Draft letter from Sir Isaac Newton probably to John Chamberlayne, defending Keill Isaac Newton c. 5,563 words The Newton Project Oxford 2020 Newton Project, University of Oxford

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 c. 1710-1720, in English and Latin, c. 5,563 words, 18 ff. 18 ff.

in English and Latin

 MS Add. 3968, ff. 438r-443v, Cambridge University Library, Cambridge, UK UKCambridgeCambridge University Library Portsmouth Collection MS Add. 3968, ff. 438r-443v c. 1710-1720 England English Latin Isaac Newton Unknown Hand (11) Unknown Cataloguer (1) Unknown Cataloguer (2) Unknown Cataloguer (3) Unknown Cataloguer (5) Unknown Cataloguer (6) ScienceMathematics Mint Transcription by Marie Soulier Transcription by Michelle Pfeffer Transcription continued by Robert Ralley Transcription completed by Robert Ralley. Code audited by Michael Hawkins.
120438

SrSir

The papers in yethe Acta Leipsica wchwhich gave occasion to the controversy wthwith Mr Keil I did not see till I the last summer when I was told of Mr Leibnitz's Letter against him, & as I therefore had no hand in beginning theise controversiyes so I desire to be excus have always desired to avoid all controversies of this kind. The controversy is between the author of those papers & Mr Keil Mr Leibnitz thinks that one of his age candor & reputation should not enter into a dispute wthwith Mr Keil. & I am of the same opinion, & therefore with his Letters against him had not been written. I think that it is improper for me to enter into a dispute wthwith the author of those papers. & therefore cannot be induced to set pen to paper against him cannot be induced to enter into this con enter into this co write against him hope that Mr leibnits will approve of my forbearing to write For The controversy is between that author & Mr Keil

But Mr Leibnitz seems to say that I know how the matter stands & can put an end to the controversy if I would declare my opinion knowledge. If he means would have me declare that he is the authorinventor of the differential method so far as that method differs from yethe method of fluxions: all men, even Mr Keil himself, will allow him that. If he would have me declare that he is the author inventor author of yethe met differential method e even where the methods do agree, that is, the author inventor author of the method called by him the differential method & by me the method of fluxions: the author inventor author is the first author, & I am not yet convinced that he was the first author of that method. If he would have me declare that the Papers approve the papers in the Acta Leipsica wchwhich gave occasion to yethe controversy wthwith Mr Keil, I know not what he means by that: for those papers call my candor in question. as much as Tis there said After that author instead of describing hatd asserted the invention of the Differential method to Mr Leibnits & fortified the assertion by the credit of those that used it; he adds. Pro differentijs igitur Leibnitianis Dn. Newtonus adhibet semperque adhibuit fluxiones quæ sint quamproxime ut fluentium augmenta æqualibus temporis particulis quamminimis genita; ijsque tum in suis Principijs Naturæ Mathematicis tum in alijs postea editis eleganter est usus &c. There is some ambiguity in the words but the most proper sense is that from yethe beginning I always used fluxions instead of the differences of Mr Leibnitz. And is not this to make the readers beleive that I always used fl knew the differential method, & used yethe method of fluxions instead of it. of Mr Leibnitz & invented the method of fluxions by using fluxions instead of his differences. This gave occasion to Mr Keil to represent on yethe contrary that Mr Leibnitz used his differences instead of fluxions. And if this derogates from the candor of Mr Leibnitz the contrary derogates from my candour. & that unjustly. For By the Letters wchwhich passed between him & me in the years 1676 & 1677 he knows that I wrote a treatise of the methods of converging series & fluxions above five six years before I heard of his method of fluxions differential method. And I beleive he will allow tha And I beleive he will allow that I found out the method before I wrote of it In the generation of f lines & figures by motion, & Fermat Barrow & Gregory considered the small partsicles by wchwhich the quantities increased every moment of time & thereby drew tangents. Dr Barrow called those partsicles moments & from him I had the language of momenta & incrementa momentanea & this language I have always used & still use as may be seen in my Analysis per æquationes infinitas communicated by Dr Barrow to Mr Collins A.C. 1669 & & in my Principia mathematica & Quadratura Curvarum. And by considering the putting the velocities of yethe increase & decrease of quantities proportion all to the incrementa momentanea I found out the demonstration of yethe method of fluxi wchwhich I of moments & thence called it the method of fluxions. This demonstration you have in the end of yethe Analysis per æquationes infinitas & in the first Proposition of yethe book of Quadratures. But I do not know that Mr Leibnitz has demonstrated the differential method. So then the method of moments & the method of Fluxions is one & the same method variously named in several respects as I have always used it & Mr Leibnits by calling the moments differences has given it the name of the differential method. But when & by what steps he found out his way of explaining it I do not know In yethe year 1664 I leartnt Fermats method of drawing tangents.

Add. 3968 No. 30439439

365y 000001506+1. 2094 06 2190d 0 174500 0 8760h 0 349000 0 525600′ +349′+1251′ 575594 0 1742400′ −4y. 0 0

To Leibnitz SrSir Since you shewed me the passage of in Mr Leibnitz letter concerning Mr Keil I have discoursed the matter wthwith Mr Keil & he represents to me the injustice done partic unfair injustice done to me in the Acta Leipsica gave him occasion to write the words complained of. That by your Letter to Mr Collins dated & published by Dr Wallis by Dr Wallis he was convinced that you knew nothing at that time in did not then use the methodus differentialis. That the next year in my letter dated I represented that I I represented that I had a method of solving Problem drawing tangens solving direct & inverse Problemes of tangents, & others more difficult, & that this method stuck not at fractions & surd quantities & that I had written a treatise of it five years before & had invented it some years fbefore that, whereas you had t (wchwhich method was that of fluxions:) & that you before you received this Letter from Mr Oldenburg you dicovere made no dicoverydiscovery of your knowing the methodus differentialis. That in my first Letter ta of wchwhich was sent to you by Mr Oldenburg & dated 1676, I described at once the foundation of the differentiall method by this series & of the method of infinite series by setting down this series A P+P⁢Qmn=Pmn+mn⁢A⁢Q+m−n2⁢n⁢B⁢Q+m−2⁢n3⁢n⁢C⁢Q+m−3⁢n4⁢n⁢D⁢Q+&c I do For if Pmn be any fluent or indeterminate or fluent Quantity & PQ be its fluxion the difference difference, th mn⁢A⁢Q will be its first difference m−n2⁢n⁢B⁢Q will be its second difference, m−2⁢n3⁢n⁢C⁢Q will be its third difference & so on perpetually. As for instance. If Or if any fluent quantity simple or compound Or if you take the fourth exam d+e43=d43 Or if according to yethe sixt last examples of this rule set down in that Letter you put d+emn=dmn+mnd⁢m−nn×e+m⁢m−m⁢n2⁢n⁢n⁢e⁢e⁢dm−2⁢nn+&c, & suppose d to be any in fluent quantity & e its f indeterminate quantity simple or compound whose difference is e & desire the difference or fluen of any power quotient or radical of d whose index is m dmn that is the index difference of any power quotient or radical of d whose index is mn, the second quantity of the series vizt mn⁢e⁢dm−nn will be the first difference & the third quantity m⁢m−m⁢nn⁢e⁢e⁢dm−2⁢nn will be the second difference & so on. As for example if the first differences of the quantity a⁢a+x⁢xm be desired & the the letter o be put for the first difference of the fluent quantyquantity x, then by this rule x2⁢o will be the difference of a⁢a+x⁢x & putting df a⁢a+x⁢x for d & for e, the Rule will give you a⁢a+x⁢xm×a⁢a+x⁢x× the first difference m2⁢x⁢o×m⁢mna⁢a+x⁢xm And & the second difference m2⁢m−m2⁢m×e⁢e×a⁢a+x⁢xm−2 the second And thus this method is done without sticking at fractions or surds. Thus the foundation of the method wchwhich I described in my second Letter being laid in my Thus in the first of my two letters wchwhich you Mr Leibnitz received from Mr Oldenburg before you he discovered any thing to me of the you his knowing thaet methodus differendtialis the foundation of that method bein foundation of that medthod being described & in the second of those letters the nature & use thereof being to hb, & mentioned, tho he might not be able to decipher the sēentences in wchwhich the name of [Data æquatione: quotcunque fluentes quantitates involvente fluxiones invenire & vice versa] set down in that the second letter: yet Mr Keil thinks that he might had more reason to say write what he was published in the transactions concerning this matter, then the author of yethe papers published in the Acta Leipsica against me had to tell the world that what I published in yethe Treatise de Quadratura Curvarum was methodus the either the either the methodus differentialis of wchwhich Mr Leibnitz was the author, ofr some other things what had been published before by Mr Sheen & Mr Craig. I desire therefore that These are to certify that the bearer of hereof Edward Carter is Waterman & Servant to the Mint & one of that corporation & on that account is exempted from all Parrochial Services by the Charters of the Mint that he may attend her MajestitisMajestites sericeservice therein. These Charters b These Charters bein And as these Charters have hather are & have been constanly allowed whenever whenever there has been occasion in such cases & privileges are of auncient standing in such cases of this nature Charters are of ancient standing are of ancient standing & have ever been allowed & were granted in our Charters for enabling the Officers & Ministers of her MajtsMajesties Mint to carry on perform their duties without interruption & to carry on the coinage wthwith dispatch & are of very ancient standing &in cases of this nature have ever been allowed; for the publick service, so & their services are not wanted in their respective Parishes, there being great choise of other people to perform parish duties: s & so tis hoped that her Majesties Officers in all other stations will have that regard to her Majesties government & to the rights of the Crown as not & services under it as not to give her Office of yethe Mint any trouble in this matter. For if the constitution of the Mint be disturbed in this case, we shall our servants will we ill our service will the people imployed in the coinage will soon become liable to be taken from ourthe service at such times as we cannot spare them without putting a stop to the coinage till we can get others capable of serving in their room. For if any one part Which W I For if the coinage be stopt in any one part thereof it must stand still in every part till it can be pu to the dammage of the crown Merchant & the disabling of yethe Master & Worker to perform his contract wthwith her Majesty. & the And hereof all her MajtisMajesties Iustices of the peace & all other her Officers herein concerned are desired to take notice Is. Newton Master & Worker of her MajtsiesMajesties Mint. And in an earliernother MS composed in October 1666 in which the Elements of solving Problems by motion were laid down in eight Propositions the sevent of which conteined the method now described of deducing fluxions from fluents [& the eitghth the contrary method of deducing fluents from fluxions: And in the end of the seventh of] in the end of wchwhich I added to the seventh this improvement thereof: Si in æquatione &c y4=a⁢a⁢x⁢x−x4 4⁢q⁢y3=a2⁢a⁢p⁢x−4⁢p⁢x3. 2⁢q⁢y=a⁢a⁢p−2⁢x⁢x⁢pa⁢a−x⁢x=p⁢a⁢a−x⁢x−p⁢x⁢x0 Quæ quidem ea vero per Regulam tertiam id est per methodum serierum, ex hoc tempore generalis evasit. Has methodos anno 1665 inveni, anno proximo conjunxi et ex utraque per mutuum subsidium methodum Analysin unam generalem conflavi. And in an earlier Manuscript composed in Autum October 1666 I mentio there is this Precept. Si in æquatione quavis occurrat quantitas aliqua vel fracta vel surda, vel mechanica (id est quæ Geometrice inveniri non potest sed per ream curvæ ali aliquam curvilineam definitur aut prper longitudinem Curvæ alicujus aut solidum contentum figuræ superficiem curvam habentis, aut per gravitates eorum ) ut inveniatur in qua proportione in qua quantitates indeterminatæ augentur vel crescunt, ita procedas. Assum Litera aliqua (qualis ξ) designetur quantitas illa fracta vel surda vel mechanica, & litera alia (qualis π) designetur quantitatis illius motus incrementi vel decrementi seu velocitas qua augetur vel diminuitur. Et facta æquatione inter literam ξ & quantitatem quam significat: quære (per Prop. 7 si quantitas illa sit Geometrica, vel per alias methodos si mechanica sit idque per Prop. 7. si quanti) valorem literæ alterius π. Deinde in æquatione prima pro quantitate per ξ significata nova involvens incrementorum velocitates. Et in hac nova æquatione pro literis ξ illis ξ et π substituantur earum valores. et habebitur æquatio quam invenire oportuit. Exempl. 1. Invenire Si Quantitatum x et y quarum relatio ad invicem per hanc æquationem y⁢y=x⁢a⁢a−x⁢x designatur invenire quaruntur motus seu crescendi velocitates p et q: Primo sit ξ=a⁢a−x⁢x , seu a⁢a−x⁢x−ξ⁢ξ=0 ξ⁢ξ+x⁢x−a⁢a=0. Et inde per Prop. 7, prodibit 2⁢π⁢ξ+2⁢p⁢x=0 seu −p⁢xξ=π=−p⁢xa⁢a−x⁢x. Deinde in æquatione prima y⁢y=x⁢a⁢a−x⁢x pro a⁢a−x⁢x scribatur ξ et habebitur æquatio y⁢y=x⁢ξ & inde (per Prop. 7) prodibit æquatio relationem velocitatum p, q, et π definiens, vizt 2⁢q⁢y=p⁢ξ+π⁢x. in qua si pro ξ et π scribantur earum valores, proveniet æquatio quæsita 2⁢q⁢y=p⁢a⁢a−x⁢x−p⁢x⁢xa⁢a−x⁢x. Exempl. 2. Si quantitatum x et y quarum crescendi velocitates p et q per hanc æquationem desiderantur relatio ad invicem definiatur per hanc æquationem, et x3−a⁢y⁢y+b⁢y3a+y−x⁢x⁢a⁢y+x⁢x=0, & quæratur relatio velocitatum p et q quibus quantitates illæ crescunt augentur vel diminuatur ponantur x3−a⁢a⁢y=τ, b⁢y3a+y=φ, et −x⁢x⁢a⁢y+x⁢x=ξ. Et si velocitates quibus τ, φ, et ξ440 τ, φ, et ξ mutantur nominantur p, q et r β γ et δ respective æquatio prima per R Prop. x3−a⁢y⁢y=τ (per Prop. 7) dabit 2⁢r⁢z+r⁢x⁢x+p⁢a⁢z 3⁢p⁢x⁢x−q2⁢a⁢y=β, secunda by3=a⁢φ+y⁢φ dabit 3⁢q⁢b⁢y⁢y=a⁢γ+y⁢γ+q⁢γ seu 3⁢q⁢b⁢y⁢y−q⁢φa+y=γ=3⁢q⁢a⁢b⁢y⁢y+2⁢q⁢b⁢y3a⁢a+a2⁢y+y⁢y tertia a⁢y⁢x4+x6=ξ⁢ξ dabit q⁢a⁢x4+4⁢p⁢a⁢y⁢x3+6⁢p⁢x5=2⁢δ⁢ξ, seu −q⁢a⁢x⁢x−4⁢p⁢a⁢y⁢x−6⁢p⁢x32⁢a⁢y+x⁢x=δ. Denique −q⁢a⁢x⁢x+4⁢p⁢a⁢y⁢x+6⁢p⁢x32⁢a⁢y+x⁢x=0 æquatio est quam invenire oportuit. Exempl. 3. Sit Curvaæs AC A cujus vis AC sit Abscissa s AB=x Et Ordinata BC=y rectangula BC=y=a⁢x−x⁢x & superficies curviline inclusa ABC dicatur z, et relation inter x, y et z definiatur per æquationem z⁢z+a⁢x⁢z−y4=0 et quæratur relatio inter ipsarum motus seu crescendi vel decrescendi velocitates sint p, q & r respective, et quæratur relatio inter p et q: Æquatio z⁢z+a⁢x⁢z−y4=0 (per Prop 7) dat æquationem novam 2⁢r⁢z+r⁢a⁢x+p⁢a⁢z−4⁢q⁢y3=0. TAd ipsius AB terminos A et B erigantur perpendicula unitati æqualia et longitudine data AD et BH et compleatur parallelogrammum ADHB. Et si abscissa AB augeatur, superficies duæ ADHB et ACB augebuntur in ratione Ordinatarum BH et BC id est ita ut BH=1 sit ad p sit ad r ut BH=1 ad BC=a⁢x−x⁢x, adeoque r=p⁢a⁢x−x⁢x. Quo ipsius r valore in æquationem 2⁢r⁢z+r⁢a⁢x+p⁢a⁢z−4⁢q⁢y3=0 substituto prodit æquatio quam invenire oportuit 2⁢p⁢z+p⁢a⁢x×a⁢x−x⁢x+p⁢a⁢z−4q⁢y3=0. And in the eighth Proposition wchwhich is for deducing fluents from equations involving their fluxions I laid down these Rules. Sunto quantitates indeterminatæ x et y et velocitates incrementorum p et q In octava Propositione docebam vicissim quomodo ex æquatione velocitates augmentorum vel decrementorum involvente quantitates crescentes vel decrescentes deduci possent, idque reducendo Problema ad quadraturam Curvæarum, et quadranda Curvam per sequentes Regulas. Primo tres illas Regulas quas etiam postea etiam descripsi in principio Tractatus de Analysi per Æquationes numero terminorum infinitas, ut et per Catalogum Curvarum quæ vel quadrari possent vel cum Conicis Sectionibus comparari & quarum Ordinatas posui postea in Epistola mea ad Oldenburgum 24 Octob. 1676 data. Et ex his intelligi potest. Epistola Co quid sibi voluit Barrovius noster Collinius noster scribendo ad D. Thomam Strode 26 Iulij 1672 in hæc verba. Mense septembri 1678 Mercator Logarithmotechniam edidit suam, quæ specimen hujus methodi in unica tantum figura, nempe quadraturam Hyperbolæ continet. Haud multo postquam prodierat liber, exemplar ejus ad Cl. Wallisio Oxonium misi (qui suum de eo judicium in Actis Philosophicis statim fecit,) aliudque Barrovio Cantabrigiam, qui quasdam Newtoni chartas [qui jam Barrovium in Mathematicis Prælectionibus publicis excipit)] extemplo remisit: e quibus ete ET ALIIS EX ALIIS, QVÆ QVÆ OLIM ab Authore cum Barrovio AB AVTHORE CVM BARROVIO COMMVNICATA fuerant FVERANT, patet illam Methodum a dicto Newtono ALIQVOT ANNIS ANTEA EXCOGITATAM & modo universali applicatam fuisse: ita ut ejus ope in quavis . . . . . . . obtineri quæant. Hactenus Collinius. & Et hæc est Methodus a Leibnitio summatoria dicta, a me vero inversa methodus fluxionum (vel et momentorum) mererca dicta. Quinetiam ex his manifestum est quod methodum fluxionum et methodum serierum ab anno 1666 in unam methodum conjunxi [et ex utraque methodum unum generalem s per methodum et quod ope methodo vel ope method serierum inversam methodum fluxionum ab eo tempore generalem reddidit et vicissim ope methodi fluxionum methodus serierum ab eo tempore generalis evasit. Quæ in hacis Manuscriptais brevioribus derbis brevioribus complexus sum fusius explicui in Tractatu de quem anno 1671 composui SrSir Mr Leibnits referring the difference between him & Mr Keel to me, I have spoke wthwith Mr Keel about it & he represents that by the Letter of Mr L. to Mr C. dated Since you shewed me the passage in Mr Leibnitz letter wchwhich relates to Mr Keil, I have spoke to him about it & he represents to me that inby your the letter of Mr L. to Mr Collins dated & published by Dr Wallis he was of opinion satisfied that Mr Leibnitz did not then underst at that time use the Metho differential method & that in my two Letters sent to the next year by Mr Oldenburg to Mr Leibnitz I& since printed by Dr Wallis I sufficiently described the characters properties nature uses & elements of that method of fluxions. [ [wchwhich Mr Leibnits d has since callsed the differential method in that Letter & of wchwhich I & of wchwhich I there represented that I had written a treatise five years before. for in resolving any Binomium into an infinite series if & that the method called the diffe of fluxions published in the introduction & book of Quadratures is that very method the first term be the indeterminate or fluent quantity &the second will be the first difference, the third will be the second difference the fourth will be the third difference & so on in infinitum. Mr Keil further shewed me some passages in the Acta Leipsica in wchwhich I find my self & my friends t dted very constantly libelled sleighted & injured & represents that that injuries there done to him & me put him upon writes what Mr Leibnitz complains of.. And since in those Acta the method f in dispute is in dispute is taken from me & gi ascribed given to Mr Leibnits as th I th & yet I never could learn that he himselfe ever pretended to be the first author, I desire that you would send him this] And since I there say that I had written a treatise of it above five years before & set down the first proposition of yethe book of Quadratures in letters put out of due order, he beleives that yethe very method of fluxions described in that book was is that very method & therefore was invented by me long before Mr Leibnitz knew any thing of began to use the methodus differentialis. SrSir Vpon speaking wthwith Mr Keil about yethe complaint of Mr Leibnitz he shewed me some passages in the Acta Leipsica whe concerning his what he had inserted into the transa Ph. Trasnsactions, he represented to me that I my friends were used w what he there said was to obviate the bad usage which I & my friends met with in the Acta Leipsica, & shewed me some passages wchwhich in those Acta, by wchwhich I was convinced of the unfair usage I there met with by wchwhich to justify what he said. And I had not seen those passages before, but upon reading them I found that I have more reason to complain of the collectors of yethe mathematical papers in those Acta then Mr Leibnitz hath to complain of Mr Keil. For the collectors of those papers every where endeavour to insinuate to their readers that yethe methodus differentialis of fluxions was ded is very d deduced from to is the the differential method of Mr Leibnitz th making in & do it in such a manner as if he was the true author & I had taken it from him, & particularly in the account that they give of ttst the enumeratio curvarum & Q the Quadra give such an account of the booke of Quadratures as if it was nothing else then an improvement of what had been found out before by Mr Leibnitz Mr C Dr Sheen & Mr Craig. Whereas he that compares that book wthwith the LettesLetters wchwhich passed between me & Mr Leibnitz before by means of Mr Oldenburg before Mr Leibnitz began to discover his knowledge of the differential method will find reason to beleive see ytthat the things conteined in this book were invented before the writing of those Letters. For the first Proposition is set down in those letters ænigmatically, In alio autem schediasmate MS Maij 16 1666 composito methodum solvendi Problema per motum complexus fui Propositionibus septem quarum ultima fuit Regula jam descripta deducendi augmentorum incementorumincrementorum vel decrementorum velocitates ex æquatione quantitates augcreescentescrescentes vel decrescentes involvente. Et mense Octobri ejusdem anni in alio schediasmate quod anno mense Octobri ejusdem anni composui descripsi easdem Propositiones septem & septimum anni & octavam addidi [deducendi quantitates augescentes vel decrescentes ex æquatione velocitates crescendi vel decrescendi involvente] Quæ ad septimam addidi ita se habent Septimam vero auxi additis ijs quæ sequunte 133441 The papers in yethe Acta Lipsica wchwhich gave occasion to yethe controversy wthwith Mr Keil I did not see till the last summer & therefore had no hand in beginning this controversy. Mr Leibnitz thinks that one of his age & reputation should not enter into a dispute wthwith Mr Keil, & have as much reason to forbeare disputing wthwith the author of those papers. The controversy is between the author of those papers & Mr Keil. And I have as much reason to complain of that author for questioning my candor as Mr to desire that Mr Leibnitz would set the matter right without engaging me in a dispute wchwhich ytthat author as Mr Leibnitz has to complain of Mr Keil for questioning his candor & to desire that I would set the matter right without engaging him in a controversy wthwith Mr Keil. For after that author had asserted the invention of the Differential method to Mr Leibnitz & fortified the assertion by the credit of those that used it he adds. Pro differentijs igitur Leibnitianis Dn. DNewtonus adhibet semperque adhibuit fluxiones quæ sint quamproxime ut fluentium augmenta æqualibus tempusoris particulis quamminimis genita; ijsque tum in suis Principijs Naturæ Mathematicis tum in alijs postea editis eleganter est usus &c. The words are ambiguous but imost properly import that I always knew of the Differential method & borrowed every thing from it: whereas Mr Leibnitz by the Letters wchwhich passed between us in the years 1676 & 17677 I knows that I wrote a treatise of the methods of fluxions & infinite series Six years before I heard of the method of differential method. The truth is, I always used & still use the language of particula momenta, & augmenta & or incrementa momentanea & particulæ nascentes [as may be seen in my Analysis per æquationes infinitas communicated by Dr Barrow to Mr Collins A.C. 1669 & in my Principia Philosophiæ & Quadratura Curvarum.] other Tracts And where I had by putting the meth velocities of the increase of quantities proportial to the incrementa prima nascentia momentanea, I demonstrated the method of moments & from thence called it the method of veloci fluxions giving the name of fluxions to those velocities. Those momenta Mr Leibnitz calls differences & thence came the name of the Differential method. [The Demonstration you have in the end of the my Analysis per æquationes infinitas & in the first ProposionProposition of my Treatise of Quadratures.] Those momenta or augmenta Mr Leibnits calls Differences & thence came the name of named the method the Differential method but has not yet demonstrated it. And I know that Mr Leibnitz he had that method in yethe year 1677, but when & how he found it I do not know. That must come from himself. Fermat I had the hint of this method from Fermats way of drawing Tangents & by applying it to abstracted Æquations in general I extended it to directly & invertedly I made it general. Mr Gregory & Dr Barrow had & improved the same method in drawing of Tangents. Vpon my communicating some things A paper of mine gave occasion of this kind to Dr Barrow he to shewed me his method of Tangents before he ad inserted it into his 10th Gemetrical Lecture. For I am that friend wchwhich he there mentions. For Mr Leibnitz affirms that that Augthor has every where given every man his due own & for that author in giving an account of my book of Quadratures to give every man his due own is to tell the world that I have borrowed from other people men & thereby to tax my candor. as much as Mr Leibnitz Kei Whereas in theat book of Quadratures there is nothing but what I had invented some years before I heard of the Differential method name of Mr Leibnitz, or of any of those authors from whom it can be pretended that I have borrowed any thing. For Mr Leibnits 0000000000006636,66 334,0000003∟350.0∟01520000,051000,025506656,68 000,06031000,202000,10100673676 000,13397460∟449000,22440685689 00233955600,783800,39190702706 00357212401,196660,59307.20726 00500000001,674160.835087.49750 06579864798002,176501,088207.72775 000,82635202,839401,41970801804 00100000003,348001,67400830833 0000000 00826352.002,7500013750000 000066300000 000080050000 0000000862 0000000891 0000000916 0000000940 0000000960 00000009,77 00000009,90 00000009,98 000000010∟00 131.1∷438,63(3,3503=3,34832 (3,3−. 633000 524000 109000 104800 4200 3930 270 39300,00 45,6300 63300 65500 2200 2620 420 16610 8305 4560 456 76 0∟05092 100500 20102005 20202005 6,630 3348 9∟98 136 663 799 40192 4019 665 44876 701867 70187 11698 783752 107160 10720 1786 119666 19491 1949 325 21765 228.3∷56875 245329 34533 4089 283951 2479056 247906 42318 2769280 138464 65798 197394 19739 2632 219765 131.1∷438,63 3930,00 456,00 633,0 655,0 −220 +073 109880 666000 36(3∟3500 72(3∟3500 432(3∟3500 6(3∟3−25 3∟3485 9,9800 83500 666000 749,500 5000 170625(748∟35500 159600(748∟35500 11020(748∟35500 9120(748∟35500 1905(748∟35500 1824(748∟35500 810(748∟3550 684(748∟3550 126(748∟3550 120(748∟355 0112 57623.115 110.113 224791 802479 12330 1727500 210 2426000 280 290 290 74800 15200 7,4800 37400 1496 11,3696 5687357299 57621 748374187 57248374 137 74800 0060310 44∟6800 2234 0000075 449100 65 13 74800 133⁤34 74,8000 224400 22440 6732 570 100∟214 74800 0,23400 149,600 22440 2992 175∟032 0,357210 3740 142884 25005 1072 168961 65800∟00 3740∟00 26320∟00 460600∟00 1974000∟00 1429311∟892 25005000246 1071630000∟00 13359700000∟00 82635 374 247905 57844 3305 309∟054 11,45∟70 44,922∟45 100,250∟10 175,087,50 0133∟60 374,017.00 0246,00 0309,00 0374,00 diametri Lunæ. . . . . . . id est hexapedarum Parisiensium 57303 Quoniam Cassinus intervallum omnium maximum mensuravit idque maxima ut opinor cum diligentia mensura its hujus mensuram in Tabula præcedente condenda adhiberi et invenit gradum unum esse haxapedarum 570293 et medium intervalli ab 10 mensurati incidat in latitudinem 45.′41′ in condenda Tabula superiore posui mensuram gradus in Latitudine illa esse hexapedarum 57292, Et in latitudine vero 45gr esse hexapedarum 57284. Et hæc est mensura gradus unius in æquatore pergendo ab oriente in occidentem. Et propterea diameter Terræ a centro ejus ad æquatorem ducta æqualis est hexapedis 328213146⁤12 æqualis est seu pedibus Parisiensibus 19692789 2⁢r⁢x−x⁢x=y⁢y. 2⁢r⁢x.−2⁢x⁢x.=2⁢y⁢y.. r⁢r⁢x+z⁢z⁢x=2⁢r3. r⁢r⁢x.+z⁢z⁢x.+2⁢z⁢z.⁢x=0. x⁢z=r⁢y. x.⁢z+x⁢z.=r⁢y.. x.⁢y=z.⁢t. t=x.⁢yz.=x.⁢x⁢zr⁢z.. x.⁢z+x⁢z.r=r⁢x.−x⁢x.y=r⁢r⁢x.−r⁢x⁢x.x⁢z. 2⁢r⁢x−x⁢x=x⁢x⁢z⁢zr⁢r. 2⁢r3−r⁢r⁢x=x⁢z⁢z. r⁢x.−x⁢x.y.=x⁢zr x.⁢x⁢z2+x2⁢z⁢z.=r3⁢x.−r⁢r⁢x⁢x.. x⁢x.⁢z=r⁢z.⁢t. r⁢x−x⁢x=z⁢z x.⁢x⁢z2+x3⁢x.⁢z2r⁢t=r3⁢x.−r⁢r⁢x⁢x.. x⁢z2+x3⁢z2r⁢t=r3−r⁢r⁢x. r3+x3⁢z2r⁢t=0. t+x3⁢z2r4=0. −t=8⁢r5⁢z2r⁢r+z⁢z3 2⁢r3r⁢r+z⁢z=x. 2⁢r⁢r⁢zr⁢r+z⁢z=y. 2⁢r3=r⁢r⁢x+z⁢z⁢x. r⁢r⁢x.+z⁢z⁢x.+2⁢z⁢z.⁢x. r⁢r+z⁢z+z2⁢x⁢yt=0. −t=2⁢z⁢x⁢yr⁢r+z⁢z. t+r5⁢z3r⁢r+z⁢z3=0. 442 For he claims it as And by these letters it seems to me that Gallia was at this time subject to the Pope & that the bishop of Rouen was his Vicar or one of them. For he directs him to refer the greater causes to yethe sea of Rome according to custome. But the BpBishop of Arles soon after was made his Vicar over all Gallia: for Honored SrSir. Your saying yesterday that your coming to the miday or twesday was I shewed your Letter to SrSir Is. Newton at a meeting before of yethe R. Society & report discoursing the matter wthwith Mr Keil, & Mr Keil shewed showing somethings in the Acta Leipsica wchwhich gave occasion to what he wrote in the Transactions & chose rather to write the inclosed answer to yours then to retract & beg pardon he was thereupon desired by the Society to draw up an Account of that matter, wchwhich account I herewith send you 443Add. 3968 No 30 Copy of Leibnitzs Letter to Mr Chamberlayn SrSir I am obliged to you as well for the communication of the Letter of the excellent Mr Wotton (who is more favourable to me then I could hope, & I pray return my thanks for his good sentiments opinion) as for your obliging offer to mediate a good understanding between me & Mr Newton & me. It was not I that interrupted it. One named Mr Keil inserted something against me in one of the your Philosophical Transactions. I was much surprized at it & demanded reparation by a Letter to Dr Sloane Secretary of the Society. Dr Sloane sent me a discourse of Mr Keil where he justified what he said after a manner wchwhich attaqued my reputation reflected even upon my integrity. I took this for an private animosity peculiar to thisat personage, without having the least suspicion that the Society & the said even Mr Newton himself took part with him therein And not finding it material judging it worth the while to enter into a dispute wthwith a man ill instructed in former affairs & supposing also that Mr Newton himself being better informed of that wchwhich had happened passed would do me justice, I continued only to demand that satisfaction wchwhich was due to me. But I know not by whoat chicanry & foul play some brought it about that this matter was taken as if I was pleadeding before the Society, & submitted my self to their jurisdiction, wchwhich I never thought of. And according to justice they should have let me know that the Society would examin the bottom of this affair & have given me opportunity to declare if I would propose my reasons & if I did not hold any of the Iudges far suspected. So they have given sentence, one side only being heard, in such a manner that the nullity is visible. Also I do not at all beleive not at all that the judgment wchwhich is given can be taken for a final judgment of the Society. Yet Mr Newton has caused it to be published to the world by a book intituled printed expresly for discrediting me, & sent it into Germany, into France, & into Italy as in the name of the Society. This pretended judgment, & this affront done without cause to one of the most ancient members of the Society it self & who has done it no dishonnour, has will founindd but few approvers in the world. And in the Society it self I hope that all the members will not agree to it. The ableet men among the French, Italians, & others disapprove highly of this proceeding & are astonished at it, & their I have several Letters upon it in my hands. The proofs produced against me appear to them very short. As for me I have always carried my self with the greatest respect that could be towards Mr Newton. And tho it appears now that there is the greatest room to doubt whether he knew my invention before he had it from me; yet I have spoken ~ ~ as if he had of himself found something like my method: but being abused by some flatterers ill advised, he has taken the liberty to carry himself in my affair attaque me in a mann in a manner very sensibley. Iudge you notwithstanding now, SrSir from what side that should necessarily come principally come wchwhich is requisite to terminate this controversy. I have not yet seen the book published published against me, being at Vienna which is in the extremity furthest part of Germany where such books come very slowly, & I have not designe to make it come speedily thought it worth the while to send for it by the Post. So I have not yet been able to make such an Apology as the affair requires. But others have already took care of my reputation. I abhor disobliging disputes among men of Letters & have always avoyded them. but at present all meanes possible have been taken to engage me in them. If the evil could be redressed, SrSir, by your means intercession interposition, towhich you offer so obligingly, I shallould be glad very glad, & I am already very much obliged to you for advancing it b already for it.