Add. 3968
An Account of the
Commercium Epistolicum.
22
This book is composed of ancient Letters & Papers written by Dr Barrow Mr Newton Dr Barrow, Mr Greg. Mr Collins Mr Newton Mr Leibnitz
Mr Oldenburg Mr Gregory Mr Collins, Mr Newton & others, the originals of still extant
in their hands or in the looks of the R. Society as they were coppied in the
days of Mr Oldenburg. And copp And the truth thereof has been exami
ned & reported by a Committee of the R. Society & the Society acquiesce in the
Report.
The first occasion of this Committee was this. Mr Leibnitz In a Pi
In the Acta Eruditorum published at Leipsic for yethe month of Ianuary 1704
in Account was given of two Tracts of Mr Newton pub concerning the
species & magnitude of figure fig Curvilinear Figures. And in this Account
Mr Newton was represented to have al substituted fluxions for the Leibnitian
differences as Honoratus Faber had substituted motion for the method of Ca
vellerius, that is, that Mr Newton is saying that he had had not found the
method of fluxions in the years 1665 & 1666 as he had affirmed, but had
received deduced it from the method of Mr Leibnitz & put only put it into another form.
Mr Keil observing that this paper wthwith divers others published in the name
of the editors, was writ in the style of Mr Leibnitz, & that Mr Newton was hereby sued with plagiary & falshood took occasion
a paper b published in the Transactions A.C. 1708 to write that the Arithmetic
of fluxions Mr Newton without all doubt was the first who found out the
Arithmetick of fluxions as will easily appear to any man who shall read
his Letters published by Dr Wallis : but Mr Leibnitz afterwards having
changed the name & manner of notation, published the same in the
Acta Eruditorum. Mr Leibnitz afterwards in a letter to Dr Sloan dated
4 March 1711 complained of this to yethe R. Society, & that he knew no never heard
of the name nor saw the characters used by Mr Newton before they were
published by Dr Wallis, & proposed to yethe Society that he did not Mr Keil
should testify openly that he had no d intention to erup cast an imputation
upon Mr Leibnitz as if he had learnt his differential method from another
& attributed it to himself. Mr Keil refused to recant & chose rather
to return an Answer whereas he represented that what he wrote was occasioned by the Acta Lipiensia & explained his meaning to be
not that Mr Leibnitz had knew the name or notation used by Mr
Newton but that only that Mr Newton was the first inventor &
that in two Letters written to Mr Oldenburg & by Mr Oldenburg commu
nicated to Mr Leibnitz, had sufficiently described it to a man of Mr a
Leibnitzes perspications understanding with from whence Mr L. had or might have the
principles of his calculus, & not being able to learn the modes of
speaking & the Notation names & notation used by Mr Newton
might se That he was induced to write this by the editors of the Acta Leipsiensia w affirmed that Mr Newton had substituted his did impose his own. And for this his opinion Dr Keil subjoyned
his reasons grounded upon several passages taken out of Mr Newtons said
Letters & saout of his Letter of 10 Decemb. 1672 & his Analysis per æquationes nume
ro terminurum infinitas r Iohn Collins A.C. 1669. And
Mr Leibnitz replied that his candor was more openly attact then before; that
no prudent & just man wchwhich that he in such an old age after so many proofs of
his life should defend wthwith an Apology, & contend wthwith a man who was learned
but a novice, & unacquainted wthwith what what had formerly been done, nor authorized
by the person concerned, no prudent or just man would approve of. That
the A what was in the Acta Leipsiensia detracted from no man, but every where
gave every man his due. That he did not make hast to assert the differential method68 method to himself but kept it secret above nine years before he began to
publish it, that he left it to yethe equity of the Society to restrain the
the vain & unjust clamours, that he was persuaded that Mr Newton
would willingly which he beleived were displeasing to Mr Newton himself.
& was persuaded that Mr Newton would declare as much.
The R. Society being thus twice appealed unto, & there being no witnesses
now alive for Mr Newton & Mr Leibnitz(For Mr Leibnitz could not be witness for hirmself nor Newton for Mr Keil), the Society appointed
a Committe to search out ancient Letters & papers & letter books extant or either
in their own archives or in left by Mr Oldenburg or Mr Collins in the custody of the society or
otherwise extant, & to examin what might relate to this matter
& report their opinion thereupon: & the Committe made the following Report.
Whe have consulted to be made publick
Whereupon the R. Society ordered the collection of epistles & many
scripts & th wthwith the Report of their Committee to be made pu & what else
might occurr in the Acta Eruditorum proper to clear up this matter to be
made publick. And the same were accordingly published under the Title
of Commercium Epistolicum.
The Letters & MSS themselvers are of unquestionable authority
the Originals of most of them & authentic copies of the rest being still
preserved. or long since published by Dr Wallis. And the first paper in this collection is a Tract written And the Questions to be decided by them are these.
Mr Leibnitz in his lettter of Iune 21th 1677 wrote that he agreed
wthwith Mr Newton that Slusius'es method of Tangents was not yet perfect
& that he himself had handled the of a long time handled the business
of Tangents much more generally, namely by the difference of the
Ordinates, & then described his Differential method as if he had found it
long before : & in his letter of 29 Decemb 1711 that he had found
it above 9 years before he began to publish it, that is before that he
found it before Octob. 1675. For he began to published it in Octob. 1684.
by Mr Newton communicated by Dr Barrow to Mr Collins in the year 1669 as appears by some three letters of Dr Barrow
still extant & by the Letters of Mr Collins who cites several p Mr Grego
ry, Mr Bertet, Mr Borellus, Mr Vernon, Mr Strode, & by Mr Oldenburgs
Letter to Mr Slusius dated 14 Sept 1669 & by a copy of this Tract
found by the hand of Mr Collins. This Tract was first first printed by
Mr Iones from a copy found in the hand writing of Mr Collins, wchwhich of then
reprinted in the Commercium from the same copy. It conteins an account
of the method of reducing æquations into infinite series, & of of applying
those series to the quadrature of curves by meanes of three Rules set down in
the beginning & to other Problems by meanes of Quadratures & of the method
of motions or fluxions & moments, & to the inverse of those Problemes by extracting the affected roots transforming the series
& to Mechanical curves in finding their Ordinates tangents, areas, lengths, &c. And
And after Mr Newton had shewed how it extended to these Problems he
adds that he knows nothing to wchwhich this method doth not extend itself
& that after various manners. And by its universality & the exactness
of its reasoning justifies his giving it the name of Analysis : especi
ally since by the help of it, the Areas & lengths of Curves &c may be
determined exactly & Geometrically when it is practicable. But
he How to do this he describes not here but gives an instance of it
in his letter of Octob. 24. 1676 then communicated to Mr Leibnits & printed
in this Commercium p.72, 73. Thus Mr Newton before the year
1669 that is above 44 years ago had carried his Analysis to a higher pitch & made it of a more
universal extent, then any other person what so ever hath been able
to hitherto able to carry the differential method. without Mr Newton
For the method of series & moments together make and Analysis much69 much more universal then yethe method of moments alone. And the
conclusions are as exact if you continue the series in infinitum
as the conclusions arein vulgar Analysis by finite equations if you
extract the roots of those Equations in infinitum.
Now tho it was not Mr Newton designe in this Tract to
explain the Analysis of moments any further then in relation
to infinite seri its use in the working by infinite series, yet by what is
here said of it, it appeares that he was then master of it. F he considers
the Area of a curve as generated & described by the Ordinate moving upon the
Abscissa with an uniform motion & thereby describing it & calls the
Abscissa Ordinate the moment of the area, & where several areas are so
described by several Ordinates he considers the several Ordinates as
the moments of the several areas, & where an abscissa is of a increases uni
given length formly he represents the moment by a given quantity & particular
ly by an unit, & proposes to deduce the find the areaat all time by having by having the
moment thereof at all times. That is, by having an equation expressing the relation of yethe moments to yethe abscissa to find
a equation expressing the relation of the area to the Abscissa or to
the rectangle of the Abscissa & a given moment or to And then adds
that by the same method that an Area may be deduced from its
moment any other quantity may be deduced from its moment. Where
by he it is manifest that he had at that time the method of deducing
quantities of any sort from their moments. Vide pag 14 Commercy :
An instance of this he gives in finding the lengths of Curves
& adds that yethe like may be done for their superficies & solid
contents & centers of gravity, & on the contrary he gives examples of deducing the Abscissa may be deduced
from the Area or Length of a Curve & of extending the method to
Mechanical Curves, & of finding such curves as may be squared. All
wchwhich shew that Mr Newton had at that time the method of
moments direct & inverse so far as relates to this Analysis
by infinite equations series. And in his letter of Decem. 10
1672 wher he represents that the method of Tangents there
described was but a particular or rather a Corollary of a gene
ral method wchwhich without any troublesome calculation extended
it self not only to yethe drawing tangents to any Curves Geometrical
or Mechanical or in any manner whatever related to right lines
but also to other more obstruse sorts of Problemes concerning
the curvities, areas, lengths, centers of gravities &c of curvilinear
figures, & this without sticking at æquations affected wthwith surd
quantities. All wchwhich is as much as to say that Mr Newton had
extend his method of moments whereof his method of TangtsTangents was extended a Corrollary extended to all sorts of æquations whether
finite or consisting of an infinite number of terms, rational
or affected wthwith surd quantities, & dispatcht the known sorts of
Problems in all sorts of the Mechanical Curves as well as in
those A called Geometrical. And
So The And in the first of his two letters written to Mr Oldenberg A.C. 1676
& by him communicated to Mr Leibnitz, Mr speaking again of this
method said that this Analysis by the help of series was so much
inlarged as to extend to almost all sorts of problems (except yethe numeral
ones like those of Diophantus,) but did not become general wthwithout some further
methods of reducing problems to series : wchwhich methods he comprehended in the70 the following sentences exprest ænigmatically. Data æquatione
fluentes quotcunque equati quantitates involuente inventre fluxiones
invenire et vice versa. And in his second Letter making mention
of the method of Tangents by wchwhich he drew tangents determined
maxima & minima, squared curves, &c, he exprest this comprehended
this method in the following sentence exprest A œnigmatically. Data
æquatione quotcung fluentes quantitates involvente, fluxiones
invenire, & vice versa. And this is the method general method
desc mentioned in his aforesaid Letter of 1672. And by his saying
there that he had this method he determined the curvitures of
curves it is manifest that he had then extended it to the second
fluxions
Mr Newton had therefore in those days
of considering indeterminate quanties as growing by continual increase, & putting
some symbols for those quantities, & others for the velocities
of their increase which velocities he calls their fluxions & perhap the rectangle of their fluxions & their
others the moment o for their particles generated in every moment of time
wchwhich particles he calls their moments, & deriving the fluxions
from the quantities or the quantities from the fluxions as the
nature of the Question should require. For that he had symbols
for these quantities in the manifest by his Analysis above mentioned
For towards the end of that Analysis (pag. 19) he puts some letters as x &
z for fluents & others as y or v for fluxions & where the fluxion is uniform he represents it by an unit & drawing the fluxions into
the an indefinitely small quantity o, he puts the rectangles oy, ov for ox
the moments, & where the fl or o for the moments. And a little before he
puts where is the Ordinate of a Curve he puts for the area generated by that Ordinate. And by the like notation if z is the area of any curve whose ordinate is z, & z is the ordinate of any curve whose area is z & z is the fluxion & oz the moment of any quantity whose fluent quantity z & if z be put for any fluent represented by that area or proport an area or capable of being so represented its fluxion will be z & its moment oz, & for the fluxi of z or second fluxion of of z may be put v or any other letter or mark.
And this much appear at least concerning the method of fluxions & & series appears from the papers extant in out of the Commercium & written by Mr Newton & communicated to his friends before Mr Leibnitz began to speak of his differential method. late lately published in the Commercium Epistolicum
In the year 1671 Mr Leibnitz was in London & there printed his Hypothesis Physica nova, & in his Dedication thereof to yethe R. Society mentions his correspondence with Mr Oldenburgh, & in the beginning of the year 1673 being reprehended by Dr Pell for pretending to yethe differential method of Monsr Mouton he wrote an Apology for himself to Mr Oldenburg as is mentioned above in the Report of the Committee. And soon after went from London to Paris. being about that time chosen fellow of the R. Society. And the next year in summer Iuly & there applied himself to yethe study of infinite series began to write to Mr Oldenburg as if he had found the Area of a circle by a certain series of rational numbers continually produced in infinitum.
When Mr Collins had received the Analysis above mentioned from Dr Barrow he sent communicated the series therein set down to his friends & particularly to Dr Barrow Mr Iames Gregory within the spa who thereby within the space of about a year found out the method & sent back to Mr Collins several other series found thereby, & Mr Collins communicated those also very freely to his friends having Mr Gregories leave to do so. All wchwhich appears by the Letters of Mr Collins & Mr Gregory printed in the Commercium.
In the mean time Mr Leibnitz spent some time at London, & in the year 1671 printed there his Hypothesis Physica nova, & in his Dedication thereof to the R. Society mentions his correspondence with Mr Oldenburg; & in the beginning of the71 of the year 1673 being reprehended by Dr Pell for pretending to the differential method of Monsr Mouton he wrote an apology for himself to Mr Oldenburg extant in the Commercium, & soon after went from London to Paris being about that time chosen fellow of the R. Society. And the next year in Iuly began to & October wrote to Mr Oldenburg representing that Vicount Brounker & N. Mercator had found an infinite series of rational numbers equal to yethe area of an Hyperbola, but no man had hitherto found a progression of rational numbers which being continued in infinitum became exactly equal to the Circle or in the circumference But he had now found such a summ series & durst venture to say that he was the first who had done it. And that by the same method there wasere a certain harmonyies between with the quadrature of the circle & that of the Hyperbola this series had discovered certain harmonies between the circle & Hyperbola & as he had found the proportion of the circumference to the Diameter in such a series so that by the same method the valor of any Arch whose sine was given might be exhibited Geometrically by a s such a series without knowing the length of the whole circumference, or the proportion of the Arc to the whole circumference. All wchwhich is as much as to say that he had a series wchwhich by the sine gave the arc & one & the same method from the sine gave the arc & from the whole sine or radius gave the By one & the same method therefore he found either the whole circumference or any part of it & by consequence had got Mr Newtons series described pag 15 of in his Analysis pag 15 pag 15, viz if x be the sine of an arc in a circle in a circle whose radius is 1, the arch will be &c. For this series gives the arch of & this series multiplied by 3 gives the Quadrantal arch, wchwhich multiplied by gives the area of a circle whose diameter in an unit. And must be the series of rational numbers boa here boasted of by Mr Leibnitz the summ of whose terms is exactly equal to the area of a circle. For this series is found by the same method whereby the Arch is given by the sig sine, & Mr Leibnitz at that time boasted of no more series then one for the area of the circle. Mr Leibnitz had therefore obteined Mr Newtons series of find the arc by the sine, but the meth Demonstration of Mr Newton"s series he had not yet or method of finding it he had not yet obteined : but for he wrote to Mr Oldenburg for it about two years after.
In the mean time Mr Oldenburg in his Letter dated Aprill 15 1675 sent eight or nine series to Mr Leibnitz wchwhich he had received from Mr Collins. One of them was Mr Newtons series for finding the arc by the sine. Another was his series for finding the sine by the arc. And the two last were Mr Gregories for finding the arc by the Tangent & the tangttangent by the arc & the arc by the tangent. This Letter came to the hands of Mr Leibnitz, for he wrote an Answer to it dated 20 May 1675, & the asnwer was in these words Literas tuas multa fruge Algebraica refertas accepi pro quibus tibi et doctissimo Collinio gratias ago. Cum nunc præter ordinarias Curas Mechanicis imprimis negotijs distrahar non potui examinare series quas misistis ac cum meis comparare. Vbi fecero per scribam tibi sententiam meam : nam aliquot jam anni sunt quod inveni meas via quadam sic satis singulari. Collinium ipsum magni facio quoniam omnes pura Matheseos partes ab ipso egregie cultas video. Thus Mr Leibnitz know none of the series sent72 sent to him by Mr Oldenburg to be his own, but pretended to have some others found some years before : wchwhich others he has not been able to produce to this day. Hence forward therefore he quitted his claim to M the series for finding the sine arc by the sine : for he wanted the Demonstration to support his claim. But the next year having rec having received the by another hand the series for finding the arc by the sine & the inverse thereof for finding the by sine by the arc he wrote back to Mr Oldenberg for the Demonstration or method of finding them. His Letter was dated 12 May 1676 & began in this manner Cum Georgius Mohr Danus in Geometria & Analysi versatissimus nobis attulerit communicatam sibi a Doctissimo Collinio vestro expressionem relationis inter arcum et sinum per infinitas series sequentes. Posito sinu x, Arcu z, Radio 1 erit
+
+ − &c Hæc, inquam, cum nobis attulerit ille quæ mihi valde ingeniosa videntur, & posterior imprimis series elegantiam singularem habeat, ideo rem gratam mihi feceris, Vir Clarissime, si Demonstrationem transmiseris. Habebis vicissium me ab his longe diversa circa hanc rem meditata de quibus jam aliquot abhinc annis ad te perscripsisse credo, demonstratione tamen non addita quam nunc polio.
Mr Leibnitz had now so far forgot that Mr Oldenbergs letter of April i15th 1675 that he did not so much as remember that he had seen these two series before they received them from Mr Mohr, & tho they were sent him in that Letter. Nor did he remember that the series wchwhich he was now polishing & adorning with a demonstration, was his sent him in that letter & that he did not then know it to be his own. But after Or else whe perhaps when he had found the Demonstration & by transmutation of figures & kew how to deduce that series from such a transmutation, he began might reccon himself a collateral inventor as in the case of Mounton's differential method. For he began the last year 1675 to communicate to his friends a discourse wchwhich he had written upon this subject Quadrature as I found appears by his words published in the Acta Eruditorum ffor April 1691. The words are these. Iam anno 1675 compositum habebam opusculum Quadraturæ Arithmeticæ ab Amicis ab illo tempore lectum, sed quod materia sub manibus erescente, limare ad editionem non vacavit postquāam aliæ occupationes supervenere præsentim com nunc prolixius exponere. more vulgari more quæ Analysis nostra nova paucis exhibet non satis operæ pretium videatur. By these words it is evident These words are a confession that he did not find out his M Differential Analysis till he had done polishing this Opusculum & began to be taken up with business of another nature. He was polishing it when he wrote his letter of May 12 1676 & sent it to Mr Oldenburg in his letter of 27 Aug 27th 1676, & returned home by London in October & Holland in October & November to be imployed in publick affairs
And that the Differential method was not yet found out is demonstrated also by what Mr Leibnitz wrote to Mr Oldenberg in his Letter of 27 Aug 1676 73 Neither doth he describe his method way of considering quantities as binomials & reducting their powers & radicals & the reciprocals thereof into infinite series by one & the same general Rule : but yet by he his sett Down the beginning of those series & thereby it appears that he had then found out that Rule. see pag the Commercium pag. 19 lin 19, 20 & pag 4599 lin 22.
In his Analysis therefore for making his Method universal, he applyes his method universal by these se universal, he applis these series to the solution of all sorts of Problems by considering quantities as growing or increasing by continual motion or flux, & giving the name of moments to their parts generated in equal moments of time, he puts any symbols for the flowing increasing quantities & any others for the velocities of their increase & by drawing or multipliying these velocities into a moment of time he represents their moments. And when he considers any one of the increasing quantities as increasing uniformly or in proportion to time, he puts an given quantity unit or some other given for the velocity of its increase & that unit multiplied by a moment of time for its moment. So when he represents the growing quantities by the areas of Curves Figure, he represents the velocities of their growth the velocities of their growth or flux by the ordinates of the curve figures & time by the Abscissa drawn into a given ordinate, & for the length of wchwhich he puts an unit, & the moments of these flowing quantities he represents by the Ordinate drawn into yethe moment of the Abscissa the exponent of time. for in which they are generated. So in this Tract when he puts any sym the Ordinate or velocity of increase or flux is , x or or y, or v, the are he represents the area or fluent by the symbol or x or y or u & their moments by or ox or oy or ou. Or else he represents the fluents by some letters as & where the fluxion is 1 oy v or yo he represents the moment by o, oy, ov or oy z oe v or z & their fluxions by other letters. Or where the fluents are represented by some letters he represents the fluxions by other in the Tract by other letter & the moments by other letter or marks & the moments And by the like notation The fluents may be repres if the Ordinates or fluxions be v, x, y, z the fluents will be v, x, y, z & the fluxions moments ov, ox, oy, oz. Or whe Also when he fluent is x or z he puts y or v for the fluxion & oy or ov for yethe moment, & this way of notation he demonstrates the first of his three Rules above mentioned. And by these same ways of notation he reserves to himself a liberty of putting any symbols for the fluents, any other symbols for the fluxions & the symbols of the fluxions multiplied by o for the moments By this means he performed all his computations in this method wthwithout any other any other infinitely small quantity then the moment o (as in demonstrating the first of the said three Rules in this Tract & in demonstrating the first Proposition in his book of Quadratures) he performed the wole operation by vulgar considered the symbol o as infinite quantity & performed the whole operation without by vulgar Geometry without considering any one quantity as infinitely small & when the computation was brought to an end & the equation reduced : he considered the quantity o as decreasing till it vanish ed & thereby had a new equation wchwhich determined & what he sought. And this way of working he calls his methods rationum primarum & ultimarum. But if he was minded not to work demonstratively to make dispatch by such approximations as were of would breed no error in the conclusion, he supposed the quantity o to be infinitely little, & this way of working his method of Moments And both these ways of working are his general method wchwhich he calls his method of fluxions. And by these methods he applied his series so generally to the solution of Problemes as to say in his Analysis that he knew of no Problems Geo about figures Geometrical or Mechanical to wchwhich this74 this method did not extend & that after various manners
And as by this Tract of Analysis it appeasr that Mr Newton had then applied this method generally to the solution of Problemes by seires : so by his Letter of to Mr Collins dated Decem. 10 1672 dated to Mr Collins & printed in the Commercium it appears that he had then applied it generally to the solution of Problems by finite equations. For in this Letter having Mr Newton having described his method of Tangents wchwhich proved to be the same wthwith that of Slusius, he added that this method was but a particular or rather a Corollary of a general method wchwhich without any toublesome calculation extended itself not only to the drawing of Tangents to any Curves Geometrical or Mechanical or in any manner whatever related to right lines, but also to other more abstrase sorts of Problemes concerning the curvities areas lengths centers of gravities &c of curves &c & this without any sticking at equations affected with such irrational quan tities. And thus much concerning the method of series & fluxions appears out of what Mr Newton wrote before the end of the year 1672 : at wchwhich time Mr Leibnitz had not begun to write any thing about either series or the Differential method.
Mr When Mr Collins had received Mr Newtons Analysis he communicated one of the series to Mr Iames Gregory who after a years consideration fell upon the same method & sent some other series to Mr back to Mr Collins in the beginning of the year 1671 & gave Mr Collins leave to communicate them to whom he pleased. And Mr Collins was very free in communicating the series of Mr wchwhich he had received from them both as appears by the Letters published in the Commercium.
Mr Leibnitz had been was in England at London in the year 1671 & there printed his Hypothesis Physica nova & in his Dedication thereof the R. Society mentioned his correspondence wthwith Mr Oldenburg, & in the beginning of the year 1673 claimed the differential method of Monsr Mouton pretending that he had found it apart wthwithout the assistance of Mouton & had much improved it, & soon after went from London to Paris & the next year A.C. 1674 wrote to Mr Oldenburg that he had found a series of rational numbers equal to yethe circle & by the same method the any arch might be found in such a series wthwithout knowing its proportion to the whole circumference. Which is as much as to say that by the sine of an arch whose proportion the to the whole circumference was known he had found a series of rational numbers for the whole circumference & by the same method could find a series for any arc whose tho its proportion to the whole circumference was not known. By the method of finding these particular series in rational numbers he means a ge Mr Newtona general series for find wchwhich by assuming the arc sine in a rational numbers gives the arc in a series of rational numbers. And this general series was Mr Newwtons not found out by Mr Leibnitz for he wrote afterwards to Mr Oldenburg to procure from Mr Collins the Demonstration thereof meaning Mr Newtons method for finding it. And all this
The next year vizt A.C. 1675 in April Mr Oldenburg having received eight series from Mr Collins sent the to Mr Leibnitz then at Paris & Mr Leibnitz Two of them were Mr Newtons for finding the Arc by the sine & the sine by the arc, & other two were Mr Gregories for finding the arc by the tangent & the tangent by the Arc.75 arc. But Mr Leibnitz knew none of yethe eight series to be his own For his answer dated at Paris May 20 1675 began in these words : Literas tuas multa fruge Algebraica refertas accepi, pro quibus tibi et doctissimo Collinio gratias ago. Cum nunc præter ordinarias curas Mechanicis imprimis negotijs distrahar, non potui examinare series quas misistis ac cum meis comparare. Vbi fecero, perscribam tibi sententian meam: nam aliquot jam anni sunt quod inveni meas via quadam sic satis singulari. These series were in general terms those of Mr Leibintz in particular numbers. And tho he had got the general series for finding the arc whose sine was given, yet he wanted the Demonstration. But having by found a demonstration of the series for fo the arc whose tangent was given he published began this year to communicate it to his friends at Paris as he himself has told us in the Acta Eruditorum mense Aprili anni 1691 pag 178 in these words 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 postquam aliæ occupationes supervenere; præsertim com nunc prolixius exponere vulgari more, quæ Analysis nostra nova paucis exhibet, non satis operæ pretium videatur. Mr Leibnitz therefore had not yet found his new Analysis nor did it out till after he had done filing & polishing this Opusculum in order to publish it.
The next year when Mr Leibnitz had again received two of the aforesaid eight sries by another hand, he took occasion from thence as if he had forgot that he had receved them before, to write to Mr Oldenburg for the Demonstration of them in these words Parisijs 12 Maij anno 1676: Cum Georgius Mohr Danus in Geometria et Analysi versatissimus, nobis attulerit communicatam sibi a Doctissimo Collinio vestro expressionem relationis inter Arcum et sinum per infinitas series sequentes: Posito sinu x, arcu z, Radio 1.
+ &c Hæc, inquam cum nobis attulerit ille, quæ mihi valde ingeniosa videntur, et posterior imprimis series elegantiam quadam singularem habeat ideo rem gratam mihi feceris, Vir Clarissime si Demonstrationem DEMONSTRATIONEM transmiseris. Habebis vicissium mea ab his longe diversa circa hanc rem meditata, de quibus jam aliquot abhinc annis ad te perscripsisse credo, demonstratione non addita quam nunc polio POLIO. By the Demonstration he understands Mr Newtons method: & by his continuing still to polish his Quada inven Quadrature by the transmutation of figures, it appears that he had not yet found out his new Analysis.
This summer between the 14th of Iune & 11th of August Mr Oldenburg sent to Mr Leibnitz at Paris a collection of the Letters & papers of Mr Iames Gregory newly dead. The Collection was made by Mr Collins newly dead w under this title. Extracts from the Mr Gregories Letters to be lett lent to Mons Leibnitz to peruse who is desired to return the same to you. And Mr Collins in a letter to Mr David Gregory the brother of Mr Iames dated 11th Aug. 1711, wrote that they were such. And Mr Tscaurnhause then at Paris new them af in a letter to Mr Oldenburg dated 1 Sept. 1676 has this pas sentence relating to them : Similia porro quæ in hac re præstitit eximius ille Geometra Gregorius [sc. quod series infinitas memoranda certe sunt & quidem optimæ famæ ipsius consultiri, qui ipsus relicta Manuscripta luci publicæ ut exponantur operam navabunt. The collection was therefore received at PAris & sent back, being now in the custody of Mr lately found of Mr Iames Gregory dated 15 Feb. 16701 in wchwhich was the series for finding the arc by the Ta whose from the Tangent given & another for the tangent from the Arc given. [And yet Mr Leibnitz goes on still to claim the first of these two series as76 as having found them himself by himself wthwithout by the transmutation of figures. And] There was also a copy of Mr Newton's Letter of 10 Decemb. 1672, in wchwhich this the extent genera of this general method method Mr his general method was desc of fluxions was described in gene at lapingral terms, & the method a & b & hi & his method of tangents wchwhich proved to be the same with that of Slusius, was described fully plainly & said to be a branch or Corollary of that gener Meth general method. And about the same time vizt 13 Iune 1676 a letter of Mr Newtons dated 13 Iune 1676 was sent to Mr Leibnitz in wchwhich he he at the requese of Mr Leibnitz he describes his method of infinite series & in the end of wchwhich he subjoyns: 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 extendit: non tamen omnino universalis evadit nisi per ulteriores quasdam methodos eliciendi series infinitas
And to this Mr Leibnitz returned answer in his Letter of 27 Aug. returned answer : Quod dicere videmini, plerasque difficultates exceptis Problematibus Diophantæis) ad series infinitas reduci, id mihi non videtur sunt enim multa usque adeo mira et implexa ut NEQVE ABQVADRATVRIS ÆQVATIONIBVS pendeant neque a PENDEANT NEQVE EX QVADRATVRIS, qualia sunt (ex multis alijs) Problemata mathodi tangentium inversa. From wchwhich words its most evident that Mr Leibnitz did not yet understand the differential method.
Mr Newton in his Letter of 24 Octob 1676 replied that that the inverse Problemes of Tangents were in wthwithin the compass reach of his & others more difficult were in his power: for solving of wchwhich he used a double method; one more concise the other more general & expa comprised least they
Mr Newton in his Reply dated 24 Octob 1676 making mention of a Tract composed 5 years before but not finished, & of the method of Tangents set down therein wchwhich & communicated to Mr Collins in his letter of 10 Decemb 1670 & communicated to Mr Collins two or three years before (vizt in yethe Letter of 10 Decemb. above mentioned) & how this method wchwhich was the same wthwith that of Slusius, flowed readily from his Principles & stuck not at surd quantities questions de, & that yethe same Principles extended in like manner to the determining Maximas & Minimas & some others & to the Quadratures of Curves wchwhich thereby became more easy : he gave some instances of the power of this method in squaring Curves & se in letters put out of order, set down the foundation of this Analysis in this sentence, Data Æquatione fluentes quotcunque quantitates involvente fluxiones invenire & vice versa. And towards the end of the Letter he adds that the inverse problemes of Tangents & other more difficult were in his power: for resolving of wchwhich he used a double method, one more concise the other more general, wchwhich methods he set down in two sentences in transposed letters comprehending these sentences Vna methodus consistit in extractione fluentis quantitatis ex æquatione fluxionem ejus involvente: altera tantum in assumptione seriei pro quantitate qualibet incognita ex qua cætera commodederi vari possunt, et in collatione terminorum homologorum æquationis resultantis ad eruendos terminos assumptæ seriei.
And now Mr Leibnitz being convinced that Mr Newtons method was77 was much more general then he had apprehended, & extended to inverse problems of Tan tangents & others more difficult & stuck not at surd, & the method of Tangents of S in direct problems of Tangents was of the same kind wthwith the Method of Slusius; wrote back in a letter & in squaring the Curves gave very general T & useful Theoremes he wrote back in his Letter of 21 Iune 1677, Clarissimi Slusij methodum tangentium nondum esse absolutam Celeberrino Newtono assentior, et jam a multo tempore rem tangentium longe generalius tractovi scilicat per differentias Ordinotarum.
Iacobus Gregorius in Prop 7 Geometriæ suæ universalis anno 1668 impressæ rem tangentium tr
Mr Iames Gregory in yethe 7th Proposition of his Geometria Vniversalis printed in yethe year 1668 handed the business of Tangents by the differences of the Ordinates. Dr Barrow in his tenth Lecture printed in the year 1669 did the like in a manner something more general. Slusius founded his method of Tangents in this Lemma Differentia duarum dignitatum data ehusdem generis applicata ad Differentiam laterum dat partes singulares gradus inferioris ex binomio laterum, ut = yy + yx + xx. And these are the differences of Mr Leibnitz. in the characters of Mr Leibnitz is = 3yy, or = 3yydy & so of other dignities , , ,
The differ Mr Newton's admonition that the Method of Tangents of Slusius was but a branch or coll corollary of a very generall method & easy method was sufficient to put him upon con sidering the method of Slusius wthwith the Lemmas upon wchwhich it was grounded. The first Lemma was sufficient to give him the elements & notation of & theevery the name of the differential method & the universality of Mr Newtons Method was sufficient to put him upon applying this differential method to thequadratures determin to the abstruser Problems, & Mr Newton's saying that his method extended to the finding of curvature of curves was sufficient to put him upon considering the second differences.
✝ And that Mr Leibintz received light into the differential method from what Mr Newton wrote of his general method whereof the method of Tangents of Slusius was but a branch or corollary may be gathered even from his own words. For in his Letter to Dr Wallis dated 28 May 1697 he wrote Mihi consideratio Differentiarum & summarum primam lucem affuderat cum animadverterem Differentias Tangentibus & Summas Quadraturis respondere. Vidi mox Differentias Differentiarum in Geometria Osculis exprimi. Dixerat Mr Newtonus (in Epistola his Letter dated 10 Decem 1672 data wrote) curvitates curvarum (id est circulosrum deosculantesium) per methodum suam determinari, respondet id est circulos ejusdem Curvaturæ cum Curvis il d inveniri respondet Leibnitius Dicit Leibnitius, Mr Leibnitz writes of his invention anno 1677 Vidi mox Differentias Differentiarum in Geometria Osculis exprimi. Hoc vitdit
Mr Newton in the beginning of his Letter dated 13 Iune 1676 sent Mr Leibnitz a Theorem for reducing any dignity of aBinomials into a series Let the Binomials be x+o, its digniy (x+0)n whe the indew of yethe dignity n, the dignity & the series will be . Here if Therefore (x-o)n−. That is in the language of Mr Leibintz &c. In this series if all the terms multipli except the first be omitted as being infinitely less then the first, there will remain . Or At And this is the first Lemma of Slusius. Mr Leibnitz therefore had not only the notice from Mr Newtons Letters the notice that the metho Mr Newtons general method was of the same kind founded upon the same principle wthwith the Method of Tangents of Slusius, the one being a breanch or corollary of the other, but also sent him a series the two first terms of which was the very principle upon wchwhich his general method was founded.
And its further observable that Mr by the two first terms of this series Mr Newton in his Analysis per æquationes numero terminorum infinitas, demonstrated the first Rule in that Analysis. Which Rule is the inverse off the first Lemma of Slusius, & the
‖‖ If it be said that Mr Leibnitz might find his ne differential some other way wthwithout the help of Mr Newton, thats not the question, but whether he had not light enough into it from Mr Newton, & whether Mr Newton had it not before him. Now when Mr Leibnitz had said, as above : Et jam a multo tempore rem generalius tractavi, scilicet per differentias Ordinatarum he goes on &c When therefore he had said Et jam a multo tempore rem tangentium longe generalius tractavi scilicet per differentias ordinatarum, he does on to describe his method by the difference of two Ordnates & the difference of two abscissas & then adds. Hinc nominandnominando in posterum dy differentiam duarum proximarum y & dx duarum proximarum x And by the words nominando in posterum its manifest that patet esse 2ydy & esse & ita porro: which is the very Lemma of Slusius. And here by the words hinc nominando posterum it may be concluded that he began at this very time to communicate his differential method in writing.
Then he sets down as example of drawing Tangents by this Method & concludes wthwith these words : Quod coincidit cum Regula Slusij, ostendite eam statim occurrere hanc methodum intelligenti. And this is the first argument that he gives for the similitude of his of this differential method & Mr Newtons.
Then he explains how his method is like that of Mr Newtons in that it sticks not at surd quantities, & subjoyns Arbitror quæ celare voluit Newtonus de Tangentibus ducendis ab his non abludere. Quod 78 addit ex hoc eodem fundamento quadraturas quoque reddi faciliores me in sententia hac confirmat, nimirum semper figuræ illæ sunt ad æquationes quadrabiles quæ sunt ad æquationesm differentialesm
And in October 1684 when Mr Leibnitz first published the elements of his calculus differentialis, he made this the title of his paper Nova methodus pro Maximis et Minimis itemque Tangentibus quæ nec irrationales moratur & after he had described the elements of his calculus he subjoyned : Et hæc quidem initia sunt tantum Geometriæ cujusdam multo sublimioris ad difficillima & pulcherrima quæque problemata etiam mistæ matheseos pertingentis quæ sine calculo nostro differentiali, aut simili SIMILI, non temere quisquam pari facilitate tractabit. Vide Acta Eruditorum Compare this description of the differential method wthwith the like description of the l wchwhich Mr Newton in his three Letters of 10 Decemb 1672, 13 Iune 1676 & 24 Octob 1676 had given of his general method; & then you will understand what method Mr Leibnitz meant by his words aut SIMILI & wthwith the comparison wchwhich Mr Leibnitz in his Letter of 21th of Iune 1677 makes between Mr Newtons method & his own. & then you will understand see what method Mr Leibinitz understands understood by his words, aut SIMILI.
And two years after in another paper published in the Acta Eruditorum in Iune 1686 wthwith under this title De GemetriaGeometria recondita & Analysi eruditorum indivisibilium atque eruditorum infinitarum Addenda. he has these words Porro quoniam ad problemata transcendentia, ubicunque dimensiones tangentesque occurrunt, calculo tractanda, vix quicquam utilius brevius universalius fingi potest calculo meo differentiali seu Analysi indivisibilium atque infinitorum, cujus exiguum tantum velut specimen sive Corollarium continetisur in methomethodo mea Tangentium in Achis Octob. 84 edita. Which description agrees so exactly wthwith the description wchwhich Newton had given of his in his letter 10 Decemb. 1672, had given of his method, as if that Letter had then been before thise eyes of Mr Leibnitz
In the same paper he writes that before he was acquainted with yethe vulgar Algebra he found out many Theoremes a part of wchwhich he afterwars met wthwith in the works of the Gregories & Dr Barrow. And after he was acquainted wthwith the Vulgar Algebra he found out his Arithmetical Quadrature, & at length he found out his calculus differentialis or Analysis indivisibilium aut infinitorum wchwhich being once discovered whatsoever he had before admired in these matters, became a play & jeast. Which agrees wthwith what was said above of his having the advantages of the words of Mr Gregory & Dr Barrow, of his beginning to communicate Gregories quadrature in his own name in the year 1675 & of his finding the differential method in yethe year 1677.
In the same paper of his writes in speaking of his way of notation he writes. Malo autem dx et similia adhibere quam literas pro illis, quia istud dx est modificatio quædam ipsius x, et ita ope ejus fit, ut sola quando id fieri opus est litera x cum suis scilicet potestatibus & differentialibus calculum ingrediatur, et relationes transcendentes inter x et aliud exprimantur. Qua ratione etiam lineas transcendentes æquatione explicare licet. Here Mr Leibnitz allows that li in his differential method he might have represented the differences by letters (as Dr Barrow did) but he chose rather to represent them by the symbols dx, dy &c for this being more convenient79 convenient for explaining representing transcental curves by an equation By his own confession therefore, the differential method is one & the same method whether the differences be represented by letters or any oth by the by any other marks such as are dx, dy, dz or ox, oy, oz. And therefore if Mr Newton might use any other sort of Notation without using another method. The invention does not lye the method does not lye in the invention of this or that sort of notation but any sort of notation may be used without varying the method, & the method called by Mr Leibnitz the differential method may be Mr Newtons tho the Notation dx, dy, dz be Mr Leibnitz. The area of a curve whose Ordinate is Mr Leibnitz in his Analysis represents by this mark , Mr Mr Newton in his Analysis by this . The difference of Notation makes no difference in the method. But Mr Newtons notation being much older then that of Mr Leibnitz makes him the first inventor of the method. Mr When the fluents of a curve is repr are represented by the areas of Curves & their fluxions by the Ordinates, if the let any letters as v, x, y, z be put for yethe Ordinates the fluents will be x, y v, x, y, z & by this way of notation transcendental curves may be exprest in equations as well as by the Notation of Mr Leibnitz. In the eno Mr Newton in the end of his Analysis put some letters for fluents & others for their fluxions, Mr Leibnitz allows that he could have done the same thing in his differential method : but Newtons being used by him in & perhaps before yethe year 1669 gives him preference priority of invention. Mr Newton Leibnitzhas now confined his met the method to a particular sort of Notation, Mr Newton has not confined it to any sort of Notation but left it all liberty to use what symbols any man shall think fit. In his Tract of Quadratures & some other papers published by Dr Wallis he has used the note symbols , , , ; but he has not confined his method to those symbols. He leaves every man at liberty to use what symbols he In his Procipies he has put smaller greater letters for fluents & pleases & small ones for fluxions. And whatever symbols he uses, if are used, if their signification be ones described in the beginning, he may by them they will be sufficient to express transcendent Curves by equations, & whatever else can be be exprest be the character symbols of Mr Leibnitz.
But its to be observed that in the differential method there are no symbols for fluxions, there the fluxions are finite quantities & the differences dx dy dz are infinitely small. B
But its to be observed that in Mr Newtons method there are symbols for fluents & others for fluxions & that the put symbols for fluxions multiplied by the letter o wchwhich he uses for a moment of time become the his symbols of moments or differences: but in Mr the Differential method there are no symbols of fluxions & there but moments differ (for fluxions are finite quantities) & therefore Mr Newtons notation is more complete & comprehensive. Its further to be observed that Mr Newton can use the letter o as a finite quantity whenever he pleases & there by perform the whole computation in finite quantities by the Geometry of Euclide, & make his conclusions demonstratively certain: Mr Leibnitz's his method wants this advantage his differences dx, dy, dz being always considered as infinitely small. And its still further to be considered that Mr Leibnitz when Mr Newton's method preceeds not in finite equations he can work in infinite series & thereby apply resolve apply his method to almost all sort of Problems: wh whereas the working by infinite series is no part of the Analysis of Mr Leibnitz80 Mr Leibnitz So Then the method of Mr Newton in the year 1669 when he communicated it to Dr Barrow his Analysis to Dr Barrow & by Dr Barrow to Mr Collins was be more extensive & complete in its notations, more demonstrative in its way of working, & more universal in its application to Problems then that of Mr Leibnitz is at present, or perhaps can even be made wthwithout borrowing from Mr Newton. And further, Mr Newton's way of considing quantities as growing increasing by various degrees of velocity is agreable to yethe nature of things; that of considering them as composed of indivisibles is forced ungeometrical, there being then no indivisibles in Geometry. [His differences arefrequently incommensurable & by consequence divisible in infinitum] In his Theoria motus abstract (pag 8) he gives this definition of a point. Punctum, saith he, non est cujus pars nulla est, nec cujus pars non consideratur, sed cujus extensio nulla est, seu cujus partes sunt indistantes, cujus magnitudo est inconsiderabilis, inassignabilis, minor quam quæ ratione, nisi infinita ad aliam sensibilem exponi possit, minor quam quæ dari potest atque hoc est fundamentum Methodi Cavallerianæ. Whether Euclid or Mr Leibnitz are in the right you may understand by this considering that two solids touch one another in a superfici no in a superficies without penetrations of dimensions, two surfaces cur one another in Euclids line & two lines cut one another in a point Euclids point. Cer But according to Eucl Mr Leibnitz two solids touch one another in a skin whose depth or thickness is indivisible & penetrate one anothers dimensions to the depth of that skin. And upon this notion of indivisibles he has founded the method of Cavallerious & his own. Cavallerius & Leibnitz might found their methods upon such an Hypothesis, but the Hypotheses is not Geometrical. The differences of Mr Leibnitz wchwhich he calls indivisibles, are in various proportions to one another, & frequently they are incommensurable & by consequence divisible in infinitum. For two unequal incommensurable quantities may be subducted from one another in finitum. The language of indivisibles is not Geometrical & a language not Geometrical ought not to be introduced into Geometry.
The symbols by which Mr Newton represents fluxions are the oldest symbols for that sort of quantities: for Mr Leibnitz has not yet any symbols for fluxions. The symbols by wchwhich Mr Newton represents moments are (vizt the rectangles of fluxions & a moment of time) are older then those of Mr Leibnitz for differences, being used by Mr Newton in his Analysis communicated to Mr Collins in the year 1669. And Mr Leibnitz in his giving an account of this AnalysAnalysis in yethe Acta Eruditorum for February 16 1712 represents the notation by the letter o was used before by Mr Fermat & others in such cases B Mr Newton in honour of Mr Fermat, so far as Gregory Barrow Gregory & other who used it has kept to their notation as far as was practicable convenient consistant wthwith the generality of his method Mr Leibnitz has endeavoured to extinguish the notation of those that went before him by a new sort of Notation has departed from the old way of Notation without any necessity & endeavoured to brought in a new one tending to extinguish the memory of those that went before him & make himself the sole inventor. And therefore its an Act of justice to the memory of Fer those that went before him to keep yethe use of the letter o, especially since its more convenient & proper & make the method more demonstrative