Pag. 186. lin. 3. add. For by the Demonstration of finding these two series wchwhich he desired Mr Oldenburg to procure from Mr Collins, saying that Mr Collins could easily supply him therewith, he meant the method of finding them conteined in the Analysis per series numero terminorum infinitas Pag. 197. lin. 15. add. And while he knew th by Mr Newtons Letters that Mr Newton had such a method before the year 1677, he ought not to have published the differential method as his own before he ha without mentioning that correspondence & mention making a candid an acknowledgement as he made in that Letter, of Mr Newtons having in before those days a method like the differential, as he made thereof in that Letter. In Feb. 1682 Mr Leibnitz published the series of Mr Leibnitz Iames Gregory in the Acta Eruditorum as his own without making any mention of his having received it from Mr Oldenburg & Mr Collins In NovemberOctober 1682 ––– & 1676. But it was impossible for foreigners to understand this. Mr Leibnits ought to have acknowledged that he undestood by my Letters that I had a Methodus similis wchwhich did all this. In the Acta Eruditorum of Iune Pag 198 lin. 32. post Lemmate add. The designe of this Lemm Scholium was not to give away Lemma but to put Mr Leibnitz in mind of the correspondence by wchwhich he making a publick & candidacknowledgment of this correspondence by wchwhich he had wthwith wthwith Mr Newton in the year 1696 by means of Mr Oldenburg [& of what he had learnt by that correspondence], before he l proceeded any further to claim the differential mathod exclusively of Mr Newton For in these Letters wchwhich in the year 1676 passed between them, & in another Letter dated 10 Decemb 1672, a copy of wchwhich was sent to Mr Oldenburg in Leibnitz in the year 1676, as is mentioned above Pag. 199. lin.5. Insert. Dr Halley & Mr Ralpson had the Book of Quadratures in MS in their hands in the year 1691 as Mr Ralpson has attested publickly & Dr Halley still attests Pag. 1998. lin 51 add insert. Mr Leibnitz published in the Acta Eruditorum for 1689 In Iune anno 1689 Mr Leibnitz Anno 1684 1683 ad finem vergente Newtonus cum Societate Regia (Rogante Halleio) In may 1684 Mr Newton published not made known that he had demonstrated the Proposition of Kepler from the principle of gravity wchwhich & in autum following sent demonstrationem Propositionis Kepleri communicavit Quod nempe Planetæ moventur in Ellipsibus & radijs ad Solem in inferiore foco positum ductis areas describunt temporibus proportionales; & insub Autumno subsequente Demonstrationem ad Halleium misit qui eandem cum Societate Regia mox communicavit. et Hookius noster qui mecum cum Newtono hac de re contendisse dicitur, nunquam protulit Demonstrationem aliquam. Mathematicus enim non erat. Anno 1686 mense Maio circiter Newtonus Princip Philosophiæ Naturalis Principia Mathematica ad Societatem Regiam misit ut imprimeretur Et liber ille mense Martio anni proximi proximi lucem vidit. This book is full of such Problems as – – – – – – – he acknowledged the same thing Pag. [198. lin 1]. or Pag. 197 lin 15 insert. In spring 1684 Mr Newton made known th to some Mathematicians that he had demonstrated from the Principle of Gravity the Proposition of Gravity fro the Proposition of Kepler that that the Planets move in Ellipsis & with rays drawn to the Sun placed in the lower focus of the Ellipsis describe areas proportionall to the times. And in Autum following he sent it the Demonstration to Dr Halley who communicated it to the R. Society with some other Propositions concerning the heaven: & the R. S. desired that the MDr Hook said there upon to apermigh have be printed; & Mr Newton thereupon began to write his Book of Principles Mr Hook is contended with Mr Hook Newton about this matter: but he never produced any Demonstration of the Proposition. For he was not skilled in Mathematicks & the Vpon the receipt of the Paper which conteined the Demonstration of Keplers was not to be found without the Method of fluxions. In November 1684 Mr Leibnitz published in his letters of 1672 & 1676. But it was impossible from foreigners to understand this by the words here published. Mr Leibnitz ought to have acknowledged the in express words that he understood by his late correspondence with Mr Oldenburg that Mr Newton had a methodus similis. ☉+ ☉+ He sa On the contrary He ☉+ On the contrary he published in the Acta Eruditorum for May 1700 pag 203, that when he published the Elements of his Calculus He In the Acta Eruditorum of Iune 1686 Mr Leibnitz acknowledged pag. 297 Mr Leibnitz added more brevity. Anno 1686 mense Maio Newtonus Philosophiæ naturalis PrincipioMathematica ad Societatem Regiam misit ut imprimeretur. Et liber ille Mense Martio anni proximi lucem vidit. This book is full of Problems he acknowledged the same thing. Certe, saith he, cum edidi calculi mei edidi anno 1684 ne constabat quidem mihi aliud de inventis ejus in hoc genere quod olim ipse significaverat in literis, posse se tangentes invenire147 invenire non sublatis irrationalibus, quod Hygenius quoque se posse mihi significavit postea, etsi cœterorum illius Calculi ad huc expers: sed majore multo consecutum Newtonum, viso denum libro Principiorum ejus satis intellexi. And a little after, pag. 206 lin. 5. Non hic de problemate menti valdeque diffusa circa maxima et minima fuit actum: quam ante Newtonum et me nullus, quod sciam, Geometra habuit, uti ante hunc maximi nominis Geometram nemo specimine publice dato se habere probavit This method is a principal b part of the Method of fluxions being the method by wchwhich the Curva cellerimi descensus was found out by Mr Newton & Probleme proposed by Mr Bernoulli Probleme de Curva celerimi descensus proposed by Mr Bernoulli to all the world, was solved, & Mr Leibnitz here acknowledges that Mr Newton by finding the solid of least resistance had proved that he had this method when he wrote his book of Principles. Vide Scholium in Prop. XXXIV Lib.II. Pag. 199. lin 54. add. Dr Halley & Mr Ralpson had the Book of Quadratures in MS in their hands in the year 1691 as Mr Ralpson has attestested publick ly & Dr Halley still attests 148 2 Of the Method of fluxions & moments If any equation involve tow unknown quantities suppose x & y, to find when either of them, suppose y is greater or least, Fermat put the letter o fir the indefinitely small difference of two valors of the other quantity x, & substituting x+o for x, has a knew equation, & by reducting these two equations & making the difference o decrease & vanish finds the greatest or least quantity y. This was his method de maximis & minimis; & by this method he drew Perpendiculars to curve lines. A specimen of this method was published by Mr I Schooten in his Commentary upon yethe Geometry of Des Cartes 1659. Mr Iames Gregory in yethe 7th Proposition of his Geometriæ Pars universalis published in 1668 puts the letter o for the difference of two Abscissas & draws a right line through the ends of the Ordinates, & that this line may become the tangent of the Curve makes the difference o decrease & vanish. Dr Barrow in his 10th Geometrical Lecture published Anno 1670, to find Tangents to curves puts the Letters a & e for the indefinitely small Differences of the Abscissas & Ordinates, brings the Probleme to an equation, rejects all the terms of the Equation in wchwhich a & e are either wanting or of more dimensions then one, & by the proportion of a to e draws the Tangent, & this method readily gives the method of Slusius. Newton at the request of Mr Collins h sent him his method of Tangents in a Letter dated 10 Decem. 1672. It proved to be the same with the method of Slusius, but was sent as a Corollary of a general method of solving Problems: wchwhich method in drawing of tangents agreed with those of Gregory & Barrow. Slusius sent his method to Mr Oldenburg in Ianuary 1673. It was founded on three Lemmas, the first of wchwhich was this. Differentia duarum Dignitatum ejusdem gradus applicata ad dDifferentiam Laterum dat partes singulares gradus inferioris ex binomio Laterum; ut $\frac{{\mathrm{y}}^3-{\mathrm{x}}^3}{\mathrm{y}-\mathrm{x}}=\mathrm{y}\mathrm{y}+\mathrm{yx}+\mathrm{x}\mathrm{x}$ And Mr Leibnitz in his Letter dated 21 Iune 1677 drew tangents by putting the symbols dy & dx for the letters a & e of Dr Barrow. [And by this notation the example in the first Lemma of Slusius vizt $\frac{{\mathrm{y}}^3-{\mathrm{x}}^3}{\mathrm{y}-\mathrm{x}}=\mathrm{y}\mathrm{y}+\mathrm{yx}+\mathrm{x}\mathrm{x}$ became $\frac{d{\mathrm{y}}^3}{d\mathrm{y}}=3\mathrm{y}\mathrm{y}$, or $d{\mathrm{y}}^3=3\mathrm{y}\mathrm{y}d\mathrm{y}$ & in general the Lemma became $d{\mathrm{y}}^{\mathrm{n}}=\mathrm{n}{\mathrm{y}}^{\mathrm{n}-1}d\mathrm{y}$; & in this notation is the convers of this is the first Rule of the Analysis per Æquationes numero terminorum infinitas] When Mr Newton in his Letter of 10 Decem. 1672 had described his method of Tangents, he added. Hoc est unum particulare vel Corollarium potius Methodi generalis quæ extendit se citra molestum ullum calculum non modo ad ducendum Tangentes ad quasvis Curvas sive Geometricas sive mechanicas vel quomodo cunque rectas lineas aliasve Curvas respicientes verum etiam ad resolvendum alia abstrusioraora Problematum genera de Curvitatibus Areis Longitudinibus, centris gravitatis Curvarum &c. Neque (quemadmodum Huddenij methodus de maximis et minimis) ad solae restringitur æquationes illas quæ surdis quantitatibus sunt immunes. And in his Letter dated 13 Iune 1676, after he had described his method of series he added: Ex his videre est quantum fines Analyseos per hujus modi Æquationes infinitas 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 Sed quomodo in istis casibus procedendum sit non vacat dicere: ut neque alia quædam tradere quæ circa reductionem serierum infinitarum in finitas ubi rei natura tulerit, excogitavi. And in his Letter dated 24 Octob. 1676, he represented that a Tract which he wrote five years before upon the method of Series, was for the most part taken up with other things.That there was in it the method of Slusius built upon another foundationwchwhich gave that method readily, even without a particular demonstration, & made it more general so as not to stick at surdes; the Tangent not withstanding surdes being speedily drawn without any reduction of the Equation wchwhich would often render the work immense. And that the same manner of working held in Questions de Maximis & Minimis & some others wchwhich in that Letter he forbore to speak of. And that upon the same foundation the Quadrature of Curves became more easy, an example of wchwhich he gave in a series which brake off & became finite when the Quadrature might be done by a finite equation. And that this method extended to inverse problemes of Tangents & others more difficult. But the foundation of this method he concealed in sentences set down enigmatically, the first of wchwhich was this: Data æquatione quotcunque fluentes quantitates involvente, fluxiones invenire, & vice versa For Here Mr Newton considers not quantities as composed of indivisibles but as generated after the manner proposed by local motion, after the manner used by of the Ancients. They considered rectangles as generated by drawing one side into the th other that is by moving one side upon the other to describe the area of the rectangle: & in like manner Mr Newton considers the areas of curves as generated by drawing the Ordinate into the Abscissa, & all in determinate quantities he considers as generated by continual increase And from the flowing of time & the moments thereof, he gives the name of floxing quantities to all quantities wchwhich increase in time, & that of fluxions to the velocities of their increase & that of moments to their parts generated in equal moments of time. T He considers time as flowing uniformly, & exposes or represents it by any other quantity wchwhich is considered as flowing uniformly: & its fluxion by an unit. And the And for Andth The moments of time or of its exponent he considers as equal to one another, & for d represents one of t this such a moment by the Letter o or by any other letter or mark drawn into an unit. The other other flowing quantities he represents by any other letters or marks & most commonly by the letters at the end of the alphabet. Their fluxions he represents by any other letters 149 or marks, or by the same letters in a different form or di magnitude or distin otherwise distinguished. T And their moments he represents by their fluxions drawn into a moment of time. Fluxions are not moments but finite quantities of another kind. They are motions & to make them become moments Mr Newtons multiplies them by th a moments of the exponent of time. When Mr Newton is demonstrating any Proposition he considers the moment of time in the sense of the vulgar as indefinitely small but not infinitely small, & by that means performs the whole work in finite figures or schemes by the Geometry of lucid & Apollonius with exactly without any approximation: And when he has brought the work to an equation & reduced the equation to the simplest form, he supposes the moment to decrease & vanish, & from the terms wchwhich remain, he deduces the Demonstration. An example of this you have in his Demonstration of the Construction of the first Proposition of his Book of Quadratures. vizt Data æquatione fluentes quotcunque quantitates involvente invenire fluxiones And another example you have in his demonstration of the first of the three Lemmas upon which he grownded his Treatise intituled Analysis per æquationes numero terminorum infinitas. [And by these examples it is sufficiently manifest that he had found out this method of fluxions before he wrote the composed those Demonstrations] But when he is only investigating any truth or the solution of any Problem he supposes the moment of time to be infinitely little in the sense of Philosophers, & uses works in figures or schemes infinitely small & uses any approximations, wchwhich he conceives will make no error in the conclusion as by putting the arc the & sine chord, sine & tangent equal to one another, & for the more greater dispatch he neglects to write down the moment o. In thie Analysis per æquat above mentioned wchwhich Dr Barrow sent to Mr Collins in the year 1669, his principall designe was to describe the method of series: but that method being inseparably conjoyned with the method of fluxions, he touches upon this method in the following manner When he described the three Lemmas upon wchwhich he founded this Analysis in squaring of Curves, he proceeded touched upon the method of fluxions in the following manner. Et hæc de areis Curvarum investigandis Dr Wallis by various steps arrived at thie 59th Proposition of his Arithmetica Infinitorum published 1655. And that Proposition (in other language) is this. Let the Abscissa of any curvilinear figure be x & the Ordinate b erected at right angles by y, & let m & n be numbers & $\frac{\mathrm{m}}{\mathrm{n}}$ be the index of thesany dignity of the abscissa; & if the Ordinate y be equal to this dignity the area of the curve will be $\frac{\mathrm{m}}{\mathrm{n}}{\mathrm{x}}^{\frac{\mathrm{m}-\mathrm{n}}{\mathrm{n}}}$ $\frac{\mathrm{n}}{\mathrm{m}+\mathrm{n}}{\mathrm{x}}^{\frac{\mathrm{m}+\mathrm{n}}{\mathrm{n}}}$ And This is the first Proposion ofLemma Rule upon wchwhich in Mr Newtons Analysis per æquationes numero terminorum infinitas. And the second is that when yethe Ordinate of a Curve is composed of several such Ordinates the Area is composed of several such areas. Dr Wallis published his Arithmetica infinitorum in yethe year 1655 & by the 59th Proposition of that Book if m & n be numbers & the Abscissa of any Curve Curvilinear figure be called x & putting m & n forbe numbers) & the Ordinate erected at right angles by ${\mathrm{x}}^{\frac{\mathrm{m}}{\mathrm{n}}}$ the area of the figure shall be $\frac{\mathrm{n}}{\mathrm{m}+\mathrm{n}}{\mathrm{x}}^{\frac{\mathrm{m}+\mathrm{n}}{\mathrm{n}}}$. This is assumed by Mr Newton as the first Rule upon wchwhich he founds his Analysis. And by the 108th Proposition of the same book & several other Propositions wchwhich follow therein, if the Ordinate be composed of two or more such Ordinates taken wthwith their signes + or − the area will be composed of two or more such areas taken wthwith their signes + or −. and this is assumed by Mr Newton as the second Rule upon wchwhich he founds his Analysis. And in the same Arithmetica Infinitorum Dr Wallis squared a series of Curves whose ordinates were $1. 1-{\mathrm{x}}^2. {\overline{)1-{\mathrm{x}}^2}}^2. {\overline{)1-{\mathrm{x}}^2}}^3. {\overline{)1-{\mathrm{x}}^2}}^4$. & shewed that if the series of their Areas could be interpoled in the middle places the interpolation would give the Quadratures of yethe circle. And in his Opus Arithmeticum published A.C. 1657 cap. 33 Prop. 68, he reduced the fraction $\frac{\mathrm{A}}{1-\mathrm{R}}$ by perpetual division into yethe series $\mathrm{A}+\mathrm{A}\mathrm{R}+\mathrm{A}{\mathrm{R}}^2+\mathrm{A}{\mathrm{R}}^3+\mathrm{A}{\mathrm{R}}^4+\mathrm{\&c}$ Mr Newton A.C. 1665 Vicount Brunker squared yethe Hyperb. . . . . in April 1668. Mercator soon after published a demonstration . . . . numero terminorum infinitas For Mr Newton A.C. 1665 upon reading the Arithmetica infinitorum of Dr Wallis & considering how to interpole the series of Areas above mentioned, found the infiniteconverging series for yethe Arc whose sine is given & pursuing the method of interpolation he found also the Quadrature of interpolation all Curves whose Ordinates are the dignities of binomials affected wchwhich indices whole or fract or surd, affirmative or negative, together with the resolution of a binomial into a converging series; as at the request of Mr Leibnitz he has explained at large in his Letter dated 24 Octob. 1686 & long since printed by Dr Wallis The two first terms This resolution of a binomial into a converging series together wthwith the quadrature of the Curve whose Ordinate is yethe the binomial he has explained at large in his Letter dated 13 Iune 1676 & set down the two first terms of the series in his Analysis above mentioned follows from this general Quadrature, but Mr Newton supposes that Myr L' Brounker might find & Mercator demonstrate some two or three years that Quadrature above or three four some years before they published their performance. [This Analysis is the first piece published in printed in the Commercium. It is the It was sent to Mr Collins in Iuly 1669 as appears by the dates of three of Dr Barrows Letters still extant. And Mr Newton in his Letter dated 24 Octob. 1676 mentions it in this manner. Eo ipso temporepore quo Liber [Mercatoris] prodijt communicatum est per amicum D. Barrow (hunc matheseos Professorem Cantab) cum D. Collinio, compendi um quoddam Methodi harum serierum, in quo significaveram areas & Longitudines Curvarum omnium & solidorum superficies & con tenta, ex datis Rectis; et vice versa ex his datis Rectas determinary posse: et methodum illustraveram diversis seriebus. And Mr CollinsOldenburg in a Letter to Mr Strode Slusius dated 14 Sept. 1669, cites sev & entred in the b Letterbook of yethe R. Society, in giving an account of it cites several sentences out of it. And Mr Collins in a Letter to Mr Strode makes this mention dated 26 Iuly 1672 mention of it. Exemplus ejus [Logarithmotechnia] si cum meridiana clasitate conferatur. Now This Analysis or Compendium conteins a general method composed of two methods the one converging series, the other of moments & fluxions.] In this Compendium. 150 An Account of the Analysis per Quantitatum Series Fluxiones ac Differentias cum Enumeratione Linearum tertij Ordinis, published by Mr Iones. 151 If any Equation contein two unknown indeterminate quantities suppose x & y; to find either of them suppose x when yethe other is greatest or least. Des Cartes teaches that the quantity to be found suppose x will in that case have two of its roots become equal. Fermat for the difference of those two roots before they become equal, puts the letter o, & thereby has two equations in both wchwhich the other two other quantitys y ought to be one & the same. THen by exterminating that other quantity & reducing the equations & putting the two roots suppose x & x+o equal, that is, by putting the difference o equal to nothing he finds the quantity x, & by this method draws perpendiculars to Curves & resolves other Problemes by maxima & minima. 16 12 Iames Gregory in the 7th Proposition of his Geometriæ pars Vniversalis published 1678 puts the letter o upon for the difference of the Ordinates, & that this line may become the Tangent of the Curve, makes the difference o vanish. 173 Barrow in his 10th Lecture published 1669, to find Tangents, puts the letters a & e for the differences of the two Abscissass & two Ordinates, conceives a right line to be drawn through the ends of the Ordinates, brings the Probleme to an equation rejects all the terms of the equation in wchwhich either a & e are either wanting or of more dimensions then one & by the proportion of a to e draws the Tangent. A method of Tangents was communicated by Mr Hudde to Mr Schortem in November 1659 to be kept secret, & Slusius also & Newton fell upon the same method 14 Slusius founded his method of Tangents on three Lemmas the first of wchwhich was this. Differentia duarum dignitarum ejusdem gradus applicata ad Differentiam laterum dat partes singulares gradus inferioris ex binomio laterum, ut $\frac{{\mathrm{y}}^3-{\mathrm{x}}^3}{\mathrm{y}-\mathrm{x}}=\mathrm{y}\mathrm{y}+\mathrm{y}\mathrm{x}+\mathrm{x}\mathrm{x}$. That is, in the language of Mr Leibnitz, $\frac{d{\mathrm{y}}^{\mathrm{n}}}{d\mathrm{y}}=\mathrm{n}{\mathrm{y}}^{\mathrm{n}-1}$ or $d{\mathrm{y}}^{\mathrm{n}}=\mathrm{n}d\mathrm{y}{\mathrm{y}}^{\mathrm{n}-1}$ 165 A month before Slusius sent his Method to Mr Oldenburgh Newton at yethe request of Oldenburgh Mr Collins sent his in a letter dated 10 Decem. 1672, wchwhich method proved the same wthwith that of Slusius; & added in the latter part of his Letter, & added these wrods concerning & added Hoc est unum particulare vel Corollarium potius methodi generalis, quæ extendit se citra molestum ullum calculum non modo ad decendum Tangentes ad quasvis Curvas sive Geometricas sive Mechanicas vel quomocunque rectas lineas aliasve Curvas respicientes; verum etiam ad resolvendum alia abstrusiora Problematum genera de Curvitatibus, Areis Longitudinibus, Areis Centris gravitatis Curvarum &c. Neque quemadmodum Huddenij Methodus de Maximis et Minimis ad solas restringitur quantitates illas quæ quantitatibus surdis sunt immunes. T And a copy of this Letter was sent to Mr Leibnitz among the extracts of Gregories Letters above mentioned, Iun 26. The same method of Tangents was communicated by Hudde to Schooten in the year 1659 to be kept secret, & accord 16 Mr Leibnitz being then at London in the beginning of the next year 1673 pretended to yethe invention of the Differential method of Mouton & being reprehended for it by Dr Pell, persisted in making himself coinventor of that method, & it appears not that he had any other Differential method at that time. When Mr Leibnitz wrote 26 B 28 Mr Leibnitz in a letter to Mr Oldenburg dated 2818 Novem 1676 Methodus Tangentium as lusip pe publicata nondum rei fastigium tenet potest aliquid amplius prœstari in es genere quod maximi foret usus ad omnis generis Problemata: etiam ad meam (sine extractionibus Æquationum ad Series Reductionem. Nimirum posset brevis calculari quæ calculari circa Tangentes Tabula, eouque continuanda donec progressio Tabulæ apparet, ut eam scilicet quisque, quousque libuerit, sine calculo continuare possit. 152 3. Of the Method of Fluxions & Moments All quantities being indivisible in finitum Mr Newton conceived it more agreable to nature to consider quantities as increasing by continual motion then by opposition of indivisible parts, & from the fluxion & moments of time, he gave the names of fluxions to the velocityies wherewith quantities increase & that of moments to their indefinitely small parts by wchwhich generated by motion in moments of time. The fluxion of time or of any quantity increasing or flowing uniformely by which the fluxion of time is represented or exposed, he represents by a given quantity & most commonly by an unit & for the moment of such a s time or its exponent he usualy puts the letter o.For the other increasing or flowing quantities he puts orwhich he calls fluents he puts any symbol & for their fluxions any other symbols & for their moments the symbols of their fluxions drawn into the moment o. Fluxions are finite quantities & to make them signify indefinitely or infinitely small parts of fluents he multiplies them by the indefinitely or infinitely small moments of time. When he is demonstrating any Proposition he uses this Notation the letter o in this sense & t considers the moment of time it as indefinitely small & performs the whole operation in finite figures by the Geometry of Euclide. But when he is only investigating a Proposition, he usually considers the moment o as infinitely little & for making dispatch neglects to write it down, & proceeds in the calculation by any in figures in by any approximations But as for the Method of fluxions it was certainly known to Mr Newton when he wrote his Letter to Mr Oldenburg dated 24 Octob. 1676. For in that Letter he exprest it of œnigmatically by these sentences comprehended it in these sentences. Data æquatione quotcunque fluentes quantitates involvente fluxiones invenire et vice versa. EtAnd said that this was the foundation of the method upon wchwhich in conjunction wthwith the method of Series he had writ a treatise five years before vizt in the year 1671 In the mean time time it is be remains to be considered whether Mr Leibnitz after Mr Newton had told him that in his Letter of 24 in three several Letters sent to him described several the charactedrs of his method & told him that in the year 1671 he had wrote a treatise of it In the mean time, when Mr Leibnitz Newton Leibnitz had seen three of Mr Newtons Letters in wchwhich the characters & universality of his met Newton had told Mr Leibnitz that he in yethe year 1671 he had wrote a treatise of the Method & of the method of Series together, when & of another method together, when he had concealed this metho in three Letters wchwhich came to the hands of Mr Leibitz described the characters & universality of this method other method so far as to make enable Mr Leibnitz understand that it did not abludere from yethe differential method but was a methodus similis to compare it with the Differential & see that they did not ab invicem abludere but were similes, when he had concealed the foundation of this method so far a sym in an enigmatical sentence c it in an Ænigma to hide it not from honest men but from plagiaries Now if after Mr Newton had told Mr Leibnitz that in the year 1671 he had a method of wrote a treatise of the method of Series & of another method together, after he had in three Letters wchwhich came to the hands of Mr Leibnitz described the characters & universality of this other method so far as to enable Mr Leibnitz to compare it with the differrential method & say that they did not abinvicem abluder but were similes, after Mr Newton had concealed the foundation of it in an Ænigma to hid it not from honest men but from plagiaries: the Question is, whether Mr Newton should not have Leibnitz should not rather have invited Mr New & encouraged Mr Newton to have published his method then have rivalled him & claimed the method from him by saying & My computation of the time between yethe burning of the first Temple & building of yethe 2d is this. Pharoah Nechoh reigned over Phenicia & & Syrea cava & Namath as far as Euphrates the three first years of Iehojakim 2 King 33 In yethe year Nebuchadnezzar (having newly conquered Assyria) came against him & besieged Ierusalem & in the 4th year beat him at Euphrates & took from him all Syria & Phœnicia from yethe river of Egypt to yethe river Euphrates & reigned in his stead, the first year of Nebuchadnezzar over those countries being yethe 4th of Iehojakim. But whether he took Ierusalem in the thrid or fourth year doth not appear. Dan. 1,1. Ier 46, 2 And Nebuchadnezzar continued wthwith his army in those parts to conquer the nations round about (Ier 25.9,11) & to f recover to Babylon whatever had lately belonged to Nineveh Assyria & settle his new conquests untill he heard of the death of his father wchwhich was in yethe fift or sixt year of Iehojakim, & then hasted to Babylon to succeed his father in the whole kingdom leaving his army captains to follow wthwith the captives, & in the 43th year of his reign counted from the death of his father he his army & the captives. died Eupolamug & Berosus apud Euseb l.9 c.39,40 By yethe Canon of Ptolomy he succeded his father A. Nabonass 144 &died A. Nabonass 187. But the Iews recconed not by the in years of Nabonassar Iehojakim reigne eleven years & Iehojakim three months being captivated in yethe end of yethe eleventh year or beginning of yethe 12th (2 Chrom 36. 10.) And the 12th year was the first year of his successor Ie Zedekiah & the first year of the captivity of Iehojakin, F the years of this captivity beginned wthwith the reigh of Zedekiah. The Iews recconed by Lumisolar years the Babylonian Astronomers by the years of Nibonassar. In the 37th year of this captivity, that is in yethe 45th year of yethe reign of Nebuchadnezzar recconned by Lumisolar years from the 4th year of Iehojakim inclusively, Nebuchadnezzar died, & was succeeded by his son Evilmerodach. 2 King. 25. 27 Ier. 52. 31 And in yethe end of the year 25th day of yethe 12th month of the year brought his friend Iehojakim out of prison 2 King. 25. 27 Ier. 52. 31. It is not likely that after he came to yethe throne he would lett his great friend Iehojakim stay long in prison & therefore it's reasonable to beleive that Nebuchadnezzar died in the end of winter neare the end of the 35 45th year of his reign accordingly & by consequence the recconing of yethe Iew And this was in yethe beginning of yethe year of Nebuchadnez Nabonassar 1867 And by consequence therefore Iehojakim therefore began his reign in the year of Nabonassar 139, & Nebuchadnezzar in yethe year year of Nabonassar 142 according to yethe Iewish account, & one or about about two years after that he succeded his father at Babylon & reigned 43 years form yethe death of his father, according to the recconing of the Chaldeans, & Canon of Ptolomy, & 44 years & some months according to yethe recconing of the Chaldeans Iews. And according to this recconing account There Iews had fasted just 70 in yethe 5t month for yethe burning of yethe City just 7 & in yethe 7th month for yethe death of Gedaliah just 70 years befoer yethe 9th month of yethe 4th year of Darius Hystaspis 2 King. 25. 1 Zech 7. 1 And in yethe end eleventh month of yethe 2d year of Darius there had had Gods indignation against yethe cities of Iudah had lasted just 70 years, the indignation beginning in the tenth month of the ninth year of Zedekiah And so it really was on the 10thmon See 2 King 25. 2 & Ier 34. 7 & Zech 1.12, & 7.1. This recconing therefore And these agrees two recconingagrees with exactly with scripture & concerned thereby the computation here set down And as for yethe DrsDoctors Not being able to yethe digest the opinion that Zechary might begin to prophesy wthwithin 16 years after his grand fathers death; can he not digest yethe opinions that 4 generations have been alive together in France, the King & three Dauphins? that And Can he digest the opinions that Zembbabel & Ioshuah might govern I the Iews 118 years together, & that there might be men alive in yethe 2d year of Darius Hystaspis Nothus, who remembered the first Temple wchwhich had been burnt 165 years before? The Iews were to serve the King of Babylon 70 years. They be & after 70 years were accomplished at Babylon they were to return to their own land (Ier 25. 12 & 29. 10). They began to serve him in yethe first year of his reign over them which was the fourth year of Iehojakim A. Nabonass. 142. They returned from remained in captivity until the reign of the kingdom of Persia, & in yethe first year of Cyrus king of Persia (A. Nabonass. 212) they returned home (2 Chron. 36. 20,21,22) But Dr Alex places yethe 4th year of Iehojakim & 1st of Nebuchadnezzar two years later. My computation is therefore favored by 4 argumtsarguments taken from scripture 1st yethe death of Nebuchadnezzar in yethe 37th year of Iehojakins captivity. 2dly the fasting 70 years. 3dly the indignation upon yethe cities of Iudah 70 years & 4thly the serving the king of Babylon 70 years. 153 Ex mente Newtoni, primus annus Nebuchadnezzaris juxta Iudœos fuit 4tus Iehojakimi. juxta Chaldæos Iudæos et non secundus sed sextus junta Chaldæos. Et regnavit Nebuchadnezzar ille annos 48 a morte Patris, annos vero 44 & menses aliquot a quarto Iehojakim inclusive. Obijt enim & filio Euilm erodacho regnum reliquit anno 37mo captivitatis Iehojachin, sev hoc est, anno 37 regni Zedekiæ ed est. anno 48vo regni Iehojakim, ed est et propterea anno 45to regni proprij cum ab quod capit anno 4to Iehojakimi incipientis inclusive. Et intra annum ung unum generationes quator Iaclvus conquer surisse eatitisseo quam ( (sc Regem Galliæ scilicet & tres Delphinos) concoquer possum probabiliest est & quam aliquos in vivis esse fuisse qui Regem nostrum Henricum Octavum de facie norant. Observations upon yethe Notes of yethe RndReverend Dr Alix. The Dr in yethe firs 2d Paragaph supposes ytthat I place yethe 1st year of Nebuchadnezzar according to the Chaldees, upon yethe 2d of Iehojakim, disputes against this opinion & in the 6t Paragraph concludes that I erred in placing the destruction of Ierusalem upon the 17th year of Nebuchadn. wchwhich fell upon according to the Chaldees. But if I placed yethe taking of Ierusalem upon yethe 17th year of Nebuchadn. dated from yethe death of his father according to the Chaldees, I placed the death of his father & first year of his reign upon the not upon yethe 2d but uopn yethe sixt year of Iehojakim. Evilmerodach succeeded his father Nbuchadnezzar in yethe 37th year of Iehojakins captivity (2 King 25. 27) The years of this captivity & the years of the reign of Zedekiah have yethe same epocha. Add yethe eleven years of Iehojakim & ther the death of Nebuchnezzar will fall upon the 48th of year of Iehojakim recconed from the first year of his reign inclusively. Take away the three first years of Iehojakins &the death of Nebuchadnezzar will fall upon yethe 45th year of his reign recconed from yethe 4th year of Iehojakim inclusively. Whereas Nebuchadnezzar reigned but 43 years from yethe death of his father according to yethe Canon. During the three first years of Iehojakim, the king of Egypt reigned over Palestine & Cœlosyria. Nebuchadnezzar came against him in yethe 3d year of Iehojakim & captivated some of the Iews & the next year beat him at Carchemish & took from him all Syria & Palestine from Euphrates to yethe borders of Egypt (2 King. 24.7) & reigned in his stead. And when the war was fully ended Nebuchadnezzar heard of the death of his father & returned in hast to Babylon to succeed him, leaving his Captains to follow with his army & the captives according to Berosus. His reign therefore might have a double beginning, the first when he succeeded Pharaoh Nechoh in Syria & Palestine & the neighbouring coasts of Arabia & Hamath, the second when he succeeded his father at Babylon the first commonly used by the people of Syria & Phenicia, the second by these in Balonia. Lin 4. lige, annis tardius scil. a 6to Iehojakimi (vide lin 28) Lin 9, lege A. Nabonass 160 Ne lin: 11 An Ezekiel in Chaldœa inter captivos usus sit annis Nabonassari? Lin 13 Annois 10 dantur regno Iehojakimi, 12 regno Zedekiœ. Nam regnum Zedeckiæ cœpit cum annis captivitatis Iehojakin. Lin 14 lije 1 37 ad finem vergente ab ipsius deportatione, Lin A ultimo Nebuchadnezzaris quo Evilmerodach regnare cœpit. Lin 6. Nebuchadnezzar ex er citur anno 3o Iehojakimi Hieroslyma obsedisse dicitur, eodem anno cœpissa non dicitur. L. 22. non Suppano, Pharaonem cœsum anno quarto Iehojakimi & Nebuchadnezzarum eidem in regno Syria et Palestine tune successisse, L. 30 patri vero mortua successisse post bellum finitum. Lin 50 Quatuor fuerunt generationes eodem tempore in Gallia Rex ipse et tres Delphini. Iddo In the first year of Cyrus Iddo might be ass old as the presentpresent king of France & Z solary as oldn then his Grandson the King of Spain. But could You have seen how Mr Newton in his Letters of 10 Decemb. 1672 13 Iune 1676 & 24 Octob. 1676 represented that he had a very general method by wchwhich he drew Tangents after the manner of Slusius determined maxima & minima, squared curvilinear figures determined their center of gravity found the lengths of crooked lines & the quantity of their crookedness, measured the surfaces & solid contents of round solids, determined the inverse Problemes of tangents & other more difficult, & wrought in transcendent curves as well as others reduced difficulties to infinite equations where tall could they could not be overcome in finite ones, & applied equations finite or infinite to the solution of almost all Problemes except perhaps some numeral ones like those of Diophantus. Vpon this method & the method of Slusius together he said that he in his Letter of 24 Octob. 1676 that he had wrote a Tract five years before, & in the same Letter he said that comprehend wrote down the foundation of this method ænigmatically in this sentence Data æquatione quotcunque fluentes quantitates involvente thinking that he had said too much of it & described it too plainly, that it might not be taken from him he wrote down the foundation of ænigmatically in this sentence Data æquatione quotcunque fluentes quantitates involvente Fluxiones invenire; & vice versa. And in the second Lemma of the second book of his Principia Philosophiæ when he had demonstrated Geometricaly the elements of this method he added in a Scholium. In literis quæ mihi cum Geometra peritissima annis abhinc decem interecedebant cum significarem me compotem esse methodi determinandi maximas & minimas ducendi Tangentes & similia peragendi quæ in terminis surdis æque ac in rationalibus procederet, & literis transpositis hanc sententiam involventibus [Data æquatione quotcunque fluentes quantitates involvente fluxiones invenire, & vice versa] eandem celarem, rescripsit Vir Clarissimus se quoque in ejusmodi methodu incidisse, & methodum suam communicavit a mea vix abludentem præterquam in verborum et notarum formulis. Vbiusque fundamentum contineter in hoc Lemmate The method of fluxions was therefore known to Mr Newton when he wrote the said three Letters And in the last of those Letters he represents that he wrote of Trac it in a Tract wchwhich he composed upon the method of series five years before, that is, in the year 1671. And in the Compendium wchwhich he wrote communicated to Dr Barrow & Mr Collins two years before that there sufficient footsteps of his knowing it at that time. 154 An account of the Analysis per quantitatum series fluxiones ac differentias cum enumeratione linearum tertij Ordinis published by Mr Iones This Analysis is f If any equation conteining two indeterminate quantities fial the less suppose x & y to find eonither of them * when the other * is greatest or least, Fermat supposes D. Cartes teaches that yethe first the quantity to be found will in that case will have two roo of its roots become equal. Fermat For the difference of thos two roots before they become equal Fermat puts the letter o, & thereby has two equations in both wchwhich the other quantity ought to be one & the same. Then by this putting this exterminating that other quantity & reducing the equations, & putting the two roots suppose x & $\mathrm{x}+\mathrm{o}$ equal, that is, by putting the difference o equal to nothing, he finds the quantity x. And this method he applies to y dra And by this method draws perpendiculars to curves & & resolves such other Problemes as are to be resolved by finding when quantities are greatest or least Iames Gregory in his the 7th Proposition of his Geometriæ pars Vniversalis published 1678 puts the letter o for the difference of two Abscissas & thereby finds the difference of two Ordinates & by proportion of these differences & draws a line right line through the ends of the Ordinates, & that this line may become the Tangent of the curve makes the difference o vanish Barrow in his 10th Lecture published 1669 to fin to find Tangents puts the letters a & e for the differences of the Abscissa & Ordinate, & draws a right line through the ends of the Ordinates, & then makes brings the probleme to an equation, rejects all the terms of the Equation in wchwhich a & e are either wanting or of more dimensions then one, & by the proportion of a to e draws the Tangent. Leibnits in the year 1677, chan for yethe letters a & e, substituted the symbols dx & dy, & drew tangents after the same manner with Barrow. This method he Newton in Iuly the year 1669 published his comm afterwards published in the Acta Eruditorum & two years aft mensis Octobris & 1684, & in the conclusion added. Et hæc quidem initia sunt tantum Geometriæ cujusdam multo sublimioris ad difficillima & pulcherrima quæque Problemata etiam mistæ matheseos Problemata, quæ sine calculo nostro differentiali, aut simili, non temere quisquam pari facilitate tractabit. By the words aut simili he means a Newtons method as is evident by his letter of 21 Iune 1677. For he had notice of this method by three of Mr of Mr Newtons letters dated 10 Decem. 16673. 13 Iune 1676 & 24 Octob 1676 by Mr Oldenberg & communicated to him by Mr Oldenburgh. Whe he changed the letters dx a & e into dx & dy he tells us two years after in yethe Acta Eruditorum 16 mensis Iunij 1686 M Malo autem, dx & saith he, dx et similia ad hibere quam literas pro illis quia istud dx est modifactio quædam ipsius x, & ita ope ejus fit ut sola quando id fieri opus est litera x, cum suis scilicet potestatibus & differentijs calculum ingrediatur & relationes transcendentes inter x et aliud exprimantur. Qua ratione etiam lineas transcendentes inter x et ali æquatione explicare licet. That is, If he had used letters he must have defined their signification upon partial every new occasion, & to avoyd that trouble he chose rather to use yethe symbols dx & dy & define them once for all. For there is nothing that can be done by the symbols dx, dy, &c dz but may be done by letters or any other symbols after their signification is defined. He also put the letter d before the ordinate of a Curve to signify the area of the Curve or summ of the Ordinates in the method of Cavallerius. For Fluxions he has no proper symbol. In the meane time Mr Newton in Iuly 1669 communicated to Dr Barrow & Dr Barrow to Mr Iohn Collins a short Tract intituled Analysis per æquatione numero terminorum infinitas wchwhich is the first Tract in the Collection of Mr Iones. In wchwhichthis Tract he shews how to reduce finite equations to infinite ones when there shall be occasion, & by the help of the moments of quantities to apply these æquations both finite & infinite to the solution of the harder Problems. And this is the Geometria multo sublimior ad difficillima et pulcherrim ad quæque Problemata, spoken of by Mr Leibnits seven fifteen years after: excepting that it is more universal & more demonstrative then yethe one of Mr L. In this Analysis Mr Newton represents the area of a curve by inclosing the Ordinate in a square & considers quantities as increasing or flowing by continual motion in time, & represents time by any quantity wchwhich increases or flows uniformely or in proportion to time & from the fluxion & moments of time gives the names of fluxions & moments to the velocities of flowing & momentaneous increases of other quantities in time. For the flowing quantities he puts any symbols letters or symbols & for their fluxions he puts any other letters or symbols & for the fluxion of time or of its exponent he usually puts an unit, & for the a moment of time whether infl infinitely little or only indefinitely little he puts frequently usually puts the letter o, & for the moments of other quantities generated in that moment of time, he puts the fact or content under their fluxions, & that moment of time And for an area of any Curve described by the Ordinate of the Curve he puts the Ordinate with a square about it. Mr L drawn into one another. When he is demonstrating any Proposition he always uses the letter o or some other symbol, for an indefinitely small quantity quantity or particle of time & so soon as the proceeds in finite quantities & finite figures by Euclides Geometry to the end of the calculation without any error or approximation or error, & then supposes the indefinitely small quantitys to be decrease in finitum & vanish or become nothing. But when he is only investigating a truth or the solution of a Probleme he supposes the moment o to be infinitely little & proceeds in the calculation by such approximations as he thinks will create no error in the conclusion, as by putting arches & their chords sines & tangents for one another. And for making dispatch he forbeares to write down the letter o putting the symbol of the fluxion alone for the moment of the fluxion, but understanding that symbol to be multiplied by the moment o wheneve to make it infinitely little whenever it signifies a moment. For fluxions or velocities are finite quantities but moments are infinitely little. In the book of Quadratures & some other papers Mr Mr Leibnitz uses no proper symbols for fluxions, but when he considers any quantity as the ordinate of a Curve he puts the letter s before it to signify yethe area described by that Curve & Mr Newton long before did the same thing This Tract of Analysis is founded on three Newton uses th any letters for sy fluents & the same letters with pricks up above them for their fluxions. In other Paper Mr Newton uses sometimes the same letters in a different magnitude or form sometimes other letters, sometimes lines represented by two capital letters. For he doth not confine his method to any particular sort of symbol, of fluxions. Mr Leibnitz hath no proper symbols for of fluxions. A But methods consist not in names & symbols. What ever be the symbols the methods may be are the same if the do the same things & after the same & after the same manner of working. This Tract of Analysis is founded on three Rules, The two first of wchwhich are equipollent . . . . . 155 2 Off the Differential Method Inf any equation involveing two unknown quantities suppose x & y, to find when either of them suppose y is greatest or least, Fermat puts o for the difference of two roots valors of the other quantity x, &substituting x+o for x has a new equation, & by reducing these two equations & making the quan difference o become infinitely little & vanish finds the greatest or least quantity y. This was his method de maximis & minimis, & by this method he drew Perpendiculars to Curve Lines. Mr Iames Gregory in the 7th Proposition of his Geometriæ pars universalis published 1667 puts the Letter o for the difference of two Abscissas & draws a right line through the ends of the Ordinates & that this line may become the Tangent of the Curve makes the difference o vanish. Dr Barrow in his 10th Lecture published 16670, to find Tangents to Curves, puts the Letters a & e for the indefinitely small Differences of the Abscissas & Ordinates draws a right line through the ends of the Ordinates, by the conditions of the Curve brings the Probleme to an Equation, rejects all the terms of the Equation in wchwhich a & e are either wanting or of more dimensions then one & by the proportion of a to e draws the Tangent. And this method readily gives the method of Slusius. Slusius founded his method of Tangents on three Rules 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 $\frac{{\mathrm{y}}^3-{\mathrm{x}}^3}{\mathrm{y}-\mathrm{x}}=\mathrm{y}\mathrm{y}+\mathrm{y}\mathrm{x}+\mathrm{x}\mathrm{x}$ Mr Newton at the request of Mr Collins sent him his method of Tangents in a Letter dated 10 Decem. 17672. It proved to be the same with the method of Slusius, but was sent as a Corollary of a general method of solving Problems: wchwhich method in drawing of Tangents agreeds with thatose of Dr Barrow Gregory & Barrow. Slusius for sent his method to Mr Oldenburg in Ianuary 1673. It was founded on 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 $\frac{{\mathrm{y}}^3-{\mathrm{x}}^3}{\mathrm{y}-\mathrm{x}}=\mathrm{y}\mathrm{y}+\mathrm{y}\mathrm{x}+\mathrm{x}\mathrm{x}$ Mr Leibnitz for $\frac{{\mathrm{y}}^3-{\mathrm{x}}^3}{\mathrm{y}-\mathrm{x}}=\mathrm{y}\mathrm{y}+\mathrm{y}\mathrm{x}+\mathrm{x}\mathrm{x}$, wrote $\frac{d{\mathrm{y}}^3}{d\mathrm{y}}=3\mathrm{y}\mathrm{y}$, putting dy & dx for the a & e of Dr Barrow. For in his Letter of 21 Iune 1677 in wchwhich he first proposed his method of Tangents, he wrote thus Clarissimi Slusij methodum Tangentium nondum esse absolutam celeberrimo Newtono assentior: et jam a multo temporerem Tangentium longe generalius tractavi, scilicet per differentias Ordinatarum. And a little after he added. Hinc nominando dy differentiam duarum proximarum y & dx differentiam duarum proximarum x; patet $d{\mathrm{y}}^2$ esse $2\mathrm{y}d\mathrm{y}$, & $d{\mathrm{y}}^3$ esse $3{\mathrm{y}}^2$ et ita porro. Which is the first Lemma of Slusius. Then putting y for the Abscissa & x for the Ordinate of a Curve he feigns assumes an equation expressing the relation between them, & to find the Tangent to this Curve he substitutes in this equation the Abscissa xy+ dxy for xy & the Ordinate $\mathrm{x}+d\mathrm{x}$ for x, & in doing this he writes down first those terms in wchwhich dy & dx are not, & draws a line under them. Then under that line he writes down those terms in wchwhich dy & dx are but of one dimension, & draws a line under them. And under that line he writes down those terms in wchwhich dy & dx are either severally or joyntly of more dimensions then one. And then he adds: Vbi abjectis illis quæ sunt supra lineam primam lineam, quippe nihilo æqualibus per æquationem primam; et abjectis illis quæ sunt infra secundam quia in in illis duæ infinitæ parvæ in se invicem ducuntur, restabit tantum quicquid reperitur inter lineam primam et secundam. Then after he had shewn by what remained between the lines to draw the tangent, he added: Quod coincidit cum Regula Slusiana, ostenditque eam statim occurrere hanc methodum intelligenti. By hanc methodum therefore he did not understand the method of Slusius but another method wchwhich readily gave the method of Slusius; & this was the method of Dr Barrow. For Dr Barrow thus described his own method. He proposes to compute an equation from any conditions of the Curve & in doing this prescribes these Rules saith: Primo inter computandum omnes abjicio terminos saith he in quibus ipsarum a vel e potestas habetur, vel in quibus ipsæ ducuntur in se. Etenim isti termini nihil valebunt. Secundo post æquationem constitutam omnes abjicio terminos literis constantes quantitates notas seu determinatas designantibus, aut in quibus non habentur a vel e. Etenim illi termini semper ad unam æquationis partem adducti nibilum adæquabunt. These were Dr Barrows Rules & these Rules are followed by Mr Leibnitz who sets between two lines the terms that are to be retained & the term above the upper line & below the lower those two sorts of terms that by Dr Barrows Rules are to be rejected, & rejects them accordingly. And that this was the original of Mr Leibnitz method of Tangents is further confirmed by what he wrote in the Acta Eruditorum mensis Iunij 1686 pag 299. Malo autem, saith he, dx et similia adhibere quam literas pro illis quia istud dx est modificatio quædam ipsius x et ita ope ejus fit ut quando sola quando id fieri opus est litera x cum suis scilicet potestatibus & differentialibus calculum ingrediatur & relationes transcendentes æquatione explicare licet. Dr Barrow used the letters a & e. Mr Leibnitz allows that Letters he might have used Letters but tells us that for certain reasons he chose rather to use the symbols dx & dy. for the reasons here set down But he should have told us that he us whence he had the method. He should have acknowledged that he used Dr Barrows method of Tangents, excepting that for certain reasons he had changed the letters a & e used by Dr Barrow, into yethe symbols dx & dy. For he had seen Dr Barrows Lectures. Mr Leibnitz first published his method of Tangents in the Acta Eruditorum mensis Octobris An. 1684, pag.467, with this Title Nova Methodus pro maximis et minimis itemque tangentibus quæ nec fractas nec irrationales quantitates moratur, & singulare pro illis calculi genus, per G.G.L. And in the end of it he added Et hœc quidem initia sunt tantum Geometriæ cujusdam multo sublimioris ad difficillima et pulcherrima quæque etiam mistæ matheseos problemata pertingentis, quæ sine calculo nostro differentiali, aut simili, non temere quisquam pari facilitate tractabit. It remains that we enquire how Mr Leibnitz came to know that this method of tangents stuck not at fractions or surds & that it was the found conteined the was the principles of a far more sublimer Geometry reaching to all the most difficult & curious Problems in Mathematicks & what was the Calculus similis here hinted at. At the request of Mr Collins Mr Newton sent to him his Method of Tangents in a Letter dated 10 Decem. 1672. It proved to be the same with that wchwhich Slusius about five weeks after sent to Mr Oldenburg but was derived from a more general Principles. For when Mr Newton had described it, he subjoyned in the same Letter Hoc est unum particulare vel Corollarium potius Methodi generalis quæ extendit se citra molestum ullum calculum non modo ad ducendum Tangentes ad quasvis Curvas sive Geometricas sive Mechanicas, vel quomodocunque rectas lineas aliasve Curvas respicientes; verum etiam ad resolvendum alia abstrusiora Problematum generade curvitatibus, Areis, Longitudinibus centris gravitatis Curvarum &c Neque (quemadmodum Huddenij methodus de maximis & minimis) ad solas restringitus æquationes illas quæ surdis quantitatibus sunt immunes. And a Copy of this Letter was sent by Mr Oldenburg Iune 26th 1676, to Mr Leibnitz at Paris amongst the extracts of Mr Gregories Letters collected by Mr Collins as above. 156 And Mr Newton in his Letter of dated 13 Iune 1676 & sent by Mr Oldenburg to Paris Iune 26, taught how to resolve any dignity of an binomium into a converging series, the second terme of wchwhich Series by the method of Dr Barrow readily gives the first term Lemma of Slusius together with his wole method. And after he had in that Letter described his method of Series, he subjoyned: Ex his videre est quantum fines Analyseos per hujusmodi æquationes infinitas 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 methodus eliciendi series infinitas . . . . . . Sed quomodo in istis casibus procedendum sit non vacat dicere: ut neque alia quædam tradere quæ circa reductionem serierum infinitarum in finitas ubi rei natura tulerit, excogitabi. And in his Letter dated 24 Octob. 1676, he represented how the Tract wchwhich he wrote five years before upon the method of Series, was for the most part taken up wthwith other things. That there was in it the method of Slusius built upon another foundation wchwhich gave that method readily, even without a particular Demonstration, & made it more general so as not to stick at surdes; the tangent, not withstanding surdes, being speedily drawn without any reduction of the Equation wchwhich would often render the work immense. And that the same manner of working held in Questions de Maximis & Minimis & some others, wchwhich in that Letter he forbore to speak of. And that upon the same foundation the Quadratures of Curves became more easy, And he set down an instance of the force of this method in an infinite series for squaring of Curves wchwhich brake off & became a finite equation when the light be example of wchwhich he gave in a Series wchwhich brake of and became finite when the Quadrature might be done by a finite equation. And that that this method extended to inverse problems of Tangents & others more difficult. But the foundation of this method he concealed in sentences set down œnigmatically: the first of wchwhich was this. Data æquatione fluentes quotcunque æquationes involvente fluxiones invenire, & vice versa. Thus Mr Newton in these three Letters represented that his method was very universal, that it gave the method of Slusius as an obvious Corollary, & that it proceeded wthwithout sticking at surds & facilitted Quadratures. And after all this information Mr Leibnitz in his Letter of 27 Iune 1677 proposed his differential calculus in these words Clarissimi Slusij Methodum tangentium nondum esse absolutam Celeberrimo Newtono assentior. Et jam a multa tempore rem Tangentium longe generalius tractavi, scilicet per differentias Ordinatarum. Then he defines his new Notation, saying: Hinc in posterum nominando in posterum dy differentiam duarum proximarum y, &c. Then he gives an example of drawing Tangents by the method changing the a & e of Dr Barrow into dy & dx, & observes how the method of Slusius follows from it, & how it is to be improved so as not to stick at surds, & then adds Arbitror quæ celare voluit Newtonus ab his non abludere. Quod addit ex hoc eodem fundamento Quadraturas quoque reddi faciliores me in sententia hac confirmat; nimirum semper figuræ illæ sunt quadrabiles quæ sunt ad æquationem differentialem. Thus he concludes that he had now got a method like that of Mr Newton, & therefor in the Acta Eruditorum by yethe calculus similis meant Mr Newtons method. But But he tells us: Et jam a multo tempore rem tangentium longe generalius tractavi, scilicet per differentias Ordinatarum. If he means that he had used Dr Barrows method of Tangents jam a multo tempore, tis nothing to his purpose. But if he means that he had improved it into a general method jam a multo tempore, it lies upon him to prove it. For by the law of all nations, in cases of controversy no man can be a witness for himself. And for any man to insist upon his own candour with a designe to be admitted a witness for himself is a demonstration of his want of candour. If there had been no competition in the case he might have been credited without doing injustice to any man: but he is here putting in his claim to the methods of Dr Barrow & Mr Newton, & therefore by the law of all nations it lies upon him to prove his assertion. In the mean time these Arguments make against him. In the beginning of the year 1672 he claimed the differential method of Mouton as his own & was reprehended for it by Dr Pell, & yet persisted in maintaining that he had invented it apart & much improved it, but he did not yet pretend to any other Differential method. In the year 1675 he composed a small work upon the Quadrature of the circle vulgari more because he had not yet found out his new Analysis. For in the Acta Eruditorum mensis Aprilis 1691 pag 178 he wrote thus. Iam anno 1675 compositum habebam opusculum Quadraturæ Arithmeticæ ab amicis ab illo tempore lectum, sed quod materia sub manibus crescente limare ad editionem non vacavit post quam aliæ occupationes supervenere; præsertim cum nunc prolixius exponere vulgari more quæ Analysis nostra nova paucis exhibet non satis operæ pretium videatur. The matter grew under his hands till other affairs came on, that is, till he was called hom to be imployed in publick affairs which happened in October & November 1676,; & after that when he had found his new Analysis wchwhich exprest that Quadrature in few words, he did not think it worth his while to o on with his composition vulgari more. In his Letter to Mr Oldenburg dated 12 May 1676 he wrote that he was polishing the Demonstration of this Quadrature; & he sent it to him in his Letter of 27 August. 1676 composed vulgari more without the help of his new Analysis: & therefore he had not yet found out that method. In the same Letter of 27 August 1676, when Mr Newton had said that his Analysis by the help of infinite equations extended to the solution of almost all Problems, he replied: Id mihi non videtur. Sunt enim multa usque adeo miro et implexa ut neque ab Æquationibus pendeant neque ex Quadraturis. Qualia sunt (ex multis alijs) Problemata methodi Tangentium inversæ. Which is a Demonstration that he had not yet found out the Differential method After he had received Mr Newtons a copy of Mr Netons Letter of 10 Decem. 1672 whereby he had notice that the method of Tangents published soon after by Slusius was but a branch or Corollary of a general method for solving of Problems; his mind ran upon improving that Method, as appears by his Letter to Mr Oldenburg from Amsterdam dated $\frac{18}{28}$ N Novemb. 1676 For there he wrote: Methodus Tangentium a Slusio publicata nondum rei fastigium tenet. Potest aliquid amplius præstari in eo genere quod maximi foret usus ad omnis generis Problemata. Nimirum posset brevis quædam calculari circa Tangentes Tabula, eousque continuanda donec progressio Tabula apparet. Amstelodami cum Huddenio locutus sum. Amplior Methodus tangentium a Slusio publicata dudum illi fuit nota. Amplior ejus methodus est quam quæ a Slusio fuit publicata. And these were the improvements of the Method of Slusius wchwhich then occured to Mr Leibnitz. But when he had received Mr Newtons Letter dated 24 Octob. 1676, which gave him further light into the true improvement, he wrote back: Clarissimi Slusij methodum Tangentium nondum esse absolutam Celeberrimo Newtono assentior. And: Hinc nominando in posterum dy differentiam duarum proximarum y &c He had now fixed his Notation & began here to communicate it: And if he would have his method of an earlier date, he is in point of candor to prove it & by yethe law of all nations to prove it. 157 Mr Leibnitz for $\frac{{\mathrm{y}}^3-{\mathrm{x}}^3}{\mathrm{x}-\mathrm{y}}=\mathrm{y}\mathrm{y}+\mathrm{y}\mathrm{x}+\mathrm{x}\mathrm{x}$, wrote $\frac{d{\mathrm{y}}^3}{d\mathrm{y}}=3\mathrm{y}\mathrm{y}$, or rather finding the method of Dr Barrow to be more v founder upon clearer & & more general principles changed his a & e into dx & dy. In the year 16 beginning of the year 1673 he claimed the differential method of Mouton as his own but being & was reprehended for it by Dr Pell amp; yet persisted in making him maintaining that he had invented it apart & much improved it,; but did not yet pretend to any other differential method. In the year 1675 he composed a compend small work upon the Quadrature of the Hyperbola Circle vulgari more because he had not yet found out his new Analysis. For in the Acta Eruditorum mensis Aprilis 1691 pag 178, he writes thus. Iam anno 1675 compositum habebam opusculum Quadraturæ Arithmetiticæ ab amicis ab illo tempore lectum, sed quod materia sub manibus crescente limare ad editionem non vacavit postquam alijæ occupationes supervenere; præsertim cum nunc prolixius exponere vulgari more quæ Analysis nostra nova paucis exhibet, non satis operæ pretium videatur. Interim insignes quidam Mathematic The matter grew under his hands till other business aff affairs, & after that, when other business came on he had fonud his new Analysis by wchwhich exprest it in few words he did not think it any longer worth his while to to expound propose it prolixly in the vulgari more He returned home by England & Holland in November & October & November & December 1676 & therefore found the Differential method after that time. In his Letter to Mr Oldenburg dated 12 May 1676 he wrote to Mr Oldenburg that he was polishing the Demonstration of this method, & he sent it it to him in his Letter of 27 Aug. 1676 composed more vulgari without the help of his Analysis nove: A therefore he had not yet found out that method. In his Letter of 27 Aug In the same Letter of 27 Aug 1676 he when Mr Newton had said that his Analysis by the help of co infinite equations extended to the solution of almost all Problemes except those of num he replied Id mihi non non videtur. Sunt enim multa usque adeo mira & implexa ut neque ab Æquationibus pendeant neque ex Quadraturis. Qualia sunt (ex multis alijs) problemata methodi tangentium inversæ. Which is a Demonstration that he had not yet found out the Differential method. In his Letter to Mr Oldenburg $\frac{18}{28}$ Novem. 1676 he was upon improving the method of Slusius by getteng a Table of Tangents to be computed, which was another method of improving it. t In his Letter to Mr Oldenburg $\frac{18}{28}$ Novem. 1676, he wchwhich was about four months after he had received a copy of Mr Newtons letter of 10 Decem. 1672 concerning the method of Tangents [In Iuly or Aug 1676 he received a copy of Mr Newtons letter of 10 December 1672 concerning the representing that the method of Tangents there set down (wchwhich proved to be yethe same wthwith that of Slusius) was a Co branch or Corollary of a very general method &] in his letter from to Mr Oldenburg from Amsterdam dated $\frac{18}{28}$ Novem. 1676, he wrote Methodus tangentium a Slusio publicata nondum rei fastigium tenet. Potest aliquid amplius prœstari in eo genere quod maximi foret usus ad omnis generis Problemata: [etiam ad meam (sine extractionibus) Æquationum ad series reductionem]. Nimirum posset brevis quædam calculari circa Tangentes Tabula eousque continuanda donec progressio Tabulæ apparet. And this was the improvement of the method of Slusius wchwhich his mind then ran upon. The next year upon his arrival at Hannover he fell into public And Mr Newton in his Letter of . . . . after taught how to resolve the any dignity of a binomium into a converging series the second, term of which series by the method of Dr Barrow gives the first Lemma of Slusius together wthwith his whole method. And after he had described this method of series he subjoyned: Ex his videre est the reduction of infinite equations into finite ones when it might be, And whereas he had said in his Letter of 13 Iune 1676 that his method of series became not universal wthwithout some other methods, & that wchwhich he then forbore to describe as also what he had invented concerning he here set down the foundation of those methods in sentences exprest enigmatically wchwhich & gave a series for squaring of figures wchwhich brake off & gave the quadrature in a finite equation wchwhich it might be. Mr Newton at the request of Mr Collins sent him his method of Tangents in a Letter dated 10 Decem. 1672. It proved to be the same wthwith that of wchwhich Slusius but founded upon another Principle sent to Mr Oldenburg about five weeks after but founded upon another principle the method of fluxions wchwhich in drawing of Tangents agrees with the method of Dr Barrow.. [If x be the Abscissa & v the o xn=y the ordinate & x the Abscissa be increased by an indefinitely small quantity o so as to become x+o the ordinate will be ${\overline{)\mathrm{x}+\mathrm{o}}}^{\mathrm{p}}$ wchwhich being reduced into an infinite series become ${\mathrm{x}}^{\mathrm{p}}+\mathrm{p}\mathrm{o}{\mathrm{x}}^{\mathrm{p}-1}+\mathrm{\&c}$ as is set down in Mr Newton Analysis pag 19. And by Dr Barrows rules if all the terms be rejected in wchwhich o is either wanting or of more dimensions then one there will remain $\mathrm{p}\mathrm{o}{\mathrm{x}}^{\mathrm{p}-1}$ for the diffe increase of the ordinate. Therefore the subtangent is to the Ordinate as o the increase of the abscissa to $\mathrm{p}\mathrm{o}{\mathrm{x}}^{\mathrm{p}-1}$ the increase of the Ordinate that is as 1 to px $\mathrm{p}{\mathrm{x}}^{\mathrm{p}-1}$, & by consequence the subtangent is is $\frac{1}{\mathrm{p}}\mathrm{x}$ $\frac{\mathrm{y}}{\mathrm{p}\mathrm{x}\mathrm{p}-1}$ or $\frac{\mathrm{y}\mathrm{x}}{\mathrm{p}\mathrm{x}\mathrm{p}}$ If z be yethe Abscissa of a Curve & x the Ordinate, & ${\mathrm{z}}^{\mathrm{n}}=\mathrm{x}$ the equation, & Mr Newton by interpolation of series This method readily gives the method of Slusius] Mr Newton in his Letter Mr O. stated of 24 Novem. 171 1676. wrote that the had explained his method of Tangents in a Tract written 5 years before vizt A. 1671 that it flowed readily was the same with the method communicated by Slusius, that it but flowed from a fountain wchwhich gave it readily without needing a D particular Demonstration, & that it stuck not at radicals, or surds & by wchwhich exten & wchwhich in like manner extended to the solution of determining of maxima & minima & some other P sorts of Problems & rendred Quadratures of Curves more easy & stuck not at surds & was comprehended concealed in this sentence exprest enigmatically: Data æquatione quotcunque fluentes quantitates involvente fluxiones invenire, & vice versa. Thus Mr Newton in these three Letters described represented that his method to as very universal, theat gave that it gave the method of Slusius as an obvious Corollary, & not to st & that it proceeded in Problems of Tangents & of maxima & minima with &c some others wthwithout sticking at surds & to faciliated quadratures. And after all this information Mr Leibnits in his Letter of 21 Iune 1677 proposed his differentiall calculus in these words Clarissimi Slusij Methodum Tangentium nondum esse absolutam New Celeberrimo Newtono assentior In the meane time these arguments make against him. business, wchwhich hindered him from finishing his Nova Analys Arithmetical Quadrature of the circle compos for the Press until he found his New Analysis wchwhich made him not think it worth the while to pub finish what he had composed been composing vulgari more. But after when he had received Mr Newtons Letter dated 24 Octob. 1676 wchwhich gave him further light into the true improvement: he [wrote back Clarissimi Slusij methodum tangentium nondum esse absolutam Celeberrimo Newtono assentior. And & And: Hinc nominando imposterum dy differentiam duarum] And fixed his differential no] beg notation & began to communicate it writing Hinc nominando wrote back: Clarissimi Slusij methodum tangentium nondum esse absolutam celeberrimo Newtono assentior. And Hinc nominando in posterum dy differentiam duarum proximarum y &c: [This is the first mention of this method in his Letters & if he would have his method differential method it of an earlier date he is bound in point of candour to prove it.] Here he fixed his notation & began to communicate it. And if he would have his method of an earlier date he is bound to communicate it in point of candour to prove it. 158 Mr Leibn as may appear by the following comparison The calculation by the method of Mr Leib Dr Barrow $\begin{array}{c}\underline{\mathrm{a}+\mathrm{b}\mathrm{y}+\mathrm{c}\mathrm{x}+\mathrm{d}\mathrm{y}\mathrm{x}+\mathrm{e}\mathrm{y}+\mathrm{f}{\mathrm{x}}^2+\mathrm{g}{\mathrm{y}}^2\mathrm{x}+\mathrm{h}\mathrm{y}{\mathrm{x}}^2\text{ \&c}}\\ \phantom{\mathrm{a}}\phantom{+}\mathrm{b}\mathrm{a}+\mathrm{c}\mathrm{e}+\mathrm{d}\mathrm{y}\mathrm{e}+2\mathrm{e}\mathrm{y}\mathrm{a}+2\mathrm{f}\mathrm{x}\mathrm{e}+2\mathrm{g}\mathrm{x}\mathrm{y}\mathrm{e}+2\mathrm{h}\mathrm{x}\mathrm{y}\mathrm{e}\text{ \&c}\\ \underline{\phantom{\mathrm{a}}\phantom{+}\phantom{\mathrm{b}\mathrm{a}}\phantom{+}\phantom{\mathrm{c}\mathrm{e}}+\mathrm{d}\mathrm{x}\mathrm{a}\phantom{+}\phantom{2\mathrm{e}\mathrm{a}\mathrm{a}}\phantom{+}\phantom{2\mathrm{f}\mathrm{x}\mathrm{e}}+\mathrm{g}{\mathrm{y}}^2\mathrm{a}+\mathrm{h}{\mathrm{x}}^2\mathrm{a}\text{ \&c}}\\ \phantom{\mathrm{a}}\phantom{+}\phantom{\mathrm{b}\mathrm{a}}\phantom{+}\phantom{\mathrm{c}\mathrm{e}}+\mathrm{d}\mathrm{a}\mathrm{e}+\mathrm{e}\mathrm{a}\mathrm{a}+\mathrm{f}\mathrm{e}\mathrm{e}+\mathrm{g}\mathrm{x}\mathrm{a}\mathrm{a}+\mathrm{h}\mathrm{y}\mathrm{e}\mathrm{e}\phantom{\text{ \&c}}\\ \phantom{\mathrm{a}}\phantom{+}\phantom{\mathrm{b}\mathrm{a}}\phantom{+}\phantom{\mathrm{c}\mathrm{e}}\phantom{+}\phantom{\mathrm{d}\mathrm{a}\mathrm{e}}\phantom{+}\phantom{\mathrm{e}\mathrm{a}\mathrm{a}}\phantom{+}\phantom{\mathrm{f}\mathrm{e}\mathrm{e}}+2\mathrm{g}\mathrm{y}\mathrm{a}\mathrm{e}+2\mathrm{h}\mathrm{x}\mathrm{e}\mathrm{a}\text{ \&c}\\ \phantom{\mathrm{a}}\phantom{+}\phantom{\mathrm{b}\mathrm{y}}\phantom{+}\phantom{\mathrm{c}\mathrm{e}}\phantom{+}\phantom{\mathrm{d}\mathrm{a}\mathrm{e}}\phantom{+}\phantom{\mathrm{e}\mathrm{a}\mathrm{a}}\phantom{+}\phantom{\mathrm{f}\mathrm{e}\mathrm{e}}+\mathrm{g}\mathrm{e}\mathrm{a}\mathrm{a}+\mathrm{h}\mathrm{a}\mathrm{e}\mathrm{e}\phantom{\text{ \&c}}\end{array}\}=0$ The calculation by the Method of Dr Mr Leibnitz $\frac{\mathrm{a}+\mathrm{b}\mathrm{y}+\mathrm{c}\mathrm{x}+\mathrm{d}\mathrm{y}\mathrm{x}+\mathrm{e}{\mathrm{y}}^2+\mathrm{f}{\mathrm{x}}^2+\mathrm{g}{\mathrm{y}}^2\mathrm{x}+\mathrm{h}\mathrm{y}{\mathrm{x}}^2\text{\&c}}{\begin{array}{c}\mathrm{b}\mathrm{d}\mathrm{y}+\mathrm{c}\mathrm{d}\mathrm{x}+\mathrm{d}\mathrm{y}\mathrm{d}\mathrm{x}+22\mathrm{y}\mathrm{d}\mathrm{y}+2\mathrm{f}\mathrm{x}\mathrm{d}\mathrm{x}+2\mathrm{g}\mathrm{x}\mathrm{y}\mathrm{d}\mathrm{y}+\\ +\mathrm{d}\mathrm{x}\mathrm{d}\mathrm{y}+\mathrm{g}{\mathrm{y}}^2\mathrm{d}\mathrm{x}+\end{array}}$ This put Mr Leibnitz upon considering the method of Slusius & how it might be improved. For in his Letter to Mr Oldenburg dated from Amsterdam $\frac{18}{28}$ Novem 1676 he wrote thus. Methodus tangentium a slusijo publicata nondum rei fastigium tenet. Potest aliquid amplius præstari in eo genere quod maximi foret usus ad omnis generis Problemata. Nimirum posset brevis quædam calculari circa Tangentes Tabula, eousque continuanda donec progressio Tabulæ apparet. And a little after: Methodus tangentium a slusijo publicata dudum Huddenio fuit nota Amplior ejus methodus est quam quæ a slusio fuit publicata. B By it appears that he He was not yet master of the right way of improving it, but this winter or in spring following began to understand it. Mr Newton in his Letter dated — — — others more difficult. And after all this description of an universal method, Mr Leibnitz at length wchwhich stuck not at surds & whereof the method of Tangents published by Slusius was but a branch or Corollary Mr Leibnitz at length found but wchwhich was derived from a bette more general principle Mr Leibnitz [at length [fell upon the differential method of dawing Tangents & found that it was capable of these b those improvements mentioned by Mr Newton &] in his Letter of 21 Iune 1677 proposed his differential method in these words Clarissimo Newtono Slusij Methodum Tangentium nondum esse absolutam Celeberrimo Newtono assentior. Et jam a multo tempore rem Tangentium longe generalius tractavi, scilicet per differentias Ordinatarum. And Then explaining what he meant by these differences he added proposes his notation new notation Hinc nominando in posterum dy differentiam duarum proximarum y &c This And this was the beginning of his notation. A And Then he goes on to shew how by this method with Dr Barrows method shewing how Tangents may be drawn, allowing to yethe ons keeping thereby Rules of Dr Barrow above mentioned & how the Method of Slusius follows from it & how it as Mr Newton had notified, how & as Mr Newton had notified & how it is to be improved so as not to sticks not at Tang surds, & then adds Arbitror quæ celare voluit Newtonus ab his non abludere. Quod addit, ex hoc eodem fundamento Quadraturas quoque reddi faciliores me in sententia hac confirmat, nimirum semper figuræ illæ sunt quadrabiles quæ sunt ad æquationem differentialem. Thus he concludes that he had now got a method like that of Mr Newton & therefore in the Acta Eruditorum by the caculuscalculus similis meaant me the like method wchwhich Mr Newtons method had had partly described openly & partly concealed Mr Newtons method. But he tells us: Et jam a multo tempore rem Tangentium longe generalius tractavi scilicet per differentias Ordinatarum. If he means that he had used Dr Barrows so method of Tangents jam a multo tempore, tis nothing to his purpose. But if he means that he had improved it into a general method jam a multo tempore: it lies upon to prove it. For by the law of all nations, in cases of controversy no man can be a witness for himself. And for any man to insist upon it upon it upon any fuetence of d integrity r candor his own candor with a designe to be a of integrity witnes for himself, is a demonstration of his want of candor integ candor. & integrity If there were had been no competition in the case, he might have been credited without doing injustice to any body: but he is here putting in his claim to the methods of Dr Barrow & Mr Newton & therefore by the laws of all nations must put in his prove his assertion p. 88. pro nota * Idem fecit &c scribe * Gregorius in Prop. 7 Geometriæ universalis anno 1668 impressæ et Barrovius rem langentium tractavit per differentias ordinatarum. Barrovius in ejus Lect 10 anno 1669 impressa idem fecit, sed, paulo generalius. Methodus T Slusius methodum suam tangentium fundavit in hoc Lemmate: Differentia duarum Dignitatum ejusdem gradus applicata ad differentiam laterum [id est Differentia duarum Ordinatam dat partes singulares gradus inferioris ex binomio laterum, ut $\frac{{\mathrm{y}}^3-{\mathrm{x}}^3}{\mathrm{y}-\mathrm{x}}=\mathrm{y}\mathrm{y}+\mathrm{y}\mathrm{x}+\mathrm{x}\mathrm{x}$. Et hic applicando hoc Theore Lemma ad rem Tangentium subintelligat appliunt per differentias infinite parvas. Newtonus in Literis Epistola 10 Decem. ad Collinium datis c missa data et hoc an a anno superrore circa mensem inter 14 Aug Iun & 11 Aug. d circa mensem Iulium an a circa mensem Iulium ad D. Leibnitium missa fuit fuerat ad Leib superiore data cujus exemplar inter Collectiones Gregorianas anno D. Oldenburgus anno superiore ad D. Leibnitium miserat, scripsit methodum Tangentium Slusij esse particulare quoddaam vel potius Corollarium potius Methodi generalis quæ extenditeret se citra molestum ullum calculum ad Tangentes curvarum tam Mechanicaru resolvendum alia abstrusiora Problematum genera de Curvitatibus, Areis, Longitudinibus, centris gravitatis curvarum &c et ad quantitates surdas & Curvas Mechanicas minime hæreret. Et in epistola 13 Iunij 1676 ad D. Leibnitium itidem missa scripserat Analysin suam beneficio serierum ad omnia pene Problemata sese extendere. D. Leibnitius tandem respondit id sibi non videri; esse enim multa usque adeo mira & implexa ut neque ab æquationibus pendeant neque ex quadraturis, qualia sunt (ex multis alijs) Problemata methodi tangentiūum inversæ. Newtonus rescripsit inversa tangentium Problemata esse in potestate aliaque illis difficiliora, I Et methodum Tangentium flusij a suis principijs statim consequi idque generalius cum methodus sua quantitates surdas minime moraretur, & eodem modo se rem habere in quæstionibus de æ maximis et minimis alijsque quibusdam & eodem fundamento quadraturm Curvarum faciliorem reddi simpliciorem reddi & cujus exempla quædam dedit sed fundamentum ipsum literis transpositis celavit hanc sententiam involventibus. Data Æquatione quotcunque fluentes quantitates invol vente fluxiones invenire; et vice versa. Et D. Leibnitius his omnibus admonitus ne a Newtono aliquid didicisse videretur, tandem respondit in hæc verba: Clarissimi Slusij methodum Tangentium nondum esse absolutam Celeberrimo Newtono assentior; et jam a multo tempore rem Tangentium longe generalius tractavi, scilicet per differentias ordinatarum. Et in epistola 29 Decem. 1711 data, addidit, se inventum plusquam nonum in annūum pressisse: Quasi habuisset ante mensem OctobrenOctobrem anni 1675 ideoque a Newtono nil didicisset me nil didicisset. Ad Notam * pag 90 adde. Certe D. Leibnitius similitudinem methodorum non tantum jam intellexerat & sed etiam postea anno 1684 mense Octobri postea ubi methodum differentialis elementa in lucem emisit sub hoc titulo: Nova methodus pro maximis et minimis, itemque tangentibus, quæ nec fractas nec irrationales moratur, & subjiunxit: Et hæc methodus quidam initia sunt tantum Geometriæ cujusdam multo sublimioris ad difficillima & pulcherrima quæque etiam mistæ Matheseos pertingentis quæ sine calculo nostro differentiali, aut SIMILI, non temere quisquam pari facilitate tractabit. Vide Acta Eruditorum Mensis Octob. pag 467, 473. Conferatur hæc methodi Ldifferentialis descriptio cum descriptione consimili methodi fluxionem in Epistolis tribus Newtoni ad D. LeibnititumLeibnitium missis, ut similitudinem methordorum Leibnitio cognitam videas. 159 The History of the Method of Moments or Differenc called Differences by Mr Leibnitz. In the Introduction to Book of Quadratures published A.C. 1704 Mr Newton wrote that he found the Method of fluxions gradually in the years 1665 & 1666, & tho this was not so much as Dr Wallis said nine years before in the Preface to the first volume of his works without being then contradicted, yet in the Acta Eruditorum for Ianuary 1705, in giving an Account of this Book Mr Leibnitz is called the Inventor; & from thence is deduced this conclusion: Pro differtijsdifferentijs igitur Leibnitianis Newtonus adhibet semperque [pro ijsdem] adhibuit fluxiones — iijsque tum in Principijs nNaturæ Mathematicis tum in alijs postea editis [pro differentijs illis] eleganter est usus quemadmodum et Honoratus Fabrius in sua Synopsi Geometrica motuum progressus Cavallerianæ methodo substituit. Dr Wallis was homo vetus & rerum anteactarum peritissimmus informed himself of these matters from the beginning, being very inquisitive in Mathematicall affairs, & having received from Mr Oldenburg Mr cópies of Mr Newtons my two Letters of 13 Iune & 24 Octob. 1676 when they were newly written. and in the said Preface he said that in those Letters I Mr Newton I had explained to Mr Leibnitz the Method found by hime ten years before or above; meaning that Mr NewtonI had found the Method above ten years before Mr Leibnitz, & that he I had so far discovered it to Mr Leibnitz him by those Letters, as to leave it easy to find out the rest. And even before Mr NLeibnitz had the Method, Mr Newton I said in one of those Letters (that of 24 Octob 1676) said that the foundation of the Method was obvious, & therefore, since he I had not then leasure to describe it at large, he I would conceale it in an Ænigma. This he I did, not to make a mystery of it, but to prevent its being taken from him me because it was obvious. And in that Ænigma he I set down the first Proposition of the book of Quadratures because it was obvious in the very words of the Proposition; & therefore had this Proposition with the Method founded upon it where he I wrote that Enigma: or rather, because the very words of the Ænigma Proposition are copied in the Ænigma, it argues that the Book of Quadratures was then before hime, & so was written before Mr Leibnitz had the Method. This Book is said to have bee was extracted out of older papers. In the said Letter of 24 Octob. 1676 Mr Newton I set down a series for squaring of figures wchwhich in some cases breaks off & becomes finite & illustrated it with examples & said that he I found this & some other Theorems of the same kind by the method whose foundation was comprehended in that Ænigma, that is, by the method of fluxions. And how I found them I explained in the first six Propositions of the Book of Quadratures, & do not know any other method by which they could be found: & therefore when I wrote that Letter I had the Method of fluxions so far as it is conteined in those six Propositions. After I had finished the Book & while the 7th 8th 9th & 10th Propositions were fresh in memory I wrote upon them to Mr I. Collins that Letter wchwhich was dated 8 Novem 1676 & being found amongst his Papers was published by Mr Iones. The Theoremes160 Theoremes at the end of the tenth Proposition for comparing curvilinear figures with the Conic sections were known to m are mentined in th my said Letter of 24 Octob 1676, & all the Ordinates of the figures are there copied in the second part of the Table are there copied from the Book. And therefore the Book was then before me. To understand the two Letters of Octob 24 & Novem 8 1676 & how to perform the things & find the Theorems mentioned in them requires skill in all the Method of fluxions so far as it is comprehended in all the first ten Propositions of the Book. And the eleventh & last Proposition depends upon a series of first second third & fourth fluxions. The first Proposition of this Book & its solution illustrated with examples in first & second fluxions &c was at the request of Dr Wallis sent to him almost verbatim in a Letter dated 27 Aug. 1692, & printed by him that year in the second Volume of his works, which came abroad the next year, A.C. 1693. And thus the Rule for finding second third & fourth fluxions there set down was published some years before the Rule for finding second third & fourth differences & was at least seventeen years before in manuscript before it was published. In the Indtroduction to this Book the method of fluxions is taught without the use of prickt letters; for I seldom used prickt Letters when I considered only first fluxions: but when I considered also second third & fourth fluxions I distinguished them by the number of pricks. And this notation is not only the oldest but is also the most expedite, tho it was not known to the Marquess de l'Hospital when he recommended the differential Notation. In my Analysis per æquationes numero terminorum infinitas communicated by Dr Barrow to Mr Collins in Iuly 1669, I said that my Method by series gave the areas of Curvilinear figures exactly when it might be, that is, by the Series breaking off & becoming finite: & thence it appears that when I wrote that Analysis, I had the Method of fluxions so far, at least, as it is conteined in the first six Propositions of the Book of Quadratures; tho those Propositions were not then drawn up in the very words of the Book. In that Tract of Analysis I represented time by a line increasing or flowing uniformly & a moment of time by a particle of the line generated in the moment of time, & thence I called the particles a moment of the line; & the particles of all other quantities generated in the same moment of this line time I called the moments of those quantities; & the fact under the rectangular Ordinate & a moment of the Abscissa I considered as the moment of the rectiline cursa (rectilinear or curvilinear) area described by that Ordinate moving uniformly upon the Abscissa. For a moment of time I put the letter o, & thence computed the moments of the other quantities generated in the moment of time, & for those moments put any other symbols. And for the Area of a figure I sometimes put the Ordinate included in a square. And by considering how to deduce moments from increasing quantities & quantities from their moments, I deduced areas Ordinates of figures from their Ordinates Areas & Ordinates the Areas from the areas Ordinates: wchwhich is the same thing with deducing fluxions161 fluxions from fluents & fluents from fluxions. And in the end of the Book I demonstrated by this sort of calculus the first of the three Rules set down in the beginning thereof. And ain this Rule for the index of a Dignity I put an indefinite quantity affirmative or negative integer or fract, for the inden & thereby introduced indefinite indices of Dignities into Analysis. And applying this Method of Moments not only to finite equations but also to equations involving converging series I gave this Tract the name of Analysis per æquationes numero terminorum infinitas. 163 The history of the Differential Method. Mr Collins having received from me & Mr Iames Gregory several series for squaring the Circle & Conic Sections, was very free in communicating them to the Mathematicians both at home & abroad in the years 1670, 1671 & 1672 & Mr Leibnitz was in London in the beginning of the year 1673 & went from thence to Paris in the beginnin end of February carrying Mercators Logarithmotechnia along with him & kept a correspondence wthwith Mr Oldenburg till Iune following about Arithmetical Questions, being not yet acquainted with the higher Geometry. Then he intermitted his correspondence till Iuly 1674 & in the mean time studied the higher Geometry beginning with the Horologium oscillatorium of Mr Huygens wchwhich came abroad in April 1673. His following correspondence was about converging series till spring 1676. And I And then upon the news of Mr Iames Gregories death, he wrote for a collection of M Gregories Papers, & the Demonstration of my Series, meaning my Method of finding them & p promised Mr Oldenburg a reward for my Method & directed him to Mr Collins for the same told him that Mr Collins could help him to it. I suppose he meant my Analysis per Series numero terminorum infinitas. For that was the only Paper in wchwhich I had sent my Method of Series to Mr Collins. Thereupon Mr Collins drew up as well Extracts out of Gregories Letters, & the Collection was sent to Paris In Iune following to be returned & it is now in the Archives of the R. Society; but instead of sending a copy of my Analysis, he & Mr Oldenburg jointly sollicited me to send what the Method wchwhich Mr Leibnitz desired, & thereupon I wrote my Letter to Mr Olde of 13 Iune 1676, wchwhich & this was sent to him at the same time with the Collection. In this Collection was a copy of a Letter of Mr Iames Gregory to Mr Collins dated 15 Feb. 16768 16$\frac{70}{71}$ wchwhich conteins several Series one of wchwhich was that famous one for finding the Arc whose sin tangent is given: wchwhich series had been also sent by Mr Oldenburg to Mr Gregory Leibniz the year before & the receipt thereof acknowledged. There was also in the same Collection a copy of a Letter of Mr Gregory to Mr Collins dated Sept 5. 1670 in wchwhich Gregory wriotes that by improving the method of Tangents of Barrow he had found a method of Tangents wthwithout calculation. There was also in the same Collection a copy of a Letter which I had writ to Mr Collins above thre years before. The Letter was dated 10 Decem 1672, & is as follows. Ex animo gaudeo – – me grave ducas. And copies of these two last Letters had be were communicated also by Mr Oldenburg to Mr Thurn Tschunhause in Iune 1675. In my letter of 13 Iune 16756 I had said (with relation to the Method described in my Analysis per æquationes numero terminorum infintas,) Ex his videre est quantum fines Analyseos per hujusmodi series infinitas æquationes ampliantur: quippe quæ earūum beneficio, ad omnia pene dixerim problema (si numeralia Diophanti & similia excipias) sese extendit. And Mr Leibnitz in his Answer dated 27 Aug. 1676, replied: Quod dicere videmini plerasque difficultates (exceptis Problematibus Diophænteis) ad series Infinitas reduci; id mhihi non videtur. Sunt enim multa usque adeo mira et implexa ut neque 164 ab Æquationibus pendeant, neque ex quadraturis. Qualia sunt (ex multis alijs) Problemata methodi Tangentium inversæ. In the same Letter he placed the perfection of Analysis not in the Differential method but in another Analysis method composed of Analytical Tables of tangents & the Combinatory Art. Nihil est, said he, quod norim in totāam Analysi momenti majoris. And a little after: Ea vers nihil differt ab Analysisi illa SVPREMA, ad eujus intima Cartesius non pervenit. Est enim ad eam constituendam opus Alphabeto cogitationum humanarum. This was the top of his skill at that time & therefore he had not yet found out the differential method nor had hitherto used fluxions for his differences. In October 1676 he canecame to London a second time & there met with Dr Barrows Lectures, & saw my Letter of Octob. 24. 1676 & therein had fresh notice of my Compendium of Series or Analysis com & consulting Mr Collins saw in his hands many of mine & Gregories Letters, especially those relating to series, & therein had fresh notice of my Compendium of Series or Analysis communicated by Dr Barrow to Mr Collins, & consulting Mr Collins saw in his hands many of mine & Gregories Letters, especially those relating to series & in his way home from London was meditating how to improve the method of Tangents of Slusius as appears by his Letter to Mr Oldenburgh dated from Amsterdam $\frac{18}{28}$ Novem. 1676. And the next year in a letter to Mr Oldenburgh dated 21 Iune he sent hither his new Method with this Introduction. Clarissimi Slusij Methodum Tangentium nondum esse absolutam celeberrimo Newtono assentior. And in describing this Method he abbreviated Dr Barrows method of Tangents by new symbols & shewed how it might be improved by the so as to the Method of Slusius (wchwhich was the same with that of Gregory) & to proceed in equations involving surds; & then subjoyned: Arbitror quæ celare voluit Newtonus de tangentibus ducendis ab his non abludere. Quod addit, ex eodem fundamento Quadraturas reddi faciliores me in sententia hac confirmat. This was the first time that he began to communicate his differential Method & therefore I had not hitherto used fluxions for his Differences; nor can the assertion be true Pro differentijs Leibnitianis Newtonus semper adhibuit fluxiones. Mr Iames Bernoulli in the Acta eruditorum for December 1691 pag. 14 said that the Calculus of Mr Leibnitz was founded in that of Dr Barrow & differed not from it except in the notation of differentials & some compendium of operation. And the Marquess de l'Hospital in the Preface to his Analysis of infinite petits published A.C. 1696, represented that where Dr Barrow left off Mr Leibnitz proceeded, & that the improvement wchwhich he made to the Doctors Analysis consisted in excluding fractions & surds: but the Marquiss did not then know that Mr Leibnitz had notice of this improvement from me by two Letters above mentioned, dated 10 Decemb. 1672 & 24 Octob. 1676, a copy of the first being sent to h im in Iune 1676. After he had notice that such an improvement was to be made, he might find it proprio Marte, but by that notice knew that I had h it before him. And in his Letter of 21 Iune 1677 he confessed that he had such notice. In the Acta Eruditorum for October 1684 Mr Leibnitz published the Elements of the differential Method as his own without mentioning the correspondence wchwhich he had formerly had with the English me about these matters. He mentioned indeed a Methodus similis; but whose that Method was or what he knew of it he did not say, as he should have done. And this his silence put me upon a necessity of writing the Scholium upon the second Lemma of the second Book of Principles, least it should be thought that I borrowed that Lemma from Mr Leibnitz — but said that I had interwoven it with the method of infinite series & that being tyred wthwith these speculations I had absteined from them five years, that & therefore had this method above five years before & had that is before the year 1671. And that it was so general as to reach almost all Problems except perhaps some numeral ones like those of Diophantus. And & therein he was again told that the foundati I had a my Method of working wchwhich readily gave me the method of tangets of Slusius directly & immediatediatelyimmediately so as to need no demonstration thereof & that stuck not at equations equations affected with radicals involving one or both indeterminate quantities, & proceeded in the same manner in questions about maxima & minima & in some others wchwhich I did not there mention, & faciliated Quadratures & gave me general Theremes for that end Quadratures, one of wchwhich I there set down & illustratd wthwith examples & that I had wrote a tract of it upon thits method & the method of series together five years before with a designe to print it together wthwith a tract about light & colours. In my Letter of Octob 24 I After things Mr Leibnitz in the year 1684 published thie elements of this method of as his own without making any mention of the foregoing correspondence The first Proposition of the Book of Quadratures is certainly the foundation of the method of fluxions. This This Proposition was comprehended verbatim in the ænigma by wchwhich in my Letter of Octob. 24 1676 I set down concealed the foundation of the Method th there spoken of & therefore that method was the method of fluxions. In that Letter I said that I had written a Tract ofn this Method & the Method of series together five years before but did not finish it nor meddle any more with these things till the year 1676 being tyred with them.. And in my Letter of Iune 13 I wrote to the same purpose. And this is the Method wchwhich I described in my Letter of Decem 10th 1672. In my Letter of Octob. 24 1676 I said also that Analysis per Series Æquationes numero terminorum infinitas I said that of that method of Analysis the method there described: Denique ad Analyticam merito pertinere censeatur, cujus beneficio Curvarum areæ & longitudines &c (id modo fiat) exacte & Geometrice determinentur. Sed ista narrandi non est locus. This was done effected by series which in these some cases break off & become finite, as you may understand by the Letter of Mr Collins to Mr Strode Iuly 26, 1672. And therefore when I wrote before Dr Barrow sent that Analysis to Mr Collins, that is before Iuly 1669 I had the method of fluxions so far at the least as it is conteined in the first six Propositions of the Book of Quadratures. I sent him also the method of extracting fluents out of the method of equations involving fluxions, wchwhich to the best of my memory was composed in the year 1671. Mr Leibnitz never was in quiet possession abroad nor I out of possession in England. He has been told again & again that he was not the first iinventerorInventor & never would answer directly to this point, but only pretendeth answered indirectly by pretending that he found it apart, & that gave it to him in the Scholium upōon the 2d Lemma of yethe 2d Book of Principles In a Paper dated 29 Iuly 1713 & written by one who used the Leibnitian phrase illaudabili laudis amore & knew what Mr Leibnitz did at Paris 40 years before I was singled out & treated very reproachfully. And this was to make me appear. And in his Postscript to Mr. l'Abbe Cont written in Autumn. 16715 singled out & treated in a very provoking manner. to make me appear. And when I was prevaidled with to return an Answer to this Postscript he declared in his Letter to Abbé Conti of Apr. 9th 1716 to Mr. l' Abbé Conti that he had resolved not to would by no means enter the lists with my forlorn hope (meaning Mr Keill & Mr Coles &c) but since I was willing to appear my self he would give me satisfaction. Nothing wcould satisfy content him but to make me appear & give declare my opinion, & now I have declared it, I leave every body tato his own opinion, out of a desire to be quiet. The Editors of the Acta Eruditorum of Iune 1696 for silencing Dr Wallis it t pretendinged that I had acknowledged both publickly & privately that in the year 1676 or about 20 years before or above in the year 1676 or before Mr Leibnitz had the Differential calculus & Infinite Series & general methods for them + + & yet I never all of heard of his having the differential calculus before Iune 1677, nor in those days knew of Mr Oldenburghs letter of Apr. 15 1675 nor of a of tegories Letter of 15 Feb. 1671 sent to Mr Leibn in frane 1676 by both wchwhich he received his infinite series from England, & nor looked upon his method of Series to be general, Afterwards Mr Leibnitz in his Letters to Dr Sloan of 4 March st. n. 177 & 29 Decem St. n. 1711 declared answering Dr Keill & pressed me to declare my opinion, & declined answering Dr Keill, & ×× & in the defamatory libel dated 29 Iulij 1713 berated me very (written by one who knew what passed between Mr Leibnitz & his friends at Paris) forty years before) & in his Letter of Apr 29 to Mr. l'Abbe Contei he declared that he would not enter the lists wthwith my forlorn hope (meaning Keill & Mr Coats) but since I was willing to appear my self he would give me satisfaction. Nothing would satisfy him but to make me appear & declare my opinion & now I have declared it, I am resolved to be silent 165 The History of the Method of Moments Fluxions & Moments approaching Series. In the Introduction to Book of Quadratures published A.C. 1704 I wrote that I invented the Method of fluxions gradually in the years 1665 & 1666, & tho this was not so much as Dr Wallis said nine years before in the Preface to the first volume of his works without being then contradicted, yet the in the Acta Eruditorum for Ianuary 1705 Mr Leibnitz in giving an Account of this Book Mr Leibnitz is called the Inventor, meaning the first Inventor. For And from thence is deduced this conclusion. Pro differentijs igitur Leibnitianis Newtonus adhibet semperque [pro ijsdem] adhibuit fluxiones — iijsque tum in Principijs Naturæ Mathematicis, tum in alijs postea editis [pro differentijs illis] eleganter est usus, quemadmodum et Honoratus Fabrius in sua Synopsi Geometrica, motuum progressus Cavallerianæ methodo substituit. Dr Wallis was homo vetus & rerum anteactarum peritissimus being inquisitive in these matters & having received from Mr Oldenburg copies of my two Letters of 13 Iune & 24 Octob. when 1676 when they were newly written. And he said that I had ex in those Letters I had explained to Mr Leibnitz the method found by me ten years before or above, meaning I suppose that I had it above ten years before him & had so far discovered it to him by those Letters as to leave it easy to find out the rest. And this is nothing more than what I said in so the Letter And in one of those Letter wchwhich before he had it before he began to write of it) that of 24 Octob 1676,) we I said that the foundation of the Method was very obvious & therefore I (since I had not leasure to describe it at large) I would conceale it in an Ænigma. This I did not to make a mystery of it but to prevent its being taken from me because it was obvious. And in In That Ænigma I set down the first Proposition of the book of Quadratures in the very same words of the Proposition ‖ ‖ & therefore had this Proposition with the method founded upon it when I wrote the said Letter. Or rather; because the very words of the Proposition are copied in the Ænigma, it argues that the Book of Quadratures was then before me. For this was written that year being d Book was then before me, being newly written. It was extracted for thate most part out of a Tract wchwhich I wrote in the year 1671 but left unfinished & out of & some other older papers. In the said Letter of 24 Octob. 1676 I set down a series for squaring of figures wchwhich in some cases breaks off & becomes finite, & illustrated it with examples & said that I found thatis & some other Theorems of the same kind by the method whose foundation was comprehended in that Ænigma, that is, by the Method of fluxions. And how I found them I explained in the first six Propositions of the Book of Quadratures: & I do not know any other method by which they could be found: & therefore when I wrote that Letter I understood had the Method of fluxions so far as it is conteined in those six Propositions. When After I had finished the Book & the 7th 8th 9th & 10 Propositions were fresh in memory I wrote upon them to Mr Iohn Collins that Letter wchwhich was dated 8 Novem. 1676, & being found amongst his Papers of Mr Iohn Collins was published by Mr Iones. The Theorems at the end of the tenth Proposition for comparing curvilinear figures with the Conic sections were known to me also when before I wrote the said Letter of 24 Octob 1676, they being there mentioned & all the Ordinates of the Figures being there set down. copied from yethe Book in due order. [Those Theoremes were copied from the Tract wchwhich I wrote in the year 1671.] To understand these two Letters & how to performe the things & find the Theorems conteined in them requires skill in all the Method of fluxions so far as it is conteined comprehended in all the first ten Propositions of the Book. And the eleventh & last depends upon a series of first second third & fourth fluxions. The first Proposition of this Book with and its solution illustrated with examples was at the request of Dr Wallis communicated sent to him in a Letter dated 27 Aug. 1692 & printed by him almost verbatim that year in the second Volume of his works wchwhich came abroad the next year, A.C. 1693. And thus the Rule for finding second third & fourth fluxions there set down was published some years before the Rule for finding second third & fourth differences, & was at least eighteen seventeen years before in manuscript before it was published . In the Introduction to this Book the method of fluxions is taught without the use of prickt letters: for I seldome used prickt letters when I considered only first fluxions. But when I considered also second third & fourth fluxions I distinguished them by the number of pricks. And this notation is not only the166 the oldest but is also the most expedite, tho it was not known to the Marquess de l'Hospital when he recommended the differential Notation. In my Analysis per æquationes numero terminorum infinitas communicated by Dr Barrow to Mr Collins in the year Iuly 1669, I said that my Method by series gave the areas of Curvilinear figures exactly when it might be, (that is, by the series breaking off & becoming finite,) & thence it appears that when I wrote that Analysis I had the Method of fluxions so far at the least as it is conteined in the first six Propositions of the Book of Quadratures, though those Propositions were not then drawn up in the words of the Book. In that Tract I represented time by a line increasing or flowing uniformly & a moment of time by a particle of the line generated in the moment of time, & thence I called the particle a moment of the line, & the particles of all other quantities generated in the same moment of the Lin time I called the moments of those quantities, & the fact under the rectangular Ordinate & a moment of the Abscissa I considered as the moment of the curvilinear area described by that Ordinate moving uniformly upon the Abscissa. ☉ ☉ For a moment of time I put the letter o, & thence computed the moments of the other quantities generated in the moment of time & for those moments put any other symbols. And for the Area of a figure I sometimes put the Ordinate included in a square. And by considering how to deduce moments from increasing quantities & those quantities from their moments I deduced areas of figures from their ordinates & their Ordinates from their areas: which is the same thing with deducing fluents from fluxions from fluents & fluents from fluxions For the area I something put the Ordinate included in a square & And in the end of yethe Book I demonstrated by this sort of calculus the first of the three Rules set down in the beginning thereof. And applying this method of moments not only to finite equations but also to equations involving converging series, considered as equations consisting of an infinite number of terms, I gave this Tract the name of Analysis per æquationes numero terminorum infinitas. And ✝ And shewing also thereby to find the Ordinates & Areas of Mechanical Curves &c [that their lengths & tangets may be found by the same Method;] I said: Nec quicquam hujusmodi scio ad quod hæc methodus idque varijs modis sese noli extendit. Imo . . . & quiquid vulgaris Analysis per æquationes ex finito terminorum numero constantes (quando id sit possibile) perficit, hæc per æquationes infinitas semper perficiet. And in my Letter of 13 Iune 1676 I said of this Analysis: Ex his &c And in my Letter of 13 Iune 1676 I said of this Analysis: Ex his videre est quantum fines Analyseos per hujusmodi infinitas æquationes ampliantur: Quippe quæ earum beneficio ad omnia pene dixerim Problemata (si numeralia Diophanti et similia excipias) sese eatendit. And Mr Leibnitz in his Letter of 27 Aug. 1676 replied that he did not believe that ⊡ ⊡ my method was so general the inverse Problems of TangtsTangents & many others not being reducible to Equations or Quadratures. BAnd in the same Letter he & placed the perfection of Analysis not in this Method but in a Method composed of Analytical Tables of Tangents & the Combinatory Art, saying of the one; Nihil est quod norim in tota Analysi momenti majoris: & of the other; Ea vero nihil differt ab Analysi illa suprema, ad cujus intima, quantum judicare possum, Cartesius non pervenit. Est enim ad eam constituendam opus Alphabeto cogitionum humanarum. This was the top of his Analytical skill at that time. Mr Collins in a Letter to Mr Strode dated 26 Iuly 1672 gave this account of these Methods. Mense Septembri 1668 Mercator Logarithmotechniam edidit suam, quæ specimen hujus methodi (i.e. Serierum infinitarum) in unica tantum figura, nempe quadraturam Hyperbolæ, continet. Haud multo postquam prodierat liber, exemplar ejus Cl. Wallisio Oxonium misi, qui suum de eo judicium in Actis Philosophicis statim fecit: aliumque Barrovio Cantabrigiam, qui quasdam Newtoni Chartas – – – extemplo remisit: E quibus et ex alijs, quæ olim ab Auctore cum Barrovio communicata fuerant, patet illam Methodum a dicto Newtono aliquot annis antea excogitatam et modo universali applicatam fuisse: ita ut ejus ope in quavis figura Curvilinea proposita quæ una vel pluribus proprietatibus definitur, Quadratura vel area dictæ figuræ, accurata si possibile sit, sin minus, infinite vero propinqua; evolutio vel longitudo lineæ curvæ; Centrum gravitatis figuræ; solida ejus rotatione genita, et eorum superficies, idque non obstantibus radicalibus, quæ eam non morantur, obtineri queant. ✝ ✝ Postquam intellexerat D. Gregorius hanc methodum, a D. Mercatore in Logarithmotechnia usurpatam, et Hyperbolæ quadrandæ adhibitam quamque audauxerat ipse Gregorius, jam universalem redditam esse, omnibusque figuris applicatam; acri studio eandem acquisivit, multumque in ea enodanda desudavit. Vterque D. Newtonus & D. 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 grounded upon papers wchwhich I had communicated to him from time to time I had the method here described some years before the Doctor sent my Analysis to MrCollins, that is, some years before Iuly 1669. And this is sufficient to justify what I said in the Introduction to the Book of Quadratures, vizt that I found the Method gradually in the years 1665 & 1666. Dr Barrow then read his Lectures about motion, & that might put me upon taking these things into consideration. I found the Method of Series in the beginning of the year 1665 the method of Moments soon after * * If it be asked why I did not publish this method sooner, it was for the same reason that I did not publish the Theory of colours sooner. I found the method of series in the beginning of the year 1665 & the method of fluents & moments soon after & the theory of colours in the beginning of the year 1666, & in the year 1671 was about the Theory of colours also in the beginning of the year 1666, & in the year 1671 was about to publish it them all together with the Methods of Series & fluxions: but for a reason mentioned in my LettterLetter of 24 Octob. 1676, I desisted till the year 1704, excepting that some of my Letters were published before by Dr Wallis & Mr Oldenburg.] The first line that Mr Leibnitz began to communicate the differential Method was his Letter of 27 Iune 1677, & therefore the Editors of the Acta eruditorum calumniated me accused me falsly in saying Pro differentijs Leibnitoanis Newtonus adhibet semperque adhibuit fluxiones: And they & their adherents ought to have proved the accusation. In one of those Letters (that of 24 Octob. 1676) 167 And Account of the Method of Fluxions found until Mr Leibnitz had notice of it in the year 1676 inclusively. The method of fluxions is this. Mr Newton considers two or more quantities as flowing or growing in magnitude by continuall increase in the same line, & the velocities of their increase he calls their fluxions, & their parts generated in moments of time he calls their moments, the names of fluxions & moments being taken from the fluxion & moments of time. For the flowing quantities or fluents he puts any symbols as z, y, x, & for their fluxions he puts any other symbols or even the same symbols distinguished by their magnitude or by form or by any mark as $\stackrel{.}{\mathrm{z}}$, $\stackrel{.}{\mathrm{y}}$, $\stackrel{.}{\mathrm{x}}$. But one of the fluents he usually considers as flowing uniformly or in proportion to time, & usually puts an unit for its fluxion & the letter o for its moment. And for the moments of the other fluents he puts their fluxions multiplied b letter o multiplied by their fluxions Foras 1 the fluxion of time (or of the exponent) is to $\stackrel{.}{\mathrm{z}}$ the fluxion of z, so is o the moment of time to $\stackrel{.}{\mathrm{z}}\mathrm{o}$ the moment of z. When he is demonstrating any Proposition he takes the moment o for o in the sense of the vulgar for an indefinitely (not infinitely) small part of time, & performs the whole operation in finite figures by the Geometry of the Ancients without any approximation, & when the calculation is finished & the Equation reduced, he supposes that yethe moment van of the equation vanish in wchwhich are affected with the moment o & by from the remaining terms d draws his conclusion. But when he is only investigating a truth, he supposes the moment o to be infinitely little, & for making dispatch neglects to write it down, it works in figures infinitely small by all manner of approximations wchwhich he conceives will make no error in the conclusion. The first way is sure & exact the second expedite., & both are according natural In his Letter dated 24 Octob 1676 Mr comprehended thise method of fluxions in this sentence. Data æquatione fluentis quotcunque fluentes quantitates involvente invenire fluxiones & vice versa. And how to deduce fluxions from æquations is involving their fluents is taught in the first Proposition of his Book of Quadratures. The Rule there delivered is this Multiplicetur omnis æquationis terminus per indicem dignitatis quantitatis cujusque fluentis quam involvit & in singulis multiplicationibus mutetur dignitatis latus in fluxionem suam & aggretgatum factorum omnium sub proprijs signis erit æquatio nova. This Rule wthwith its Examples Mr Newton copied & sent to Dr Wallis in his Letters & sent to DrWallis in his two Letters dated Aug. 27, & Sep. 17, 1692 & therefore the method was invented & the book of Quadratures was written before that time. The Rule teaches the inventio Proposition relates to the 2d 3d & 4th Fluxions as well as to the first. For As every Fluent has its fluxion so every fluxion may be so every fluxion may be considered as a fluent & have its fluxion, And hence a & this is called the second fluxion of the first fluxionent, & the fluxion of the second fluxion may be called the third fluxion & so on. And the like is to be understood of moments. The moment of the first moment is the second moment of the fluent & the moment of the second moment is the third moment & so on. How to deduce fluxions from equations involving their fluents is taught in the first Proposition of Mr Newtons book of Quadratures. The Rule there described delivered is this. Multiplicetur omni æquationis terminus per indicem dignitatis quantitatis cucjusque fluentis quam involvit & in singulis multiplicationisbus mutetur dignitatis laturs in fluxionem suam, & aggregatum factorum omnium sub proprijs signis erit æquatio nova. This Rule The first multiplication gives This single Rule gives all the fluxions. The first multiplication gives the first fluxions the second gives the second fluxions & so on, as you may perceive by is explained in the examples of the Rule. This Proposition with the solution & the Examples is see was copied by Mr Newton & in his Letters dated Aug 27 & Sept. 17, 1692 sent to Dr Wallis & the next year printed by Dr Wallis in his works the second Volume of his works pag. 392 ,& 393 & therefore the book of Quadratures was writ before that time. The Mr Newton in his Letter dated 24 Octob. 1676 comprehended the method of Fluxions is this sentence Data æquatione quotcunque fluentes quantitates involvente invenire fluxiones & vice versa. And the fift Proposition of the book of Quadratures was set down at length & s in the same Letter & was & said to be deduced from the same method of Fluxions & called the first Theorem in a series of such Propositions. Whence the sixt Proposition of the book of Quadratures was also then known to Mr Newton it being the second of the series. And so were the third & fourth Propositions, how these two Propositions were found out by the method of fluxions is fully set down in the first six six Propositions of yethe book of Quadratures. And therefore These six Propositions were therefore known to Mr Newton when wh in the year 1676; & so were some others that follow in the same book as may be gathered from what is cited out of them in the same Letter & in another Letter written by Mr Newton to Mr Collins & pub Nov. 8. 1676 & published by Mr Iones. And by so many things quoted out of the book of Quadratures it may be concluded that in the year 1676, it may be concluded that the book of Quadratures (except the Introduction & last Scholium) was written before that year. For indeed the designe of the book was to make a step towards the inverse method of fluxions. Another step was made by the solution of this Probleme, Extracere fluentem quantitatem ex æquatione simul involvente fluxionem ejus. Ex æquatione fluxionem radicis involvente radicem extrahere. The solution whereof was me sent to Dr Wallis by Mr Newton in his Letter of dated Sept 17th 1692 & published by the DrDoctor in the second volume of his work pag. 394. And at the end of the example there set down it was noted Mr Newton that note affirmed added that by the same method the roots of equations might involving the second third fourth fluxions & other fluxions ($\stackrel{..}{\mathrm{y}}$ $\stackrel{.}{\stackrel{..}{\mathrm{y}}}$ $\stackrel{..}{\stackrel{..}{\mathrm{y}}}$ &c) might be extracted. And this method was known to Mr Newton in the year 1676 as appears by his Letter then dated 24 Octob. where he saith Inversa de Tangentibus Problemata sunt in potestate aliaque illis diffilicilora. Ad quæ solvenda usus sum duplici methodo; una concinniori altera generaliori. Vna methodus consistit in extractione fluentis quantiatis ex æquatione simul involvente fluxionem ejus: altera &c. And by these things its manifest that Mr Newton in the year 1676 understood the method of fluxions direct & inverse (including the second & had then extended it to the 2d 3d 4th & other fluxions. And tho he sometimes uses prickt letters for fluxions yet he doth not confine himself to those symbols. In the first Proposition of his book of Quadratures he uses such letters: in the Introduction to yethe book he desembles the method & gives examples of solving Problemes by it wthwithout making any use of such Letters. Mr Newton in his Letter dated 24 Octob 1676 represented that five years before, vizt A.C. 1671, he wrote a treatise of the method of the method of infinite series & of another method wchwhich readily gave the method of Tangents of Slusius & stuck not at surds & wchwhich was founded in this sentence Data æquatione quotcunque fluentes quantitates involvente invenire fluxiones & contra vice versa. And in a Letter to Mr Collins dated 10 Decem. 1672 (which was some weeks before Slusius sent his method of Tangets into England) he described the same method of Tangets as a Corollary of the ge or branch of the his general method wchwhich stuck not at Tangents surds & said that this general metho wchwhich was therefore his method of fluxions. And he said of this method that it extended to the solution abstruser sort of Problems about the Curvatures areas, lengths, centers of gravities of Curves &c & succeeded in Mechanical Curves as well as others. And By its succeeding in M By its determining the Curvatures of Curves you may know So then he understood the method of fluxions in those days, & by his applying it to Problems about the Curvature of Curves you may know that he had then extended it to the consideration of the second fluxions. The sentence  The sentence, Data æquatione quotcunque fluentes quantitates involvente, fluxiones invenire; & vice versa, being the foundation of the method upon wchwhich he wrote in the year 1671 & relating to the second third & following Fluxions as well as to the first, & being one & the same method in them all, it must be allowed that in the year 1671 his method extended to all the fluxions. For after the very same manner that this method, being applied to any equation gives a new æquation involving the first fluxions of the fluents, if it be applied to this new one it gives another new one involving their second fluxions, & so on perpetually. In this Analysis per Æquationes numero terminorum infinitas was founded upon three Rules. The first Rule was this. Let a be a given quantity, x the ab of a Curve, y the Ordinate, & $\frac{\mathrm{m}}{\mathrm{n}}$ , a y $\frac{\mathrm{m}}{\mathrm{n}}$ any number whole the index of a dignity whole or broken affirmative or negative. And if the Ordinate be $\mathrm{a}{\mathrm{x}}^{\frac{\mathrm{m}}{\mathrm{n}}}=\mathrm{y}$ the area of the Curve will be $\frac{\mathrm{a}\mathrm{n}}{\mathrm{m}+\mathrm{n}}{\mathrm{x}}^{\frac{\mathrm{m}+\mathrm{n}}{\mathrm{n}}}$. That is (as Mr Newton afterwards explains) if the fluxion be $\mathrm{a}{\mathrm{x}}^{\frac{\mathrm{m}}{\mathrm{n}}}$ the fluent will be a $\frac{\mathrm{n}\mathrm{a}}{\mathrm{m}+\mathrm{n}}{\mathrm{x}}^{\mathrm{m}+\mathrm{n}}$ And this Rule he demonstrates in the end of his book Analysis by the method of fluxions after the very same manner that i the Introduction to his book of Quadratues he Demonstrated found the inverse thereof by that method vizt that if the fluent be ${\mathrm{x}}^{\mathrm{n}}$ the fluxion will be $\mathrm{n}{\mathrm{x}}^{\mathrm{n}-1}$. His Demonstration is this. Let the Abscissa of any Curve ADδ be $\mathrm{AB}=\mathrm{x}$, the perpendicular ordinate $\mathrm{BD}=\mathrm{y}$ the area $\mathrm{ABD}=\mathrm{z}$, the moment of the Abscissa $\mathrm{B\beta }=\mathrm{o}$, the moment 170 of the Area $\mathrm{B\beta \delta D}=\mathrm{o}\mathrm{\upsilon }=\mathrm{BD}\times \mathrm{BK}=rectangulo\mathrm{B\beta HK}$, the side of this rectangle BK being called υ. And the Abscissa AB will be $\mathrm{x}+\mathrm{o}$ & the Area $\mathrm{AB\delta }\text{will be}=2+\mathrm{o}\mathrm{\upsilon }$. Now since if $\frac{\mathrm{n}}{\mathrm{m}+\mathrm{n}}\times \mathrm{a}\mathrm{x}\frac{\mathrm{m}+\mathrm{n}}{\mathrm{n}}=\mathrm{z}$, (as is sub or putting $\frac{\mathrm{x}\mathrm{a}}{\mathrm{m}+\mathrm{n}}=\mathrm{c}$ & $\mathrm{m}+\mathrm{n}=\mathrm{p}$, since $\mathrm{c}{\mathrm{x}}^{\frac{\mathrm{p}}{\mathrm{n}}}=\mathrm{z}$, or ${\mathrm{c}}^{\mathrm{n}}{\mathrm{x}}^{\mathrm{p}}={\mathrm{z}}^{\mathrm{n}}$, if $\mathrm{x}+\mathrm{o}$ be written for x & $\mathrm{z}+\mathrm{o}\mathrm{\upsilon }$ be written for z you will have $\mathrm{c}\mathrm{n}$ in ${\mathrm{x}}^{\mathrm{p}}+\mathrm{p}\mathrm{o}{\mathrm{x}}^{\mathrm{p}-1}+\mathrm{\&c}={\mathrm{z}}^{\mathrm{n}}+\mathrm{n}\mathrm{o}\mathrm{\upsilon }{\mathrm{z}}^{\mathrm{n}-1}+\mathrm{\&c}$ the following terms of these two series being omitted because they would vanish in the conclusion b makin diminishing o in infinitum & making it vanish. of th Now if you reject the equal quantities ${\mathrm{c}}^{\mathrm{n}}$ in ${\mathrm{x}}^{\mathrm{p}}$ & ${\mathrm{z}}^{\mathrm{n}}$ be da & divide the remainder by o the you will have c ${\mathrm{c}}^{\mathrm{n}}\mathrm{p}{\mathrm{x}}^{\mathrm{p}+1}=\mathrm{n}\mathrm{\upsilon }{\mathrm{z}}^{\mathrm{n}-1}(=\frac{\mathrm{n}\mathrm{y}{\mathrm{z}}^{\mathrm{n}}}{\mathrm{z}}=\frac{\mathrm{n}\mathrm{y}{\mathrm{c}}^{\mathrm{n}}{\mathrm{x}}^{\mathrm{p}}}{\mathrm{z}})$ or& dividing both parts of yethe equation by ${\mathrm{c}}^{\mathrm{n}}{\mathrm{x}}^{\mathrm{p}}$ you will have $\mathrm{p}{\mathrm{x}}^{\mathrm{-1}}=\frac{\mathrm{n}\mathrm{y}}{\mathrm{z}}$ or $\mathrm{p}\mathrm{c}{\mathrm{x}}^{\frac{\mathrm{p}-\mathrm{n}}{\mathrm{n}}}=\mathrm{n}\mathrm{y}$. And by restoring $\frac{\mathrm{a}\mathrm{a}}{\mathrm{m}+\mathrm{n}}$ for c & $\mathrm{m}+\mathrm{n}$ for p that is m for $\mathrm{p}-\mathrm{n}$ you & na for pc you have $\mathrm{a}{\mathrm{x}}^{\frac{\mathrm{m}}{\mathrm{n}}}=\mathrm{y}$. Whence on the contrary if $\mathrm{a}{\mathrm{x}}^{\frac{\mathrm{m}}{\mathrm{n}}}\text{ be }=\mathrm{y}$ then $\frac{\mathrm{n}}{\mathrm{m}+\mathrm{n}}\mathrm{a}{\mathrm{x}}^{\frac{\mathrm{m}+\mathrm{n}}{\mathrm{n}}}\text{ will be }=\mathrm{z}$. Q. E. D. If this Demonstration oe calculation be compared with that in the Introduction to the the book of Quadratures you will find them perfectly of the same kind They both procced by resolving the dignity of a Binomium into a converging series & shewing that the second when the second term of the binomium is the moment of the first, the second term of the series is the moment of the dignity. But third term of the series is double to yethe moment of the second & the fourth is triple to the moment of the third. The second Rule upon wchwhich Mr Newton founds his Analysis is that if the area of a Curve be described by an Ordinate composed of several parts the whole area shall be composed by all yethe Areas described by those parts. And the third is And the third Rule is to reduce the compound Ordinates of Curves into such parts wchwhich describe such areas as may be found by the first Rule. And after several examples of squaring Curves by this method, he described the method of fluxions for finding the Areas, Lengths, su convex surfaces, & solid contents & centers of gravity of curvelines & curvilinear figures. Let ABD represent a curvilinear figure & all ABKK a rectangle described wthwith the rectangular Ordinate BD moving uniformly from A towards B upon the Absciss AB,(x) & ABKH a rectangle described in the same time with the given Ordinate BK moving uniformly upon the same Absciss. And let the Ordinates BK(1) & BD(y) be considered as the moments of the by wchwhich theose Areas ABHK (x) & ABD are in continually increased And let c conceive that you can by the three foregoing Rules, from the moment BD deduce Sit ABD Curva quævis et AHKB rectangulum — dum utuntur methodis Indivisibilium. Here several things are to be observed, as first that when he takes a point for yethe moment of a line & a line for the moment of a superficies, he takes a point for an infinitely short line & a line for an infinitely narrow superficies as in the method of Cavallerius. And therefore in pulling the ordinates BKD & BDK for the moments of the areas ABD & ABKH, he takes the Ordinates for infinitely narrow parallelogramms whose bases are the infinitely short parts or moments of the Abscissa AB. Let the moment of the Abscissa (x) be called o & the moments of the Areas ABKH & ABD will be $1\times \mathrm{o}$ & $\mathrm{y}\times \mathrm{o}$, or o & $\mathrm{y}\mathrm{o}$ And this Mr Newton expresses in the end of his Analysis in demonstrating the first Rule. But where hise is only investigating the solutions of Problems he neglects to write down the moment o as was said above. Its to be further observed that in the sense here described he considers a superficies as the moment of a solid, a line as the moment of a supferficies & a point as the moment of a line: wchwhich consideration gives the notion of first second & third moments, a point being the moment of a superf line & the moment of the moment of a superficies & the moment of the moment of the moment of of a solid, that is the first moment of a line, the second moment of a superficies & the third moment of a solid. Its to be observed also that in this Analysis Mr Newton teaches how to find as as many Curves as you please wchwhich may be squared. And this is the second Proposition in the book of Quadratures, & is performed by assuming any equation expressing the relation between the Abscissa & area & of the Curve & finding the relation between their fluxion of the Area relation of their fluxions by the first Proposition, Whence & putting the fluxion of the Ordinate for the Area for the Ordinate, the fluxion of the Abscissa being an unit. And this makes it appear that the two first Propositions of the book of Quadratures were known to Mr Newton when he wrote his Analysis wchwhich was in in the in or before the year 1669 when he wrote his Analysis. It is to be observed also that Mr Newton in the same Analysis affirms that by the help of this method of this Method that Ejus beneficio Curvarum areæ & longitudines &c (id modo fiat) exacte et Geometricæ determinantur. Sed ista narrandi non est locus. And this referrs to the fift & sixt Propositions of the book of Quadratures & shews that the first six Propositions of that book were known to Mr Newton in or before the year 1669. For the fift & sixt Propositions depend upon the third & fourth & those upon the first & second. It is still further to be observed that Mr in the same Analysis Mr Newton uses the symbol $\boxed{\frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}}$ to express the area of a Curve whose Ordinate is $\frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}$ & Mr Leibnits expresses the same thing by the symbol $\int \frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}$, but this symbol is of later date And lastly Also the relation between the method of Series & the method of fluxions deserves to be observed. For let th any dignity of any binomium whose second term is the fluxion of the first term be resolved into a series the & the first term of It is also to be observed that when Mr Newton wrote the said Analysis he understood the resolution of the any dignity of any Binomium into a converging series. For in the Demonstration of the first Rule he set down the two first terms of the series. Let the dignity be ${\stackrel{-}{\mathrm{x}+\mathrm{o}}}^{\mathrm{p}}$ & the series will be ${\mathrm{x}}^{\mathrm{p}}+\mathrm{p}\mathrm{o}{\mathrm{x}}^{\mathrm{p}-1}+\mathrm{p}\times \frac{\mathrm{p}-1}{2}\times \mathrm{o}\mathrm{o}{\mathrm{x}}^{\mathrm{p}-2}+\mathrm{p}{\mathrm{x}}^{\frac{\mathrm{p}-1}{2}}\times \frac{\mathrm{p}-2}{3}\times {\mathrm{o}}^3{\mathrm{x}}^{\mathrm{p}-3}+\mathrm{\&c}$ the two first terms of wchwhich series were set down in the Analysis. And lastly further it is to be noted that when Mr Newton wrote that Analysis he had noted the observed the great relation which the method of converging series have to one ano & that of fluxions have to one another. For he every where applys the method of series to yethe solution of Problems by the method of fluxions & by the method of Series demonstrates the first Rule wchwhich is the foundation of the method of fluxions, & fi determin finds the second term of the series to be the moment of the first by reason of this relation between the methods he had then composed one general method of them both. [This relation is further manifest by comparing the terms of the series above me comparing the terms of yethe series above mentioned with the first second & third moments of the first term. For if the terms of the series (beginning wthwith the second term) be multiplied by this series of numbers, they be 1. 1×2. 1×2×3. &c they become the moments. And this affinity Mr Newton took some no For the second term of the series is the moment of the first term & the third term is half the moment moment of the second, & the 4th is a third part of the moment of the third & so on: as is manifest by the first Rule in the Analysis & by the Demonstration thereof & by what is said in several places of the book of Quadratures. And it may be observed in the last place that Mr Newton by taking putting fract & negative numbers for the indices of Dignities in the first Rule of his Analysis & in the Dignities wchwhich he teaches to resolve into a binomia converging series, reduced the operations of multiplication Division & extraction of roots to one common Rule or way of common way of considering them, & thereby very much enlarged the bounds of Analysis] 168 Mr Newton in a Letter written to Mr Collins & dated 10 Decem 1672, that is, some weeks before Slusius sent his Method of Tangents into England, described the same method of Tangents & represented it a branch or Corollary of his general method wchwhich 171 The History of the Differential Method, written by SrSir Isaac Newton. In Iuly 16769 Dr Barrow sent to Mr Collins a small Tract writ by Mr Newton & entituled Analysis per series Æquationes numero terminorūum infinitas. It was called Analysis per series Æquationes to signify that it was not only a method not only of finding such converging series, but also of considering such series as equations & applying such equations to the s as well as the vulgar ones to the solution of Problemes by means of the three Rules laid down in the beginning of the Tract & more generally by the method of Moments, & thereby enlarging the bounds of the Vulgar Analysis. In this Tract time is represented by a right line bounded at one end in a given point & increasing uniformly at the other end, & from the moments of time the particles of this line generated in these moments of time are called moments of the line, & if Ordinates erected at right angles upon the end the line if Ordinates erected at right angles upon the increasing end of the line describe the area curvilinear areas, the Ordinates drawn into the moments of the limne are considered as moments of the curvilinear areas generated. And the Problemes methods of finding the Areas whose moments are given known & mutually the Moments whose Areas are known are considered explained in this Tract so far as it may be done by the said three Rules. Mr Oldenburgh 14 Sept 1669 gave notice of this Tract to Slusius citing several things out of it And Mr I. Collins in a letter to Mr Tho. Strode dated 26 Iuly 1672 wrote thus concerning it. Mense Septembri 1668, Mercator – – – Extractione obtineri queant anticipare haud integrum ducit. In the year 1671 Mr Newton composed a larger Tract on this subject And from the fluxion of time he gave the name of fluxions to the veolocieisvelocities by wchwhich all quantities increase in the same time, & thence came the name of the Method of fluxions. This is that Method Analysis of wchwhich Mr Newton in his Letter of 10 Decem. 16672 called his general method & of wchwhich ain his Letter of 13 Iune 1676 described to be so general as to reach he said Ex his videre est quantum fines Analyseos per hujusmodi series amplio infinitas æquationes ampliantur: numeralia Diophanti et similia excipias) sese extendit. By the very words of this Letter g here cited you may understand that the Analysis here spoken of is the same wthwith the Analysis æquationes numero terminorum infinitas above mentioned. And this was the state of the Method in the year 1671 when he left of these studies till the writing of this Letter. For in this Letter when he had mentioned the universality of theis Method he added that he had not leasure to explain what related to it. Nam parcius scribo, said he, quod hæ speculationes diu mihi fastidio esse cæperūunt adeo ut ab ijsdem jam per quinque fere annos astinuerim. Mr Collins in the years 1671 ,& 16702 1673 &c was very free in comminating the series wchwhich he had received from Newton & Gregory as appears by his Letters printed in the Commercium Epistolicum. And Mr Leibnitz was in London in the beginning of the year 1673