Draft of part of the "Account of the Commercium Epistolicum" (i.e. the English "Recensio") for Philosophical Transactions (1683-1775), Vol. 29. (1714

575 To the Reader

That the following Tracts may be the better understood, it may be convenient to presmise their history.

Dr Wallis in his opus Arithmeticum published A.C. 1657. Ccap. 33. Prop. 68, reduced the fraction $\frac{A}{1 - R}$ by perpetual division, into yethe series $A + AR + {AR}^{2} + {AR}^{3} + {AR}^{4} + &c$

Vicount Brounker squared the Hyperblola by this Series

\frac{1}{1 \times 2} + \frac{1}{3 \times 4} + \frac{1}{5 \times 6} + \frac{1}{7 \times 8} + &c

. that is by this

1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \frac{1}{7} - \frac{1}{8} + &c

conjoyning every two terms into one. And thies Quadrature was published in the Philosophical Transactions in Apr 1668. Mr Mercator soon after published a demonstration of this Quadrature by the Division of Dr Wallis; & soon after that before the end of the year Mr Ia. Gregory published a Geometrical Demonstration thereof. And these books were a few months after sent by Mr Iohn Collins to Dr Barrow at Cambridge & by Dr Barrow comunicated to Mr Newton (now SrSir Isaac Newton) in Iune 1669. Whereupon Dr Barrow mutually sent to Mr Collins a Tract of Mr Newton's entituled Analysis per æquationes numero terminorum infinitas. And this Tract Mr Newton menti in his Letter of to Mr Oldenburg dated 24 Octob. 1676 mentioned by the name of Compendium Serierum in the following manner. Eo ipso tempore quo Mercatoris Logarithmotechnia prodijt, communicatum est per amicum D. Barrow (tunc Matheseos Professorem Cantab.) cum D. Collinio Compendium quoddam harum Serierum, in quo significaveram Arias & Longitudines Curvarūum omnium, & Solidorum superficies & contenta ex datis rectis; & vice versa ex his datis rectas determinari posse, & methodum indicatam illustraveveram diversis seriebus. Mr Collins in the years 1669, 1670, 1671 & 1672 gave notice of this Compendium to Mr Iames Gregory in Scotland, Mr Bertet & Mr Vernon then at Paris, Mr Alphonsus Borelli in Italy, & Mr Strode, Mr Townsend, Mr Oldenburg, Mr Dary & others in England, as appears by his Letters still extant. And Mr Oldenburg in a Letter dated 14 Sept 1669, & entred in the Letter Book of the R. Society gave notice of it to Mr Francis Slusius at Liége & cited several sentences out of it. And particularly Mr Collins in a Letter to Mr David Gregory dated 11 Aug. 1676 mentions it in this manner. Paucos post menses quam editi sunt hi libri, [viz. Mercatoris Logarithmotechnia et Exercitationes Geometricæ Gregorij] missi sunt ad Barrovium Cantabrigieænsem Ille autem responsem excogitatam fuisse quam ederetur Mercatoris Logarithmotechnia & generaliter omnibus figuris applicatam simulque transmisit D. Newtoni opus manuscriptum. And this Mr Collins in a Letter to Mr Strode dated Iuly 26 1672 confirmed by the testimony of Dr Barrow And in a Letter to Mr Strode dated Iuly 26 1672 Mr Collins wrote thus of it. Exemplar ejus [Logarithmotechniæ] misi Barrovio Cantabrigiam, qui quasdam Newtoni chartas extemplo remisit: e quibus et alijs quæ prius ab autore cum Barrovio communicata fuerant patet illam methodum a dicto Newtono aliquot annis antea cogitatam & modo universali applicatam fuisse: ita ut ejus ope, in quavis figura curvilinea proposita, quæ una vel pluribus proprietatibus definiter, Quadratura vel Area dictæ figuræ, accurata si possibile sit, sin minus infinite vero propinqua, Evolutio vel longituto Lineæ Curvæ, Centrum gravitatis figuræ, Solida ejus rotatione genita & eorum superficies: sine ulla radicum extractione obtineri576 obtineri queat. Postquam intellexerat D. Gregorius hanc Methodum a a D. Mercatore in Logarithmotechnia usurpatam & Hyperbolæ quadrandæ adhibitam, quamque adauxerat ipse Gregorius jam redditam esse universalem redditam esse, omnibusque figuris applicatam; acri studio eandem acquisivit multumque in ea enodanda desudavit. Vterque D. Newtonus & Gregorius in animo habet hanc Methodum exornare: D. Gregorius autem D. Newtonum primum ejus inventorem anticipare haud integrum ducit. So then by the testimony of Dr Barrow this Analysis was invented two or thre years before the Logarithmotechnia of Mr Mercator came abroad. And since it gave the areas of figures accurately if it might be, or else by approximation, it included the invention of such converging series as brake of & became finite when ever the area could be found by a finite equation. How this was to be done is not described in the Compendium, but it's there said: hujus [methodi] beneficio Curvarum areé & longitudines &c. (id modo fiat) exacte et Geometrice determinantur Sed ista narrandi non est locus. Mr Newton in his Letter of 24 Octob. 1676 tells us sets down one that this was done by the method of fluxions, & in his Quadratura Com the first six Propositions of his book of Quadratures sets down how it was done & there is no other way of doing it. And therefore the Method described in those six Propositions was known to him when before he wrote the said Compendium. And as MrCollins in his Letter to Mr Strode describes this method to be universal so Mr Mr Newton in the Compendium it self makes the same representation. For after he had shewn how to reduce the Quadratrix to an equation & how from that Equation to deduce the Area thereof, he adds Nec quicquam scio ad hujusmodi scio ad quod hæc Methodus, idque varisjs modis sese non extendait. Imo tangentes ad Curvas Mechanicas (siquando id non alias fiat) hujus ope ducentur. Et quiquid vulgaris Analysis per æquationes numero terminorum ex finito terminorūum numero constantes (quando id si possibile perficit, hæc per æquationes infinitas semper perficitat: Vt nil dubitaverim etiam nomen Analyseos etiam huic tribuere. But to make this method an universal Analysis something more is was requisite then the invention of Series by Division & extraction of roots. For in his Letter of 13 Iune 1676 after he had taught the invention of series by those methods he subjoyned Ex his videre est quantum fines Analyseos per hujusmod æquationes ampliantur. Quippe quæ earum beneficio ad omnia pene dixerim Problemata (si numeralia Diophanti et similia excipias) sese extendit. Non tamen omnino universalis evadit nisi per ulteriores quasdam methodos eliciendi series infinitas And whereas Mr Collins represents this Method a general one & that it proceeds with without any extraction of wroots, meaning that it proceeds without stopping at surds, this is also an argument that Mr Newton had the method of fluxions before he wrote the Compendium. For in his Letters of 10 Decem 1672 & 24 Octob. 1676 he gave this as a character of his general method that it stuck not at surds. Mr Newton in his Answer to a Letter of Mr Leibnitz dated 9 Apr 1716, has told us that he hath several Mathematical papers still in his custody several Mathematical Papers written in the years 1664, 1665 & 1666 some of wchwhich are dated happen to be dated; & that in one of them dated the 13th of Novem 1665 the direct Method of fluxions is set down in these words. PROB. An Equation being given, expressing – – – – gives the relation of p. q, r, &c. Suppose now that the equation

a x^{2} - x y^{2} + y \sqrt{a a - x x} = 0

was given & the fluxions p & q were to be found, the surd quantity

\sqrt{a a - x x}

might be extracting its root be turned into a converging series, but this is not necessary. Put

z = \sqrt{a a - x x}

, & the first equation will become

a x^{2} - x y^{2} + a y z

577

+ a y z = 0

& the second

z z = a a - x x

. & thiese gives the fluxional equations

2 a x p - p y^{2} - 2 x q y + a q z + a y r = 0

2 r z = −2 p x

; & by putting

\frac{−2 p x}{z}

, that is

\frac{- p x}{\sqrt{a a - x x}}

for r, the firs &

\sqrt{a a - x x}

for z, the first fluxional exquation becomes

2 a x p - p y^{2} - 2 x q y + \frac{a^{3} q - a q x^{3} + a y p x}{\sqrt{a a - x x}} = 0

. So you have the relation of the fluxions p & q without turning extracting the root of

a a - x x

. And this is what Mr Collins means by saying that the method gives the areas & solid contents & surfaces of figures &c without any extraction of roots. This Compendium is called Analysis per æquationes numero terminorum infinitas to signify that it is not merely a particular method of squaring figures by converging series, but en a general method of Analysis — teaching first howe to reduce finite equations & other given Data in Problemes to converging series to Equations including converging series whenever it shall be found necessary & then how to work in such Equations as well as in finite ones untill the Problem be solved. And this we are told in the Compendium it self where its said. Et quic kquid quicquid Vulgaris Analysis per æquationes ex finito terminorum numero constantes (quando id sit possibile) perficit, hæc per æquationes infinitas semper perficiat: Vt nil dubitaverim etiam nomen Analyseos etiam huic tribuere. For it teaches how to resolve finite equations & finite terms of equations into converging series whenever it shall be necessary, & then how by the Method of Moments or fluxions to apply Equations both finite & infinite to the solving solution of all Problemes. It begins where Dr Wallis left off, & founds the method of Quadratures upon these Rules. Dr Wallis published his Arithemtica Ininitorum – – – – – – – he calls the second Difference the Difference of Moments. And that you may know what kind of Calculation Mr Newton used in or before yethe year 1669 when he wrote this Compendium, you read the Demonstration of the first Rule above mentioned set down in the end of the Compendium. And by the same way of working the second Rule may be also demonstrated. And if any equation – – – – fluxions of the first Area. And as by this way of working the Ordinate may be deduced from any equation expressing the relation between the Abscissa & the Area

579 H. 2 Wallis in his opus Arithmeticūum published 1657 A. 1657 cap 33 Prop 68 by perpetual division reduced the Algebraic fraction

\frac{A}{1 - R}

by perpetual Division into the series

A + AR + {AR}^{2} + {AR}^{3} + {AR}^{4} + &c

. H 63 TIn April 1688 Vicount Brunkers Quadrature of squared the Hyperbola was published by this series

\frac{1}{1 \times 2} + \frac{1}{3 \times 4} + \frac{1}{5 \times 6} + \frac{1}{7 \times 8} + &c

that is by this

1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \frac{1}{7} - \frac{1}{8} + &c

, & the Quadrature was published in April 1668. 13. 4 Mercator about foure or five in Iune or Iuly following months after published a Demonstration of this Quadrature by the division of DrWallis. 8 Gregory soon after published a Geometrical Demonstration thereof & months after Mr Collins sent thes sending these books to Dr Barrow & Mr Collins after 14. 8 Newton having found a general method of squaring Curves & solving new Analysis composed of a double method the one of ed converging series the other of Moments; wchwhich Analysis wchwhich extended to yethe Quadrature of all Curves, & the solving of such other Problems as were reducible to Quadratures, & gave the Ordinates Tangents & Quadratures of these Curves then called Mechanical & communicated a Tract Trac small Tract upon this subject to Dr Barrow then Mathematical Professor at Cāambridge & Dr Barrow communicated it to Mr Iohn Collins in Iuly 1669. In this Tract amongst other Theorems wasere these two series. Let Theorems an Let the Radius of a circle be 1, & the sine x, & arc z &

z = x + \frac{1}{6} x^{3} + \frac{3}{40} x^{5} + \frac{5}{112} x^{7} + \frac{35}{1152} x^{9} + &c

x = z - \frac{1}{6} z^{3} + \frac{1}{120} z^{5} - \frac{1}{5040} z^{7} + \frac{1}{362680} z^{9} - &c

Mr Iames Gregory having y received one of Mr Newtons series from MrCollins after some consideration found the method of series in December 161670, & in the beginning of yethe next year 15 Feb. 1701 sent Mr Collins some other series Theorems of the same kind particularly wthwith liberty to communicate them freely. Amongst othe Theoremes was this particularly this. Let the Radius be r, the Arc a & the tangent t

a = t - \frac{t^{3}}{3 r^{2}} + \frac{t^{5}}{5 r^{4}} - \frac{t^{7}}{7 r^{6}} + \frac{t^{9}}{9 r^{9}} - &c

These Theoremes Mr Gregory gave Mr Collins full liberty to communicate to whom he pleased & Mr Collins was very free in communicating the what he had received both from Mr Newton & from MrGregory. 6 Mr Leibnitz was that year in London & in the year 16671 published there two pieces, the one dedicated to the Royal Society the other to yethe Academy at Paris, & in the dedication of the first he mentions his correspondence wthwith Mr Oldenburg. I In March 1673 he went thence to Paris In th & in the next year in Iuly & October wrote two Letters to Mr Oldenburg in yethe first of wchwhich he represented that he had a wonderfull Theoreme by the help of wchwhich the A wchwhich gave the Area of a Circle or any Sector thereof exactly in a series of rational numbers; & in the second he described this Theoreme a little further saying that it gave hi he had found the circumference of a circle in a series of very simple numbers, & that by the same method any arc whose sine was given might bybe found in a like series tho the proportion to the circumference be not known. His Theorem was therefore wchwhich gave for finding the ar him the Area of a Circle or of any Sector thereof was for finding the Arc whose sine was given. If the proportion of the Arc to the circumference was not known the Theorem or method gave him only the Arc: if it was known it gave him onl also the whole circumference. But the Demonstration of this Theorem he wanted & therefore had not invented it himself. For in his Letter of 12 May 1676 he desired Mr Oldenburg to procure the Demonstration from Mr Collins, meaning the method by wchwhich Mr Newton found had invented it. 140 7. Mr Oldenburg in a letter to Mr Leibnitz dated 15 Apr. 1675, sent him about eight of Mr Newtons & Mr Collins's Gregories series amongst wchwhich was were the Mr Newtons two series for finding the arc whose sine was given & yethe sine whose arc was given & Mr Gregories series Gregories series for finding the arc by the whose Tangent giv was given. Mr Leibnits in his answer dated 20 May 1675 acknowledged the receipt of this Letter in these words. Mu Literas tuas multa fruge Algebraica refertas accepi pro quibus tibi et doctissimo Collinio gratias ago. Cum nunc præter ordinarias curas Mechanicis imprimis negotijs distrahar, non potui examinare series quas misistis, ac cum meis comparare. Vbi fecero perscribtm tibi sententiam mean: nam aliquot jam anni sunt quod inveni meas via quadam sic satis singulari. What his own series were is unknown to this day. For he has never yet produced any other series then those wchwhich he received in this Letter. And what he did wthwith Gregories series for finding the Arc whose Tangent was is given he has told us in the Acta Eruditorum menseis Aprilis pag 178. Iam anno 1675, saith he, compositum habebam opusculum Quadraturæ Arithmeticæ ab amicis ab illo tempore lectum &c. 28 Mr I. Gregory died in the latter end of yethe year 1675 & Mr Leibnitz wrote from the next year from Paris to London Mr Oldenburg that his things at the request of Mr Leibnitz & his friends & some others of the Academy at Paris. Mr Collins drew upon extracts of his papers & Letters & the collection is still extant in the hand of Mr Collins with this litle Exactracts from Mr Gregories Letters to be lent Mr Leibnitz to peruse who is desired to return the same to you. These were accordingly sent to Paris between the 14th of Iune & a little before yethe 11th of August 1676 as appers by a letter of Mr Collins dated 11th of August 1676 of that date, ‡ ‡ & by the Answer of Mr Leibnitz dated 27 Aug. 1676, in wchwhich he writes Ad alia tuarum literarum venio quæ doctissimus Collinius communicare gravatus non est. Vellem adjecisset appropinquationis Gregorianæ linearis Demonstrationem. Credo tamen aliam haberi simpliciorem, etiam in infinitum euntem; quæ fiat sine ulla bisectione anguli, imo sine supposita circuli constructione; solo rectarum ductu. Vellem Gregoriana omnia conservari. Fuit enim his certe studijs promovendis aptissimus &c. The same thing appears also by a Letter of Mr Tscunhaouse By this Answer I gather that that yethe Extracts were sent at yethe same time wthwith Newtons Letter vizt Iune 26th. That they were sent appeasappears further bey La Letter of Mr Tschurnhause from Paris to Mr Oldenburg dated 1 Sept 1676, wchwhich & by a letter of Mr Tschurnhause from Paris to Mr Oldenburg dated 1 Sept. 1676, wchwhich ends wthwith these words. Similia porro quæ in hac re [id est in methodo serierum Newtoni] præstitit eximius Geometra Gregorius memoranda certe sunt, et quidem optimæ famæ ipsius consulturi sunt, qui ipsius relicta Manuscripta luci publicæ ut exponantur operam navabunt. In this Collection was Mr Gregories Letter of 15 Feb. 1671 wherein he communicated several series to Mr Oldenb Collins & among others the series above mentioned for finding the Arc by the whose Tangent was given. But Mr Leibnitz notwithstanding persisted in his designe of making himself yethe Inventor of this series. If an Equation contein two unknown 103 For in his Answer dated 27 Aug 1676 he sent back this series to Mr Oldenburgh to be communicated to Mr Newton as his own pretending that he had found it out three years before or above.. And he endeavoured also to claim yethe two other following series from Mr Newton. Let

1 - m

be any number less then an unit & let the Hyperbolic Logarithm be l, & on will be

= \frac{l}{1} - \frac{l 3}{1 \times 2} + \frac{l 3}{1 \times 2 \times 3} - \frac{l 4}{1 \times 2 \times 3 \times 4}

&c. Let

1 + m

be any number greater then an unit & l its m will be

\frac{l}{1} + \frac{l 3}{1 \times 3} + \frac{l 3}{1 \times 2 \times 3} + \frac{l 4}{1 \times 2 \times 3 \times 4} + &c

Let p be the Radius & a the arch of a circle, & the sine of the Complement will be

1 - \frac{a^{2}}{1 \times 2 \times 3} + \frac{a 4}{1 \times 2 \times 3 \times 4 \times 5}

&c. These series he pretended580 pretended to have found before he received Mr Newtons Letter of 13 Iune 1676, tho at his own request Mr Newton had sent him in that Letter the method of fir wthwith of finding them. And thus much concerning the method of converging Series. 140 In the Compendium above mentioned Mr Newton considered indeterminate quantities as increasing in time & from the flowing & moments of time gave the name of fluxions to the velocities wherewith quantities increased & that of moments to their parts generated in each moment of time. He exposed time by any line flowing uniformly & for the & most commonly by the Abscissa of a Curve, & for the fluxion of time or of its exponent he put an unit & for its moment the letter o, & for the other flowing quantities he put any letters or symbols & for their fluxions any other letters or symbols, & for their moments he put their fluxions multiplied by the moment o. For fluxions are finite quantities & to make them infinitely he multiplied them by the moment o. In demonstrating Propositions he it considers the moment o as indefinitely but not infinitely & performs the whole operations in finite quantities & finite figures by the Geometry of Euclide & then supposes that yethe moment o varies h decreases in infinitum & vanishes. In finding out Propositions he considers to the moment o as infinitely little, forbears to write it down, w & works in figures infinitely little by such approximations as he thinks will make no error in the conclusion. An instance of the first way of working you have in demo neare yethe end of the Compendiump 19 in demonstrating the first of the three Rules upon wchwhich the Compendium is founded. A description of the second way you have four or five pages beforep 14, 15 where he considers the Ordinate of a curve moving uniformly upon the Abscissa to describe the Area, & considers a point or infinitely short line as the moment of a line, & a line or infinitely narrow surface as the moment of a surface, & a surface or infinitely thin solid as the moment of a solid, & puts the lines BK(1) & AK(y) for the moments of two Surfaces, the coefficient o being understood to make these lines infinitely narrow surfaces. B And by this method of moments he applies æquations both finite & infinite to the solution of Problems & describes this method to be very universal & gives it the name of Analysis. 11 If an Equation contein 581 And the same is manifest also by what Mr Leibnitz wrote in the Acta Eruditorum Anno 1691 concerning this matter. Iam anno 1675, saith he, compositum habebam opusculum ab amici Quadratuæ Arithmeticæ ab ab amicis ab illo tempore lectum, sed quod materia sub manibus crescente, limare ad editionem non vacavit, postquam aliæ occupationes supervenere; præsertim com nunc prolixius exponere vulgari more quæ Analysis nostra nova paucis exhibet, non satis operæ pretium videatur. This Quadrature he composed vulgari more & began to communicate in it at Paris in the year 1675. The next year he was polishing the Demonstration thereof to send it to Mr Oldenburg as he wrote to him in his lettedr dated 12 May 1676. All this is The winter following he returned from Paris into Germany & to enter upon public business & had no longer any leasure to fit it for the press, nor thought it afterwards worth his while to explain it those things prolixly in the vulgar manner after wchwhich his new Analysis exhibited in short. He found this new method Analysis therefore after his return into Germany & by consequence not before the year 1677. At The Marquess de L'Hospital (a person of very great candor) in the Preface to his book De Analysis quantitatum infinite parvarum tells us that a little after the publication of the method of Tangents of Descartes, M Fermat found also a method wchwhich Des Cartes himself allowed at length allowed to be for the most part more simple then his own. But it was not yet so simple as D M. Barrow afterwards made it by considering more nearly the nature of polygons wchwhich offers naturally to the mind a little triangle composed of a particle of the Curve lying between two Ordinates infinitely neare one another & of the difference of these two Ordinates & of that of the two correspondent Abscissas the Ordinate & the subtangent: so that by our simple Analogy this last method saves all the calculation which was requisite either in the method of Des Cartes or in this same method before. M. Barrow stopt not here, he invented also a sort of calculation proper for this method. But this was wanting in it as well as in that of Descartes, namely, to But it was necessary in this as well as in that of Deslcartes to take away fractions & radicals for making it usefull. By Upon the defect of this calculus that of the celebrated Mr Leibnitz was added introduced & this learned Geometer began where Mr Barrow & others had left off. His calculus This his calculus went through countries led into regions hitherto unknown & there made discoveries wchwhich astonished the most able mathematicians of Europe &c. Thus far the Marquess. He had not seen Mr Newtons Analysis nor his Letters of 10 Decem 1672, 13 Iune 1676 & 24 Novem 1676: & so not knowing what Mr Newton had done, attributed recconed that Mr Leibnitz began where Mr Barrow left off, & by teaching how to apply Dr Barrow's method wthwithout sticking at fractions & surds, had enlarged that method wonderfully. Mr Newton How Mr Newton described his method of fluxions & moments in his Analysis communicated by Dr Barrow to Mr Collins in r Iuly 1669 we have has been shewed above Dr Barrow published his method of Tangents in the year 1670: Mr N. in his Letter of dated 10 Decemb. 1672 communicated his method of Tangents to Mr Collins, & then added. Hoc est unum particulare . . . . . . . . . reducendo eas ad series infinitas. Memini Memini me ex occasione aliquando narrasse D. Barrovio edendis Lectionibus suis occupato, instrctctum me esse hujusmodi methodo tangentes ducendi sed nescio quo diverticulo ab ea ipsi describenda fuerim avocatus. Slusij methodum tangentes ducendi brevi publice prodituram confido. Quamprimum advenerit exemplar ejus ad me transmittere ne grave ducas. Mr Slusius sent his method to Mr Oldenburg 17 Ian 1673 & the same was soon after published in the Transactions. It proved to be the same wthwith that of Mr Newton. It was founded upon three Lemmas the first of wchwhich was this Differentia duarum dignitatum ejusdem gradus applicata ad differentiam laterum, dat partes singulares gradus inferioris ex binomio laterum; ut

\frac{y^{3} - x^{3}}{y - x} = y y + y x + x x

. One O m That is (in the language of Mr Leibnitz)

\frac{{dy}^{3}}{dy} = 3 y y

. A copy of Mr Newton's Letter of 10 Decem. 1672 was sent to Mr Leibnitz by Mr Oldenburg amongst the papers of Mr Iames Gregory at the same time 20th Mr Newton's Letter of 13 Iune 1676. And Mr Newton having described in these two Letters that he had a very general Analysis consisting in the method of infinite converging series, partly in another method by wchwhich he applyed those series to the solution of almost all problems & found the tang tangents areas, lengths, solid contents, centers of gravity, & curvities of crooked lines curves & curvilinear figures without sticking at surds or mechanical curves & that the method of Tangents of Slusius was but a branch or corollary of this method: Mr Leibnitz in his return home through Holland was meditating upon this method the improvement of the method of Slusius For in a Letter to Mr Oldenburg dated from Amsterdam

\frac{18}{28}

Novem. 1676 he wrote thus. Methodus Tangentiusm Slusija Slusio publicata nondum rei fastigium tenet. Potest aliquid amplius præstari in eo genere, quod maximi fore usus ad omnis generis Problemata: etiam ad meam (sine extractionibus) Æquationum ad series reductionem. Nimirum, posset brevis quædam calculari circa Tangentes Tabula, eouque continuanda donec progressio Tabulæ apparet; ut eam quisque scilicet quisque, quousque libuerit, sine calculo continuare possit. This was the improvement of the Method of Slusius wchwhich Mr Leibnitz was then thinking upon & by his words posset Potest aliquid amplius præstari in eo genere quod maximi foret usus ad omnis generis problemata, it seems to be the only improvement wchwhich he had then in his mind for making the method of Slusius general. the The improvement by the differential calculus was not yet in his mind. In spring following he received Mr Newtons Letter dated 242 Octob 1676: in wchwhich MrNewton mentioned the Analysis communicated by Dr Barrow to Mr Collins & also another Tract written in 1671 about converging series & about the another method by wchwhich Tangents were drawn after the method of Slusius & maxima & minima were determined & the Quadrature of Curves was made more easy &c &c this without sticking at radicals. &And the foundation of these operations he comprehended in this sentence exprest enigmatically. Data æquatione fluentes quotcunque quantitates involvente fluxiones invenire & vice versa. Which puts it past all dispute that he was then he had invented the method of fluxions before that time. And if other things in that Letter be considered it will appear that he had then brought to it to great perfection, the Propositions in his book of Quadratures & the methods of converging series & of drawing a Curve through any number of given points being then known to them. After the receipt of this Letter Mr Leibnitz wrote back that wrote back in a letter dated 21 Iunij 1677: Clarissimie Slusij methodum Tangentium nondum esse asbsolutam Celeberrimo Newtono assentior. Et jam a multo tempore rem Tangentium generalius tractavi scilicet per differentias Ordinatarum. — Hinc nominando in posterum dy differentiam duarum proximarum y &c. Here Mr Leibnitz began first to propose his differential method: & there is not the large evidence that he knew it before this year. He affirms indeed that C jam a multo tempore rem tangentium generalius tractavi scilicet per differentias Ordinanatarum: but he canno is not a witness in his own case. A Iudge would be very unjust & act contrary to the laws of all nations who582 who should admit any man to be a witness in his own case. And therefore it lies upon Mr Leibnitz to prove that he found out this method before the receipt of Mr Leibnitz Newton's Letters. And if he cannot prove this, the Question, Who was the first Inventor of the Method, is an sed decided in favour. Dr Barrow in his method of Tangents puts the letter drawing two Ordinates indefinitely neare one another puts the letter a for the difference of the Abscissas infinitely near one another & the letter e for the difference of the Ordinates & for drawing the tangents gives these three Rules. 1 Inter computandum, saith he, omnes abjicio terminos in quibus ipsarum a vel e potestas habetur vel in quibus ipsæ ducuntur in se. Etenim isti termini nihil valebunt. 2 Post æquationem constitutāam omnes abjicio terminos literis constantes quantitates notas seu determinatas significantibus aut in quibus non habentur a vel e. Etenim illi termini semper ad ad unam æquationis partem adducti nihil adæquabunt. 3 Pro a Ordinatam & pro e subtangentem substituo. Hinc demum subtangentis quantitas dignoscetur. This Mr Leibnitz in his Letter of 21 Iune 1677 abovementioned has followed this method exactly excepting that he has changed the letters a & e of Dr Barrow into dy & dx. For he pr draw in the Example wchwhich he then gives he draws two parallel lines & sets all the terms in w below the under line in wchwhich dx & dy are but of of jo (severally or joyntly) of more then one dimension & all the terms above the upper line in wchwhich dy & dx are wanting & for the reasons given by Dr Barrow makes all these terms vanish. And by the terms in wchwhich dy & dx are but of one dimension he wchwhich he sets between the lines he determines the positi length of proportion of yethe subtangent to yethe ordinate. Well therefore did the Marquess de L' Hospital observe that where Mr Newton Barrow left off Mr Leibnitz began: for their methods of Tangents are exactly yethe same In the next place Mr Leibnitz shews how this method may become more large & co But Mr Leibnitz observes that the conclusion is coincident wthwith the Rule of Slusius & shews how that Rule presently occurrs to any one who understands this method. And in the next place — — — — — — — — — — In the next place Mr Leibnitz observes shews how this method of tangents may her v be enlarged improved so as comprehend to procead in more unknown quantities then two & not to stick at radicalls. And then in relation to what Mr Newton had told him of these improvements he adds. Arbitror quæ celare voluit Newtonus de tangentibus ducendis, ab his non abludere. Quod addit, Ex hoc eodem fundamento quadraturas quoque reddi faciliores, me in hac sententia confirmat; nimirum semper figuræ illæ sunt Quadrabiles quæ sunt ad æquationes differentialem. By wchwhich words its manifest that Mr Leibnitz at this time understood that Mr Newton had a method wchwhich would do all these things & that he was his method was either the same with Dr Barrows method of Tangents improved & made general or another like it. At length, vizt in the year November 1684, Mr Leibnitz published the Elements of his differential method in the Acta Eruditorum & illustrated it wthwith examples of drawing Tangents & determining maxima & minima, & then added: Et hæc quidem initia sunt tantum Geometriæ cujusdam multo sublimoris ad difficillima et pulcherrima quæque etiam mistæ Matheseos Problemata pertingentis, quæ sine calculo nostro differentiali, AUT SIMILI non temere quisquam pari facilitate tractabit. The words AVT SIMILI plainly relate to Mr Newtons method. And in the Acta Eruditorum of Iune 1686 pag 297 he added Malo autem dx et similia adhibere quam literas pro illis quia istud dx est modificatio quædam ipsius x &c. He knew that in this method he might have used better with Dr Barrow but chose rather to use the new symbols dx & dy, tho there is nothing wchwhich can be done by these symbols but may be done by letters wthwith more brevity. When Mr Newton wrote his Analysis he used letters for fluxions Mr Leibnitz & the rectangles a under those Letters & the moment o for moments. Mr Leibnitz has no Letters to this day symbols for fluxions & for to to this day & for moments wchwhich he calls differences he uses the same therefore al Mr Newton's symbols for fluxions are the oldest older then any of And the Woman into the true Church into the Ted first Temple wthwith seven Candlesticks the Lamb & the second Temple with two Candlesticks, the Lamb with seven eyes & seven horns in the first Temple. & the son of Man with two eyes & two horns Leggs in the seco on mount Sion, h the 144000 who in the persecution now raised by the of the remnant of the womans seed nor raised by the Dragon remain receive the mark of God in their foreheads & remain wthwith the Lamb on mount Sion & worship in the first Temple & & the And the true Church becomes divided into the first Temple where with seven golden Candlesticks where the Dragon makes war upon the rem of the womans seed & into the second Temple wthwith two Candlesticks of olive tree where the Gentiles composing the body of the Beast worship in the The Empire becomes divided into the Dragon who goes being returned back from from the woman 1 begins to make war upon the remnant of her seed left by her in the outward court of the first Temple, & the Beast who now rises out of the Sea & whose people are the Gentiles to whom the outward Court of the second Temple is given. And the church at the same timetime becomes divided into two false churches The Act of 18 Car. II chap. 5, saith that no moneys leviable & payable by this Act shall be applied or converted to any use or uses whatsoever other then to the defraying the charge or expencs of the Mint or Mints & [that] of Assaying melting down wast & coynage of Gold & Silver & the encouragement of bringing in of God & Silver into the said Mint or Mints there to be coyned &c. The first is the charge of supporting a standing Mint (if I mistake not) & the second is the further charge of carrying on the st coynage in that Mint. And by the next clause of the Act. the first is limited to 3000li for preventing extravagance & the other is left unlimited for encouraging the coynage. The words are. And it is hereby further enacted that there shall not be issued out of the Exchequer of the said moneys in any one year, for the fees & Salaries of the Officers of the Mint or Mints & towards the providing maintaining & repairing of the Houses Offices & Buildings & other [standing] necessaries for assaying melting down & coynage & buying in of gold & silver to be coyned, & not otherwise coyning above the summ of three thousand pound 583 Argumentum The Letters & Papers till the year 1676 inclusively shew that Mr Newton had a general method of solving Problems by deducing them to equations finite or infinite, whether those equations include moments (the exponents of fluxions) or do not include them, & by deducing fluents & their moments from one another by means of those equations. The Analysis printed in the beginning of this collection shews that he had such a general method in the year 1669. And by the Letters & Papers wchwhich follow, it appears that in the year 1671, at the desire of his friends he composed a larger Treatise on the same method (p. 27. l. 10, 27 & p. 71. l. 4, 26) that it was very general & easy without sticking at surds or metchanical curves & extended to the finding tangents, areas lengths centers of gravity & curvatures of Curves &c (p. 27, 30, 85) that in Problemes reducible to quadratures it proceeded by the Propositions since printed in the book of Quadratures, wchwhich Propositions are there founded upon the method of fluents (p. 72, 74, 76) that it extended to the extracting of fluents out of equations involving their fluxions & proceeded in difficulter cases by assuming the terms of a series & determining them by the conditions of yethe Probleme (p. 86) that it determined the species of the curve by the length thereof (p 24) & extended to inverse Problemes of Tangents & others more difficult, & was so general as to reach almost all Problemes except numeral ones like those of Diophantus (p. 55, 85, 86). And all this was known to Mr Newton before Mr Leibnitz began to write of the method as appears by the dates of their Letters. For in the year 1673 Mr Leibnitz was upon another differential method (p. 32) In May 1676 he desired Mr Oldenburg to procure him the method of infinite series (p. 45) & in his Letter of Aug. 27th 1676 he wrote that he did not beleive that Mr Newton's method to be so general as Mr Newton had described it. For, said he, there are many Problemes & particularly the inverse Problemes of Tangents wchwhich cannot be reduced to equations or quadratures (p. 65) wchwhich words make it evident that Mr Leibnitz had not yet the method of differential equations. And in the year 1675, he communicated to his friends at Paris a Tract written in a vulgar manner about a series wchwhich he received from Mr Oldenburg & continued to polish in the year 1676 wthwith intention to have it published584 published, but it swelling in bulk he left off polishing it after other business came upon him, & afterwards finding out the differential Analysis he did not think it worth publishing because written in a vulgar manner p. 42, 45. In all these Letters & Papers there appears nothing of his knowing the Differential method before yethe year 1677. It is first mentioned by him in his Letter of Iune 21th 1677, & there he began the description of it wthwith these words: Hinc nominando IN POSTERVM dy differentiam duarum proximarum y &c p. 88. 585 — And in the mean time I take the liberty to acquaint him, that by taxing the Royal Society with partial injustice in giving sentence against him without hearing both parties, he has transgressed one of their Statutes which makes it expulsion to defame them. The Philosophy wchwhich Mr Newton in his Principles & Opticks has pursued is experimental, & it is not the business of experimental Philosophy to teach the causes of things any further then they can be proved by Experiments. We are not to fill this Philo sophy with opinions which cannot be proved by phænomena. In this Philosophy Hypotheses have no place unles as conjectures or Questions proposed to be examined by experiments. For this reason Mr Newton in his Optiques distinguished those things wchwhich remained uncertain & wchwhich he therefore proposed in the end of his Optiques in the form of Quæres. For this reason, in the Preface to his Principles, when he had mentioned the motions of the Planets Comets Moon & Sea as deduced in this book from gravity, he added: Vtinam cætera Naturæ Phænomena ex Principijs Mechanicis eodem argumentandi genere derivare liceret. Nam multa ne movent ut nonnihil suspicer ea omnia ex viribus quibusdam pendere posse quibus corporum particulæ per causes nondum cognitas vel in se mutuo impelluntur & secundum figuras regulares cohærent, vel ab invicem fugantur & recedunt: quibus viribus ignotis Philosophi hactenus Naturam frustra tentarunt. And in the end of this book in the second Edition, he said that he forbore to describe the effects of this [electrical] attraction for want of a sufficient number of experiments to determin the laws of its acting. And for the same reason he is silent about the cause of gravity, there occurring no experiments or phænomena by wchwhich he might prove what was the cause thereof: And this he hath abundantly declared in his Principles neare the beginning thereof in these words: & Virium causes et sedes Physicas jam non expendo. And a little after: Voces attractionis, impulsus vel propensionis cujuscunque in centrum indifferenter & pro se mutuo promiscus usurpo, has vires non physice sed mathematice tantum considerando. Vnde caveat Lector ne per hujusmodi voces cogitet me speciem vel modum actionis, causamve aut rationem physicam alicubi definire, vel centris (quæ sunt puncta Mathematica) vires vere et physice tribuere, si forte aut centraa trahere aut vires centrorum esse dicero. And in the end of his Opticks: Qua causa efficiente hæ attractiones [sc. gravitas, visque magnetica et electrica aliæque] peragantur, hic non inquiro. Quam ego attractionem appello fieri sane potest ut ea efficiatur impulsu vel alio aliquo modo nobis incognito. Hanc vocem attractioneis ita hic accipi velim ut in universum solummodo vim aliquam significare intelligatur qua corpora ad se mutuo tendant, cuicunque demum causæ attribuenda sit illa vis. Nam ex phænomenis Naturæ illud nos prius idoctos oportet quænam corpora seinvicem attrahant, et quænam sint leges et proprietates istius attractionis, quam in id inquirere par sit quanam efficiente causa peragatur attractio And a little after he mentions the same attractions as forces which by phænomena appear to have a being in nature, tho their causes be not yet known, & distinguishes them from occult qualities which are supposed to flow from the specific forms of things. And in the Scholium at the end of his Principles, after he had mentioned the properties of gravity, he added: Rationem vero harum harum gravitatis proprietatem ex phænomenis nondum potui deducere et hypothesis non fingo. Quicquid enim ex phænomenis non deducitur Hypothesis vocanda est, et Hypotheses seu Metaphysicæ seu Physicæ seu qualitatum occultarum seu Mechanicæ, in philosophia experimentali locum non habent. — Satis est quod Gravitas revera existat & agat secundum leges a nobis expositas, & ad corporum cœlestium et maris nostri motus omnes sufficiat. And after all this one would wonder that Mr Newton should be reflected upon for not explaining the cause of gravity by an Hypothesis, as if it were a crime to content himself with certainties & let uncertainties alone. 2 Whether the cause of gravity be mechanical or not Mechanical he hath no where affirmed And yet the Secretary of the Editors of the Acta EruditorumaAnno 1714 mense Martio p. 142 2 hath have accused him of denying that the cause of gravity is mechanical. tho he hath not yet declared any opinion — not yet said whether mean about that cause it be mechanical or not He hath no where said that Gravity is essential to matter or an occult quality or a miracle, & yet Mr LeibnitzbIn Tractatu De Bonitate Dei & in Epistolis ad D. Hartsoeker, & alibi hath accused him of making gravity a natural or essential property of bodies & an occult quality & a miracle. It lies upon Mr Leibnitz therefore in point of candor & justice to beg Mr Newton's pardon publickly for endeavouring by such indirect & unfair practises to defame him & prejudice the learned part of Europe against him It is true that the Philosophy of these two Gentlemen differ very much in Philosophy. The one proceeds upon the evidence arising from Experiments & Phænomena, & stops where such evidence is wanting the other is taken up with Hypotheses, & propounds them not to be exaimined by Experiments, but to be beleived without examination. The one doth not presume to affirm that God cannot chuse or act by the power of his will in matters indifferent: the other affirms that nothing is done without a [fatal] reason. The one, for want of experiments to decide the Question, doth not affirm whether the cause of gravity be mechanical or not mechanical: the other, that it is a perpetual miracle if it be not mechanical. The one, by way of inquiry, attributes it to the will of the creator that the least particles of matter are hard: the other attributes the hardness of matter to conspiring motions, & calls it a perpetual miracle if the cause of this hardness be other than mechanical. The one doth not affirm that the animal motion in man is purely mechanical: the other doth a f teaches that it is purely mechanical; the soul or mind (according to the Hypothesis of an Harmonia præstabilita) never acting upon the body, or never without a miracle. so as to alter or influence its motions. The one teaches that God (the God in whom we live & move & have our being) is Omnipresent: the other that he is not as the soul of the world: the other that he is not the soul of the world but INTELLIGENTIA SVPRAMUNDANA a God thats no where in an Intelligence above the bounds of the world: Whence it may seems to follows that he cannot act upon do any thing in within the world without unless by an incredible a miracle. The one teaches that Philosophers are to argue from Phænomena & Experiments first to the causes thereof, & thence to the causes of those causes, & so on till we come to the first cause: the other that all the actions of God are miracles, & all the laws imprest on nature by thise will of God are perpetual miracles & occult qualities, & therefore not to be considered by Philosophers. But why must it go for a miracle or wonder if God if the first cause of things & the governor of the Vniverse hath intermedled with the world since the first creation, & somso hath any thing to do with the world? Why must heGod be banished removed out of the uVniverse? bounds of the Vniverse? Why must the body of a man be a mere Machin acting without the influence of his mind & nothing but matter be left within those bounds wWhy must the laws of nature be called miracles & occult qualities (that is to say, wonderful absurdities) if derived from the will power of God, be called miracles & occult qualities, that is to say, wonders & absurdities? And why must all the arguments for a God taken from the Phænomena of Nature, be exploded by new hard-names? For certainly Philosophers are rather to argue without railing, & thaton to rail without arguing. 586 But [if the first cCause of things & the Author of the Vnivers may have some times & in some places intermedled with the World since the first creation; if he is not to be removed out of the bounds of the Vniverse;] And if the body of a man be not a meer machine acting without the influence of his mind; And if God be not to be removed out of the bounds of the Vniverse; if the first cause of things & the Author of the Vnivers may have sometimes & in some places intermedled with the World since the first creation; if he created all things for certain reasons & with certain designes: why must the laws of Nature if derived from the will of God be called miracles & occult qualities, And why that is to say, wonders & absurdities? A & why must all the Phæ arguments for a Deity taken from Phænomena be exploded by hard names? For certainly Philosophers are to argue without railing & not to rail without arguitng. They are to examin things Opinions by experiments & Phænomena & not to broach & propag publish & press opinions to be believed wthwithout examination. The Editors of the Acta Eruditorum have indeed accused Mr Newton of publishing an Hypothesis in the end of his Principles about the atteions of actions caused by of a very subtile spirit. But And if he had done so, yet its more excusable to propose one Hypothesis amongst many things proved then to propose nothing but Hypotheses. But Mr Newton did not propose it by way of H an Hypothesis but in order to an inquiry He had told his friends that as by his words may appear to any unprejudiced person. After he had shewed the laws, power & effects of Gravity without medling with the cause thereof & from thence deduced all the motions of the heaven bodies & the Planets, Equinoxes, Comets, & Sea Eart great bodies in the system of the Vniverse, he added in a few words his suspicions about another sort of attraction between the small parts of bodies upon wchwhich many Phæneoman might depend, & for want of a a sufficient number of experiments left the enquiry to others who might hereafter have time & skill enough to pursue it, & to give them some light into the enquiry mentioned two or three of the principal phænomena wchwhich might arise from Phæ the actiosn of thise Spiprit or Agent by which this Attraction is performed. He has told his friends that there are sufficient Phænomena to ground an inquiry upon but not yet sufficient to bring ma determin the laws of this attraction. And why must Mr Newton be reflected upon as introducing absurdities into philosophy miraculous & the Philosophy of Mr Newton be traduced exploded as miraculous & absurd because he will has taken time to consider whether all the Phænomena in nNature could can be solved by mere mechanical causes & has not yet declared his opinion that they can, for want of experiment & the solutions proved by experiments & for want of such such a proof has not yet declared his opinion that they can. By way of inqui He hath suggested a suspicion that there is a nother sort of attraction. & that upon wchwhich many phænomena may depend & that this attraction may be electrical & of He hath suggested a suspicion that there is a subtile Spirit or Agent latent in bodies by wchwhich Electrical Attraction & many other phænomena may be performed, & that this Agent may have great effects in many Phænomena of nature, & proposed this matter is not to be beleived without proof but to be inquired into by experiments: but the Editors of the Acta Eruditorum tell us that if this Agent be not the subtile matter of teh Cartesians it will be looked upon as a trifle. [These Gentlemen are so much accustomed to uncertain Hypotheses that we must not enquire into the causes laws & properties & effects of the Agent by wchwhich electrical attraction & is performed, & of that by wchwhich light is emitted reflected & refracted, & of that by wchwhich sensation is performed &c & inform orour selves whether they be not one & the same agent; unless we have first explained by an hypothesis what this Agent may be. We must not pursue experimental Philosophy by experiments untill we founded it upon hypotheses] And by such these indirect practises they insinuate that would have it beleived that Mr Newton was unable to find the infinitesimal method. / And must experimental Philosophy be rendred uncertain by filling it with Hypotheses Opinions not yet proved by any experiments?] Inf Mr Leibnitz never found but a new experiment in all his wlife for proving any thing; If Mr Newton has by nso great multitude of new Experiments setled a theory of discovered & proved many thingthings about light & colours never to be shaken & settled a new Theory thereof never to be shaken; If Mr Newton by athe infinitessimal Analysis apply applyed to Geometry & Mechanicks, has setled the Theory of the Heavens, & Mr Leibnitz has endeavo in his Trac Tentamen de Motuum Cœlestium causis has endeavoured to imitate him, but without success for want of skill in this Analysis. If Mr Leibnitz has pretended that he had the infinitesimal Analysis before he had it in order to step before Mr Newton; If he has done the like in several other inventions. If he has concealed from the Germans world Cul what he received from Mr Oldenburg concerning this Method & concerning the series of Gregory in order to make them his own; if he when he had possessed Geromany with an opinion that they were his own he accused Mr Newton of plagiary in order to make the same opinion be received in England, but has not produced any one good argument for proving his pretense; if for want of arguments his he has endeavoured to make himself a witness in his own cause contrary to the laws of all nations insisting upon his own candor as if it were injustice to question it & in the very same Letter questioneding the candor of Mr Newton, & by doing so has insisted upon the laws of all nations endeavoured to make himself a witness in his own cause contrary to the laws of all nations; if after he had accused Mr Newton N Keill & Mr Newton he moved the R. Society to make him recant he refused to m & was put upon proving the accusation, he refus declined to do it if for want of arguments against And must Experimental Philosophy be exploded as miraculous & absurd because [we cannot yet prove by experiments that all the Phænomena in nature can be solved by mere mechanical causes, &] do not in it is the nature of this Philosophy to assent any no thing more then can be proved by experiments, &] it is the nature of this Philosophy to it asserts nothing more then can be proved by experiments, & we cannot yet prove by experimtsexperiments that all the Phænomena in Nature can be proved by Experiments solved by mere mechanical causes? Certainly these things deserve to be better considered. So then Mr Leibnitz 587 Mr Leibnitz therefore in his natur Mr Gregory being told by Mr Collins that I had a general method of series & having received one of my Series from Mr Collins, after a year's study found out the method. Mr Leibnitz had more light into the method of fluxions & the method was much easier to be found out. And perceiving by any Letters that it was readily gave the method of tangents of Slusius, he was in his journey from London through Holland into Germany studying how to make the method of Tangents of Slusius extend to all Problems as I find by a letter of his dated from Amsterdam. And at length in a Letter dated from Hannover 21 Iune 1677 he wrote back Clarissimi Slusij methodum tangentium nondum esse absolutam celeberrimo Newtono assentior. Et jam a multo tempore rem tangentium longe generalius tractavi. And then subjoyned Dr Barr a method of tangents published by Dr Barrow in the year 1670, but changed the disguised it by a new notation to make it his own, & shewed how this method readily gave the Rule of Slusius & might be improved so as not to stick at surds. And then from these characters concluded that from these of q he took it to be like my method to be like it, especially since both of them faciliated Quadratures. In the year 1684 (Mr Oldenburg & Mr Collins being newly then dead) he published this method of Tangents as his m so far as it related to tangents & maxima & minima & added that it extended to the abstruser Problems of Geometry or another like it, mꝑ such as could not be solved without this method or another like it, meaning my method, but did not yet shew how to apply it to such abstruser Problems. In the year 1687 my book of Principles came abroad, which was full of such Problems as (according to Mr Leibnitz) could not be resolved without the differential method or another like it. And Mr Leibnitz in a Letter to me dated from Hanover 17 March 1693 & still extant, acknowledged this & the same thing of this book. And in his Answer the Acta Eruditorum for May 1700 p. 306. l. 89 said further of this Book, that no man before Mr Newton had by a specimen made publick, proved that he had this method. This book was therefore by the acknowledgement of Mr Leibnitz himself in those days, the first specimen made publick of applying this method to the difficulter Problemes of Geometry. And the next specimen was that of three papers published by Mr Leibnitz in the Acta Eruditorum of the year 1689 concerning Opticks, the resistance of Mediums, & the systeme of the heavens: All wchwhich were nothing else then Mr Newton's Propositions taken from Mr Newtons my book of Principles & put into a new form of words & intermixt with some hyp physical Hypotheses & claimed by Mr Leibnitz as invented by himself before my book came abroad. And its very remarkable that Mr Leib. to make the my Principal Proposition his own, adapted to it an erroneous demonstration by wchwhich it was impossible to invent it. Hitherto the Differential Method had made no noise, but the next year began to be it was taken notice of by Mr Iames Bernoulli, & from that time to be celebrated more & more in Germany France & Holland while the method of fluxions was celebrated in England. In this state things continued till the death of Dr Wallis wchwhich happened in October 1703 And now atte upon any pub saying in the Introduction to the book of Quadratures that I found the method of fluxions gradually in the years 1665 & 1666, I was traduced in yethe Acta Eruditor.Eruditorum as a lying plagiary, & [accuse of deriving the deducing deducing v subsating theis method from the differential method of Mr Leibnitz, whom the who now beg to pretended to be the first inventor] it was pretended that when I wrote my Book of Letter Book of Principles I had no such method but deduced formed it afterwards from the differential method by who first substituted fluxions for differenses. And this accusation is still supported by Mr Leibnitz has been brought by him before the R. S. & is still supported by him therefore it lies upon him to prove it. What he saith about Philosophy is foreign to the Question & therefore I shall be very short upon it. He denys conclusions without telling the fault of the premisses. His arguments are p against me are founded upon metaphysical & precarious hypotheses & therefore do not affect me: for I meddle only with experimental Philosophy. He changes the signification of the words Miracles & Occult qualities that he may use them in railing at me universal gravity. For Miracles are so called not because they they are the actions of God but because they occat happen seldom & by happening seldom create wonder. If they happened constantly they would not be wonders. And occult qualities are decried not because their causes are unknown, but because according to the Schoolmen beleived that those things wchwhich were unknown to their Master Aristotel, could never be known. He insinuates that I make God to ascribe to God a sensorium in a literal sense, wchwhich is a mistake, & fiction hHe p presents that God must be Intelligentia supramundana least he should be the soul of the world & by the same way of reasoning argu one a man may feigne prove that the soule of a man iscannot be in his head least it should be the soul of the Images, of Objects formed in the sensorium. He represents that the God has made this world so perfect that it can last eternally without needing any amendments because God was able to make it, so, & by the same way of arguing a man may prove that matter can think. He pleads for Hypothetical philosophy because there may happen experiments to decide which of the Hypoteses are true, & yet almost all his Philosophy consists in metaphysical Hyposeses such as never were never can be decided by experiments, & one of them (that of the Harmonia præstabilita) is contrary to the daily experience of all mankind. For every man finds in himself a power of moving his body by his will And if he is happy in disciples (as he boasts,) it is because he has spent all his life in corresponding with men of all nations for propagating his opinions whilst I have rested & left truth to shift for it self Hypotheses may be propended by way of Questions to be examined by experiments: but when they are grounded as Opinions to be beleived without examination, they are turn Philosophy into a Romance. He boasts of the number of his disciples, that is of his having spent all his life in keeping a correspondance with men of all nations, whilst I keep no to make disciples whilst I keep no such correspondence but leave truth to shift for it self. 588 finding how to deduce the Method of Slusius from the Differential method of Tangents published in the year 1670 by Dr Barrow; he wrote back (21 Iune 1677) Clarissimi Slusij Methodum Tangentium nondum esse absolutam Clariss Celerrimo Newtono assentior. Et jam a multo tempore rem tangentium longe generalius tractavi, scilicet per diffentiasdifferentias Ordinatarum. And then he set down Dr Barrows method of Tangents as his own, & shewed how this method readily gave the Method of Slusius & might be improved so as not to stick at surds. And from these circumstances concluded that he took my Method to be like it, especially since both of them faciliated Quadratures. In the year 1684 (Mr Oldenburg & Mr Collins being both dead) Mr Leibnitz published this method so far as it related to Tangents & Maxima & Minima & added that it extended to the abstruser Problems of Geometry, such as could not be solved without this method or another like it, meaning my method; but did not yet shew how to apply it such abstruser Problems In the year 1687 my Book of Principles came abroad which was full of such Problems as (according to Mr Leibnitz) could not be resolved without the Differential method or another like it. And Mr Leibnitz in a Letter to me dated from Hannover 17 March 1693 & still extant, gave the same testimony of this book in these words: Mirifice ampliaveras Geometriam tuis seriebus, sed edito Principiorum opere ostendisti patere tibi etiam quæ all Analysi receptæ non subsunt. Conatus sum Ego quoque, notis commodis adhibitis quæ Differentias et Summas exhibeant, Geometriam illam quam Transcendentem appello Analysi quodammodo subjicere: nec res male processit. And in the Acta Eruditorūum for May 1700 pag. 306, lin. 89 he said further of this Book, that no man before me had by a publick specimen made publick, proved that he had this method. This Book was therefore by the acknowledgement of Mr Leibnitz himself the first specimen made publick of applying this method to the difficulter Problemes of Geometry And the next Specimen was that of three Papers published by Mr Leibnitz in the year 1689 concerning Opticks, the resistance of Mediums & the systeme of the heavens. All which were nothing else then Propositions taken from my Book of Principles & put into a new form of words & intermixt with some Physical Hypotheses & claimed by Mr Leibnitz as invented by himself long before my book came abroad. And its very remarkable that Mr Leibnitz to make my principal Proposition his own, adapted to it an erroneous Demonstration by which it was impossible to invent it. Hitherto the Differential Method had made no noise, but the next year it was taken notice of by Mr Iames Bernoulli who published an Example of this Calculation in the Acta Eruditorūum for May 1690. And from that time the Method began to be celebrated more & more in Germany France & Holland while the Method of fluxions was celebrated in England my Book of Quadratures being handed about among my friends.. And this made Mr Leibnitz, after he had said in the Acta Eruditorum of May 1700 that no man before me had proved by a published specimen that he had the method, subjoyn that no man before the Bernoullis & himself had communicated the method I first published the difficulter Problemes resolved: they afterwards published the resolution of the difficulter Problemes. For as the Ancients invented their Propositions by Analysis & then compounded them, & for preserving the certainty of of Geometry, which is the glory of this science, admitted no Propositions into it till they were demonstrated by composition: so I first invented the Propositions in the Book of Principles by Analysis & then demonstrated them by composition that they might be admitted into Geometry. And tho this Book was written by Composition (as all things in Geometry ought to be) yet the Analysis of moments shines through the Composition so clearly that the Marquis de L'Hospital wrote that this Book was presque tout de ce calcule, & Mr Leibnitz himself that it was a proof that I had this Analysis, & the first public proof which any man gave that he had it. Dr Wallis died in October 1703, the last of the old men who knew what had passed between Mr Leibnitz & me by means of Mr Oldenburg. And afterwards I was accused in the Acta Eruditorum & before the Royal Society as a Plagiary who had substituted Diff Fluxions for Differences & thereby taken the Method from Mr Leibnitz. And when the Royal Society caused the ancient Letters & Papers to be remaining in their Archives & Letter Books & in the Library of Mr Collins to be published all wchwhich are unanswerable matter of fact: instead of answering the same in a fair manner, & proving his accusation of plagiary against me, a defamatory Libel was published against me in Germany without the name of the Author or Publisher or City where it was published, & dispersed over Germany France & Italy. & the Libel it self represents that Mr Leibnitz set it on foot. In the latter part of his Postscript he departs from the Question & falls foul upon my Philosophy as if I (and by consequence the ancient Phenicians & Greeks) introduced Miracles & occult qualities. And to make this appear he gives the name of Miracles or Wonders to the laws imprest by God upon Nature tho by reason of their constant working they create no Wonder; & that of occult qualities to qualities which are not occult but whose causes are occult tho the qualities themselves be very manifest. He said that God is Intelligentia supramundana because if he were in the world he would be the soul of the world, that is, he would animate the world, & yet according to his Philosophy (that of an Harmonia præstabilita) the soul of a man doth not animate his body. He accuses me as if I affirmed that God hath a Sensorium in a litteral sense. He saith that I have not demonstrated a Vacuum nor universal gravity. but he denies Conclusions without shewing the fault of the Premisses, & seems to mean that the argument of Induction from experiments upon wchwhich experimental Philosophy is founded is not a demonstration & therefore ought to be respected. He saith also that I have not proved Atomes: but I have not affirmed them but place them among a set of Queries. He saith that Space is the order of coexistences & time the order of successive existences: I suppose he meanes that space is the order of coexistences in space, & time the order of successive existences in time, or that space is space in space & time is time in time. He calls the world Gods Watch, & insinuates that it is the fault of the workman & not of the Mater materials if a Watch will at length cease to go, & in like manner that it would be Gods fault if his Watch should ever decay & want an amendments. And by the same way of arguing a man may say that it would be Gods fault if matter doth not think. He applauds experimental Philosophy, but recommends Hypotheses to be admitted into into Philosophy in order to be589 be examined by experiments: whereas almost all his Hypotheses are uncapable of such an examination, & he should recommend not Hypoteses to be admitted & beleived before examination, but Questions to be examined & decided by experiments before they are admitted into Philosophy & proposed to be beleived. And whilst he applauds experimental Philosophy & exclaims against Miracles, he introduces an Hypothesis of Harmonia præ-stabilita which cannot be true without an Miracle incredible Miracle, & is contrary to this daily experience of all mankind. For all men find by experience that they can move their bodies by their will, & that they see heare & feel by means of their bodies. And if, notwithstanding all this, he glories in the number of disciples, you know what his disciples are in England & that he has spent his life in keeping a general correspondence for making disciples, whilst I leave truth to shift for it self. For its about 40 years since I left of all correspondence by Letters about Mathematicks & Philosophy, & about 20 since I left off these studies And for that reason I hope you will pardon me if I have been averse from writing this Letter, & continue averse from being engaged in these disputes of this kind which make nothing to the Question in hand. He sends you also Mathematical Problemes to be solved by the English Mathematicians. And all this I look upon as nothing else then an amusement to avoid proving his accusation against me & returning a fair answer to the matter of fact which has been published by order of the Royall Society.Written in 592 & was trying whether Dr Barnes Differential method of Tangents might not be extended to the same performances. And In November 1684 Mr Leibnitz published the Elements of theis differential method in the Acta Eruditorum & illustrated it wthwith examples of drawing Tangents The next year Mr Newtons Principia Philosophiæ came abroad, wchwhich a Book w was full of such such the difficulter Problemes quæ sine calculo Differentiali aut simili non poss temere erant facile tractanda. And quisquam pari facilitate tractabit. And the Marquess de L'Hospital has represented this book presque tout de ce calcul; almost wholly f to consist almost wholly of this calculus. In At the end of In the second Lemma of the second book, the elements of this calculus are demonstrated synthetically & at the end of the Schol Lemma there is a Scholium in these words. In literis quæ mihi cum Geometra peritissimo G. G. Leibnitsio annis abhinc decem . . . . . . . Vtriusque fundamentum continetur in hoc Lemmate. This was written in the year 1686. In the year 1689 Mr Leibnitz having seen the accou Mr Leibnitz [having seen an account given of this Book in yethe Acta Leipsica th] Mr Leibnitz published in the Acta Lepsica: these papers wthwith relation to this book pretending to have have found the same things in relation to about Opticks, to about the resistance of Mediums & to about yethe motions of the Planets. And henceforward the Differencial method began to be taken notice of abroad. Mr Bernoulli began to take notice of it the next year In the first of these these papers he tells us that intituled De Lineis Opticis Re writes: A me ut obiter hic dicam, methodo serierum promovendæ præter transformationem irrationalium linearum in rationales symmetras — excogitata est ratio pro curvis transcendentes datis ubi ne extractio quidem locum habet. Assumo enim seriem arbitrariam, eamque ex legibus Problematis tractando obtineo ejus coefficientes. But this was invented by Mr Newton many years before being set down in his Letter of 24 Octob. 1676 [in these words Altera [methodus consistit] tantum in extractione fluentis quantitatis ex æquatione simul involvente fluxionem ejus assumptione] Mr Leibnitz in his Letter of 27 Aug. 1676 had written Quod dicere videmini plerasque difficultates (exceptis problematibus Diophantæis) ad series infinitas reduci, id mihi non videtur. Sunt enim multa usque adeo mira et implexa ut neque ab æquationibus pendeant neque ex quadraturis Qualia sunt (ex multis alijs) Problemata mathodi tangentium inversæ. Mr Newton in his Letter of Octob. 24 1676 replied. Inversa de tangentibus Problemata sunt in potestate, aliaque illis difficiliora. Ad quæ solvenda usus sum duplici methodo: una concinniori, altera generaliori. Vna methodus consistit in extractione fluentis quantitatis ex æquatione simul involvente fluxionem ejus: altera tantum in assumptione seriei pro quantitate qualibet incognita ex qua cætera commode derivari possunt & in collatione terminorum homologorum ass æquationis resultantis ad eruendos terminos assumptæ seriei. By the words of Mr Leibnitz its manifest that he had not the method at that time, & by those of Mr Newtons, that he had it. In the second paper intituled Schediasma de Resistentia Medij Mr Leibnitz tought to do by Logarithms what Mr Newton had taught to do by the area of the Hyperbola; & in the begin represented that he had for the most part found out those things twelve years before while he was yet at Paris, that is, before he had the Differential method. And g in the end of this paper he adds Nobis nunc (as if Mr Newton had done nothin from whom he copied had done nothing) he adds Nobis nunc fundamenta Geometrica jecisse suffecerit, in quibus maxima consistebat difficultas. Et fortasse attente consideranti vias quasdem novas vel certe antea impeditas aperuisse videbimur. Omnia autem respondent nostræ Analysi infinitorum hoc est calculo summarum & differentiarum (cujus elementa quædam in his Actis dedimus) communibus quoad licuit verbis hic expresso. In the sixt section Article of this Schediasma Mr Leibnitz endeavoured to make a step beyond what Mr Newton had done, vizt in determining the Curve line wchwhich a Projectile describes with a resistance in a duplicate ratio of the velocity. But in doing this he has erred. In the Paper entituled Tentamen de motuum cœlestium causis, Mr Leibnitz layd down several Hypotheses & definitions in the first Eleven Articles. The 12th Article is true in concentric circles false in all other figures. The 15th is false. The 19th is the Principal Proposition of Mr Newton. Mr Leibnitz in attempting to demonstrate it commited two errors wchwhich correct one another, & by adapting to it this erroneous demonstration he pretended to have found it himself. demonstrated Mr Leibnitz This he said, not knowing that the Method was communicated to Mr Collins in the year 1669 in the above mentioned Compendium. & in maintaining it prenect that Mr Newton should declare his opinion refused to contend with any man but Mr Newton as if all others were novices. & according in his Letter of 27 Aug. 1676 sent it composed & polished vulgari more. The same is further manifest by the following consideration. Dr Barrow published his method of Tangents in the year 1670. Mr Newton in his Letter dated 10 Decem 1672 communicated his method of Tangents to Mr Collins & then added. Hoc est unum particulare . . . . . .. reducendo eas ad series infinitas. Mr Slusius sent his method to Mr Oldenburg 17 Ian 1673 & the same . . . . . )

\frac{{dy}^{3}}{dy} = 3 y y

. A copy of Mr Newtons Letter of 10 Decem 1672 was sent . . . . . . . . The improvement by the differential calculus was not yet in his mind. But Mr had not only invented the method of fluxions before this time but brought it to In spring following . . . . . . . that he had then brought it to perfection, the Propositions in his book of Quadratures & the methods of converging series & of drawing a Curve line through any number of given points being then known to him For when the method of fluxions proceeds not in finite equations recourse is had he reduces the equations to converging series, & & when finite equations are wanting he deduces converging series from the con- & when fluents are to be derived from fluxions & the law of the fluxions is wanting reco recourse is had to yethe method of he finds that law quam proxime by drawing a curve line through any number of given points. for finding that law quam proximæ And nothing more has been added After the receipt of this Letter, Mr Leibnitz . . . . . . . . Who was the first Inventor of the method is decided. conditions of the Probleme by assuming the terms of the Series gradually & determining them by those conditions & when fluents are to be derived from fluxions & the law of yethe fluxions is wanting, he finds that law quam proxime by drawing a Parabolick line through any number of given points. [And by these improvements Mr Leibn Newton had in those days made the differential his method of fluxions much more universal then the differential is at present method of Mr Leibnitz is at present.] This Letter of Mr Le Newton dated 24 Octob 1676 came not to the hands of Mr Leibnitz till the end of the winter following or beginning of the spring & soon after Mr Leibnitz wrote back soon after vizt in a Letter dated 21 Iune 1677 wrote back: Clarissimi . . . . before the receipt of Mr Newton's last Letter. He affirms indeed, jam a multo tempore rem tangentium generalius tractavi scilicet per differentias Ordinatarum wh he is not a witness in his o, & so he affirmed in other Letters that he had invented several converging series before direct & inverse before he had the methods of finding them & had forgot an inverse method of series before he knew what use to make of it.: but he is not no man is a witness in his own cause. A Iudge . . . decided. And Mr Iames Bernoulli in the Acta Eruditorum of Ianuary 1691 pag 13, 14 tells us writes thus. [Seques illius [sc. calculi differentialis] specimen in gratiam Lectorum nostrorum quibus calculum hunc agitare volupe fuerit, in Lucem emitto: ut si forte mentem Viri Acutissimi, ex ijs quæ in Actis 1684 de invento istas suo edi] Qui calculum Barrovianum, ut verum fatear quem decennio ante in Lectionibus suis Geometricis adumbravit Auctor, cujusque specimina sunt tota illa Propositionum inibi contentarum farrago) intellexerit [calculum] alterum a Dno Leibnitio inventum ignorare vix poterit; utpote qui in priori illo fundatus est, & nisi forte in differentialium notatione & operationis aliquo compendio ab eo non differt. [And afterwards in the Acta of Iune 1691 pag 290 he speaks thus of the Compendium Cæterum in by wchwhich the methods differ. Cæterum in his Problematibus omnibus . . . . . . maxime commendat. And Mr Iames Bernoulli in yethe Acta Eruditorum of Ianuary 1 Iune 1691 pag. 14 & 290] Now Dr Barrow in his Method of Tangents draws two Ordinates indefinitely neare one another & puts . . . . . . exactly the same [But Mr Leibnitz adds this improvement of yethe method that the conclusion of the calculus is coincident wthwith the Rule of Slusius & shews how that Rule presently occurrs to any one who understands this method. For Mr Newton had represented told him represented in his Letters that thereis was one character had f his general method. whereof the method of Slusius was but a corollary And in twhereas Mr Newton had said that his method proved in drawing of tangents, determining maxima & minima, &c st proceeded wthwithout sticking at surds: Mr Leibnitz in the next place By saying ante Dominos Bernoullios et me nullus [methodum] communicavit, he did not know that Mr Newton had communicated the above mentioned compendium of the method communicated to Mr Collins in yethe year 1669 was still extant & gwould be produced be produced & appear to be genuine. And if this Compendium had been published a little sooner by Mr Iones, it mighmight have prevented