460
Observationes
in Synopsin Analyseos per quantitatum series fluxiones
ac differentias cum enuimeratione linearum tertij Ordinis.
This Synopsis being
The style & spirit of this Synopsis discover the Author
of the synopsis of the Book of Quadratures, & therefore deserve to be come kn & therefore deserveds to noted.
— in meditationes suas inciderint. Th That So much of The principal
part of the yo Commercium Epistolicum Collinsianum has been b relating
to these the subject of theis Analysis has been lately published, And toge
ther wthwith 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 Mr
Collins in Iuly 1669 as appears by three Letters of Dr Barrow & by several other LettesLetters of Mr Collins & Mr Iames Gregory about things cotnteined therein. & therefore the things conteined therein were
invented before that time. It was the Compendium mentioned by Mr Newton in his
letter of 24
The Author of this synopsis de in the next place produces out
of the Analysis a part of the computation by wchwhich Mr Newton demon
strated the first of the three Rules upon wchwhich the Analysis was
founded. And by this Computation compared wthwith other things in the Analysis
it appears that Mr 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 wchwhich 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 wchwhich 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 & 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 wchwhich he foresees will create no error
in the conclusion. In the latter he proceeds in finite quantities exactly
by vfinis
finished the computation he supposes that the finite moment o decreas
es in infinitum & vanishes. This is Mr Newtons met And by this me
thod 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 Dr
Barrow & by Dr Barrow to Mr 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
These Rules Mr Newton illustrates wthwith 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
contents & surfaces of solids & centers of gravity may be reduced to Qua
dratures after the following manner. Sit ABD Curva quævis — dato elicietur 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 uni
formly, or in proportion to time & its BK his an unit is put for its fluxion, & by
si moment, But its to be the coefficient o wchwhich makes the moments y & 1 infi
nitely little being neglected. For
And after he had set down an example iby computing the length of the arch of a
circle from its moment he subjoyns. Sed notandum est quod
mento 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 ur
Newton understands here an infinitely shr Newton under
stands here an infinitely short line, & by a lne & 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 & the moment y is , & but the coefficient o for shortning yethe operation is not written down but understood. If o be not understood the lines 1 & y are represent the fluxions of AK the areas BK & ABD, but if o be understood, these fluentssions become multiplied by o become the moments of BK & AD. For fluxions are finite quantities but moments here are infinitely little,. Thus you see his Notation when he wrote this Analysis is of the same kind wthwith that wchwhich he uses at present.
So in demonstrating the first of his three Rules by this method, in the equation he supposes x & z to increase & be augmented by the moments o & & to become & , & & by those moments understands the f rectangles o & conteined under the fluxions 1 & v & the moment o. Then in the said æquation writing for x & for z there arises in . Where the first terms & destroy one another & the next divided by o vizt & become equal & determin the proportion of the fluxions 1 & v or of their moments o & . And its here observable that the series is the same wthwith that set d in the beginning of Mr Newtons Letter of 13 Iun 1676, & being produced becomes , &And the like is to be understood of the series in like manner to be produced. But Mr Newton foreseeing haveing shewn before that all the terms after the second would vanish with the letter moment o, neglected them. He saw there fore in those days that the second terems of these series gave the moments fluxions & moments of the dignities of any fluent quantity x or ; & And when he understood this he could not be long wthwithout seing the use of the third & fourth terms & those that follow upon by this property of these series he founded the Demonstratioedn of the first of the three The Rules upon wchwhich 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 wchwhich 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 by his method extended to the he determined yethe Curvature of Curves.
After this Demonstration Mr Newton subjoyns the following conclusion from it. Hinc in transitu notetur modus quo Curvæ tot quot placuerit quotcunque, 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 , ex calculo invenies . 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 wchwhich the Mr 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 Mr Newton
In this T Tract of Analysis Mr 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 be founbde by461 be found by the same method. And then he subjoyns Nec quicquam hujusmodi scio ad quod hæc methodus, seidque varijs modis sese non extendit Imo tangentes ad Curvas metchanicas (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 exactæ. — Denique ad Analyticam merito pertinere censeatur, cujus beneficio Curvarum areæ & longitudines &c (id modo fiat () exacte et Geometrice determinentur. Sed ista narrandi non est locus. Spectant hæc These words [id modo fiat] have respect to a certa sort of series wchwhich sometimes breakes off & give the Quadrature in finite equations. Th One of these series is set down by Mr Newton in his Letter of 24 Octob 1676 as a part o the first of certain Theoremes for Quadratures wchwhich he had found b formerly found by his method of fluxions This is the 5th 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 3d & 4th & those on of the 1st & 2d. So that the first 5 5 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 Mr Collins in his letter to Mr 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 Mr Iones has reprinted Mr Newton's Letter of 13 Iune 1676, wherein the series Rule for reducing binomials into infinite series is set down whereof 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 Mr 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, wchwhich m wthwith the approximation of Quadratures makes up the body of the Epistle. And in the next place is an extract of Mr Newtons Letter of 24 Octob 1676, conteining a further explionexplication of the method of extracting the roots of affected Equations. And next after is Mr Newto a fragment of a letter of Mr Newton written to Dr Wallis conteining in the year 1692 wherein he represents that the sentence enigmatically exprest neare the end of his Letter of 24 Octob 1676, wherein in wchwhich 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 wthwith 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 Mr 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 wthwith whose area its area might be compared. And this proves shews that he had then found out the 10th Proposition of his book of Quadratures & by consequence the 9th uppon w 7th 8th & 9th upon wchwhich it depends. And this is further confirmed by the Tables Catalogus of Quadratures in the Scholium of the 10th Proposition, the mentioned in his Letter 24 Octob 1676 & there said to have been composed long before
By these things it appears that what hads been printed about the differe method of fluxions since the year 1676 was known to Mr Newton in ytthat year when he wrote his two Letters of Iune 13th & Octob 24 1676, & even five years before: For he saith that in the first of those Letters that he had been tired with these studies & left them off five years before, & in thise second that he five yeares before he wrote a Tract of this method, five years before, designing to publish it wthwith a Tract about Colours. But upon some disputes being raised against him 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 Mr L. rivalled him in theis method as he was further discouraged
Now as he understood this method in yethe 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 Introduction to his Tract of of Quadratures he tells us that he found it gradually in the years 1665 & 1666. The method of Series he found in yethe year 1665 & the second terms of the series gave him the method of fluxions moments of quantesites & upon those moments with the method of working in the the first terms.
The next piece is Mr Newtons Tract de Quadratura Curvarum It relates to yethe 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 ais 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 find Asymptotes of the crura Hypolica of these Curves & the plagæ crurum infinitorum Parablolicorum In the eighth nithninth tenth & eleventh sections he teaches how to find the by the Asymptotes & plagæ infinitæ to find the position of the Abscissa & its angle wthwith yethe Ordinate by wchwhich the species is to b of yethe Curve is to be known. And supposing you know how from the nature of the curve by vulgar Analysis to find anthe Equation expressing the relation between that Abscissa & itsthat Ordinate, he enumerates all the cases of these Equations & in the following sections shews how many cas forms of curves there are in every case.
The last T piece sconteins 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 Mr Newtons Letter of 24 Octob 1676 that theis method was then known to Mr Newton him. It depends upon the differences of the Ordinates & the differences of these differences & therefore is called the Differential method.
Mr 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 nont 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 letterss o, a & e to become infinitely little. So Mr Newton in his Analysis in demonstrating the first rule, so soon as he has done yethe calculation, uses these words Si jam supponamus BB in infinitum diminui et evanescaere, sive o esse nihil. And so when he has finished the calculation in yethe 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 & so is demonstrative. And upon this Account Mr Iones commends 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 argumentamtur, æque demonstrationibus Apodicticis ac illa Veterum innititur munitur: utpote quæ haud innitur duriusculæ illi Hypothesi quantitatum infinite parvarum, vel Indivisibilium, quarum evanescentia obstat quo minus eas tanquam quantitates speculemur. To this But Mr L. in the Account wchwhich he has given of the Analytical Collection ofMr Iones462 Mr Iones in yethe Acta Eruditorum mensis Februarij pag 167 1712 pag 76 makes the f corrects Mr 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 utramque ad methodum Archimedeam revocari debere ut error quovis dato minor ostendatur. Cumque in calculo præcedente [i.e. in prædicta Demonstratione præ Regulæ primæ sub finem Analyseos] adhibeatur o et ov, quis non videt revera dadhiberi infinite parvas, nempe o pro dx et ov pro dy dz. Sane o jam Fermatius alijque in talibus casibus adhibuere. Sed calculo illustris Leibnitij differentiali invento &c But here Mr 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 & frequentbly using other approximations wchwhich want a rigorous Demonstrations that the erro is less then any given error. Without such rigous Demonstrations the method is not geometical, nor are the Propositions found thereby or by any AnalysssAnalysis wh to be admitted into Geometry till they are rigorously demonstrated. [Mr 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 bey an Arithmetical method. All Algebra is Arithmetia in speces. And tho for the improvement 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 wchwhich they called Synthesis or Composition. And by that means they have transmitted down to us an excellent Geometry] Nor is an Analytical Demonstration alone sufficient to make a Proposition Geometrical. 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 Mr L. Cum in calculo præcedente adhibeatur o, et ov, quis non videt revera adhiberi infinite parvas, nempe o pro dx, et ov pro dz. Whereas Mr Newtons by his words above cited supposeds 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 Mr L. saith that should have told his reader that the Analysis was written in or before sent to Mr Collins in Iune 1669 & by consequence that the dx & dz of Mr Leibnitz were used put for the o & ov of Mr Newton, & not on the contrary th that yethe o & ov of Mr Newton were put for yethe dx & dz of Mr Leibnitz.
Mr L. subjoyns: Sane o jam Fermatius alijque in talibus casibus adhibuere Sed calculo differentiali invento illustris Leibnitij diferentiali invento &c. WchWhich is as much as to say that the d invention of a new method lay in the invention of new symbols, & But Mr 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 Quadraturees, And its for the honour of Mr Fermat that it should still be used. And if Mr Newtons method was the same when he wrote his Analysis that is at present, the Question will be what Mr LeibntsLeibnits hath added to this method besides a new notation. In the Acta Eruditorum mensis Ho or what advantages have been brought to it by this notation. He He saith Mr 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: atque adeo æquationes etiam curvarum transcendentium per solarum ordinatarum abscissarumque 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 scilitcet 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 Dr Barrow used a &c in his method of tangents: but he chose rather to use dx 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 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 wchwhich 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, z for yethe Abscissas of curves, ox oy oz for yethe Ordinates, ax, ay, az for yethe areas, sx, sy, sz for yethe solid contents made by rotation about yethe 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 wchwhich they the quantities have to one another. But certainly no man would call this Notation a new method. of Analysis. Mr 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 wchwhich Method whether letters or the symbols dx; dy, dz be used.
But let us compare the symbols of Mr Newton & Mr Leibnitz & see wchwhich isare the older & the better.
Mr Newton in his Analysis uses sometimes represents fluents by the areas of Curves & their fluxions by yethe Ordinates, & moments by the Ordinates drawn into yethe letter o. So where the Ordinate is hise puts for the area. And so if the Ordinate be v or y the Area will be or . & And in this way of notation the fluent moments will be , vo, yo. Mr Leibnits instead of the Notes , , uses the notes , , . Newt Mr Newtons are much the older being used by him in or bef the year 1669. & When letters are put for fluents (as is commonly done) Mr Newton 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. Mr 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. When Mr Newton's symbols of fluxions are therefore the oldest. What O For moments Mr Newton puts the symbols of fluxions multiplied by the letter o wchwhich (as was said) represents an infinitly little quantity answering to a moment of time. Mr Leibnitz puts yethe symbols of the fluents mult wthwith the letter d before them. Mr Newtons way of Notation is the older being used in his Analysis above-mentioned.
463
In the Acta Leipsica of yethe month of February 1712, an Account Extract is given of a collection of Tracts published the year before by Mr Iones & Entituled Analysis per Quantitatum series, fluxiones ac differentias cum enumeratione linearum tertij Ordinis. And whereas in th this the beginning of this AcctAccount the author wishes that Mr Iones had given a fuller AcctAccount of the Commercium epistolicum of Mr Collins found in the study of amongst his papers. If yethe Author pleases to signify what inventions of any of his friends who corresponded wthwith Mr Collins want an inquiry into their originalls, MrIones will be desired to search Mr Collin the scrinia of Mr Collins for that purpose. Some of Something further has been The since communicated by Mr Iones, & inf Mr Leibnitz who had a correspondence wthwith Mr Oldenburg & by his means wthwith Mr Collins, would publish the Letters remainging in his custody relating to that correspondence sor such extracts of them as give light to the times may conduce to complete what is wanting in the collection of Mr 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 acctaccount of the method used in yethe Analysis wthwith yethe application thereof to the solving of Problemes & in yethe end of the extract subjoins: Cæterum quod Cl. Editor methodum rationūum primarum & ultimarum methodo quantitatum infinite parvarum præfert; sciendūum est, variari tantum in modo loquendi & pro rigorosa demonstratione utramque ad methodum Archimedeam revocari debere, ut error quovis dato minor ostendatur. By wchwhich words I perceive that yethe Extract doth not yet understand the method of the first & last ratios. For in this Cumque in calculo præcedente adhibeatur o et ov, quis non videt revera adhiberi in finite parvas nempe o pro dx et ov pro dz. By wchwhich words I perceive that the Author of yethe Abstract doth not yet understand yethe 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 wchwhich they are not exactly equal, but the whole operation is performed by Eucl exactly in finite quantities by Euclides Geometry yethe equation wchwhich remains will solve the Probleme. A And this way of working being demonstrated from the beginning to yethe end very intelligible evident exact & demonstrative as any thing in Geometry is preferred by Mr Iones to yethe method of infinitely little quantities by approximations & the vulgar Ari untill you come to an equation & then the equation is reduced by th rejecting the terms wchwhich destroy one another & dividing the residue by the finite quantity o & d making this quantyquantity o not to become infinitely little but totally to vanish. For Mr 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 wchwhich solves the Probleme. And this way of working being throughout as evident exact & demonstrative as any thing in Geometry is justly preferred by Mr Iones to yethe method of computing infinitely little quantities: by approximations wchwhich proceeding frequently by approximations is less Geometrical & more liable to errors, but yet may be usefull in some cases. & And therefore is by Mr upon both these methods Mr 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, 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 their momentaneus increases or decreases. wchwhich 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, Mr Newton preferred & being exact & demonstrative & free from approximations: Mr Newton chose to calll this sort of Analysis the method of fluxions rather then yethe method of moments, But or the method of Indivisibles or Infinitesimals. But yet he intended not thereby to exclude the consideration of working in moments moments & infinitely little figures, whenever it should be thought convenient this way of working being expedite but thippes. And this he has sufficiently explained in the Introduction to his Quadratura Curvarum.