Newton’s De Analysi and the priority dispute∗
Naoki Osada†
Abstract
In 1669 Newton wrote De Analysi to claim the priority of themethod of inﬁnite series. Newton did not want to publish his methodof ﬂuxions, so he introduced the moment instead of the ratio of ﬂuxions, which is the diﬀerential quotient in modern calculus, and expressed the ﬂuent, which is the antiderivative in modern calculus, interms of the pair of the indeﬁnite region drawn by the motion of theordinate and its signed area.
In the priority dispute, Newton suspected that Leibniz had read
De Analysi in 1676, so he used De Analysi as evidence that Newton was the ﬁrst inventor of the method of ﬂuxions and that Leibniz
had plagiarized diﬀerences from the moments of De Analysi. However Newton did not use ﬂuxions when he wrote De Analysi and themoment in De Analysi was diﬀerential quotient, not diﬀerential, inmodern calculus. In 1715, in his anonymous An Account of the Book,
Newton explained how he had represented the ﬂuxions in De Analysi
and rewrote the note he had given in De Analysi regarding the unitof moment so that moment meant diﬀerential.
Keywords: Isaac Newton, De Analysi, priority dispute, moment,
Commercium Epistolicum

1. Introduction
In 1669 Isaac Newton wrote the De Analysi per Æquationes numero Terminorum Inﬁnitas, abbreviated as De Analysi, to claim the priority of themethod of inﬁnite series against Nicolaus Mercator. When Newton wrote De
Analysi, because he did not want to publish his method of ﬂuxions, he did
not use velocities, which he later called ﬂuxions. Therefore he introduced
the moment instead of the ratio of ﬂuxions, which is the diﬀerential quotientin modern calculus, and he expressed the quantity to be sought, which helater called ﬂuent, in terms of the pair of the indeﬁnite region drawn by the
motion of the ordinate and its signed area.
∗

This paper is based on a lecture given by the author at the Newton Symposium of theHistory of Science Society of Japan on 29 May 2022.
†
Tokyo Woman’s Christian University, Email: osada@lab.twcu.ac.jp

1

In the priority dispute with Leibniz on calculus, Newton tried to provethat Leibniz had plagiarized from De Analysi and the two letters, Epistola
Prior and Epistola Posterior. Westfall1 and Guicciardini2 have described theaccusations of plagiarism from the two letters in details. In this paper wewill focus on the accusations of plagiarism from De Analysi 3 .
2. Rule I and II in De Analysi
The procedure of the method of inﬁnite series is to expand the diﬀerential
quotient of a quantity to be sought into an inﬁnite series of the form∑miai x n i ,(1)i

and then to obtain the quantity by termwise integration. In order to perform
this method it is suﬃcient thatm

1. To ﬁnd the antiderivative of ax n .
2. The possibility of termwise integration.
3. To expand the diﬀerential quotient of the quantity into an inﬁnite seriesof the form (1).
Newton gave Rule I, II, and III in De Analysi which correspond to the above
1,2, and 3, respectively. Rule I and Rule II are as follows.
To the base AB of some curve AD let the ordinate BD be perpendicular and let AB be called x and BD y. Let again a, b, c, . . .
be given quantities and m, n integers. Then
D

A

B

1

Richard Westfall, Never at Rest, Cambridge University Press, 1980, pp.712-720.
Niccolò Guicciardini, Isaac Newton On Mathematical Certainty and Method, The
MIT press, 2009, pp.373-381.
3
The main purpose of De Analysi was to show how to express y as an inﬁnite seriesof x given an implicit function f (x, y) = 0, but this is not closely related to the priority
dispute and is therefore omitted. For the series expansion of implicit functions, see theauthor’s paper: Naoki Osada, “Literal resolution of aﬀected equations by Isaac Newton,”
RIMS Kôkyûroku Bessatsu B73 (2019): 1-20.
2

2

The Quadrature of simple Curves
m+nmna
Rule I. If ax n = y, then will m+n
x n equal the area ABD.
The matter will be evident by example.
And of those compounded of simple ones
Rule II. If the value of y is compounded of several terms of thatkind the area also will be compounded of the areas which arise
separately from each of those terms.4
Since Newton said “The matter will be evident by example”, we shalllook at the fourth example.
Example 4 If x12 (= x−2 ) = y, (that is, if )
a =
1 −1
D
n = 1 and m = −2, thenx1 = −
−1
)
(
−1
A B
= αBD inﬁnitely extended in thex−1 =x
direction of α: the computation sets its sign negative because it
lies on the further side of the line BD.5α< −1, Newton deﬁned the indeﬁnite
Example 4 shows that when mnmregion described by the line segment BD = ax n by αBD, and its area ism+nna
x n . Newton made the area negative because it matched the sign ofn+m
the antiderivative.
Next we see the third example of Rule II.
Third examples. If x2 + x−2 = y, then 13 x3 −αx−1 =the surface described. But here you shouldDnote that the parts of the said surface thus foundδ
lie on opposite sides of the line BD: precisely, onFsetting BF = x2 and FD = x−2 , then 31 x3 = thesurface ABF described by BF and −x−1 = DFα
describes by(DF. Andϕ) this always happens whenm+nof the ratios of the base x
the indicesA β Bnin the values of the surface sought are aﬀected with diﬀerent
signs.6
4

D.T. Whiteside, The Mathematical Papers of Isaac Newton, Vol.II, Cambridge University Press, 1968, pp.206-209. This book will be abbreviated as MP II in this paper.
5
Ibid., pp.208-209.
6
Ibid., pp.210-211.

3

Example 4 in Rule I and the above example show that Rule I should beviewed as giving the antiderivative rather than the area of the curve. Thatmis, the antiderivative of the simple curve y = ax n corresponds to the signedmarea of the indeﬁnite region drawn by the line segment BD = ax n .
] [∫ x
m


if m
> 0,
ax n dx = ABD

n
m+n
na
0
[∫ x
]
x n =
m

m+n


ax n dx = αBD
if m
< −1.
n
∞

See our recent paper7 for details.
The equivalent of Rule I was given in the October 1666 tract as follows:
If two Bodys A&B, by their velocitys p&q describe ye lines x&y.
[...]
n+m
m
q
na
x n = y.8
As if ax n = . Then ispn+m
The ratio9 pq of velocities in the October 1666 tract was represented as theline segment BD in Rule I in De Analysi.
3. Moments in De Analysi
3.1 Introduction of moments
Newton introduced the new term and concept of moment as follows:
Let ABD be any curve and AHKB a rectangle
whose side AH or BK is unity. And consider
D
that the straight line DBK describes the areas
ABD and AK as it moves uniformly away fromB
AH; that BK(1) is the moment by which AK(x) A
K
gradually increases and BD(y) that by whichH
ABD does so; and that, when given
continuously the moment of BD, you can by the foregoing rules
investigate the area ABD described by it [...]The matter will be clariﬁed by example.10
7

Naoki Osada, “Area and moment in De Analysi by Isaac Newton,” RIMS Kôkyûroku
Bessatsu, B85 (2021): 15-34.
8
D.T. Whiteside, The Mathematical Papers of Isaac Newton, Vol.I, Cambridge University Press, 1967, p.403.
dy
q
dy
9
In modern notation, p = dx
dt and q = dt , thus p = dx .
10
MP II, pp.232-233

4

It is not clear from the above explanation alone.11
3.2 The length of an arc
Newton gave an example of ﬁnding the length of an arc using moments.
Let ADLE be a circle whoseDHL
arc length AD is to be disG
covered. On drawing the tangent DHT, completing the inT
A K
B
C
E
deﬁnitely small rectangle
HGBK and setting AE = 2AC = 1, there will then be BK or
GH (the moment of the base AB) to DH (the moment of the arc
AD)[= BK : DH]√
1= BT : DT = BD( x − x2 ) : DC( 12 ) = 1(BK) : √
(DH)
2
2
x
−
x
√
1
x − x2is the moment of the arc AD.
so that √or2(x − x2 )2 x − x2
1
1
3 32
5 52
35 27
When reduced this becomes 12 x− 2 + 14 x 2 + 16
x + 32x + 256x +63 92
x &c. Therefore by Rule II the length of the arc AD is
512
1

3

5

7

9

11

3 2
5
35
63
x 2 + 16 x 2 + 40
x + 112x 2 + 1152
x 2 + 2816
x 2 &c.12

Since HGBK is an indeﬁnitely small rectangle, the moment BK of the(base AB and the moment DH of the arc AD, denoted by AD, are indeﬁnitely
1small line segments. However Newton put BK = 1 and DH = √.
2 x − x2
The fact that BK is indeﬁnitely small and BK = 113 are contradictory.
3.3 Newton’s true intention of the concept of moment
There are two main interpretations of moment in De Analysi: one is
the ﬂuxional velocity by Whiteside14 , other is the inﬁnitesimal increments ofMalet and Panza† wrote “Nothing else is said about the nature and properties of
those moments, although it seems to be implicit that (the line DBK being in motion) they
are something else that mere Cavalierian indivisibles – in any case, whatever Newton had
in mind, here, he left to the reader the task of clarifying the matter.” (p.377)†Antoni Malet and Marco Panza, “Newton on indivisibles” in Vincent Jullien, ed.
Seventeenth-Century Indivisibles Revisited, 365-390, Birkhäuser, 2015.
https://halshs.archives-ouvertes.fr/halshs-01172653/document
12
MP II, pp.232-233
13
The author wrote “if BK = 1 is corrected to BK = o, the moment in De Analysi
becomes the diﬀerential and coincides with that in De Methodis.” (footnote 7, p.31.) If
HGBK were indeﬁnitely small rectangle, BK = o, but Newton did not treat the moments
in De Analysi in that way. See Section 3.3.
14
MP II, p.233 (101).
11

5

variable quantity by Boyer15 , Whitrow16 and Guicciardini17 , or diﬀerential18
by Dunham19 and others.
In this section, we consider Newton’s true intention of the concept of
moment from the description in De Analysi. After the example of ﬁnding
the length of an arc, Newton made the following note about the units ofmoments.
But it must be noted that unit20 which is set for the moment is asurface when the question concerns solids, a line when it relates
to surfaces and a point when (as in this example) it has to dowith lines. Nor am I afraid to talk of a unit in points or inﬁnitely
small lines inasmuch as geometers now consider proportions inthese while using indivisible methods.21(

Let us consider the area of ABD in Section 3.1. In modern notation, since
the unit of ABD is a surface, the unit of dtd ABD is a velocity of changing area,
the unit of the diﬀerential dABD is an inﬁnitesimal increment of surface andd
ABD is a line, it is appropriate to
the unit of the diﬀerential quotient dx
determine that Newton represented diﬀerential quotients by moments.
We will verify by modern calculus that Newton represented the diﬀerential
quotient by moments, using the example of the length of an arc (Section√ 3.2).
x − x2d,
It follows from AB = x that BK = dx
AB = 1. Since ̸ DCA = sin−112√x − x2
1
we have AD = sin−1
and
1
2
2
1−2x

(

√
d
1
1
x−x2
DH =
AD = √= √,2dx
2 1 − 4(x − x )2 x − x2

1
(0 < x < ),2which is consistent with Newton’s calculation.
15

Carl B. Boyer, The History of the Calculus and its Conceptual Development, Dover,
2018, (Hafner Publishing, ﬁrst published 1949), p.191.
16
G.J. Whitrow, “Newton’s Role in the History of Mathematics,” Notes and Recordsof the Royal Society of London, Vol.43, No.1, (1989), pp.71-92.
17
Footnote 2, p.155.
18
Let y = f (x) be a function. The diﬀerential dy is deﬁned by dy = f ′ (x)dx, where′f (x) is the derived function of f , and dx is the inﬁnitesimal increment of the variable x.
19
William Dunham, The Calculus Gallery: Masterpieces from Newton to Lebesgue,
Princeton University Press, 2009.
http://assets.press.princeton.edu/chapters/s7905.pdf
20
The Latin original is unitas, which Whiteside translates as “unity”, while Malet andPanza (footnote 11, p.378.) translated as “unit”.
21
MP II, pp.234-235.

6

In 1671 Newton deﬁned the moment in the De Methodis Serierum etFluxionum,22 abbreviated as De Methodis, as follows:
The moments of the ﬂuent quantities (that is, their indeﬁnitely
small parts, by addition of which they increase during each inﬁnitely small period of time) are as their speeds of ﬂow. Wherefore if the moment of any particular one, say x, be expressedby the product of its speed m and an inﬁnitely small quantity o(that is, by mo), then the moments of the others, v, y, z, will be
expressed by lo, no, ro, seeing that lo, mo, no, and ro are to one
another as l, m, n, and r.23
The symbol o is considered to represent dt in modern notation. Thus mo =dx
dt = dx and lo = dv
dt = dv are the diﬀerentials of x and v, respectively.
dt
dtoIn the sense of De Methodis, BK = o and DH = 2√x−x
2 . The representation “indeﬁnitely small” was not appropriate in De Analysi, but became an
important phrase in De Methodis.24
3.4 The proof of Rule I in De Analysi> 0. In the preparaIn De Analysi Newton proved Rule I for the case mntion of the proof he used a tool which is equivalent to diﬀerential.
Preparation for demonstrating the ﬁrst rule.
1. The quadrature of simple curves in Rule I.
Let then any curve ADδ have base AB = x,Kperpendicular ordinate BD = y and areaD
ABD = z, as before. Likewise take Bβ = o,
BK = v and the rectangle BβHK(ov) equal
to the space BβδD. It is,
A B
therefore, Aβ = x + o and Aδβ = z + ov.25

δ
H

β

dz
dz
In modern notation, since o = dx and v = dx
, ov = dx
dx = dz isthe diﬀerential of z, and v is the moment (diﬀerential quotient) of the areaz = ABD. The ordinate v = BK represents the moment of z in terms of
22

D.T. Whiteside, The Mathematical Papers of Isaac Newton, Vol.III, Cambridge University Press, 1969, pp.32-329.
23
Ibid.,, pp.78-81.
24
Kokiti Hara‡ wrote “The argument of the next work [De Methodis], which calls thediﬀerential, not velocity, the moment, has already started here.” (Early-modern Mathematics, p.307.)‡Kokiti Hara, Early-modern Mathematics (in Japanese), in A History of Mathematics,
Chikuma Shobō, 1975.
25
MP II, pp.242-245.

7

De Analysi, and the ﬂuxion of z in terms of De Methodis. If we assume thespeed of the base AB is 1,
dz
dz
dz
dt
= dx
= .
dx
dt
dt

Thus moments in De Analysi and the ﬂuxions in De Methodis have equalvalues but diﬀerent units. The physical unit of the former is length, that of
the latter is a velocity of changing area.
4. Accusations of Plagiarism in the priority dispute
Until the discovery of the Collins’ letters and papers by William Jones in
1708, Newton and mathematicians around him believed that Leibniz had plagiarized from the two letters (Epistola Prior and Epistola Posterior) in 1676.
In 1699, John Wallis published the two letters with Newton’s permission in
his Opera Mathematica.
In 1704, Newton published the Tractatus de Quadratura Curvarum (abbreviated as De Quadratura) as an appendix to his Opticks. De Quadratura
was the ﬁrst published treatise on the method of ﬂuxions, nearly 40 years
after Newton’s discovery of it. In the preface (advertisement) to the Opticks,
Newton wrote:
In a Letter written to Mr. Leibniz in the Year 1676 and publishedby Dr. Wallis, I mentioned a Method by which I had foundsome general Theorems about squaring Curvilinear Figures, orcomparing them with the Conic Sections, or other the simplest
Figures with which they may be compared.26
Newton claimed the priority of the method of ﬂuxions using Epistola Posterior as evidence, but did not directly mention Leibniz’s plagiarism 27 . And
in the introduction to De Quadratura, Newton explicitly dated the discoveryof the method of ﬂuxions to 1665-1666 as follows:
I was led to seek a method of determining quantities out of the
speeds of motion or increment by which they are generated; and,naming these speeds of motion or increment ‘ﬂuxions’ and thequantities so born ‘ﬂuents’, I fell in the years 1665 and 1666 upon
the method of ﬂuxions which I have here employed in the quadrature of curves.28
26

I. Newton, Opticks, Printed for Sam. Smith, and Benj. Walford, London, 1704
https://library.si.edu/digital-library/book/optickstreatise00newta
27
Leibniz might have taken Newton to imply Leibniz’s plagiarism in the above preface.
28
D.T. Whiteside, The Mathematical Papers of Isaac Newton, Vol.VIII, Cambridge
University Press, 1981, p.123.

8

In 1705, Leibniz published an anonymous review of the De Quadraturain the Acta Eruditorum.
when some magnitude grows continuously, as (for instance) a line
grows by the ﬂowing of a point which describes it, the instantaneous increments are called ‘diﬀerences’, [...] whose elements havebeen delivered in these Acts by its inventor Mr Gottfried Wilhelm
Leibniz, [...] In place of Leibnizian diﬀerences, accordingly, Mr
Newton employs, and has ever employed, ‘ﬂuxions’, 29
Leibniz claimed the priority of the diﬀerential calculus. He also implied that
Newton’s ﬂuxions were plagiarism from Leibniz’s diﬀerences.
In 1708, John Keill, in his paper published in the Philosophical Transactions, inserted Leibniz’s plagiarism.
All of these [propositions] follow from the now highly celebrated
Arithmetic of Fluxions which Mr. Newton, beyond all doubt,First Invented, as anyone who reads his Letters published by Wallis can easily determine; the same Arithmetic under a diﬀerent
name and using a diﬀerent notation was later published in the
Acta eruditorum, however, by Mr. Leibniz.30
In the same year 1708, William Jones obtained the letters and papers left
by John Collins, which were later called the Collins papers. The Collins papers consisted of Collins’ handwritten copy of De Analysi and correspondencewith mathematicians. In 1711, Jones published De Analysi, De Quadraturaand others with Newton’s permission.
In 1711, Leibniz learned that he had been accused of plagiarism by Keill,
so he wrote a letter of protest to Hans Sloan, the secretary of the Royal
Society, asking Keill to retract his accusation. In March 1712, the Royal
Society appointed a committee and adopted the report 31 attributing Newton
as the ﬁrst inventor, and published a book in evidence, Commercium Epistolicum D. Johannis Collins, et aliorum de Analysi promota (Printed 1712,
distributed 1713; abbreviated as Commercium Epistolicum). The report andthe Commercium Epistolicum were substantially prepared by Newton himself.

29

Ibid., p.26.
Footnote 1, pp.715-716.
31
Westfall stated “the published report did not name the committee, whose membership
remained concealed in the society’s records until the nineteenth century.”(ibid., p.725)
30

95. De Analysi in the priority dispute
The following statement appears in the report included at the end of theCommercium Epistolicum.
I. That Mr. Leibniz was in London in the beginning of the Year
1673, and went thence in or about March to Paris, where he kept
a Correspondence with Mr. Collins by means of Mr. Oldenburg,till about September 1676, and then return’d by London and Amsterdam to Hannover: And that Mr. Collins was very free incommunicating to able Mathematicians what he had receiv’d from
Mr. Newton and Mr. Gregory.32
Since Newton learned the above facts from the Collins papers, he would have
had a strong suspicion, though not conclusive evidence 33 that Leibniz had
read or transcribed De Analysi from Collins during his stopover in London in
1676. It is believed that Newton was trying to prove that Leibniz plagiarizedfrom De Analysi, Epistola Prior and Epistola Posterior.
While the two letters had already been published by Wallis in his Opera
Mathematica, De Analysi was published by Jones in 1711 and reproducedin the Commercium Epistolicum. Furthermore, Newton decided to use De
Analysi and the two letters as evidence for his claim of priority of the methodof ﬂuxions. The report appears as follows:
III. That by Mr. Newton’s Letter of the 13th of June 1676 it appears, that he had the Method of Fluxions above ﬁve Years before
the writing of that Letter. And by his Analysis per Æquationes
numero Terminorum Inﬁnitas, communicated by Dr. Barrow to
Mr. Collins in July 1669, we ﬁnd that he had invented the Method
before that time.34
With De Analysi, Newton attempted to prove that Newton was the ﬁrst
inventor of the method of ﬂuxions and that Leibniz plagiarized from moments
to make diﬀerential calculus.
However, when Newton wrote De Analysi, he did not use ﬂuxions, whichcaused two problems: First, if one reads De Analysi in contrast to the October
32

Royal Society, Commercium Epistolicum Collinii & Aliorum, De Analysi promota,
1713. p.120. https://archive.org/details/bub gb r1BCoilZ3PwC
33
Westfall writes that “Leibniz never breathed a word about De Analysi [...] Only when
Leibniz, shortly before his death, inadvertently revealed the extent of Collins’s liberality
in the fall of 1676, did Newton begin to realize that he might have seen the tract as well.”(footnote 1, p.720.)
34
Footnote 32, p.121.

10

1666 tract and De Methodis, one can understand that this paper is a paper on
the method of ﬂuxions, but if one only reads De Analysi by itself, one would
not have thought that this paper dealt with the method of ﬂuxions. Second,
the moments in De Analysi were diﬀerential quotients, not diﬀerentials, inmodern calculus.
We discuss the ﬁrst problem in the rest of this section and the second
problem in the next section. In 1715,35 Newton published anonymously the
An Account of the Book (abbreviated as An Account) in the PhilosophicalTransactions. In it, he explained how he represented the method of ﬂuxions
when writing De Analysi.
When Mr. Newton had in this Compendium explained these three
Rules, and illustrated them with various Examples, he laid down
the Idea of deducing the Area from the Ordinate, by considering the Area as a Quantity, growing or increasing by continual
Flux, in proportion to the Length of the Ordinate, supposing theAbscissa to increase uniformly in proportion to Time. 36
Mr. Newton doth not place his Method in Forms of Symbols,
nor conﬁne himself to any particular Sort of Symbols for Fluentsand Fluxions. Where he puts the Areas of Curves for Fluents,
he frequently puts the Ordinates for Fluxions, and denotes theFluxions by the Symbols of the Ordinates, as in his Analysis.37
In De Analysi, Newton used symbols that diﬀered from those in his otherworks on the method of ﬂuxions. He explained that he expressed the ﬂuent interms of the area of the curve and the ﬂuxion in terms of the ordinate. Thus,m+nmnax n = z of Rule I is the ﬂuent, and the ordinate ax n = y is
the area m+n
the ﬂuxion.
6. Diﬀerentials in the priority dispute
As mentioned in Section 4, Leibniz wrote in his review of the De Quadraturathat Newton plagiarized Leibniz’s diﬀerences (instantaneous increment) as a
ﬂuxion. On the other hand, in the Commercium Epistolicum perhaps Newton
himself added the following footnote to moments in De Analysi.
35

Derek Gjertsen, The Newton Handbook, Routledge & Kegan Paul, 1986, p.505.
Royal Society, “An Account of the Book entitles Commercium Epistolicum Collinii &Aliorum, De Analysi promota”, Philosophical Transactions, Vol. 29 (1714 - 1716), p.178,
https://www.jstor.org/stable/103050
37
Ibid., p.204.
36

11

Here is described the method of operating by ﬂuents and their
moments. These moments were afterwards called diﬀerences by
Mr Leibniz: and so came the name of diﬀerential method. 38
Furthermore, Newton essentially rewrote in the An Account the note on theunits of moments (see Section 3.3) as follows:
And from the Moments of Time he gave the Name of Moments
to the momentaneous Increases, or inﬁnitely small Parts of the
Abscissa and Area, generated in Moments of Time. The Momentof a Line he called a Point, in the Sense of Cavallerius, tho’ it
be not a geometrical Point, but a Line inﬁnitely short, and theMoment of an Area or Superﬁcies he called a Line, in the Sense
of Cavallerius, tho’ it be not a geometrical Line, but a Superﬁcies
inﬁnitely narrow.39
If the note on the unit of moment given in De Analysi is revised as above, themoment becomes an momentaneous increases, i.e., a diﬀerential in modern
calculus. The moment of AK(x) is BK(1 × dx), because BK is not a line
[segment] but an inﬁnitely narrow surface, and the moment of ABD(z) isBD(y × dx) because BD is not a line [segment] but an inﬁnitely narrowsurface. It is thought that Newton changed moments in De Analysi from
diﬀerential quotients to diﬀerentials in order to claim priority of diﬀerentials
and Leibniz’s plagiarism from moments.
7. Why did Newton choose De Analysi as evidence?
What Newton described in De Analysi was the method of inﬁnite series
and its applications. Why did Newton choose this paper, which did not
explicitly use the method of ﬂuxions, as the most important evidence for his
claim of priority of calculus?
The method of inﬁnite series is a method of ﬁnding a quantity by expanding the diﬀerential quotient (moment) of it to inﬁnite series, and thenby integrating it term by term (Rule II). Newton applied the method of inﬁnite series to the length of an arc, the area of a cycloid and the area of aquadratrix. In the arc length problem discussed in Section 3.2, the moment
1of the arc AD is DH = 2√x−x
2 . Expanding DH to an inﬁnite series gives
(
)
1
1
3 32
5 52
35 72
63 29
√1
= 21 x− 2 + 14 x 2 + 16
x + 32x + 256
x + 512x + ···
2 x−x238
39

MP II, p.232 (99).
An Account, p.178.

12

and then integrating term by term yields

1
(
√ )
3
7
9
11
3 52
5
35
63
x 2 + 16 x 2 + 40
x + 112x 2 + 1152
x 2 + 2816
x 2 + · · · = sin−1 x ,which is the arc length. In modern notation, Newton derived
∫ x
√
1
√
dx = sin−1 x.
20 2 x−x

Based on these applications, Newton concluded De Analysi as follows.
It should, ﬁnally, merit consideration as a relevant part of analysis
since by its aid the areas and lengths of curves and so on (provided
that may be done) may be exactly and geometrically determined.
But this is not the place to dwell on these matters.40
Newton regarded the method of inﬁnite series to be a powerful tool for
quadrature. Furthermore, as Guicciardini wrote41 , Newton considered themethod of integration more important than the method of diﬀerentiation, inmodern terms.
8. Conclusion
There are three possible reasons why Newton choose De Analysi as themost important evidence in the priority dispute.
First, the Collins papers made it extremely likely that Leibniz had reador transcribed De Analysi in 1676, so Newton probably thought that if hecould prove Leibniz’s plagiarism from De Analysi it would not be necessary
to publish the October 1666 tract or De Methodis.
Second, as we described in sections 5 and 7, Newton regarded De Analysi
as a treatise on the method of ﬂuxions.
Third, Newton showed De Analysi to Collins through Barrow, and Collins
circulated it to many mathematicians, so Newton would have regarded thepaper as having been published42 . In the early 18th century, there was norule established today that the ﬁrst paper submitted to an academic journal
has the right of priority.
40

MP II, pp.242-243.
Guicciardini wrote “When one analyzes the mathematical examples adduced in Commercium Epistolicum, it emerges that Newton and his acolytes who were slavishly editing
it were referring to the inverse method of ﬂuxions applied to problems of quadrature.”(footnote 2, pp.375-376.)42
For example, “And by his Analysis per Æquationes numero Terminorum Inﬁnitas,
communicated by Dr. Barrow to Mr. Collins in July 1669, we ﬁnd that he had invented
the Method before that time.” (the report in Commercium Epistolicum, p.121)
41

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As Newton suspected, Leibniz is conﬁrmed to have viewed De Analysi andmade excerpts43 from it in London in mid-October 1676. Leibniz transcribed
the parts on inﬁnite series, replacing them with his own symbols, but did
not transcribe the introduction of moments, the application of moments tothe length of an arc and the area of a cycloid, or the proof of Rule I. The
parts that Leibniz did not transcribe are important parts of the method of
ﬂuxions, but since Newton concealed the ﬂuxion, i.e., avoiding the terms andsymbols of the method of ﬂuxions, Leibniz probably did not consider it apaper on the method of ﬂuxions.

43

We can read the excerpt in MP II, pp.248-259.

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