amongst the papers of M^r Collins & in his hand-writing & \being/ |&| \was/ dated y^e 14^th of Iune 1676

In answer to M^r Leibnitz Letter of y^e 12^th of May . . . . . . . . . . . . . . was but as dawning to noon day.

M^r Newton's Letter of Iune 13^th 1676 occasioned by M^r Leibnitz Letter of y^e 12^th of May, together w^th M^r Leibnitz answer of dated at Paris $\stackrel{27}{17}$ Aug. 1676 & M^r Newtons Reply dated 24 Octob 1676 & M^r Old Leibnitz answer dated at Paris 21 Iune 1677, & a Letter of M^r Collins to M^r Newton dated 5 Mat|r|tij 167$\frac{6}{7}$, have been printed by D^r Wallis.

Out of a Letter of M^r Tschurnhause dated at Paris 1 Sept 1676, a Copy of w^ch is extant in the hand of M^r C writing of M^r Collins.

|②| Out of a Letter of M^r Collins to M^r David Gregory the brother of M^r Iames Gregory newly deceased, dated 11^th of August 1676, a copy of w^ch is extant in the hand writing of M^r Collins.

I have drawn up an account of the Letter commerce . . . . . . . . . . . . as himself acknowledgeth in his Letter of 19^th December 1670.

|①| Out of a Letter of M^r Tschunhause dated at Paris 1 Sept. 1676, a copy of w^ch is extant in the hand writing of M^r Collins

Expectabam cum desiderio Responsum . . . . . . . . . . . . . . operam navabunt.

|③| A Letter of M^r Leibnitz to M^r Oldenburgh dated \at Hanover the/ 12^th of Iuly 1677\6/ 1677 hath & extant in the hand-writing of the Author, hath been printed by D^r Wallis.

<277r>

|①| Extracts out of M^r Collins Letters founda mong the papers of M^r Iohn Collins.

'Out of a Letter of M^r Collins to M^r Iames Gregory dated the 25 of November 1669

M^r Barrow hath resigned his \Lecturers/ place to M^r Newton Lecturers place to one M^r Newton of Cambridge whom he mentioneth in his Optick Præface as a very ingenious person, one who hath (before M^r Mercators Logarithmotechnia was extant) invented the same method & applied it generally to all Curves & divers ways to the circle whereof either for the whole or the parts it more readily rendreth the Area then it doth in the Hyperbola.

|③| 'Out of a Letter \of M^r Collins/ to M^r Strode dated y^e 26 of Iuly 1672

In September 1668 M^r Mercator published his Logarithmotechnia containing a Lemma

As to Curvilinear & solid Geometry that this should at last generally fall under a calculus Analyticus s News to the World, the Æquations being series consisting of an infinite number of terms (whereof few are sufficient for use) derived from the known properties of Curves. As to the Author & e|E|xcellency of this method take this account narrative.

In September 1668 M^r Mercator published his Logarithmotechnia containing a specimen of this Method in one only figure, to wit in the quadrature of the Hyperbola. Not long after forthwith gave his sense of it in the Transactions. Another of them I sent to {illeg}|D|^r Barrow at Cambridge who forthwith sent me up some Papers of M^r Newton who is since become D^r Barrows successor in the Mathematical Lecture there, by which & former communications made thereof by the Author to y^e Doctor, it appears that the said method was invented some years before by the said M^r Newton & generally applied. So that thereby in any Curvilinear figure proposed that is determined by one or more common properties, by the said Method may be obtained the quadrature or Area of the said figure accurately when it is quadraturæ capax, but always infinitely neare, the evolution or length of the curved line, the center of gravity of the figure, its round solids made by rotation & their surfaces & all performed w^thout any extractions of roots.

Both M^r Gregor Newton & M^r Gregory intend to write of this method in Latin, but M^r Gregory will not anticipate M^r Newton the first inventor thereof.

|④| Out of a \{large}/ discourse of M^r Collins dated 1675 \written to M^r Tschunhause in y^e year \1675// to diswade him from ascribing all th Mathematical Inventions to Deschartes Out of M^r Newton's Letter of the 10^th of December 1672 for \'Out of M^r Newtons Letter of the 10^th of December 1672/ copied by M^r Collins in the following

He \M^r Newton in his letter of Deceber {sic} 10^th 1672/ takes notice that he was informed by M^r Collins that Slusius & M^r Gregory were fallen into new methods of drawing Tangents & that D^r Barrows Lectures were of high esteem amongst the learned & in answer thereto saith thus I am heartily glad at the acceptance of our Reverend friend D^r Barrows Lectures find w^th foreign Mathematicians, & it pleased me not a little to understand that they \[Slusius & Gregory]/ are fallen into y^e same method of drawing Tangents with <278r> me. What I gh{illeg}esse their method to be you will apprehend by \this/ example. Suppose CB applied to AB in any given angle be terminated at any curve line AC. Putting $\mathrm{AB}=\mathrm{x}$, & $\mathrm{Bl}=\mathrm{y}$ let the r{illeg}elation between x & y be expressed by any Æquation as ${\mathrm{x}}^3-2\mathrm{x}\mathrm{x}\mathrm{y}+\mathrm{b}\mathrm{x}\mathrm{x}-\mathrm{b}\mathrm{b}\mathrm{x}+\mathrm{b}\mathrm{y}\mathrm{y}-{\mathrm{y}}^3=0$ whereby the Curve is determined. To draw the tangent CD the rule is this.

Multiply the termes of the Æquation by any Arithmetical Progression according to the dimensions of x & y; suppose thus.
$\begin{array}{c}010023\phantom{0}\\ {\mathrm{x}}^3-2\mathrm{x}\mathrm{x}\mathrm{y}+\mathrm{b}\mathrm{x}\mathrm{x}-\mathrm{b}\mathrm{b}\mathrm{x}+\mathrm{b}\mathrm{y}\mathrm{y}-{\mathrm{y}}^3=0.\\ 322100\phantom{0}\end{array}$
The first product shall be the Numerator, & the last divided by x the Denominator of a fraction w^ch expresseth the length of BD to whose end D the tangent CD must be drawn. The l{a}|e|ngth{t} BD therefore is $\frac{\mathrm{-2}\mathrm{x}\mathrm{x}\mathrm{y}+2\mathrm{b}\mathrm{y}\mathrm{y}-3{\mathrm{y}}^3}{3\mathrm{x}\mathrm{x}-4\mathrm{x}\mathrm{y}+2\mathrm{b}\mathrm{x}-\mathrm{b}\mathrm{b}}$.

This S^r is one particular or another a Corollary of a general method w^ch extends it self without any troublesome calculation not only to the drawing Tangents to all curve Lines whether Geometric or Mechanic or however related to streight Lines or to other curve Lines, but also to the resolving other abstruser kinds of Problems about the crookedness, Areas, Lengths, centers of gravity of Curves &c. Nor is it (as Huddens method de maximis et minimis, & consequently Slusius's new method of tangents as I presume) limited to æquations w^ch are free from surd quantities. This method I have interwoven with that other of working in æquations by reducing them to infinite Series. I remember I once occasionally told D^r Barrow when he was about to publish his Lectures that I had such a method of drawing Tangents, but some divertisment or other hindred me from describing it to him.

‡ < insertion from f 277v > ‡ 'A copy of this Letter was sent to M^r Tschurnhause A.C. 1675|4| or 1675 & another copy to the Royal Academy at Paris A.C. 1676, as will appear by the two next extracts.

< text from f 278r resumes >

After M^r Collins had recited the words of this Letter, he goes on this \in his next Articles of his discourse./ M^r Newton hath invented a new method of infinite series by w^ch he performs these things. 1 Streightens all Curves. 2 Squares all Curvilinear figures. 3 finds their centers of gravity, round solids, & the surfaces of those round solids whether upright or inclined 4 t|T|he second segments of such solids. 5 apply|i|es those series for the finding the roots of all equations. And for all this, being the main things wanting in Geometry, the World is no more beholden to Des Chartes then to Plato.

|{illeg} $\overline{)10}$| 'Out of a Letter of M^r Collins to M^r Iames \David/ Gregory the brother of M^r Iames Gregory newly deceased, dated 11 Aug. 1676.

I have drawn up an account of the Letter commerce, between & of all that was mathematical, {illeg}|t|hat I at any time received from your brother either in writing or discourse, with an intent to impart the same to the Royal Society whereof he was a member, where any friend of his may peruse the same or have <278v> it transcribed upon paying for the Duplicate w^ch will be half a quire of paper or more, whereof without omitting any thing material that was not controversial w^th Hugenius &c I can safely touch the altar. The French being desirous to have an account of his {illeg} attainments & Remains, you will see by what follows that I have endeavoured the same.

'And at the end of the Letter was added.

The Learned of the Royal Academy at Paris hearing of M^r Gregories death & being desirous to know what remains he left & to have an account of the doctrine of infinite Series, invented here, the following account was transmitted to them, & since to M^r David Gregory the surviving brother of the deceased.       As

1. As to y^e doctrine of infinite series, that Mercator in his Logarithmotechnia was the first person that published any thing of it to the world, applying it only to the Quadrature of y^e Hyperbola & the making of Logarithms without extraction of roots, the w^ch doctrine of his was illustrated by D^r Wallis in the Transactions of the Royal Society, & afterwards commented on & advanced by M^r Gregory in his Exercitationes Geometricæ in 1668.

2. The said Books being sent some few months after they were published sent to M^r Barrow at Cambridge, in return he gave answer that the said doctrine of infinite series was invented by M^r Isaac Newton above two years before & generally applied to all figures; w^th w^ch answer a Manuscript of M^r Newtonw^chs pains was transmitted to Collins & communicated to the Lord Brouncker President of the Royal Society & afterwards when M^r Barrow left his place as Mathematick Professor he recommended M^r Newton to it who read Lectures on the said Doctrine w^ch were put into the Vniversity publick Library at Cambridge.

3 Collins by means of Doctor|^r| Barrow grew acquainted w^th M^r Newton & kept a correspondence w^th him: from whom Collins receiving a Letter of 10 December 1672 about drawing Tangents to Geometric Curves out of the Equation that expresseth the {illeg} relation of the Ordinates to the Base subjoyns [it] in these words.

4 A blank space was here left in the MS for the words of the Letter.

5 Collins in divers of his Letters to M^r Gregory in y^e year 1669 acquainted him w^th M^r Newtons success in the said doctrine. To which M^r Gregory replied that himself had many series for the circle & desired earnestly to have some of M^r Newton's sent to him to compare with his own. And M^r Collins accordingly sent him some; w^ch M^r Gregory finding to be wholy different from his & much more easy & proper for calculation, did about December 1670, after much search, fall into M^r Newton's method, as himself acknowledgeth in his Letter of 19^th December 1670.

<279r>

Out of a M^r Iames Gregories Letter of y^e 19^th of Decemb^r 1670

|9–| In my last to you, I had not taken notice that \M^r/ Newtons series for the Zones of a circle (w^ch ye sent me a long time ago,) together with an infinite number of series of the like nature may be a consectarie to that w^ch I sent you concerning logarithms, viz Dato logarithmo invenire ejus numerum, vel radicem potestatis cujuscun puræ in infinitam seriem permutare. I admire much my own dulness that in such a considerable time, I had not taken notice of this, nevertheless that I had taken much pains to find out that series. But the truth is, I thought always (if so be it were a series) that I might fall upon it by some combination of my serieses for the circle, seing I had such infinite numbers of them; not so much as once desiring any other method. Your series a little produced is $2\mathrm{R}\mathrm{B}-\frac{{\mathrm{B}}^3}{3\mathrm{R}}-\frac{{\mathrm{B}}^5}{20{\mathrm{R}}^3}-\frac{{\mathrm{B}}^7}{56{\mathrm{R}}^5}-\frac{5{\mathrm{B}}^9}{576{\mathrm{R}}^7}-\frac{7{\mathrm{B}}^{11}}{1408{\mathrm{R}}^9}-\frac{21{\mathrm{B}}^{13}}{6656{\mathrm{R}}^{11}}-\frac{11{\mathrm{B}}^{15}}{5120{\mathrm{R}}^{13}}-\mathrm{\&c}$. Iisdem etiam positis, erit arcus (cujus sinus B) $=\mathrm{B}+\frac{{\mathrm{B}}^3}{6{\mathrm{R}}^2}+\frac{3{\mathrm{B}}^5}{40{\mathrm{R}}^4}+\frac{5{\mathrm{B}}^7}{112{\mathrm{R}}^6}+\frac{35{\mathrm{B}}^9}{1152{\mathrm{R}}^8}$. I could give you several other serieses of this nature, but perchance ye know more of them then my self.

This is in another place

|– 11| Out of M^r Gregories Letter of 15^th Feb. 1671.

\Since my last to you I have received three of yours, one dated Decem. 15^th another Decem. 24^th, the third Ianuary 21^th./ As for M^r Newton series his universal method I imagin I have some knowledg of it both as to Geometrick & Mechanick Curves. However I thank you for the series ye sent me & send you these following in requital Sit radius=r, arcus=a, tangens=t, secans=s, et erit $\mathrm{a}=\mathrm{t}-\frac{{\mathrm{t}}^3}{3{\mathrm{r}}^2}+\frac{{\mathrm{t}}^5}{5{\mathrm{r}}^4}+\frac{{\mathrm{t}}^7}{7{\mathrm{r}}^6}+\frac{{\mathrm{t}}^9}{9{\mathrm{r}}^8}$, erit $\mathrm{t}=\mathrm{a}+\frac{{\mathrm{a}}^3}{3{\mathrm{r}}^2}+2{\mathrm{a}}^{5}15{\mathrm{r}}^4+17{\mathrm{a}}^{7}315{\mathrm{r}}^6+\frac{3233{\mathrm{a}}^9}{181440{\mathrm{r}}^8}$, et $\mathrm{s}=\mathrm{r}+\frac{{\mathrm{a}}^2}{2\mathrm{r}}+\frac{5{\mathrm{a}}^4}{24{\mathrm{r}}^3}+\frac{61{\mathrm{a}}^6}{720{\mathrm{r}}^5}+\frac{277{\mathrm{a}}^8}{8064{\mathrm{r}}^7}$. Sit nunc tangens artificialis=t, et secans artificialis=s & integer quadrans=q, erit $\mathrm{t}=\mathrm{e}+\frac{{\mathrm{e}}^3}{6\mathrm{r}}+\frac{{\mathrm{e}}^5}{24{\mathrm{r}}^4}+\frac{61{\mathrm{e}}^7}{5040{\mathrm{r}}^6}+\frac{277{\mathrm{e}}^9}{72576{\mathrm{r}}^8}$. Sit nunc secans artificialis <280r> ${45}^{\mathrm{gr}}=\mathrm{s}$, sit s+l secans artificialis ad libitum, erit ejus arcus $=\frac{\mathrm{q}}{2}+\mathrm{l}-\frac{{\mathrm{l}}^2}{\mathrm{r}}+\frac{4{\mathrm{l}}^3}{3{\mathrm{r}}^2}-\frac{7{\mathrm{l}}^4}{3{\mathrm{r}}^3}+\frac{14{\mathrm{l}}^5}{3{\mathrm{r}}^4}-\frac{452{\mathrm{l}}^6}{45{\mathrm{r}}^5}$; erit $2\mathrm{a}-\mathrm{q}=\mathrm{t}-\frac{{\mathrm{t}}^3}{6{\mathrm{r}}^2}+\frac{{\mathrm{t}}^5}{24{\mathrm{r}}^4}-\frac{61{\mathrm{t}}^7}{5040{\mathrm{r}}^6}+\frac{277{\mathrm{t}}^9}{72576{\mathrm{r}}^8}$. Ye shall here take notice that the rad{illeg}|iu|s artificalis $=0$, & that when ye find $\mathrm{q}>2\mathrm{a}$, or the artificial secant of 45^gr be greater then the given secant, to alter the signes & go on in the work according to the ordinary precepts i|o|f Algebra. Sit ellipsis cujus unus semiaxis=r, alter=c. Ex quolibet semiaxis puncto Curvæ Ellipticæ puncto demittatur in semiaxem r recta perpendicularis=a. Erit area Elliptica perpendiculari {illeg}|a| adjacens $=\mathrm{a}+\frac{{\mathrm{r}}^2{\mathrm{a}}^3}{6\mathrm{c}4}+\frac{4{\mathrm{r}}^2{\mathrm{c}}^2{\mathrm{a}}^5-{\mathrm{r}}^4{\mathrm{a}}^5}{40{\mathrm{c}}^8}+\frac{8{\mathrm{c}}^4{\mathrm{r}}^2{\mathrm{a}}^7+{\mathrm{r}}^6{\mathrm{a}}^7-4{\mathrm{c}}^2{\mathrm{r}}^4{\mathrm{a}}^7}{112{\mathrm{c}}^{112}}+\frac{64{\mathrm{c}}^6{\mathrm{r}}^2{\mathrm{a}}^9-48{\mathrm{c}}^4{\mathrm{r}}^4{\mathrm{a}}^9+24{\mathrm{c}}^2{\mathrm{r}}^6{\mathrm{a}}^9-5{\mathrm{r}}^8{\mathrm{a}}^9}{1152{\mathrm{c}}^{16}}$ Si determinetur Ellipseos species, series hæc simplicior evadet. Vt si $\mathrm{c}=2\mathrm{r}$, f{illeg}|o|ret curva prædicta $=\mathrm{a}\frac{{\mathrm{a}}^3}{96{\mathrm{r}}^2}+\frac{3{\mathrm{a}}^5}{2048{\mathrm{r}}^4}+\frac{113{\mathrm{a}}^7}{1158752{\mathrm{r}}^6}+\frac{3419{\mathrm{a}}^9}{7549747{\mathrm{r}}^8}$ Reliquis vero manentibus si curva prædicta esset Hyperbola, prædicta quo series ei inserviret, si omnium terminorum partes affirmentur, & negentur totus terminus tertius, totus quintus, septimus &c in locus imparibus. I thank you heartily for your good advice as to the publication of my notions & for your civil profer. I would be very sorry to put you to so much trouble. I have no inclination to publish any thing save only to reprint my quadrature of the circle, & to add some little trifles to it. As to my method for finding the roots of all equations; one series gives only one root, f|b|ut for every root there may be infinite numbers of serieses. There is some industry required to enter the series & know what root it belongeth relateth to: but it is like I may entertain you at more length with this matter hereafter. Ye need not be so close handed of any thing I send you. Ye may communicate them to whom ye will, for I am little concerned if they be published \under/ any anothers name or not.

<281r>

|⑧| Out of a Letter of M^r Collins to M^r Oldenberg to be communicated to M^r Leibnitz dated Apr. 10^th 1675

There is a Letter w^thout a date written in the hand of M^r Collins & corrected in some places in the hand of M^r Oldenberg to be copied & sent to M^r Oldenberg Leibnitz. It was written It mentions that M^r Kersies first Volume of Algebra was \then/ half finished & by consequence \it was written/ in the year 1673. [It mentions that M^r Leibnitz \said he/ had a method of summing up the ranks of fractions whose Denominators are figurative numbers. {illeg} And that M^r Collins could do the like] And there is another Letter in the hand of M^r Collins & corrected in some places in the hand of M^r Oldenberg to be copied & sent to M^r Oldenberg Leibnits. This \be copied & sent to M^r Leibnits in answer to one of his wherein he de/\sired to know what was doing here. This/ second Letter is dated 10^th of April 1675 & has some things in it relating to the contents of the former Letter. And these two Letters shew that M^r Oldenberg \Leibnitz/ had at that time a correspondence w^th M^r Oldenberg & by his means w^th M^r Collins. M^r Leibnitz in his Letter w^ch interceded The second Letter was directed \by M^r Collins/ to M^r Oldenberg to be sent that he might send a copy of it to M^r Leibnits \who had written to know what was doing in England \here/. For/ It begins thus /The second Letter begins thus.\

In answer to the desire of M^r Leibnitz I here impart what I observe first to be in print. M^r Gregory in his last answer to Hygens hath given a s|S|eries for finding the Semi-circumference of a circle, and it is this. Put the Radius = r, half the side of a square inscribe in a circle = d & the difference between the Radius & the side of the square = e. Then is the circumference {illeg} equal to $\frac{4\mathrm{r}\mathrm{r}}{2\mathrm{d}-\frac{\mathrm{e}}{3}-\frac{{\mathrm{e}}^2}{90\mathrm{d}}-\frac{{\mathrm{e}}^3}{756{\mathrm{d}}^6}-\frac{23{\mathrm{e}}^4}{113400{\mathrm{d}}^3}-\frac{263{\mathrm{e}}^5}{7484400{\mathrm{d}}^4}-\mathrm{\&c}}$ in infinitum. Which series may be so produced that it differs from the semicircle less then any quantity assignable.

This he published after M^r Mercators Logarithmotechnia was extant, w^ch as soon as it came forth, I sent to D^r Barrow & he observing an infinite series therein used for the making of the Logarithms writ back that the said method had been sometime before found out by his successor M^r Neton & generally applied to all Curves & their portions, as well Geometrical as Mechanical & sent up some specimina thereof

|{y, z}| Putting the Radius unit, if you give x the sine, to find z the arch, the series is $\mathrm{z}=\mathrm{x}+\frac{{\mathrm{x}}^3}{6}+\frac{3}{40}{\mathrm{x}}^5+\frac{5}{112}{\mathrm{x}}^7+\frac{35}{1152}{\mathrm{x}}^9+\mathrm{\&c}$ in infinitum. And extracting the root of this Equation in species, if you give z the arch to find x the sine the series is $\mathrm{x}=\mathrm{z}-\frac{1}{6}{\mathrm{z}}^3+\frac{1}{120}{\mathrm{z}}^5-\frac{1}{5040}{\mathrm{z}}^7+\frac{1}{362880}{\mathrm{z}}^9-\mathrm{\&c}$ And these Series are easily continued ad infinitum. |By| T|t|he first out of the sine of 30^deg. <281v> Van Coullens numbers are easily raised.

In like manner if you put the Radius R, & B the sine of an arch, the zone between the Diameter & a Chord parallel thereto is $=2\mathrm{R}\mathrm{B}-\frac{{\mathrm{B}}^3}{3\mathrm{R}}-\frac{{\mathrm{B}}^5}{20{\mathrm{R}}^3}-\frac{5{\mathrm{B}}^9}{576{\mathrm{R}}^7}-\frac{7{\mathrm{B}}^{11}}{1408{\mathrm{R}}^3}-\mathrm{\&c}$

And the same series changing the sines of the second fourth & sixt terms &c serves to give the Area of the zone of the equilateral Hyperbola, viz^t $2\mathrm{CFGA}=2\mathrm{R}\mathrm{B}+\frac{{\mathrm{B}}^3}{3\mathrm{R}}-\frac{{\mathrm{B}}^5}{20{\mathrm{R}}^3}+\frac{{\mathrm{B}}^7}{56{\mathrm{R}}^5}-\frac{5{\mathrm{B}}^9}{576{\mathrm{R}}^7}+\frac{7{\mathrm{B}}^{11}}{1408{\mathrm{R}}^9}-\mathrm{\&c}$

Again

If you give the Radius R & the versed sine, or sagitta a & would find the Area of the segment cutt off by the chord line: for $2\mathrm{R}\mathrm{a}$ and the segment {illeg} {illeg} segm^t$=\frac{4\mathrm{b}\mathrm{a}}{3}-\frac{2{\mathrm{a}}^3}{5\mathrm{b}}-\frac{{\mathrm{a}}^5}{14{\mathrm{b}}^3}-\frac{{\mathrm{a}}^7}{36{\mathrm{b}}^5}-\frac{5{\mathrm{a}}^9}{352{\mathrm{b}}^7}-\frac{7{\mathrm{a}}^{11}}{832{\mathrm{b}}^9}-\mathrm{\&c}$. And the intire Arch $=2\mathrm{b}+\frac{{\mathrm{a}}^2}{3\mathrm{b}}+\frac{3{\mathrm{a}}^4}{20{\mathrm{b}}^3}+\frac{5{\mathrm{a}}^6}{56{\mathrm{b}}^5}+\frac{35{\mathrm{a}}^8}{576{\mathrm{b}}^7}+\frac{63{\mathrm{a}}^{10}}{1408{\mathrm{b}}^9}+\mathrm{\&c}$ T These two Series are M^r Gregories since he fell into this method, w^ch was some years after he heard M^r Newton had generally applied the same. Since w^ch he hath sent up the like for attaining the natural tangents out of their arch, & the converse.

For example, put the Radius = R the arch = a & the Tangent T, then is $\mathrm{T}=\mathrm{a}+\frac{{\mathrm{a}}^3}{3{\mathrm{r}}^2}+\frac{2{\mathrm{a}}^5}{15{\mathrm{r}}^4}+\frac{17{\mathrm{a}}^7}{315{\mathrm{r}}^6}+\frac{3233{\mathrm{a}}^9}{181440{\mathrm{r}}^8}+\mathrm{\&c}$

And conversly out of the tangent to find the arch thereto
$\mathrm{a}=\mathrm{t}-\frac{{\mathrm{t}}^3}{3{\mathrm{r}}^2}+\frac{{\mathrm{t}}^5}{5{\mathrm{r}}^4}-\frac{{\mathrm{t}}^7}{7{\mathrm{r}}^6}-\mathrm{\&c}$

And now you see this done you may easily observe \beleive/ that by y^e same method you may as easily out of the arch find the Logarithm sine or tangent without finding the natural. You may also beleive that this method is applied to the{illeg} streightning or rectification of all kinds of curve lines, as particularly to the Linea Q quadratrix & to the finding the area of that figure, w^ch hath not been performed before by any method whatsoever. With some further toyle in the calculation it is applied to the finding the areas of the surfaces of inclined round solids, as also to the finding the solidities of the second segments of round solids. For instance, if a Conoid were cut by a plane passing through the Base thereof, this may be called the first segment, & if this portion be cut again by a plane erect to the former cutting plane, the portion so cut is hereby intended to be a segment.

It is also applied to the finding the roots of pure powers or high affected equations, so that out of any number without aid of Logarithms you may raise any power per saltum, & out of any power tho never so high find the root thereof or any mean assigned between it & Vnit. M^r Gregory hath with much labour prepared infinite Series to be generally fitted to the respective adfected Potestates of any Equation proposed, so that an Algebraist being furnished with his stock, will quickly fit a series for the finding of any one of the roots of any equation proposed after <282r> it be known on which side of which Limit the root falls. But this he hath not yet communicated, nor \[been]/ pressed to do it, he willingly leaving it to M^r Newton to make the first discovery of this new method of infinite s|S|eries to the world.

The next year \(May 12^th)/ M^r Leibnitz wrote back an Answer to this Letter, & M^r Collins having signified that the series of M^r Newton were easily continued, he desired \in this Answer/ to know how they were continued, & signified \& had found a method of reducing all quantit/ also that he had contemplations \also/ on the same argument \& had found a method of reducing any quantities into the like series./. {sic} Whereupon both M^r Oldenberg, & M^r Collins wrote pressing Letters to M^r Newton to communicate his method in writing at large in writing to be made publick least he should be prevented by M^r Leibnitz. And M^r Collins wrote a \the following/ Letter to M^r Oldenberg to be sent to M^r Leibnits in answer to his of May

|⑨.| Iune 14 1676

M^r Oldenburgh

S^r

In answer to M^r Leibnits Letter of y^e 12 of May be pleased to return that as to the first Series the Coefficients whereof are $\frac{1}{6}\frac{3}{40}\frac{5}{112}\frac{35}{1152}$, they are thus made $\frac{1\times 1}{2\times 3}=\frac{1}{6}$, & $\frac{1}{6}\times \frac{3\times 3}{4\times 5}=\frac{3}{40}$, and $\frac{3}{40}\times \frac{5\times 5}{6\times 7}=\frac{5}{112}$, and $\frac{5}{112}\times \frac{7\times 7}{8\times 9}=\frac{35}{1152}$, and $\frac{35}{1152}\times \frac{9\times 9}{10\times 11}=\frac{567}{25344}$, and so ad infinitum; so that he may conceive this series not to fall short in its elegancy to the converse w^ch he the rather commends. His contemplations on the same argument being on other foundations, will be very acceptable to us, & [we] wish they may exceed our beleif as to the excellency of this method, w^ch is so extensive as to sweep away all difficulties, so that I apprehend M^r Gregory &c to be of opinion that all that was known before it was but as dawning to noon day. &c.

At the same time that M^r Collins wrote this Letter, M^r Newtō sent another to M^r Oldenburg dated the {illeg} {illeg} Iune 13^th in answe{r} to the his Letter & the Letter of M^r Collins above mentioned \the Letter above mentioned sent him of him & M^r Collins./ And in this Letter he described the his method of infinite series giving many instanc|l|es \examples/ thereof. M^r Leibnits was directed to communicate the same to M^r Tschurnhause, & about the about the middle \tenth/ of Nove September M^r Newton received their answers \directed to M^r Oldenburgh/ & the answer of M^r Tschurnhause \to M^r was directed to M^r Oldenburgh & {illeg}/ began in this manner

Parisijs Primo Septembris 1676.

Nobilissime Vir

Expectabam cum desiderio Responsum cum aliquot abhinc mensibus ad te literas meas transmiseram. Sed nec ex modo datis colligere licet has receptas fuisse. Interim admodum oblectatus fui hisce conspectis quæ ad D. Leibnitium exaratæ, maxime me tibi devinxisti quod me participem volueris facere tam ingeniosarum <282v> inventionum et promotionis Geometriæ tam pulchræ quam utilis statim cursim eas pervolvi ut viderem num forte inter hasce series infinitas existeret ea qua ingeniosissimus D. Leibnitius Circulum, imo quamvis Sectionem Conicam (Centro in finita distantia gaudentem) quadravit, tali ratione ut mihi{illeg} persuadeam simpliciorem viam nec quoad linearem Constructionem nec numeralem expressionem nunquam visum iri; qui hisce porro insistens generalem adiuvenit methodum figuram quamvis datam in talem rationalem transmutandi, quæ per solum Inventum (admodum præstans meo judicio) D. Mercatoris ad seriem infinitam posset, reduci. Sed hac de matera cum ipse non ita pridem mentem suam declaravit non opus [est] ut prolixior sim. Verum ut ad specimina perquam ingeniosa Newtoni revertar, hæc non potuere non mihi placere tam ob utilitatem qua se tam tale ad quarumvis quantitatum dimensiones ac alia diffilia {sic} enodanda in Mathematicis extendunt, quam ob deductionem harum a fundamentis non minus generalibus quam ingeniosi|e|s derivatam. Non obstante quod existimem ad quantitatem quamvis ad infinitam seriem æquipollentem reducendi, fundamenta adhuc dari et simpliciora et universaliora &c.

M^r Leibnitz in his Answer dated              described his method of transmutations for reducing all quantities into infinite series

After the arrival of these Letters M^r Leibnitz came to London for a week \in October/ & returned back before the receipt of M^r Newton's L next Letter w^ch was dated Octob 24^th Octob. 1676, & received not this Letter till the spring following. In this Letter M^r Newton mentioned the Tract communicated by D^r Barrow to M^r Collins & lately published by M^r Iones & another \larger/ tra{illeg}|c|t written soon after \about y^e year 1671/ i|o|n y^e same subject but never method of infinite series & the method of fluxions together. & d represented that the method of fluxions was the same w^th that of Slusius f{illeg}|o|r drawing of tangents but much more general, not stopping at fractions or surds s {sic} & extending to abstruser Problems, & of this method he gave some examples, but concealed the name of fluxions. And this gave occasion to M^r Leibnits in his Letter of

to signify that he had also such a method & to describe the same. And This was the first discovery of the Differential method made \by M^r Leibnitz/ to us in England|,|. The two L & whether he found it before his journey to London is a question The two Letters of M^r Newton w^th the Answers of M^r Leibnitz were published by D^r Wallis in the third volume of his works.

<283r>

In the year 1684 M^r Leibnitz published the elements of his Calculus without taki making any m

As M^r Leibnitz in the year 1682 published the Series of Gregory without for the circle without acknowledgi{t}{illeg}|ng| \mentioning/ that he had received it from M^r Oldenburg in y^e year 1675 & again in the year 1676; \& that in the year 1676 he had received it again in a copy of Gregories Letter dated 15 Feb 1671:/ so in the year 1684 he published the elements of his Calculus differentialis without aski mentioning the letters w^ch had passed between him & me in the years 1676 & 1677 about this method. Which omission put me upon writing the Scholium in my second upon the second Lemma of the second book of the Principia Philo\so/phiæ Mathematica fo{illeg}|r| asserting that Lemma to my self. M^r Leibnitz a{illeg} in the year 1684 published only the Elements of the Calculus Differentialis & applied them to – – – –

Out of the Account given of D^r Wallis's works by the Editors of \works in/ y^e Acta eruditorum pr for Iune 1696 pag 257, 258.

Cæterum ipse Newtonus non minus candore quam præclaris in rem mathematicam meritis insignis, publice et privatim agnovit, Leibnitium tum cum (interveniente celeberrimo viro Henrico Oldenburgio . . . reciprocum.

NB Nunquam agnovi D. Leibnitium ante ante annum 1676 nedum anno 1676 calculum suum differentialem habuisse. Nihil unquam de hoc re agnovi nisi quod in ejus Epistola 21 Iunij 1677 data inven{illeg}{illeg} {illeg} annum 1677{:} nihil \agnovi de impentione hujus calcul{i} ante annum 1677: nihil amplius de hoc methodo D^no Leibnitio per/ cum interea semper assererem me anno 1676 ad ipsum \D. Lei/ de L Methodo fluxionum \ad D. Leinitium/ scripsisse; Seriem {illeg} quam|vis| rem non tanti esse crederem ut de ea contenderem, ne {de} {illeg} de hac re jam scripsisses|m| nisi accusatus. D Leibnitium series|m| Seriem pro arcu \inveniendo/ ex Tangente assumpta Gregorianam esse et \anno 1675/ ab Oldenburgo {eum} \ad/ Leibnitio|um| communicatā \missam/ per ea tempora minims noveram. Leibnitium vero metho pro seriebus method methodum generalem habuisse per ea tempora habuisse nunq{q}|u|am agnovi. Methodus per transmutationem figurarum non est methodum|s| \generalis/ serierum nedum methodus serierum proprie dicta, sed præpartio tantum qua series p^r divisiones|m| ab \{illeg} |nonnunquam|/ obtineri queant quæ melius abs per extractionem radicum prodeunt.

NB I never allowed that M^r Leibnitz had the Differential Method either before in|or| in the year 1677, nor that he then \in those days/ knew more of it then what he described \to me/ \wrote of it |described| that year/ in his Letter of 13 Iune \13/: 167{2} but that [& \I/ always affirmed \represented/ that in the year 1676 I wrote to him about the method of fluxions. And when he sent me the Series of Grego\r/ies|ry| as his own I did not know that M^r Oldenburg had sent to him that series the year before. And did I I never acknowledged that \nor accounted/ the method of Transmutations was a general m [w^ch he sent me in his Letter of 27 Aug. 1676, was] a general method for Series, nor kno|e|w that the series w^ch he then sent to me was sent to him the year before by M^r Oldenburg & invented in the beginning of the year 170|67|1 by M^r Iames Gregory. Here The Editors \{illeg}/ complain that D^r Wallis made \|forbad| said|yi|ng/ so little mention of the Differential Method while he had said so much of the method of fluxions, But nei And & attributed said & yet excuse{d} his|m| D^r because perhaps he might not know all this. And indeed the D^r had recc|e|id|v|ed from M^r Old. in y^e year 1676 \copies/ my two Letters dated \that year in/ {13} Iune |13| & 21 Octob. {illeg} but had \not/ yet seen the Letter of M^r Leibnitz dated 27 Iune 1677 He procured it afterwards from the Library of M^r Collins \wherin{g} the differential method was described/.

Part of a Letter of D^r Wallis &c

Neque Calculi – – – – inseruerim

Part of a Letter of M^r Leibnitz to the Editors of the Acta Erudiorum {sic} published in these Acta for Iune 1697 pag. 24|5|4.

Dⁿ Iohannes Wallisius, Vir magnis in Geometriam meritis insignis celeberrimus – – – – si non omnia sit sciscitatus.

NB. After this, s|S|everal Letters passed \afterwards/ between M^r Leibnitz & the Doctor & in a friendly manner. And in all this neither the Editors nor the Doctor M^r Leibnitz himself found fault with the Doctor for saying that in my two Letters writ in the <283v> year 1676 I had explained to M^r Leibnitz the Method found by me ten years before that time or above, nor contradicted it what he thus affirmed. And what D^r Keill hath since affirmed amounts to nothing more. Neither did M^r Leibnitz was any answer was given

Part of the book of M^r Nicolas Fatio de Duillier {illeg} intituled Investigatio Geometrica solidi rotundi in quod minima fiat resistentia & published in the year 1699

Quæret forsan Cl. Leibnitius, unde mihi cognitus sit iste Calculus quo utor Ejus equidem fundamenta universa – – – – – – – quæ ipse evolvi, instrumenta.

NB The method of converging series by divion {sic} & extraction of roots was found by me in the beginning of the year 1665 & the Theoreme for extraction reducing the digniti powers, & roots dignities of of & & roots of converging Series \Binomials/ into series, set down in my Letter of 13 Iune 166|7|6, I found at the sme time. M^r D^r Barrow soon after save notice of this to M^r Collins. M^r Mercator My Lord Brounker for in the beginning of the year 1668 published the squaring of the Hyperbola by this series $1+1$ $\frac{1}{\mathrm{x}}\pm \frac{1}{2\mathrm{x}}$ $\mathrm{x}\pm \frac{\mathrm{x}\mathrm{x}}{2}+\frac{{\mathrm{x}}^3}{3}\pm \frac{{\mathrm{x}}^4}{4}+\frac{{\mathrm{x}}^5}{5}\pm \mathrm{\&c}$. M^r Mercator {illeg} {in|a|} few months after published the same Series de a Demo the invention of the same series by \a/ division of D^r Wallis, but went no further. M^r Collins in the y ear 1669 gave notice of my M{illeg} Method to M^r Iames Gregory who after a years study found it out & sent back many series to M^r Collins in a Letter dated 15 Feb. 167$\frac{0}{1}$. M^r Collins communicated \these & mine/ freely to his friends both at home & abroad. M^r Leibnitz in the year 1673 carried Meractors {sic} series with him into France but was not yet acquainted w^th the higher Geometry. In the year 1674 he pretended to have found a series for the arc whose sine was given. The next year he M^r Oldenburg sent to him the Series of Gregory for the Arc whose tangent was given. The third year in Iune {illeg} M^r Oldenburg sent to him the Le a copy of the Letter of Gregory dated 15 Feb. 1671 wherein this series, th & the same year in August M^r Leibnitz sent back \his method of Transmutations with/ that Series to O M^r Oldenburg to be communicated to me as a recompense for my method of series. & in the year 1682 printed it a|i|n \the/ Acta Lips. without letting us know that he had received it \twice/ from England.|;| In his Letter & soon after claimed my method of arbitrary Series, a method which he \might/ f{illeg}ind out by himself but not so early. And is the history of the method of Series. M^r Bernoulli observes that these series are for {illeg}dert{illeg}s|o{m}| Determining such quantities as transcend v{illeg}|u|rgar equations & therefore are by M^r Leibnitz called transcendent, & M^r N{e} Leibnitz himself boasts that he has reduced transcendent equations to quantites {sic} to æquations by the Differential method: So The name of transcendent is new but the methods of reducing them to equations are fifty years old. And because they are the subject bot {sic} of the Method of series & of that of fluxions therefore I began about fifty years ago to joyne those two methods & make but one \general/ Analysis of them {illeg}|both|, putting

<284r>

And And if you read the latter part of the Letter of M^r Leibnitz dated 28 May 1697 you will find that M^r Leibnitz claimed the differences|i|mprovements \to himself/ & therefore \left/ the first invention to me. For he claimed the expression of transcendent equations quantities {illeg} & of such Curves a Mechanical Curves by equations & then added Primus autem, ni fallor etiam Exponentiales æquationes introduxi. And thus M^r Leibnitz & the Doctor were agreed to allow me the first invention where \so far as/ the methods agreed & that the improvements made by M^r Leibnitz should be his; t|a|s the Methods of C improvements made by Cartes to the Method of Vieta, belonged to Cartes. For my part

<284v>

P.S.

When the Committe of the Royall Society published the Commercium Epistolicum, the Letters & Papers in my custody were not produced. Among them were the following Letter of M^r Leibnitz \dated 7 Mar 1693/ & a Letter of D^r Wallis dated Apr. 18. 1695 both which upon a fresh occasion have been \were two years ago/ produced, & delivered into th \examined &/ left in the Archives of the Royal Society. The first shews what was his opinion of this matter before he knew \my symbols or/ any thing more of the Method of fluxions then what he learnt from my Letters & Papers writ in or before the year 1676 or from the Principia Philosophiæ Mathematica, the second \(compared with the Pr{illeg}|e|face to the D^rs works)/ shews what opinion the English \Mathematicians/ had of the|se| same method \these/ the|i|ngs when the differential method began \first/ to be celebrated in Holland under {illeg} as invented by M^r Leibnitz. The{re}\se/ are \These Letters are/ hereunto annexed. The first of th{e}se two Letters & part of the second are hereunto annexed.

Illustri Viro ISAACO NEWTONO Godefridus Gulielmus Leibnitius S.P.D.

Quantum tibi scientiam rerum Mathematicarum – – – – – – Vale. Dabam Hannoveræ $717$ Martij 1693.

{illeg} |1.| Part of a Letter of D^r Wallis to M^r Newton S^r \M^r/ Isaac Newton dated Ap from Oxford Apr. 10. 167|9|5

I wish you would print the two large Letters of Iune & October 1676. I had intimation from Holland as desired there by yo^r friends that somewhat of that kind were done, because your notions of fluxions pass there with great applause by the name of Leibnitz's Calculus differentialis. I had this intimation when all but part of the Preface to this Volume was printed off: so that I could only insert (while the Press stayd) that short intimation thereof which you there find. You are not so kind to yo^r reputation (& that of the Nation) as you might be when you let things of worth lye by you so long till others carry away the reputation which is due to you. I have endeavoured to do you justice in that point; & am now sorry that I did not print those two Letters verbatim.

3. Part of a Letter of D^r. Wallis to M^r Leibnitz dated from Oxford Decem. 1. 1696 .|&| |published in the third Volume of the Doctors works. pag. 653, 654.|

{illeg} Neque Calculi Differentialis vel nomen audivisse me memini, nisi postquam utrum Volumen absolverant operæ, erat Præfationis (præ figendæ) postremum folium sub Pr{illeg}\æ/lo, ejus typos jam posuerant typothetæ. Quippe tum me monuit amicus quidam (harum rerum gnarus) qui peregre fuerat, tum talem methodum in Belgio prædicari, tum illam cum Newtoni methodo Fluxionum quasi coincidere. Quod fecit ut (transmotis typis jam positis) id monitum inseruerim.

1|2|. Part of the Præface to the two first Volumes of D^r Wallis's Works published \at Oxford/ in April 1695.

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NB. The complaint of the Editors against {illeg} of the Acta Eruditorum against D^r Wallis was that the said so much of the Method of fluxions & so litte|l|e of the Differential Method. But they excuse him because perhaps he might not muc know but little of that Method. And indeed he had received copies of My two Letters of Iune & October 1676 from M^r Oldenburg but but {sic} had not yet seen the Letter of 13|27| Iune 1677 wherein M^r Leibnitz began first to describe his method. The D^r app in his Letter to M^r Leibnitz approves of this excuse & M^r Leibnitz sends the Doctors approbation to the Editors of the Acta. But neither the Editors nor M^r Leibnitz himself complained of the Doctor for saying that in my Letters of Iune & October 1676 I explained to M^r Leibnitz the Method of fluxions found by me ten years before those days or above, nor contradicted the D^r therein.

NB. \In/ All this relates to a complaint of the Editors of \of {sic} the Acta Eruditorum/ that D^r Wallis with in the second book of his works \& Preface to the first/ had said so much of the Method of fluxions & so little nothing of the Differential M. & in his Preface had but just mentioned it as {illeg}dent with the D \& represented it/ the same method with that of fluxions. And indeed the D^r \in the year 1676/ had received from M^r Oldenburg copies of my two Letters of Iune & October \whereby h{e} had notice of the method of Fl./ but had not \yet/ seen the Letter of M^r Leibnitz dated 27 Aug|Iun|ust \Iune/ 1677, nor heard of the Differential method till his Preface was the last sheet of his Preface was in the Press. But neither the Editors nor M^r Leibnitz himself complained of the Doctor for saying that in my Letters of Iune & October 1676 I had explained to to {sic} M^r Leibnitz the Method found of Fluxions found by me ten years before those days or above. {illeg} {S} {illeg} And whereas the Doctor|^r| Wallis had represented his Letter \On the contrary in answer to his saying that the Differential method/ was the same w^th that of Fluxions excepting the f{illeg} different forms of Speaking, M^r Leibnitz replied that as the Analysis of Vie{illeg}ta & Cartes {illeg}|a|greed in the main but had some differences so mine & his might differ in some things. {illeg} He did not claim the whole, nor det|n|y the antiquity of mine, but only contended that there were some different|c|es. In And whether he did this with a designe least the whole should be ascribed to me And who was the interloper let the Reader judge. And whether his meaning was that he |I| had improved hi{illeg}|s| method or \that/ {illeg} he had improved mine let the Reader judge. For my part I do not know that I have improved this method since the year 1676 1676 1672, & I have writ an this to avoid the title of an Interloper. or used it \but on the contrary I have been forgetting it ever since/ since the writing of the Principia Philosophiæ except when others have put me upon it, & In And \& meddle not with M{illeg}/ & should not have meddled with this dispute had it not been for wiping off the odious reflexion cast upon me in point of \probity &/ candor & particularly that of being an interloper. {illeg}

NB In these Extracts, \In these Extracts/|,| D^r Wallis represented that in his Preface that the \two/ Methods were the same exceps in forms of speaking & that in my Letters of Iune & October 1676 I had explained to M^r Leibnitz the|i||s| method of fluxions found by me ten years before or above & that he inserted this admonition nequis causetur de calculo differentiali nihil ab ipso dictum esse. The Editors of the Acta Eruditorum com reciting these last words complained that |y^e| Doctor had said that I {illeg} in the year 1676 I explained to M^r Leibnitz the method of fluxions found by me ten years before those days or above. M^r Leibn This was not denyed \by them/ in those days|.|; b|B|ut whereas the Doctor had said that the Me Differential Method was the same with mine except in forms of words M^r Leibnitz replied that the Methods indeed were of the same kind & therefore they he called them by the common name of the infinitesimal method; but yet as the Analysis of Vietas & that of Cartes are called by the common name of Analysis speciosa & yet differ in some things, so the perhaps M^r N my method & his may differ in some things. For my part \I had fluxional Equations & knew how to apply them to such Curves as Cartes Calls mechanical & to resolve them vnto converging Series fifty years ago &|I| had then the methods both direct of fluxions both inverse & I d./ I do not know that I have improved this method since the year 1672|1|, but on the contrary I have been forgetting it ever since I wrote the Principia Philosophiaæ mathematica, & [should not have medled with this dispute had it not been for wiping off the reflexion cast upon me in point of probity] was {illeg}|ve|ry unwillingdly \have been very unwillingly/ drawn into this dispute{illeg} {illeg}.

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Out of the Account given of the Works of D^r Wallis in the Acta Eruditorum for Iune 1696 pag. 257, 258.

Cæterum ipse Newtonus non minus candore quam præclaris in rem Mathematicam meritis insignis, publice & privatim agnovit, Leibnitzium tum cum (interveniente celeberrimo Viro Henrico Oldenburgio Bremensi Societatis Regiæ Anglicanæ tunc Secretario) inter ipsos (ejusdem jam tum Societatis Socios) commercium intercederet, id est jam fere ante annos viginti et amplius calculum suum differentialem series infinitas & pro ijs quo methodos generales habuisse; quod Wallisius, in Præfatione Operum factæ inter eos communications|i|s mentionem faciens, præterijt, quoniam de eo fortasse non satis ipsi constabat. Cæterum differentiarum consideratio Leibnitziana, cujus mentionem facit Wallisius (ne quis scilicet, ut ipse ait, causaretur de calculo differentiali nihil ab ipso dictum fuisse) meditationes aperuit, quæ aliunde non æque nascebantur. Est enim differentia Analyticum quiddam & calculi capax, & quod rei caput est, summæ reciprocum.

NB. |In| also my Letters of 13 Iune & 24 October I {illeg} 1676 I affirmed that I had the method of fluxions some years before that time, but never allowed that M^r Leibnitz had the Differential Method before the year 1677, nor in those days \did I/ know an more of it then what he represented that year in his Letter of 13 Iune, nor \did I/ allowed the method of Transmutations to be a general method of series, nor \nor do \now/ look upon it to be a method of Series or any thing more then a particular method of Transmutations/ \like those of Gregory & Barrow, nor/ did I then know that the series w^ch he then sent to me was sent to him by M^r Oldenburg the year before & invented by M^r Iames Gregory in the year 1671.

Part of a Letter of D^r Wallis to M^r Leibnitz dated from Oxford Decem. 1. 1696, & published in the Doctor's Works. Vol. 3. pag. 653, 654.

Ne calculi Differentialis vel nomen audivisse memini, nisi postquam utrum Volumen absolverant operæ, erat Præfationis (præfigendæ) postremum folium sub Prelo ejus typos jam posuerant typothetæ. Quippe tum me monuit amicus quidam (harum rerum gnarus) qui peregre fuerat, tum talem methodum in Belgio prædicari, tum illam cum Newtoni methodo fluxionum quasi coincidere Quod fecit ut (transmotis typis jam positis) id monitum inseruerim. |✝ Sed| < insertion from f 285v > ✝ Sed et ante monueram, Algebræ Prop. 95 pag. 389 (quod solum potui) Leibnitium & Tschurnhausium talia meditatos; sed quæ ego non videram. (Nudum vidi.) Et sicubi forte viderim literas G.G.L. nesciebam quem illæ virum indicabant. Extant credo plura in Actis Lipsicis, sed quæ ego non vidi Vti nec credo vidisti Brunkeri quadraturam Hyperbolæ, quæ extat in Transactionibus Londinensibus. Mihi condonari potest h ac ætate (qui annum Octagesimum superavi) si non omnia{illeg} sciscitarer. Noveram quidem jam{illeg}|d|udum (et indicavi) de rebus hujusmodi nonnulla Te meditatum esse Tibi cum Newtono (mediante Oldenburgio) intercessisse Literas quasdam tuas sed quas Ego nondum vidi, nec scio quales fuerint: erat Oldenburgius diu mortuus, ut non potuerim ab illo sciscitari. Rogabam quidem per literas Newtonum nostrum, ut si eas penes se haberet earum mihi copiam faceret literarum; sed retulit ille se non habere. (Et quidem perijsse credo flammis inopinato correptas, cum plurimis Newtoni scriptis, meliori luce dignis: et nisi per me stetisset, perijssent etiam Newtoni literæ.) Eo animo rogabam ut tuas illas cum Newtoni literis junctim ederem. Id etiamnum, si ferat occasio, facturus forte sum, modo mihi dignaberis earum copiam facere.

NB. At the bottom of the leaf \page/ D^r Wallis noted made this Note. Obtinui tandem ex Schediasmatis Collinij. Copies of my two Letters he had received long \twenty years/ before from M^r Oldenburgh himself, as he mentions|e||d| in the second Volume of his works, pag. 368, & thence \he/ learnt what he said of those two Letters in his Preface. < text from f 285r resumes >

Part of a Letter of M^r Leibnitz to the Editors of the Acta Eruditorum, published in these Acta for Iune 1697. pag. 254.

Dn. Ioh. Wallisius, Vir magnis in Geometriam meritis celeberrimus, occasione quarundam suarum ad me literarum, significavit nonnulla, circa recensionem operum suorum Actis Iunij superioris anni insertam monenda, occurrisse. Nempe in ea videri oblique insinuari, quasi nostrorum inventorum parcius meminerit; quæ autem viderit candide meminisse \retulisse./ Calculi autem differentialis ne nomen quidem audivisse antequam absoluto utro Volumine operum, pro ultimo præfigendæ Præfationis folio jam typos posuissent typothetæ. Ac tum demum ab amico qui peregre fuerat, <286r> mentionem ejus apud se factam & typis transmotis a se inser{t}am, sibi autem hac ætate, cum annum octagesimum superaverit, condonavi posse, si non omnia sit sciscitatus.

Part of a Letter of D^r Wallis to M^r Leibnitz Apr. 6. 1697, to D^r Wallis 1|2|9 Mar. 1697. st. n

— Novo calculi genere effeci ut etiam Algebram transcendentia Analysi subjicientur, et Curvas quas Cartesius a Geometria male excluserat, suis quibusdam æquationibus explicare docui. —  Quæ inter \De te/ < insertion from f 285v > \—  De te autem queri nunquam mihi in mentem venit; quem facile apparet/ \nostra, in Actis Lipsicis, prodita, non satis/ |vidisse|  Quæ inter Oldenburgum et me commutatæ sunt l|L|iteræ, quibus aliqua accesserant a Domino Newtono, excellentis ingenij Viro, varijs meis itineribus & negotijs ab hoc studiorum genere plane diversis, vel periere, ut alia multa, vel jacent in mole chartarum aliquando excutienda digerenda, ubi a necessarijs occupationibus vacatio erit; quam mihi tam subito, quam vellem, promittere non possum. < text from f 286r resumes >

Part of a Letter of D^r Wallis to M^r Leibnitz Apr. 1697

Quis sit ille \tuus/ calculus Differentialis

Veniam interim exorare debeo si locorum distantia fecerit ut eruditissima tua scripta & inventa minus ego sciverim aut intellexerim quam vellem; et quidem quis sit ille tuus Calculus Differentialis non satis mihi comportum sit; nisi quod mihi nuper nunciatum {illeg}|e|st, eum \cum/ Newtoni Doctrina Fluxionū quasi coincidere.

Part of a Letter of M^r Leibnitz to D^r Wallis 28 May. 1697

Methodum Fluxionum profundissimi Newtoni, cognatam esse methodum meæ Differentiali, non tantum animadverti postquam opus ejus et tuum et tuum prodijt; sed etiam professus sum in Actis Eruditorum, et alias quoque monui. Id enim candori meo convenire judicavi non minus quam ipsius merito. Ita communi nomine designare soleo Analyseos Infinitesimalis, quæ latius quam Tetragonistica patet. Interim. quem{illeg}|a|dmodum et Vietæ et Cartesiana methodus Analyseos speciosæ nommine venit; discrimina tamen nonnulla supersunt; ita fortasse et Newtoniana et Mea differunt in nonnullis. Mihi consideratio differentiarum et summarum in seriebus numerorum primam lucem affuderat cum animadverterem Differentias Tangentibus et Summas Quadraturis respondere.

NB. For understanding the last words, they may be compared with the beginning of his Letter of A 27 Aug. 1677, where he writes thus

Part of a Letter of D^r Wallis to M^r Leibnitz Iuly 30 1697.

Optaverim item ut tibi vacet tuum Calculum Differentialem & Newtono suum fluxionum methodum, justo ordine exponere; ut quid sit utri commune, & quid intersit discriminis, & utram distinctius, intelligamus.

NB. In these Extracts, D^r Wallis represented that the two Methods were the same except in forms of speaking & that in my Letters of Iune & October 1676 I explained to M^r Leibnitz this Method found by me ten years before or above & that he inserted this admonition nequis causetur de calculo differentiali nihil ab ipso dictum esse. The Editors of the Acta Eruditorum reciting these last words complained that the Doctor had said so little of the Differential method; but neither they nor M^r Leibnitz himself complained that the Doctor had said that in the year 1676 I explained to M^r Leibnitz the|i|s method found by me ten years before that year or above. This was not then denyed by them M^r Leibnitz only put a further difference between the methods then the Doctor did, saying that they were indeed of the same kind, but & therefore he called them \both/ the infinitesimal method: but yet as the Analysis of Vieta & that of Cartes are are called by the common name of Analysis speciosa & yet differ in some things; so perhaps my method & his may differ in some things. By this its manifest that he did not yet begin to claim the whole. And its as pl{illeg} manifest that the Doctor claimed for me not the differences but the common method, & was not contradicted in his claim. For my part I had fluxional Equations & \fifty years ago and then/ knew how to apply them to such Curves as Cartes calls Mechanical \& to almost all sorts of Problems/ & to red|s|olve them into infinite series converging Series, fifty years ago & have \neglected &/ been forgetting this method ever since I wrote the Principia Philosophiaæ Mathematica, & \am now so \very/ we are of these things that I/ should never have resumed the consideration of these things \thereof/ had it not been to avoid the \wipe of/ aspersions of being an interloper & something worse {illeg} in matter of fact.

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|⑤|Out of a Manuscript written in the hand of M^r Collins & corrected in some places in y^e hand of M^r Oldenburge A.C. 1675 \in order to be/ \to be copied to be &/ & sent to M^r Tschurnhause A.C. 1675 \sometime before Iune 1675/ to diswade him from ascribing \too much/ to Des Cartes. all {illeg} mentions in Geometry

NB. Des Cartes was likewise accused of Arrogance as asserting that of all ways & methods possible he had fallen on y^e best. How true soever the relation is I know not but tis certain the method of tangents hath been since much advanced by M^r Newton & M^r Gregory. Thus saith M^r Newton in his Letter of 10 December 1672. [Here was a large blank in the MS for entring the words of the Letter & a reference to another page where the Letter w{illeg}|a|s entred, & after the blank the MS thus proceeds.]

N. M^r Newton hath invented a new method of infinite series by w^ch he performs these things. 1 Streightens all Curves. 2 Squares all curvilinear figures. 3 Finds their centers of gravity, round solids, the surfaces of those round solids whether upright or inclined. 4 The second segments of such solids. 5 Applies those series for finding the roots of all Equations. And for all this, being the main things wanting in Geometry the World is no more beholden to Des Cartes then to Plato. Hei {sic} in pag 39 saith that Ratio quæ inter rectas & curvas existit non cognita est, nec etiam ab hominibus ut arbitror cognosci queat. But by his good favour D^r Barrow hath given a ratio between infinite. And before him M^r William Neile the first of mortals streigned the quadratick Parabolick Line on supposition of the quadrature of the Hyperbola. And S^r Christopher Wren streighned the Cycloidical line on supposition of the Quadrature of the circle. And M^r Newton in the Circle hath performed this. If any sine chord tangent secant or versed sine be given he hath fitted series for finding the length of the arch, w^ch are derived from the touch line of a circle & Cavaleirs method of indivisibles. Yea & giving the arch he finds the converse of the Logarithm Sine Tangent or Secant thereto without finding the natural. And this converse work for the natural a|A|dscripts or Inscripts ariseth from Vietas general method of extracting the roots of adfected Æquations not in numbers but in species: a Doctrine that will be difficult to those that understand it not. And to ascribe this to Desc|C|artes were very injurious to others; Du{illeg}|{L}|aurens acknowledging in his Præface that to apply Algebra to those parts of Mathematicks there had been no medium yet found.

|⑥|The answer of M^r Tschurnhause to this paper is entred{illeg} in the same MS in the hand of M^r Collins with a note y^t it was received the 8^th of Iune 1675. The title of the answer is Responsum ad Scriptum Dⁿ Collinij de Cartesij inventis For M^r Oldenburg used to note the day when he rec{ip} For when M^r Oldenburg received papers of moment he used to note the day write upon them the day of his receiving them. The title to this Answer is Responsum ad scriptum Dⁿ. Collinij de Cartesij inventis. And the Article w^ch relates answer of M^r Tschurnhause to the three \four/ Articles above recited is in these words

NB. Hic rursum nonnulla circa superius a me dicta confirmantur et fateor egregium esse illud Inventum Newtoni et ejusmodi promotionem Geometriæ qualem primo loco superius tradidi \[i.e. facilem]/, proinde nihil contrarium sententiæ meæ afferri. Esse vero eam ejusmodi promotionem qua nempe speciali ex generali methodo Geometriæ Cartesianæ eliciuntur, ipse Objectionum author affirmat dum ait, Hanc si non solvisset forte infecta mansisset. Vnde inferri jure potest nec Dⁿ Newtonum forte egregium illud inventum exinde nobis exhibere <288r> potuisse, proinde constare quantum Cartesio nostro debeamus.

|Ad| {B}|M|. Si intelligatur hic per præstantiam alicujus rei præ alia ad quam detegendam plus virium ingenij est opus, loquamur solum de Methodo quam ex Calculo indagare nobis licet circa curvas Geometricas (qua de re solum locutus est Cartesius) probare possem, ejus methodum esse præstantissimam omnium, imo universalissimam ex qua infinitæ aliæ regulæ poterunt facillime deduci.

|Ad| C|N|. Ante jam divi concessi hanc esse promotionem Geometriæ Cartesianæ. Interim \{illeg}/ candide fassus est Heuratius se eo non pervenisse nisi instructus Cartesij Principijs fuisset, quibus adjutus, omnium Curvarum Geometricarum in rectas transmutationem generalem quo tradidit. Verum cætera quæ hic habentur, fateor si hæc inventa fuerint mathematice esse sane promotionem quam optarem. Alias quæ absolute mathematice possunt inveniri circa dimensionem ea poterunt omnia expedite methodo Cartesij explicari. Nec veritati quicquam derogat quod Du Laurens hæc ignoraverit.

Ad D. Hunc medio proportionalio

Ad D. Quoad inventa Gregorij fateor a tanto Viro non nisi præclara esse expectanda. Et sane si hæ quæ referuntur inveniri possint Mathematice, lubens concedam promotionem Geometriæ factam esse eo sensu quo eam secundo loco accepi [i.e. particulariter] Sed hæc mihi ignota sunt, nec nisi de ijs locutus |[sum]| quæ publice extant.

|⑦| Out of a Letter of M^r Iames Gregory to M^r Collins dated 20 Aug 1675

I can leave but little charity for any person that so overvalues Des Cartes method as to think all since discovered to be but its Consectaries.

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B As to Cartes's Probleme of 3, 4, 5, 6, 7 or more lines out of Pappus & his describing the nature of a Locus, it is the best thing in his works, worthy the Author. And perchance if he had not solved it, it had remained hitherto undone. But since him (tho not yet extant) M^r Newton hath solved the said Probleme w^th much more ease & variety; namely by five points in an Ellipsis \{illeg}/ & four in a circle or Hyperbola & four in a Parabola he describes the figure by aid of two moveable angles. w|W|hilst one pair of leggs move in a right line the other intersect in a Conic section: Without w^ch aid the actual description of those Sections in constructions of A for æquations would be unpleasa|i|nt|g|.

D As to his curves in pag 20 for finding of mean proportionals they are estemed of little use, forasmuch as there may be a Logarithmic Curve, or a Spiral line made by aid of a rank of lines in continual proportion standing at equal parallel distances or issuing at equal angles from y^e same pole point, by help of either of w^ch ungraduated lines, any number of means may be found between any two extremes given. But if it be the Spiral, then the help of a Sector or moveable Angle is required, w^ch may have lines graduated upon it to divide an angle in ratione data. — From the nature of this Spiral M^r Gregory hath derived an infinite series for finding in numbers any one, upon demand, of the means that shall be required between any two extremes given numbers given, being herein more fertile then any thing that can be pretended to be derived from Des Cartes's doctrine.

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Extracts of y^e MS Papers of M^r Iohn Collins concerning some late improvements of Algebra

By the proportion of the increase of lines Archimedes drew tangents to Spirals & M^r Fermat applied the consideration \method/ to Equations for determining maxima & minima & thereby drew tangents to curve lines. The method of Fermat was published by Herigon in his Cursus A.C. 1631, & an instance of it \was/ inserted by Schooten into his Commentary on the 2^d Book of Cartes's Geometry. M^r Newton acknowledges that he first met w^th this method upon reading that Geometry w^th these Commentaries|tors| \thereon/ & by applying it to abstracted Equations found the proportions of the increases of indeterminate quantities in the manner described in the first Proposition of his book of Quadratures. Those increases \or augmenta momentanea/ he called moments as app may be seen in his Analysis per æquationes infinitas & in his Principia Philosophiæ Lib. II. Lem II & Prop. VIII, IX, X, & in his Book of Quadratures: & the proportion of a veloci or velocity or proportion of those increases {illeg} pro \the/ velocity|ie||s| by w^ch the \the/ quantities increased|d| he called \motions, velocities of increas &/ fluxions; but it was three or four years before he used the name of fluxions {illeg}|M|^r Iames Gregory & M^r Barrow \also/ improved the method of Fermat \for drawing of Tangents/ & published their improvements, \viz^t/ M^r Gregory, \A.C. 1668/ in the 7^th Proposition of his Geometria universalis & M^r Barrow A.C. 1670 in thi|e| end of his 10^th Gemetrical Lecture|.|, {sic} And but they applied it only to the drawing of tangents. And M^r Newton's Analysis \aforesaid/ conteining his general method|s| of endless series & moments together, being \was/ communicated by M^r Barrow to M^r Collins A.C. 1669, & between|l|ately published by M^r Iames Out of this Analysis M^r {illeg} & other Letters of M^r N M^r Collins communicated several things \gave occasion to several Letters between M^r Collins & others/, as will appear by the following extracts out of \his/ papers written in his own his {illeg} & lately fallen into the hands of M^r Iones: w^ch extracts have been lately compared w^th the originals by a about the new improves about their improvements of Analysis, & found genuine & & lately published by M^r Iones. This Analysis gave occasion to several things in the Epistolary commerce between M^r Collins & others, the following extracts of w^ch have been made in the hand taken out of the Letters & papers written in the hand of M^r Collins & left to his Executors, & {illeg}fl{illeg} carefully examined & examined by some who know the hand.

D^r Wallis having published in the third Volume of his works several Letters between M^r Collins M^r Iames {Gre} himself M^r Oldenburg M^r Collins M^r Leibnitz & M^r Newton, & some others \letters rela{iin} {sic} rel{anging} to that Collection/ being lately found among the papers of M^r I. Collins, relating to the same subject, it has been thought to make them publick|.| for completing that collection

Extracts of Letters found among the papers of M^r Iohn Collins relating to the collection of \mathematical/ Letters p{illeg} published by D^r Wallis in the third volume of his works.

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|②| Out of a Letter of M^r Iames Gregory to M^r Collins date the 5^th of September 1670.^[1]

I have re{illeg}|a|d over M^r Barrow's Geometrical Lectures with much pleasure & attention wherein I find him to have infinitely transcended all that ever writ before him. I have discovered from his method of drawing Tangents together with some of my own a General Geometrical \method/ without calculation, of drawing tangents to all Curves comprehending not only M^r Barrows particular methods but also his general a|A|nalytical method in y^e end of the 10^th Lecture. My method contains not above 12 Propositions.

N. 6. pag 35.^[2]

A Letter of M^r Leibnitz dated {illeg} \to M^r Oldenb. dated at London/ 3^d Feb 167$23$ concerning Moutons differences|tia|l method in w^ch M^r Leibnitz had made progress, & for clearing himself from plagiary

Another Letter of M^r Leibnitz \to M^r O./ dated at London $\underset{20}{10}$ Feb 1673 desiring to be a member of y^e Society. N. 6. pag. 34.

Another of $\underset{26}{16}$ April 1673 from Paris, concerning the summs of fractions whose denominators are triangular {illeg} figurate numbers. In this Letter he saith: Subtilissimo Collinio tam præclara communicanti obligatum me profiteor. N 6 p. 101

Another from Paris May $\frac{14}{24}$ 1673.

Another from Paris 1 Iune 1673 returning thanks for electing him felllow of the R. S. N 6. p 137

Oldenberg to Slusius reciting 10 Feb 167$\frac{2}{3}$ reciting M^r Newtons Letter Th{illeg}g{illeg}ll{an} N 6 p 1

Another of Leibnitz to Oldenburg^[3] dated at Paris Iul. 15. 1674 concerning his arithmetical instrument & the squaring a portion of y^e Cycloid. Impress. a Wallisio. N 6|7| p 93.

|N 7 p 110^[4]| Another of Leibnitz to Oldenberg of $\frac{16}{26}$ Oct 1674 from Paris, published by Wallis. Etsi enim Illi Brunkerus & Wallisius dederint numeros rationales magis magis approprinquantes: Nemo tamen dedit progressionem numerorum rationalium cujus in infinitum continuatæ summa sit exacte æquatis circulo. Id vero mihi tandem fœliciter successit Inveni enim Seriem numerorum valde simplicium cujus summa exacte æquatur circumferentiæ circuli posito diametrum esse unitatem. [Impress a Walis.]

Another of Leibnitz to Oldenburgh^[5] w^thout a date \12 Iuly 1675 in y^e Original/ in w^ch Parisius \[M. Darius/ is twice mentioned. Printed by Wallis \w^thout a date/. Impress. N. 7. p. 11|4|9.

|N. 7. p. 189.^[6]| Another of Leibnitz to Oldenburg 28 Decemb 1675|4| \[or 75 as tis printed]/ from Paris He was then taking a short journed|y| from Paris. Printed (I think) by Wallis

|N. 7. pag 213.^[7]| Another of Leibnitz to Oldenburg 30 March 1675 from Paris. In this he writes. Scribis Iacobum Gregorium vestrum nescio quid demonstraturum se minari quod Tu meæ de quadratura circuli Arithmetico demonstrationi adversum putas putas {sic}. Sed Gregorius hoc tantum demonstrare pollicitus est, rationem diametri ad <290r> circumferentiam nulla æquatione sive relatione exacta analytica exacte exprimi posse, quod mihi non est adversum qui seriem infinitam sed numerorum rationalium, eam valde simplicem affero, cujus summa exacte æquali circulo, posito diametrum esse unitatem. &c. Scribis Cl. Newtonum vestrum habere methodum exhibendi quadraturas omnes, omnium curvarum superficium et solidorum dimensiones ex revolutione genitorum g|d|imensiones & centrorum gravitatis dimensiones inventiones, per appropinquationes scilicet, ita enim interpretor. Quæ methodus si est universali et commoda, meretur æstimari, nec dubito fore ingeniosissimo dignam Authorè dignam. Addis tale quid Gregorio innotuisse &c

M^r Oldenberg|s| \Answer/ to M^r Leibnitz 30|15| March \April/ 1675.^[8] In this Letter are set down several series of M^r Newton & M^r Gregory & de the general extent of M^r Newtons method is described, as also the description of Curves by two angles. |N 7 p. 216.|

|N. 7. p. 235.| Another of M^r Leibnitz to M^r Oldenberg 20 May 1675^[9] from Paris

Oldenberg to Leibnitz 8 Decem 1674^[10] in answer to Leibnitz of Octob. 16, 1674. – Quod de profectu in curvilearum dimensione memoras bene se habet, sed ignorare te nolim Curvarum dimetiendarum rationem & methodum a laudato Gregorio nec non ab Isaaco Newtono ad Curvas quaslibet tum Mechanicas tum Gem|o|metricas {illeg}|q|uin et Circulum se extendere, ita scilicet ut si in aliqua Curva Ordinatam dederis, istius methodi beneficio possis lineæ Curvæ longitudinem, aream figuræ, ejusdem centrum gravitatis; solidum rotundum ejus superficiem sive erectam sive inclinatam, solidi rotundi segmenta secunda horum omnium conversa invenire; quin et, dato quolibet arcu in quadrato Logarithmicum sinum tangentem vel secantem non cognito naturali, & conversum computare. Quod vero ais neminem hactenus dedisse progressionem numerorum rationalium, cujus in infinitum continuatæ summa sit exacte æqualis circulo, id vero Tibi tandem fœliciter successisse; de eo quidem tibi gratulor &c. N 7 p. 1{illeg}|1|9 p. 119.

Oldenburgh to Leibnitz 30^th Sept. 1675.^[11] Tschunhause then newl{illeg}|y| gone to Leibnitz at Paris, having received at London one of Gregories Series for y^e circle. |N 7 p 159.|

Oldenberg to Leibnitz 24 Iun 1675 [N 7 p. 243]^[12] conteining several things in relation to Cardans rules called y^e cubicus malleus of Cartes & Daries considerations thereon: in answer to w^ch was the letter above mentioned w^thout a Date (printed by wallis) in w^ch Davy is called Parius.

Letters in y^e hand of M^r Leibnitz. 3 Feb. 170$\frac{2}{3}$. March {3}0 1675 from Paris, Hannover 30 May 1677, $\frac{20}{30}$ March 1677, $\stackrel{22}{12}$ Iuly 1677, 21. 20 May 1675. 28 Ianuar 1677 from Hanover. 28 Decemb 1675 12 Iuly 1675, Franckford 29 Apr. 1671. Mente $\stackrel{28}{18}$ Septemb 17|6|{6} 1670, 15 Iuly 1674, $\stackrel{20}{10}$ Feb. 1673. 8 Iune 1673 Gratias reddit Societati Regia de electione sua. $\stackrel{24}{14}$ May 1673 from Paris. $\stackrel{26}{16}$ Apr 1673. 16 {illeg} Octob. 1674.