Newton's observations on the Synopsis given in the Leipzig Acts of Jones's "Analysis per quantiatum Series" (London, 1711)

Observationes
in Synopsin Analyseos per quantitatum series fluxiones ac differentias cum enui|m|eratione linearum tertij Ordinis.

This Synopsis being

The style & spirit of this Synopsis {illeg}|s|hew that it was writ by the Author \discover the Author/ of the synopsis of the Book of Quadratures, & therefore deserve to be come k{n} \& therefore deserved|s| to noted./

{illeg}|It| begins thus: Gulielmus Iones edita synopsi a celebribus — — — — in meditationes suas inciderint. Th That So much of The principal part of the {illeg}y{o} Commercium Epistolicum Collinsianum has been b relating to these {illeg} the subject of the|i||s| Analysis has been lately published, And together w^th the dates of the Letters for discovering the times when things were invented. And because \noth/ the time when the Analysis per æquationes numero terminorum infinitas was written is omitted \not described/ in this synopsis, it may may not be amiss to tell the Reader, that it was communicated to M^r Collins in Iuly 1669 |as appears by three Letters of D^r Barrow & by several other Lettes {sic} of M^r Collins & M^r Iames Gregory about things cot|n|teined therein.| & therefore the things conteined therein were invented before that time. It was the Compendium mentioned by M^r Newton in his letter of 24 {illeg}|O|ctob. 1676.

The Author of this synopsis de in the next place produces out of the Analysis a part of the computation by w^ch M^r Newton demonstrated the first of the three Rules upon w^ch the Analysis was founded. And by this Computation compared w^th other things in the Analysis it appears that M^r Newton when he wrote that Analysis used the very same method of fluxions which he uses at present. He puts represents time by any quantity w^ch folows uniformly, the fluxion of time by an unit, a moment of time by the letter o, other flowing quantities & their fluxions by any other \other/ symbols, & their \del symbo/ moments by their fluxions drawn into a moment of time, the name of moments being taken from the moments of time in w^ch they are generated \& that of fluxions from the fluxion of time/. If he is enquiring after truth or resolving a Probleme, he uses the letter o for an infinitely little moment & commonly for the greater dispatch neglects to express it, putting the symbol of the fluxion \alone as well both for the fluxion &/ for the moment also, but & {illeg} but {subunderstanding} \but usually neglecting to express/ the letter coefficient o when it signifies the fluxion moment. If he is demonstrating any Proposition he makes the he uses the letter \o/ \he always expresses the letter o & uses it/ for \a/ finite moment of \an/ indefinitely (not infinitely) little part of time. In the former case he uses any approximations w^ch he foresees will create no error in the conclusion. In the latter he proceeds in finite quantities exactly by v{illeg}|u|lgar Geometry without any approximations & when he has finis finished the computation he supposes that the finite moment o decreases in infinitum & vanishes. This is M^r Newtons met And by this method of fluxions he applies æquations both finite & infinite to the resolution of Problemes. This was hi Th is his method at present, this was his method when he wrote his \two/ Letters of 1676 & five years before when he wrote the Tract mentioned in the latter of those two letters & \And that this was his method/ in the year 1669 when he wrote his A communicated his Analysis to D^r Barrow & by D^r Barrow to M^r Collins, will appears by the Analysis it self.

This Analysis is founded on three Rules the two first of which are equipollent to the solution of this Problem, Data æquatione fluentes duas quantitates involvente \se invicem non multiplicantes/ fluxiones invenire; & contra. The third Rule {illeg}|d|irects the resolution of finite equations infinit{illeg}|e| ones when there is occasion. These Rules M^r Newton illustrates w^th various examples & then proceeds applies them to the quadrature of curves & then adds /that\ Et hæc de areis curva all Problems concerning the length of curves & the{illeg} contents & surfaces of solids & centers of gravity may be reduced to Quadratures after the following manner. Sit ABD Curva quævis — dato elicietur <460v> Here the rectangle fluents are exposed f \represented/ by the Areas ABD & AK & their fluxions \& moments/ by the Ordinates BD\=y/ & BK=1 & the area AK is supposed to flow uniformly, or in proportion to time & its BK his an unit is put for its fluxion, & {by} si{illeg} moment, But its to be the coefficient o w^ch makes the moments y & 1 infinitely little being neglected. For And after he had set down an example i|b|y computing the length of the arch of a circle from its moment he subjoyns. Sed notandum est quod {illeg}|u|nitas quæ pro momento ponitur est superficies cum de solidis & linea cum de superficiebus & punctum cum de lineis agitur. Nec vereor loqui de unitate in punctis, sive lineis infinite parvis, siquidem proportiones ibi jam contemplantur Geometræ dum u{illeg}|t|untur methodis indivisibilem. {illeg} Therefore so then by a point M^r Newton understands here an infinitely sh So then by a point M^r Newton understands here an infinitely short line, & by a l{ne} & by a line an infinitely narrow surface & by a s & when he calls these moments & represents them by an unit it is to be understood that this unit is multiplied by an infinitely \small/ quantity o, or moment of time, to make it infinitely little. The moment 1 is $1\times \mathrm{o}$ & the moment y is $\mathrm{y}\times \mathrm{o}$, & but the coefficient o for shortning y^e operation is not written down but understood. If o be not understood the{illeg} \lines/ 1 & y are \represent/ the fluxions of A{K} the areas BK & ABD, but if o be understood, these fluents|sion|s become multiplied by o become the moments of BK & AD. For fluxions are finite quantities but moments \here/ are infinitely little{,}. {illeg} |Thus you see his Notation when he wrote this Analysis is |of| the same kind w^th that w^ch he uses at present.|

So in demonstrating the first of his three Rules by this method, in the equation ${\mathrm{c}}^{\mathrm{n}}{\mathrm{x}}^{\mathrm{p}}={\mathrm{z}}^{\mathrm{n}}$ he supposes x & z to increase & be augmented by the moments o & $\mathrm{o}\mathrm{v}$ & to become $\mathrm{x}+\mathrm{o}$ & $\mathrm{z}+\mathrm{o}\mathrm{v}$, & & by those moments understands the f rectangles \$\mathrm{o}\times 1${o} & $\mathrm{o}\times \mathrm{v}$ conteined/ under the fluxions 1 & v & the moment o. Then in the said æquation writing $\mathrm{x}+\mathrm{o}$ for x & $\mathrm{z}+\mathrm{o}\mathrm{v}$ for z there arises ${\mathrm{c}}^{\mathrm{n}}$ in ${\mathrm{x}}^{\mathrm{p}}+\mathrm{p}\mathrm{o}{\mathrm{x}}^{\mathrm{p}-1}+\mathrm{\&c}={\mathrm{z}}^{\mathrm{n}}+\mathrm{n}\mathrm{o}\mathrm{v}{\mathrm{z}}^{\mathrm{n}-1}+\mathrm{\&c}$. Where the first terms ${\mathrm{c}}^{\mathrm{n}}{\mathrm{x}}^{\mathrm{p}}$ & ${\mathrm{z}}^{\mathrm{n}}$ destroy one another & the next divided by o viz^t ${\mathrm{c}}^{\mathrm{n}}\mathrm{p}{\mathrm{x}}^{\mathrm{p}-1}$ & $\mathrm{n}\mathrm{v}{\mathrm{z}}^{\mathrm{n}-1}$ become equal & determin the proportion of the fluxions 1 & v \or of their moments o & $\mathrm{o}\mathrm{v}$./ And its here observable that the series ${\mathrm{x}}^{\mathrm{p}}+\mathrm{p}\mathrm{o}{\mathrm{x}}^{\mathrm{p}-1}+\mathrm{\&c}$ is the same w^th that set d in the beginning of M^r Newtons Letter of 13 Iun 1676, & being produced becomes ${\mathrm{x}}^{\mathrm{p}}+\mathrm{p}\mathrm{o}{\mathrm{x}}^{\mathrm{p}-1}+\frac{\mathrm{p}\mathrm{p}-\mathrm{p}}{2}\mathrm{o}\mathrm{o}{\mathrm{x}}^{\mathrm{p}-2}+\frac{{\mathrm{p}}^3-3\mathrm{p}\mathrm{p}+2\mathrm{p}}{6}{\mathrm{o}}^3{\mathrm{x}}^{\mathrm{p}-3}+\mathrm{\&c}$, &|A||nd| the like is to be understood of the series ${\mathrm{z}}^{\mathrm{n}}+\mathrm{p}\mathrm{o}{\mathrm{z}}^{\mathrm{n}-1}+\mathrm{\&c}$ in like manner to be produced. But M^r Newton foreseeing \have|i|ng shewn before/ that all the terms after the second would vanish with the letter moment o, neglected them. He saw there |fore| in those {illeg}|d|ays that the second ter{illeg}e|m|s of these series gave the moments fluxions & moments of the dignities of any fluent quantity x or $\mathrm{x}+\mathrm{o}$; & And when he understood this he could not be long w^thout seing the use of the third & fourth terms & those that follow upon \by/ this property of these series he founded the Demonstratio|ed|n of the first of the three The Rules upon w^ch he founded his Analysis. \After this/ And when he understood this he could not be long without seing the use of the third terms of these series & of the rest of the terms w^ch follow the third. |For that he knew the use of the third terms in those days is evident from his letter of 10 Dec. 1672, where he saith that {illeg} |by| his method extended to the he determined y^e Curvature of Curves.|

After this Demonstration M^r Newton subjoyns the following conclusion from it. Hinc in transitu notetur modus quo Curvæ tot quot placuerit \quotcun/, quarum Areæ sunt cognitæ possunt inveniri; sumendo nempe quamlibet æquationem pro relatione inter aream z & basem [vel abscissam] x ut inde quæratur [ordinatim] applicata y. Vt si suppunas $\sqrt{\mathrm{a}\mathrm{a}+\mathrm{x}\mathrm{x}}=\mathrm{z}$, ex calculo invenies $\frac{\mathrm{x}}{\sqrt{\mathrm{a}\mathrm{a}+\mathrm{x}\mathrm{x}}}=\mathrm{y}$. Et sic de reliquis. And this \is the second Proposition of his book of Quadratures, & it/ is as much as to say that the method by w^ch the M^r Newton had now demonstrated the first of his three Rules was general, & extended to the determination of this Problem Data Æquatione fluentes duas quantitates involvente fluxiones invenire. And that he extended it also to more then two fluents appears by his letter of 10 December 1672 where he saith that his \method/ stuck not at surds. For surds are in this method considered as fluents. |So then the two first Propositions of the book of Quadratures were then known to M^r Newton|

In this {T}{illeg} \Tract of/ Analysis M^r Newton writes also that his Method extends to such Curve lines as were then called Mechanical. And instances in the Quadratrix by shewing how to find the Ordinate & Area of this Curve & adding that its length may <461r> be found by the same method. And then he subjoyns Nec quicquam hujusmodi scio ad quod hæc methodus, se|id| varijs modis sese non extendit Imo tangentes ad Curvas me{t}|c|hanicas (siquando id non alias fiat) hujus ope ducuntur. Et quicquid vulgaris Analysis per æquationes ex finito terminorum numero constantes (quando id sit possibile) perficit, hæc per æquationes infinitas semper perficit: Vt nil dubitaverium nomen Analysis etiam huic tribuere. Rationcinia nempe in hac non minus certa sunt quam in illa, nec æquationes minus exact{illeg}|æ|. — Deni ad Analyticam merito pertinere censeatur, cujus beneficio Curvarum areæ & longitudines &c (id modo fiat ({illeg}) exacte et Geometrice determinentur. Sed ista narrandi non est locus. Spectant hæc {illeg} These words [id modo fiat] have respect to a certa \sort of/ series w^ch sometimes breakes off & give the Quadrature in finite equations. Th One of these series is set down by M^r Newton in his Letter of 24 Octob 1676 as a part o the first of certain Theoremes |for| Quadratures w^ch he had found b formerly found by his method of fluxions This is the 5^th Proposition in his book of Quadratures \& the first of those for squaring a given curve/ & the sixt is \the second/ of the \second of/ same kind, & these two depend on the 3^d & 4^th & those on of the 1^st & 2^d. So that the first \{5}{illeg}/ 5{illeg} \six/ Propositions \of that book/ were known to him when he wrote his Letter of 24 Octob 1676, or rather \& even/ when he wrote his Analysis as /M^r Collins in his letter to M^r Strode dated 26 Iuly 1672 explains in these words: By the same method may be obteined the Quadrature or Area of the figure accurately. when it can be done, but always infinitely near.\

Next after the Analysis M^r Iones has {re}printed M^r Newton's Letter of 13 Iune 1676, wherein the series \Rule/ for reducing binomials into infinite series is set down whereof {illeg} is set down at large & explained by Examples. The two first terms of thsi Rule being set down in Analysis, it And all this was known to M^r Newton when he wrote his Analysis, (the two first terms of this Rule being there set down,) as was also the extraction reduction of finite æquations into infinite series by extraction of roots out of affected æquations, & the Quadratures of Curves by those series, w^ch m{illeg} w^th the approxi\mati/on of Quadratures makes up the body of the Epistle. And in the next place is an extract of {illeg} M^r Newtons Letter of 24 Octob 1676, conte\in/ing a further explion {sic} of the method of extracting the roots of affected Equations. And next after is M^r Newto a fragment of a letter \of M^r Newton written/ to D^r Wallis conteining in the year 1692 {illeg} wherein he represents that the sentence enigmatically exprest {illeg}|ne|are the end of his Letter of 24 Octob 1676, wherein \in w^ch/ he setts down his double method for solving the Problem of determining Curves by the conditions of their Tangents & others more difficult, was this: Vna methodus consistit in Extractione fluentis quantitatis ex æquatione simul involvente fluxionem ejus Altera tantum in assumptione seriei pro quantitate qualibet incognita ex qua cætera commode derivari possunt; et in collatione terminorum homologorum seriei æquationis resultantis ad eruendos terminos assumptæ seriei And then he sets down his method of Extracting a fluent out of an equation involving its fluxion \representing it to be an operation of the same kind w^th the extraction of roots out of affected equations./ This method was therefore known to him when \before/ he wrote his Letter of 24 Octob 1676. Then fo

Then follows an extract a fragmen a part of a letter writ by Newton to M^r Collins Nov. 8. 1676 wherein he represents that he had then a m if any Curve be propose defined by an equation of no more than three terms expressing the relation between its abscissa & Ordinate, he could presently find the simplest figure w^th whose area its area might be compared. And \this proves/ shews that he had then found out the 10^th Proposition of his book of Quadratures & \by consequence/ the 9^th uppon w 7^th 8^th & 9^th upon w^ch it depends. And this is further confirmed by the Tables \Catalogus/ of Quadratures in the Scholium of the 10^th Proposition, the mentioned in his Letter 24 Octob 1676 & \there/ said to \have/ be|en| composed long before

By these things it appears that what had|s| been printed about the differe method of fluxions since the year 1676 was known to M^r Newton in y^t year when he wrote his two Letters of Iune 13^th & {illeg}|O|ctob 24 1676, & even five years before: For he saith that in the first of those Letters that he {illeg}|ha|d been tired with these studies & left them off five years before, & in this|e| second that {illeg}{he} five yeares before he wrote a Tract of this method, five years before, designing to publish it w^th a Tract about Colours. But upon some disputes being raised against <461v> him about the nature of colours, he laid aside his designe for the sake of quiet before he had finished the Tract upon this Method, & when M^r L. rivalled him in the|i||s| method as {illeg} he was further discouraged

Now as he understood this method in y^e year 1671 when he wrote the said Tract upon it, so, the Analysis shews that he understood it when he wrote that Tract & in the {illeg} Introduction to his Tract of {illeg} of Quadratures he tells us that he found it gradually in the years 1665 & 1666. The method of Series he found in y^e year 1665 & the second terms of the series gave him the method of fluxions moments of quantes|it||es| & upon those moments with the method of working in the {illeg} the first terms.

The next piece is M^r Newtons Tract de Quadratura Curvarum It relates to y^e direct & inverse method of fluxions. The reason It was published before at the end of his Opticks. The reason why it was published no sooner {a}|i|s given above. In the Scholium at the end thereof the word ut is accidentally omitted. It is expressed in the sentence: Hæ fluxiones sunt ut termini serierum. It should have been expressed in the next sentence in app applying this sentence \afterwards in repeating the sense of this sentence & applying it/ to the particular terms /of the series.\

The next piece is his enumeratio Curvarum secundi \Linearum tertij/ Ordinis In the seventh section he shews how to {illeg} find \Asymptotes of/ the crura Hypolica of these Curves & the plagæ crurum infinitorum Parablolicorum In the eighth \nith {sic} tenth & eleventh/ sections he teaches how to find the by the Asymptotes & plagæ infinitæ to find the position of the Abscissa & its angle w^th y^e Ordinate by w^ch the species is to b of y^e Curve is to be known. And supposing you know how from the nature of the curve by vulgar Analysis to find an|th|e Equation expressing the relation between that Abscissa & its|th|at Ordinate, he enumerates all the cases of \these/ Equations & in the following sections shews how many c{as}{illeg} forms of curves there are in every case.

The last {T} piece s|c|onteins a method of drawing a Curve line of a Parabolic kind through the ends of any number of Ordinates of any Curve, for squaring the Curve quamproxime. It appears by M^r Newtons Letter of 24 Octob 1676 that the|is| method was then known to M^r Newton \him/. It depends upon the differences of the Ordinates & the differences of these differences & therefore is called the Differential method.

M^r Iones in his Preface to this Collection of A Analytical Tracts

Fermat in his method de maximis & minimis & Gregory in his method of Tangents, \& Newton method of the first and last ratios/ suppos uses the letter o to signify a quantity non|t| infinitely but {de} indefinitely small & Barrow use in his method of tangents uses the letters a & e in the same manner. For they all make the letter{s}\s/ o, {illeg}|a| & e to become infinitely little. So M^r Newton in his Analysis in demonstrating the first rule, so soon as he has done y^e calculation, uses these words Si jam supponamus BB in infinitum diminui et evanesc{a}|e|re, sive o esse nihil. And so when he has finished the calculation in y^e Demonstration of the first Proposition in his Book of Quadratures he saith: Minuatur jam quantitas o in infinitum. By this means the whole calculation is done in infinite figures by the Geometrical Propositions of Euclide, b{illeg} & so is demonstrative. And upon this Account M^r Iones commends {illeg} in the method in the Preface to his Collectio Analytical Collection. Hujus Geometriæ Newtonianæ, saith he, non minimam laudem esse duco quod dum per limites Rationum primarum et ultimarum argumentam|t|ur, æque demonstrationibus Apodicticis ac illa Veterum innititur munitur: utpote qu{illeg}|æ| haud innitur duriusculæ illi Hypothesi quantitatum infinite parvarum, vel Indivisibilium, quarum evanescentia obstat quo minus eas tanquam quantitates specul{illeg}|e|mur. To this But M^r L. in the Account w^ch he has given of the Analytical Collection of <462r> M^r Iones in y^e Acta Eruditorum mensis Februarij pag {167} 1712 pag 76 makes the f corrects M^r Iones in the following manner. Cæterum quod Cl. Editor \[Ionesij]/ methodum rationum primarum et ultimarum methodo quantitatum infinite parvarum præfert; sciendum est quod differunt variari tantum in modo loquendi, et pro rigorosa demonstratione utram ad methodum Archimedeam revocari debere ut error quovis dato minor ostendatur. Cum in calculo præcedente [i.e. in \prædicta/ Demonstratione \præ/ Regulæ primæ sub finem Analyseos] adhibeatur o et ov, quis non videt revera d|a|dhiberi infinite parvas, nempe o pro dx et ov pro dy dz. Sane o jam Fermatius alij in talibus casibus adhibuere. Sed calculo illustris Leibnitij differentiali invento &c But here M^r L. has misrepresented both the methods. For in the method of first & last ratios the error is not proved to be less then any given ratio but to be none at all. First In this method the error is be there are no errors at all. The letter o represents a finite quantity indefinitely small till the operations are calculation be finished & the whole operation calculation is performed in finite quantities by the Geometry of Euclide without any error. And {th} when the operation then the quantity o vanishes beca decreases in infinitum & becomes nothing & leaves the ratio ultima without any of quantities without any error In the other method, the calculation is not grounded upon Euclides Geometry There is not one Proposition in Euclide concerning quantities or figures infinitly small. The calculation proceeds by approximations putting the infinitely small arcs of curves & their chords, sines & tangents equall to one another & frequent{b}ly using other approximations w^ch want a \rigorous/ Demonstrations that the erro{illeg} \is/ less then any given error. Without such rigous Demonstrations the method is not geometical, {illeg} nor are the Propositions found thereby \or by any Analys{illeg}|{s}|s {sic} {w}h{illeg}/ to be admitted into Geometry till they are rigorously demonstrated. [M^r Newton found many \of the/ Propositions in his Principia Mathematica Philosophiæ by this infinitesimal Analysis, but he did not propose them as Geometrical Propositions till he had rigorously demonstrated them. And in order to demonstrate them he spent a whole section in demonstrating Lemmas by the Method of first & last Ratios. For Geometry is not to be laid aside or corrupted be|y|{illeg} an Arithmetical method. {illeg} All Algebra is Arithmetia in speces. And tho for the improve\me/nt of invention it be applied to {Ge} magnitudes yet is nothing more then an Arithmetical Anaylsis of Geometrical Problems & The Ancients admitted nothing into Geome no Propositions into Geometry before they were demonstrated in words at length by the direct method w^ch they called Synthesis or Composition. And by that means they {illeg}|h|ave transmitted down to us an excellent Geometry] Nor is an Analytical Demonstration alone sufficient to make a Proposition Geometrical. {illeg} Algebra is Arithmetia It ought to be demonstrated synthetically & that in words at length to be read by people not skilled in Analysis.

But, saith M^r L. Cum in calculo præcedente adhibeatur o, et ov, quis non videt revera adhiberi infinite parvas, nempe o pro dx, et ov pro dz. {illeg} Whereas M^r Newtons by his words above cited supposed|s| implies that \they/ are considered as finite & indefi \only/ indefinitely small till the calculation is ended, & he supposes o to be diminished in infinitum & become nothing. And while M^r L. saith that should have told his reader that the Analysis was written in or before sent to M^r Collins in Iune 1669 & by consequence that the dx & dz of M^r Leibnitz were used \put/ for the o & ov of M^r Newton, & not on the contrary th that y^e o & ov of M^r Newton were put for y^e dx & dz of M^r Leibnitz.

M^r L. subjoyns: Sane o jam Fermatius alij in talibus casibus adhibuere Sed calculo differentiali invento illustris Leibnitij diferentiali invento &c. W^ch is as much as to say that the d invention of a new method lay in the invention of new symbols, & {illeg} But M^r Newtons next \still/ uses the letter o to this day And its use ought to be continued both in honour to Fermat & others who used it before & for the convenience of the Notation. \as may be seen in his book of Quadrature{e}|s|, And its for the honour of M^r Fermat that it should still be used. And if M^r Newtons method was the same when he/ wrote his Analysis that is at present, the Question will be what M^r Leibn^ts hath added {illeg} to this method besides a new notation. In the Acta Eruditorum mensis Ho or what advantages have been brought to it by this notation. <462v> He saith M^r L. Saith: dx vel dz est quantitas specialiter ad quantitatum x vel z relata, seu affectio quædam ipsius x vel ipsius z, nempe duarum x vel duarum z differentia, sed nullescens. Et ita non multiplicantur quantitates quarum affectionibus ad Curvas exprimendas est opus: at adeo æquationes etiam curvarum transcendentium per solarum ordinatarum abscissarum relationem habentur. This is the great advantage of the differential notation. And so in the Acta Eruditorum mensis Iunij A. 1697 pag 297 he saith Malo autem dx et similia 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 scilit|c|et potestatibus & differentialibus calculum ingrediatur et relationes transcendentes inter x et aliud exprimantur. Qua ratione etiam lineas transcendentes æquatione explicare licet. He acknowledges |here| that he used in the differential method he might have used letters for the differences as D^r Barrow used a &c in his method of tangents: but he chose rather to use dx {illeg} the symbol dx & the like for the sake of the advantages here set down mentioned. If he had used letters he must have defined their significations in every {illeg} new Problem or at least in every Tract or Mathematical Book, but by using dx & dy, it suffices to define them once for all: & this is the great advantage for the sake of w^ch he chose to use these symbols. [By his own confession therefore 'tis one & the same method whether he uses letters or synthese Symbols] The case is as if a man should put x, y, {illeg} z for y^e Abscissas of curves, ox oy oz for y^e Ordinates, ax, ay, az for y^e areas, sx, sy, sz for y^e solid contents made by rotation about y^e Abscissa, tx, ty, tz for the lengths of the Tangents, stx, sty, stz, for the subtangents, rx, ry, rz for the Radij of curvatures, & magnify himself upon the inventiō of these symbols, because it saves him the pains of definining the symbols of these quantities oftner then once & expresses the relation w^ch they \the quantities/ have to one another. But certainly no man would call this \Notation/ a new method. of Analysis. M^r Leibnitz confesses that in the Differential method he might have put letters for the differences & therefore by his own confession he might have put it is one & the same w^ch Method whether letters or the {illeg} symbols dx; dy, dz be used.

But let us compare the symbols of M^r Newton & M^r Leibnitz & see w^ch is|a|re the older & the better.

M^r Newton in his Analysis uses sometimes represents fluents by the areas of Curves & their fluxions by y^e Ordinates, & moments by the Ordinates drawn into y^e letter o. So where the Ordinate is $\frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}$ his|e| puts $\boxed{\frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}}$ for the area. And so if the Ordinate be v or y the Area will be $\boxed{\mathrm{v}}$ or $\boxed{\mathrm{y}}$. & {illeg} And in this way of notation the {illeg} fluent moments will be $\frac{\mathrm{a}\mathrm{a}\mathrm{o}}{64\mathrm{x}}$, {illeg} vo, yo. M^r Leibnits instead of the Notes {illeg} $\boxed{\frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}}$, $\boxed{\mathrm{v}}$, $\boxed{\mathrm{y}}$ uses the notes $\int \frac{\mathrm{a}\mathrm{a}}{64\mathrm{x}}$, $\int \mathrm{v}$, $\int \mathrm{y}$. {Newt} M^r Newtons are much{illeg} the older being used by him in or bef the year 1669|.| & {illeg} When letters are put for fluents (as is commonly done) M^r Newton{illeg} puts \for the fluxions/ sometimes other letters, sometimes the same letters with a prick, sometimes the same letters in a greater \different/ form or magnitude \& still uses any of these notations without confining his method to any one of them/. M^r Leibnits has no \proper/ symbols for fluxions; for fluxions are finite \these being finite quantities & fluxions being infinitely little the/ quantities being velocities of motion & differences are \dx dy &c being/ infinitely little \ones/. {illeg} When M^r Newton's symbols of fluxions are therefore the oldest. Wh{at O} For moments M^r Newton puts the symbols of fluxions multiplied by the letter o w^ch \(as was said)/ represents an infinitly little quantity answering to a moment of time. M^r Leibnitz puts y^e symbols of the fluents mult w^th the letter d before them. M^r Newtons way of Notation is the older being \used/ in his Analysis above-mentioned.

In the Acta Leipsica of {illeg} y^e month of February 1712, an Account \Extract/ is given of a collection of Tracts published the year before by M^r Iones & Entituled Analysis per Quantitatum series, fluxiones ac differentias cum enumeratione linearum tertij Ordinis. And \whereas in/ th this the beginning of this Acc^t the author wishes that M^r Iones had given a fuller Acc^t of the Commercium epistolicum of M^r Collins found in the study of amongst his papers. If y^e Author pleases to signify what inventions of any of his friends who corresponded w^th M^r Collins want an inquiry into their originalls, M^rIones will be desired to search M^r Collin the {sic} scrinia of M^r Collins for that purpose. Some of Something further has been T{he} since communicated by M^r Iones, & in|f| M^r Leibnitz who had a correspondence w^th M^r Oldenburg & by his means w^th M^r Collins, would publish the Letters \remainging {illeg}/ in his custody relating to that correspondence s|o|r such extracts of them as give light to the times {illeg} may conduce to complete what is wanting in the collection of M^r Iones, & make & make a fuller discovery of the times when any inventions were he would equally oblige the world.

In the next place the Author of the Extract gives an acc^t of the method used in y^e Analysis w^th y^e application thereof to the solving of Problemes & in y^e end of the extract subjoins: Cæterum quod Cl. Editor methodum rationū primarum & ultimarum methodo quantitatum infinite parvarum præfert; sciendū est, variari tantum in modo loquendi & pro rigorosa demonstratione utram ad methodum Archimedeam revocari debere, ut error quovis dato minor ostendatur. By w^ch words I perceive that y^e Extract doth not yet understand the method{illeg} of the first & last ratios. For in this Cum in calculo præcedente adhibeatur o et ov, quis non videt revera adhiberi in finite parvas nempe o pro dx et ov pro dz. By w^ch words I perceive that the Author of y^e Abstract doth not yet understand y^e Method of the first & last ratios. For in this method quantities are never considered as infinitely little nor are right lines \ever/ put for arches nor \neither are/ any lines \or quantities/ put by approximation for any other lines \or quantities/ to w^ch they are not exactly equal, but the whole operation is performed by Eucl exactly \in finite quantities/ by Euclides Geometry y^e equation w^ch remains will solve the Probleme. {A} And this way of working being demonstrated \from the beginning to y^e end very intelligible evident/ exact & demonstrative /as any thing in Geometry\ is preferred by M^r Iones to y^e method of infinitely little quantities {illeg} by approximations \& the vulgar A{ri}/ untill you come to an equation & then the equation is reduced by th rejecting the terms w^ch destroy one another & dividing the residue by {illeg} the finite quantity o & d making this quanty {sic} o not to become infinitely little but totally to vanish. For |M^r Newtons words in explaining this method, are: Iam supponamus BB in infinitum diminui et evanescere, sive o esse nihil. Had BB or o been considered as infinitely little before as infinitely little, he would not have said Iam supponamus BB in infinitum diminui. \Now/ By the vanishing of o there will remain an Equation w^ch solves the Probleme.| And this way of working being throughout as evident exact & demonstrative as any thing in Geometry is \justly/ preferred by M^r Iones to y^e method of \computing {illeg}/ infinitely little quantities: by approximations w^ch proceeding frequently by approximations is less Geometrical & more liable to errors, but yet may be usefull in some cases|.| & And \therefore is by M^r {illeg}/ upon both these methods M^r Newton founded his method of fluxions as is manifest by this Analysis written in the year 1669, where he gives the name of moments to infinitely little quantities. & represents them by the same characters, {illeg} putting o f & ov for infinitely those moments putting o for an infinitely little quantity, sometimes considers quantities as increasing or decreasing by continual motion or fluxion & gives the name of moments to the|ir| momentaneus increases or decreases|.| \w^ch are/ or infinitely little particles. The Fluxions or motions being finite quantities & the method of first & last ratios co nothing being considered \consisting in the consideration/ \of nothing/ but finite quantities, M^r Newton preferred & being exact & demonstrative & free from approximations: |M^r| Newton chose to calll this sort of Analysis the method of fluxions rather then y^e method of moments|,| But \or the method of/ Indivisibles or Infinitesimals. But yet he intended not thereby to exclude the consideration of \working in/ <463v> moments \& infinitely little figures/, whenever it should be thought convenient \this way of working being expedite but {thi}ppe{illeg}s/. And this he has sufficiently explained in the Introduction to his Quadratura Curvarum.

p. 44. Libri hujus Propositio quanita et sexta pendent a Propositionibus quatuor primis. In Tractatu Analysi per series numero terminorum infinitas quam Barrovius noster mense Iulio anni 1669 ad Collinium misit, dixi quod methodi ibi expositæ beneficio, Curvarum areæ & longitudines &c (id modo fiat) exacte & Geometrice determinantur: sed ista \ibi/ narrandi \ibi/ non esse locum. Et Collinius in Epistola ad Thomam Strode, 26 Iulij anno 1672 data, \ / < insertion from f 464r >  scripsit quod haud multo post quam in publicum prodierat Mercatoris Logarithmotechnia exemplar ejus — Barrovio Cantabrigiam misit, qui quasdam Newtoni chartas — extemplo remisit: E quibus, et \ex/ alijs, quæ olim ab Autore cum Barrovio communicata fuerant, patebat illam methodū a dicto Newtono aliquot annis antea [i. e. ante quam \editam illam/ Logarithmotechniam] excogitatam et modo universali applicatam fuisse. < text from f 464r resumes > his in {illeg} scripsit quod ex hic|as| Analysi \Analysi/ et alijs quæ olim \a me/ cum Barrovio communicata fuerant, patebret illam methodum a me aliquot annis antea (i.e. ante mensem Iulium anni 1669) excogitatam & methodo universali {illeg}|a|pplicatam fuisse: ita ut ejus ope in quavis figura Curvi linea proposita quæ una vel pluribus proprietatibus definitur, Quadratura vel Area dictæ figuræ, accurata si possibile sit, sin minus, infinite vero propinqua — obtineri queat. Et in Epistola mea ad Oldenburgum Octob. 24. 1676 data posui fundamentum harum operationum hac sententia Propositione sequente [Data Æquatione quotcun fluentes quantitates involvente fluxiones invenire; & vice versa.] deinde addidi quod hoc fundamento conatus essem Theorema primum \inde derivatum/ quo Curvæ Geometrice quadrantur ubi fieri potest. Et hoc idem fit per Theo{illeg}{m} primum \Proposit. quintam/ hujus Libri. a qu{illeg} Hæc quæ Propositio pendet a quatuor Propositionibus quatuor prioribus{illeg} primis: ideo Methodus fluxionum quatenus pendet a Propositionibus \{a}/ quatuor primis exponitur in Propositionibus quin primis hujus libri, mihi innotuit annis aliquot ante mensem Iulium anni 1669 Septembrem anni 1668 quo Mercatoris Logarithmotechnia prodijt.

Pag 45. In Tractatu de Quadratura Curvarum Analysi per series numero terminorum infinitas, pro fluxionibus posui lite{illeg}|ra|s quascun (ut x|z| vel y; pro momentis, literas easdam muliplicatas {sic} per {illeg}|l|iteram o, \&/ pro fluentibus \vel |vel|/ literas alias quascun ut vel vel |et| x, {illeg} {illeg} s{illeg} vel fluxiones in quadrato inclusas ut $\boxed{\mathrm{x}}$ et $\boxed{\mathrm{y}}$. Et sub finem Libr Tractatus illius specimen dabam calculi p{illeg} demonstrando Propositionem primam illius Tractatus.

Pag. 46. Hanc Propositionis hujus solutionem pos Wallisius noster in lucem edidit anno 1693, {illeg} cum exemplis in fluxionibus primis & secundis, Wallisius noster in lucem edidit anno 1693. Et hæc fuit Regula omnium prima quæ lucem vidit pro fluxionibus secundis \tertijs/, & alijs omnibus inveniendis.

Pag. 48 Extat hujus \hæc/ Propositio et ejus solutio in Analysi per Æquationes numero terminorum infinitas prope finem. Et ejus solutio eadem est cum solutione Propositionis primæ.

Pag 52 Prop. VI. Hanc Propositionem anno 1671 mihi innotuisse patet ex Epistola mea prædicta ad D. Oldenburgum 24 Octob. data ubi dicitur Pro trinomijs etia{illeg} etiam et alijs quibusdam Regulas quasdam concinnavi.

Pag 62|1|. Prop. X. Corol. II Ad hoc Corollarium spectabat Epistola mea Novem 8. 1676 ad Collinium s{illeg}t{illeg}|cript|a his verbis. Nulla extat Curva cujus Æquatio — haud tamen adeo generaliter ^‡ < insertion from f 464r > ‡ Scripsi uti hanc Epistolam u|V|bi Librum hunc de Quadratura Curvilinearum ex chartis meis antiquioribus tum modo ad us hoc Corrollarium composueram. Scripsi ad Collinium nostrum Epistolam sequentem \Octob. 24 1626 {sic} datam./ Nullam extat Curva – – – – haud tamen adeo generaliter. < text from f 464r resumes >

Pag 62. Hanc Tabulam diu{illeg} ante annum 1676 proinde anno 1671 mihi innotuisse \compositam fuisse/ patet per ordinatas Curvarum \positam/ in Epistola mea prædicta Octob. 24 \1676/ ad Oldenburgū data positas.

Pag 61. Prop X Corol {illeg}|11|.

Ad Lectorem.

Interea dum componerem Philosophiæ naturalis Principia Mathematica, plura Problemata solvi per quadraturas figurarum quadraturas quas Liber de Quadratura figurarum mihi suppeditavit Alia proposui solvenda concessis figurarum quadraturis. Et plurima demonstravi invertendo ordinem ivnentionis Analyticæ. Ideo \Et propterea/ l|L|ibrum de Quadratura Curvarum subjungere visum est quo figuras vel quadravi vel quadrandas proposui, et qui Analysin meam momentorum \exhibet/ quo sæpissime usus sum. Et cum in exponenda Cometarum Theoria usus sim methodo mea differentiali, visum est etiam eandem methodum Libro de Quadraturis subjungere